Using mathematical models to understand metabolism, genes, and disease
© Nijhout et al. 2015
- Published: 23 September 2015
Mathematical models are a useful tool for investigating a large number of questions in metabolism, genetics, and gene–environment interactions. A model based on the underlying biology and biochemistry is a platform for in silico biological experimentation that can reveal the causal chain of events that connect variation in one quantity to variation in another. We discuss how we construct such models, how we have used them to investigate homeostatic mechanisms, gene–environment interactions, and genotype–phenotype mapping, and how they can be used in precision and personalized medicine.
- Tyrosine Hydroxylase
- Substrate Inhibition
- Metabolic System
- Homeostatic Mechanism
We began more than a dozen years ago creating mathematical models in order to understand the systems behavior of a variety of metabolic networks that are important for human health. In doing so, we were amazed at the plethora of regulatory and control mechanisms that have evolved to keep these systems functional in the face of genetic mutations and large changes in environmental inputs. The study of these regulatory mechanisms has led us, in turn, to devise methods to study four interlocking complex ideas: homeostatic plateaus, cryptic genetic variation, predisposition to disease, and precision or personalized medicine. The article is not intended as a review of the field, but as a description of our approach, the methods we have created, and our journey towards a practical understanding of these fundamental biological ideas.
A model gives voice to our assumptions about how something works. Every biological experiment is designed within the context of a conceptual model and its results cause us to confirm, reject, or alter that model. Conceptual models are always incomplete because biological systems are very complex and incompletely understood. Moreover, and as a purely practical matter, experiments tend to be guided by small conceptual models of only a very small part of a system, with the assumption (or hope) that the remaining details and context do not matter or can be adequately controlled.
Mathematical models are formal statements of conceptual models. Like conceptual models, they are typically incomplete and tend to simplify some details of the system. But what they do have, which experimental systems do not, is that they are completely explicit about what is in the model, and what is not. Having a completely defined system has the virtue of allowing one to test whether the assumptions and structure of the model are sufficient to explain the observed, or desired, results. This is one of the main points made by Jeremy Gunawardena in his essay that initiated this series of expository articles .
Mathematical models should not be ends in themselves. If they are to be of use, they should illuminate interesting things about the biology of a system or allow the user, by in silico experimentation, to discover things that would be difficult (e.g., severely reduce nutrient input), unethical (e.g., knock out or modify a gene in humans) or expensive (e.g., change the expression levels of different combinations of genes), or impractical to do in vivo or in vitro.
Ideally, a well-validated mathematical model is a tool, just like a microscope. It is a tool that complements other tools used in biological investigations, and it operates best (or at least most usefully) when there is an active interaction between modelers and experimenters. Experiments provide parameter values, functions and interactions that are essential for constructing the topology and kinetics of a model. A model, in turn, can suggest new experiments or help explain unexpected results. New experimental results improve a model and a model can guide the next round of experiments. Reciprocal illumination between the two should allow one to advance understanding more quickly and more accurately than would be possible with experimentation alone.
We have focused most of our modeling studies on metabolic networks that are relevant to human health [2–13]. These systems are inherently interesting, and there are always large amounts of data and observations that need to be understood and explained. Sometimes data are contradictory or inconsistent, and that can lead to controversies; mathematical modeling can help provide explanations for the observed differences. More importantly, because these systems have been studied for a very long time there are a lot of structural and quantitative data that can be used to determine functional relationships and parameters. Metabolic systems are very complex and typically have many regulatory mechanisms (see below), so knowledge of the kinetics of individual reactions, though important, does not by itself explain the behavior of the whole system. In such a situation, mathematical models are essential to gain understanding at the systems level.
Since the purpose of our modeling is not to produce a model but to produce an exploratory and explanatory tool, we focus on systems where the kinetics of individual reactions or transporters have been well-studied experimentally. The aim of our work is to develop models that can help explain puzzling, conflicting, or contradictory experimental or clinical findings, and that can also be used to deduce the causal chain between genetic variation and phenotypic variation.
We normally work on metabolic systems in which the number of molecules of the species of interest is large enough that we can model the system by ordinary differential equations (ODEs) for the concentrations of the species (the well-mixed assumption). These differential equations simply reflect mass balance. The rate of change of a metabolite is the sum of the rates of the reactions by which it is made minus the rates of the reactions in which it is used. These rates are complicated, usually highly non-linear, formulas that express the rates as functions of the current values of one or more metabolite or allosteric regulator concentrations. These functions, the reaction kinetics, and the associated parameter values are obtained from the literature and from a large network of collaborators who perform experimental or clinical studies. The time courses of solution curves for different choices of inputs or parameter values are obtained by solving the ODEs in MatLab (Mathworks, Natick, MA, USA). If a system is large, we start by making mathematical models of small subsystems. Only when the subsystems are well understood do we study the whole system.
Models are inevitably limited, and can in fact be biased, by the data used to derive the equations and parameter values. This is especially problematic when a model is based on a small amount of data. It is essential, therefore, to continually test a model against new data that did not go into its construction. If the model cannot reproduce the basic trends in the data, then one knows that some new biological or biochemical ideas need to be added. On the other hand, if the model performs well with the new data, one’s confidence is increased that the model represents physiological reality. In this sense, the ‘model’ is not a fixed object, but continually evolves through testing it against data and revising it accordingly.
Of necessity one has to make a choice about the parameter values in a deterministic mathematical model. They are initially selected from the literature, although those values often vary considerably from publication to publication. The values are then gradually refined by testing the model against a broad diversity of data from experimental perturbations, dose–response studies, effects of mutations, and clinical studies. A good model should eventually reproduce, with some accuracy, a broad diversity of experimental data without having to fiddle with the parameter values. The result is a kind of ‘average’ or ‘normal’ model. Of course it cannot be expected to give an accurate representation of individual variability. To do that, we create population models based on our deterministic models (see below).
Metabolic systems are subject to hourly, daily, and seasonal variation in input and demand. Moreover, individuals and populations have different genetic makeups that can affect the expression and activity of enzymes and transporters in metabolic networks. Yet there are critical reactions and metabolite levels that must be maintained in the face of this environmental and genetic variation, or that must be adjusted to meet temporary increases in demand without compromising other important reactions in the system.
All the metabolic systems that we have studied contain homeostatic mechanisms that accomplish this task. These mechanisms include product inhibition (as one might expect), but also substrate inhibition and allosteric regulation, both positive and negative, of enzymes by metabolites in distant parts of the network. We will illustrate these mechanisms through several specific examples.
Substrate inhibition is an interesting phenomenon: as the concentration of substrate increases the rate of the reaction decreases because the substrate binds at allosteric sites that reduce the activity of the enzyme that uses it . In the folate cycle, for instance, many enzymes are subject to substrate inhibition [3, 15–17]. As a consequence much of each enzyme is catalytically inactive under normal folate concentrations. When folate levels decline, some of the inactive enzyme–folate complexes dissociate, releasing new active enzyme and folate. Modeling has shown that in the absence of substrate inhibition flux through the pathway declines linearly with folate concentration, but in the presence of substrate inhibition reaction rates stay nearly constant until folate drops to less than 20 % of its normal level . Folate substrate inhibition probably evolved as a buffering mechanism against seasonal variation in folate availability from fresh leafy greens in order to maintain critical reactions in the folate cycle, such as early steps in the synthesis of purines and pyrimidines for DNA synthesis. Substrate inhibition was emphasized by Haldane  in 1930, but it has been treated as a chemical curiosity in much of the literature even though some 20 % of all known enzymes show substrate inhibition . In all cases that we have examined, substrate inhibition serves a biological purpose .
A quite different homeostatic mechanism occurs in the stabilization of dopamine (DA) in synapses. The concentration of DA in the extracellular space is maintained by a balance between release from the terminal and reuptake by the DA transporters (DATs), and by autoreceptors that sense the DA concentration in the extracellular space [21, 22]. When the concentration goes down the autoreceptors increase the activity of tyrosine hydroxylase (TH), the first enzyme in the two-step conversion of tyrosine to dopamine , and the rate of release of DA from synaptic vesicles [24–26]. If extracellular DA is too high, the opposite occurs. We developed a mathematical model for DA synthesis and release [5, 27, 28], and showed that the autoreceptor mechanism is very effective at stabilizing extracellular DA.
Common large-effect mutations in FOCM and dopamine metabolism, their effects on the activities of the respective enzymes, and frequencies in selected populationsa
Activity relative to wild type
9 % (C), 16 % (US), 20 % (EU)
17 % (J), 55 % (US)
51 % (I), 34.5 % (ME), 35 % (US)
33 % (I), 33 % (US)
48 % (US), 40 % (EU), 8 % (C)
76 % (US), 33 % (C)
4.5 % (AA)
0.6 % (S)
14–75 % (H)
We have shown only a few genotype–phenotype surfaces, but, of course, we can create such surfaces for all the genotypic variables (enzyme and transporter activities) and phenotypic variables (concentrations, reaction velocities) in our models and place an individual who has been genotyped at a particular spot on each of the surfaces. If the individual is near the edge of a cliff, medical advice could suggest lifestyle changes (diet) or drug therapy that would make the region around the genotypic point flatter and more homeostatic. We give two examples to explain how our mechanistic models could be used to do just that.
Suppose that an individual has very low MS activity and therefore very low AICART activity (Fig. 4c). To intervene wisely one has to know what the causal connection is between MS and AICART; it’s not obvious since they are not near each other in the network (Fig. 1). It’s easy to see in the model (simulations not shown) that when MS has very low activity then the concentration of 5mTHF gets very high and the other folates get very low (a situation know as the folate trap, because all folates are eventually converted to 5mTHF), so 10-formyltetrahydrofolate (10f-THF) declines and cannot drive the AICART reaction, which results in a diminished ability to synthesize nucleotides. Thus, the goal of the therapy is to reduce the folate trap. This could be done by giving folate, or it could be done by giving vitamin B12, which is a co-factor for the MS reaction. High homocysteine in the plasma is a biomarker for the folate trap, so in either case one can test the efficacy of the strategy by measuring the decrease in homocysteine.
As a second example, consider the genotypes that are close to the DA cliff in Fig. 5 because they have low TH activity and are therefore at risk for dystonia. A good therapeutic strategy should move them away from the edge of the cliff, or make the region around the defective genotype flatter. Our simulations (not shown) indicate that this can be done by increasing the strength of the DA autoreceptor effect, either by giving autoreceptor agonists or by increasing the expression level of the DA autoreceptors.
What does biological variation mean? It means, for instance, that no two individuals have exactly the same folate metabolisms in their liver cells. There are many reasons for this. Their genotypes are different and that affects the activity of various folate enzymes. The expression levels of the genes that code for these enzymes vary in time depending on what the cell is doing, and are influenced by endocrine factors. Furthermore, during the day, enormous changes in amino acid inputs occur due to meals. Finally, many of the enzymes require vitamin co-factors, and so the reactions depend to some extent on individual dietary histories over months and years.
Such a virtual population database can now be studied to discover interesting statistical relationships between phenotypic, genetic, and environmental variables. The population model is based on a deterministic model, so we can use the deterministic model to discover the causal chain of events that leads to the statistical relationships seen in the database. In this way, statistical analyses, which deduce the correlated variation among variables, and deterministic mathematical models, which can show how those variable are causally linked, can work together to elucidate underlying biological function.
All methods and tools have limitations and that is true of our approach too. Since we stay very close to the underlying biology and biochemistry, we can only study metabolic systems where there is a great deal of information about the component parts and the kinetics of how they affect each other. Nevertheless, our approach can be used in other areas besides cell metabolism. There are some outstanding examples in physiology, such as the regulation of blood pressure , and kidney function [40, 41]. Other examples can be found in epidemiology [42, 43], immunology , cancer biology [45, 46], regulation of the cell cycle [47, 48], insulin and diabetes [49–51] and various examples worked out in .
For biological systems in which little is known about details or kinetics, phenomenological models that capture some aspect of the data may be helpful in clarifying ideas and suggesting targets for experimentation. For example, this is surely the case in studying the brain, where the chasm between experimentation at the neural level and behavior is so vast that models using hypothetical intermediate variables can be valuable. However, we remark again that experimentation with such models will tell us things about the underlying biology only if the parameters have biological meanings and could potentially be measured by experimentalists.
An approach, favored by some systems biologists, is to use correlations between phenotypic variables measured in a population (people or cells) to reverse engineer the network itself and then to assign strengths to the interactions between the variables by using machine learning and parameter estimation. We believe such methods are unlikely to succeed because the underlying biological processes are highly nonlinear, and nonlinear in many different ways, whereas statistical learning approaches to parameter estimation depend on linear models or on models that allow only a restricted type of nonlinearity. It is unlikely that such approaches could find the roles of allosteric activation, substrate inhibition, and dynamic stability to short-term perturbations, robustness to genetic variation, and gene–environment interactions that we see in biology, and in our models. Our virtual population databases could, of course, be used to test whether a reconstruction method can accurately deduce the structure and kinetics of the network that generated the data.
Statistical approaches on large data sets can be useful for revealing interesting correlations and relationships between phenotypic and genetic variables. As we indicated above, we can derive interesting and useful information by statistical analyses on our virtual population databases. And in systems where the biology is largely unknown, statistical methodology may be the only useful set of tools that one has. However, one must always remember that correlation is not causality. Until one understands mechanism, designing experiments or intervention strategies (whether via drugs, life style changes, or surgery) based on the model is difficult and risky.
The main point of this article is to explain that mathematical models are a useful tool for investigating a large number of questions in metabolism, genetics, and gene–environment interactions. If the model is based on the underlying biology and biochemistry, then it becomes a platform for in silico biological experimentation and it can also reveal the causal chain of events that connect variation in one quantity to variation in another. The variables and parameters in the model must be related, directly or indirectly, to quantities that biologists measure, so that experiments with the model have biological meaning. The metabolic systems that have evolved are very complicated, subtle, and difficult to understand. There is no substitute for detailed biological experimentation on the biology and the biochemistry of the parts. But mathematical models, based on the real biology, can shed light on how the parts work together and the causal relationships between them, and suggest strategies for interventions in disease states.
This work was supported in part by NSF grant EF-1038593 (HFN, MCR), the Mathematical Biosciences Institute and the NSF under grant DMS-0931642 (JAB, MCR), and an NSF CAREER Award DMS-0956057 (JAB).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
- Gunawardena J. Models in biology: ‘accurate descriptions of our pathetic thinking’. BMC Biol. 2014;12:29.PubMed CentralView ArticlePubMedGoogle Scholar
- Nijhout HF, Reed MC, Anderson DF, Mattingly JC, James SJ, Ulrich CM. Long-range allosteric interactions between the folate and methionine cycles stabilize DNA methylation reaction rate. Epigenetics. 2006;1:81–7.View ArticlePubMedGoogle Scholar
- Nijhout HF, Reed MC, Budu P, Ulrich CM. A mathematical model of the folate cycle: new insights into folate homeostasis. J Biol Chem. 2004;279:55008–16.View ArticlePubMedGoogle Scholar
- Reed M, Thomas R, Pavisic J, James S, Ulrich C, Nijhout H. A mathematical model of glutathione metabolism. Theor Biol Med Model. 2008;5:1–16.View ArticleGoogle Scholar
- Reed MC, Best JA, Nijhout HF. Passive and active stabilization of dopamine in the striatum. Biosci Hypotheses. 2009;2:240–4.View ArticleGoogle Scholar
- Reed MC, Nijhout HF, Neuhouser ML, Gregory JF, Shane B, James SJ, et al. A mathematical model gives insights into nutritional and genetic aspects of folate-mediated one-carbon metabolism. J Nutr. 2006;136:2653–61.PubMedGoogle Scholar
- Ben-Shachar R, Chen Y, Luo S, Hartman C, Reed M, Nijhout HF. The biochemistry of acetaminophen hepatotoxicity and rescue: a mathematical model. Theor Biol Med Model. 2012;9:55.PubMed CentralView ArticlePubMedGoogle Scholar
- Lawley S, Cinderella M, Hall M, Gamble M, Nijhout HF, Reed M. Mathematical model insights into arsenic detoxification. Theor Biol Med Model. 2011;8:31.PubMed CentralView ArticlePubMedGoogle Scholar
- Lawley S, Yun J, Gamble M, Hall M, Reed M, Nijhout H. Mathematical modeling of the effects of glutathione on arsenic methylation. Theor Biol Med Model. 2014;11:20.PubMed CentralView ArticlePubMedGoogle Scholar
- Reed MC, Nijhout HF, Best JA. Mathematical insights into the effects of levodopa. Front Integrative Neurosci. 2012;6:21.View ArticleGoogle Scholar
- Best JA, Oakley G, Reed MC, Nijhout HF: Mathematical models: interactions between serotonin and dopamine in Parkinson’s Disease. In: Rana AQ, editor. Etiology and pathophysiology of Parkinson's disease. InTech; 2011. p. 405–20.Google Scholar
- Duncan TM, Reed MC, Nijhout HF. A population model of folate-mediated one-carbon metabolism. Nutrients. 2013;5:2457–74.PubMed CentralView ArticlePubMedGoogle Scholar
- Reed MC, Nijhout HF, Sparks R, Ulrich CM. A mathematical model of the methionine cycle. J Theor Biol. 2005;226:33–43.View ArticleGoogle Scholar
- Reed MC, Lieb A, Nijhout HF. The biological significance of substrate inhibition: a mechanism with diverse functions. Bioessays. 2010;32:422–9.View ArticlePubMedGoogle Scholar
- Wagner C. Symposium on the subcellular compartmentation of folate metabolism. J Nutr. 1996;126:1228S–34S.PubMedGoogle Scholar
- Zamierowski MM, Wagner C. Identification of folate binding proteins in rat liver. J Biol Chem. 1977;252:933–8.PubMedGoogle Scholar
- Zamierowski M, Wagner C. High molecular weight complexes of folic acid in mammalian tissues. Biochem Biophys Res Commun. 1974;60:81–7.View ArticlePubMedGoogle Scholar
- Haldane JBS. Enzymes. London: Longmans, Green & Co.; 1930.Google Scholar
- Chaplin M, Bucke C. Enzyme technology. Cambridge: Cambridge Univ. Press; 1990.Google Scholar
- Wagner C, Briggs WT, Cook RJ. Inhibition of glycine N-methyltransferase activity by folate derivatives: implications for regulation of methyl group metabolism. Biochem Biophys Res Commun. 1985;127:746–52.View ArticlePubMedGoogle Scholar
- Bergstrom BP, Garris PA. ‘Passive stabilization’ of striatal extracellular dopamine across the lesion spectrum encompassing the presymptomatic phase of Parkinson's disease: a voltammetric study in the 6-OHDA-lesioned rat. J Neurochem. 2003;87:1224–36.View ArticlePubMedGoogle Scholar
- Garris PA, Walker QD, Wightman RM. Dopamine release and uptake rates both decrease in the partially denervated striatum in proportion to the loss of dopamine terminals. Brain Res. 1997;753:225–34.View ArticlePubMedGoogle Scholar
- Feldman R, Meyer J, Quenzer L. Principles of neuropharmacology. Sunderland, MA: Sinauer Associates; 1997.Google Scholar
- Walters JR, Roth RH. Dopaminergic neurons: drug-induced antagonism of the increase in tyrosine hydroxylase activity produced by cessation of impulse flow. J Pharmacol Exp Ther. 1974;191:82–91.PubMedGoogle Scholar
- Tissari AH, Lillgäls MS. Reduction of dopamine synthesis inhibition by dopamine autoreceptor activation in striatal synaptosomes with in vivo reserpine administration. J Neurochem. 1993;61:231–8.View ArticlePubMedGoogle Scholar
- Benoit-Marand M, Borrelli E, Gonon F. Inhibition of dopamine release via presynaptic D2 receptors: time course and functional characteristics in vivo. J Neurosci. 2001;21:9134–41.PubMedGoogle Scholar
- Best JA, Nijhout HF, Reed MC. Homeostatic mechanisms in dopamine synthesis and release: a mathematical model. Theor Biol Med Model. 2009;6:21.PubMed CentralView ArticlePubMedGoogle Scholar
- Best JA, Nijhout HF, Reed MC. Models of dopaminergic and serotonergic signaling. Pharmacopsychiatry. 2010;43:561–6.View ArticleGoogle Scholar
- Nijhout HF, Reed MC, Lam S-L, Gregory JF, Shane B, Ulrich CM. A day in the life of cell metabolism. J Biol Theory. 2008;2:2124–7.Google Scholar
- Best JA, Nijhout HF, Reed MC. Serotonin synthesis, release and reuptake in terminals: a mathematical model. Theor Biol Med Model. 2010;7:34.PubMed CentralView ArticlePubMedGoogle Scholar
- Thomas D, Conti D, Baurley J, Nijhout F, Reed M, Ulrich C. Use of pathway information in molecular epidemiology. Hum Genomics. 2009;4:21–42.PubMed CentralView ArticlePubMedGoogle Scholar
- Ulrich CM, Curtin K, Potter JD, Bigler J, Caan B, Slattery ML. Polymorphisms in the reduced folate carrier, thymidylate synthase, or methionine synthase and risk of colon cancer. Cancer Epidemiol Biomarkers Prevention. 2005;14:2509–16.View ArticleGoogle Scholar
- Kealey C, Brown K, Woodside J, Young I, Murray L, Boreham C, et al. A common insertion/deletion polymorphism of the thymidylate synthase (TYMS) gene is a determinant of red blood cell folate and homocysteine concentrations. Hum Genet. 2005;116:347–53.View ArticlePubMedGoogle Scholar
- Kennedy DA, Stern SJ, Matok I, Moretti ME, Sarkar M, Adams-Webber T, et al. Folate intake, MTHFR polymorphisms, and the risk of colorectal cancer: a systematic review and meta-analysis. J Cancer Epidemiol. 2012;2012:24.View ArticleGoogle Scholar
- Nazki FH, Sameer AS, Ganaie BA. Folate: metabolism, genes, polymorphisms and the associated diseases. Gene. 2014;533:11–20.View ArticlePubMedGoogle Scholar
- Nijhout HF, Best J, Reed MC. Escape from homeostasis. Math Biosci. 2014;257:104–10.View ArticlePubMedGoogle Scholar
- Nijhout HF, Reed MC. Homeostasis and dynamic stability of the phenotype link robustness and plasticity. Integr Comp Biol. 2014;54:264–75.View ArticlePubMedGoogle Scholar
- Knappskog PM, Flatmark T, Mallet J, Lüdecke B, Bartholomé K. Recessively inherited L-DOPA-responsive dystonia caused by a point mutation (Q381K) in the tyrosine hydroxylase gene. Hum Mol Genet. 1995;4:1209–12.View ArticlePubMedGoogle Scholar
- Guyton AC, Coleman TG, Granger HJ. Circulation: overall regulation. Annu Rev Physiol. 1972;34:13–44.View ArticlePubMedGoogle Scholar
- Sgouralis I, Layton A. Control and modulation of fluid flow in the rat kidney. Bull Math Biol. 2013;75:2551–74.View ArticlePubMedGoogle Scholar
- Sgouralis I, Layton A. Theoretical assessment of renal autoregulatory mechanisms. Am J Renal Physiol. 2014;306:F1357–71.View ArticleGoogle Scholar
- Grassly NC, Fraser C. Mathematical models of infectious disease transmission. Nat Rev Micro. 2008;6:477–87.Google Scholar
- Luo S, Reed M, Mattingly JC, Koelle K. The impact of host immune status on the within-host and population dynamics of antigenic immune escape. J R Soc Interface. 2012;9:2603–13.PubMed CentralView ArticlePubMedGoogle Scholar
- Andrew SM, Baker CTH, Bocharov GA. Rival approaches to mathematical modelling in immunology. J Comput Appl Math. 2007;205:669–86.View ArticleGoogle Scholar
- Alarcón T, Byrne HM, Maini PK. A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells. J Theor Biol. 2004;229:395–411.View ArticlePubMedGoogle Scholar
- Johnston MD, Edwards CM, Bodmer WF, Maini PK, Chapman SJ. Mathematical modeling of cell population dynamics in the colonic crypt and in colorectal cancer. Proc Natl Acad Sci U S A. 2007;104:4008–13.PubMed CentralView ArticlePubMedGoogle Scholar
- Sible JC, Tyson JJ. Mathematical modeling as a tool for investigating cell cycle control networks. Methods. 2007;41:238–47.PubMed CentralView ArticlePubMedGoogle Scholar
- Tyson JJaBN. Regulation of the eukaryotic cell cycle: molecular antagonism, hysteresis and irreversible transitions. J Theor Biol. 2001;210:249–63.View ArticleGoogle Scholar
- Sedaghat AR, Sherman A, Quon MJ. A mathematical model of metabolic insulin signaling pathways. Am J Physiol Endocrinol Metab. 2002;283:E1084–101.View ArticlePubMedGoogle Scholar
- Topp B, Promislow K, Devries G, Miura RM, Finegood DT. A model of β -cell mass, insulin, and glucose kinetics: pathways to diabetes. J Theor Biol. 2000;206:605–19.View ArticlePubMedGoogle Scholar
- Chew YH, Shia YL, Lee CT, Majid FAA, Chua LS, Sarmidi MR, et al. Modeling of glucose regulation and insulin-signaling pathways. Mol Cell Endocrinol. 2009;303:13–24.View ArticlePubMedGoogle Scholar
- Keener J, Sneyd J. Mathematical physiology, Volumes 1 and 2. 2nd ed. New York: Springer; 2009.Google Scholar