Department of Mathematics

Plenary Speakers:

Stochasticity is a hallmark of cellular processes, and different classes of genes show large differences in their cell-to-cell variability (noise). To decipher the sources and consequences of this noise, we systematically measured pairwise correlations between large numbers of genes, including those with high variability. We find that there is substantial pathway variability shared across similarly regulated genes. This induces quantitative correlations in the expression of functionally related genes such as those involved in the Msn2/4 stress response pathway, amino acid biosynthesis, and mitochondrial maintenance. Our results argue that a limited number of well-delineated ''noise regulons'' operate across a yeast cell and that such coordinated fluctuations enable a stochastic but coherent induction of functionally related genes. We show that pathway noise is a quantitative tool for exploring pathway features and regulatory relationships in un-stimulated systems. These fluctuation-based investigations reveal many system level properties, which we explore mechanistically for PKA signaling.

Bio: Hana El-Samad is a faculty member in the department of Biochemistry and Biophysics at the University of California, San Francisco and the California Institute for Quantitative Biosciences (QB3), where she holds the Grace Boyer Junior Endowed Chair in Biophysics and is the deputy director of the UCSF Systems and Synthetic Biology Center. She is a 2009 Packard Fellow, a recipient of the 2011 Donald P. Eckman Award and the 2012 CSB2 prize in Systems Biology. Dr. El-Samad joined UCSF after obtaining a doctorate degree in Mechanical Engineering from the University of California, Santa Barbara, preceded by a Masters Degree in Electrical Engineering from the Iowa State University. Dr. El-Samad's research group emphasizes the role of control theory and dynamical systems in the study of biological networks. Her research interests include the investigation of stress responses and biological stochastic phenomena, in addition to the establishment of computational and technological infrastructures that allow for their quantitative probing in single cells.

There is often a striking variability in behavior as cells respond to stimuli. This is seen in the response of bacteria to antibiotics, the response of cancer cells to therapies, or even in the response of precursor cells to differentiation-inducing stimuli. For example, when clonal human cancer cells are treated with death-inducing ligands cells that die do so at different times and some cells survive. For clonal cells in particular, genetic variations can be ruled out as the cause of the cell-to-cell variability and instead we have found that heterogeneity in the expression level of key regulatory proteins could explain much of the variability in response. Using ligands of the Tumor necrosis factor (TNF) superfamily as a model system, we combine single-cell measurements and analysis of computational models of the relevant signal transduction networks to uncover how cells integrate their response. In this talk I will discuss some of our recent results as well as the computational approaches that we use to understand cell-to-cell variability.

Bio: Suzanne Gaudet obtained her Ph.D. in Biochemistry from Harvard University. She then trained as a postdoctoral associate in Peter Sorger's laboratory at MIT participating in the launch of a project to study how TNF, EGF and insulin affect the cell death decision, using a systems approach. From 2003 to 2008, she worked as a research scientist and scientific coordinator at the Cell Decision Processes Center at MIT and Harvard. She is now assistant professor at the Department Genetics at Harvard Medical School and the Department of Cancer Biology at the Dana-Farber Cancer Insitute. Her research focuses on the quantitative understanding of how ligands, including TNF-superfamily ligands, control cell fate.

Gaining an improved understanding of the molecular mechanisms involved in the acute phase response (APR) in the liver upon trauma or injury can lead to improved treatment of complications arising from inflammatory disorders. The dynamics of expression and interaction of the IL-6 signaling pathway molecules is a key factor of the phenotypical characteristic of the APR, as IL-6 has been identified as one of the systemic inflammatory mediators involved in the regulation of the hepatic APR.

This work develops and analyzes a comprehensive mathematic model for signal transduction through the JAK/STAT and the MAPK signaling pathways in hepatocytes stimulated by IL-6. Interactions among the two signaling pathways are systematically investigated using sensitivity analysis in order to ultimately derive and validate an improved model. An important aspect of this work is the novel use of sensitivity analysis for determining which parts of the model may benefit from further model refinement, whereas traditionally sensitivity analysis has been applied to determine the contribution of parameters of an existing model to the dynamic behavior, i.e., such that the important parameters should be estimated from experimental data. While the exact nature of the additional mechanisms to include depends upon biological insight into the model, sensitivity analysis indicates which parameters may be masking more detailed mechanisms of importance to the model's predictions.

In this work, results from the sensitivity analysis are used to determine a location for including a (previously) hidden feedback loop between twice phosphorylated ERK and SOS as parameters contributing to reactions affecting these proteins were computed to be important. Additionally, experiments with GFP reporter cells were carried out where the amount of observed fluorescence is quantified to determine a profile for the concentration of GFPs. An inverse problem is formulated and solved that determines the transcription factor concentration from the measured fluorescence intensity profiles. These experimental results are compared to simulation data with the original and the newly developed model and were found to be in excellent agreement with the model derived in this work.

Bio: Juergen Hahn was born in Grevenbroich, Germany, in 1971. He received his diploma degree in engineering from RWTH Aachen, Germany, in 1997, and his MS and Ph.D. degrees in chemical engineering from the University of Texas, Austin, in 1998 and 2002, respectively. He was a post-doctoral researcher at the chair for process systems engineering at RWTH Aachen, Germany, before joining the department of chemical engineering at Texas A&M University, College Station, in 2003. He joined the Rensselaer Polytechnic Institute as a professor in 2012 and currently holds appointments in the department of biomedical engineering and the department of chemical & biological engineering. His research interests include systems biology and process modeling and analysis with over 60 articles and book chapters in print. Dr. Hahn is a recipient of a Fulbright scholarship (1995/96), received the Best Referee Award for 2004 from the Journal of Process Control, the CPC 7 Outstanding Contributed Paper Award in 2006, and was named the 2010 CAST Outstanding Young Researcher. He is currently serving as an associate editor for the journals Automatica, Control Engineering Practice, and the Journal of Process Control.

Synthetic biology can be broadly parsed into the ``top-down'' synthesis of genomes and the ``bottom-up'' engineering of relatively small genetic circuits. In the genetic circuits arena, toggle switches and oscillators have progressed into triggers, counters and synchronized clocks. Sensors have arisen as a major focus in the context of biotechnology, while oscillators have provided insights into the basic-science functionality of cyclic regulatory processes. A common theme is the concurrent development of mathematical modeling that can be used for experimental design and characterization, as in physics and the engineering dis- ciplines. In this talk, I will describe the development of genetic oscillators over increasingly longer length scales. I will first describe an engineered intracellular oscillator that is fast, robust, and persistent, with tunable oscillatory periods as fast as 13 minutes. Experiments show remarkable robustness and persistence of oscillations in the designed circuit; almost every cell exhibits large-amplitude fluorescence oscillations throughout each experiment. Computational modeling reveals that the key design principle for constructing a robust oscillator is a small time delay in the negative feedback loop, which can mechanistically arise from the cascade of cellular processes involved in forming a functional transcription factor. I will then describe an engineered network with intercellular coupling that is capable of generating synchronized oscillations in a growing population of cells. Microfluidic devices tailored for cellular populations at differing length scales are used to demonstrate collective synchronization properties along with spatiotemporal waves occurring on millimeter scales. While quorum sensing proves to be a promising design strategy for reducing variability through coordi- nation across a cellular population, the length scales are limited by the diffusion time of the small molecule governing the intercellular communication. I will con- clude with our recent progress in engineering the synchronization of thousands of oscillating colony ``biopixels'' over centimeter length scales through the use of redox signaling that is mediated by hydrogen peroxide vapor. We have used the redox communication to construct a frequency modulated biosensor by coupling the synchronized oscillators to the output of an arsenic sensitive promoter that modulates the frequency of colony-level oscillations due to quorum sensing.

Bio: Jeff Hasty received his Ph.D. in physics from the Georgia Institute of Technology in 1997, where he worked with Kurt Wiesenfeld. He was a postdoc with Jorge Vinals at the Supercomputing Research Institute ('97-'98), and a postdoctoral fellow with Jim Collins in the Applied BioDynamics Lab at Boston University ('98-'01). Somewhere during his postdoctoral stay at Boston University, he mutated into a hybrid computational/molecular biologist. He is currently at the University of California, San Diego, where he is a Professor in the Departments of Molecular Biology and Bioengineering, and the Director of the BioCircuits Institute. His main interest is the design and construction of synthetic gene-regulatory and signaling networks.

Chemotaxis, the directed motion of cells in response to chemical gradients, requires the coordinated action of three different and separable processes: motility, gradient sensing and polarization. Much effort has been expended understanding each of these processes, and numerous mathematical models have been proposed that explain each one. In this talk I will present a comprehensive model that explains all three aspects of chemotaxis. The central element is the presence of a of a biased excitable system. This model takes into account reports that the actin cytoskeleton and other signaling elements in motile cells have many of the hallmarks of an excitable medium, including the presence of propagating waves. This excitable behavior can account for the spontaneous migration of cells. We suggest that the chemoattractant-mediated signaling can bias excitability, thus providing a means by which cell motility can be directed. We also provide a mechanism for cell polarity that can be incorporated into the existing framework.

Bio: Pablo A. Iglesias was born in Caracas, Venezuela. He received the B.A.Sc. degree in Engineering Science from the University of Toronto in 1987, and the Ph.D. Degree in Control Engineering from Cambridge University in 1991. Since then he has been on the faculty the Johns Hopkins University, where he is currently the Edward J. Schaefer Professor of Electrical Engineering. He also holds appointments in the Departments of Biomedical Engineering, and Applied Mathematics & Statistics. He has had visiting appointments at Lund University, The Weizmann Institute of Science, the California Institute of Technology and the Max Planck Institute for the Physics of Complex Systems. His current research interests focus on the use of control and dynamical system theory to study biological signal transduction pathways, particularly those involved in regulating directed cell motion and cell division.

Mathematical modeling and computer simulation provide robust and powerful tools for the study of biological complexity. Through constructing mechanistic models of current knowledge and using appropriate mathematical and computational methods, new insights into the workings of biomedically important systems can emerge, as well as novel design approaches. This talk will discuss examples from synthetic and systems biology. Current molecular therapeutics generally act as inhibitors, antagonists or, less frequently, agonists that exert control over the biology with which they interact. Results probing how and where to intervene in cellular systems through an understanding of network properties will be discussed. Looking forward, therapeutics able to sense and respond appropriately to their setting may be useful to distinguish diseased from normal tissue and to generate a type-appropriate response. In this talk we explore the potential of such therapeutics using biochemical pathways of signaling processes together with appropriate optimization techniques. The results show great potential for effecting programmed responses that operate robustly across wide ranges of conditions.

Bio: Bruce Tidor received his Ph.D. (1990) in Biophysics from Harvard University under the supervision of Professor Martin Karplus, where he studied protein folding and binding with free energy simulations and normal mode calculations. In 1990 he moved to the Whitehead Institute for Biomedical Research, where he started his independent research as a Whitehead Fellow. In 1994 he was appointed to the faculty at MIT. He is currently Professor of Biological Engineering and Computer Science. His research focuses on the analysis of complex biological systems at the molecular and cellular level. Using molecular modeling, theory, and computation, he explores the structure, function, and interactions of proteins and the roles played by specific chemical groups in defining the stability and specificity of molecular interactions. Using cell-level models his group is exploring the relationship between network structure and biological function. He is actively applying knowledge from modeling studies to rational design.

Contributed Speakers:

Gene expression plays a central role in the orchestration of cellular processes. The use of inducible promoters to change the expression level of a gene from its physiological level has significantly contributed to the understanding of the functioning of regulatory networks. However, from a quantitative point of view, their use is limited to short-term, population-scale studies to average out cell-to-cell variability and gene expression noise and limit the nonpredictable effects of internal feedback loops that may antagonize the inducer action. Here, we show that, by implementing an external feedback loop, one can tightly control the expression of a gene over many cell generations with quantitative accuracy. To reach this goal, we developed a platform for real-time, closed-loop control of gene expression in yeast that integrates microscopy for monitoring gene expression at the cell level, microfluidics to manipulate the cell's environment, and original software for automated imaging, quantification, and model predictive control. By using an endogenous osmostress responsive promoter and playing with the osmolarity of the cells environment, we show that long-term control can, indeed, be achieved for both time-constant and time-varying target profiles at the population and even the single-cell levels. Importantly, we provide evidence that real-time control can dynamically limit the effects of gene expression stochasticity. We anticipate that our method will be useful to quantitatively probe the dynamic properties of cellular processes and drive complex, synthetically engineered networks.

Mitochondria can take up enormous amounts of Ca2+ in an energy dependent manner while they maintain their matrix Ca2+ concentration in the low micromolar range. As a result, when Ca2+ efflux is initiated by the addition of Na+, the Ca2+ concentrations in the extra-mitochondrial and matrix compartments display asymmetrical dynamics. To elucidate this phenomenon, we developed and characterized a mechanism of the mitochondrial Ca2+ sequestration system from an experimental data set obtained using isolated guinea pig cardiac mitochondria. The sequestration model is integrated into a previously corroborated model of oxidative phosphorylation including the Na+/Ca2+ cycle. This integrated model reproduces the Ca2+ dynamics observed in both compartments of isolated mitochondria respiring on pyruvate after a bolus of CaCl2 followed by ruthenium red and a bolus of NaCl. The computational results support the conclusion that the Ca2+ sequestration system is composed of at least two classes of Ca2+ buffers. The first class represents prototypical buffering, and the second class encompasses the complex binding events associated with the formation of amorphous calcium phosphate. Moreover, model analysis rules out simple calcium phosphate precipitation as the primary mechanism of Ca2+ sequestration. The integrated model was corroborated by simulating the set-point phenomenon. With the Ca2+ sequestration system in mitochondria more precisely defined, computer simulations can aid in the development of innovative therapeutics aimed at addressing the myriad of complications that arise due to mitochondrial Ca2+ overload.

Bio: Jason Bazil received his Ph.D. in biomedical engineering from Purdue University in 2010. He worked under Ann Rundell and Greg Buzzard to develop a quantitative description of the mitochondrial permeability transition phenomenon and advance an algorithm capable of generating an optimal sequence of experiments to inform an experimentalist using the model-based design of experiments approach. He is currently a post-doctoral fellow in the Biotechnology Center for Computational Medicine at the Medical College of Wisconsin. He works with Ranjan Dash investigating the relationship between mitochondrial bioenergetics and oxidative stress by merging experiment and theory in a consistent modeling framework. He also works with Dan Beard to develop an algorithm designed to reverse engineer gene regulatory networks from gene expression data and to elucidate the gene program for cardiogenesis in order to predict the effect of gene dosing in congenital heart disease patients.

Reconstructing cellular networks at the genome scale provides a mechanistic understanding of the genotype-phenotype relationship and enables accurate prediction of cellular responses to perturbations. A forefront challenge in this process is the integration of the gene regulatory network with the corresponding metabolic network. I will discuss approaches that enable the automated and quantitative integration of these networks at the genome scale. The integrated models allow us to simultaneously interrogate the transcriptional and metabolic states of the organism, thereby capturing the complex, multifaceted interactions between these two systems. Prediction of such metabolic changes has wide-ranging applications in biotechnology, drug discovery and diagnostics.

Bio: Sriram Chandrasekaran is a PhD candidate in Biophysics and Computational Biology at the University of Illinois at Urbana-Champaign. Working with Dr. Nathan Price at the Institute for Systems Biology, he is developing new systems approaches for analyzing gene regulatory and metabolic networks. He has applied these methods to study gene expression changes in the brain, and for identifying drug targets for microbial infections like Tuberculosis. Sriram earned a Bachelor of Technology degree in Biotechnology from Anna University (First Class with Distinction) in 2008. He is also a recipient of the 2011 Howard Hughes Medical Institute (HHMI) Predoctoral Fellowship and a finalist for the 2012 Lemelson-MIT Illinois Student Award for innovation.

Mathematical models of the electrical system of cells are based on experimental data collected from different cells under different experimental conditions and sometimes even from different species. The parameters that go into the models are then necessarily average values. What we cannot learn from these experiments is how different currents co-vary in individual cells. We recently developed the ``Onion Peeling'' method that allows recording multiple (7 so far) currents from a single cell. We find surprisingly wide cell-to-cell variability in the magnitudes of the currents yet the action potential shape remains virtually identical. This immediately begs the question of how the different currents conspire to maintain a fixed action potential. The answer, still elusive, might be found in coordinated changes of multiple kinetic parameters in different currents. The challenge then is to identify and quantify the relevant parameter set encompassing multiple channels and transporters. The parametric signature of a single cell afforded by the Onion Peeling data will enable models to better predict the electrical responses to drugs and hormones.

In developing tissues, when cells expressing two different factors are juxtaposed under specific conditions, they give rise to a boundary of specialized cells. This phenomenon has been identified at the dorsoventral cell boundary during the development of drosophila wing disc, at the Apical Ectodermal Ridge (AER) of the vertebrate limb, and at the boundary between neural and non-neural ectoderm in neural crest formation. Specialized cells form at the boundary of Fringe expressing and Fringe non-expressing cells by a specific type of Serrate -> Notch / Delta -> Notch interaction, called inductive signaling. The presence of Fringe is said to inhibit the binding ability of Serrate ligand to Notch and enhance that of Delta to Notch. Although several of the signaling elements have been identified experimentally, it remains unclear how the inter-cellular interactions can give rise to such a boundary of specialized cells. Here we present a simple ordinary differential equation (ODE) model involving Delta -> Notch and Serrate -> Notch interactions between juxtaposed Fringe expressing and Fringe non-expressing cells. When this ODE model is incorporated into a 2D spatial arrangement of cells using cell-based modeling environment - Compucell3D and SBML based ODE solver called Bionet solver, it shows that a boundary of specialized cells forms which expresses a higher level of Notch than the others. We analyze this model both analytically and numerically showing the conditions under which such a boundary is formed as observed in living systems. In addition, we also incorporate this model into a 3D cellular arrangement and show the formation of Apical Ectodermal Ridge in vertebrate limb.

Circadian rhythms are responsible for coordinating 24 hour oscillations in physiology, and are coordinated in many species through genetic regulatory networks. In mammals, transcription factors Clock and Bmal1 activate the expression of Cryptochrome (Cry1 and Cry2) and Period (Per) genes. PER and CRY protein products subsequently enter the nucleus and repress Clock-Bmal1 mediated activation, resulting in rhythmic gene expression. KL001, a recently discovered small molecular stabilizer of CRY, was found to lengthen circadian period. Since existing experimental results have linked perturbations to CRY to both period lengthening and shortening, a mathematical model was employed to predict the mechanism by which KL001 affected circadian rhythms. These predictions were then confirmed experimentally, suggesting a mathematical approach to designing pharmacological treatments to common circadian disorders may be feasible.

Bio: Peter St. John received his bachelor's degree in chemical and biological engineering from Tufts University in Medford, Massachusetts in 2010. He is currently a PhD candidate in chemical engineering at the University of California, Santa Barbara. His research interests lie in dynamical systems and computational biology, with applications to the mammalian circadian clock.

Blebbing occurs when the cytoskeleton detaches from the cell membrane, resulting in the pressure-driven flow of cytosol towards the area of detachment and the local expansion of the cell membrane. Recent experiments involving blebbing cells have led to conflicting hypotheses regarding the timescale of intracellular pressure propagation. The interpretation of one set of experiments supports a poroelastic cytoplasmic model which leads to slow pressure equilibration when compared to the timescale of bleb expansion. A different study concludes that pressure equilibrates faster than the timescale of bleb expansion. To address this, a dynamic computational model of the cell was developed that includes mechanics of and the interactions between the intracellular fluid, the actin cortex, the cell membrane, and the cytoskeleton. Results show the relative importance of cytoskeletal elasticity and drag in bleb expansion dynamics. Our results also show that a poroelastic model of the cytoplasm can explain experimental data and support the hypothesis that pressure equilibrates relatively fast.

Bio: I received my doctorate in Mathematics from the University of North Carolina at Chapel Hill under Timothy Elston (Department of Pharmacology) and David Adalsteinsson (Department of Mathematics). My thesis work involved developing novel algorithms for simulating spatiotemporal biochemical reaction networks in moving cells. I am currently a postdoctoral researcher at the UC Davis. I will be an assistant professor in the Department of Mathematics at Case Western Reserve University beginning August 2013.

The chemokine CXCL12 is known to promote growth and metastasis in breast cancer. Two receptors for CXCL12, CXCR4 and CXCR7, bind &beta-arrestin, causing receptor internalization and signaling. To identify distinct and integrated effects of CXCR4 and CXCR7 on signaling, we developed an ordinary differential equation model to describe the kinetics of &beta-arrestin binding. Model parameters were determined from imaging of &beta-arrestin binding in cells expressing only one receptor type. The model predicted that coexpression of CXCR4 and CXCR7 would decrease CXCL12-regulated &beta-arrestin recruitment to CXCR4 because CXCR7 wins the ``competition'' with CXCR4 for CXCL12 and recruitment of &beta-arrestin. These predictions were experimentally validated, establishing the effectiveness of combining quantitative imaging with mathematical modeling to analyze the CXCL12/CXCR4/CXCR7 pathway. This work will ultimately reveal new targets and help develop new therapies for the treatment of breast cancer.

Bio: Danielle Trakimas is a graduate student in the department of Chemical Engineering at the University of Michigan, under the advisement of Professor Jennifer J. Linderman. Danielle's research currently focuses on modeling of G-protein coupled receptor dynamics and signaling pathways involved in metastatic breast cancer.

Cells are continuously faced with the challenge of sensing signals in their environment and eliciting intracellular programs accordingly. While changes in some environmental cues produce proportional responses, others induce decisive action whereby a cell exhibits a binary (on or off) phenotypic change. In this talk, I will describe a combination of experiments and computational modeling that are used to analyze the mechanism for bimodality in the switch-like activation response of the S. cerevisiae galactose gene-regulatory network. We show that bistability underlies the observed bimodality and that a synergistic collaboration between two unique positive feedback loops established by Gal1p and Gal3p, that both regulate network activity by molecular sequestration of Gal80p, induces this bimodality. Using a simple mathematical model, we demonstrate that the sequestration binding affinity is a critical parameter that can tune the range of conditions for bistability. We next characterize the dynamics of this bistable genetic switch in an environment with conflicting signals from glucose (repression) and galactose (activation). We identify that similar concentrations of these two sugars can trigger a dynamic bimodal transition to full GAL activation involving two subpopulations that exhibit an early and delayed activation response. We demonstrate that the observed delays in GAL gene expression scale linearly with the log of the glucose concentration and with the initial number of cells. Our results indicate that activating the GAL pathway can generate a fitness cost in this mixed sugar environment.

Bio: Ophelia Venturelli is a graduate student in Biochemistry and Molecular Biophysics at the California Institute of Technology in the group of Richard M Murray. She received her B.S. in Biological Sciences from Stanford University. Ophelia is spending the last year of her PhD in the lab of Prof. Hana El-Samad at UCSF. Her current research interests include a combination of experiments and computational modeling of single-cell dynamic metabolic responses and promoter architecture.

Understanding signal transduction in cellular processes is a central goal in systems biology. Identification of key signaling components can provide important insights into the fragility of a cellular process. In this talk, I will present a structural method for predicting the essentiality of signaling components. This method integrates the signs of interactions and the synergy among components into graph-theoretical analysis and simulates both knockout and constitutive activation of components as node disruptions. We introduce the concept of elementary signaling mode (ESM) and rank the importance of signaling components by the effects of their perturbation on the ESMs of the network. The applications of this method to several signaling networks demonstrate its ability to uncover key signaling components mediating cellular phenotypes and candidate therapeutic targets in disease networks.

Noise affects cellular processes at a fundamental level, in some cases causing radical changes of state and function. While the response of a system to noise has been studied extensively, there is only nascent understanding of how to control this response and exploit it to produce a desired outcome. Here we present a quantitative, scalable method based upon large deviation theory to predict and control the response to noise in a biological system. As a set of examples, for genetic regulatory network models of cell differentiation we analyze the robustness to noise of each differentiated state in relation to a set of external control factors. We show how small changes in these factors can cause large changes in the response to noise throughout a system, and hypothesize interventions to induce lineage changes towards a desired cell state. Such analysis offers a new, systems approach to predicting novel reprogramming strategies and to related applications within systems biology.

Bio: Danny Wells is a third year graduate student in applied math at Northwestern University, advised by William Kath and Adilson Motter. His research is focused in computational systems biology, particularly in understanding and controlling the effects of biological noise. He graduated with a degree in math from Carleton College in 2010, and is a National Science Foundation Graduate Research Fellow.

Co-authors: Adilson Motter (Northwestern Physics and Astronomy, Northwestern Institute on Complex Systems); William Kath (Northwestern Engineering Sciences and Applied Mathematics, Northwestern Institute on Complex Systems)

Both engineering and evolution are constrained by tradeoffs between efficiency and robustness, but theory that formalizes this fact is limited. For a simple two-state model of glycolysis, we explicitly derive analytic equations for hard tradeoffs between robustness and efficiency that have oscillations as an inevitable side effect. The model describes how the tradeoffs arise from individual parameters, including the interplay of feedback control with autocatalysis of network products necessary to power and catalyze intermediate reactions. We then use control theory to prove that the essential features of these hard tradeoff ``laws'' are universal and fundamental, in that they depend minimally on the details of this system, and generalize to the robust efficiency of any autocatalytic network. The theory also suggests worst-case conditions which are consistent with the initial single cell experiments we performed in a microfluidic device.

The chemokine CXCL12 is identified as a major chemotactic proponent in breast cancer. CXCL12 gradients within the tumor environment are thought to drive CXCR4+ cancer cells to invade and extravasate. A second receptor to CXCL12, CXCR7, is also present in the tumor environment. Understanding how CXCR7 can shape ligand gradients on a molecular level, and influence cancer cell chemotaxis within the tissue scale, is unclear. In addition, CXCL12 exists as multiple isoforms, yet studies examining its role in cancer have only focused on one isoform. These isoforms have varying affinities for cell surface receptors and glycosaminoglycans, as well as experimental device surfaces, ultimately changing the presentation of the ligand to the receptor.

In order to investigate the molecular mechanisms that regulate gradient formation and cancer cell responses to these gradients, we built a data-driven multi-scale agent-based model that simulates chemotaxis within a microfluidic assay. CXCR4+, CXCR7+, and CXCL12-secreting cells are agents that move and interact on a lattice. Each agent contains a set of ordinary differential equations that describe the internalization, recycling, and degradation of CXCL12 and its receptors. Therefore, the cells update and respond to their environment. We constructed and validated the model based on experimental data. We find that the presence of CXCR7+ cells significantly alters CXCL12 gradients, thus controlling CXCR4+ cell migration. Using sensitivity analysis, we identify key events in the CXCL12/CXCR7 pathway that can be targeted to inhibit CXCR4+ cell migration.