Spurred by recent discussions with +Jacob Scott about preprints in biology and fed up with the slow review process of some journals I've decided to upload my most recent paper on brain tumour modelling on arXiv (and continue to do so with future papers).
This paper is quite technical (at least by my standards) and contains the mathematical analysis of a model of glioblastoma growth that was published last year in PLoS Computational Biology. In this model the cancer cells switch between a proliferative and migratory phenotype, and it was previously shown that the dynamics of the cell-based model can be captured by two coupled partial differential equations, that exhibit (like the Fisher equation) travelling wave solutions. In this paper I have analysed this PDE-system and shown the following things:
1. With a couple of assumptions on model parameters one can obtain an analytical estimate of the wave speed.
2. In the limit of large and equal switching rates the wave speed equals that of the Fisher equation (which is what you'd expect).
3. Using perturbation techniques one can obtain an approximate solution to the shape of the expanding tumour.
4. In the Fisher equation the wave speed and the slope of the front are one to one (faster wave <--> less steep front). This property does not hold for our system.
Here's a link to the submission at arXiv.
Thursday, 23 May 2013
Wednesday, 22 May 2013
Mathematical biology or Bioinformatics
Conversation overheard in an inter-disciplinary research centre of unknown location.
Molecular Biologist: So, what kind of research do you do?
Mathematical Biologist: I do modelling, mathematical modelling of cancer.
Molecular Biologist: I see, interesting. So you mean bioinformatics?
Mathematical Biologist (trying to be polite): Well, not quite. My work is more about building mechanistic models that help our understanding of different steps of tumour progression.
Molecular Biologist: I never quite understood all those statistical methods and hypothesis testing, but I'm glad someone likes it!
Mathematical Biologist (slowly losing patience): Well actually….I'm not very good with statistics either, my work is more about understanding the mechanisms at work in cancer, using mathematics.
Molecular Biologist: Oh, I think I understand now. By the way, I have some microarray-data that maybe you can have a look at.
Mathematical Biologist (squeezing through the opening elevator doors): Ok…..drop me an email.
This dialogue is fictional but draws inspiration from the many encounters and discussions I've had about my research with biologists. Usually the conversations last a bit longer than the above, and end in some sort of understanding of what my work is really about.
It's not that I'm easily offended when people think that I'm a bioinformatician, but mathematical/theoretical biology and bioinformatics are fundamentally different lines of research, with different methods and goals, and I'll try to explain why I think that is the case.
In order to illustrate my point we need to take a step back from biology and look at science from a broader perspective. The process of doing science and producing new knowledge about the world is usually termed the scientific method and can roughly be divided into: Hypotheses, Experiments, Results and Conclusions/Findings (I'm sure many philosophers of science will disagree on this, but this basic subdivision will do for my argument). The process is circular in that we start with some idea about how a certain system or phenomena is structured (i.e. a hypothesis), we then transform that hypothesis to a statement that is experimentally testable, carry out the experiment, and from the data determine if the hypothesis was true or false. This fact is added to our knowledge of the world and from our extended body of knowledge we produce new hypotheses.
In order to structure our knowledge about a phenomenon we construct theories that in a more or less formal manner codify our knowledge within a coherent framework. Mathematics is such a framework that was applied successfully first in physics, and then later in chemistry and most other natural sciences. In the language of mathematics we can transform statements in a rigorous, truth-preserving manner, moving from things that are certainly true (based on observation) to things that are possibly true (to be decided by experiment).
It is in this part of the scientific method that mathematical biology fits in. In a mathematical model we incorporate known facts, and maybe add some hypothetical ones, analyse the model and produce hypotheses that hopefully are testable in experiments (disclaimer: this is highly idealised. A lot of mathematical biology is far disconnected from experiments and more concerned with mathematical analysis, but where to draw the line between mathematical biology and applied analysis is at least to me a pointless exercise). Another equally important task for mathematical biologists is to form new theoretical constructs, and define new properties that are of relevance. An example of this is R0, the 'basic reproduction number' of a pathogen, that quantifies the number of cases one case generates on average over the course of its infectious period. It was defined by Ronald Ross when studying malaria with the aid of mathematical modelling, work that later was awarded with the Nobel Prize in Medicine in 1902.
If the role of mathematical biology is to define new concepts and generate hypotheses, where does bioinformatics fit into the process of scientific discovery? The role of bioinformatics is to structure and make sense of the other side of the scientific method; to design experiments and aid us in interpreting the outcomes. In molecular biology the days of simple experiments when measuring a single quantity was enough to prove or disprove a hypothesis are almost gone. With todays measurement techniques such as microarray, methylation probes or SNP-analysis, one is presented with quantities of data that are far beyond the reach of the human intellect. In order to decode the data, and draw conclusions we need algorithms developed by bioinformaticians. Apart from this they are also involved in the step between hypotheses and experiments, designing the most efficient and accurate ways of carrying out experiments (e.g. determining how much coverage we get with a given sequencing technique).
In my view mathematical biology and bioinformatics serve as two independent and non-overlapping disciplines that both aid the actual biologists (i.e. the experimentalists) in making the scientific method spin.
The inspiration for this post (and the figure) came from +Jacob Scott who came up with the idea when writing a recent review on mathematical modelling of metastases. Thanks!
Molecular Biologist: So, what kind of research do you do?
Mathematical Biologist: I do modelling, mathematical modelling of cancer.
Molecular Biologist: I see, interesting. So you mean bioinformatics?
Mathematical Biologist (trying to be polite): Well, not quite. My work is more about building mechanistic models that help our understanding of different steps of tumour progression.
Molecular Biologist: I never quite understood all those statistical methods and hypothesis testing, but I'm glad someone likes it!
Mathematical Biologist (slowly losing patience): Well actually….I'm not very good with statistics either, my work is more about understanding the mechanisms at work in cancer, using mathematics.
Molecular Biologist: Oh, I think I understand now. By the way, I have some microarray-data that maybe you can have a look at.
Mathematical Biologist (squeezing through the opening elevator doors): Ok…..drop me an email.
This dialogue is fictional but draws inspiration from the many encounters and discussions I've had about my research with biologists. Usually the conversations last a bit longer than the above, and end in some sort of understanding of what my work is really about.
It's not that I'm easily offended when people think that I'm a bioinformatician, but mathematical/theoretical biology and bioinformatics are fundamentally different lines of research, with different methods and goals, and I'll try to explain why I think that is the case.
In order to illustrate my point we need to take a step back from biology and look at science from a broader perspective. The process of doing science and producing new knowledge about the world is usually termed the scientific method and can roughly be divided into: Hypotheses, Experiments, Results and Conclusions/Findings (I'm sure many philosophers of science will disagree on this, but this basic subdivision will do for my argument). The process is circular in that we start with some idea about how a certain system or phenomena is structured (i.e. a hypothesis), we then transform that hypothesis to a statement that is experimentally testable, carry out the experiment, and from the data determine if the hypothesis was true or false. This fact is added to our knowledge of the world and from our extended body of knowledge we produce new hypotheses.
In order to structure our knowledge about a phenomenon we construct theories that in a more or less formal manner codify our knowledge within a coherent framework. Mathematics is such a framework that was applied successfully first in physics, and then later in chemistry and most other natural sciences. In the language of mathematics we can transform statements in a rigorous, truth-preserving manner, moving from things that are certainly true (based on observation) to things that are possibly true (to be decided by experiment).
It is in this part of the scientific method that mathematical biology fits in. In a mathematical model we incorporate known facts, and maybe add some hypothetical ones, analyse the model and produce hypotheses that hopefully are testable in experiments (disclaimer: this is highly idealised. A lot of mathematical biology is far disconnected from experiments and more concerned with mathematical analysis, but where to draw the line between mathematical biology and applied analysis is at least to me a pointless exercise). Another equally important task for mathematical biologists is to form new theoretical constructs, and define new properties that are of relevance. An example of this is R0, the 'basic reproduction number' of a pathogen, that quantifies the number of cases one case generates on average over the course of its infectious period. It was defined by Ronald Ross when studying malaria with the aid of mathematical modelling, work that later was awarded with the Nobel Prize in Medicine in 1902.
If the role of mathematical biology is to define new concepts and generate hypotheses, where does bioinformatics fit into the process of scientific discovery? The role of bioinformatics is to structure and make sense of the other side of the scientific method; to design experiments and aid us in interpreting the outcomes. In molecular biology the days of simple experiments when measuring a single quantity was enough to prove or disprove a hypothesis are almost gone. With todays measurement techniques such as microarray, methylation probes or SNP-analysis, one is presented with quantities of data that are far beyond the reach of the human intellect. In order to decode the data, and draw conclusions we need algorithms developed by bioinformaticians. Apart from this they are also involved in the step between hypotheses and experiments, designing the most efficient and accurate ways of carrying out experiments (e.g. determining how much coverage we get with a given sequencing technique).
In my view mathematical biology and bioinformatics serve as two independent and non-overlapping disciplines that both aid the actual biologists (i.e. the experimentalists) in making the scientific method spin.
The inspiration for this post (and the figure) came from +Jacob Scott who came up with the idea when writing a recent review on mathematical modelling of metastases. Thanks!
Monday, 13 May 2013
Technology is in the driver seat and we're heading to petaland
There is little doubt that technology has had a large impact on the course of scientific discovery in the past. For example it was Tycho Brahe's development of more precise measurement techniques for astronomical observation that paved the way for Johannes Kepler's identification of regularity in the motion of the planets (i.e. Kepler's three laws of planetary motion), that eventually led to Newton's formulation of the laws of celestial (and terrestrial) mechanics. While Brahe made his observations with the naked eye, the invention of the telescope by Galileo Galilei, and his subsequent observation of the moons of Jupiter, were important events that, together with other empirical evidence
eventually toppled the Aristotelian world-view. During the same period there was a improvement in observing not only the very distant, but also the very small. The first microscope appeared at the turn of the 16th and 17th century, and within 50 years the technology had improved to such an extent that single cells were visible to the human eye. In fact the word 'cell' was coined by Robert Hook in his book Micrographia (1665). It would take another 200 years of observation before the dynamics of cell division were observed, and another 50 years until it was understood that the chromosomes, that were being shuffled between the two daughter cells, were the carriers of hereditary information.
Since the days of the scientific revolution in the 17th century technology has advanced enormously and every aspect of human life is influenced by technological artefacts. Most of us don't have the faintest idea of how they are constructed or even how they operate, but this is not really an issue since someone knows how they are built and how to fix them if they fail. More disconcerting is the fact that we are often told that only the latest gadgets are worth owning, and that each epsilon change to a piece of technology will revolutionise its use. The need for the latest gadget might be a fundamentally human capacity, and rather of economical and political interest, but what happens when this need for new technology enters into the scientific method and our ways of doing science?
The last 30 years of biological research has been heavily influenced by advances in technology, which have lead to a massive increase in knowledge about the living world. DNA sequencing, RNA-expression measurements (i.e. microarrays) and DNA methylation measurements, just to mention a few, have allowed biologists to address questions that for a long time remained unanswered. But technology doesn't just answer existing questions it also poses new ones. Microarray measurements made it possible to map out the RNA expression levels of all the genes in the genome at once. Since all processes that occur within a human cell are affected by RNA-expression of this or that gene(s), it soon became standard practice within all branches of molecular biology to perform microarray measurements, and basically required if you wanted to publish your findings in a respected journal. The data that emerged was high-dimensional, complex, and to this date we don't have a precise understanding of how RNA-expression relates to gene regulation and protein expression. Completely ignorant of this lag between measurement, and theory and concept formation the biotech industry has continued to develop techniques with even higher coverage and larger depth. The scientific community has become drawn into this technological whirlwind and today, when we still don't have a full understanding of the microarray-data, using it is basically frowned upon, and we are asked why we didn't make use of RNA-seq, or 'Next generation sequencing' in our study.
New technology lets us probe further and deeper into living matter, and doubtless this has helped our understanding of these systems. However, considering how little we have learned about human biology from the Human Genome Project (sequencing of the entire human genome), it's tempting to speculate about where we would have been today if instead all that effort and money that went into sequencing was spent on dedicated experiments and hypothesis-driven inquiry.
Today we are producing terabytes upon terabytes of data from biological systems, but at the same time we seem to know less and less what that data actually means. I think it's time to focus on the scientific problems at hand, and try to understand and build theories of the data produced by current technology, before we rush into the next next-generation piece of technology, that in the end will just make us forget what we initially were asking. If not, it won't be long until we count our datasets in petabytes.
eventually toppled the Aristotelian world-view. During the same period there was a improvement in observing not only the very distant, but also the very small. The first microscope appeared at the turn of the 16th and 17th century, and within 50 years the technology had improved to such an extent that single cells were visible to the human eye. In fact the word 'cell' was coined by Robert Hook in his book Micrographia (1665). It would take another 200 years of observation before the dynamics of cell division were observed, and another 50 years until it was understood that the chromosomes, that were being shuffled between the two daughter cells, were the carriers of hereditary information.
Since the days of the scientific revolution in the 17th century technology has advanced enormously and every aspect of human life is influenced by technological artefacts. Most of us don't have the faintest idea of how they are constructed or even how they operate, but this is not really an issue since someone knows how they are built and how to fix them if they fail. More disconcerting is the fact that we are often told that only the latest gadgets are worth owning, and that each epsilon change to a piece of technology will revolutionise its use. The need for the latest gadget might be a fundamentally human capacity, and rather of economical and political interest, but what happens when this need for new technology enters into the scientific method and our ways of doing science?
The last 30 years of biological research has been heavily influenced by advances in technology, which have lead to a massive increase in knowledge about the living world. DNA sequencing, RNA-expression measurements (i.e. microarrays) and DNA methylation measurements, just to mention a few, have allowed biologists to address questions that for a long time remained unanswered. But technology doesn't just answer existing questions it also poses new ones. Microarray measurements made it possible to map out the RNA expression levels of all the genes in the genome at once. Since all processes that occur within a human cell are affected by RNA-expression of this or that gene(s), it soon became standard practice within all branches of molecular biology to perform microarray measurements, and basically required if you wanted to publish your findings in a respected journal. The data that emerged was high-dimensional, complex, and to this date we don't have a precise understanding of how RNA-expression relates to gene regulation and protein expression. Completely ignorant of this lag between measurement, and theory and concept formation the biotech industry has continued to develop techniques with even higher coverage and larger depth. The scientific community has become drawn into this technological whirlwind and today, when we still don't have a full understanding of the microarray-data, using it is basically frowned upon, and we are asked why we didn't make use of RNA-seq, or 'Next generation sequencing' in our study.
New technology lets us probe further and deeper into living matter, and doubtless this has helped our understanding of these systems. However, considering how little we have learned about human biology from the Human Genome Project (sequencing of the entire human genome), it's tempting to speculate about where we would have been today if instead all that effort and money that went into sequencing was spent on dedicated experiments and hypothesis-driven inquiry.
Today we are producing terabytes upon terabytes of data from biological systems, but at the same time we seem to know less and less what that data actually means. I think it's time to focus on the scientific problems at hand, and try to understand and build theories of the data produced by current technology, before we rush into the next next-generation piece of technology, that in the end will just make us forget what we initially were asking. If not, it won't be long until we count our datasets in petabytes.
Monday, 6 May 2013
The principles of metastasis
I have, since the beginning of my scientific career, been under the somewhat naive impression, that science moves relentlessly forward, and that we (as scientists) slowly accumulate more and more knowledge about the world. Things are discovered, communicated and never forgotten within the realms of science. Of course this cannot be true in every single case, but reading this book by Leonard Weiss put me into first-hand contact with the loss of scientific knowledge, which struck me particularly hard since it is about a topic I myself am concerned with. This book was published for everyone to read in 1985, so what I talk about is not a straight denial of scientific fact, but rather a collective amnesia or erosion of knowledge.
Let me be a bit more specific, and mention two things that caught me by surprise when I read this book. Firstly, it was established in the 40's that cancer cells from primary tumours in the breast and prostate can travel to the vertebrae and pelvis without first passing through the capillary bed of the lung. This passage is mediated by a structure known as Batson's venous plexus, which allows for a reversal of flow in the veins, in particular during coughing of sneezing. This, at least in part, explains the predilection for breast and prostate tumours to metastasise to the bone, yet these organ pairs are today often mentioned as prime examples of the seed-soil hypothesis, i.e. the idea that cancer cells thrive in certain organs with favourable 'soil'. This is often talked about in terms of micro-environmental compatibility, which compared to physical blood flow is quite a far fetched and complex explanation.
My second moment of surprise came when Weiss discussed the literature on cancer cell clump size distribution, meaning the size of the cancer cell clusters that enter the circulation and eventually arrest in capillary beds. I'm far from an expert in the field of metastasis but the view that I have acquired from reading review articles and papers on the topic is that cancer cells travel on their own, never collectively. This mistake from my side (or of the authors I've previously read) is made even more serious since it turns out that the success of cancer cells when they arrest in foreign organs depend very much on the clump size; larger clump of cancer cells means higher chance of forming a metastases.
I sincerely hope that the above mentioned 'surprises' are isolated occurrences, but the impression I get from reading this book is that many fundamental insights into the dynamics of the metastatic process have been lost since the 80's. When did you last hear about the transit times and arrest patterns of circulating cancer cells in different organs, intravascular growth prior to extravasation, the rate at which metastatic cells appear in the primary tumour and the number of CTCs that pass through a capillary bed?
In my view these questions lost their appeal when we entered the 'gene-centric' era with its expectation that the answer to every question in biology lies in the genome, and in deeper and yet deeper sequencing. (On a side note I think the gene-centricity in turn is driven by the love of new technology which implies that any data that is acquired with the previous technology is worthless and does not need any explanation.) When focus moved to the genome a lot of the knowledge about the physical aspects of metastasis was ignored, and when those facts had been forgotten they were re-examined in genetic terms. That said, I'm happy to witness how genetic and proteomic techniques have advanced since the publication of this book, and I can honestly say that the chapters that deal with the biochemistry of cancer have not aged well. Today we know a lot more about the biochemistry of the metastatic cascade, but on the other hand possibly less about its physical aspects.
Apart from providing a new (or rather old) perspective on the process of metastasis the book also contains something that I find lacking in the current scientific literature: the critical review. Dr. Weiss takes the time to dissect the experimental setup of the studies he reviews and identifies loop-holes and mistakes in the deduction and hence conclusions in many of the referenced papers. The typical review article of today tells a pleasant story (via numerous studies) of an important topic and sets the agenda for future research. I think this mainly happens since the author often is an authority on the topic, and hence has much to loose on criticising the techniques or methods used within the field. It is much more tempting then to show off your research field as a successful endeavour where experiments are successfully carried out and theory is steadily improving. What I would like to see is a much more critical stance when writing review articles: most experiments are good, but some are bad, and just repeating the conclusions drawn by the authors does not in my opinion advance science.
These are some of the thoughts that passed through my mind when reading Leonard Weiss' 'The principles of metastasis', an excellent companion for those who are interested in the dynamics of metastatic spread. I'll leave you with a figure from the book that I think summarises the authors view of metastases.
Let me be a bit more specific, and mention two things that caught me by surprise when I read this book. Firstly, it was established in the 40's that cancer cells from primary tumours in the breast and prostate can travel to the vertebrae and pelvis without first passing through the capillary bed of the lung. This passage is mediated by a structure known as Batson's venous plexus, which allows for a reversal of flow in the veins, in particular during coughing of sneezing. This, at least in part, explains the predilection for breast and prostate tumours to metastasise to the bone, yet these organ pairs are today often mentioned as prime examples of the seed-soil hypothesis, i.e. the idea that cancer cells thrive in certain organs with favourable 'soil'. This is often talked about in terms of micro-environmental compatibility, which compared to physical blood flow is quite a far fetched and complex explanation.
My second moment of surprise came when Weiss discussed the literature on cancer cell clump size distribution, meaning the size of the cancer cell clusters that enter the circulation and eventually arrest in capillary beds. I'm far from an expert in the field of metastasis but the view that I have acquired from reading review articles and papers on the topic is that cancer cells travel on their own, never collectively. This mistake from my side (or of the authors I've previously read) is made even more serious since it turns out that the success of cancer cells when they arrest in foreign organs depend very much on the clump size; larger clump of cancer cells means higher chance of forming a metastases.
I sincerely hope that the above mentioned 'surprises' are isolated occurrences, but the impression I get from reading this book is that many fundamental insights into the dynamics of the metastatic process have been lost since the 80's. When did you last hear about the transit times and arrest patterns of circulating cancer cells in different organs, intravascular growth prior to extravasation, the rate at which metastatic cells appear in the primary tumour and the number of CTCs that pass through a capillary bed?
In my view these questions lost their appeal when we entered the 'gene-centric' era with its expectation that the answer to every question in biology lies in the genome, and in deeper and yet deeper sequencing. (On a side note I think the gene-centricity in turn is driven by the love of new technology which implies that any data that is acquired with the previous technology is worthless and does not need any explanation.) When focus moved to the genome a lot of the knowledge about the physical aspects of metastasis was ignored, and when those facts had been forgotten they were re-examined in genetic terms. That said, I'm happy to witness how genetic and proteomic techniques have advanced since the publication of this book, and I can honestly say that the chapters that deal with the biochemistry of cancer have not aged well. Today we know a lot more about the biochemistry of the metastatic cascade, but on the other hand possibly less about its physical aspects.
Apart from providing a new (or rather old) perspective on the process of metastasis the book also contains something that I find lacking in the current scientific literature: the critical review. Dr. Weiss takes the time to dissect the experimental setup of the studies he reviews and identifies loop-holes and mistakes in the deduction and hence conclusions in many of the referenced papers. The typical review article of today tells a pleasant story (via numerous studies) of an important topic and sets the agenda for future research. I think this mainly happens since the author often is an authority on the topic, and hence has much to loose on criticising the techniques or methods used within the field. It is much more tempting then to show off your research field as a successful endeavour where experiments are successfully carried out and theory is steadily improving. What I would like to see is a much more critical stance when writing review articles: most experiments are good, but some are bad, and just repeating the conclusions drawn by the authors does not in my opinion advance science.
These are some of the thoughts that passed through my mind when reading Leonard Weiss' 'The principles of metastasis', an excellent companion for those who are interested in the dynamics of metastatic spread. I'll leave you with a figure from the book that I think summarises the authors view of metastases.
Saturday, 20 April 2013
ESTRO Forum 2013
I'm currently attending the European Society for Radiotherapy & Oncology in Geneva, Switzerland. I feel slightly out of place amongst all radiologists and well-dressed oncologist, not to mention the salesreps trying to sell me massive MRI-machines and devices for brachytherapy (I admit I had to google that one). In any case I've been invited to speak about my work on modelling of metastatic spread and delivered the talk this morning. I focused on my recent work with Jacob Scott et al. from Moffitt on self-seeding, and I believe it was well received (at least I got a bunch of questions and comments).
Wednesday, 27 March 2013
Principles of Metastasis
I have recently become interested in the dynamics of metastatic spread, and together with colleagues at Moffitt Cancer Center, I have started to work on mechanistic models that look at the impact of the topology of the vascular network (e.g. here and here). When coming into this field I was surprised by the lack of work along these lines, and found that most people were delving deep into the genome of cancer cells to find the answers of why, where and how metastases appear.
I was therefore positively surprised to hear of the work of the late Dr. Leonard Weiss, who did a lot of work on metastases throughout his long career. In his work we find the physical perspective of metastatic spread that is almost completely absent in this gene-centric era. The other day I finally received his book "Principles of Metastases" from 1985, which I still believe to be highly relevant. A review will be posted when I've finished reading it, which should be soon.
The dynamics of cross-feeding
Many functions carried out by microbes, such as degradation of man-made toxic compounds, require the joint metabolic effort of many bacterial species. In such bacterial communities the success of each species depends on the presence or absence of other species and chemical compounds, effectively joining the components of the community into a microbial ecosystem. A common mode of interaction in such ecosystems is cross-feeding or syntrophy, whereby the metabolic output of one species is being used as a nutrient or energy source by another species.
I have together with my colleague Torbjörn Lundh formulated and analysed a mathematical model of cross-feeding dynamics. We show that under certain assumptions about the system (e.g., high flow of nutrients and time scale separation), the governing equations reduce to a second-order series expansion of the replicator equation. By analysing the case of two and three species we derive conditions for co-existence and show under which parameter conditions one can expect an increase in mean fitness.
The paper was recently published in Bulletin of Mathematical Biology:
http://link.springer.com/content/pdf/10.1007%2Fs11538-013-9828-3
I have together with my colleague Torbjörn Lundh formulated and analysed a mathematical model of cross-feeding dynamics. We show that under certain assumptions about the system (e.g., high flow of nutrients and time scale separation), the governing equations reduce to a second-order series expansion of the replicator equation. By analysing the case of two and three species we derive conditions for co-existence and show under which parameter conditions one can expect an increase in mean fitness.
The paper was recently published in Bulletin of Mathematical Biology:
http://link.springer.com/content/pdf/10.1007%2Fs11538-013-9828-3
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