Math-Blog » Applied Math

Recent Data For Planning Mathematical Software Projects

John F. McGowan, Ph.D. — Thu, 19 Jan 2012 13:35:49 +0000

This article is a follow up to the previous article Estimating the Cost and Schedule of Mathematical Software. In the previous article, the author advocated using software engineering expert Barry Boehm’s Basic COCOMO Embedded Mode cost model to estimate the cost and schedule of mathematical software projects, with the important qualification that there are substantial variations between actual effort and estimated effort using this model. By Boehm’s own account, Basic COCOMO estimates are within a factor of two of actual effort only 60 percent of the time.

The formula for Basic COCOMO Embedded is:

where SM is Staff Months, the politicaly correct term formerly known as the Mythical Man Month, and KSLOC is thousand (kilo) source lines of code.

Basic COCOMO is based on a database of sixty-three software projects at TRW, Boehm’s then employer, during the 1970s. The Embedded Mode model is based on twenty-eight (28) of these projects that Boehm classified as Embedded projects. The projects were written in FORTRAN (24), COBOL (5), Jovial (5), PL/I (4), Pascal (2), Assembly (20), and miscellaneous other languages (3). None of these is commonly used today. Nonetheless, in the author’s experience, Basic COCOMO Embedded gives a rough order of magnitude (ROM) estimate of the effort for mathematical software projects such as implementing a video codec in C/C++ today (2012).

The Measurement Free Zone

Remarkably, despite the growing cost and importance of software, it is difficult to find publicly available information on the cost, schedule, and effort of software projects. There are a number of consulting firms and proprietary cost and schedule estimation tools but these do not disclose their databases of historical data. Indeed, many organizations, including many commercial businesses, do not seem to use historical data on the cost and schedule of software development to plan projects!

Donald Reifer’s 2004 Software Productivity Data

In 2004, software engineering expert Donald J. Reifer of Reifer Consultants, a colleague of Barry Boehm, published an article in The DoD SoftwareTech News, now The Journal of Software Technology, “Industry Software Cost, Quality and Productivity Benchmarks” giving the software productivity numbers, broken down by categories such as “Scientific” or “Web Business” for the most recent 600 of 1800 projects in his database of projects. These were projects from the last seven years prior to 2004 (about 1997 to 2004).

Table One below is a subset of Reifer’s data from Table 1 in his paper. These are the categories — Command and Control, Military – Airborne, Military – Ground, Military – Missile, Military – Space, and Scientific — that are similar (Command and Control, Military) or the same (Scientific) as mathematical software. The category “Web Business” is included as a point of reference.

Reifer uses equivalent source lines of code (ESLOC). For new code, ESLOC is equivalent to a line of code. For “legacy” code that is modified or reused, ESLOC applies a weighting factor to the line of code such as 0.4. This way data on maintenance or modifications of existing software can be combined with writing new software. Reifer uses equivalent source lines of code as defined by the Software Engineering Insitute.

Application Domain Number	Projects	Size Range (KESLOC)	Avg. Productivity (ESLOC/SM)	Range (ESLOC/SM)	Example Application
Command & Control	45	35 to 4,500	225	95 to 350	Command centers
Military -All	125	15 to 2,125	145	45 to 300	See subcategories
Airborne	40	20 to 1,350	105	65 to 250	Embedded sensors
Ground	52	25 to 2,125	195	80 to 300	Combat center
Missile	15	22 to 125	85	52 to 175	GNC system
Space	18	15 to 465	90	45 to 175	Attitude control system
Scientific	35	28 to 790	195	130 to 360	Seismic processing
Web Business	65	10 to 270	275	190 to 985	Client/server sites
Totals	600	10 to 4,500		45 to 985

Table 1 (Abridged): Software Productivity (ESLOC/SM) by Application Domains

Note that productivity in KESLOC (One Thousand Equivalent Source Lines of Code) is significantly higher for the Web Business category. This actually understates the difference because the “Web Business” projects, as indicated elsewhere in Reifer’s article, are usually written in so-called Fourth Generation Languages (4GLs), scripting languages such as Python, Perl, PHP, and so forth, whereas the other software categories are typically written in lower level languages such as C/C++. A single line of a 4GL language such as Python often corresponds to several lines of a language such as C/C++.

Scientific software has an average productivity of 195 ESLOC per Staff Month (SM). Note that there is a wide range of variation: 130 to 360 ESLOC per Staff Month (SM). This is for fairly large projects ranging from 28,000 lines of code to 790,000 lines of code.

Basic COCOMO Embedded predicts a productivity of 142 lines of code per Staff Month for a project with 28,000 lines of code. It predicts a productivity of 73 lines of code per Staff Month for a project with 790,000 lines of code. It predicts a productivity of about 280 lines of code per Staff Month for a project with 1,000 lines of code.

Basic COCOMO Embedded is quite similar to the numbers for Military Airborne, Missile, and Space.

Software productivity numbers are close to meaningless without an associated measure of the quality of the software. Reifer uses the number of bugs/errors/defects per thousand equivalent source lines of code (KESLOC). The error rates upon delivery to the customer show the difference between Web Business and the other categories. When the quality must be high, ideally no errors for mission critical life/death software such as airplane control software (avionics), then the number of lines of code per Staff Month is correspondingly lower.

Application Domain	Number Projects	Error Range (Errors/KESLOC)	Normative Error Rate (Errors/KESLOC)	Notes
Command & Control	45	0.5 to 5	1	Command centers
Military — All	125	0.2 to 3	< 1.0	See subcategories
— Airborne	40	0.2 to 1.3	0.5	Embedded sensors
— Ground	52	0.5 to 4	0.8	Combat center
— Missile	15	0.3 to 1.5	0.5	GNC system
— Space	18	0.2 to 0.8	0.4	Attitude control system
Scientific	35	0.9 to 5	2	Seismic processing
Web Business	65	4 to 18	11	Client/server sites

Table 8 (Abridged): Error Rates upon Delivery by Application Domain

Quality Requirements for Mathematical Software

The required quality for many types of mathematical software is often very high, meaning less than one error per thousand lines of code. For example, a video codec such as used by YouTube or Skype, generates the output, the video, seen and used by the customers. Almost any bug in a video codec will result in visible artifacts at best and often completely destroys the video. Many readers have probably noticed occasional blurriness or other anomalies in YouTube or other Web video; these are problems that remain after extensive debugging of the video software.

Many video, image, and audio processing applications have similar quality requirements to video codecs. Similarly, encryption and decryptions such as the Advanced Encryption Standard (AES) usually requires extremely high quality since even a single bit error will result in gibberish. Many other types of mathematical software require similarly high levels of quality. Many seem to have quality requirements in practice similar to avionics and other demanding applications modeled by Basic COCOMO Embedded.

Where Are All The Super Programmers?

It is not uncommon in verbal conversations or comments on Web blogs to encounter programmers who claim to routinely write five to ten-thousand lines of code per month. Nonetheless, Reifer’s data shows little evidence of this performance level. With some exceptions, studies of software productivity usually show much smaller numbers.

There is tremendous variation in software projects. The author once implemented the Advanced Encryption Standard (AES) in about one week. This is about 1500 lines of code. This would translate to 6000 lines of code per month if naively extrapolated. However, this was clearly unusual and stands out in the author’s memory precisely because the project went so quickly and smoothly.

It is probably possible to write many lines of working usable code for certain kinds of simple straight-forward business and user interface software. For example, the top productivity for the Web Business category in Reifer’s published data is 985 lines of code/month.

It is clear though that the average performance for the vast majority of software engineers, including most exceptional software engineers, is much less than 5000 lines of code per month for most categories of software projects, with the possible exception of some types of business and user interface software, if one requires reasonable quality.

Conclusion

In the author’s experience, it is common to encounter extremely optimistic ideas about the size, scope, and difficulty level of mathematical software projects. Many people appear to be genuinely unaware of how complex, how many lines of code, many commonly used examples of mathematical software such as video codecs actually are. Similarly, many people seem to be unaware of the quality level needed to produce an acceptable end-user/customer experience such as an enjoyable streaming video. Many people, even technical people who should know better, often seem to consciously or unconsciously use software productivity numbers like 5-10,000 lines of code per Staff Month even though these are not supported by most historical experience.

How should one use models like Basic COCOMO Embedded that are based on historical data or historical software productivity numbers like Donald Reifer’s data? These are good for rough order of magnitude (ROM) estimates including basic sanity checks. If one only has resources for a two week project and Basic COCOMO says the project is a six month project, one should probably reevaluate one’s plans. On the other hand if one has the resources for a six month project and Basic COCOMO says seven months, the difference is probably not meaningful given the large variation between actual effort and estimated effort. The same applies to blindly plugging in numbers like Reifer’s average 195 lines of code per Staff Month for Scientific software.

These models and data are not good for precise scheduling. There is substantial variation between actual and estimated effort. Software seems to inherently involve large variations in effort that are difficult or impossible to predict in advance.

Suggested Reading/References

Barry Boehm, Software Engineering Economics, Prentice-Hall, Englewood Cliffs, NJ, 1981

About the Author

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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Estimating the Cost and Schedule of Mathematical Software

John F. McGowan, Ph.D. — Tue, 10 Jan 2012 19:50:34 +0000

Mathematics and mathematical software combined with today’s powerful computers can deliver large improvements in speed and efficiency as well as new useful features. Mathematical software is in widespread use: digital video such as YouTube and Skype, digital audio such as MP3 files, JPEG images, speech recognition such as Apple’s Siri, computer generated images in movies and video games, and the Global Positioning System or GPS that tells people where they are, all multi-billion dollar markets today.

Mathematical software may offer solutions to many pressing problems such as curing cancer by helping develop systems of smart drugs that perform mathematical or logical calculations to identify and selectively kill cancer cells (see the previous post Animations of a Possible Cure for Cancer).

Mathematical software can also solve many small problems such as the need to relax with entertaining new audio/video effects for computer games and movies (see the previous post, Creating Cartoon Voices with Math).

The successful solution of problems using mathematics and mathematical software usually requires estimating the cost and schedule of mathematical software projects based on historical experience. A good coach and quarterback plan the strategy and plays for a successful football team based on what the players and team can actually do.

Estimating the Cost and Schedule of Mathematical Software

Mathematical software development is an uncommon area, unlike mainstream software development such as business web sites and user interface software development. User interface software, for example, is often extremely easy today. With modern scripting languages like Python and GUI (Graphical User Interface) Builders, it is possible to create working user interfaces rapidly with little risk if one sticks with standard GUI components such as buttons, sliders, and data entry fields. Mathematical software development is usually much harder than modern user interface software development — taking longer per line of code, involving much more debugging — and much less predictable.

Individuals and groups often have extremely optimistic ideas about the size and scope of mathematical software projects. Many people appear to be unaware of the size and complexity of various commonly used examples of mathematical software. For example, the widely used open-source x264 h.264 video encoder is over 62,000 lines of code developed between 2004 and 2011 with at least 18 contributors. The H.264 video compression standard is the video compression used in many YouTube videos, BluRay discs, and other high performance video systems.

The popular open source Independent JPEG Group JPEG image encoder/decoder used by many commercial and open-source image editors and viewers is over 52,000 lines of code developed between 2000 and 2011 with at least 13 contributors.

The LAME MP3 audio encoder, best known as the MP3 encoder plugin for the Audacity audio editor, is over 87,000 lines of code (about 40,000 lines of algorithmic C/C++ code and about 47,000 lines of Bourne shell installer code) developed between 1998 and 2011 with at least nine primary developers and at least 21 total contributors.

A line of code is comparable to at least one moving part in a physical machine. As a point of reference, the Space Shuttle Main Engine, one of the most sophisticated engines in the world, has about 50,000 moving parts. These mathematical programs are comparable in complexity to the most sophisticated physical machines in the world and often fail catastrophically due to tiny errors just as a rocket engine will.

Software engineering expert Barry Boehm’s original software cost estimating model COCOMO (Embedded) which stands for the Constructive Cost Model for Embedded Software Development — appears to give a rough estimate of the time it takes to develop these low level mathematical programs, although there is substantial variation between estimates and actual effort.

The model is:

where Boehm’s Man Month is 152 man-hours (19 man-days) and KDSI is 1000 (Kilo) Delivered Source Instructions (lines of code). Blank lines and comments are not counted.

A few quick numbers from COCOMO Embedded:

1000 lines of code 3.6 man months
2000 lines of code 8.3 man months
5000 lines of code 24.8 man months

Basic COCOMO (Embedded)

If the project is using consultants at an hourly rate, one should multiply the number of man months times 152 hours times the hourly rate of the consultants. If the project is using direct employees, one should use the cost of the employees per month. Boehm’s model omits one man-day per month for the average paid time off of the employee.

Cost = (Estimated Man-Months)*(152 hours)*(hourly rate)

Cost = (Estimated Man-Month)*(Monthly Salary and Overhead)

There is substantial variation between the actual and estimated effort from this simple model. In his book Software Engineering Economics (p. 84), Boehm notes:

From a practical standpoint, it is important to note that Basic COCOMO estimates are within a factor of 1.3 of actuals only 29% of the time, and within a factor of 2 only 60% of the time.

Barry Boehm advises against using his model for projects smaller than 2000 lines of code.

In the author’s experience, shorter projects, such as 1000 lines of code, are still in the same ballpark, on average, as predicted by Basic COCOMO Embedded but there is even more variation between the estimates and actual effort. The author once implemented the Advanced Encryption Standard (AES), about 1500 lines of code, in one week which was much faster than the model would predict or the author’s experience with other projects. The effort varies even more on small projects depending on the details of the algorithm and other factors that are hard to know in advance.

The free open-source CLOC (Count Lines of Code) utility is available for the major programming platforms: Windows, Mac OS X, Linux, and other common forms of Unix. There are now many free open-source programs that implement known mathematics and algorithms such as x264 and the other examples cited above. It is thus often possible to get a rough estimate of the size and scope of mathematical software projects that involve known mathematics and algorithms.

Limitations of Lines of Code

It is important to keep in mind that a line of code (LOC) is a rough estimate of the size and complexity of a computer program. In the C or C++ programming languages, these are both a single line of code:



a = 1;
if ( (a > b && a < c) || d < e) { a = sin(b+c) } else { a = tanh(a + b + c)/(d+e) };

The second example line of code would usually require more actual effort than the first example line. This is one of the reasons cost and schedule estimates based on counts of the lines of code vary a lot compared to the actual effort.

Because of the many problems with using lines of code for cost and schedule estimation, other methods such as function points have been developed. Function points are currently popular in books and articles on software cost and schedule estimation. However, function points were developed for business and user interface software. Function point estimation generally involves counting the number of inputs and outputs to the program such as data entry fields in a business program. This often predicts the actual effort well because many business and user interface programs have simple internal logic or mathematics and the effort is proportional to the number of inputs and outputs of the program. Business software usually uses only basic arithmetic, adding columns of numbers and similar simple operations.

Mathematical programs are generally extremely complex internally but often appear as only a few inputs and outputs. For example, a video compression program takes one input, the uncompressed raw video, and returns one output, the compressed video. Thus, methods such as function points tend to grossly underestimate the size and scope of mathematical software projects.

This weakness of function points has been recognized for many years and there are more advanced versions of the function point method that attempt to better estimate the size and complexity of complex algorithms hidden from the end user. However, it is still better to rely on lines of code for estimating the size and scope of mathematical software, despite the obvious limitations of using lines of code for cost and schedule estimation.

Scripting Languages (Matlab) Versus Low-Level Compiled Languages (C/C++)

One well known way to speed up the development of mathematical software is to use mathematical scripting languages such as Matlab, Mathematica, Octave (a free open-source program that is mostly compatible with Matlab), and many others. These are scripting languages similar to Python or PHP that have large well-integrated libraries of mathematical function combined with a list (e.g. Mathematica) or numerical array/matrix data type (e.g. Matlab).

In the author’s experience, the speed of development of mathematical software using Octave, MATLAB, or similar tools is generally 2-3 times faster on average than C/C++. This is mostly because the number of lines of code is reduced by a factor of 2-3. The Basic COCOMO Embedded model still gives a useful rough estimate of the actual effort required, but the number of lines of code input to the cost model is reduced!

There is a lot of variation in the increased speed of development from using Octave, Matlab, or similar tools, depending on the details of the algorithm and just plain luck. Some algorithms are well adapted to implementation in Octave/MATLAB and the speed of development gain can be 10-20 times the speed to develop in C/C++. Mathworks, which markets MATLAB, plays up cases like this. There are also some algorithms where there is no gain; the Octave/MATLAB code is pretty much the same as the C/C++ code.

Unfortunately, the speed of execution of the programs in Matlab, Octave, and similar tools is often significantly less than compiled code written in C, C++, or similar programming languages. With languages like Matlab that use numerical arrays, the penalty is not as great as it was a decade ago. Some operations such as the Fast Fourier Transform (FFT) often seem to be just as fast in Matlab or similar tools as compiled versions. However, one should generally plan for a penalty of 2-3 times in speed of execution. It is still often not practical due to speed of execution and memory usage problems to develop computationally intensive mathematical software such as video compression programs using tools such as Octave, Matlab, or Mathematica.

Conclusion

Mathematics and mathematical software can deliver large improvements in speed and efficiency as well as new useful features. Success is much more likely with estimates of the size and scope of mathematical software development based on historical experience.

Suggested Reading/References

Barry Boehm, Software Engineering Economics, Prentice-Hall, Englewood Cliffs, NJ, 1981

About the Author

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Animations of a Possible Cure for Cancer

John F. McGowan, Ph.D. — Mon, 31 Oct 2011 10:00:37 +0000

This article is the third in a series on possible ways to use mathematics to cure or treat cancer, that began with Can Mathematics Cure Cancer?. It presents the Bathtub Mechanism, a possible way to kill cells with abnormal numbers of chromosomes, a common characteristic of many cancer cells, in greater detail and presents several animations of the mechanism.

Cancer is the second leading cause of death in the United States. Over five-hundred thousand people died from cancer in 2007. If current trends continue, about one in three of readers will die from cancer.

Since 1971 the United States has spent about $200 billion on research into cancer. The National Cancer Institute has an annual budget of over $5 billion. This is comparable to the Manhattan Project that invented the atomic bomb and the first nuclear reactors continued for forty years. The results have clearly been quite disappointing. Is there a way to get better results from the many years of hard work, billions of dollars, and mountains of knowledge collected? Are there ways to apply today’s powerful computers and mathematics to defeat this disease?

Cancer is now thought to be caused by mutations of genes, cancer genes or oncogenes and tumor suppressor genes, that control complex networks of proteins that regulate the division, growth, and differentiation of cells in the body. Differentiation refers to the process by which cells turn into specialized kinds of cells such as skin, blood, and nerves. As we age, we accumulate mutations of these genes in some cells. It requires several mutations of several different genes to produce most forms of cancer. Many different sets of mutated genes cause cancer.

While a medical doctor or pathologist may identify a cancer as breast cancer or skin cancer, at a molecular and genetic level, skin cancer is thought to be many different cancers caused by many different sets of mutated genes. In total, cancer is now thought to be thousands of different diseases. This makes finding a single chemical similar to penicillin, for example, that can kill all cancers either impossible or very difficult, at least by starting from the individual cancer genes and the proteins they produce.

Even worse, cancer cells are generally thought to become genetically unstable and mutate much more rapidly than normal cells. Hence, the cancer cells begin to evolve in the body and can develop immunity to anti-cancer drugs such as chemotherapy agents.

While cancer varies enormously at the level of genes and proteins, the part level, cancer cells may have common system-level features. For example, pathologists can identify cancer cells or tissues from biopsies under an optical microscope as cancer. Another common characteristic is that many, perhaps all, cancer cells have an abnormal number of chromosomes, often too many. This article considers targeting the abnormal number of chromosomes.

The Bathtub Mechanism, developed by the author several years ago, is an algorithm, which can be implemented by a relatively simple set of molecules, that may be able to selectively destroy cell with an abnormal number of chromosomes. This system of drugs is like a bathtub with several running faucets, one for each chromosome, and a single drain. If there are too many faucets, chromosomes, the water level, the concentration of the cell killer, will rise and overflow the bathtub. If there are the right number, forty-six, or too few, less than forty-six, faucets, the drain can remove the water being added and the water level never rises. The water level remains almost zero; the concentration of the cell killer is far too low to harm the cell.

One can kill cells with too few chromosomes (less than forty-six) by swapping the roles of the drain and the source. The drain is now a feature of the chromosomes. The source is the constant numerical feature of the cells. Thus, if there are too few chromosomes, there are not enough drains to remove the cell killer produced by the source. The bathtub has one big faucet and many small drains, one for each chromosome. The water level, the concentration of the toxin, rises if there are too few drains/chromosomes.

It may be possible to create proteins that react directly with the source and drain features in the cell. On the other hand, it may be necessary to use a source and a drain catalyst that bind to the source and drain features and become active catalysts only when binding to the source or drain features. In this article the first case is considered. The source and drain catalysts are discussed in more detail in the previous two articles.

Molecular Building Blocks of the Bathtub Mechanism

(A (BC)) harmless Precursor
(BC) Cell Killer
B harmless fragment
C harmless fragment

IN Inhibitor Precursor
I Bacteriophage Inhibitor
N harmless fragment

D Drain
IS Inhibitor Source
S Source (on or associated with chromosome, may be a DNA sequence)

The bathtub mechanism requires two features in the cell: a numerical or quantitative feature that is proportional to the number of chromosomes and a feature that is constant in all cells, both normal and cancerous. Some obvious features that probably vary with the number of chromosomes are the telomeres at the end of the chromosomes and the centromeres at the center of the chromosomes.

There are many molecular structures in the chromosomes and associated with the chromosomes. It seems probable, although not certain, that one can find a numerical or quantitative feature that varies with the number of chromosomes that could be used. A more serious problem with the bathtub mechanism is the constant feature that is the same in both healthy cells and cancer cells, especially since cancer cells are thought to be constantly mutating and changing. This may be a show-stopper.

Since the cancer cells may be mutating, it may be impossible to find a constant feature in the cancer cells. The feature could disappear entirely or change in size or number. There is at least one possible way to add such a feature artificially to the cells, both healthy and malignant.

A bacteriophage is a kind of virus that attaches to the exterior membrane of a cell and injects its genetic material into the cell. The bacteriophage’s genetic material then takes over the machinery of the cell and directs it to make more bacteriophages. The bacteriophage consists of a protein sheath that looks something like a science fiction bug with several arms (see animations below) that grab the surface of the cell and a spherical or polyhedral chamber that carries the genetic material.

In principle, one could modify the genetic material of the bacteriophage to create cells (the commonly used E. Coli bacteria, for example) that make not the virus, but the protein sheath with a payload of other proteins or non-coding DNA sequences, in particular DNA sequences that regulatory proteins bind to. These pseudo-bacteriophages would inject their protein or non-coding DNA payloads into cells instead of the genetic material of the naturally occurring bacteriophage. They would not be infectious like a normal virus.

If, and this is a big if, one could modify the protein sheath so it would only inject the protein or non-coding DNA payload into a cell without an inhibitor protein I that is generatd by inhibitor sources (IS) in the payload, one could inject a payload that contained an artificial constant drain feature D and the inhibitor sources IS into the cell. The inhibitor protein I might work, for example, by blocking the arms of the bacteriophage from attaching to the exterior membrane of the cell, which presumably triggers the injection of the payload.

Once the new drain feature was added to the cell, the pseudo-bacteriophages would stop injecting payloads into the cell because it now also contained the inhibitors. Thus, a constant number of features could be added to each cell, both healthy and cancerous.

The Pseudo-Bacteriophage Payload is either a string of protein units or non-coding DNA with repeated sequences of regulatory protein binding sites, drains D and inhibitor sources IS

Series of reactions:

ABC (Precursor) ==> Source (S) (telomere or other chromosome feature) ==> A + BC (Cell Killer)

BC (Cell Killer) ==> Drain (D) ==> B (Harmless Fragment) + C (Harmless Fragment)

IN (Inhibitor Precursor) ==> IS (Inhibitor Source) ==> I (Bacteriophage Inhibitor) + N (Harmless Fragment)

The pseudo-bacteriophage payload is:

DDDDDDDDD(IS)(IS)(IS)(IS)(IS)(IS)(IS)

In the animations below:

The inhibitor I and the inhibitor source IS are represented by the blue spheres in the payload string

The drain D is the orange spheres in bacteriophage payload

The bacteriophage payload is shown as a string of blue and orange spheres in the first four animations below, mostly clearly in the fourth closeup animation. The inhibitors are shown in the second animation as blue spheres on the surface of the cell that prevent the bacteriophage from injecting a second payload string (drain) into a cell.

The payload is a single strand of protein sub-units or non-coding DNA. When the cell divides, the payload should end up in only one daughter cell. The other daughter cell will lack the payload and the inhibitor sources. The pseudo-bacteriophages will then add another payload string with the drain to the drainless daughter cell.

Alternatively, if the payload is a non-coding DNA string, not proteins, it may be possible to integrate the DNA string into the cell’s DNA, the chromosomes, as a single inherited drain. In this case, the drain will be inherited by both daughter cells when the cell divides.

Animations

The following animations illustrate the Bathtub Mechanism, a basic concept. The animations were created by the author using the free POV-Ray (Persistence of Vision Ray Tracing Program) for Windows 3.62 on a PC running Windows XP Service Pack 2. The POV-Ray scene description files contain a very simple mathematical model of the bathtub mechanism. The rendered frames were combined into MPEG-4 video files using the free, open-source ffmpeg video encoding utility. These animations illustrate a basic concept. They are not a quantitative mathematical model or simulation of cells, even at low fidelity.

This animation shows a pseudo-bacteriophage injecting a drain payload into a cell:

This animation shows a pseudo-bacteriophage prevented from injecting a second drain payload into a cell that already has a drain. The blue spheres are the inhibitors that prevent the pseudo-bacteriophage legs from attaching to the cell membrane.

This animation shows a wide angle view of the harmless precursor (red cone with green sphere cap) converted to the cell killer (red cone) by the telomere (yellow end of cylinder) of a single chromosome and then neutralized by the drain payload (shown as a string of orange drain spheres and blue inhibitor source spheres):

This animation shows a closeup view of the harmless precursor (red cone with green sphere cap) converted to the cell killer (red cone) by the telomere (yellow end of cylinder) of a single chromosome and then neutralized by the drain payload (shown as a string of orange drain spheres and blue inhibitor source spheres):

This animation shows a normal cell with forty-six chromosomes (represented by a simple blue sphere for clarity). The drain is represented by a simple green and gray sphere for clarity. The drain is green when it can process a cell killer, converting it to a harmless fragment (represented by a white sphere for clarity) which is excreted by the cell. The drain is black when it is processing a cell killer and cannot convert another. The drain has a maximum throughput. In a normal cell, the drain can remove as many cell killers as are added by the sources, the chromosomes. The concentration of the cell killer, the number in the lower right corner of the animation, remains low, never reaching the lethal level of two-hundred.

This animation shows the cell killer concentration rising and killing a cancer cell with too many chromosomes (represented by two blue spheres for two sets of chromosomes). The cell killer concentration is the number displayed in the lower right corner. The drain cannot remove the cell killers as rapidly as they are added. The concentration rises to the lethal level of two-hundred and the cell disintegrates. The membrane is shown decaying by making it more and more transparent as the cell killer concentration rises.

Future Steps

Many technical details and difficulties have been omitted to present the idea. While it might be possible to research and develop the bathtub mechanism entirely empirically at a laboratory bench through extensive trial and error, it should be possible to substantially accelerate the development process by simulating the molecular mechanisms using today’s powerful computers. In practice, it would probably require careful tuning of the chemical reaction rates in the cell to produce the desired selective destruction of cells with abnormal numbers of chromosomes or other features associated with cancer.

The next logical step is to construct a mathematical model and simulation of the bathtub mechanism in real cells, iteratively increasing the level of fidelity. This would enable evaluation of the feasibility of the concept and of specific variants of the concept, as many variations are possible and more will become evident with detailed simulation and working through of the concept. Perhaps more importantly a detailed simulation would make it easier for specialists in various fields of biology and organic chemistry — chromosomes, bacteriophages, proteins, many others — to see where their expertise could fit into the concept or resolve otherwise intractable problems.

Naturally occurring networks of proteins and other molecules in cells seem to be able to perform many complex mathematical and logical calculations, such as the feedback control networks that seem to malfunction in cancer. While one cannot be certain, it is not unlikely that a relatively simple network of proteins and other molecules can implement the bathtub mechanism or something similar.

Even engineering a single molecule such as genetically engineered insulin for diabetics is a daunting task at present. So a system of even a few molecules would be a substantial and difficult undertaking. Nonetheless it is probably doable now or in the near future.

However, the underlying biology is unknown. Even though there are over one-million research papers on cancer, it is difficult to get a clear picture of the role of aneuploidy in cancer. Most modern cancer research is conducted within the framework of the oncogene theory and an implicit assumption that the way to cure or treat cancer is to target either a protein generated by a cancer gene or the gene directly.

Chromosomal anomalies, both abnormal numbers of chromosomes and the rearrangements of chromosomes that are common in many cancers, are usually discussed as an aside to the putative cancer genes. This translocation of chromosome X mutated the key cancer gene ABC, or the duplication of chromosome X resulted in two copies of the key cancer gene ABC.

It could be that killing cancer cells with the wrong number of chromosomes would have no effect on the disease. It would simply result in a cancer with the correct number of chromosomes in the surviving cancer cells. It could slow the disease if the abnormal number of chromosomes is related to the malignancy of the cancer cells. In the best case, it might cure the disease, if the abnormal number of chromosomes is either the cause of cancer or essential in some way to the malignant characteristics of the cancer cells.

Conclusion

Everyone faces about a one in three chance of dying from cancer. Cancer researchers would like more impressive results to show policy makers and the general public, especially when seeking continued or increased funding. Pharmaceutical and biotechnology companies should desire improved anti-cancer drugs and treatments to maintain and increase their profits. Defeating cancer would free up resources and researchers to tackle other diseases of old age and even the aging process itself.

The bathtub mechanism discussed in this article is one possible example of such a system of smart drugs. Mathematics and computers can enable or greatly accelerate the development of such systems of smart drugs.

Given the multitude of cancer genes and tumor suppressor genes that have been discovered in the last forty years, we should look at other aspects of cancer such as possible system level features for a cure or effective treatment. Today’s powerful computers, mathematics, and physics combined with the vast biological knowledge acquired in the last forty years may make it possible to attack cancer successfully in ways that were not practical even a few years ago.

About the Author

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Tackling Cancer with Math

John F. McGowan, Ph.D. — Fri, 07 Oct 2011 08:28:33 +0000

The recent death of Apple founder Steve Jobs from pancreatic cancer at the young age of fifty-six highlights the dismal progress in the War on Cancer, despite over $200 billion, over one million published research papers, and the efforts of hundreds of thousands of highly qualified, hard working, committed researchers since 1971.

Steve Jobs inspiring commencement address to Stanford University is also a poignant reminder of the ephemeral nature of words like “cured” and “curable” in cancer research and treatment. Steve Jobs may well have believed his rare form of pancreatic cancer was “cured” or “curable” as he claimed his doctors told him.

Steve Jobs death also highlights the limited benefits of today’s extremely powerful computers and electronics in fields outside of computers and electronics. Despite the frequently hyped promise of multi-Gigahertz and multi-core CPUs, these impressive chips have rarely translated into substantial progress in medicine, power, propulsion, and other essential areas.

One need only consider the many tragic deaths from cancer, the current rising energy prices, and the seeming wars over dwindling supplies of inexpensive oil and natural gas that plague the world today. Steve Jobs and his team at Apple have created many impressive gadgets such as the iPhone and iPad, but they were unable to exploit their computer expertise to defeat cancer. Is there a better way? Can we harness the unused power of today’s computers to solve these pressing problems?

The enormous power of today’s computers is useless without concepts, mathematics, and algorithms that use this power to solve real problems.

There has been impressive progress in some areas including video compression such as the H.264 and related standards used by YouTube, Skype and many other tools, audio compression such as MP3, image compression such as the widely used JPEG standard, computer generated images for movies, television, and computer games, the Global Positioning System or GPS that tells people where they are, and even speech recognition which is slowly finding some practical use despite many difficulties.

There is currently a fad to develop and implement recommendation engines such as Netflix’s Cinematch system to recommend purchases to customers using advanced statistical methods.

At best recommendation engines can increase sales by only a tiny amount, a few percent, and can never solve critical, trillion dollar market size, problems such as cancer, the diseases of old age, and energy shortages. Improved video compression in the form of video conferencing tools such as Skype may well help solve the current energy crisis. Video conferencing, however, cannot substitute for most energy needs. Other advances are needed. As Steve Jobs death shows, many major problems have not been solved at all.

This article discusses some ways that math and computers might be used to develop a cure for cancer. It is a follow-on article to the previous article Can Mathematics Cure Cancer?

This article discusses ways that mathematics might be used to identify and selectively destroy cancer cells. It discusses a specific approach and algorithm, “The Bathtub Mechanism,” that may be able to selectively kill cells with an abnormal number of chromosomes, a common feature of many cancers, and presents a sketch of some ways this algorithm might be implemented using cellular and molecular building blocks that may be known to present day biology, avoiding the need to construct nanorobots, something still far in the future.

All About Cancer

The current prevailing theory of cancer is the oncogene or “cancer gene” theory. This is viewed as a proven fact by many molecular biologists. Cancer is now said to be hundreds, even thousands of different diseases. While a medical doctor or pathologist may identify something as “breast cancer” or “skin cancer” or a similar general category, at a molecular and genetic level, “breast cancer” is actually many different diseases.

It is thought that cancer is caused by the accumulation of many mutations of many different oncogenes and tumor suppressor genes that control complex networks of proteins that direct the growth, functioning, and differentiation of cells. In biology, differentiation refers to the process by which cells “differentiate” during growth into various specialized types of cells such as neurons in the brain, blood cells, and skin cells with different specific properties and functions.

There may be system level features of cancer cells that identify them. Traditional chemotherapy drugs were designed to kill dividing cells on the theory that cancer cells divide rapidly. However, healthy cells divide also and traditional chemotherapy has very limited benefits for most cancers. Only surgical removal of a tumor before it spreads — becomes metastatic in cancer jargon — appears to be able to cure cancer using the common sense definition of “cure”. While targeting cell division largely does not work, targeting other system level characteristics of cancer may work.

It may be possible, with great difficulty, to produce a small system of interacting drugs that perform a mathematical or logical calculation in the cell and selectively kill cancer cells or probable cancer cells while sparing normal cells. It is here that mathematics may be of use. To achieve success in the near future, the simpler the mathematics the better. Even engineering a single molecule such as genetically engineered insulin for diabetics is a daunting task at present. So a system of even a few molecules would be a substantial and difficult undertaking.

The Selective Destruction of Cells with Abnormal Numbers of Chromosomes

(NOTE: This section largely repeats the section with the same title in the previous article Can Mathematics Cure Cancer? If you are familiar with the concept, you may skip this section and jump to the following section which discusses how to implement the bathtub mechanism.)

One historical theory, now out of favor, is that the abnormal number of chromosomes causes cancer. This theory is usually credited to the German biologist Theodor Boveri. The most prominent modern advocate of the role of aneuploidy and chromosomes in cancer is the extremely controversial researcher Peter Duesberg who has published some articles on his theories in cancer research journals and a popular article in Scientific American in 2007 (“Chromosomal Chaos and Cancer”, Scientific American, May, 2007).

A number of other researchers such as Angelika Amon at MIT have been investigating the role of chromosomes and aneuploidy in cancer in recent years; references are given in the previous article Can Mathematics Cure Cancer?.

The abnormal number of chromosomes or the other chromosomal anomalies often seen in a wide range of cancers may be a system-level characteristic of cancer that could be targeted despite the extreme variation in gene-level mutations (part-level characteristics of cancer).

Chromosomal anomalies, both abnormal numbers of chromosomes and the rearrangements of chromosomes that are common in many cancers, are usually discussed as an aside to the putative cancer genes. This translocation of chromosome X mutated the key cancer gene ABC, or the duplication of chromosome X resulted in two copies of the key cancer gene ABC.

It could be that killing cancer cells with the wrong number of chromosomes would have no effect on the disease. It would simply result in a cancer with the correct number of chromosomes in the surviving cancer cells. It could slow the disease if the abnormal number of chromosomes is related to the malignancy of the cancer cells. In the best case, it might cure the disease, if the abnormal number of chromosomes is either the cause of cancer, essential to the malignant nature of the cancer cells, or simply always associated with malginancy for some other reason.

It may be possible to kill cells with an abnormal number of chromosomes using a system of five molecules: a harmless precursor A, a source catalyst S, a cell killer B, a drain catalyst D, and a neutralized cell killer C that the cell can safely digest or excrete.

The source catalyst S is inactive until it bonds to a numerical or quantitative feature on the chromosomes such as the telomeres at the ends of the chromosomes or the centromeres at the center. It becomes an active catalyst S* when it bonds to the chromosomes. Then the activated catalyst S* catalyzes the conversion of a harmless precursor A into a cell killer B. The activated catalyst S* has a maximum throughput. If the concentration of the precusor A is high enough in the cells, the catalyst S* will add the cell killer to the cell at a rate proportional to the number of chromosomes in the cell.

The cell killer B is relatively harmless in low concentrations. It needs to build up to a high level to kill the cell. So far, this will happen in all cells. However, if there is a drain catalyst D that bonds to a numerical feature in the cell that is the same in both normal cells and abnormal cells (cancer cells) and becomes an active drain catalyst D* that removes the cell killer B by converting it to the neutralized cell killer C, then the concentration of B can be engineered to rise to lethal levels only in cells with too many chromosomes.
A ==>S*==> B
B ==>D*==> C

This system of drugs is like a bathtub with several running faucets, one for each chromosome, and a single drain. If there are too many faucets, chromosomes, the water level, the concentration of the cell killer B, will rise and overflow the bathtub. If there are the right number, forty-six, or too few, less than forty-six, faucets, the drain can remove the water being added and the water level never rises. The water level remains almost zero; the concentration of the cell killer B is way too low to harm the cell.

One can kill cells with too few chromosomes (less than forty-six) by swapping the roles of the drain and the source. The drain catalyst bonds to the chromosomes. The source catalyst bonds to the constant numerical feature of the cells. Thus, if there are too few chromosomes, there are not enough activated drains to remove the cell killer B produced by the source catalyst. The bathtub has one big faucet and many small drains, one for each chromosome.

In principle, one could eliminate all cells with either too many or too few chromosomes by first treating the patient with a system of drugs that kills cells with too many chromosomes and then a system of drugs that kills cells with too few chromosomes. Cancer cells are frequently reported to have too many chromosomes, but sometimes too few is also reported.

A computational system of this type would now (2011) be easy to implement using mechanical components like the gears and springs used in traditional mechanical clocks, vacuum tubes and other traditional analog electronics components, or an integrated circuit. The problem is that as simple as such a computational system is, it is extremely challenging to implement using our current ability to engineer proteins and molecular biological systems in the cell.

How to Implement the Bathtub Mechanism

The bathtub mechanism requires two features in the cell: a numerical or quantitative feature that is proportional to the number of chromosomes and a feature that is constant in all cells, both normal and cancerous. It is sometimes reported that cancer cells have abnormal numbers of antigens on the membranes of the cells. Hence, the bathtub mechanism may not require a feature that varies with the number of chromosomes, but this article is about targeting abnormal numbers of chromosomes rather than antigens.

Some obvious features that probably vary with the number of chromosomes are the telomeres at the end of the chromosomes and the centromeres at the center of the chromosomes. These are both involved in cell division. There should be concern that the source or drain catalyst binding to the telomere or centromere may interfere with cell division. The bathtub mechanism must kill all the cancer cells and spare most or all of the healthy cells. It may be possible to use the telomeres or centromeres, but it could be impossible.

A more promising feature may be some of the non-coding sequences in the chromosome DNA, the so-called “junk DNA.” It is currently thought that the vast majority of DNA in the chromosome has no function. On theoretical grounds, the author finds this implausible as do many. However, the genes that appear to code for the proteins in the body seem to comprise only a few percent of the DNA in the chromosomes. The rest seems to do nothing. Sequences of non-coding DNA are used in DNA profiling, for example. Depending on the actual function of the junk DNA, if any, it may be possible to safely bind a source or drain catalyst to non-coding sequences that vary in quantity with the number of chromosomes.

Bacteriophage P2 using Transmission Electron Microscope

A bacteriophage is a kind of virus that attaches to the exterior membrane of a cell and injects its genetic material into the cell. The bacteriophage’s genetic material then takes over the machinery of the cell and directs it to make more bacteriophages. The bacteriophage consists of a protein sheath that looks something like a science fiction bug (see pictures) with several arms that grab the surface of the cell and a polygonal chamber that carries the genetic material.

3D Model of T4 Bacteriophage

In principle, one could modify the genetic material of the bacteriophage to create cells (the commonly used E. Coli bacteria, for example) that make not the virus, but the protein sheath with a payload of other proteins. These pseudo-bacteriophages would inject their protein payloads into cells instead of the genetic material of the naturally occurring bacteriophage. They would not be infectious like a normal virus.

If, and this is a big if, one could modify the protein sheath so it would only inject the protein payload into a cell without an inhibitor protein I that is part of the payload, one could inject a payload that contained an artificial constant feature F and the inhibitors I into the cell. Once the new feature that the drain or source catalysts would bind to was added to the cell, the pseudo-bacteriophages would stop injecting payloads into the cell because it now also contained the inhibitors. Thus, a constant number of features could be added to each cell, both healthy and cancerous.

Math and Computers

This is a simplified sketch of the bathtub mechanism, a basic concept. Many technical details and difficulties have been omitted to present the idea. While it might be possible to research and develop the bathtub mechanism entirely empirically at a laboratory bench through massive trial and error, it should be possible to substantially accelerate the development process by simulating the molecular mechanisms using today’s powerful computers. In practice, it would probably require careful tuning of the chemical reaction rates in the cell to produce the desired selective destruction of cells with abnormal numbers of chromosomes or other features associated with cancer.

One should not expect the computer simulations to be perfect. They would probably be far from perfect at first. Rather, the use of mathematical models and computers should be part of an iterative process in which the models and simulations are continuously compared to laboratory bench experiments and improved. The basic concept may also need to be modified iteratively as new data is collected. This has been the usual process in most genuine breakthroughs.

Conclusion

It may be possible to cure or effectively treat cancer with a system of smart drugs that perform a simple mathematical or logical calculation to selectively destroy cancer cells or probable cancer cells while sparing normal healthy cells. These systems of smart drugs may be able to identify system level features of cancer cells independent of the confusing plethora of cancer genes and tumor suppressor genes. The bathtub mechanism discussed in this article is one possible example of such a system of smart drugs. Mathematics and computers can enable or greatly accelerate the development of such systems of smart drugs.

The author suggests that cancer researchers, business leaders, and policy makers should direct a significant amount of time and resources to the investigation of such systems of smart drugs. This should be a diversified effort not focusing on any one particular approach such as the bathtub mechanism. While there should be some redundancy, there is probably no point in having dozens of competing research groups all trying the same basic approach as seems to be the case with the current attempts to apply differential equations to modeling the growth and spread of cancer, the major current example of applying mathematics to cancer research and treatment. A more diverse effort that is willing and able to question more assumptions is more likely to succeed based on the history of scientific research and technological development.

The successful application of mathematics and computers to cancer and biology requires a professional working relationship based on mutual respect between experts in several fields: computers, mathematics, physics, and traditional biology. The recent appearance of extremely powerful computers presages a sea change in biology and many other fields where computers and mathematics play a much more important role than in the past. Computer experts, mathematicians, and physicists need to respect the hard earned experience of traditional biologists. There is no way the bathtub mechanism could be implemented successfully, if possible, without the expertise of molecular biologists, cell biologists, organic chemists, and others familiar with the detailed structure and function of the chromosomes and cells in the human body. The same can be said of other possible systems of smart drugs and algorithms that may be able to selectively kill cancer cells.

So too, biologists need to respect the expertise of computer experts, mathematicians, and physicists. Successful mathematical modeling is usually a tedious, time consuming process taking months or years, typically longer than many quick few week biology experiments. Even a Nobel Prize in molecular biology or other impressive credentials does not make one an expert in mathematical modeling or other techniques that will be needed to apply mathematics and computers successfully to cancer and other problems. Management level issues such as technical feasibility, scope, difficulty, and complex technical issues will arise in a collaboration between biologists and mathematicians. These will need to be discussed freely in an adult manner to succeed.

There are many pressing problems in the world today like cancer. As current headlines attest, we are doing a poor job solving many of these problems. For the most part, the enormous power of today’s computers has not been applied successfully to these problems. In some cases, there has been no attempt. In other cases, the favored approaches have failed despite decades of effort and genuinely new or simply unpopular ideas should be tried. The War on Cancer is probably an example of the latter case.

Steve Jobs will be remembered for entertaining gadgets like the iPad, the iPhone, and the Macintosh. What an accomplishment it would be if these gadgets went on to successfully solve major problems like the cancer that felled their creator.

© 2011 John F. McGowan
About the Author
John F. McGowan, Ph.D. solves problems by developing complex algorithms that embody advanced mathematical and logical concepts, including video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

Credits

http://commons.wikimedia.org/wiki/File:Bacteriophage_P2.jpg

English: Bacteriophage P2 using Transmission Electron Microscope
Author: Mostafa Fatehi
This file is licensed under the Creative Commons Attribution 3.0 Unported license.

http://commons.wikimedia.org/wiki/File:T4_rendered.jpg
An artist’s rendering of a T4 bacteriophage.
Source Self-modeled in Blender.
Author: Mysid
This file is in the public domain

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The Cold Hit Problem

John F. McGowan, Ph.D. — Sun, 25 Sep 2011 22:31:13 +0000

The previous article Are Fingerprints Unique? discussed the case of Brandon Mayfield, a Muslim American attorney from the Portland, Oregon area who was wrongly identified as one of the Madrid train bombers in 2004 by the FBI based on an erroneous fingerprint identification.

The Mayfield case is probably the most famous case of an incorrect fingerprint identification. The Mayfield case is an example of a “cold hit” in which a huge biometric database was searched for a possible match to an unknown fingerprint taken from a crime scene. Unlike suspects with plausible links to the crime, there was nothing specific to connect Mayfield to the crime other than the database search match.

There are subtle and serious mathematical and statistical problems with cold hits, which occur with both DNA profiling and fingerprint identification. This article explores in detail the mathematics and statistics of the cold hit problem.

The cold hit problem is closely related to a well-known problem in probability and statistics known as the birthday problem. Imagine a room full of people: Bob, Frank, Mary, Estelle, and others. Each person has a birthday: May 1, December 13, March 11, July 17, and so on.

Not knowing the birthdays of the people in the room, what is the probability that at least two people in the room have the same birthday? How many people need to be in the room for there to be an even (50/50) chance that at least two people in the room have the same birthday?

A naive and incorrect answer would be to reason as follows. There are three-hundred and sixty-five (365) days in the year. The probability that two people have the same birthday is 1/365. Therefore, the probability that at least one pair of people in a room with N people have the same birthday is about N/365. Thus the room needs about 183 people for an even chance of a match. The actual answer is twenty-three (23) people, much smaller than 183!

Let us consider the problem in detail. First, what is the probability that Bob and Frank have the same birthday? There is a 1/365 chance that Bob was born on January 1. There is a 1/365 chance that Frank was born on January 1. Thus, there is a 1/(365*365) chance that both Bob and Frank were born on January 1. There are, however, three hundred and sixty-five days in the year, so the probability that Bob and Frank were born on the same day is 365/(365*365) or 1/365.

We need to find the probability that at least one pair of people in the room (Bob and Frank, Bob and Mary, Bob and Estelle, Frank and Mary, Frank and Estelle, Mary and Estelle, and all other possible distinct pairs) have the same birthday. If there are N people in the room, there will be distinct possible pairs of people. Each pair will have a probability of 1/365 of having the same birthday.

The probability that at least one pair of people have the same birthday is:

P = 1.0 - (Probability that the Pair Does Not Have the Same Birthday)^(Number of Distinct Pairs of People)

which is

P = 1.0 - (Number of Distinct Pairs of People)(Probility that the Pair Does Not Have the Same Birthday)

or 

P = 1.0 - (1.0 - 1/365)^(N(N-1)/2)

It turns out that P is 0.50048, almost exactly even, for N = 23. The number of distinct pairs of people in the room is proportional to the square of the number of people in the room , not the number of people in the room (N). Hence, it takes far fewer people in the room than one would naively expect for there to be an even chance that at least two people in the room have the same birthday.

Probability At Least Two People in Room Have Same Birthday

The plot of the probability of at least two people in a room having the same birthday was generated using the two Octave scripts below: birthday.m and plot_bday.m.

Octave is a free open-source numerical programming environment that is mostly compatible with MATLAB.

birthday.m


function [p] = birthday(n, m, bTrace)
% p = birthday(n [, m, bTrace])
% probability that at least one pair of members of set of N have same birthday (M days in year)
% n  number of people
% m  number of "days" in year (default value = 365)
% bTrace flag to trace operation of function (default value = false)
%
% (C) 2011 John F. McGowan
% E-Mail: jmcgowan11@earthlink.net
% 

if nargin < 2
	m = 365;
	bTrace = false;
end

if nargin < 3
	bTrace = false;
end

p = 0.0;

p_no_pair = 1.0; % probability no pair of people in the sample have the same birthday

% loop over pairs of people in the sample (room full of people)
% brute force
% for i = 1:n
	% for j = i+1:n
	% p_pair = m*(1/m)*(1/m); % probability i and j have same birthday
	% p_no_pair = p_no_pair*(1.0 - p_pair);
	% end
% end

% fast
number_pairs = n * (n-1)/2;
p_pair = m*(1/m)*(1/m);
p_no = 1.0 - p_pair;
if bTrace
	printf("number_pairs: %d  p_pair: %f p_no: %f\n", number_pairs, p_pair, p_no);
	fflush(stdout);
end % if

p_no_pair = p_no_pair*power( p_no, number_pairs);

p = 1.0 - p_no_pair;

end % function

plot_bday.m


% plot probability of at least two people having the same birthday
% in a room full of N people
%
% (C) 2011 John F. McGowan, Ph.D.
% E-Mail: jmcgowan11@earthlink.net
%

p = zeros(1,100);

for i=1:100
	if mod(i, 10) == 0
		printf("processing %d people in the room\n", i);
	end
	p(i) = birthday(i);
end

printf("displaying graph");
fflush(stdout);

figure(1);
plot(p);
title('Probability At Least Two People Have Same Birthday');
ylabel('P');
xlabel('Number of People in Room');

printf("writing plot to file prob_bday.jpg");
fflush(stdout);

print('prob_bday.jpg');

What does the birthday problem have to do with fingerprint identification, DNA profiling, or other forms of biometric identification? Replace the people in the room with fingerprints or other biometric identifiers (DNA profiles, iris images, faces,...) in a database.

Replace the three-hundred and sixty-five distinct birthdays with thousands, millions or more distinct biometric identification codes derived from the fingerprint, DNA profile, iris, or other form of identification. The pairs of people with the same birthday become pairs of people with the same fingerprint or other biometric identifier: the actual criminal who commits a crime and at least one other innocent person.

What happens if a fingerprint database has 100 million people and the chance of two people having the same fingerprint (we are referring to the same partial prints such as a thumb print lifted from a crime scene) is only one in a trillion ().

Astonishingly, the probability of at least two people in the database having the same fingerprint is almost one (1.0). This is because there are (100,000,000)(99,999,999)/2 possible pairs of people in the database — about five quadrillion (1,000 trillion) possible pairs. Even though the probability of any two people having the same fingerprint is extremely low, at least one misidentification occurring somewhere in the system is almost certain (probability 1.0).

The FBI fingerprint database contains about 200 million people, accumulated since the 1920s, and the probability of two people having identical or indistinguishable partial fingerprints (or even all ten fingerprints) is unknown.

DNA profiles are currently claimed to have a probability of two people having the same profile of about one in ten trillion. With cold hits, with a search of a large database of DNA profiles such as are currently being collected, it is actually likely that there will be incorrect matches somewhere in the system.

Brandon Mayfield probably fell victim, in part, to the counter-intuitive statistics of the birthday problem. As the size of biometric databases collected by governments, law enforcement agencies, intelligence agencies, and private companies grows, the cold hit problem will grow — as the square of the number of entries in the databases.

If everyone, all of the nearly seven billion people on Earth, was in the databases, one could produce a list of all possible suspects based on fingerprint or other biometric identification alone. This could easily be hundreds or thousands or even more people.

How does one handle possible suspects who lack an adequate alibi and could have flown to a crime? How many of those possible suspects will have some tenuous seven degrees of separation connection to the crime? Brandon Mayfield was a Muslim American who had represented an alleged Islamic terrorist in a child custody case: a tenuous but possible connection to the terrorists responsible for the Madrid train bombings. This is the crux of the cold hit problem.

About the Author

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Bad Mathematics: A Trillion Dollar Problem

The Magical Mathematics of Numb3rs

John F. McGowan, Ph.D. — Mon, 29 Aug 2011 11:00:51 +0000

Numb3rs is a television show that ran for six seasons on CBS from 2005 to 2010 about FBI agent Don Eppes and his brother Charles, a child math prodigy turned math professor at CalSci (a thinly disguised Caltech), who fight crime with mathematics in a sunny, smog-free TV version of Los Angeles filled with an astonishing number of extremely attractive young women. All six seasons are now available on DVD. Numb3rs features some real mathematics used in solving some real crimes as well as pure science fiction. In many respects, Numb3rs is a techno-thriller that features a mix of real present-day technology, advanced technology that may exist, and technology and mathematics that might plausibly exist in the near future. Mathematics and science is especially well integrated into many episodes in the first and second seasons of the show. Unlike over-the-top science fiction shows like Eureka, a viewer could believe that Numb3rs is a realistic presentation of mathematics and science used in crime fighting and other applications today.

Actor David Krumholtz (Charlie Eppes in Numb3rs)

Real life mathematicians including Caltech statistics professor Gary Lorden consulted for the show. Gary Lorden is listed in the credits for each episode as “Math Consultant”. In later seasons, Stephen Wolfram’s Wolfram Research also provided consulting advice to the series. There are scenes with references to Mathematica, Wolfram’s flagship product, and close-up shots of Wolfram’s magnum opus A New Kind of Science on Charlie Eppes desk. Wolfram also has a Caltech connection; he received his Ph.D. in Physics from Caltech in 1979. Gary Lorden and fellow mathematician Keith Devlin published a popular book The Numbers Behind Numb3rs: Solving Crime with Mathematics in 2007: “A companion to the hit CBS crime series Numb3rs presents the fascinating ways mathematics is used to fight real-life crime.” The shadowy National Security Agency (NSA), probably the largest patron of mathematics and mathematicians in the United States and the world, makes several appearances in Numb3rs.

Millikan Library at Caltech (appears often in Numb3rs)

CBS, Texas Instruments (a leading maker of digital signal processor or DSP chips), and the National Council of Teachers of Mathematics (NCTM) developed the “We All Use Math Every Day” initiative, sometimes abbreviated WAUMED, to inspire students to achieve more in math by showing how the subject is relevant to their lives:

Using the hit CBS television show, Numb3rs, the “We All Use Math Every Day” initiative provides free classroom activities online at cbs.com/Numb3rs that help students understand how the math they are learning in the classroom applies to the real world. The activities explore the math derived from the concepts used to solve cases in the FBI crime-solving show.

The show used the “We All Use Math Every Day” tagline in the opening introduction to the show in the first and second seasons.

How Realistic is Numb3rs?

Although explicitly fiction, in many respects Numb3rs paints a picture of mathematics and science that is similar to ostensibly factual popular science such as Scientific American articles, PBS/Nova video programs, Congressional testimony by leading scientists, and informal discussions at fundraising cocktail parties — unless the scientists or mathematicians are in the rare and unusual position of having to explain an obvious failure to a lay audience:

“Well, Senator, as everyone knows, science is a risky enterprise. Eighty to ninety percent of our research projects fail. Surely your staff briefed you on that; the proposal committee mentioned this clearly in italics in footnote 83 in Appendix C of the Proposal for the New Manhattan Project that Will Produce Miraculous Results by the Next Election.”

In several important respects, Numb3rs is very unrealistic. It is also the case that many people ranging from Silicon Valley executives trying to use mathematical methods for their businesses — for example, the current fad trying to use machine learning for recommendation engines in social networking and search businesses — to practicing scientists and engineers who one might think would know better often have expectations similar to what is portrayed in Numb3rs. These misconceptions almost certainly contributed in a major way to the multi-trillion dollar global housing bubble and crash through the widespread use of invalid mathematical models for the valuation of mortgage-backed securities. With business and political leaders seemingly floundering in the current economic difficulties, these misconceptions may wreak even greater havoc.

Before launching into a critique of Numb3rs, it is important to realize that there have been many successes in applied mathematics and mathematical software including impressive advances in video compression such as used by YouTube and Skype, audio compression such as the widely used MP3 standard, still image compression such as JPEG images, computer generated imagery in movies and video games, the Global Positioning System (GPS) that tells people where they are, and even speech recognition which is finally finding some practical use. Modern computers are extremely powerful, comparable to the supercomputers of previous decades; this power is mostly unused because we do not have the mathematics to put this power to practical use. Today’s powerful computers and new mathematics probably can solve or help solve many pressing problems, even trillion dollar problems such as energy shortages or major diseases such as cancer. Success in solving problems with mathematics requires realistic expectations, realistic planning, and adequate time and resources.

In The Numbers Behind Numb3rs (page 208), the mathematicians Keith Devlin and Gary Lorden, a full professor at Caltech, write:

One thing that is completely unrealistic is the time frame. In a fast-paced 41-minute episode, Charlie has to help his brother solve the case in one or two “television days.” In real life, the use of mathematics in crime detection is a long and slow process. (A similar observation is equally true for the use of laboratory-based criminal forensics as depicted in television series such as the hugely popular CSI franchise.)

Also unrealistic is that one mathematician would be familiar with so wide a range of mathematical and scientific techniques as Charlie. He is, of course, a television superhero — but that’s what makes him watchable. Observing a real mathematician in action would be no more exciting than watching a real FBI agent at work! (All that sitting in cars waiting for someone to exit a building, all those hours sifting through records or staring at computer screens… boring.)

It’s also true that Charlie seems able to gather masses of data in a remarkably short time. In real-life applications of mathematics, getting hold of the required data, and putting it into the right form for the computer to digest, can involve weeks or months of labor-intensive effort. And often the data one would need are simple not available.

(Emphasis Added)

In their discussion of the episode “Manhunt” (Airdate: May 13, 2005,The Numbers Behind Numb3rs, page. 78), in which Charlie Eppes uses Bayesian statistics to predict the actions and location of an escaped killer, Devlin and Lorden also write:

As is often the case with dramatic portrayals of mathematics or science at work, the length of time available to Charlie to produce his ranking of the reported sightings [of the escaped killer] is significantly shortened, but the idea of using the mathematically based technique of Bayesian analysis is sound.

(Emphasis Added)

Real-life mathematics and mathematical software development involves much more time, much more trial and error, much more debugging, and much more risk than depicted in Numb3rs. Scientists often claim an eighty to ninety percent failure rate in their research projects, frequently when explaining an obvious failure to disappointed graduate students, donors, policy makers, and others who expected more. Charlie Eppes almost never fails! There is historical evidence that the failure rate in genuine “breakthroughs” is higher, quite possibly ninety-nine percent or worse. Some of the mathematics that Charlie whips up in a few “television days” in the show would actually qualify as breakthroughs in real-life, notably some mathematics and algorithms for artificial intelligence and pattern recognition (see below). Historically, genuine breakthroughs have usually involved at least five years of effort when successful. To give a recent example, Grigoriy Perelman’s proof of the Poincare Conjecture took him at least seven years. There appear to have been about one hundred failed published attempts to prove the conjecture by mathematicians prior to Perelman’s success.

The reality is, in fact, worse than Devlin and Lorden concede in their book. Numb3rs has several episodes that portray artificial intelligence (AI), pattern recognition, machine learning, and similar technologies far superior to reality at the time the show aired (2005-2010) or even today (2011). In one episode, Charlie whips up an image/object recognition algorithm in a few hours to enable the NSA to track a yellow truck carrying a contraband missile guidance system through their satellite images of LA to a terrorist (“Finders Keepers,” Original Air Date: January 12, 2007). Similarly, remarkably effective face recognition algorithms play a role in several episodes. Many of Charlie’s AI and pattern recognition algorithms and the other pattern recognition technology shown in Numb3rs works much better than the real algorithms and math.

The Specter of 9/11

Numb3rs is a fast-paced entertaining show with sexy, idealistic, highly effective heroes and heroines. Although it is sometimes critical of security agencies like the CIA and powerful institutions like pharmaceutical companies, in many respects it is Hollywood product placement for the post 9/11 world of massive, expensive high-tech surveillance and security measures both overseas and at home — in which mathematics plays an important and growing role. It reminds one of President Eisenhower’s speeches during the 1950′s:

The worst to be feared and the best to be expected can be simply stated.

The worst is atomic war.

The best would be this: a life of perpetual fear and tension; a burden of arms draining the wealth and the labor of all peoples; a wasting of strength that defies the American system or the Soviet system or any system to achieve true abundance and happiness for the peoples of this earth.

Every gun that is made, every warship launched, every rocket fired signifies, in the final sense, a theft from those who hunger and are not fed, those who are cold and are not clothed. This world in arms is not spending money alone.

It is spending the sweat of its laborers, the genius of its scientists, the hopes of its children.

The cost of one modern heavy bomber is this: a modern brick school in more than 30 cities.

It is two electric power plants, each serving a town of 60,000 population.

It is two fine, fully equipped hospitals. It is some 50 miles of concrete highway.

We pay for a single fighter plane with a half million bushels of wheat.

We pay for a single destroyer with new homes that could have housed more than 8,000 people.

This, I repeat, is the best way of life to be found on the road the world has been taking.

This is not a way of life at all, in any true sense. Under the cloud of threatening war, it is humanity hanging from a cross of iron.

Chance for Peace (April 16, 1953)
President Dwight David Eisenhower (shortly after the death of Joseph Stalin)

President Dwight D. Eisenhower

Eisenhower and his advisers were no shrinking violets. They were well aware the world can be a nasty, dangerous place. They presided over a massive military buildup and controversial covert operations in Guatemala, Iran, Vietnam, and other countries. By the end of his Presidency Eisenhower and his advisers found that it was never enough. Even thousands of nuclear weapons, ships, tanks, spies, and what we now know was a massive lead over the Soviet Union was not enough to satisfy what he famously labeled the “military industrial complex” in his Farewell Address. Eisenhower found himself attacked by Republicans and Democrats alike for not spending even more money on guns and preparations for war!

Osama Bin Laden: The Trillion Dollar Man

Following the reported death of Osama Bin Laden, Andrea Millen Rich, writing in the libertarian Reason magazine, estimated the direct cost of getting Bin Laden at $1.1 trillion. Tim Fernholz and Jim Tankersley, writing in The Atlantic estimated the total cost at $3 trillion over fifteen years. Sam Stein of the Huffington Post, citing a Congressional Research Service report of March 29, 2011, put the cost at at least 1.283 trillion.

According to the United States Centers for Disease Control, the leading causes of death in the United States in the calendar year 2007 were:

Number of deaths for leading causes of death

* Heart disease: 616,067
* Cancer: 562,875
* Stroke (cerebrovascular diseases): 135,952
* Chronic lower respiratory diseases: 127,924
* Accidents (unintentional injuries): 123,706
* Alzheimer’s disease: 74,632
* Diabetes: 71,382
* Influenza and Pneumonia: 52,717
* Nephritis, nephrotic syndrome, and nephrosis: 46,448
* Septicemia: 34,828

All homicides, of which terrorist attacks are a small fraction even in 2001, do not make the top ten. In 2007, the Centers for Disease Control listed all homicides as the 15th leading cause of death:

All homicides

* Number of deaths: 18,361
* Deaths per 100,000 population: 6.1
* Cause of death rank: 15

It is worth noting that the US invasion of Iraq in 2003 resulted in a dramatic drop in Iraqi oil production, undoubtedly contributing substantially to the large increases in oil and energy prices in the last decade. So too the US invasion of Afghanistan in 2001 seems to have scuttled any chance of constructing a pipeline for natural gas from Turkmenistan to the Indian Ocean, also undoubtedly contributing to high energy prices.

It is difficult to improve on President Eisenhower’s words today. Bayesian statistical analyses that predict terrorist attacks, even if they work, don’t make up for dwindling supplies of inexpensive oil and natural gas. They don’t feed people. They don’t cure diseases like cancer or prevent heart attacks. How much more could have been and could still be accomplished if today’s powerful computers and new mathematics were applied to substantive problems such as energy, food, and health instead of the will-o’-the-wisp of perfect security or the pseudo-scientific financial engineering that helped cause the current Great Recession? Mathematicians, scientists, business leaders, and policy makers can do better than we have done.

Conclusion

Numb3rs is a fun, entertaining show. If you are a mathematician, it will probably make you feel great about your profession unless you are in the unfortunate position of dealing with an employer, client, investor, or funding agency that expects you to do what Charlie Eppes does in every episode of Numb3rs. Some of the math and science in Numb3rs is completely realistic. Some of the math is somewhat exaggerated. Some of the math is pure science fiction even though it generally seems very real and believable. As Devlin and Lorden admit in their book, the time frame is, in most cases, completely unrealistic.

The world is presently confronted with serious and worsening problems, possibly due to a dwindling supply of inexpensive oil and natural gas. The political and economic leadership of the world appears paralyzed and unable to deal with the problems, bickering over debt ceilings and other silliness. We do have vast unused resources in the computational power of hundreds of millions of computers and other devices. With the proper mathematics and creative thinking, we may be able to harness this power to resolve many of the current problems, without waiting for paralyzed governments or blundering Too Big To Fail banks to act wisely.

Most mathematics and mathematical software has been developed by individuals and small teams working over periods of several months to several years with total costs of tens of thousands to a few million dollars per project. Success requires realistic expectations about the size, scope, difficulty level, and risks of developing and implementing mathematics and mathematical software. In these difficult times, mathematicians and scientists must gain support for realistic projects that can find real solutions to our pressing problems, and honestly reject the fantasy elements of Numb3rs.

Suggested Reading/References
The Numbers Behind Numb3rs: Solving Crime with Mathematics
Keith Devlin, Ph.D. and Gary Lorden, Ph.D.
Penguin Books, New York, 2007

The Shadow Factory: The Ultra-Secret NSA from 9/11 to the Eavesdropping on America
James Bamford
Doubleday, New York, 2008

The Cost of Iraq, Afghanistan, and Other Global War on Terror Operations Since 9/11
Amy Belasco, Congressional Research Service, Washington, D.C, March 29, 2011

Credits
The picture of actor David Krumholtz at the Serenity Premiere is from Wikimedia Commons, licensed under the Creative Commons Attribution 2.0 Generic license.

This image was originally posted to Flickr by RavenU at http://flickr.com/photos/36330825119@N01/45967991. It was reviewed on 10:00, 30 April 2007 (UTC) by the FlickreviewR robot and confirmed to be licensed under the terms of the cc-by-2.0.

Millikan Library at Caltech Image from Wikimedia Commons.

Official Portrait of President Dwight D. Eisenhower, May 29, 1959

(from Wikipedia) This image is a work of an employee of the Executive Office of the President of the United States, taken or made during the course of the person’s official duties. As a work of the U.S. federal government, the image is in the public domain.

The image of Usama Bin Laden (Osama Bin Laden) is from the FBI Ten Most Wanted Poster.

About the Author

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Creating Cartoon Voices with Math

John F. McGowan, Ph.D. — Mon, 15 Aug 2011 11:00:33 +0000

Have you ever wanted to create a humorous or entertaining voice like a cartoon character’s voice for a get-well video, a Valentine’s video, the narration for a DVD of home videos, an advertisement for your business or some other application? This article tells how to create cartoon voices using mathematics to shift the pitch of normal voices. The article includes the Octave source code for an Octave function chipmunk that applies pitch shifting to audio.

The standard audio pitch shifting incorporated in many commonly used audio editors such as the free open-source Audacity editor is presented in detail. The article also shows the results of using a more sophisticated algorithm that produces a more natural sounding pitch-shifted voice similar to the voice of the famous cartoon character Mickey Mouse.

One of the basic concepts and methods of signal and speech processing is the Fourier transform, named after the French mathematician and physicist Joseph Fourier. The basic concept is that any real function can be represented as the sum of the trigonometric sine and cosine functions. For example, a function defined on the region can be expanded as the sum of sines and cosines:

where the coefficients and are known as Fourier coefficients. This is a continuous Fourier Transform.

There is a discrete version of the Fourier Transform, often used in digital signal processing:

where is the index of an array of discrete values such as audio samples, is the value of the th audio sample, is the index of the discrete Fourier coefficients and is the number of discrete values such as the number of audio samples in an audio “frame”. The index is essentially the frequency of the Fourier component. This version of the discrete Fourier Transform uses the mathematical identity:

where

to combine the cosine and sine function components into complex functions and numbers.

In audio signal processing such as speech or music, the Fourier Transform has a straightforward meaning. The sound is broken up into a combination of frequency components. In most instrumental music, this is very simple. The music is a collection of notes or tones with specific frequencies. Percussion instruments and certain other instruments can produce more complex sounds with many frequency components. A spectrogram of a signal such as speech or music shows time on the horizontal axis and the strength of the frequency component on the vertical axis. This is the spectrogram of a pure 100 Hertz (cycles per second) tone:

Spectrogram of 100 Hz Tone

The spectrogram is generated using the specgram function in the Octave signal signal processing package by dividing the signal into a series of overlapping audio frames. Overlapping audio frames are frequently used to achieve better time resolution during signal processing in the Fourier domain. Each audio frame is windowed using the Hanning window to reduce aliasing effects.

The Fourier transform is applied to each windowed audio frame, giving a series of frequency components, which are displayed on the vertical dimension of the spectrogram. Each frequency component is a bin in frequency covering a frequency range equal to the audio sample rate divided by the number of samples in the audio frame. This frequency bin size or frequency resolution of the Fourier transform is about 20 Hz in the spectrogram above (44100 samples per second/2048 samples in an audio frame = 21.533 cycles per second). Because the 100 Hz tone in the example is not perfectly centered in the frequency bin spanning 100 Hz, the tone spreads out in the spectrogram, contributing to other bins as can be seen above. This is a limitation of the discrete Fourier transform which can lead to problems with signal processing such as pitch shifting.

Speech has a much more complex structure than a pure tone. In fact, the structure of speech remains poorly understood which is why current (2011) speech recognition systems perform poorly in realistic field conditions compared to human beings. This spectrogram shows the structure of the introduction to United States President Barack Obama‘s April 2, 2011 speech on the energy crisis: “Hello everybody. I’m speaking to you today from a UPS customer center in Landover, Maryland where I came to talk about an issue that is affecting families and businesses just like this one — the rising price of gas and what we can…”.

President Obama on the Rising Price of Gas

The spectrogram below shows the region from 0 to 600 cycles per second (Hertz). One can see a series of bands in the spectrogram. These bands are located at integer multiples (1, 2, 3, …) of the lowest frequency band, which is often referred to as F0 in the scholarly speech literature. The bands are known as the harmonics. F0 is known as the fundamental frequency. This is the frequency of vibration of the glottis which provides the driving sound for speech and is located in the throat. The glottis vibrates at frequencies ranging from as low as 80 cycles per second (Hertz) in some men to as high as 400 cycles per second (Hertz) in some women and children. This fundamental frequency appears to be loosely correlated with the height of the speaker, higher for short speakers such as children and lower for taller women and men.

The fundamental frequency F0 fluctuates in a rhythmic pattern that is not well understood as people speak. In some languages such as Mandarin Chinese, the changing pitch conveys meaning; a word with rising pitch has a different meaning from an otherwise identical word with falling pitch. In English, a rising pitch at the end of a phrase or sentence indicates that a question is being asked. “The chair.” is pronounced with falling pitch whereas “The chair?” is pronounced with a rising pitch at the end. It is difficult and even sometimes impossible to understand English if the rhythmic pattern of the fundamental frequency or pitch is abnormal.

President Obama on the Rising Price of Gas (to 600 CPS)

This spectrogram shows President Dwight David Eisenhower saying “in the councils of government we must guard against the acquisition of unwarranted influence, whether sought or unsought, by the military industrial complex” from his Farewell Address, January 17, 1961, probably his most famous phrase and his most famous speech today.

Eisenhower on the Military Industrial Complex

This spectrogram shows the spectrogram in the range 0 to 600 Hertz (cycles per second). Again, one can easily see the repeating bands.

Eisenhower on the Military Industrial Complex (to 600 CPS)

Human beings perceive something which we call “pitch” in English which appears closely related to or identical to the center frequency of the F0 band in the spectrogram. The F0 band will be higher in higher pitched speakers such as many women and most children. Both President Obama and President Eisenhower have similar pitches, varying between 200 and 75 Hertz with an average of about 150 Hertz. Nonetheless, their voices sound very different. The F0 band can be as low as 70 or 80 Hertz (cycles per second) in a few speakers. Former California governor and actor Arnold Schwarzenegger used an extremely low pitched voice while playing the Terminator, his most famous role.

In general, low pitched voices tend to convey seriousness and sometimes menace whereas high pitched voices tend to convey less seriousness, although there are exceptions. The voice of the genocidal Daleks in the BBC’s Dr. Who series is both high pitched and menacing at the same time. Cartoon style voices can be created by shifting the pitch of normal speakers. This has been done for the Alvin and the Chipmunks characters created by Ross Bagdasarian Sr.. It is probable that some form of pitch shifting has been used over the years to create some of the voices of the Daleks on Dr. Who. Some robot voices have probably been created by combining pitch shifting with other audio effects.

Traditional Pitch Shifting

Pitch shifting predates the digital era. In the analog audio era, one could shift the pitch of a speaker by playing a record or tape faster or slower than normal. This shifts the pitch but also changes the tempo — speed or rate of speaking — as well. One can achieve a pure pitch shift by, for example, recording a voice performer speaking at half normal speed and then playing the recording back at twice the normal rate. In this case, the pitch will be shifted up by a factor of two and the tempo or rate of speaking will be normal. One can create the Alvin and the Chipmunks high pitched voice in this way using analog tapes or records. One can also create lower pitched voices by appropriately combining the tempo of the original voice and the playback rate of the recording. Although these voices are easily understandable, they have artificial, electronic qualities not found in normal low or high pitched speakers or voice performers intentionally creating a low or high pitched voice. The voice of Walt Disney’s Mickey Mouse was performed by a series of voice artists starting with Walt Disney himself. This high pitched voice sounds much more natural than the Alvin and the Chipmunks voice.

In digital audio, it is possible to shift the pitch of the voice without changing the tempo of the speech. This can be done by manipulating the Fourier transform of the speech, the spectrogram, and converting back to the “time domain,” the actual audio samples. One can simply shift the Fourier components from their original frequency bin in the spectrogram to an appropriate higher or lower frequency bin. For example, if a Fourier component is in the 100 Hz bin, one shifts this Fourier component value to the 200 Hz bin to double the pitch. This must be done for each and every non-zero Fourier component. In general, this will produce a recognizable pitch shifted voice. If the Fourier components are not centered in each bin, which is normally the situation, this pitch shifted voice will have an annoying beat or modulation. It is necessary to perform some additional mathematical acrobatics to compensate for these effects to produce a relatively smooth pitch shifted voice similar to the output of the analog processing described above.

This video is President Obama’s original introduction from his April 2, 2011 speech on the energy crisis. Click on the images below to download or play the videos.

This video is President Obama speaking with his pitch doubled by shifting the Fourier components but without the mathematical acrobatics to compensate for un-centered frequency components:

This video is President Obama speaking with a chipmunked voice; his pitch has been doubled.

This video is President Obama speaking with a deep voice; his pitch has been reduced to seventy percent of normal.

Octave is a free open-source numerical programming environment that is mostly compatible with MATLAB. The Octave source code below, the Octave function chipmunk, implements the standard pitch shifting algorithm in widespread use. The Octave code requires both Octave and the Octave Forge signal signal processing package for the specgram function which computes the spectrogram of the signal.

The videos in this article were created by downloading the original MPEG-4 videos from the White House web site and splitting the audio and video into a MS WAVE file and a sequence of JPEG still images using the FFMPEG utility. Presidential speeches and video are in the public domain in the United States. The original still images were reduced in size by half using the ImageMagick convert utility. The audio was pitch shifted in Octave using the chipmunk function below. The new audio and video were recombined into the MPEG-4 videos in this article by again using the FFMPEG utility. Variants of this pitch shifting algorithm can be found in many programs including the widely used free open-source Audacity audio editor (the Audacity pitch shifting algorithm may be slightly different from the algorithm implemented below):

function [ofilename, new_phase, output] = chipmunk(filename, pitchShift, fftSize, numberOverlaps, thresholdFactor)
% [ofilename, new_phase, output] = chipmunk(filename [,pitchShift , fftSize, numberOverlaps, thresholdFactor]);
%
% chipmunk audio effect (as in Alvin and the Chipmunks)
%
% ofilename -- name of output file with pitch shifted audio
% new_phase -- the recomputed phases for the pitch shift audio (for debugging)
% output -- the pitch shifted audio samples
%
% arguments:
%
% filename -- input file name (MS Wave audio file)
% pitchShift -- frequency/pitch shift (default=2.0)
% fftSize -- size of FFT (default = 2048)
% numberOverlaps -- number of overlaps (default = 4)
% thresholdFactor -- threshold factor for zeroing silence frames
%
% $Id: chipmunk.m 1.44 2011/08/04 01:25:35 default Exp default $
% (C) 2011 John F. McGowan, Ph.D.
% E-Mail: jmcgowan11@earthlink.net
% Web: http://www.jmcgowan.com/
%

if nargin < 2
	pitchShift = 2.0; % frequency shift
end
nPitchShift = uint32(pitchShift*100); % to write output file

if nargin < 3
	fftSize = 2048; % size of audio blocks/FFT size
end

if nargin < 4
	numberOverlaps = 4; % number of overlaps
end

if nargin < 5
	thresholdFactor = 0.002;
end

printf("pitchShift: %f fftSize: %d numberOverlaps: %d thresholdFactor: %f\n", pitchShift, fftSize, numberOverlaps, thresholdFactor);
fflush(stdout);

stepSize = fftSize/numberOverlaps;
phaseShift = 2.0*pi*(stepSize/fftSize);

printf("loading %s\n", filename);
fflush(stdout);

result = char(strsplit(filename, '.'));
filestem = result(1,:);
ext = sprintf("_oct_%d_%d_%d.wav", nPitchShift, fftSize, numberOverlaps);
ofilename = [filestem ext];

[data, sampleRate, bits] = wavread(filename);

freq_resolution = sampleRate / fftSize; % frequency resolution = sample rate / fft size

if columns(data) > 1
	raw_data = data(:,1); % input is stereo with 2 channels in 2 columns of array
else
	raw_data = data; % mono sound input
end
data = [];
clear data; % free memory

mx_input = max(abs(raw_data(:)));

printf("applying fft\n");
fflush(stdout);
%spectrogram = fft(spectrogram);

overlap = fftSize - stepSize;
printf("stepSize: %d overlap is %d\n", stepSize, overlap);
fflush(stdout);

nsamples = length(raw_data);

% hanning window
window = hanning(fftSize); % window the output
window = (numel(window)/sum(window(:)) )*window; % normalize the window

% use Octave signal package specgram function to apply fft to windowed overlapping frames
% [] indicates default window (hanning)
%
[spectrogram, f, t] = specgram(raw_data, fftSize, sampleRate, window, overlap);

printf("spectrogram has dimensions %d %d\n", rows(spectrogram), columns(spectrogram));
fflush(stdout);

% free memory
raw_data = [];
clear raw_data;

intensity = dot(spectrogram, spectrogram, 1); % each column is an audio frame
max_intensity = max(intensity(:));
threshold = thresholdFactor*max_intensity;

speech_frames = intensity > threshold;

printf("speech_frames has dimensions: %d %d \n", rows(speech_frames), columns(speech_frames));
fflush(stdout);

printf("zeroing silence frames...\n");
fflush(stdout);

speech_frames = repmat(speech_frames,rows(spectrogram), 1);

spectrogram = spectrogram .* speech_frames; 

printf("dimensions spectrogram are now: %d %d \n", rows(spectrogram), columns(spectrogram));
fflush(stdout);

printf("computing phase...\n");
fflush(stdout);

% spectrogram is half-array without duplicate fft coefficients
% 1:fftSize/2 rows, number time steps columns
% each row is an fft coefficient
%
magn = 2.*abs( spectrogram ); % magnitude of fft coefficients
phase = arg( spectrogram ); % phase of fft coefficients

previous_phase = zeros(size(phase));
previous_phase(:,2:end) = phase(:,1:end-1);

phaseShifts = (0:(fftSize/2)-1)*phaseShift; % expected phase shift if frequency component is centered in bin
phaseShifts = repmat(phaseShifts', 1, columns(phase));

spec_buf = phase - previous_phase; % change in phase from previous time step
spec_buf = spec_buf - phaseShifts; % difference between change in phase and expected phase change
         % if frequency component is centered in frequency bin

printf("computing phase adjustment\n");
fflush(stdout);
									% handle mapping to -pi to pi range of atan2/arg (below)
phase_adjust = uint32(spec_buf./pi); % 0 if spec_buf between -pi and pi
phase_adjust = phase_adjust + ((phase_adjust >= 0).*(2) - 1).*bitand(phase_adjust,1);

spec_buf = spec_buf - pi*double(phase_adjust);
spec_buf = numberOverlaps*spec_buf./(2*pi);

printf("computing corrected frequencies\n");
fflush(stdout);
% compute corrected frequency
frequencies = repmat(f',1,columns(spectrogram)); % f is row vector when returned by specgram

spec_buf = frequencies + spec_buf*freq_resolution;

corrected_freq = spec_buf;

printf("applying frequency shift\n");
fflush(stdout);

shifted_magn = zeros(size(magn));
shifted_freq = zeros(size(corrected_freq));

oldTime = time;
for k = 1:fftSize/2
	ind = uint32((k-1)*pitchShift) + 1;
	if (ind <= fftSize/2)
		shifted_magn(ind,:) += magn(k,:);
		shifted_freq(ind,:) = corrected_freq(k,:) * pitchShift;
	end
	newTime = time;
	deltaTime = newTime - oldTime;
	if (deltaTime > 1)
		pct = (k / fftSize)*100.0; % percent progress
		printf("frequency shift: processed %3.1f%% %d/%d\n", pct, k, fftSize);
		fflush(stdout);
		oldTime = time;
	end % end if
end

%shifted_freq = corrected_freq * pitchShift;

% now convert from mag and freq to mag and phase
%
printf("computing new phase\n");
fflush(stdout);

spec_buf = zeros(size(spectrogram)); % make sure start with zeros

printf("new phase: assigning shifted frequencies\n");
fflush(stdout);

spec_buf(2:end,:) = shifted_freq(2:end,:);

printf("new phase: subtracting center frequencies\n");
fflush(stdout);

spec_buf(2:end,:) = spec_buf(2:end,:) - (frequencies(2:end,:) );

printf("new phase: dividing by frequency resolution\n");
fflush(stdout);

spec_buf(2:end,:) /= freq_resolution;

printf("new phase: adjusting for overlap\n");
fflush(stdout);

spec_buf(2:end,:) = 2.*pi*spec_buf(2:end,:)/numberOverlaps;

printf("new phase: computing delta phase\n");
fflush(stdout);

delta_phase = spec_buf + phaseShifts;

%delta_phase = phaseShifts;

new_phase = delta_phase;

printf("new phase: adding delta phase\n");
fflush(stdout);

%new_phase = spec_buf;
new_phase = zeros(size(spec_buf));
% % %new_phase(:,1) = spec_buf(:,1);
% % %dc coefficient has no phase (always a non-negative real)
oldTime = time;
ncols = columns(spec_buf);
for i = 2:ncols
	new_phase(2:end,i) = new_phase(2:end,i-1) + delta_phase(2:end,i-1);
	newTime = time;
	deltaTime = newTime - oldTime;
	if (deltaTime > 1)
		pct = (i / ncols)*100.0; % percent progress
		printf("new phase: processed %3.1f%% %d/%d\n", pct, k, fftSize);
		fflush(stdout);
		oldTime = time;
	end % end if
end

spec_buf = [];
clear spec_buf; % free memory

new_spectrogram = zeros(fftSize, columns(spectrogram)); % allocate full fft array for inverse fft

new_spectrogram(1,:) = shifted_magn(1,:); % dc coefficient
new_spectrogram(2:fftSize/2,:) = shifted_magn(2:end,:).*cos(new_phase(2:end,:)) + i*shifted_magn(2:end,:).*sin(new_phase(2:end,:));

new_spectrogram(fftSize/2 + 2:end,:) = conj(flipud(new_spectrogram(2:fftSize/2,:))); % reflect fft coefficients

spectrogram = [];
clear spectrogram;

% INVERSE FFT
%
printf("applying inverse fft\n");
fflush(stdout);

new_data = real(ifft(new_spectrogram))/fftSize; 

printf("dimensions new_data are %d %d\n", rows(new_data), columns(new_data));
fflush(stdout);

new_spectrogram = [];
clear new_spectrogram;

% each column is an audio frame which may overlap with previous audio frame by overlap samples
%

iframe = 1; % start at frame 1
it = 1; % start at first sample of output
output = zeros(nsamples,1); % all rows, 1 column

printf("applying overlap and add...\n");
fflush(stdout);

while( (it+fftSize-1) < nsamples)
	update = (new_data(:,iframe).*window)/numberOverlaps; % row of audio data
	output(it:it+fftSize-1) = output(it:it+fftSize-1) + update(1:fftSize);
	it = it + stepSize; % advance to next time
	iframe = iframe + 1; % advance to next audio frame (column of new_data)
end % while

new_data = [];
clear new_data;

mx = max(abs(output(:)));
%mean = sum(abs(output(:)))/numel(output);

if mx > 1.0
	scale_factor = mx / mx_input;
	printf("scaling output by %f\n", 1.0/scale_factor);
	fflush(stdout);
	output = output / scale_factor;
end

printf("writing shifted audio to %s\n", ofilename);
fflush(stdout);
%
wavwrite(output, sampleRate, bits, ofilename);

disp('ALL DONE');
end % function
%

The screenshot below shows running the chipmunk function in Octave 3.2.4 on a PC under Windows XP Service Pack 2 (Click on the screenshot image to see the full size screenshot). This screenshot shows the function called from the Octave prompt using the default values of the function’s arguments. The argument numberOverlaps controls the mathematics to compensate for the uncentered frequency components. If numberOverlaps is one, there is no compensation. The larger numberOverlaps, the more effective the compensation. The more overlaps, the more computer time and resources required by the pitch shifting. A value of numberOverlaps of thirty-two (32) was used to pitch shift President Obama’s voice in the video above.

Running the Chipmunk Function in Octave

Although easily understandable, these pitch-shifted voices sound somewhat artificial. Indeed, this artificial quality is part of the appeal of the Alvin and the Chipmunk voice.

Pitch Shifting Gets Better

Pitch shifting algorithms have improved. It is now possible to produce voices that sound much more like natural voices at the desired new pitch, very similar to the voice of Mickey Mouse. This video is President Obama speaking with a voice similar to the voice of Mickey Mouse:

This particular pitch shifting algorithm does better with producing natural sounding high pitched voices than low pitched voices.

Conclusion

There are many ways to manipulate voices using mathematics. One of the most common is pitch shifting, which has been described in detail including working source code above. Traditional pitch shifting algorithms give artificial qualities to the pitch-shifted voice. There are now new, improved algorithms that can create more natural sounding pitch-shifted voices. These voices can be used for humor, entertainment, or emphasis in movies, television, video games, video advertisements for small businesses, personal and home video, and in many other applications.

About the Author

Possibly related articles:

Can Mathematics Cure Cancer?

John F. McGowan, Ph.D. — Mon, 11 Jul 2011 21:12:00 +0000

“I will also ask for an appropriation of an extra $100 million to launch an intensive campaign to find a cure for cancer, and I will ask later for whatever additional funds can effectively be used. The time has come in America when the same kind of concentrated effort that split the atom and took man to the moon should be turned toward conquering this dread disease. Let us make a total national commitment to achieve this goal.”

President Richard M. Nixon (State of the Union Address, 1971)

This year, 2011, is the fortieth anniversary of the War on Cancer. President Richard Nixon signed the National Cancer Act on December 23, 1971 inaugurating the “War on Cancer.” Since 1971 the War on Cancer has consumed an astonishing $200 billion, with the annual budget of the National Cancer Institute alone now at over $5 billion. This is comparable to the annual rate of expenditure of the Manhattan Project which invented the first atomic bombs and nuclear reactors between 1939 and 1945, with most of the work and expenditure between 1942 and July 16, 1945, the date of the first atomic bomb test known as Trinity, continued for forty years. The Manhattan Project consumed about $20 billion in 2011 dollars, about $2 billion in 1940′s dollars. The War on Cancer is roughly comparable to ten Manhattan Projects. The results have clearly been very disappointing. Since cancer is one of the leading causes of death, almost everyone would almost certainly like to see much more impressive results than achieved so far.

The War on Cancer was inspired in part by the spectacular success of the wartime Manhattan Project, the subsequent development of the hydrogen bomb (1945-1952), and the then (1971) recent spectacular success of the Apollo Program (1962-1969). This inspired not only the War on Cancer but many other “new Manhattan Projects” such as research into tokamaks and inertial confinement fusion devices for fusion power. Like the War on Cancer, most of these “new Manhattan Projects” have yielded disappointing results, certainly nothing on the scale of the Manhattan Project or the Apollo Program. As discussed in the previous article “The Manhattan Project Considered as a Fluke,” the Manhattan Project appears to have been a fluke, atypical of major inventions and discoveries especially in the success of the first full system tests: the Trinity test explosion on July 16, 1945 and the atomic bombings of Hiroshima and Nagasaki. Theoretical mathematical calculations and primitive numerical simulations seem to have been unusually successful in the case of the Manhattan Project compared to other breakthroughs. The Manhattan Project was probably quite unusual among major inventions and discoveries in other ways as well, but this is less clear. In terms of funding levels and the serious life and death nature of the goal, the War on Cancer is one of the closest analogs to the Manhattan Project among the many “new Manhattan Projects” of the last forty years.

With the widespread availability of extremely powerful computers, there are increasing attempts to apply mathematics and computational methods to biology and to cancer. There is a burgeoning field of “quantitative biology,” which includes its own section on the popular arxiv.org electronic preprint server. In many respects, this is an attempt to replicate the apparent success of theoretical mathematical calculations and early computer simulations in the Manhattan Project (1939-1945), the development of the hydrogen bomb (1945-1952), and the Apollo Program (1962-1969).

This article discusses the application of mathematics to the cure of cancer, the possible use of systems of smart drugs to perform simple mathematical calculations to identify and kill cancer cells, and presents a possible mechanism, developed by the author several years ago, to selectively destroy cells with abnormal numbers of chromosomes (aneuploidy), something common in many forms of cancer.

What does forty years of failure mean?

Such long periods of repeated failure are common in the history of scientific and technological breakthroughs. In most cases, this repeated failure has reflected either a lack of fundamental knowledge or an incorrect assumption or group of assumptions that was widely, even universally, held. While these two categories are not sharply defined and blur together, in general a lack of fundamental knowledge means that the state of knowledge was simply far too primitive to solve the problem. An example of this would be the failure of alchemists for thousands of years to transform base metals into gold or produce an elixir of life, the two major goals of both Western and Eastern alchemy. Today, we can with great difficulty and at great cost convert base metals into gold and the elixir of life remains a distant dream. It is clear in retrospect that the alchemical theory that metals were a composition of mercury and sulphur was grossly in error as were many other concepts of alchemy. Nonetheless, alchemists made many significant technological advances including methods for creating alloys similar to gold, gold and gold-colored coatings, and the discovery of a wide variety of useful materials. These successes, unappreciated today, probably gave the alchemists a false confidence in their theories and knowledge.

The blurriness of the two categories is illustrated by asking what might have happened if the alchemists had questioned and abandoned the mercury-sulphur theory which in various forms was widely held for many centuries. Loosely, the mercury-sulphur theory of metals postulated that metals were composed of mercury and sulphur in varying proportions; gold being mostly mercury, for example. This theory is frequently attributed to the Islamic alchemist Jabir ibn Hayyan (born in about 721 in Tus, Iran, died in about 815 in Kufa, Iraq) also known as Geber in Latin. Based on current knowledge, the alchemists would have had to have abandoned the mercury-sulphur theory and isolated a radioactive material such as uranium or invented batteries and other electrical technologies leading to particle accelerators to have had any hope of achieving their goal, both of which require performing very different experiments from the ones alchemists typically did. Batteries, in particular, could have been invented many centuries before they came into widespread use in the early nineteenth century.

In many cases, in retrospect, it is clear that this pronounced lack of progress in solving a scientific or technological problem was due to an assumption or group of assumptions that were incorrect and widely held. Indeed, often the assumption was something viewed as self-evident, something “everyone knows,” obvious, firmly established by extensive evidence, and so forth. Only in retrospect is it “obvious” that the assumption or assumptions were in error and not well supported by evidence, experience, or logic as most believed. Hence, one should ask whether some widely held, seemingly sensible belief or group of beliefs, supported by “overwhelming evidence” in modern scientific jargon, in biology is not, in fact, wrong.

In principle, the Internet and specific new technologies such as HTML or wiki’s should make it easier for researchers to collaborate and to list all assumptions, both stated and unstated, in a research field and their interdependencies, including links to all supporting raw data, experiments, and logical arguments: something like a Biology and Cancer Assumptions Wikipedia, but more rigorous than Wikipedia. Identifying and evaluating assumptions can be done much more systematically and thoroughly with hypertext, databases, and other Internet and computer software than was possible a few years ago using books, research papers, and conference presentations.

Questioning assumptions, especially assumptions “everyone knows,” is a social and political process. In modern Big Science, certain usually foundational assumptions are closely associated with high status individuals and institutions and are routinely presented with few qualifications or even as proven fact beyond any rational questioning to the public, business leaders, and policy makers: in Scientific American articles, PBS/Nova video programs, congressional testimony, private informal discussions where decisions are often actually made, and so forth.

In modern scientific research, there is pervasive rhetoric about “questioning assumptions” and “thinking outside the box,” but this usually does not apply to the foundational assumptions mentioned above. This rhetoric usually refers to subsidiary assumptions such as which protein to use in a biology lab experiment and similar non-threatening technical minutia. Indeed, many fields that have shown little or no practical results for decades, like cancer research, are periodically swept by fads and fashions in which subsidiary assumptions are replaced, modified, or added.

The discussion of mathematical approaches to curing cancer below generally assumes that modern biology has it right. A discussion of assumptions that might be in error or unorthodox biological concepts is mostly beyond the scope of this article, although the theory that abnormal numbers of chromosomes might play a more important role in cancer than generally thought is discussed. This general acceptance of current assumptions in biology is an important qualification that readers should keep in mind. At least historical experience with many “hard problems” in science and technology would suggest otherwise — that modern biology is “missing something” as physicists like to say.

Current Mathematical Approaches

There are a number of current attempts to apply mathematics to the cure or treatment of cancer. Quite a number are attempts to use differential equations to model the growth and spread of cancers and their response to various treatments and the immune system, fairly similar to the work of Dr. Swanson and Professor Levy described in detail below.

A well known example is Dr. Kristin Swanson’s work at the University of Washington:

We specialize in the mathematical modeling of pathological biosystems – specifically, primary brain tumors known as gliomas. We are currently working on several collaborative projects utilizing both clinical and experimental imaging techniques such as MRI and PET.

Our focus is on: 1) predicting patient-specific tumor growth, 2) seeking patient-specific markers of tumor progression, and, 3) identifying predictors of response to therapy in individual patients.

In plain English, this is an attempt to predict the growth and spread of certain brain tumors using a mathematical model, usually differential equations, in order to carefully target the spreading cancer with radiation or other methods.

Another well known example is the work of Professor Doron Levy at the University of Maryland, College Park. From Professor Levy’s web site:

Together with Peter Lee (Hematology, Stanford University) and Peter Kim (University of Utah) we have been working on combining new experimental data and mathematical models to develop new methods for treating leukemia patients. The type of leukmia that we have extensively studied is chronic myelogenous leukemia (CML).

Our research emphasizes the role of the immune system in the progression of the disease. By now it is known that most patients have an anti-leukemia immune response. It remains a mystery as of why this immune response is incapable of providing a sufficient response to the disease.

Our main work in this field was published in the June 2008 issue of PLOS Computational Biology. In this paper we proposed to vaccinating CML patients using their own blood in order to boost their anti-leukemia immune response. Using mathematical models we showed that the key issue is to time the cancer vaccine based on the dynamics of the immune response of the individual patient. A vaccination that is provided to early or too late in the process (i.e. after diagnosis and the initiation of drug-therapy) will have no noticeable effect. Our calculations suggest that such a procedure may ultimately be used to cure the disease. The work assumes that patients are treated with Gleevec (imatinib) starting from the diagnosis of the disease. A timed vaccine may allow them to stop the drug therapy.

There are many other attempts to apply similar mathematics to the cure or treatment of various cancers including work by Larry Norton at Memorial Sloan-Kettering Cancer Center, Lisette de Pillis at Harvey Mudd College, Vito Quaranta and Alexander “Sandy” Anderson at the Vanderbilt Integrative Cancer Biology Center and the University of Dundee in Scotland, Franziska Michor at Memorial Sloan-Kettering Cancer Center, Sofia Merajver at the University of Michigan, Paul Macklin at the University of Southern California, and many others. An Internet search for “Mathematics and Cancer” will turn up many matches of this type.

The author is not too optimistic about these approaches, although they certainly have merit. It seems that one still would really prefer something that could identify and either safely destroy or somehow render harmless the actual cancer cells. Can mathematics do this or assist in doing this?

Smart Systems of Drugs

The current prevailing theory of cancer is the oncogene or “cancer gene” theory. This is viewed as a proven fact by many molecular biologists. The current (2011) Director of the National Cancer Institute, Dr. Harold Varmus, shared the Nobel Prize in Medicine with J. Michael Bishop for early work on this theory.

Cancer is now said to be hundreds, even thousands of different diseases. While a medical doctor or pathologist may identify something as “breast cancer” or “skin cancer” or a similar general category, at a molecular and genetic level, “breast cancer” is actually many different diseases. It is thought that cancer is caused by the accumulation of many mutations of many different oncogenes and tumor suppressor genes that control complex networks of proteins that direct the growth, functioning, and differentiation of cells. In biology, differentiation refers to the process by which cells “differentiate” during growth into various specialized types of cells such as neurons in the brain, blood cells, and so forth with different specific properties and functions.

One type of breast cancer may have genes A,B,C, and D mutated while another has genes W, X, Y, and Z mutated. Not only this, but the cancers are thought to be continually mutating and evolving in the body, developing immunity to chemotherapy drugs for example. Thus, there does not seem to be a common molecular target that an anti-cancer drug can target in the way that penicillin or other antibiotics can kill a wide range of different bacteria, for example. As a result the latest favored concept in cancer research and treatment has been “personalized” targeted drugs. If one can determine that patient X has genes A, B, C, and D mutated, then in principle one can select or produce a drug specific to this particular type of cancer with A, B, C, and D mutated. This, of course, requires developing not one or a few anti-cancer drugs, but hundreds or even thousands of anti-cancer drugs possibly further personalized for each patient. See, for example, this recent article (“Targeted drugs: the future of cancer treatment?”, The Huffington Post, June 9, 2011).

Personalized cancer treatment has suffered a recent high profile black eye with the retraction of several papers from a research group at Duke University and a front page article in the Friday, July 8, 2011 New York Times (“How Bright Promise in Cancer Testing Fell Apart“, by Gina Kolata, The New York Times, July 8, 2011, page A1). Cancer research has long been characterized by a series of research and treatment fads, with one wonder drug or treatment after another heavily touted for a time, followed by disappointment, and then replaced by a new wonder drug: interferon, interleukin-2, and many others in the last forty years.

Nonetheless, medical doctors and pathologists going back to Hippocrates seem to have been able to identify a single disease as cancer long before modern genetic methods. It may be that there are system level features of cancer cells that do identify them as cancer cells. Traditional chemotherapy drugs were designed to kill dividing cells on the theory that cancer cells divide rapidly. However, healthy cells divide also and traditional chemotherapy has very limited benefits if at all. Only surgical removal of a tumor before it spreads — becomes metastatic in cancer jargon — appears to be able to cure cancer using the common sense definition of “cure”. While targeting cell division largely does not work, targeting other system level characteristics of cancer may work.

Many readers are probably familiar with the concept of nanotechnology and nanorobots, usually associated with K. Eric Drexler. One can envision tiny robots, nanorobots, that enter the blood stream, analyze each cell in turn, and selectively kill the cancer cells. The fabrication of such nanorobots is far beyond our current or near future technology. We are nowhere near implementing a computer central processing unit (CPU) or a robot at a molecular level. Even if we could, we do not know how to program a nanorobot to recognize a cancer cell and distinguish it from a normal healthy cell.

What we might be able to do, with great difficulty, is produce a small system of interacting drugs/molecules that perform some mathematical calculation in the cell and selectively kill cells identified as cancer cells or probable cancer cells while leaving the normal healthy cells alone. It is here that mathematics may be of use. To achieve success in the near future, the simpler the mathematics the better. Even engineering a single molecule such as genetically engineered insulin for diabetics is a daunting task at present. So a system of even a few molecules would be a substantial and difficult undertaking.

There are some current attempts to pursue this approach, notably the Cure Cancer Project associated with Dr. Arnold Glazier, Dr. Emil Frei and others. The Cure Cancer Project accepts that the current main assumptions of biology and cancer research are correct. The notion is to identify an unchanging property of cancer cells that can be targeted by a system of smart drugs. In specific terms, this seems to refer to targeting certain patterns among proteins that are thought to be associated with certain general properties, proliferation and invasivenes, of malignant cancer cells. Dr. Glazier has written a book and made some presentations on his ideas.

The author’s educated guess is that an approach based on a system of drugs, perhaps along the lines of the Cure Cancer Project, is the most likely approach to produce an effective cure or treatment for most common forms of cancer in the near future. (This discussion is for educational and informational purposes only and is not an endorsement of the Cure Cancer Project.) An important caveat is that such approaches generally assume that current biological assumptions and theories such as the somatic mutation theory of cancer, the oncogene theory (a specific instance of the somatic mutation theory) and the Central Dogma of Molecular Biology (DNA is the boss) are correct, making a cure for cancer “just engineering.” Next, this article presents a possible method to destroy cells with abnormal numbers of chromosomes, a condition known as aneuploidy which is common in many cancers.

The Selective Destruction of Cells with Abnormal Numbers of Chromosomes

One common characteristic of many cancers is an abnormal number of chromosomes, known as aneuploidy. This is often an excess number of chromosomes. A normal healthy human cell has forty-six (46) chromosomes. Cancer cells often have more than forty-six chromosomes. This was discovered long before the modern genetic era. One historical theory, now out of favor, is that the abnormal number of chromosomes causes cancer. This theory is usually credited to the German biologist Theodor Boveri. The most prominent modern advocate of the role of aneuploidy and chromosomes in cancer is the extremely controversial researcher Peter Duesberg who has published some articles on his theories in cancer research journals and a popular article in Scientific American in 2007 (“Chromosomal Chaos and Cancer”, Scientific American, May, 2007). A number of other researchers such as Angelika Amon at MIT have been investigating the role of chromosomes and aneuploidy in cancer in recent years; references are given below. The abnormal number of chromosomes or the other chromosomal anomalies often seen in a wide range of cancers may be a system-level characteristic of cancer that could be targeted despite the extreme variation in gene-level mutations (part-level characteristics of cancer).

Even though there are over one-million research papers on cancer, it is difficult to get a clear picture of the role of aneuploidy in cancer. Most modern cancer research is conducted within the framework of the oncogene theory and an implicit assumption that the way to cure or treat cancer is to target either a protein generated by a cancer gene or the gene directly. Chromosomal anomalies, both abnormal numbers of chromosomes and the rearrangements of chromosomes that are common in many cancers, are usually discussed as an aside to the putative cancer genes. This translocation of chromosome X mutated the key cancer gene ABC, or the duplication of chromosome X resulted in two copies of the key cancer gene ABC.

A ==>S*==> B

B ==>D*==> C

This video shows the build up of cell killer B in a cell with seven (7) chromosomes where the normal number of chromosomes is five (5). A smaller number of chromosomes than forty-six is used for demonstration purposes. The cell membrane is represented by a sphere which begins to distort when the cell killer concentration reaches the lethal level. The cell killer molecules are indicated by small green spheres that turn red when the lethal concentration is reached. The membrane disintegrates, killing the cell, and the cell killer disperses.

cancer_07

This graph shows the concentration of the cell killer B as a function of time:

Cell Killer Concentration

This video shows the lack of accumulation of cell killer B in a cell with five (5) chromosomes where the normal number of chromosomes is five (5).

cancer_05

This video shows the accumulation of cell killer B in a cell with three (3) chromosomes where the normal number of chromosomes is five (5). The sources and drains have been swapped as discussed above to kill cells with too few chromosomes.

cancer_03

This is the Octave script that simulates the bathtub mechanism and was used to make the videos above. This script runs successfully on a PC with Windows XP (Service Pack 2) using Octave 3.2.4. GNU Octave is a free, open-source high-level interpreted language, primarily intended for numerical computations that is mostly compatible with MATLAB.

cancer.m

function [result, command] = cancer(N_chromosomes, nsteps, bdisplay, lethal_level, cutoff, bTooFew, N_normal)
% [result, command] = cancer( [N_chromosomes, nsteps, bdisplay, lethal_level, cutoff, bTooFew, N_normal ])
%
% N_chromosomes - number of chromosomes in cell (default 50)
% nsteps - number of time steps to simulate
% bdisplay - boolean flag to display simulation (default false)
% lethal_level - concentration at which cells die (default 200)
% cutoff - number of time steps for cell death (default 10)
%          cell lives this many steps after the lethal concentration is reached
% bTooFew - simulate process to destroy cells with too few chromosomes
%           by using drains that attach to chromosomes and a source somewhere
%           in the cell. (default false)
% N_normal - number of chromosomes in a normal healthy cell (default 46)
%            sometimes set to small values such as 5 for demonstration or debugging
%
% Returns:
%      result -- time series of concentration of cell killer B
%      command -- ffmpeg command to create video of simulation
%
% Examples:
%
% [r, c] = cancer(7, 20, true, 10, 10, false, 5);  % kill cell with more than 5 chromosomes
% [r, c] = cancer(3, 20, true, 10, 10, true, 5);   % kill cell with less than 5 chromosomes
%
% Description:
%
% Basic Demonstration of Concept of Killing Cancer Cells by Chromosome Counting
% "Bathtub Mechanism" (conceived by John F. McGowan, Ph.D.)
% This is an Octave script (see http://www.gnu.org/software/octave/).
%
% N_chromosomes is the number of chromosomes in the cancer cell
%(46 is the number of chromosomes in a normal, healthy human cell).
%
% A (harmless poison precursor) =S=> B (poison) =D=> C (neutralized poison)
% A is harmless (ideally consumed orally or injected into bloodstream of patient).
% S is source on chromosome that catalyzes conversion of A to B
% B is toxic at concentration lethal_level (default 200)
% D is drain somewhere in cell that catalyzes conversion of B to C
% C is harmless.  Cell can safely digest or excrete C molecules.
%
% The sources on the chromosomes convert type A molecules to type B molecules.
% The drain (somewhere in the cell) converts type B molecules to type C molecules.
%
% Just like a bathtub drain, the drain in the cell has a limited throughput
% so if there are too many sources (too many chromosomes), the concentration of
% the toxic molecule B increases until it reaches the deadly lethal_level level and kills
% the cell.
%
% In most cancers, the cancer cells are aneuploid, they have the wrong number of
% chromosomes, usually too many.  If one can target the number of chromosomes in a cell
% one may be able to kill a wide range of cancers using a system of "smart drugs".
%
% In this example, the system of "smart drugs" consists of the poison precursor A, the
% sources (S) which bind to sites on the chromosomes, and the drain (D).  The sources only become active
% and able to catalyze the conversion of A to B when they bind to sites on the chromosomes.
%
% This does not require a complex Drexler style nanorobot or sophisticated pattern recognition
% to identify the cancer cells.
% The protein engineering is, of course, quite challenging, but not necessarily beyond present day
% or near future capabilities.
%
% Actual therapy would probably consist of infusing the source and drain molecules (S and D) into the patient.
% Once the sources and drains had attached to the cells (both healthy and cancerous), the precursor A
% would be introduced for a time until the cancer cells die.
%
% From "Can Math Cure Cancer?" by John F. McGowan, Ph.D., at The Math Blog
%
% Author: John F. McGowan, Ph.D.
% (C) 2011 by John F. McGowan, Ph.D.
% E-Mail: jmcgowan11@earthlink.net
%
%
concentration_a = 1.0;  % concentration of precursor A molecules
concentration_b = 0.0;  % concentration of cell killer B molecules
concentration_c = 0.0;  % concentration of neutralized C molecules

if nargin < 1
	N_chromosomes = 50;  % number of chromosomes in cell (46 is number in most normal, healthy human cells)
end

if nargin < 2
	nsteps = 100;  % number of time steps in the simulation
end

if nargin < 3
	bdisplay = false;  % display graphics of simulation
end

if nargin < 4
	lethal_level = 200;  % lethal concentration of B
end

if nargin < 5
	cutoff = 10; % number of time steps for cell death
end

if nargin < 6
	bTooFew = false;  % kill cells with too many chromosomes
end

if nargin < 7
	N_normal = 46;
end

N = N_normal; 46;  % normal number of chromosomes in a human cell (46)

printf("Normal Number of Chromosomes: %d\n", N);
fflush(stdout);

N_b = 0.0;  % number of molecules of cell killer b in cell
N_c = 0.0;  % number of neutralized molecules C in cell

volume = 1.0;  % volume of cell in arbitrary units

conc = [concentration_b];  % time series of concentration of cell killer B in cell

if bdisplay
	figure(1)  % figure 1 shows the B molecules in the cell
	[x_sphere,y_sphere,z_sphere] = sphere();  % get sphere to represent outer membrane of cell
end

delta_membrane = 0.5/cutoff;

nsteps_lethal = 0;  % number of steps since concentration reached lethal level

for i = 1:nsteps  % main simulation loop

	if bTooFew
		N_b += concentration_a * N;  % source S adds cell killer B to the cell
		delta_c = min(concentration_b, N_chromosomes);  % the drain D removes some cell killer B molecules (B =D=> C)
	else
		N_b += concentration_a * N_chromosomes;  % A =S=> B  (the sources on the chromosomes catalyze conversion of A to the cell killer B)
		delta_c = min(concentration_b, N);  % the drain D removes some cell killer B molecules (B =D=> C)
	end

	N_b = max(0, N_b - delta_c);  % drain converts B molecules to harmless C molecules
	N_c += delta_c;  % update number of neutralized C molecules in cell (cell can safely digest or excrete these molecules)
	concentration_c = N_c / volume;  % compute concentration of neutralized C molecules in cell
	concentration_b = N_b / volume;  % compute concentration of cell killer B molecules in cell

	if concentration_b >= lethal_level
		nsteps_lethal = nsteps_lethal + 1;
	end

	if nsteps_lethal > cutoff
	% cell dies so concentration of B starts to drop
	% membrane destroyed so molecule B disperses
	%
		if nsteps_lethal == cutoff
			printf('CELL DIES AT STEP %d', i);
			fflush(stdout);
		end
		N_b = floor(N_b / 2);
		concentration_b = N_b / volume;
	end

	conc = [conc concentration_b]; 	 % add new concentration of B to time series

	% display the B molecules
	% use ffmpeg (for example) to convert image sequence to video (e.g. mp4)
	% ffmpeg -r 10 -i cancer_50_chromosome_%03d.jpg cancer_50_chromo.mp4
	%
	if bdisplay
		if nsteps_lethal < cutoff  % membrane is destroyed for illustrative purposes (cell dies)
			mesh(x_sphere,y_sphere,z_sphere);
			% hard code limits of axes
%			xlim([-1.0 1.0]);
%			ylim([-1.0 1.0]);
%			zlim([-1.0 1.0]);
			hold on;
		else
			hold off
			% stop displaying the cell membrane after death
			clf(1);  % make sure figure is cleared
		end

		bpos = rand(N_b,3);
		x = bpos(:,1).*cos(2*pi*bpos(:,2)).*cos(2*pi*bpos(:,3));
		y = bpos(:,1).*sin(2*pi*bpos(:,2)).*cos(2*pi*bpos(:,3));
		z = bpos(:,1).*sin(2*pi*bpos(:,3));

		if concentration_b < lethal_level   % cell will die at concentration lethal_level
			scatter3(x, y, z, 12, [0 1 0]);  % use green color when concentration is below the lethal level
		else % lethal concentration level reached
			scatter3(x, y, z, 12, [1 0 0]);  % use red color when concentration is at or above the lethal level
			% distort membrane to indicate it is breaking down
			%
			x_sphere = (1.0 + delta_membrane) * x_sphere + delta_membrane*2*(rand(size(x_sphere))-0.5);
			y_sphere = (1.0 + delta_membrane) * y_sphere + delta_membrane*2*(rand(size(y_sphere))-0.5);
			z_sphere = (1.0 + delta_membrane) * z_sphere + delta_membrane*2*(rand(size(z_sphere))-0.5);
		end
		% hard code limits of axes
%		xlim([-1.0 1.0]);
%		ylim([-1.0 1.0]);
%		zlim([-1.0 1.0]);

		%view(60.0, 322.5);  % hard code view

		mytitle = sprintf('%3d B Molecules  Time Step: %03d', N_b, i);
		title(mytitle);
		frame_name = sprintf('cancer_%02d_chromosomes_%03d.jpg', N_chromosomes, i);
		print(frame_name);
		hold off;
	%	pause(1);  % optional pause of 1 second
	end % display
end % main simulation loop

result = conc;  % return time series of concentration of cell killer molecule B in cell
command = sprintf('ffmpeg -r 10 -i cancer_%02d_chromosomes_%%03d.jpg cancer_%02d.mp4', N_chromosomes, N_chromosomes);

% concentration of B increases without limit in cells with more than 46 chromosomes, reaches lethal level and kills cell.
% concentration of B stabilizes quickly at low level in cells with 46 or fewer chromosomes
%

if bdisplay
	figure(2)  % figure 2 shows the time series of the concentration of cell killer B molecules in the cell
	plot(conc);  % display time series
	mytitle = sprintf('Concentration of Cell Killer B (%d Chromosomes)', N_chromosomes);
	title(mytitle);
	fname_conc = sprintf('cancer_%02d_conc.jpg', N_chromosomes);
	printf("writing concentration time series to %s\n", fname_conc);
	fflush(stdout);
	print(fname_conc);
end

end % function cancer

In principle, the bathtub mechanism may be adapted to selectively kill cells with any abnormal quantity of any cellular feature, such as for example features on the exterior membrane of the cell (in which case the proteins must be trapped in the membrane surface). Thus, it may be adapted to targeting other numerical or quantitative abnormalities of malignant cancer cells than the number of chromosomes or the quantity of chromosomal features such as telomeres.

The presentation in this article is obviously simplified. For example, cells are multiplying and dividing. During cell division, a cell goes through a period where it has two sets of chromosomes before it splits into two cells. However, so long as the ratio of the sources to the drains is sufficiently constant integrated over time, the bathtub mechanism will still work and selectively destroy the dividing cells with too many or too few chromosomes.

This article does not discuss the specifics of how the five different molecules in the system might be implemented, most likely by modifying known molecules. This may be done in future articles or publications; the author does have some ideas how to implement the system in detail. Astute biologists may see how to implement some or all of the steps. It is likely that specialists from different sub-fields of molecular and cell biology will be needed to successfully implement the different parts of the system. Please feel free to contact the author if you have some good ideas.

Conclusion

Mathematics may be able to cure cancer through systems of smart drugs that perform a relatively simple calculation within a cell or on the membrane of the cell and kill or neutralize cells that are or probably are cancer cells. There are some attempts to develop a method along these lines. It does not seem to be the mainstream approach at the moment, which currently appears fascinated with personalized treatment, something of a legacy of the Human Genome Project and other gene-oriented research projects.

The author suspects that some common assumption or assumptions in biology and cancer research are in error or incomplete. This may be reflected not only in the evident lack of practical progress in cancer but also in the mostly failed attempts to cure or treat many other diseases in recent decades including most of the diseases currently identified as autoimmune disorders: Type I diabetes, multiple sclerosis (MS), systemic lupus, rheumatoid arthritis (RA), and so forth. Depending on what assumptions may be in error, all current approaches to curing cancer may be dead ends. If the unorthodox theory that abnormal numbers of chromosomes actually cause cancer is correct or a less drastic variant in which the abnormal number of chromosomes is essential to the malignant nature of cancer is correct, then a method that targets cells with an abnormal number of chromosomes, such as the method outlined in this article, may be successful. It is not clear which assumption or assumptions may be incorrect or incomplete; many other possibilities exist.

It may be possible to use modern Internet and computer software to collaboratively build an on-line database of biology and cancer research assumptions, both stated and unstated, their interdependencies, supporting data, and logical arguments to better identify and reevaluate current assumptions. Questioning assumptions, particularly those which "everyone knows," is a social and political process. It is not at all clear that we have the institutions and the mental outlook among researchers necessary to properly reevaluate assumptions in cancer or many other research fields. The rhetoric of "questioning assumptions" and "thinking outside the box" is easy to find in modern research. It is not "talking the talk," but rather "walking the walk" that is in question. On the other hand, most people and most cancer researchers would surely like to see major progress in the cure or treatment of the disease, since everyone faces a substantial risk of dying from cancer.

About the Author

John F. McGowan, Ph.D. solves problems by developing complex algorithms that embody advanced mathematical and logical concepts, including video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

Suggested Reading/References

Some Questions about Immortal Cell Lines

The Immortal Cell [Paperback]

Gerald B. Dermer (Author)

# Paperback: 212 pages

# Publisher: Avery (January 1, 1995)

# Language: English

# ISBN-10: 0895295822

# ISBN-13: 978-0895295828

# Shipping Weight: 12.8 ounces

A Conspiracy of Cells: One Woman's Immortal Legacy and the Medical Scandal It Caused [Hardcover]

Michael Gold (Author)

# Hardcover: 170 pages

# Publisher: State University of New York Press (January 1986)

# Language: English

# ISBN-10: 9780887060991

# ISBN-13: 978-0887060991

# ASIN: 0887060994

# Shipping Weight: 12.8 ounces

The Immortal Life of Henrietta Lacks [Paperback]

Rebecca Skloot (Author)

# Paperback: 400 pages

# Publisher: Broadway; Reprint edition (March 8, 2011)

# Language: English

# ISBN-10: 9781400052189

# ISBN-13: 978-1400052189

# ASIN: 1400052181

The Tissue Organizing Field Theory (TOFT) of Cancer

Wiley-Blackwell (2011, April 14). Controversial TOFT theory of cancer versus SMT model: Experts debate. ScienceDaily. Retrieved July 9, 2011, from http://www.sciencedaily.com /releases/2011/04/110415083217.htm

Ana M. Soto, Carlos Sonnenschein. The tissue organization field theory of cancer: A testable replacement for the somatic mutation theory. BioEssays, 2011; 33 (5): 332 DOI: 10.1002/bies.201100025

The Society of Cells: Cancer and Control of Cell Proliferation [Paperback]

Carlos Sonnenschein (Author), Anne Marie Soto (Author)

# Paperback: 154 pages

# Publisher: BIOS Scientific Publishers; 1 edition (January 1999)

# Language: English

# ISBN-10: 0387915834

# ISBN-13: 978-0387915838

Chromosomes, Aneuploidy, and Cancer

This is a collection of references, links, and abstracts of article on chromosomes, aneuploidy, and cancer.

Yuen, Karen Wing Yee(Oct 2010) Chromosome Instability (CIN), Aneuploidy and Cancer. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1002/9780470015902.a0022413]

Cancer Research

Published OnlineFirst December 1, 2009; doi: 10.1158/0008-5472.CAN-09-2802 Cancer Res December 15, 2009 69; 9245

Cancer Stem Cells and Aneuploid Populations within Developing Tumors Are the Major Determinants of Tumor Dormancy

1. Anjali P. Kusumbe, and

2. Sharmila A. Bapat

http://cancerres.aacrjournals.org/content/69/24/9245.abstract

Human Molecular Genetics

Constitutional aneuploidy and cancer predisposition†

1. Ithamar Ganmore,

2. Gil Smooha and

3. Shai Izraeli*

+ Author Affiliations

Sheba Cancer Research Center, Sheba Medical Center, Tel-Hashomer, Ramat Gan; Sackler Faculty of Medicine, Tel-Aviv University, Tel Aviv, Israel

1. *To whom correspondence should be addressed at: Pediatric Hemato-Oncology, Functional Genomics and Childhood Leukemia Research Section, Sheba Medical Center, Tel Hashomer 52621, Israel. Tel: +972 35303037; Fax: +972 45303031; Email: shai.izraeli@sheba.health.gov.il

* Received January 19, 2009.

* Accepted February 17, 2009.

Hum. Mol. Genet. (2009) 18 (R1): R84-R93. doi: 10.1093/hmg/ddp084

http://hmg.oxfordjournals.org/content/18/R1/R84.short

Reference:

Identification of aneuploidy-selective antiproliferation compounds. Tang, Y-C., Williams, B.R., Siegel, J.J., and Amon, A. (2011). Cell. 144(4):499-512.

http://www.hfsp.org/frontier-science/awardees-articles/aneuploidy-new-anticancer-target

Mol Cell Biol. 2009 September; 29(17): 4766–4777.

Published online 2009 July 6. doi: 10.1128/MCB.00087-09.

PMCID: PMC2725711

Loss of GATA6 Leads to Nuclear Deformation and Aneuploidy in Ovarian Cancer †

Callinice D. Capo-chichi,1,2 Kathy Q. Cai,2 Joseph R. Testa,2 Andrew K. Godwin,2 and Xiang-Xi Xu1,2*

Sylvester Comprehensive Cancer Center, Department of Medicine, and Department of Obstetrics and Gynecology, University of Miami School of Medicine, Miami, Florida 33136,1 Ovarian Cancer, Human Genetics, and Tumor Cell Biology Programs, Fox Chase Cancer Center, Philadelphia, Pennsylvania 191112

*Corresponding author. Mailing address: University of Miami School of Medicine, Rm. 417, Papanicolaou Building, 1550 NW 10th Ave. (M710), Miami, FL 33136. Phone: (305) 243-1750. Fax: (305) 243-5555. E-mail: xxu2@med.miami.edu

Received January 19, 2009; Revised February 23, 2009; Accepted June 25, 2009.

This article has been cited by other articles in PMC.

* Other Sections?

o Abstract

o MATERIALS AND METHODS

o RESULTS

o DISCUSSION

o Supplementary Material

o REFERENCES

Abstract

A prominent hallmark of most human cancer is aneuploidy, which is a result of the chromosomal instability of cancer cells and is thought to contribute to the initiation and progression of most carcinomas. The developmentally regulated GATA6 transcription factor is commonly lost in ovarian cancer, and the loss of its expression is closely associated with neoplastic transformation of the ovarian surface epithelium. In the present study, we found that reduction of GATA6 expression with small interfering RNA (siRNA) in human ovarian surface epithelial cells resulted in deformation of the nuclear envelope, failure of cytokinesis, and formation of polyploid and aneuploid cells. We further discovered that loss of the nuclear envelope protein emerin may mediate the consequences of GATA6 suppression. The nuclear phenotypes were reproduced by direct suppression of emerin with siRNA. Thus, we conclude that diminished expression of GATA6 leads to a compromised nuclear envelope that is causal for polyploidy and aneuploidy in ovarian tumorigenesis. The loss of emerin may be the basis of nuclear morphological deformation and subsequently the cause of aneuploidy in ovarian cancer cells.

http://ukpmc.ac.uk/articles/PMC2725711

[Proc Amer Assoc Cancer Res, Volume 45, 2004]

Cellular, Molecular, and Tumor Biology 89: Mouse Models of Prostate and Gastrointestinal Cancers

Abstract #4310

The role of the Bub1 gene in aneuploidy and cancer progression

Danaise V. Carrión, Marie Lia, Joerg Heyer, Patrick McDonald, Weijia Zhang, Kan Yang, Martin Lipkin, Ronan O’Hagan, Lynda Chin and Raju Kucherlapati

Harvard Medical School-Partners Center for Genetics and Genomics, Boston, MA, Albert Einstein College of Medicine, Bronx, NY, Strang Cancer Prevention Center, Rockefeller University, New York, NY, Dana-Farber Cancer Institute, Boston, MA

Human colorectal tumors can be classified on the basis of their genomic stability. Some CRC tumors show chromosomal instability (CIN) that is manifested by abnormal chromosome numbers while others show microsatellite instability. The cause for microsatellite instability is considered to result from mutations in DNA mismatch repair genes but the genetic basis for CIN is not well understood. We hypothesized that CIN might result from mutations in genes involved in the mitotic checkpoint. Bub1 is a gene involved in mitotic checkpoint. Mutations in Bub1 have been identified in CRC tumors, thus making this gene an excellent candidate to be involved in CIN and in colorectal cancer. To examine the role of Bub1 in carcinogenesis and chromosomal instability, we generated mice that carry a null mutation in this gene. Mice that are heterozygous for a null mutation in Bub1 are normal. When the heterozygotes were intercrossed, we failed obtain any homozygous offspring suggesting that Bub1 homozygosity leads to embryonic death. Bub1–/– embryos were detected at E3.5 but not at E8.5. We examined chromosome segregation in Bub1+/– and wild type (WT) ES cell lines. At different passages the Bub1+/– line showed a higher percentage of aneuploid cells. To confirm this observation FISH analysis was performed in blood cells of WT and Bub1+/– mice using probes for chromosomes 9 and 17. Modest chromosomal instability was observed in the Bub1+/– samples analyzed. Bub1+/– mice are fertile and susceptible to develop tumors very late in their lives. When Bub1+/– mice were bred to Apc1638N and Msh2 mutant mice, there was no difference in the incidence or multiplicity of tumors suggesting that the genomic instability provided by Bub1 heterozygosity is insufficient to significantly decrease the tumor latency or increase the incidence of tumors in the double mutants. Taken together, these results suggest that the Bub1 gene is essential for normal survival; and that heterozygosity of the gene leads to a mild chromosomal instability phenotype. This degree of chromosomal instability does not significantly affect the phenotype of Apc1638N and Msh2–/– mutant mice.

http://aacrmeetingabstracts.org/cgi/content/abstract/2004/1/994-d

Trends Cell Biol. 2005 May;15(5):241-50.

Aurora kinases, aneuploidy and cancer, a coincidence or a real link?

Giet R, Petretti C, Prigent C.

Source

CNRS UMR6061 Université de Rennes 1, Groupe Cycle Cellulaire, Equipe Labellisée LNCC, Université de Rennes 1, IFR140 GFAS, Faculté de Médecine, 2 Avenue du Pr Léon Bernard, CS 3417, Rennes cedex, France.

Abstract

As Aurora kinases are overexpressed in a large number of cancers, and ectopic expression of Aurora generates polyploid cells containing multiple centrosomes, it has been tempting to suggest that Aurora overexpression provokes genetic instability underlying the tumorigenesis. However, examination of the evidence suggests a more complex relationship. Overexpression of Aurora-A readily transforms rat-1 and NIH3T3 cells, but not primary cells, whereas overexpression of Aurora-B induces metastasis after implantation of tumors in nude mice. Why do polyploid cells containing abnormal centrosome numbers induced by Aurora not get eliminated at cell-cycle checkpoints? Does this phenotype determine the origin of cancer or does it only promote tumor progression? Would drugs against Aurora family members be of any help for cancer treatment? These and related questions are addressed in this review (which is part of the Chromosome Segregation and Aneuploidy series).

PMID:

15866028

[PubMed - indexed for MEDLINE]

http://www.ncbi.nlm.nih.gov/pubmed/15866028

Professor Angelika Amon (MIT)

The Cell Cycle and Cancer, Angelika Amon, June 7, 2006 (Video of Presentation to a General Audience)

Exploiting cancer cells' weaknesses

Team identifies potential drugs that enhance stress caused by too many chromosomes.

Anne Trafton, MIT News Office

http://web.mit.edu/newsoffice/2011/cancer-drugs-aneuploidy-0307.html

http://www.sciencedirect.com/science/article/pii/S0092867411000560

Volume 144, Issue 4, 18 February 2011, Pages 499-512

doi:10.1016/j.cell.2011.01.017 | How to Cite or Link Using DOI

Cited By in Scopus (7)

Permissions & Reprints

Article

Identification of Aneuploidy-Selective Antiproliferation Compounds

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$ 31.50

References and further reading may be available for this article. To view references and further reading you must purchase this article.

Yun-Chi Tang1, Bret R. Williams1, Jake J. Siegel1 and Angelika Amon1, Corresponding Author Contact Information, E-mail The Corresponding Author

1 David H. Koch Institute for Integrative Cancer Research and Howard Hughes Medical Institute, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 4 August 2010;

revised 22 November 2010;

accepted 17 January 2011.

Published online: February 10, 2011.

Available online 10 February 2011.

Referred to by: Targeting Aneuploidy for Cancer Therapy

Cell, Volume 144, Issue 4, 18 February 2011, Pages 465-466,

Eusebio Manchado, Marcos Malumbres

PDF (141 K) |

Summary

Aneuploidy, an incorrect chromosome number, is a hallmark of cancer. Compounds that cause lethality in aneuploid, but not euploid, cells could therefore provide new cancer therapies. We have identified the energy stress-inducing agent AICAR, the protein folding inhibitor 17-AAG, and the autophagy inhibitor chloroquine as exhibiting this property. AICAR induces p53-mediated apoptosis in primary mouse embryonic fibroblasts (MEFs) trisomic for chromosome 1, 13, 16, or 19. AICAR and 17-AAG, especially when combined, also show efficacy against aneuploid human cancer cell lines. Our results suggest that compounds that interfere with pathways that are essential for the survival of aneuploid cells could serve as a new treatment strategy against a broad spectrum of human tumors.

Modeling the Aneuploidy Control of Cancer

Li, Yao and Berg, Arthur and Wu, Louie R. and Wang, Zhong and Chen, Gang and Wu, Rongling (2010) Modeling the Aneuploidy Control of Cancer. BMC Cancer, 10 . Art. No. 346. ISSN 1471-2407 http://resolver.caltech.edu/CaltechAUTHORS:20100817-090307829

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Abstract

Background: Aneuploidy has long been recognized to be associated with cancer. A growing body of evidence suggests that tumorigenesis, the formation of new tumors, can be attributed to some extent to errors occurring at the mitotic checkpoint, a major cell cycle control mechanism that acts to prevent chromosome missegregation. However, so far no statistical model has been available quantify the role aneuploidy plays in determining cancer. Methods: We develop a statistical model for testing the association between aneuploidy loci and cancer risk in a genome-wide association study. The model incorporates quantitative genetic principles into a mixture-model framework in which various genetic effects, including additive, dominant, imprinting, and their interactions, are estimated by implementing the EM algorithm. Results: Under the new model, a series of hypotheses tests are formulated to explain the pattern of the genetic control of cancer through aneuploid loci. Simulation studies were performed to investigate the statistical behavior of the model. Conclusions: The model will provide a tool for estimating the effects of genetic loci on aneuploidy abnormality in genome-wide studies of cancer cells.

Possibly related articles:

Protecting Valuable Intellectual Property in Octave

John F. McGowan, Ph.D. — Mon, 27 Jun 2011 16:21:15 +0000

Octave is a free, open-source high-level interpreted language, primarily intended for numerical computations that is mostly compatible with MATLAB. Octave is an excellent tool for the rapid research and development of new algorithms as well as performing simulations and data analysis. A mathematical software developer can often prototype a new algorithm in Octave two to three times faster than in a compiled programming language such as C or C++. Octave is free both as in beer and as in speech unlike MATLAB. Anyone can download Octave and run an Octave program at no cost on the three major computing platforms: MS Windows, Mac OS X, and other forms of the Unix operating system. Because Octave is open-source, there is much less concern that the vendor will suddenly cease support as Microsoft did with Visual FoxPro or redesign the language into something unusable in order to sell yet another “upgrade.” End users can always build the language from source and create a development “fork” that preserves the compatibility with existing code and the elegance of the original language.

The Problem

A major problem with Octave and many other scripting languages is that it is an interpreted, human-readable scripting language. Potential and actual customers and other third parties can see what is being done in detail. It is easy to reverse engineer or steal programs and algorithms written in scripting languages such as Octave.

Imagine that you are small company operating on a shoe string budget in a loft in West Hollywood that has developed a breakthrough video special effect in Octave. You want to win a contract from a Hollywood movie studio to do the effect in the next blockbuster science fiction movie starring Angelina Jolie and Brad Pitt as quarreling lovers caught in an alien invasion. The famous Hollywood movie studio wants to evaluate the algorithm in-house, make sure you are not cheating with Photoshop on the glamor shot of Angelina in a skin-tight black leather jumpsuit that they sent you. The problem is that the famous Hollywood studio that you are pitching to would steal your algorithm in a microsecond if they could. You are confronted with the cost, time, and general difficulty of converting your hot new video special effect algorithm into a compiled language such as C or C++. Meanwhile your competitors at Really Cool FX in Pasadena may come out with the same algorithm while you are struggling to convert it to C or C++.

You could be a quantitative finance wizard operating out of a poorly ventilated office in Jersey City, New Jersey with a spectacular view of scenic downtown Jersey City visible through your tiny west facing window. You would like to sell your hot new nanosecond trading algorithm to a Too Big Too Fail bank so you can move to a plush well ventilated corner office across the Hudson River in New York City’s financial district, but the bank insists they must thoroughly evaluate the algorithm in-house. Probably enough said right there.

You might be an idealistic junior faculty member at a prestigious, but very low paying major research university in San Francisco. You have developed the breakthrough algorithm in quantitative biology that will cure cancer — in Octave. Now, you are completely above crass materialistic concerns and plan to follow the illustrious example of Jonas Salk in refusing to patent the polio vaccine , donate regularly to the Free Software Foundation, and have an autographed poster of Richard Stallman in your tiny cramped office, but nonetheless you would like to get tenure and move out of your landlady’s attic. You know full well that the eminent full professor down the hall who got passed over for last year’s Nobel Prize would steal your idea in a picosecond if he could; it is common knowledge in the department that his didn’t-quite-get-the-Nobel-Prize work was actually stolen from his former graduate student who is now driving a taxicab in New York City. How do you demonstrate your breakthrough algorithm without giving away the secret and get tenure?

The Solution

Fortunately, one can obfuscate Octave code, removing nearly all human-readable information, much as a compiler does when it translates a program written in C or C++ into a machine-readable binary executable. This raises the bar for stealing your ideas and algorithms considerably. In general, code obfuscation removes all comments, indentation and other formatting that clarifies what is going on, and replaces all human readable variable and function names with random strings of characters that convey no meaning to a human reader. Note that the human readable information is completely removed from the obfuscated code. Some schemes to protect programs written in scripting languages use encryption. The program is encrypted but if someone can find or determine the encryption key, they can recover the entire original program including comments, human-readable names, and so forth.

A Simple Example

This is a simple script in Octave.

mytest.m

% test script

disp('hello world'); % test comment

myflag = 1;

printf(\
"this is a \
test\n");
fflush(stdout);

myflag = myflag + 1;
myflag2 = myflag++;
printf("myflag2 is %d\n", myflag2);
fflush(stdout);

if flag > 1
disp('hi');
else
disp('no');
end

for counter = 1:10
disp(counter); % test
end

pivalue = pi;
disp(pivalue)

disp('ALL DONE');

This script generates the following output under Octave 3.2.4 running on a Windows XP Service Pack 2 PC:

octave-3.2.4.exe:18> mytest
hello world
this is a test
myflag2 is 2
no
1
2
3
4
5
6
7
8
9
10
3.1416
ALL DONE

Here is an obfuscated version of the same Octave script generated by an obfuscation function written by the author in Octave:

mytest_obfuscated.m

disp ( 'hello world' ); ; UQWSKDTZQWRO=1 ; ; printf ( "this is a test\n" ); ; fflush ( stdout ); ; UQWSKDTZQWRO=UQWSKDTZQWRO+1 ; ; BSJRZMSBRYXD=UQWSKDTZQWRO++; ; printf ( "myflag2 is %d\n" , BSJRZMSBRYXD ); ; fflush ( stdout ); ; if flag>1 ; disp ( 'hi' ); ; else ; disp ( 'no' ); ; end ; for RBVZQAHJSNWB=1:10 ; disp ( RBVZQAHJSNWB ); ; end ; VIENISLJPENX=pi ; ; disp ( VIENISLJPENX ) ; disp ( 'ALL DONE' ); ;

Note: On a Windows PC using Firefox, one can select the obfuscated code above by selecting the first few characters at the start of the line above (e.g. disp) and then hitting Shift-End on the keyboard. Then copy and paste to Octave to run the obfuscated code.

This script generates the following output (the same as the original script) under Octave 3.2.4 running on a Windows XP Service Pack 2 PC:

octave-3.2.4.exe:22> mytest_obfuscated
hello world
this is a test
myflag2 is 2
no
1
2
3
4
5
6
7
8
9
10
3.1416
ALL DONE

Note that the reserved keywords such as “if” and built-in Octave functions such as “printf” are not obfuscated. It is actually possible to make the obfuscated code even more unreadable than the example above. This is intended as a simple illustration. The obstacles to reverse engineering and theft introduced by code obfuscation are greater for longer programs and more complex algorithms.

Conclusion

A major problem with Octave and other scripting languages is that it is easy for potential or actual customers or other third parties to reverse engineer or steal algorithms or other sensitive information from a program written in a human readable scripting language. This can be a serious problem for algorithm developers using Octave. This is much less of a problem with compiled languages such as C or C++ in which, however, it is usually slower and more costly to develop algorithms than Octave. Compilers generate unreadable binary files which are difficult to reverse engineer (not impossible).

Computer programs can obfuscate Octave code, automatically removing human readable information such as comments, variable and function names, indentations, and so forth. This is very close to the same information that is removed by compilers when they convert a program written in a compiled programming language such as C or C++ to a binary executable. In some ways, this is more secure than encrypting the code since the information is actually removed entirely from the obfuscated code; the encryption can be broken, often by simply stealing the encryption key. Code obfuscation raises the bar substantially for reverse engineering or stealing an algorithm or other critical intellectual property implemented in Octave. The same comments apply to other scripting languages such as Python, Perl, and Ruby.

About the Author

Possibly related articles:

Genius, Breakthroughs, and the Manhattan Project

John F. McGowan, Ph.D. — Mon, 20 Jun 2011 19:53:20 +0000

In an enterprise such as the building of the atomic bomb the difference between ideas, hopes, suggestions and theoretical calculations, and solid numbers based on measurement, is paramount. All the committees, the politicking and the plans would have come to naught if a few unpredictable nuclear cross sections had been different from what they are by a factor of two.

Emilio Segre (Nobel Prize in Physics, 1959, key contributor to the Manhattan Project) quoted in The Making of the Atomic Bomb by Richard Rhodes (Simon and Schuster, 1986)

Introduction

It is widely believed that invention and discovery, especially breakthroughs, revolutionary technological advances and scientific discoveries, are largely the product of genius, of the exceptional intelligence of individual inventors and discoverers. This is one of the lessons frequently inferred from the success of the wartime Manhattan Project which invented the atomic bomb and nuclear reactors. It is often argued that the Manhattan Project succeeded because of the exceptional intelligence of the physicists, chemists, and engineers who worked on the atomic bomb such as Emilio Segre, quoted above. The scientific director J. Robert Oppenheimer is often described as a genius, as are many other key contributors.

Since World War II, there have been numerous “new Manhattan Projects” which have recruited the best and the brightest as conventionally defined and mostly failed to replicate the astonishing success of the Manhattan Project: the War on Cancer, tokamaks, inertial confinement fusion, sixty years of heavily funded research into artificial intelligence (AI), and many other cases. As discussed in the previous article “The Manhattan Project Considered as a Fluke,” the Manhattan Project appears to have been a fluke, atypical of major inventions and discoveries, especially in the sucess of the first full system tests, the Trinity test explosion (July 16, 1945) and the atomic bombings of Hiroshima and Nagasaki (August 6 and 9, 1945) which cost the lives of over 100,000 people and which are, fortunately, so far the only examples of the use of atomic weapons in war.

With rising energy prices, possibly due to “Peak Oil,” a dwindling supply of inexpensive oil and natural gas, there have already been many calls for “new new Manhattan Projects” for various forms of alternative energy. If “Peak Oil” is correct, there is an urgent and growing need for new energy sources. Given the long history of failure of “new Manhattan Projects,” what should we do? This article argues that the importance of genius in breakthroughs is heavily overstated both in scientific and popular culture. Much more attention should be paid to other aspects of the breakthrough process.

To a significant extent, the issue of human genius in inventions and discovery overlaps the topic of the previous article “But It Worked in the Computer Simulation!” which argues that computer simulations have many limitations at present. Frequently, when people refer to human genius they are referring to the ability of human beings to simulate their ideas in their head without actually building a machine or performing a physical experiment. Many of the limitations that apply to theoretical mathematical calculations and computer simulations apply to human beings as well.

One important difference at present is that human beings think conceptually and computers at present cannot. This article argues that many historical breakthroughs were due to an often unpopular contrarian mental attitude that is largely uncorrelated with “genius” as conventionally defined — not due to exceptional conceptual reasoning skills. The success of this contrarian mental attitude is often dependent on the acceptance, which is usually grudging at first, of society at large.

A Note to Readers: The issue of genius and breakthroughs is highly relevant to invention and discovery in mathematics, both pure and applied. This article discusses many examples from applied mathematical fields such as physics, aerospace, power, propulsion, and computers. Nonetheless, it is not a mathematics specific article.

What is Genius?

Genius is difficult to define. It is usually conceived as an innate ability, often presumed to be genetic in origin, to solve problems through reasoning better than most people. It is often discussed as if it referred to a simple easily quantifiable feature of the mind such as the speed at which people think consciously (in analogy to the clock speed of a computer) or the number of items that one can keep track of in the conscious mind at once (in analogy to the number of registers in a CPU or the amount of RAM in a computer). People have tried to quantify a mysterious “general intelligence” through IQ tests. In practice, genius is often equated with a high IQ as measured on these tests (e.g. an IQ of 140 or above on some tests is labeled as “genius”).

Genius is an extremely contentious topic. Political conservatives tend to embrace genius and a genetic basis for genius. Political liberals tend to reject genius and especially a genetic basis for genius. Some experts such as the psychologist K. Anders Ericsson essentialy deny that genius exists as a meaningful concept. The science writer Malcolm Gladwell who has heavily popularized Ericsson’s ideas stops just short of “denying” genius in his writings and public presentations.

Many people, including the author, have a subjective impression that some people are smarter than other people. The author has met a number of people that the author considered clearly smarter than the author. This seemed difficult to explain in purely environmental terms. It is extremely difficult in practice to separate environment from possible genetic factors or other as yet unknown factors that may contribute to perceived or measured “intelligence.” Sometimes really smart people do extremely dumb things: why?

Genius is almost always conceived as an individual trait, similar to height or hair color, something largely independent of our present social environment. Geniuses are exceptional individuals independent of their friends, family, coworkers and so forth. Genius may be the product of environment in the sense of better schooling and so forth. Rich kids generally go to better schools or so most people believe. Nonetheless, in practice, in the scientist’s laboratory or the inventor’s workshop, “genius” is viewed as an individual trait. This conception of individual genius coexists with curious rhetoric about “teams” in business or “scientific communities” in academic scientific research today.

In particular, genuine breakthroughs usually take place in a social context, as part of a group. Historically, prior to World War II and the transformation of science that occurred during the middle of the twentieth century, these were often small, loose-knit, informal groups. James Watt collaborated loosely with some professors at the University of Glasgow in developing the separate condenser steam engine. Octave Chanute and the Wright Brothers seem to have collaborated informally without a written contract or clear team leader. Albert Einstein participated in a physics study group while at the patent office and worked closely at times with his friend and sometimes co-author the mathematician Marcel Grossmann. In his work on a unified field theory, in a different social context at the Institute for Advanced Study at Princeton, Einstein largely failed.

After success, there were often bitter fallings out over credit: “I did it all!” The “lone” inventor or discoverer that is now remembered and revered is typically the individual who secured the support of a powerful institution or individual as James Watt did with wealthy industrialist Matthew Boulton, the Wright Brothers (minus Octave Chanute) did with the infamous investment firm of Charles Flint and Company, and Einstein did with the powerful German physicist Max Planck and later the British astronomer and physicst Arthur Eddington. In a social context, the whole can be greater than the sum of the parts. A group of mediocrities that work well together (whatever that may mean in practice) can outperform a group of “stars” who do not work well together. There may be no individual genius as commonly conceived.

This article accepts that individual genius probably exists as a meaningful concept, but genius is poorly understood. It argues that genius is not nearly as important in genuine scientific and technological breakthroughs as generally conceived.

Genius and Breakthroughs in Popular Culture

In the United States, popular culture overwhelmingly attributes scientific and technological breakthroughs to genius, to extreme intelligence. This is especially true of science fiction movies and television such as Eureka, Numb3rs, Star Trek, The Day the Earth Stood Still (1951), The Absent Minded Professor (1961), Real Genius (1985), and many others. Movies and television frequently depict extremely difficult problems being solved with little or no trial and error very quickly, sometimes in seconds. It is common to encounter a scene in which a scientist is shown performing some sort of symbolic manipulation on a blackboard (sometimes a modern white board or a see-through sheet of plastic) in seconds on screen and then solving some problem, often making a breakthrough, based on the results of this implied computation or derivation. This is also extremely common in comic books. There are a number of materials in popular culture aimed specifically at children such as the famous Tom Swift book series and the Jimmy Neutron movie and TV show (The Adventures of Jimmy Neutron: Boy Genius) which communicate the same picture. Many written science fiction books and short stories convey a similar image.

Many of these popular culture portrayals are extremely unrealistic, particularly where genuine breakthroughs are concerned. In particular, most genuine breakthroughs took many years, usually at least five years, sometimes decades, even if one only considers the individual or group who “crossed the finish line.” Most genuine breakthroughs, on close examination, have involved large amounts of trial and error, anywhere from hundreds to tens of thousands of trials or tests of some sort.

Ostensibly factual popular science is often similar. It is extremely common to find the term “genius” in the title, sub-title, or cover text of a popular science book or article as well as the main body of the book or article. The title of James Gleick’s biography of the famous physicist Richard Feynman (Nobel Prize in Physics, 1965, co-discoverer of Quantum Electrodynamics aka QED) is… Genius. Readers of the book remain shocked to this day to read that Feynman claimed that his IQ had been measured as a mere 125 in high school; this is well above average but not what is usually identified as “genius.” A genius IQ is at least 140. Feynman scoffed at psychometric testing, perhaps with good reason. One should exercise caution with Feynman’s claims. Richard Feynman was an entertaining storyteller. Some of his accounts of events differ from the recollections of other participants (not an uncommon occurrence in the history of invention and discovery). Feynman’s non-genius IQ is not as surprising as it might seem. One can seriously question whether a number of famous figures in the history of physics were “geniuses” as commonly conceived: Albert Einstein, Michael Faraday, and Niels Bohr, for example.

Popular science often creates a similar impression to the science fiction described above without, however, making demonstrably false statements. Often, the long periods of trial and error and failure that precede a breakthrough are simply omitted or discussed very briefly. The reported flashes of insight, the so-called “Eureka moments,” which can be very fast and abrupt if the reports are true, are generally emphasized and extracted from the usual context of years of study and frequent failure that precede the flash of insight. Popular science books tend to focus on personalities, politics, the big picture scientific or technical issues, and… the genius of the participants. The discussions of the trial and error, if they exist at all, are extremely brief and easy to miss: a paragraph or a few pages in a several hundred page book for example. In the 886 page The Making of the Atomic Bomb, the author Richard Rhodes devotes a few paragraphs to the enormous amount of trial and error involved in developing the implosion lens for the plutonium atomic bomb (page 577, emphasis added):

The wilderness reverberated that winter to the sounds of explosions, gradually increasing in intensity as the chemists and physicists applied small lessons at a larger scale. “We were consuming daily,” says (chemist George) Kistiakowsky, “something like a ton of high performance explosives, made into dozens of experimental charges.” The total number of castings, counting only those of quality sufficient to use, would come to more than 20,000. X Division managed more than 50,000 major machining operations on those castings in 1944 and 1945 without one explosive accident, vindication of Kistiakowsky’s precision approach.

While a close reading of The Making of the Atomic Bomb reveals an enormous amount of trial and error at the component level, it is easy to miss this given how short and oblique the references are, buried in 886 pages. The term “trial and error” is not listed in the detailed 24 page index of the book. The index on page 884 lists Tregaskis, Richard, Trinity, tritium, etc. in sequence — no “trial and error”.

In most cases, popular science books don’t point out the obvious interpretation of these huge amounts of trial and error. One is not seeing the results of genius, certainly not as frequently depicted in popular culture, but rather the results of vast amounts of trial and error. This trial and error is extremely boring to describe in detail, so it is either omitted or discussed very briefly. Where the popular science has the goal of “inspiring” students to study math and science, a detailed exposition of the trial and error is probably a good way to convince a student to go play American football (wimpy American rugby with lots of padding) or soccer (everybody else’s football) instead.

On a personal note, the author read The Making of the Atomic Bomb shortly after it was first published and completely missed the significance of Segre’s quote and the passage above. After researching many inventions and discoveries in detail, it became apparent that the most common characteristic of genuine breakthroughs is vast amounts of trial and error, usually conducted over many years. What about the Manhattan Project? Rereading the book closely reveals occasional clear references to the same high levels of trial and error, in this case at the component level. The Manhattan Project is quite unusual in that the first full system tests were great successes: worked right the first time. Many of the theoretical calculations appear to have worked better than is typically the case in other breakthroughs.

Remarkably, the Manhattan Project appears to have been unusually “easy” among major scientific and technological breakthroughs. The first full system tests, the Trinity, Hiroshima, and Nagasaki bombs, were spectacular successes which ended World War II in days. This is very unusual. Attempts to replicate the unusual success of the Manhattan Project have mostly failed. It may well be that even in most successful inventions and discoveries the equivalents of the critical nuclear cross sections that Segre mentions in the quote above are less convenient than occurred in the Manhattan Project.

The Rapture for Geeks

In 1986, the science fiction writer and mathematician Vernor Vinge published a novel length story “Marooned in Real Time” in the Analog Science Fiction/Science Fact science fiction magazine which was shortly thereafter published as a book by St. Martin’s Press/Bluejay Books. This novel introduced the notion of a technological singularity to a generation of geeks.

The basic notion that Vinge presented in the novel was that rapidly advancing computer technology would increase or amplify human intelligence. This in turn would accelerate both the development of computer technology and other technology, resulting in an exponential increase, eventually reaching a mysterious “singularity” somewhat in analogy to the singularities in mathematics and physics (typically a place in a mathematical function where the function becomes infinite or undefined). In the novel, most of the human race appears to have suddenly disappeared, possibly the victims of an alien invasion. A tiny group of survivors have been “left behind.” By the end of the novel, it is strongly implied that the missing humans have transcended to God-like status in a technological singularity.

Vinge’s notion of a technological singularity has had considerable influence and it probably also helps sell computers and computer software. It has been taken up and promoted seriously by inventor, entrepreneur, and futurist Ray Kurzweil, the author of such books as The Age of Spiritual Machines and The Singularity is Near. Kurzweil is, for example, the chancellor of the Singularity University which charges hefty sums to teach the Singularity doctrine to well-heeled individuals, likely Silicon Valley executives and zillionaires. Kurzweil’s views have been widely criticized, notably by former Scientific American editor John Rennie and others. The recent movie “Transcendent Man,” available on NetFlix and iTunes, gives a friendly but fair portrait of Ray Kurzweil.

The Singularity concept implicitly assumes the common notion that intelligence and genius drive the invention and discovery process. It also assumes that computer technology can amplify or duplicate human intelligence. Thus, increase intelligence and automatically the number and rate of inventions and discoveries will increase. An exponential feedback loop follows logically from these assumptions.

If invention and discovery is largely driven by large amounts of physical trial and error (for example), none of this is true. To be sure, fields such as computers and electronics with small scale devices where physical trial and error can be performed rapidly and cheaply will tend to exhibit higher rates of progress than fields with huge, expensive, time-consuming to build devices such as modern power plants, tokamaks, particle accelerators and so forth. This is, in fact, what we see at the moment. But there will be no Singularity.

There is now over forty years of experience in fundamental physics and aerospace, both early adopters of computer technology, in using computers to supposedly enhance human intelligence and accelerate the rate of progress. Both of these fields visibly slowed down around 1970 coincident with the widespread adoption of computers in these fields. This is particularly noticeable in aviation and rocketry where modern planes and rockets are only slightly better than the planes and rockets of 1971 despite the heavy use of computers, computer simulations, computer aided design, and so forth. NASA’s recent attempt to replicate the heavy lift rocket technology of the 1970s (the Saturn V rocket), the modern Ares/Constellation program, has foundered despite extensive use of computer technologies far in advance of those used in the Apollo program, which quite possibly owed much of its success to engineers using slide rules.

Similarly, if one looks at the practical results of fundamental physics, comparable to the nuclear reactors that emerged from the Manhattan Project, the results have been similarly disappointing. It is even possible the prototype miniature nuclear reactors and engines of the cancelled nuclear reactor/engine projects of the 1960′s exceed what we can do today; knowledge has been lost due to lack of use.

Are computers and computer software amplifying effective human intelligence? If one looks outside the computer/electronics fields, the evidence for this is generally negative, poor at best. Are computers and computer software accelerating the rate of technological progress, invention and discovery, increasing the rate of genuine breakthroughs? Again, if one looks outside the computer/electronics fields, the evidence is mostly negative. This is particularly noticeable in the power and propulsion areas, where progress appears to have been faster in the slide rule and adding machine era. Rising gasoline and energy prices reflect the negligible progress since the 1970s. The relatively high rates of progress observed in some metrics (e.g. Moore’s Law, the clock speed of CPU’s until 2003, etc.) in computers/electronics can be attributed to the ability to perform large amounts of trial and error rapidly and cheaply combined with cooperative physics, rather than an exponential feedback loop.

Genius and Breakthroughs in Scientific Culture

“Hard” scientists like physicists or mathematicians tend to act as if they believe in “genius” or “general intelligence”. In academia, such scientists tend to be liberal Democrats in the United States. Probably consciously they do not believe that this genius is an inborn, genetic characteristic. Nonetheless, the culture and institutions of the hard sciences are built heavily around the notion of individual measurable genius.

Many high school and college math and science textbooks have numerous sidebars with pictures and brief biographical sketches of famous prominent mathematicians and scientists. These often include anecdotes that seem to show how smart the mathematician or scientist was. A particularly common anecdote is the account of the young Gauss figuring out how to quickly add the numbers from 1 to 100 (The trick is 1 plus 100 is 101, 2 plus 99 is 101, 3 98 is 101, etc. so the sum is 50 times 101 which is 5050).

Much of the goal of the educational system in math and science is ostensibly to recruit and select the best of the best, in the supposed spirit of the Manhattan Project. There are tests and exams and competitions all designed to select the very best. In modern physics, for example, this means that the very top graduate programs such as the graduate program at Princeton are largely populated by extreme physics prodigies: people who have done things like publish original papers on quantum field theory at sixteen and who, by any reasonable criterion, could, in principle, run rings around historical figures like Albert Einstein or Niels Bohr. But, in practice, they usually don’t.

Psychologists like K. Anders Ericsson, sociologists, anthropologists, and other “softer” scientists indeed are more likely to seriously question the notion of genius and its role in invention and discovery, at least more broadly than most physicists or mathematicians. Even here though, Ericsson’s theory, for example, attributes breakthroughs to individual expertise acquired through many years of deliberate practice.

Circular Reasoning

It is common in discussions of breakthroughs to find circular reasoning about the role of genius. How do you know genius is needed to make a breakthrough? Bob discovered X and Bob was a genius! How do you know Bob was a genius? Only a genius could have discovered X!

The belief that genius is the essential driving force behind breakthroughs — the more significant the breakthrough, the more brilliant the genius must have been — is so strong and pervasive that the inventor or discoverer is simply assumed to have obviously been a genius and any contrary evidence dismissed. Richard Feynman’s claim to have had a measured IQ of only 125 often provokes incredulity. It is simply assumed that the discoverer of QED had to have been a genius. James Gleick titled his biography of Feynman Genius in spite of knowing Feynman’s claim.

So too Albert Einstein is almost always assumed to have been a remarkable genius. The author can recall a satirical practice at Caltech, a celebration of a special day for a high school teacher who allegedly flunked Einstein: “What an idiot!” But, Einstein in fact was an uneven student. He made many mistakes both in school and in his published papers. He ended up at the patent office, working on his Ph.D. part time at the less prestigious University of Zurich, because he was not so good. His erstwhile professor Minkowski was famously astounded that Einstein accomplished such amazing things. Einstein seems to have worked on his discoveries over many years and he seems to have had the contrarian mental attitude so common among people who make major breakthroughs. He also probably would have gone nowhere had not Max Planck become intrigued with several of his papers and heavily promoted them.

Niels Bohr was infamously obscure in his talks and writings. He had very limited mathematical skills and relied first on his brother Harald, a mathematician, and later younger assistants like Werner Heisenberg. Many of his papers and writings are impenetrable. His response in Physical Review to Einstein, Podolsky, and Rosen’s 1935 paper, which is now taken to clearly identify the non-local nature of quantum mechanics in the process of questioning the foundations of quantum theory, is complete gibberish. Yet Bohr acquired such a mystique as a brilliant physicist and genius that many of these dubious writings were uncritically accepted by his students and many other physicists — even to this day.

It is clear that if breakthroughs were usually the product of a short period of time, such as six months or less, and little or no trial and error, as often implied in popular science and explicitly portrayed in much science fiction, something like real genius would be absolutely necessary to explain the breakthroughs. But this is not the case. Almost all major breakthroughs took many years of extensive trial and error. Most inventors and discoverers seem to have been of above average intelligence, like the IQ of 125 that the physicist Richard Feynman claimed, but not clearly geniuses as conventionally defined. Some were definitely geniuses as conventionally defined.

Intelligence or Social Rank?

In discussions of intelligence or genius, one needs to ask the question and be aware whether one is really talking about intelligence, whatever it may be, or social rank. Most societies rely heavily on a hierarchical military chain of command structure. This structure is found equally in government, academia, business, capitalist nations, socialist nations, and communist nations. In military chains of command there is almost always an implicit concept of a simple linear scale of social rank or status as well as specific roles. A general outranks a colonel even though the colonel may not report to the general. A four star general outranks a three star general and so forth. One of the practical reasons for this is so that in a confused situation such as a battle, it is always clear who should assume command, the ranking officer.

In many respects, in the United States, the concept of intelligence is often used as a proxy or stand in for social rank or status. In academic scientific research, the two are often equated implicitly. An eminent scientist such as Richard Feynman must be a genius, hence astonishment at his claim to a mere 125 IQ. England in 1776 had a very status conscious society. Everyone was very aware of their linear rank in society. To give some idea of this, in social dances, the dance would be chosen in sequence starting with the most ranking woman at the dance choosing the first dance, followed by the second ranking woman, and so forth. Somehow everyone knew exactly how each person was ranked in their community. When the United States broke away from England, this notion of rank was questioned and even rejected. Americans actually deliberately drew lots at dances as to who would choose the dances in an explicit rejection of the English notions of status. This is not to portray the early United States as some eqalitarian utopia; surely it was not. Nonetheless, from the early days, the United States tended to reject traditional notions of social status and rank, and substituted notions like “the land of opportunity.”

But the United States and the modern world has social ranks and status, sometimes by necessity, sometimes not. How to justify this and perhaps also disguise the reality? Aha! Some people are smarter than other people and their position in society is due to their innate intelligence, which (surprise, surprise) is a linear numeric scale, and hard work! All animals are equal, but some animals are more equal than others.

Genius or Mental Attitude?

Clearly there is more to breakthroughs than pure trial and error. Blind trial and error could never find the solution to a complex difficult problem in even hundreds of thousands of attempts. It is clear that inventors and discoverers put a great deal of thought into what to try and what lessons to derive from both failures and successes. Many inventors and discoverers have noted down tens, even hundreds of thousands of words of analysis in their notebooks, published papers, books, and so forth. Something else is going on as well. There is often a large amount of conceptual analysis and reasoning, as well as the trial and error. Can we find real genius here? Maybe.

However the most common situation and best understood conceptual reasoning leading to a genuine breakthough does not particularly involve recognizable genius. Actually, one can argue the inventors and discoverers are doggedly doing something rather dumb. In many, many genuine breakthroughs the inventor or discoverers try something that seems like it ought to work over and over again, failing repeatedly. They are often following the conventional wisdom, what “everyone knows”: the motion of the planets is governed by uniform circular motion, rockets have always been made using powdered explosives, Smeaton’s coefficient (aviation) is basic textbook know-how measured accurately years ago for windmills, etc. How smart is it to try something that fails over and over and over again for years? How much genius is truly involved in finally stopping and saying: “you know, something must be wrong; some basic assumption that seems sensible can’t be right.”

At this point, one should make a detailed list of assumptions, both explicit and implicit, and carefully examine the experimental data and theory behind each assumption. Not infrequently in history this process has revealed that something “everyone knew” was not well founded. Then, one needs to find a replacement assumption or set of assumptions. Sometimes this is done by conscious thought or yet more trial and error: what if the motion of the planets follows an ellipse, one of the few other known mathematical functions in 1605 when Kepler disovered the elliptical motion of Mars?

Sometimes the new assumption or group of assumptions seems to pop out of nowhere in a “Eureka” moment. The inventor or discoverer often cannot explain consciously how he or she figured it out. This latter case raises the possibility of some sort of genius. But is this true? Many people experience little creative leaps or solutions to problems that they cannot consciously explain. This usually takes a while. For everyday problems the lag between starting work on the problem and the leap is measured in hours or days or maybe weeks. The lag is generally longer the harder the problem. Breakthroughs involve very difficult, complex problems, much larger in scope than these everyday problems. In this case, the leap takes longer and is more dramatic when it happens. This is a reasonable theory, although there is currently no way to prove it. Are we seeing genius, exceptional intelligence, or a common subconscious mental process operating over years — the typical timescale of breakthroughs?

Is the ultimate willingness to question conventional wisdom after hundreds or thousands of failures genius or simply a contrarian mental attitude, which, of course, must be coupled with a supportive environment? If people are being burned at the stake either figuratively or literally for questioning conventional wisdom and assumptions, this mental attitude will fail and may be tantamount to suicide. In this respect, society may determine what happens and whether a breakthrough occurs.

Historically, inventors and discoverers often turn out to have been rather contrarian individuals. Even so it often took many years of repeated failure before they seriously questioned the conventional wisdom — despite a frequent clear propensity on their part to do so. Is it correct to look upon this mental attitude as genius or something else? In many cases, many extremely intelligent people as conventionally measured were/are demonstrably unwilling to take this step, even in the face of thousands of failures. In the many failed “new Manhattan Projects” of the last forty years, the best and the brightest recruited in the supposed spirit of the Manhattan Project, in the theory that genius is the driver of invention and discovery, are often unwilling to question certain basic assumptions. Are genuine breakthroughs driven by individual genius or by a social process which is often uncomfortable to society at large and to the participants?

The rhetoric of “thinking outside the box” and “questioning assumptions” is pervasive in modern science and modern society. The need to question assumptions is evident even from a cursory examination of the history of scientific discovery and technological invention. It is not surprising that people and institutions say they are doing this and may sincerely believe that they are. Many modern scientific and technological fields do exhibit fads and fashions that are presented as “questioning assumptions,” “thinking outside the box,” and “revolutionary new paradigms.” In fact some efforts that have yielded few demonstrable results such as superstrings in theoretical physics or the War on Cancer are notorious for rapidly changing fads and fashions of this type. On the other hand, on close examination, certain basic assumptions are largely beyond question such as the basic notion of superstrings or the oncogene theory of cancer. In the case of superstrings, a number of prominent physicists have publicly questioned the theory including Sheldon Glashow, Roger Penrose, and Lee Smolin, but it remains very dominant in practice.

Conclusion

The role of genius as commonly defined in genuine breakthroughs appears rather limited. Breakthroughs typically involve very large amounts of trial and error over many years. This alone can create the illusion of exceptional intelligence if the large amounts of trial and error and calendar time are neglected. There is clearly a substantial amount of conceptual analysis and reasoning in most breakthroughs. Certainly some kind of genius, probably very different from normal concepts of genius, may be involved in this. Unlike common portrayals in which geniuses solve extremely difficult problems rapidly, the possible genius in breakthroughs usually occurs over a period of years. While inventors and discoverers usually appear to have been above average in intelligence (like Richard Feynman who claimed a measured IQ of only 125), they are often not clearly geniuses as commonly defined. The remarkable flashes of insight, the “Eureka” experiences, reported by many inventors and discoverers may well be examples of relatively ordinary subconscious processes but operating over an extremely long period of time — the many years usually involved in a genuine breakthrough.

The most common and best understood form of conceptual reasoning involved in many breakthroughs is not particularly mysterious nor indicative of genius as commonly conceived. Developing serious doubts about the validity of commonly accepted assumptions after years of repeated failure is neither mysterious nor unusual nor a particular characteristic of genius. Actually, many geniuses as commonly defined often have difficulty taking this step even with the accumulation of thousands of failures. This is more indicative of a certain mental attitude, a willingness to question conventional wisdom and society. Identifying and listing assumptions, both stated and unstated, and then carefully checking the experimental and theoretical basis for these assumptions is a fairly mechanical, logical process; it does not require genius. Most people can do it. Most people are uncomfortable with doing it and often avoid doing so even when it is almost certainly warranted. This questioning of assumptions is also likely to fail if society at large is too resistant, unwilling even grudgingly to accept the results of such a systematic review of deeply held beliefs.

In the current economic difficulties, which may be due to “Peak Oil,” a dwindling supply of inexpensive oil and natural gas, there may well be an urgent and growing need for new energy sources and technologies. This has already led to calls for “new new Manhattan Projects” employing platoons of putative geniuses to develop or perfect various hoped for technological fixes such as thorium nuclear reactors, hydrogen fuel cells and various forms of solar power. The track record of the “new Manhattan Projects” of the last forty years is rather poor and should give everyone pause. The original Manhattan Project was certainly unusual in the success of the first full system tests and perhaps in other ways as well. This alone argues for assuming that many full system tests, hundreds probably, will be needed in general to develop a new technology. Success is more likely with inexpensive, small scale systems of some sort where the many, many trials and errors usually needed for a breakthrough can be performed quickly and cheaply.

But what about genius? Many breakthroughs may be due in part to powerful subconscious processes found in most people but operating over many years rather than genius as commonly defined. Genius of some kind may be necessary, but if the contrarian mental attitude frequently essential to breakthroughs is lacking or simply rejected by society despite the pervasive modern rhetoric about “questioning assumptions” and “thinking outside the box,” then failure is in fact likely, an outcome which would probably be bad for almost everyone, perhaps the entire human race. It is not inconceivable that we could experience a nuclear war over dwindling oil and natural gas supplies in the Middle East or elsewhere — certainly an irrational act but really smart people sometimes do extremely dumb things.

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