### DIMACS Workshop on Control Theory and Dynamics in Systems Biology

#### May 18 - 20, 2009 DIMACS Center, CoRE Building, Rutgers University

Organizers:
Eduardo Sontag, Rutgers University, sontag at math.rutgers.edu
Patrick De Leenheer, University of Florida, deleenhe at math.ufl.edu
Presented under the auspices of the DIMACS/BioMaPS/MB Center Special Focus on Information Processing in Biology.

This special focus is jointly sponsored by the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), the Biological, Mathematical, and Physical Sciences Interfaces Institute for Quantitative Biology (BioMaPS), and the Rutgers Center for Molecular Biophysics and Biophysical Chemistry (MB Center).

#### Abstracts:

Title: The deficiency zero theorem for stochastically modeled systems

The dynamics of chemical reaction networks can be modeled either deterministically or stochastically. The deficiency zero theorem for deterministically modeled systems gives conditions under which a unique equilibrium value with strictly positive components exists within each stoichiometric compatibility class (invariant manifold). The conditions of the theorem actually imply the stronger result that there exist concentrations for which the network is "complex balanced." That observation in turn implies that the standard stochastic model for the reaction network has a product form stationary distribution.

Murat Arcak, University of California-Berkeley

Title: An Additive D-Stability Condition and Application to Reaction-Diffusion Systems

Matrix A is said to be additively D-stable if A -D remains Hurwitz for all nonnegative diagonal matrices D.  In reaction-diffusion models, additive D-stability of the matrix describing the reaction dynamics guarantees stability of the homogeneous steady-state, thus ruling out the possibility of diffusion-driven instabilities. We present a new criterion for additive D-stability using the concept of compound matrices. We first give conditions under which the second additive compound matrix has nonnegative off-diagonal entries. We then use this Metzler property of the compound matrix to prove additive D-stability with the help of an additional determinant condition. This result is then applied to investigate the stability of several biological circuits in the presence of diffusion.

Murad Banaji, University of Essex and University College London

Title: Predicting injectivity in interaction networks from their structure

A variety of biological systems including gene networks, biochemical reaction networks, population models, etc. can be seen as "interaction" networks, composed of elements which interact according to qualitative rules. These qualitative interaction rules, described by words such as "activation", "inhibition", "competition", "cooperation", etc. can be given mathematically precise definitions, and a variety of predictions using analytical and combinatorial tools can be made based on knowledge of these rules. A rigorous framework for describing such networks, going beyond systems with signed Jacobian, and allowing multiple or complicated interactions between elements, is presented. The main matrix- and graph-theoretic results on injectivity and the possibility of multiple equilibria are discussed. These include matrix-theoretic conditions which are sufficient and, with suitable assumptions, necessary for injectivity, and a very simple sufficient graph-theoretic condition. The proofs are outlined.

This is joint work with Gheorghe Craciun from the University of Wisconsin at Madison.

Title: Structure and qualitative dynamics in an apoptosis network

The dynamical behaviour of a biological network is strongly related to the structure of interconnections among the network's components. For large networks, Boolean models and piecewise linear systems are convenient formalisms to describe the possible qualitative dynamical trajectories. In this context, one would like to identify groups of variables and interactions associated with different dynamics (for instance, multistability or oscillatory behaviour). We will present a method for identification of "operational interactions", based on the asynchronous transition graph of the Boolean network.

For the apoptosis (or programmed cell death) network, two core groups of variables and interactions are identified: they correspond to two different mechanisms responsible for the decision between apoptosis or cell survival.

Title: Bifurcations in Systems with Mass Action Kinetics

Systems of Ordinary Differential Equations with a high number of variables and, mostly unknown, parameters are common in Systems Biology. Moreover, measurement data is almost always restricted to very few components, often qualitative and very noisy. Thus the question, whether or not a given system of ODEs can - for some conceivable parameter vector - exhibit a certain observed qualitative dynamical behaviour (e.g. multistationarity, limit cycles, ...) arises naturally. To answer this question one can, for example, try to identify parameter regions where certain bifurcation phenomena occur (e.g. saddle-node bifurcations if one is interested in multistationarity, ...). Numerical methods, however, are only of limited help due to the high number of parameters.

To overcome this limitation, an analytic approach is proposed: the Jacobian derived from a mass-action system usually has several zero eigenvalues due to the conservation relations. Thus one is interested in 'critical states' and 'critical parameters', where the Jacobian has additional zero eigenvalues. These states and parameters are then candidates for bifurcation points (given such a critical state and parameter vector, one can, for example, formulate additional conditions that guarantee a saddle-node bifurcation). The existence of critical states and parameters is equivalent to the existence of Jordan Blocks of certain dimension in the Jordan canonical form of the Jacobian. This observation, together with the rich structure of the Jacobian defined by a mass action system can be used to obtain conditions on the sign patterns of certain matrices derived from the Jacobian, which can be determined by solving linear inequality systems. If one of the sign patterns corresponds to what is called an L+-matrix in the theory of sign-solvable systems, then critical points and parameters are guaranteed, however the converse is not true.

Title: Identifiability of Chemical Reaction Networks

We consider the dynamics of chemical reaction networks with mass-action kinetics. We show that there exist reaction networks for which the reaction rate constants are not uniquely identifiable, even if we are given complete information on the dynamics of concentrations for all chemical species. Also, we show that there exist pairs of distinct reaction networks such that their dynamics are identical under appropriate choices of reaction rate constants, and present theorems that characterize the properties that make this possible. We use these facts to show how we can determine dynamical properties of some chemical networks by analyzing other chemical networks. This is joint work with Casian Pantea.

Domitilla Del Vecchio, University of Michigan

Title: Modular Cell Biology: Retroactivity and Insulation

Modularity plays a fundamental role in the prediction of the behavior of a system from the behavior of its components, guaranteeing that the properties of individual components do not change upon interconnection. Just as electrical, hydraulic, and other physical systems often do not display modularity, nor do many biochemical systems, and specifically, genetic networks. Here, we study the effect of interconnections on the input/output dynamic characteristics of transcriptional components, focusing on a property, which we call "retroactivity," that plays a role similar to impedance in electrical circuits. In transcriptional networks, retroactivity is large when the amount of transcription factor is comparable to, or smaller than, the amount of promoter binding sites, or when the affinity of such binding sites is high. In order to attenuate the effect of retroactivity, we propose to design insulation devices based on a feedback mechanism inspired by the design of amplifiers in electronics. We introduce a bio-molecular realization of an insulation device based on a phosphorylation/dephosphorylation mechanism. This mechanism enjoys a remarkable insulation property, due to the fast time scales of the phosphorylation and dephosphorylation reactions. We briefly argue how the faster time scale of a device with respect to its input can be viewed as a general insulation mechanism peculiar of bio-molecular systems.

Vincent Fromion, INRA Jouy-En-Josas

Title: Recent progresses on the metabolism modelling of Bacteria: definition of local and global modules and a first explanation of their emergence

Joint work with A. Goelzer, F. Bekkal Brikci, C. Tanous, and G. Scorletti

Recently, the reconstruction and analysis of genome-scale genetic and metabolic regulatory networks have become an area of active research. Such a work requires the integration of existing knowledge as a first step towards System Biology. We define the "System Biology" field as the study of the interactions between the components of biological systems to understand how these interactions give rise to the function and behavior of the system. The reconstruction of the metabolic network and its associated regulatory network contributes to the resolution of various problems such as unraveling how the bacterium coordinates its genetic and metabolic networks to adapt to environmental changes or such as elucidating the global organization of the regulatory network. Such works will also help to develop tools and concepts to handle and to analyze the inherent complexity of biological functions. Here, we present the recent reconstruction and the mathematical analysis of the genetic and metabolic regulatory network for the Gram-positive bacterium Bacillus subtilis presented in [1]. B. subtilis has been studied for over 40 years and is one of the best-characterized bacteria, easily amenable to genetic and physiological studies. Our model includes the biochemical reactions of the metabolic network and all the known levels of regulation involved in metabolic pathways: transcriptional, translational, post-translational and modulation of enzymatic activities. To obtain the most complete view of the interplay between the metabolic network and the genetic regulation, the description of each regulatory mechanism includes the known roles of metabolite concentrations, ions, and any other quantities related to the state of the metabolic network. The entire model (reactions, enzymes, genes and regulations) has been curated manually, using published data and expert knowledge [1].

Using this model, we were able to examine various aspects of the general organization of the metabolic regulation. We find that metabolite pools are strongly involved in the regulation of the central metabolism of Bacillus subtilis, in agreement with the findings from an analysis of the genetic regulatory network of Escherichia coli. Moreover, by introducing the notion of local and global regulation, we reveal that the complex regulatory network can be broken down into sets of locally regulated modules, which are coordinated by global regulators. Local regulations ensure that the control of elementary pathways through the genetic and/or enzymatic regulations depends on the level of key metabolites. By contrast, global regulations ensure the coordination between these elementary pathways in response to environmental changes. The integration of these local/global levels and the use of the classification of sensing signals lead to recover the main physiological aspects of the metabolism of Bacillus subtilis. Finally, we have shown that the metabolic network regulation is highly structured through the definition of a rigorous and well defined notion of modules.

The existence of such a strong structure in the regulatory network led us to focus on the constraints acting on bacteria that could explain its emergence. So, we then present another recent result about the problem of resource management in bacteria [2]. Actually, for a fixed growth rate, we formalized the problem of resource management into a non-differentiable convex constraint-based feasibility problem through the integration of three structural constraints. This feasibility problem can be easily transformed into an equivalent Linear Programming (LP) feasibility problem for which many classical polynomial-time solvers are available. The resolution of the LP feasibility problem leads to predict not only the flux distribution and the maximal growth rate achievable for a given extracellular medium, but also the concentration of ribosomes, and of the proteins involved in the metabolic network, and thus the composition of the cell for different growth rates. Moreover, the modular structure of the metabolic network can also be predicted with respect to the medium composition. Thus, the proposed method extends the well-established FBA (Flux Balance Analysis) while avoiding many drawbacks.

[1] A Goelzer, F  Bekkal Brikci, I Martin-Verstraete, P Noirot, P Bessières, S Aymerich, and V  Fromion, Reconstruction and analysis of the genetic and metabolic regulatory networks of the central metabolism of Bacillus subtilis, BMC Systems Biology
(2)  1--20, 2008.
[2] A Goelzer, C Tanous, G Scorletti  and V Fromion,  Towards a systemic prediction of all cell components: the Resource Balance Analysis (RBA), submitted.

Tomas Gedeon, Montana State University

Title: Relaxation oscillations and a cell cycle oscillator

We study a finite dimensional monotone system coupled to a slowly evolving scalar differential equation which provides a negative feedback to the monotone system. We use a theory of multi-valued characteristics to show that this system admits a relaxation periodic orbit if a simple model system in $R^2$ does. Our construction can be used to prove existence of periodic orbits in slow-fast systems of arbitrary dimension.

We apply our theory to a model of a cell cycle in {\em Xenopus} embryos. Abrupt changes in signals upon entry to mitosis suggests that the cell cycle is generated by a relaxation oscillation. Our results show that the cell cycle orbit is not a relaxation oscillator. However, we construct a closely related system that exhibits relaxation oscillations and that approximates the cell cycle oscillator for an intermediate range of negative feedback strenghts. We show that the cell cycle oscillation disappears if the negative feedback is too weak or too strong.

Title: Complete Networks of Reversible Binding Reactions

A binding reaction is a chemical reaction that transforms two or more reactants into a single product. Networks of reversible binding reactions describe many pathogenic and therapeutic mechanisms that are studied in pharmacology. Determining the equilibrium state is a recurrent issue in that context. Toward an effort to do so systematically, we propose the class of complete networks of reversible binding reactions and characterize their equilibrium states. Completeness consists of structural and kinetic requirements that are applicable in the motivating context. The structural requirement is that there is a notion of composition that is intrinsic to the network and defines species, and that reactions preserve composition. The kinetic requirement is that the law of mass action applies, and that certain coherence equations constrain rate constants along reaction pathways with same outcome. In a complete network, the nonnegative stoichiometric compatibility classes are convex polytopes defined by equations that express the conservation of composition. Within each class, there exists a unique equilibrium state; it is detailed-balanced and characterized by a positive polynomial system with rather interesting features, and it is globally attracting. Global attraction is obtained in part from the fact that the boundary of the class is weakly repelling, a property which is proved with the Vol'pert Strict Positivity Theorem.

Jeremy Gunawardena, Harvard Medical School

Title: Using Algebraic Geometry to Study Protein Phosphorylation

Biochemical reaction networks based on massaction kinetics give rise to polynomial dynamical systems, whose steady states are real algebraic varieties. Curiously, this property has rarely been exploited in modelling biological systems. We will discuss posttranslational modification networks, such as those involved in multisite phosphorylation, for which we show that algebraic geometric methods yield novel predictions, now being tested in our laboratory.

Joao Hespanha, University of California - Santa Barbara

Title: Analysing Stochasticity in Regulatory Networks: The Evolvability of Gene Auto-regulation in the Presence of Noise

Gene regulatory mechanisms typically involve a large number of distinct chemical species, but it is common for some of these species to be represented by just a few molecules, which can invalidate models based on the deterministic chemical rate equation. Our work has been using tools developed for Stochastic Hybrid Systems to construct differential equations that accurately model the stochastic effects present in biochemical networks. In this talk, we review some of these tools and discuss their use in the context of understanding whether or not the introduction of a negative feedback mechanism can be beneï¬?cial in terms of maintaining protein numbers above a critical threshold, in the presence of stochastic fluctuations. Our results show the existence of a trade off, as introducing feedback reduces stochastic ï¬?uctuations around the mean, but also decreases the average number of protein molecules, which drives this number closer to the critical threshold. Through this work we seek to understand how negative feedback can be introduced through random mutations in the gene sequence, as such mutations can only persist if there is a positive change in cell ï¬?tness.

Brian Ingalls, University of Waterloo

Title: Input-output behaviour of stoichiometric systems

The structure of a biochemical system is characterized by the stoichiometry of its reaction network. These stoichiometric descriptions are particularly useful for addressing metabolic systems, which typically exhibit complex mass-transfer networks. Since stoichiometry is linear, it allows a tractable analysis of system behaviour. This talk will address the impact of stoichiometry on two aspects of steady-state behaviour: 1) the local parametric sensitivity analysis of "Metabolic Control Analysis," and 2) "steady-state equivalent" networks, which can provide a condensed description of a network as an input-output module. In both cases a control-theretic approach has proven insightful.

A. Agung Julius, Rensselaer Polytechnic Institute

Title: Metabolic Flux Balance Analysis and Related Computational Challenges

Metabolic networks map the biochemical reaction fluxes in a living cell to the flow of various chemical substances in the cell, which are called metabolites. The metabolic network of an organism can be thought of as production lines in a large scale biochemical plant. They capture the metabolic reactions in which metabolic products are made, and the reactants that are involved. In metabolic Flux Balance Analysis (FBA), transient dynamics in the network are ignored, and one typically focuses on the space of all possible steady-state reaction fluxes in a given metabolic network topology. In this talk, I will discuss some computational challenges in metabolic FBA. In particular, two types of problems are considered: computation of minimal (sub)networks, and computation of minimal knockout sets.

Sergei Pilyugin, University of Florida

Title: Kinetics of the cell cycle

Various dye dilution essays can be used to determine the number of divisions a given cell has undergone in vitro and in vivo. The CFSE dilution essay has become popular among those trying to estimate the kinetics of cell turnover in vivo. In this talk, I will discuss the mathematical approaches to the CFSE data, describe the current results and the challenges of the dilution essays.

Herbert Sauro, University of Washington

Title: Stochastic control analysis

We have begun the development of a theoretical analysis method for biological reaction networks that allows us to understand how noise propagation affects system sensitivities, how the sensitivities are related to one another, and how the sensitivities change due to the perturbations of genotype or the environment. We propose the beginnings of a stochastic version of metabolic control analysis for biological reaction networks (not limited to metabolic systems). We provide new stochastic versions of summation theorems and show that among theorems the summation theorems for flux fluctuation strength gives an insight for controlling flux fluctuations and reveals an interesting connection to the scaling law recently found by de Menezes and Barabasi.

Joint with Kyung H. Kim, University of Washington.

Liming Wang, University of California-Irvine

Title: Asymptotic behaviors in multisite phosphorylation-dephosphorylation cycles

Multisite phosphorylation-dephosphorylation cycles appear ubiquitously in cellular signaling pathways. One very important instance is that of Mitogen-Activated Protein Kinase (MAPK) cascades, which regulate primary cellular activities such as proliferation, differentiation, and apoptosis. Asymptotic behaviors of the MAPK cascades include monostability, multistability, and oscillations. In this talk, I will discuss the number of steady states in $n$-site phosphorylation-dephosphorylation cycles and a convergence result of the double phosphorylation-dephosphorylation cycles. The latter result features applications of monotone systems and singular perturbation theories.

Posters

Title: Permanent Coexistence for an Intraguild Predation Model with Predator Stage Structure

Using permanence theory, we investigate a unique blend of predation and competition known as intraguild predation or IGP?a top predator feeds on an intermediate predator and competes with that predator for a shared resource. We use a differential equations model of IGP with nonlinear functional and numerical responses in which the top predator's life history includes stage structure. We show under what conditions our system is permanent based upon underlying assumptions and a reformulation of the permanence definition.

Eduardo Jordao Neves, University of Sao Paulo, Brazil

Title: Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

We introduce jump processes in $\Rk$, called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with $k$ types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on the multi-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation with explicit bounds on the distance between the stochastic and deterministic trajectories.

As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model, associated to a synthetic biology model known as repressilator, which leads to a dynamical system with Hopf and pitchfork bifurcations. Depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in $\Rk$. We also discuss a simple signaling pathway related to cancer research, called p53 module.

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