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
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:
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.
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.
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.
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.
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 most 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.
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.
© 2011 John F. McGowan — this article originally posted at Math Blog.
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 firstname.lastname@example.org.