Zero Deficiency Research Articles

Kinetic feedback design for polynomial systems

New computational methods are proposed in this paper to construct polynomial feedback controllers for the stabilization of polynomial systems with linear input structure around a positive equilibrium point. Using the theory of chemical reaction networks (CRNs) and previous results on dynamical equivalence, a complex balanced or weakly reversible zero deficiency closed loop realization is achieved by computing the gain matrix of a polynomial feedback using optimization. It is shown that the feedback resulting in a complex balanced closed loop system having a prescribed equilibrium point can be computed using linear programming (LP). The robust version of the problem, when a convex set of polynomial systems is given over which a stabilizing controller is searched for, is also solvable with an LP solver. The feedback computation for rendering a polynomial system to deficiency zero weakly reversible form can be solved in the mixed integer linear programming (MILP) framework. It is also shown that involving new monomials (complexes) into the feedback does not improve the solvability of the problems. The proposed methods and tools are illustrated on simple examples, including stabilizing an open chemical reaction network.

Computing zero deficiency realizations of kinetic systems

In the literature, there exist strong results on the qualitative dynamical properties of chemical reaction networks (also called kinetic systems) governed by the mass action law and having zero deficiency. However, it is known that different network structures with different deficiencies may correspond to the same kinetic differential equations. In this paper, an optimization-based approach is presented for the computation of deficiency zero reaction network structures that are linearly conjugate to a given kinetic dynamics. Through establishing an equivalent condition for zero deficiency, the problem is traced back to the solution of an appropriately constructed mixed integer linear programming problem. Furthermore, it is shown that weakly reversible deficiency zero realizations can be determined in polynomial time using standard linear programming. Two examples are given for the illustration of the proposed methods.

Tensor Approximation of Stationary Distributions of Chemical Reaction Networks

We prove that the stationary distribution of a system of reacting species with a weakly reversible reaction network of zero deficiency in the sense of Feinberg and separable propensity functions admits the tensor-structured approximation in the quantized tensor train (QTT) format. The complexity of the approximation scales linearly with respect to the number of species and logarithmically in the maximum copy numbers as well as in the desired accuracy. Our result covers the classical mass-action and Michaelis--Menten kinetics which correspond to two widely used classes of propensity functions and also to arbitrary combinations of those. New rank bounds for tensor-structured approximations of the probability density function (PDF) of a truncated one-dimensional Poisson distribution are an auxiliary result of the present paper which might be of independent interest. The present work complements recent results obtained by the authors jointly with Khammash and Nip on the tensor-structured numerical simulation of the evolution of system states distributions, driven by the Kolmogorov forward equation of the system, known also as the chemical master equation (CME). For the two kinetics mentioned above and a general reaction network, we also quantify the error of the low-rank tensor-structured approximation of the CME operator in the QTT format.

Stability in Chemical Reaction Networks

In this contribution an alternative overview of CRNT (Chemical Reaction Network Theory) concepts is presented. It is oriented to explore the stability properties of chemical reaction networks. In particular emphasis will be placed on the notion of complex balance solutions and its underlying stability.The structure of complex balance solutions will be studied with the help of an analogy with linear mass transport networks. The uniqueness and stability properties of these solutions will be investigated by convex analysis and a Lyapunov function candidate quite connected with chemical potentials.Most methods I make use of in the discussion are familiar, when not standard, to process control what might help devise possible implications (and applications) in the context of robust or decentralized control of process networks. In this concern versions of the zero deficiency theorem adapted to reaction-separation networks might become a promising tool for control design.

Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiency

Mass-action kinetics is frequently used in systems biology to model the behavior of interacting chemical species. Many important dynamical properties are known to hold for such systems if their underlying networks are weakly reversible and have a low deficiency. In particular, the Deficiency Zero and Deficiency One Theorems guarantee strong regularity with regards to the number and stability of positive equilibrium states. It is also known that chemical reaction networks with distinct reaction structure can admit mass-action systems with the same qualitative dynamics. The theory of linear conjugacy encapsulates the cases where this relationship is captured by a linear transformation. In this paper, we propose a mixed-integer linear programming algorithm capable of determining the minimal deficiency weakly reversible reaction network which admits a mass-action system which is linearly conjugate to a given reaction network.

On second- order symmetric duality for a class of semigroups with zero deficiency

This article is concerned with a pair of second-order symmetric duals in the context of non-differentiable multiobjective fractional programming problems. We establish the weak and strong duality theorems for the new pair of dual models. Discussion on some special cases shows that results in this paper extend previous work in this area.

On The Geometry of Equilibrium Solutions of Kinetic Systems Obeying the Mass Action Law

In this contribution the problem relating dynamic behavior and parameter values for weakly reversible chemical reaction networks (CRNs) that obey the mass action law is revisited. The approach is founded on previous methods and results discussed in [15, 16] for weakly reversible CRNs. Two new research directions are however undertaken in this study, one of them exploits an alternative factorization of the Kernel of the Kirchhoff matrix Ak which in a natural way links complex balance with zero deficiency. The other one intends to use the convex structure of the so-called family of solutions described in [15] on the space of reaction channels.

On the structure of generalized polyhedral semigroups with zero deficiency

On the structure of generalized polyhedral semigroups with zero deficiency

On the structure of generalized polyhedral semigroups with zero deficiency

The polyhedral or triangle groups and some generalizations of them, such as binary polyhedral groups, have been studied by several authors. In this paper, we introduce two classes of semigroups that have the same presentation as these generalizations of polyhedral groups and investigate their structures, such as finiteness, their relationship with the groups presented by the same presentation.

On the structure of generalized polyhedral semigroups with zero deficiency

The polyhedral or triangle groups and some generalizations of them, such as binary polyhedral groups, have been studied by several authors. In this paper, we introduce two classes of semigroups that have the same presentation as these generalizations of polyhedral groups and investigate their structures, such as finiteness, their relationship with the groups presented by the same presentation.

Parametric Condition for Multistationarity in Biochemical Reaction Networks

Chemical Reaction Network theory allows us to decide whether many classes of networks have the capacity for multiple positive equilibria, based on their structural properties. In this way, the deficiency zero theorem asserts that every weakly reversible network of zero deficiency has a unique equilibrium, for any choices of parameter values. We make use of CRNT, aiming not only to discriminate whether a (positive deficiency) network can exhibit multiple steady states, but also to characterize the whole space of the parameters regarding to their capability to produce multistationarity. In this work, we provide a condition, on the parameters of biochemical networks, for the appearance of multistationarity.

Multivariable Control with Generalized Decoupling for Disturbance Rejection

In this paper, a systematic procedure to design multivariable controllers that have options for selective decoupling of different structures (e.g., full or partial decoupler) is proposed. The controller structure is determined based on an analysis of relative load gain (RLG). For controller design, the adjoint of a template matrix is provided to design the MIMO decoupler that is needed for a given process. In the case where decoupling is desirable, according to the number of zero deficiency that the adjoint of the template matrix has, a pole-zero compensator is incorporated to preserve the properness of the decoupling controller and to define the decoupled open-loop dynamics. By incorporating the decoupler, the controller in the main loop is synthesized for disturbance rejection. Stability robustness of the system is tuned using measures of modeling errors in the decoupled open-loop process. Simulation examples are used to illustrate the proposed method and show its effectiveness in disturbance rejection in MIMO systems.

Modeling and analysis of mass-action kinetics

Mass-action kinetics are used in chemistry and chemical engineering to describe the dynamics of systems of chemical reactions, that is, reaction networks. These models are a special form of compartmental systems, which involve mass- and energy-balance relations. Aside from their role in chemical engineering applications, mass-action kinetics have numerous analytical properties that are of inherent interest from a dynamical systems perspective. Because of physical considerations, however, mass- action kinetics have special properties, such as nonnegative solutions, that are useful for analyzing their behavior. With this motivation in mind, this article has several objectives. First, a general construction of the kinetic equations based on the reaction laws is provided in a state-space form. Next, the nonnegativity of solutions to the kinetic equations is considered. The realizability problem, which is concerned with the inverse problem of constructing a reaction network having specified essentially nonnegative dynamics, is also considered. In particular, an explicit construction of a reaction network for essentially nonnegative polynomial dynamics involving a scalar state is provided. Next, the reducibility of the kinetic equations is considered as well as the stability of the equilibria of the kinetic equations. Lyapunov methods are applied to the kinetic equations, and semistability is guaranteed through the convergence to a Lyapunov- stable equilibrium that depends on the initial concentrations. Semistability is the appropriate notion of stability for compartmental systems in general, and reaction networks in particular, where the limiting concentration maybe nonzero and may depend on the initial concentrations. Finally, the zero deficiency result for mass-action kinetics in standard matrix terminology is presented and semistability is proven.

Observability and observer design for kinetic networks

In this paper, we deal with the observability and observer design of a certain class of reactors (batch reactors) which are globally asymptotically feedback stabilisable. We show how to design two types of high-gain observers: a Luenberger-like high-gain observer and a high-gain extended Kalman filter. Results of convergence of these estimators follow from our observability analysis. We prove that the non-linear feedback controller using these high-gain observers stabilise such reactors. In fact, we develop an observer-based control structure for a non-linear model of batch reactors. Two examples of such reactors that correspond to strongly connected networks and that have zero deficiency are presented in this paper. Simulations show the good performance of such an observer-based control structure.

Patterns of Stochastic Behavior in Dynamically Unstable High-Dimensional Biochemical Networks

The question of dynamical stability and stochastic behavior of large biochemical networks is discussed. It is argued that stringent conditions of asymptotic stability have very little chance to materialize in a multidimensional system described by the differential equations of chemical kinetics. The reason is that the criteria of asymptotic stability (Routh-Hurwitz, Lyapunov criteria, Feinberg's Deficiency Zero theorem) would impose the limitations of very high algebraic order on the kinetic rates and stoichiometric coefficients, and there are no natural laws that would guarantee their unconditional validity. Highly nonlinear, dynamically unstable systems, however, are not necessarily doomed to collapse, as a simple Jacobian analysis would suggest. It is possible that their dynamics may assume the form of pseudo-random fluctuations quite similar to a shot noise, and, therefore, their behavior may be described in terms of Langevin and Fokker-Plank equations. We have shown by simulation that the resulting pseudo-stochastic processes obey the heavy-tailed Generalized Pareto Distribution with temporal sequence of pulses forming the set of constituent-specific Poisson processes. Being applied to intracellular dynamics, these properties are naturally associated with burstiness, a well documented phenomenon in the biology of gene expression.

Input-to-State Stability of Rate-Controlled Biochemical Networks

In this paper, the study of the class of biochemical systems known as zero deficiency networks is extended to the case of time-varying kinetic parameters. We show that the resulting class of nonlinear systems with inputs satisfies a notion of input-to-state stability uniformly over a set of parameters. In particular, the input-to-state stability estimates allow us to characterize the robustness of zero deficiency networks with respect to perturbations in the parameters as well as study their stability when the reaction rates are controlled by an independent process.

On semigroup presentations and efficiency

The purpose of this paper is twofold. First we determine some forms of the relations in a finite semigroup presentation with zero deficiency which does or does not define a group. Moreover, we conclude that a finite Rees matrix semigroup M[G; I, Λ; P] is efficient when G is efficient and the index sets I, Λ are finite.

State-estimators for Chemical Reaction Networks of Feinberg-Horn-Jackson Zero Deficiency Type

This paper provides a necessary and sufficient condition for detectability for chemical reaction networks of the Feinberg-Horn-Jackson zero deficiency type. Under this condition, an explicit construction of globally convergent observers is obtained based on ISS techniques. The observers are easy to implement, and several robustness aspects are tested in numerical simulations.

Discussion on ‘State-estimators for Chemical Reaction Networks of Feinberg-Horn-Jackson Zero Deficiency Type’ by M. Chaves and E.D. Sontag

Discussion on ‘State-estimators for Chemical Reaction Networks of Feinberg-Horn-Jackson Zero Deficiency Type’ by M. Chaves and E.D. Sontag

An Alternative Observer for Zero Deficiency Chemical Networks

This paper provides an explicit observer for chemical reaction networks of the Feinberg-Horn-Jackson zero deficiency type. This observer is obtained by a construction different from the one employed in a previous paper. The new observer exhibits slower performance, but this is traded for more robustness to additive disturbances, expressed through an ISS property.