New approach to the stability and control of reaction networks

A new system-theoretic approach for studying the stability and control of chemical reaction networks (CRNs) is proposed, and analyzed. This has direct application to biological applications where biochemical networks suffer from high uncertainty in the kinetic parameters and exact structure of the rate functions. The proposed approach tackles this issue by presenting "structural" results, i.e. results that extract important qualitative information from the structure alone regardless of the specific form of the kinetics which can be arbitrary monotone kinetics, including Mass-Action. The proposed method is based on introducing a class of Lyapunov functions that we call Piecewise Linear in Rates (PWLR) Lyapunov functions. Several algorithms are proposed for the construction of these functions. Subject to mild technical conditions, the existence of these functions can be used to ensure powerful dynamical and algebraic conditions such as Lyapunov stability, asymptotic stability, global asymptotic stability, persistence, uniqueness of equilibria and exponential contraction. This shows that this class of networks is well-behaved and excludes complicated behaviour such as multi-stability, limit cycles and chaos. The class of PWLR functions is then shown to be a subset of larger class of Robust Lyapunov functions (RLFs), which can be interpreted by shifting the analysis to reaction coordinates. In the new coordinates, the problem transforms into finding a common Lyapunov function for a linear parameter varying system. Consequently, dual forms of the PWLR Lyapunov functions are presented, and the interpretation in terms of the variational dynamics and contraction analysis are given. An other class of Piecewise Quadratic in Rates Lyapunov function is also introduced. Relationship with consensus dynamics are also pointed out. Control laws for the stabilization of the proposed class of networks are provided, and the concept of control Lyapunov function is briefly discussed. Finally, the proposed framework is shown to be widely applicable to biochemical networks.Open Acces

Tropicalization and tropical equilibration of chemical reactions

Systems biology uses large networks of biochemical reactions to model the functioning of biological cells from the molecular to the cellular scale. The dynamics of dissipative reaction networks with many well separated time scales can be described as a sequence of successive equilibrations of different subsets of variables of the system. Polynomial systems with separation are equilibrated when at least two monomials, of opposite signs, have the same order of magnitude and dominate the others. These equilibrations and the corresponding truncated dynamics, obtained by eliminating the dominated terms, find a natural formulation in tropical analysis and can be used for model reduction.Comment: 13 pages, 1 figure, workshop Tropical-12, Moskow, August 26-31, 2012; in press Contemporary Mathematic

Structurally robust biological networks

Background: The molecular circuitry of living organisms performs remarkably robust regulatory tasks, despite the often intrinsic variability of its components. A large body of research has in fact highlighted that robustness is often a structural property of biological systems. However, there are few systematic methods to mathematically model and describe structural robustness. With a few exceptions, numerical studies are often the preferred approach to this type of investigation. Results: In this paper, we propose a framework to analyze robust stability of equilibria in biological networks. We employ Lyapunov and invariant sets theory, focusing on the structure of ordinary differential equation models. Without resorting to extensive numerical simulations, often necessary to explore the behavior of a model in its parameter space, we provide rigorous proofs of robust stability of known bio-molecular networks. Our results are in line with existing literature. Conclusions: The impact of our results is twofold: on the one hand, we highlight that classical and simple control theory methods are extremely useful to characterize the behavior of biological networks analytically. On the other hand, we are able to demonstrate that some biological networks are robust thanks to their structure and some qualitative properties of the interactions, regardless of the specific values of their parameters

Persistence and stability of generalized ribosome flow models with time-varying transition rates

In this paper the qualitative dynamical properties of so-called generalized ribosome flow models are studied. Ribosome flow models known from the literature are generalized by allowing an arbitrary directed network structure between the compartments and secondly, by assuming a general time-varying rate function describing the compartmental transitions. Persistence of the dynamics is shown using the chemical reaction network (CRN) representation of the system. We show the stability of different compartmental structures including strongly connected ones with an entropy-like logarithmic Lyapunov function. The L1 contractivity of solutions is also studied in the case of periodic reaction rates having the same period. It is also shown that different Lyapunov functions may be assigned to the same model depending on the factorization of the reaction rates.Comment: 28 pages, 8 figure

Qualitative modeling in computational systems biology

The human body is composed of a large collection of cells,\the building blocks of life". In each cell, complex networks of biochemical processes contribute in maintaining a healthy organism. Alterations in these biochemical processes can result in diseases. It is therefore of vital importance to know how these biochemical networks function. Simple reasoning is not su±cient to comprehend life's complexity. Mathematical models have to be used to integrate information from various sources for solving numerous biomedical research questions, the so-called systems biology approach, in which quantitative data are scarce and qualitative information is abundant. Traditional mathematical models require quantitative information. The lack in ac- curate and su±cient quantitative data has driven systems biologists towards alternative ways to describe and analyze biochemical networks. Their focus is primarily on the anal- ysis of a few very speci¯c biochemical networks for which accurate experimental data are available. However, quantitative information is not a strict requirement. The mutual interaction and relative contribution of the components determine the global system dy- namics; qualitative information is su±cient to analyze and predict the potential system behavior. In addition, mathematical models of biochemical networks contain nonlinear functions that describe the various physiological processes. System analysis and parame- ter estimation of nonlinear models is di±cult in practice, especially if little quantitative information is available. The main contribution of this thesis is to apply qualitative information to model and analyze nonlinear biochemical networks. Nonlinear functions are approximated with two or three linear functions, i.e., piecewise-a±ne (PWA) functions, which enables qualitative analysis of the system. This work shows that qualitative information is su±cient for the analysis of complex nonlinear biochemical networks. Moreover, this extra information can be used to put relative bounds on the parameter values which signi¯cantly improves the parameter estimation compared to standard nonlinear estimation algorithms. Also a PWA parameter estimation procedure is presented, which results in more accurate parameter estimates than conventional parameter estimation procedures. Besides qualitative analysis with PWA functions, graphical analysis of a speci¯c class of systems is improved for a certain less general class of systems to yield constraints on the parameters. As the applicability of graphical analysis is limited to a small class of systems, graphical analysis is less suitable for general use, as opposed to the qualitative analysis of PWA systems. The technological contribution of this thesis is tested on several biochemical networks that are involved in vascular aging. Vascular aging is the accumulation of changes respon- sible for the sequential alterations that accompany advancing age of the vascular system and the associated increase in the chance of vascular diseases. Three biochemical networks are selected from experimental data, i.e., remodeling of the extracellular matrix (ECM), the signal transduction pathway of Transforming Growth Factor-¯1 (TGF-¯1) and the unfolded protein response (UPR). The TGF-¯1 model is constructed by means of an extensive literature search and con- sists of many state equations. Model reduction (the quasi-steady-state approximation) reduces the model to a version with only two states, such that the procedure can be visual- ized. The nonlinearities in this reduced model are approximated with PWA functions and subsequently analyzed. Typical results show that oscillatory behavior can occur in the TGF-¯1 model for speci¯c sets of parameter values. These results meet the expectations of preliminary experimental results. Finally, a model of the UPR has been formulated and analyzed similarly. The qualitative analysis yields constraints on the parameter values. Model simulations with these parameter constraints agree with experimental results