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Cancer cells exhibit an altered metabolic phenotype, known as the Warburg effect, which is characterized by high rates of glucose uptake and glycolysis, even under aerobic conditions. The Warburg effect appears to be an intrinsic component of most cancers and there is evidence linking cancer progression to mutations, translocations, and alternative splicing of genes that directly code for or have downstream effects on key metabolic enzymes. Many of the same signaling pathways are routinely dysregulated in cancer and a number of important oncogenic signaling pathways play important regulatory roles in central carbon metabolism. Unraveling the complex regulatory relationship between cancer metabolism and signaling requires the application of systems biology approaches. Here we discuss computational approaches for modeling protein signal transduction and metabolism as well as how the regulatory relationship between these two important cellular processes can be combined into hybrid models.

Conformational diseases result from the failure of a specific protein to fold into its correct functional state. The misfolded proteins can lead to the toxic aggregation of proteins. Protein misfolding in conformational diseases often displays a threshold behavior characterized by a sudden shift between nontoxic to toxic levels of misfolded proteins. In some conformational diseases, evidence suggests that misfolded proteins interact with bystander proteins (unfolded and native folded proteins), eliciting a misfolded phenotype. These bystander isomers would follow their normal physiological pathways in absence of misfolded proteins. In this article, we present a general mechanism of bystander and misfolded protein interaction which we have used to investigate how the threshold behavior in protein misfolding is triggered in conformational diseases. Using a continuous flow reactor model of the endoplasmic reticulum, we found that slight changes in the bystander protein residence time in the endoplasmic reticulum or the ratio of basal misfolded to bystander protein inflow rates can trigger the threshold behavior in protein misfolding. Our analysis reveals three mechanisms to rescue bystander proteins in conformational diseases. The results of our model can now help direct experiments to understand the threshold behavior and develop therapeutic strategies targeting the modulation of conformational diseases.

One of the main challenges in the biomedical sciences is the determination of reaction mechanisms that constitute a biochemical pathway. During the last decades, advances have been made in building complex diagrams showing the static interactions of proteins. The challenge for systems biologists is to build realistic models of the dynamical behavior of reactants, intermediates and products. For this purpose, several methods have been recently proposed to deduce the reaction mechanisms or to estimate the kinetic parameters of the elementary reactions that constitute the pathway. One such method is MIKANA: Method to Infer Kinetics And Network Architecture. MIKANA is a computational method to infer both reaction mechanisms and estimate the kinetic parameters of biochemical pathways from time course data. To make it available to the scientific community, we developed a Graphical User Interface (GUI) for MIKANA. Among other features, the GUI validates and processes an input time course data, displays the inferred reactions, generates the differential equations for the chemical species in the pathway and plots the prediction curves on top of the input time course data. We also added a new feature to MIKANA that allows the user to exclude a priori known reactions from the inferred mechanism. This addition improves the performance of the method. In this article, we illustrate the GUI for MIKANA with three examples: an irreversible Michaelis–Menten reaction mechanism; the interaction map of chemical species of the muscle glycolytic pathway; and the glycolytic pathway of Lactococcus lactis. We also describe the code and methods in sufficient detail to allow researchers to further develop the code or reproduce the experiments described. The code for MIKANA is open source, free for academic and non-academic use and is available for download (Information S1).

In the endoplasmic reticulum, protein production occurs through a complex set of reaction networks which synthesize and fold protein polypeptides. Proteins can misfold due to mutation or cellular stress. Accumulation of misfolded proteins can lead to the formation of toxic protein aggregates. Cells have evolved the unfolded protein response which removes misfolded proteins from the endoplasmic reticulum before aggregation can occur. Failure of protein synthesis and folding quality control is associated with disease. We are beginning to comprehend how these processes are controlled. Here, we develop two mathematical models to investigate how cellular quality control regulates protein homeostasis. Using parametric sensitivity analysis, we identify pathway parameters involved in increasing folded protein production.

Morphological and functional changes during ameloblast and odontoblast differentiation suggest that enamel and dentin formation is under circadian control. Circadian rhythms are endogenous self-sustained oscillations with periods of 24 hours that control diverse physiological and metabolic processes. Mammalian clock genes play a key role in synchronizing circadian functions in many organs. However, close to nothing is known on clock genes expression during tooth development. In this work, we investigated the expression of four clock genes during tooth development. Our results showed that circadian clock genes Bmal1, clock, per1, and per2 mRNAs were detected in teeth by RT-PCR. Immunohistochemistry showed that clock protein expression was first detected in teeth at the bell stage (E17), being expressed in EOE and dental papilla cells. At post-natal day four (PN4), all four clock proteins continued to be expressed in teeth but with different intensities, being strongly expressed within the nucleus of ameloblasts and odontoblasts and down-regulated in dental pulp cells. Interestingly, at PN21 incisor, expression of clock proteins was down-regulated in odontoblasts of the crown-analogue side but expression was persisting in root-analogue side odontoblasts. In contrast, both crown and root odontoblasts were strongly stained for all four clock proteins in first molars at PN21. Within the periodontal ligament (PDL) space, epithelial rests of Malassez (ERM) showed the strongest expression among other PDL cells. Our data suggests that clock genes might be involved in the regulation of ameloblast and odontoblast functions, such as enamel and dentin protein secretion and matrix mineralization.

Systems biology takes an interdisciplinary approach to the systematic study of complex interactions in biological systems. This approach seeks to decipher the emergent behaviors of complex systems rather than focusing only on their constituent properties. As an increasing number of examples illustrate the value of systems biology approaches to understand the initiation, progression, and treatment of cancer, systems biologists from across Europe and the United States hope for changes in the way their field is currently perceived among cancer researchers. In a recent EU-US workshop, supported by the European Commission, the German Federal Ministry for Education and Research, and the National Cancer Institute of the NIH, the participants discussed the strengths, weaknesses, hurdles, and opportunities in cancer systems biology. Cancer Res; 70(1); 12–3

The traditional experimental practice in enzyme kinetics involves the measurement of substrate or product concentrations as a function of time. Advances in computing have produced novel approaches for modeling enzyme catalyzed reactions from time course data. One example of such an approach is the selection of appropriate chemical reactions that best fit the data. A common limitation of this approach resides in the number of chemical species considered. The number of possible chemical reactions grows exponentially with the number of chemical species, which makes difficult to select reactions that uniquely describe the data and diminishes the efficiency of the methods. In addition, a method's performance is also dependent on several quantitative and qualitative properties of the time course data, of which we know very little. This information is important to experimentalists as it could allow them to setup their experiments in ways that optimize the network reconstruction. We have previously described a method for inferring reaction mechanisms and kinetic rate parameters from time course data. Here, we address the limitations in the number of chemical reactions by allowing the introduction of information about chemical interactions. We also address the unknown properties of the input data by determining experimental data properties that maximize our method's performance. We investigate the following properties: initial substrate-enzyme concentration ratios; initial substrate-enzyme concentration variation ranges; number of data points; number of different experiments (time courses); and noise. We test the method using data generated in silico from the Michaelis-Menten and the Hartley-Kilby reaction mechanisms. Our results demonstrate the importance of experimental design for time course assays that has not been considered in experimental protocols. These considerations can have far reaching implications for the computational mechanism reconstruction process.

Pancreatic beta-cell failure is increasingly recognized as central to the progression of diabetes mellitus. Different causes are implicated in the onset of beta-cell stress, dysfunction or death. Recent genetic and biochemical evidence both in humans and mice shows that failure in modulating the capacity and quality of the endoplasmic reticulum protein-folding machinery leads to beta-cell death. The quality control of protein-folding is regulated by several signaling pathways, which are collectively termed the unfolded protein response. In beta-cells proinsulin represents up to 50% of the total protein synthesis, and the rate of glucose-stimulated proinsulin translation is approximately 1 million molecules per minute per cell. When proinsulin folding is disrupted, the high burden imposed by proinsulin synthesis on the unfolded protein response is considered the leading cause of certain diabetes phenotypes. We have developed a model of the unfolded protein response to investigate the factors that can prevent misfolded protein accumulation. We also present predictions for therapeutic strategies to ameliorate pancreatic beta-cell death in diabetes patients.

In this article we will discuss the integration of developmental patterning mechanisms with waves of competency that control the ability of a homogeneous field of cells to react to pattern forming cues and generate spatially heterogeneous patterns. We base our discussion around two well known patterning events that take place in the early embryo: somitogenesis and feather bud formation. We outline mathematical models to describe each patterning mechanism, present the results of numerical simulations and discuss the validity of each model in relation to our example patterning processes.

Since the discovery of gene products oscillating during the formation of vertebral segments, much attention has been directed toward eluciating the molecular basis of the so-called segmentation clock. What research has told us is, that even in the most simple vertebrates, enormously complicated gene networks act in each cell to give rise to oscillations, and that cell-cell communication synchronizes these oscillations between neighboring cells. A number of theories have been proposed to explain both the initiation and maintenance of oscillations in a single cell and the synchronization of such oscillations between cells. We discuss these theories in this Commentary.

Many of the chapters in this volume are concerned with processes or structures inside the nucleus, and it is relevant to consider the properties of their environment, or rather of the multiple different and specific environments that must exist in local regions of the highly heterogeneous intranuclear space. Relatively little is known about the fundamental physical properties of these environments, and theoretical treatments of phenomena in such concentrated mixtures of charged macromolecules are complex and as yet poorly developed. Some of the phenomena that occur at the molecular level are unexpected and counterintuitive for biologists, although well known to colloid and polymer scientists; for example, the existence of short-range attractive forces between macromolecules or structures with like charges. As a background for the chapters that follow, we consider here some of the particular features of intranuclear environments, how they may influence processes and structures in the nucleus, and their implications for working with nuclei.

Food allergies affect an estimated 3 to 4% of adults and up to 8% of children in developed western countries. Results from in vitro simulated gastric digestion studies with purified proteins are routinely used to assess the allergenic potential of novel food proteins. The digestion of purified proteins in simulated gastric fluid typically progresses in an exponential fashion allowing persistence to be quantified using pseudo-first-order rate constants or half lives. However, the persistence of purified proteins in simulated gastric fluid is a poor predictor of the allergenic status of food proteins, potentially due to food matrix effects that can be significant in vivo. The evaluation of the persistence of novel proteins in whole, prepared food exposed to simulated gastric fluid may provide a more correlative result, but such assays should be thoroughly validated to demonstrate a predictive capacity before they are accepted to predict the allergenic potential of novel food proteins.

In the application of the quasi-steady-state approximation, it is generally assumed that there is an initial transient during which the substrate concentration remains approximately constant while the complex concentration builds up. In this paper, we address the assumption that the substrate concentration does not change significantly during this initial transient and name it the reactant stationary approximation. For the single enzyme, single substrate reaction, the reactant stationary approximation is generally considered a sufficient condition to apply the quasi-steady-state approximation. Studying the dynamic behavior of this reaction with endogenous substrate, we show that the quasi-steady-state approximation and reactant stationary approximation are two separate approximations. We discuss the consequence of this result for the determination of reaction parameters in enzyme catalyzed reactions.

During gastrulation in vertebrates, mesenchymal cells at the anterior end of the pre-somitic mesoderm (PSM) periodically compact, transiently epithelialize and detach from the posterior PSM to form somites. In the prevailing clock-and-wavefront model of somitogenesis, periodic gene expression, particularly of Notch and Wnt, interacts with an FGF8-based thresholding mechanism to determine cell fates. However, this model does not explain how cell determination and subsequent differentiation translates into somite morphology. In this paper, we use computer simulations of chick somitogenesis to show that experimentally-observed temporal and spatial patterns of adhesive N-CAM and N-cadherin and repulsive EphA4–ephrinB2 pairs suffice to reproduce the complex dynamic morphological changes of somitogenesis in wild-type and N-cadherin (−/−) chick, including intersomitic separation, boundary-shape evolution and sorting of misdifferentiated cells across compartment boundaries. Since different models of determination yield the same, experimentally-observed, distribution of adhesion and repulsion molecules, the patterning is independent of the details of this mechanism.

Many of the chapters in this volume are concerned with processes or structures inside the nucleus, and it is relevant to consider the properties of their environment, or rather of the multiple different and specific environments that must exist in local regions of the highly heterogeneous intranuclear space. Relatively little is known about the fundamental physical properties of these environments, and theoretical treatments of phenomena in such concentrated mixtures of charged macromolecules are complex and as yet poorly developed. Some of the phenomena that occur at the molecular level are unexpected and counterintuitive for biologists, although well known to colloid and polymer scientists; for example, the existence of short-range attractive forces between macromolecules or structures with like charges. As a background for the chapters that follow, we consider here some of the particular features of intranuclear environments, how they may influence processes and structures in the nucleus, and their implications for working with nuclei.

In the past decade, advances in molecular biology such as the development of non-invasive single molecule imaging techniques have given us a window into the intricate biochemical activities that occur inside cells. In this chapter we review four distinct theoretical and simulation frameworks: (i) non-spatial and deterministic, (ii) spatial and deterministic, (iii) non-spatial and stochastic and (iv) spatial and stochastic. Each framework can be suited to modelling and interpreting intracellular reaction kinetics. By estimating the fundamental length scales, one can roughly determine which models are best suited for the particular reaction pathway under study. We discuss differences in prediction between the four modelling methodologies. In particular we show that taking into account noise and space does not simply add quantitative predictive accuracy but may also lead to qualitatively different physiological predictions, unaccounted for by classical deterministic models.

Somitogenesis is the process of division of the anterior-posterior vertebrate embryonic axis into similar morphological units known as somites. These segments generate the prepattern which guides formation of the vertebrae, ribs and other associated features of the body trunk. In this work, we review and discuss a series of mathematical models which account for different stages of somite formation. We begin by presenting current experimental information and mechanisms explaining somite formation, highlighting features which will be included in the models. For each model we outline the mathematical basis, show results of numerical simulations, discuss their successes and shortcomings and avenues for future exploration. We conclude with a brief discussion of the state of modeling in the field and current challenges which need to be overcome in order to further our understanding in this area.

There are two simple solutions to reaction–diffusion systems with limit-cycle reaction kinetics, producing oscillatory behaviour. The reaction parameter μ gives rise to a ‘space-invariant’ solution, and μ versus the ratio of the diffusion coefficients gives rise to a ‘time-invariant’ solution. We consider the case where both solution types may be possible. This leads to a refinement of the Turing model of pattern formation. We add convection to the system and investigate its effect. More complex solutions arise that appear to combine the two simple solutions. The convective system sheds light on the underlying behaviour of the diffusive system.

We consider a diffusion model with limit cycle reaction functions, in the presence of convection. We select a set of functions derived from a realistic reaction model: the Schnakenberg equations. This resultant form is unsymmetrical. We find a transformation which maps the irregular equations into model form. Next we transform the dependent variables into polar form. From here, a travelling wave analysis is performed on the radial variable. Results are complex, but we make some simple estimates. We carry out numerical experiments to test our analysis. An initial `knock' starts the propagation of pattern. The speed of the travelling wave is not quite as expected. We investigate further. The system demonstrates distinctly different behaviour to the left and the right. We explain how this phenomenon occurs by examining the underlying behaviour., 20 pages, 5 figures

Metabolic reprogramming is increasingly recognized as a fundamental hallmark of cancer. While the Warburg effect and normal proliferative metabolism are similar, they are not equivalent. We hypothesize that there are key drivers of malignant metabolism that can be modulated to impede cancer proliferation without substantial effects on normal tissue growth. Using 13C-labeled glucose and glutamine tracers in combination with mass spectrometry and measurements of extracellular glucose, lactate, and glutamine flux, we have characterized system level differences in a series of breast cancer cell lines as well as normal-like breast epithelial cells. We observed an increase in the reductive carboxylation of glutamine-derived citrate and alpha-ketoglutarate in the triple-negative inflammatory breast cancer cell line SUM149. We also observed that the SUM149 exhibit high levels of HIF-1α and low levels of oxygen consumption under normoxia, suggesting that the cell line is highly adapted to hypoxia. Surprisingly, the stable depletion of HIF-1α via shRNA had no significant effect on the metabolic profile of these cells. Previous work by our lab and others has demonstrated that the GTPase RhoC is a driver of the metastatic phenotype exhibited by inflammatory breast cancer. Activation of RhoC is known to induce cytoskeletal rearrangements and increase invasive potential. The Rho GTPase family of proteins has also recently been linked to metabolism, specifically regulation of glutaminase activity. Here we show that stable knockdown of RhoC in SUM149 cells results in a marked decrease in the rate of both glutamine uptake and intracellular reductive carboxylation. This work reinforces the role of RhoC as an important driver of inflammatory breast cancer metastatic potential. We conclude that RhoC remains an important clinical target with the potential to alter patient outcomes. Citation Format: Joel A. Yates, Michelle L. Wynn, ZhiFen Wu, Charles R. Evans, Charles Burant, Santiago D. Schnell, Sofia D. Merajver. Metastasis-associated oncogene RhoC as a regulator of glutamine metabolism in the inflammatory breast cancer cell line SUM149. [abstract]. In: Proceedings of the 106th Annual Meeting of the American Association for Cancer Research; 2015 Apr 18-22; Philadelphia, PA. Philadelphia (PA): AACR; Cancer Res 2015;75(15 Suppl):Abstract nr 1203. doi:10.1158/1538-7445.AM2015-1203

Complex signaling cascades involve many interlocked positive and negative feedback loops which have inherent delays. Modeling thesecomplex cascades often requires a large number of variables and parameters. Delay differential equation models have been helpful in describinginherent time lags and also in reducing the number of governing equations. However the consequences of model reduction via delay differentialequations have not been fully explored. In this paper we systematically examine the effect of delays in a complex network of phosphorylation–dephosphorylation cycles (described by Gonze and Goldbeter, J. Theor. Biol., 210, (2001) 167–186), which commonly occur in manybiochemical pathways. By introducing delays in the positive and negative regulatory interactions, we show that a delay differential model canindeed reduce the number of cycles actually required to describe the phosphorylation–dephosphorylation pathway. In addition, we find some ofthe unique properties of the network and a quantitative measure of the minimum number of delay variables required to model the network. Theseresults can be extended for modeling complex signalling cascades.© 2006 Elsevier B.V. All rights reserved.

Vertebrate segmentation has been proved to be under a strict temporal control governed by a biological clock, known as the segmentation clock. The present experimental evidence suggests that the segmentation clock initiates and maintains its periodic cycle by the periodic activation or inhibition of the Notch signaling pathway as well as the periodic autoregulation of the cyclic genes themselves. In this paper, we investigate the structural and evolutionary properties of the Notch pathway proteins involved in the mice segmentation clock and computationally identify the interaction map within the Notch signaling pathway. The results of our analysis strongly indicate that most of the pathway proteins are intrinsically disordered and that the mechanism of their interaction likely involves helical molecular recognition elements, short loosely structured segments within disordered regions which are directly involved in protein-protein interactions. Predicted interactions are in agreement with gene knock-out studies available in the literature.

Somites are transient blocks of cells that form sequentially along the antero-posterior axis of vertebrate embryos. They give rise to the vertebrae, ribs and other associated features of the trunk. In this work we develop and analyse a mathematical formulation of a version of the Clock and Wavefront model for somite formation, where the clock controls when the boundaries of the somites form and the wavefront determines where they form. Our analysis indicates that this interaction between a segmentation clock and a wavefront can explain the periodic pattern of somites observed in normal embryos. We can also show that a simplification of the model provides a mechanism for predicting the anomalies resulting from perturbation of the wavefront.

A theoretical analysis of the distinguishability problem of two rival models of the single enzyme-single substrate reaction, the Michaelis-Menten and Henri mechanisms, is presented. We also outline a general approach for analysing the structural indistinguishability between two mechanisms. The approach involves constructing, if possible, a smooth mapping between the two candidate models. Evans et al. [N.D. Evans, M.J. Chappell, M.J. Chapman, K.R. Godfrey, Structural indistinguishability between uncontrolled (autonomous) nonlinear analytic systems, Automatica 40 (2004) 1947-1953] have shown that if, in addition, either of the mechanisms satisfies a particular criterion then such a transformation always exists when the models are indistinguishable from their experimentally observable outputs. The approach is applied to the single enzyme-single substrate reaction mechanism. In principle, mechanisms can be distinguished using this analysis, but we show that our ability to distinguish mechanistic models depends both on the precise measurements made, and on our knowledge of the system prior to performing the kinetics experiments.

present a mathe-matical model of a peptide-major histo-compatibility complex (pMHC)-nanoparticle(NP)-based therapy for Type 1 diabetes(T1D) to understand and optimize treat-ment conditions in the non-obese diabetic(NOD) mice. This model exemplifies boththe surprising behavior of nonlinear systemsand the power of quantitative biology inmedicine.Defects in memory autoregulatory CD8

Time series data can now be routinely collected for biochemical reaction pathways, and recently, several methods have been proposed to infer reaction mechanisms for metabolic pathways and networks. In this paper we provide a survey of mathematical techniques for determining reaction mechanisms for time series data on the concentration or abundance of different reacting components, with little prior information about the pathways involved.

We review recent evidence illustrating the fundamental difference between cytoplasmic and test tube biochemical kinetics and thermodynamics, and showing the breakdown of the law of mass action and power-law approximation in in vivo conditions. Simulations of biochemical reactions in non-homogeneous media show that as a result of anomalous diffusion and mixing of the biochemical species, reactions follow a fractal-like kinetics. Consequently, the conventional equations for biochemical pathways fail to describe the reactions in in vivo conditions. We present a modification to fractal-like kinetics following the Zipf-Mandelbrot distribution which will enable the modelling and analysis of biochemical reactions occurring in crowded intracellular environments.

The Society for Mathematical Biology (SMB) owns the Bulletin of Mathematical Biology (BMB). This is an international journal devoted to the interface of mathematics and biology. At the 2003 SMB annual meeting in Dundee the Society asked the editor of the BMB to produce an analysis of impact factor, subject matter of papers, submission rates etc. Other members of the society were interested in the handling times of articles and wanted comparisons with other (appropriate) journals. In this article we present a brief history of the journal and report on how the journal impact factor has grown substantially in the last few years. We also present an analysis of subject areas of published papers over the past two years. We finally present data on times from receipt of paper to acceptance, acceptance to print (and to online publication) and compare these data with some other journals.

The linearization of the Michaelis-Menten reaction by pseudo-first-order kinetics is revised. A phase-plane analysis allows the derivation of a new condition for its validity that is directly linked to the reaction efficiency, and contrary to widely established knowledge, is independent of the initial enzyme concentration.

Disruption of normal vertebral development results from abnormal formation and segmentation of the vertebral precursors, called somites. Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. We review the most popular models of somite formation: Cooke and Zeeman's clock and wavefront model, Meinhardt's reaction-diffusion model and the cell cycle model of Stern and co-workers, and discuss the consistency of each in the light of recent experimental findings concerning FGF-8 signalling in the presomitic mesoderm (PSM). We present an extension of the cell cycle model to take account of this new experimental evidence, which shows the existence of a determination front whose position in the PSM is controlled by FGF-8 signalling, and which controls the ability of cells to become competent to segment. We conclude that it is, at this stage, perhaps erroneous to favour one of these models over the others.

Somitogenesis, the process by which a bilaterally symmetric pattern of cell aggregations is laid down in a cranio-caudal sequence in early vertebrate development, provides an excellent model study for the coupling of interactions at the molecular and cellular level. Here, we review some of the key experimental results and theoretical models related to this process. We extend a recent chemical pre-pattern model based on the cell cycle Journal of Theoretical Biology 207 (2000) 305-316, by including cell movement and show that the resultant model exhibits the correct spatio-temporal dynamics of cell aggregation. We also postulate a model to account for the recently observed spatio-temporal dynamics at the molecular level.

It is increasingly likely that the high incidence of off-target effects associated with targeted inhibitors is due, in part, to the highly complex and dysregulated intracellular molecular networks associated with cancer. Ignoring this complexity can lead to suboptimal results and subsequent loss of life through ineffective therapies. The phosphoinositide 3-kinase (PI3K) and mitogen-activated protein kinase (MAPK) pathways are two of the most dysregulated pathways across all cancers. Several regulatory mechanisms have been proposed to explain the apparent cross-talk between them. For example, RAS to ERK signaling in the MAPK pathway has been proposed as an important metastases “escape mechanism” when PI3K is inhibited. In order to rationally develop precise therapeutic avenues to target these oncogenic pathways, it is critical to understand how the pathways are wired as an integrated network, in both normal and tumor cells. We have developed a systems biology approach that integrates measurements of protein activation under diverse experimental conditions, including inhibition of MEK and PI3K, with a novel network inference computational model that predicts the causal connectivity of a network from experimental data. The network inference model utilizes a genetic algorithm to search for the “optimal” network configuration that most closely matches the experimental data used as input. We validated the approach using in silico data generated from a set of randomized test networks. Next, we applied this approach to breast epithelial cell lines (MFC10A, MCF7, MDA-MB-231, and SUM149) by performing a set of experiments using a series of pathway specific inhibitors with and without growth factor stimulation. From phospho western blot readouts of several proteins in the PI3K and MAPK pathways, we have predicted network configurations most likely responsible for the distinct experimental output of each of the four cell lines. In some cases, our network inference model predicted multiple optimal networks for a given cell line. Our method predicted the next experiments needed to optimally distinguish between the set of possible candidate networks. Our results suggest that some proposed interactions and feedback mechanisms attributed to MAPK and PI3K cross-talk in the literature may not be valid. This approach has important implications for the prospect of effective personalized cancer treatments and targeted molecular inhibition. Elucidating the mechanisms of cross-talk between the MAPK and PI3K pathways in cells collected from patient tumors will permit rational discovery of the optimal combination of targeted therapies needed to treat a specific cancer, based on the actual network that is operational in the tumor. Moreover, our methodology and predictive network model can be applied to any set of signaling pathways. Citation Format: Megan E. Egbert, Michelle L. Wynn, Zhi Fen Wu, Rabia A. Gilani, Santiago Schnell, Sofia D. Merajver. A systems biology approach for rational molecular network inference. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 357. doi:10.1158/1538-7445.AM2014-357

Lattice gas automata simulations of diffusion-limited reactions in heterogeneous media exhibit fractal-like kinetics, which is a generalised mass action kinetics with time-dependent rate constants. We develop a two dimensional lattice gas automata simulation of the Michaelis–Menten mechanism in diffusion-limited conditions to investigate the effect of density and size of obstacles on reactant diffusion and rate coefficients. In order to simulate more physicochemical realistic conditions, reactants rotate and interact according to their specific orientation. We also model weak interaction forces between reactants and obstacles. Our results show that obstacle density and size affect diffusion, first- and second-order rates. We also find that particle rotations and weak force interactions among particles lead to a significant decay in the fractal-like kinetic exponent h. These results suggest that the effects of fractal-like kinetics disappear under less restricted conditions than previously believed in lattice based simulations.

After many years of research, the mechanisms that generate a periodic pattern of repeated elements (somites) along the length of the embryonic body axis is still one of the major unresolved problems in developmental biology. Here we present a mathematical formulation of the cell cycle model for somitogenesis proposed in Development105 (1989), 119-130. Somite precursor cells in the node are asynchronous, and therefore, as a population, generate continuously pre-somite cells which enter the segmental plate. The model makes the hypothesis that there exists a time window within the cell cycle, making up one-seventh of the cycle, which gates the pre-somite cells so that they make somites discretely, seven per cycle. We show that the model can indeed account for the spatiotemporal patterning of somite formation during normal development as well as the periodic abnormalities produced by heat shock treatment. We also relate the model to recent molecular data on the process of somite formation.

After many years of research, somitogenesis is still one of the major unresolved problems in developmental biology. Recent ex- perimental findings show a novel type of pattern formation in which a signal sweeps along the presomitic mesoderm and narrows simulta- neously as a new somite is formed. The signal then residues in the posterior half of the new somite, and another wave begins to sweep up from the caudal end. This behaviour is not easily explained by the existing theoretical models. We present a new model for somitogenesis that can account for this behaviour and is consistent with previous experimental observations. Dev Dyn 2000;217:415- 420. © 2000 Wiley-Liss, Inc.

Targeted molecular inhibitors have emerged as a leading anti-cancer strategy; however, despite promising pre-clinical data, many targeted inhibitors induce undesirable off-target effects in the clinic. The large number of off-target effects associated with molecular inhibitors was recently termed the ‘‘whack a mole problem’’ because inhibiting one molecular target often unintentionally activates another molecule. It is increasingly clear that the high incidence of off-target effects associated with targeted inhibitors is related to the complex interactions and emergent behaviors inherent to the highly complex and dysregulated intracellular networks of cancer. Both the mitogen activated protein kinase (MAPK) and phosphatidylinositol-3 kinase (PI3K) pathways are known to be dysregulated in cancer. In previous work, we and others have demonstrated that the MAPK pathway promotes motility, invasion, and angiogenic factors while the PI3K pathway plays an important role in controlling anchorage independent growth. In addition, the PI3K pathway plays an essential role in stimulating glucose metabolism and the Warburg effect. We hypothesize that robust interactions exist between these two pathways that influence efficacy and potentially also acquired resistance to targeted therapies. Using a combination of experimental and theoretical techniques, we developed a predictive network model linking growth factor signaling to the MAPK and PI3K pathways as well as to glucose metabolism. Specifically, we constructed a logic-based network of the cross-talk between MAPK and PI3K signaling that relied on a detailed literature survey to identify known molecular interactions as well as proposed interactions and regulatory feedback connections in the literature. We next performed a set of experiments using a normal-like breast epithelial cell line and a series of pathway specific inhibitors with and without growth factor stimulation to validate our model. Finally, we repeated these experiments using a diverse set of breast cancer cell lines and integrated this data to produce a series of cancer networks representative of different stages of breast cancer progression. Our model was able to recapitulate both our own experimental data and published data in the literature using a smaller subset of regulatory feedback mechanisms than we started with. Together, our results suggest that some proposed interactions and feedback mechanisms attributed to MAPK and PI3K cross-talk in the literature may not be valid. Citation Format: Megan E. Egbert, Michelle L. Wynn, Zhi Fen Wu, Santiago Schnell, Sofia D. Merajver. Elucidating the complex cross-talk between the MAPK and PIK3 pathways. [abstract]. In: Proceedings of the 104th Annual Meeting of the American Association for Cancer Research; 2013 Apr 6-10; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2013;73(8 Suppl):Abstract nr 5223. doi:10.1158/1538-7445.AM2013-5223

Cancer cells exhibit a metabolic phenotype characterized by high rates of glucose uptake and lactate production, known as the Warburg effect. While the Warburg effect and normal proliferative metabolism appear similar, important molecular differences exist. We hypothesize that molecular and metabolic drivers of the Warburg effect can be modulated to impede cancer proliferation without substantial effects on normal tissue growth. Intracellular networks exhibit a variety of emergent non-linear behaviors and, as a result, the use of experimental intuition alone will not be enough to identify these drivers. Using a combination of experimental and theoretical methods, we developed a model of breast cancer progression that includes metabolism and the phosphatidylinositol-3 kinase (PI3K) signaling pathway, an important regulator of carbon metabolism. A key component of our model is a detailed logic network of molecular interactions associated with PI3K signaling as well as regulatory connections to central carbon metabolism, including the ATP/AMP ratio, GLUT receptor activation, hexokinase activation, and changes in the catalytic activity of pyruvate kinase. To validate our model, a series of phospho Western blot analyses were performed using a normal-like breast cell line and a diverse set of breast cancer cell lines exposed to PI3K pathway inhibitors. From these data, a series of predictive network models were constructed representing distinct stages of breast cancer progression. We also generated detailed metabolic flux maps for each cell line using metabolic flux analysis (MFA), a method that relies on carbon-13 tracers, mass-spectrometry, and measurements of extracellular flux to infer intracellular flux. In agreement with recent studies, we found an increase in the reductive carboxylation of glutamine derived alpha-ketoglutarate in cells constitutively adapted to hypoxia. We also identified a potentially important metabolic vulnerability in aggressive breast cancers. Moreover, we found important PI3K network differences at the RNA and protein levels, some of which were isoform specific. Together our data indicate that very different system-level properties are associated with distinct stages of breast cancer progression and metabolic transformation. Our model is suitable for performing in silico molecular perturbations to predict a normal as well as tumor level response to a targeted therapy or combination of therapies. Our approach also serves as a prototype for the use of systems biology methods in personalized medicine where molecular and metabolic data collected from a patient's biopsied tumor is input into a predictive model designed to develop a strategic treatment plan for the patient. The use of predictive models to integrate data from an individual patient will have a profound impact on cancer care decisions and patient outcomes in the future. Citation Format: Michelle L. Wynn, Megan Egbert, Lauren D. Van Wassenhove, Zhi Fen Wu, Firas Midani, Charles Evans, Charles F. Burant, Santiago Schnell, Sofia D. Merajver. Unraveling the complex regulatory relationship between PI3K signaling and metabolic transformation in breast cancer. [abstract]. In: Proceedings of the 104th Annual Meeting of the American Association for Cancer Research; 2013 Apr 6-10; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2013;73(8 Suppl):Abstract nr 5239. doi:10.1158/1538-7445.AM2013-5239

Motifs are repeating patterns that determine the local properties of networks. In this work, we characterized all 3-node motifs using enzyme commission numbers of the International Union of Biochemistry and Molecular Biology to show that motif abundance is related to biochemical function. Further, we present a comparative analysis of motif distributions in the metabolic networks of 21 species across six kingdoms of life. We found the distribution of motif abundances to be similar between species, but unique across cellular organelles. Finally, we show that motifs are able to capture inter-species differences in metabolic networks and that molecular differences between some biological species are reflected by the distribution of motif abundances in metabolic networks.

Highly complex molecular networks, which play fundamental roles in almost all cellular processes, are known to be dysregulated in a number of diseases, most notably in cancer. As a consequence, there is a critical need to develop practical methodologies for constructing and analysing molecular networks at a systems level. Mathematical models built with continuous differential equations are an ideal methodology because they can provide a detailed picture of a network’s dynamics. To be predictive, however, differential equation models require that numerous parameters be known a priori and this information is almost never available. An alternative dynamical approach is the use of discrete logic-based models that can provide a good approximation of the qualitative behaviour of a biochemical system without the burden of a large parameter space. Despite their advantages, there remains significant resistance to the use of logic-based models in biology. Here, we address some common concerns and provide a brief tutorial on the use of logic-based models, which we motivate with biological examples.

In this work we present an analysis, supported by numerical simulations, of the formulation of the cell-cycle model for somitogenesis proposed in Collier et al.(J. Theor. Biol. 207 (2000), 305–316). The analysis indicates that by introducing appropriate parameter constraints on model parameters the cell-cycle mechanism can indeed give rise to the periodic pattern of somites observed in normal embryos. The analysis also provides a greater understanding of the signalling process controlling somite formation and allows us to understand which parameters influence somite length.

The formation of spatial patterns is intrinsic to developmental biology. From an initial growing mass of cells, a robust, yet intricate, spatial design emerges. These patterns pre-figure the form of the adult organism, but despite their importance, we have very few answers to the most fundamental questions about how patterning arises. In this issue of Cell Systems, Cotterell et al. (2015)xCotterell, J., Robert-Moreno, A., and Sharpe, J. Cell Syst. 2015; 1: 257–269Abstract | Full Text | Full Text PDF | Scopus (1)See all ReferencesCotterell et al. (2015) call a 40-year-old model of patterning into question.The earliest mathematical model for self-organizing patterns was proposed in 1952 by Turing, who suggested that “a system of chemical substances, called morphogens, reacting together and diffusing through a tissue is adequate to account for the main phenomena of morphogenesis” (Turing, 1952xTuring, A. Philos. Trans. R. Soc. Lond., B. 1952; 237: 37–72CrossrefSee all ReferencesTuring, 1952). This model showed the highly non-intuitive result that diffusion—a phenomenon previously thought to homogenize any spatial heterogeneity—could drive the emergence of patterns from a spatially uniform state. Extensive mathematical and computational investigation of this mechanism has revealed its ability to generate a bewildering diversity of patterns (Murray, 2003xMathematical biology II: Spatial models and biomedical applications. Murray, J.D. See all References, Meinhardt, 2009xThe algorithmic beauty of seashells. Meinhardt, H. CrossrefSee all References). However, despite years of investigation, the identification of morphogen systems that behave precisely as Turing suggested remains beyond us.An alternative to this type of emergent patterning is the “positional information” model of Wolpert proposed in 1969. This class of model proposes that a specialized region of tissue (for example, the zone of polarizing activity in the chick limb bud) acts as a morphogen source, while the rest of the tissue acts as a morphogen sink (Wolpert, 1969xWolpert, L. J. Theor. Biol. 1969; 25: 1–47Crossref | PubMedSee all ReferencesWolpert, 1969). Together, they set up a gradient in morphogen concentration across the embryo that ensures that cells differentiate appropriately. Like Turing models, experimental validation of this model has proven problematic. In particular, we have failed to identify signals which are robust to noise in parameter values and extend long distances.Nevertheless, positional information is a central concept in the prevailing model used to explain a classic phenomenon in development, the formation of somites, which are aggregates of cells established within the presomitic mesoderm, in an anterior-to-posterior sequence. Somitogenesis has attracted huge interest because it is crucial to setting up the primary body axis.Somitogenesis also exemplifies the power of interdisciplinary research. Modeling and experiment have been closely integrated for 40 years, each informing the other in an iterative predict-test-refine-predict cycle. From a modeling point of view, somitogenesis can be very reasonably abstracted to the problem of understanding the sequential formation of periodic structures along a one-dimensional domain (Figure 1Figure 1A). From an experimental point of view, the system is amenable to manipulations that rule out whole classes of underlying mechanisms. These observations can be easily translated into corresponding model perturbations, allowing modelers and experimentalists to iterate the predict-test-refine-predict cycle further. In 1976, Cooke and Zeeman proposed that somite patterning could arise from the coupling of a propagating wave (an “arrest front” providing positional information) with an intrinsic oscillator (clock) within cells (Cooke and Zeeman, 1976xCooke, J. and Zeeman, E.C. J. Theor. Biol. 1976; 58: 455–476Crossref | PubMed | Scopus (363)See all ReferencesCooke and Zeeman, 1976). This is referred to as the clock and wavefront model and is the commonly accepted model explaining somite formation.Figure 1Unbiased Exploration of Three-Node Networks Reveals Motifs that Potentially Underlie a New Model for Progressive Somite Patterning(A) Cotterell et al. implemented an approach to explore the minimal network motifs that can reproduce the somite patterning of at least two stripes of gene expression.(B) They discovered four minimal motifs in their Network Design Space. Two of the network motifs are versions of the clock and wavefront model, but these networks are not robust to extrinsic noise. The minimal and more robust somite-patterning network is composed of two nodes: activator molecule (green) and diffusible repressor (red).View Large Image | Download PowerPoint SlideIn the present issue, Cotterell et al. propose a new model for somite formation, focusing only on the arrest front. They challenge the prevailing clock and wavefront model. In the clock and wavefront model, it is proposed that long-range morphogen gradients of FGF and WNT signaling drive a traveling wave of differentiation that arrests the clock, which happen to be synchronized in neighboring cells. In contrast to this, Cotterell et al. propose that the traveling arrest front arises as an emergent phenomenon from the local interactions of a reaction-diffusion mechanism, which they term a progressive oscillatory reaction-diffusion (PORD) model.To build the PORD model, Cotterell et al. use a computational framework to explore the different ways in which gene regulatory networks can be wired to produce the (striped) segmentation patterns observed during somitogenesis (Figure 1Figure 1B). They analyze the entire space of all possible networks with three nodes (interacting components) because networks with three nodes have the potential for generating a wide range of behaviors (Tyson et al., 2003xTyson, J.J., Chen, K.C., and Novak, B. Curr. Opin. Cell Biol. 2003; 15: 221–231Crossref | PubMed | Scopus (779)See all ReferencesTyson et al., 2003).Their search of possible network topologies identified 210 topologies with four motifs capable of producing striped patterns. Two of the motifs that produced stripes are essentially the clock and wavefront model. However, those motifs are not robust to extrinsic noise and represent only 14% of the possible segmentation-producing gene regulatory networks. The majority of the other topologies can be reduced to a two-node network motif comprising an activator molecule and a diffusible repressor. This simple reaction-diffusion motif is robust to extrinsic noise and produces the stable periodic patterns of gene expression observed during somite formation. The authors then carried out a series of experiments in chick embryos designed to distinguish between the clock and wavefront and PORD models and demonstrated that their PORD model is consistent with the resultant observations.The PORD model also provides a new explanation for another striking feature of somitogenesis: the regulation of somite size. The size of somites is tightly associated with body size in vertebrates; this observation is consistent with a global positional information model which scales with body length. However, Cotterell et al. show that this observation is also consistent with their new model. Although the basic PORD model does not require long-range gradients to make regular somites, in principle the traveling wave of FGF signal can couple growth to the dynamics of the reaction-diffusion model, allowing feedback between body size and pattern wavelength, which allows somite size to scale with body size.The work by Cotterell et al. is a truly intra- and inter-disciplinary study that throws open to debate the long-held clock and wavefront model paradigm. The latter model has been hugely influential in developmental biology, guiding decades of experimentalists to look for—and find—molecules that participate in the oscillations and propagating signals on which the model is predicated. The PORD model casts these molecules in a new light. It proposes that the somite pattern observed at the arrest front emerges from a local reaction-diffusion system rather than a dynamic global positional gradient.There are, of course, limits to the theoretical work by Cotterell et al. For example, the computational search across the swathes of potential gene regulatory networks is still restricted to a limited region of parameter space. Nevertheless, the approach pursued by Cotterell et al. is highly innovative and provides the foundation for future investigations into pattern formation in developmental biology. Just as the clock and wavefront model has profited from many years of experimental work, so too may the PORD model, as researchers look for its activator molecule and diffusible repressor. In addition, it is plausible that reaction-diffusion and positional information mechanisms could work together to robustly pattern the embryo, as shown for limb digit patterning (Maini et al., 1992xMaini, P.K., Benson, D.L., and Sherratt, J.A. IMA J. Math. Appl. Med. 1992; 9: 197–213Crossref | Scopus (44)See all ReferencesMaini et al., 1992). The proposal of a new model should generate much excitement, discussion, and, hopefully, a healthy controversy among both theoreticians and experimentalists that will enhance, even further, our understanding of one of the most important patterning events in embryology.