Schedule for: 17w5099 - Mathematical Analysis of Biological Interaction Networks

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Deterministic reaction networks equipped with a kinetics are termed absolute concentration robust (ACR) if one or more species have the same value at any positive steady state of the system. This property is particular interesting from a biological point of view, since it implies that the expression of certain molecules at a stationary regime does not change under modification of the total mass of the system. Structural conditions implying absolute concentration robustness have been found for mass action models. Surprisingly, under the same structural conditions it has been shown that the associated stochastically modeled systems undergo an extinction event with probability 1. This leads to a discrepancy between the deterministic model, where some species have always the same value at equilibrium, and the stochastic model, where the same species eventually absorb the total mass of the system. I will present a result which solves the discrepancy between the two models at typical time frames. The result implies that, under certain conditions, the averages of the ACR species counts tend to their ACR equilibria. Specifically, this holds up to any fixed time point, when the total number of molecules tends to infinity in an appropriate multiscale fashion. Finally, I will show by examples that absolute concentration robustness does not necessarily imply an extinction event for the stochastically modeled system, contrary to what could be believed.

The graph-related symmetries of a reaction network give rise to certain special equilibria (such as complex balanced equilibria) in deterministic models. Correspondingly, in stochastic models these symmetries give rise to certain special stationary measures. Previous work by Anderson, Craciun and Kurtz identified stationary distributions of a complex balanced network; meanwhile Cappelletti and Wiuf developed the notion of complex balancing for stochastic systems. We define and establish the relations between reaction balanced measure, complex balanced measure and reaction vector balanced measure and prove that with mild additional hypotheses, the former two are stationary distributions. We develop the idea of decomposing both deterministic and stochastic systems into so-called ``factor systems'' and we establish the correspondence between factors of a complex balanced deterministic system and those of the corresponding stochastic system. Furthermore, in spirit of earlier work by Joshi, we give additional conditions for when detailed balance of Markov chain theory implies detailed balance of reaction network theory. This is joint work with Daniele Cappelletti.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will highlight different algebraic tools for the analysis of reaction networks taken with mass-action kinetics. I will briefly describe some joint recent results with Mercedes Pérez Millán about structural properties of many common biochemical networks. I will also briefly introduce ongoing results joint with Frédéric Bihan and Magalí Giaroli on new conditions to describe multistationarity parameters, based on ideas from real algebraic geometry.

This talk highlights algebraic techniques for analyzing reaction networks taken with mass-action kinetics. Specifically, we describe how steady-state parametrizations arising from elimination (usually of intermediates) can be harnessed to analyze bistability and oscillations. These techniques allow us to investigate the dynamics of certain biological signaling networks, namely, multisite phosphorylation networks: the emergence of bistability in a network underlying ERK regulation, and the capacity for oscillations in a mixed-mechanism processive/distributive phosphorylation network.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Stochastic Chemical Reaction Networks (CRNs) can be viewed as programs whose instructions are reactions; these instructions execute asynchronously and in parallel to produce a number of output molecules that is a function of the initial counts of molecular species in a well-mixed solution. For example, the simple program "X + Y --> Z + Z", executing in a solution that initially contains copies of two molecular species X and Y, eventually produces a number of copies of molecule Z that is exactly twice the minimum of the initial counts of X and Y, thereby computing 2min{#X,#Y}. How fast does this program run, as a function of the initial species counts (assuming fixed conditions such as volume)? Are there faster programs that produce the same output? More generally, what can and cannot be computed by CRN programs? These questions are attracting much attention in light of significant success in "compiling" CRN programs into real molecular controllers that can sense and respond to conditions in a chemical environment. A beautiful emerging theory of computing with CRNs is providing sharp answers to such questions. The theory and underlying computing models have their roots partly in distributed computing, where population protocols and Petri nets - essentially CRNs in disguise - shed light on the computing power of massively parallel systems of distributed computing agents, interacting asynchronously. In this talk I'll introduce some stochastic CRN computing models, as well as results on their computational power that are due to Angluin, Aspnes, Doty, Soloveichik and others, along with open questions and directions for future work.

Stochasticity is a defining feature of biochemical reaction networks, with molecular fluctuations influencing cell physiology. In principle, master probability equations completely govern the dynamic and steady state behavior of stochastic reaction networks. In practice, a solution had been elusive for decades, when there are second or higher order reactions. A large community of scientists has then reverted to merely sampling the probability distribution of biological networks with stochastic simulation algorithms. Consequently, master equations, for all their promise, have not inspired biological discovery. We will present a closure scheme that solves chemical master equations of nonlinear reaction networks [1]. The zero-information closure (ZI-closure) scheme is founded on the observation that although higher order probability moments are not numerically negligible, they contain little information to reconstruct the master probability [2]. Higher order moments are then related to lower order ones by maximizing the information entropy of the network. Using several examples, we show that moment-closure techniques may afford the quick and accurate calculation of steady-state distributions of arbitrary reaction networks. With the ZI-closure scheme, the stability of the systems around steady states can be quantitatively assessed computing eigenvalues of the moment Jacobian [3]. This is analogous to Lyapunov’s stability analysis of deterministic dynamics and it paves the way for a stability theory and the design of controllers of stochastic reacting systems [4, 5]. In this seminar, we will present the ZI-closure scheme, the calculation of steady state probability distributions, and discuss the stability of stochastic systems, including oscillatory ones. 1. Smadbeck P, Kaznessis YN. A closure scheme for chemical master equations. Proc Natl Acad Sci U S A. 2013 Aug 27;110(35):14261-5. 2. Smadbeck P, Kaznessis YN. Efficient moment matrix generation for arbitrary chemical networks, Chem Eng Sci, 2012, 84, 612-618, 3. Smadbeck P, Kaznessis YN. On a theory of stability for nonlinear stochastic chemical reaction networks. J Chem Phys. 2015 May 14;142(18):184101. doi: 10.1063/1.4919834. 4. Smadbeck P, Kaznessis YN. Solution of chemical master equations for nonlinear stochastic reaction networks. Curr Opin Chem Eng. 2014 Aug 1;5:90-95. 5. Constantino P, Vlysidis M, Smadbeck P, Kaznessis YN. Modeling stochasticity in biochemical reaction networks. Journal of Physics D: Applied Physics, 2016, 49 (9), 093001

Among the central questions in systems biology are those of understanding the roles of, and interactions among, signal transduction pathways and feedback loops. This talk focuses on “dynamic phenotypes” characterized by input/output responses to external inputs in addressing such issues, using fold-change detection and monotone architectures as case studies. An ubiquitous property of sensory systems is "adaptation": a step increase in stimulus triggers an initial change in a biochemical or physiological response, followed by a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain essential variables within acceptable bounds and allows organisms to readjust themselves to an optimum and non-saturating sensitivity range when faced with a prolonged change in their environment. It has been recently observed that some adapting systems, ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes, enjoy a remarkable additional feature: scale invariance or "fold change detection" meaning that the initial, transient behavior remains approximately the same even when the background signal level is scaled. This talk will review the biological phenomenon, and formulate a theoretical framework leading to a general theorem characterizing scale invariant behavior by equivariant actions on sets of vector fields that satisfy appropriate Lie-algebraic nondegeneracy conditions. The theorem allows one to make experimentally testable predictions, and the presentation will discuss the validation of these predictions using genetically engineered bacteria and microfluidic devices, as well their use as a "dynamical phenotype" for model invalidation. The talk will also include some speculative remarks about the role of the shape of transient responses in immune system self/other recognition and in cancer immunotherapy, as well as a brief discussion of how control-theoretic structures such as differential positivity (monotonicity) have been experimentally employed together with experimental data in order to elucidating mechanisms for stress responses and chemosensing.

We consider chemical reaction networks operating on discrete state spaces, i.e., where we keep track of molecule counts. We give a sufficient syntactic condition on chemical reaction networks for the impossibility of certain reactions to take place in the long term. This result is a statement about the reachability relation and is therefore independent of stochastics. As such, it can equivalently be formulated in terms of Petri nets, which is a well-studied model of concurrency.

The Chemical Reaction Network (CRN) model is a language designed to describe the behavior of chemical or biological molecules. Determining whether, in a given semantics, two CRNs have the same behavior is an interesting problem both in itself and for its uses in practice. Such practical uses that have been demonstrated include understanding biological systems by comparison to simple, well-understood CRNs, and verifying that physical implementations of abstract CRNs correctly implement their intended specifications. We defined a concept of CRN equivalence based on bisimulation as explored in concurrent systems, and explored its implications for CRNs in the low-copy-number semantics. We then explored algorithms to check whether two CRNs satisfy this concept of equivalence, and the computational complexity of that task. I will present this definition and our results, and place them in context with other concepts and methods to check CRN equivalence. I will also touch on the uses of this area of theory in practical molecular programming.

Algebraic matroids can be used to determine all the algebraic dependency relationships among a set of polynomials, without actually calculating those corresponding polynomial relationships. I'll discuss the application of algebraic matroids in three areas of model analysis: identifiability, observability, and indistinguishability. We'll see that algebraic matroids can be particularly useful in the areas of observability and indistinguishability, especially for large nonlinear models.

Modern biology provides large networks describing regulations between bio-molecules. It is widely believed that dynamics of molecular activities based on such regulatory networks are the origin of biological functions. However, the information on the networks is not sufficient to determine the resulting dynamics. In this study we present a mathematical theory to provide an important aspect of dynamics from information of regulatory linkages alone. We show that feedback vertex set (FVS) of a regulatory network is a set of determining nodes of dynamics [1]. It assures that i) any long-term dynamical behavior of whole system, such as steady states, periodic oscillations or quasi-periodic oscillations, can be identified by measurements of a subset of molecules in the network, and that ii) the subset is determined from the regulatory linkage alone. The theory also claims that iii) dynamical behavior of whole system can be switched from one attractor to others just by controlling dynamics of FVS. We applied our theory to a real biological network to verify our prediction by collaborating with experimental biologists [2]. We analyzed a regulatory network of ascidian embryo including 90 genes, and responsible for cell differentiation in the early development. We identified five genes in minimum FVS of the network. The exhaustive artificial activation/inhibition of five genes showed that the system could be controlled just by controlling activities of five genes in FVS. [1] Mochizuki, A., Fiedler, B. et al. J. Theor. Biol., 335, 130-146. [2] Kobayashi, K., Maeda, K., Tokuoka, M., Mochizuki, A. and Satou, Y. In Review.

The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the Markov chain that represents the stochastic reaction dynamics. Often this state-space is infinite in size, making it difficult to obtain the exact solutions of the CME. Hence these solutions need to be estimated by stochastic simulations or by solving an approximate CME over a finite truncated state-space. In many applications, one is interested in finding the stationary solution of the CME which corresponds to the stationary probability distribution of the underlying Markov chain. When the state-space is infinite, this stationary distribution satisfies an infinite-dimensional linear-algebraic system which cannot be directly solved. In this talk we will discuss how the stationary distribution can be accurately estimated by solving a finite linear-algebraic system that provides the stationary distribution of a suitably constructed Markov chain over a truncated state-space. We will argue that under certain stability conditions, like the irreducibility of the infinite state-space and exponential ergodicity of the reaction dynamics, the obtained approximations are guaranteed to converge to the exact stationary solution as the truncated state-space expands to the full state-space. We will explain how these stability conditions can be computationally checked for reaction networks and also describe how the approximation errors can be quantified. The presented ideas will be illustrated through many examples.

The steady state property of absolute concentration robustness (ACR) has recently gained attention in the literature. A species is said to be ACR if its attains the same value at all positive steady states, i.e. the steady state value does not depend upon initial concentrations. This mathematical property suggests evolved structure in biochemical reaction networks which is capable of maintaining narrow ranges for certain reactants in the face of fluctuating external circumstances. In this talk, I will introduce a new method for establishing ACR in mass action systems which utilizes the recently developed theory of network translation. In addition to expanding the scope of systems which can be guaranteed to permit ACR, our method typically allows the steady state ACR value to be computed directly from a reaction network known as a translated chemical reaction network.

Recent measurements of cellular processes on the single-cell level revealed significant cell-to-cell variability even for cells that are genetically identical and share the same growth conditions. The magnitude of this variation goes beyond what can be expected just based on the stochasticity of the specific process under study. The remaining source of variation is believed to stem from the random environment in which the considered cellular process is embedded. In this talk, I will present our efforts to model biochemical networks embedded in random environments and give the corresponding master equations for them. I will also touch upon the problem of statistical inference for such models and show how they can then be used to dissect and quantify the sources of cell-to-cell variability based on measured single-cell data.

We introduce a method for simplifying the study of the steady states of biological systems that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, which we call MESSI systems. By explicit elimination of intermediate complexes, we define an important subclass of MESSI systems with toric steady states. For these systems we describe an algorithm that determines whether the system has the capacity for multistationarity, and when it does, it shows two positive steady states and reaction rate constants that witness multistationarity. We illustrate these results in the two-layer phosphorylation cascade system. This is joint work with Alicia Dickenstein.

This talk is an attempt to give a brief overview of the theory for probabilities of rare events in chemical reaction networks. I will consider what the existing theory for large deviations of Markov processes enables us to say about stochastic models for CRNs. For these Markov chain models we can describe the expected behaviour of their paths and their variability over time, even for models that involve multiple-scales. Large deviation theory further describes the events not captured by such law of large number or central limit theorem results, as they have exponentially small probabilities in terms of the scaling parameter. For models on multiple scales this theory becomes technically much more challenging, but some calculations can be made. These results can also be very useful in designing efficient algorithms for simulating rare events in CRNs.

Stochastic SIR-type epidemic processes on random graphs are a special class of interaction networks that have become of interest lately for modeling contact-type epidemics (Ebola, HIV, etc). I will discuss a particular case of the SIR epidemic evolving on a configuration model random graph with given degree distribution. In particular, I will describe the relevant large graph limit result which yields the law of large numbers (LLN) for the edge-based SIR process and is useful in building a "network-free" SIR Markov hybrid model for epidemic parameters inference.

In the literature there is a variety of models for the Calvin cycle of photosynthesis and a number of statements, based on computer calculations, about the number and stability of their steady states. Here I will discuss some progress in obtaining rigorous results on this issue. It has been shown that in a simple model with five unknowns there are parameters for which there are two positive steady states, one of them stable, and a special case where there is a continuum of steady states. A more detailed model, due to Pettersson and Ryde-Pettersson, with fifteen unknowns turns out to be a system of differential-algebraic equations (DAE), rather than a system of ODE. It has been shown that for certain parameters it has two positive steady states. The proof is related to the method of elementary flux modes. For a closely related model, due to Poolman, there are three positive steady states. The difference between the numbers of steady states in these two models is related to the biological phenomenon of overload breakdown, in which too much demand for sugar causes the system supplying it to collapse.

Multistationarity has been recognized as an important feature of dynamical systems originating in Biology. As a consequence of the usually very high parameter uncertainty numerical analysis often fails to establish multistationarity. Hence techniques allowing the analytic computation of parameter values where a given system exhibits multistationarity are desirable. I present conditions in form of polynomial systems that allow to determine parameter vectors where a mass action system exhibits multistationarity.

We give an overview of the use of sign conditions in chemical reaction network theory (in several collaborations). To begin with, we study chemical reaction networks with generalized mass-action kinetics (power-law kinetics). Existence and uniqueness of complex-balancing equilibria (in every stoichiometric class) are determined by sign vectors of the stoichiometric and kinetic-order subspaces. In fact, existence and uniqueness correspond to surjectivity and injectivity of generalized polynomial maps (with full rank), and our results can be interpreted in terms of real algebraic geometry (multivariate Descartes rule of signs). The right hand side of a generalized mass-action system is also a generalized polynomial map (not necessarily with full rank), and its injectivity can be characterized in terms of sign conditions. Clearly, injectivity is sufficient to preclude multiple equilibria. We extend this approach to even more general kinetics (e.g. monotone kinetics). Moreover, (local and global) stability of complex-balancing equilibria (in Lotka-Volterra schemes or planar S-systems) can be characterized in terms of signs of Jacobian matrices. For these systems, we also study limit cycles and the related center problem. Finally, we develop algorithms to compute sign vector conditions (for large systems) and to enumerate sign vectors of minimal and maximal support, respectively.

A common feature of pattern formation in both space and time is the destabilization of a stable equilibrium solution of an ordinary differential equation by adding diffusion or delay, or both. Here we study linear stability of general reaction-diffusion systems with off-diagonal time delays. We show that a delay-stable system cannot be destabilized by diffusion, and that a diffusion stable system is also stable with respect to delay, if the diffusion is sufficiently fast. A system with direct negative feedback which is strongly stable with respect to diffusion can be destabilized by off-diagonal delay. This is joint work with Peter Hinow, University of Wisconsin-Milwaukee.

We focus on the question of bistability, or existence of multiple (stable) positive equilibria, a dynamical property that underlies important cellular processes, and a recurring theme in recent work on reaction networks. Namely, we consider the question: "when can we conclude that a network admits multiple stable positive equilibria based on analysis of its subnetworks?” We identify a number of operations on reaction networks that preserve bistability as we build up the network, and we illustrate the power of this approach on the much-studied Huang-Ferrell MAPK cascade. This work can be thought of as a step towards a theory of “motifs”, a central theme in systems biology.

The Lyapunov-Foster criteria is one of the well known methods for verifying ergodicity of Markov processes. The "Standard Lyapunov function” has played a significant role in the analysis of deterministic reaction networks, especially in regards to deficiency 0 networks. I will provide conditions on reaction networks that guarantee the Standard Lyapunov function satisfies the Lyapunov-Foster criteria. The analysis utilizes hierarchies of intensities and rate functions (Tier structures). I will also provide some special classes of reaction networks satisfying the conditions, thus proving networks in those classes are positive recurrent. I will also provide some results pertaining to the mixing times of reaction networks.

A “permanent” dynamical system is one whose positive solutions stay bounded away from zero and infinity. The permanence property has important applications in biochemistry, cell biology, and ecology. Inspired by reaction network theory, we define a class of polynomial dynamical systems called tropically endotactic. We show that these polynomial dynamical systems are permanent, irrespective to the values of (possibly time-dependent) parameters in these systems. These results generalize the permanence of 2D reversible and weakly reversible mass-action systems.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.