Multistationarity in triple-site mixed mechanism phosphorylation network

Modeling and analysis of complex systems are important aspects of understanding systemic behavior. In the lack of detailed knowledge about a system, we often choose modeling equations out of convenience and search the (high-dimensional) parameter space randomly to learn about model properties. Qualitative modeling sidesteps the issue of choosing specific modeling equations and frees the inference from specific properties of the equations. We consider classes of ordinary differential equation (ODE) models arising from interactions of species/entities, such as (bio)chemical reaction networks or ecosystems. A class is defined by imposing mild assumptions on the interaction rates. In this framework, we investigate whether there can be multiple positive steady states in some ODE models in a given class. We have developed and implemented a method to decide whether any ODE model in a given class cannot have multiple steady states. The method runs efficiently on models of moderate size. We tested the method on a large set of models for gene silencing by sRNA interference and on two publicly available databases of biological models, KEGG and Biomodels. We recommend that this method is used as (i) a pre-screening step for selecting an appropriate model and (ii) for investigating the robustness of non-existence of multiple steady state for a given ODE model with respect to variation in interaction rates. Scripts and examples in Maple are available in the Supplementary Information. wiuf@math.ku.dk Supplementary data are available at Bioinformatics online.

In this paper, we study the possibility of the existence of a universal solution of the Cauchy problem for the partial differential equation (PDE) systems in the case if this system is overdetermined so that the new overdetermined system of PDE contains all solutions of the initial PDE system and, in addition, reduces to the ordinary differential equation (ODE) systems, whose solution is then found. To do this, the article discusses the modification of the method of finding particular solutions for any overdetermined systems of differential equations by reduction to overdetermined systems of implicit equations. In the previous papers of the authors, a method was proposed for finding particular solutions for overdetermined PDE systems. In this method, in order to find solutions it is necessary to solve systems of ordinary implicit equations. In this case, it can be shown that the solutions that we need cannot depend on a continuous parameter, i.e. they are no more than countable. In advance, there is a need for such an overriding of the systems of differential equations, so that their general solutions are no more than countable. Such an initial overdetermination is rather difficult to achieve. However, the proposed method also allows to reduce the overdetermined systems of differential equations not only up to systems of implicit equations, but also up to the PDE systems of dimension less than that of the initial systems of PDE. In particular, under certain conditions, reduction to the ODE systems is possible. It is proposed to choose solutions for the overdetermined PDE systems using the parameterized Cauchy problem, which is posed for parameterized ODE systems under certain conditions. The solution of this Cauchy problem is some function of the initial data and their derivatives. In order to find the solution of any corresponding Cauchy problem for the initial system of PDE, it is sufficient to calculate the universal solver for the reduced ODE system once. In this case, the solution will not only exist and be unique, but will also depend continuously on the initial data, since this holds for ODE systems. The purpose of this paper is to study the Cauchy problem with the possibility of its universalization and the parameterized Cauchy problem as a whole for arbitrary PDE systems.

1 - INTRODUCTION

Revealing regions of multiple steady states in heterogeneous catalytic chemical reaction networks using Gröbner basis

Solving multivariate ordinary differential equations (ODE) systems and partial differential equations (PDE) systems is the key to many complex physics and chemistry problems, such as the combustion in process of reacting flow. However, the traditional numerical methods in solving multivariate ODE and PDE systems are limited by computational cost, and sometimes its impossible to obtain the solution due to the high stiffness of ODE or PDE. Coincident with the development of machine learning has been a growing appreciation of applying neural networks in solving physics models. DeepM&M net was proposed to address complicated problems in fluid mechanics based on another neural network: DeepONet, which is used to predict functional nonlinear operators. Inspired by these two nets, a machine learning way of solving certain ODE and PDE systems is proposed with a similar framework to the DeepM&M net, which takes inputs of the initial conditions and outputs the corresponding solutions. The main ideas of this framework are first to explore the relations among solutions of the system by DeepONets and then to train a deep neural network with the assistance of trained DeepONets. The implicit operators between variables in certain ODE systems are verified to have existed and are well predicted by the DeepONet. The feasibility of the proposed framework is implied by the success in building blocks.

Symbolic analysis of multiple steady states in a MAPK chemical reaction network

We consider estimation of parameters in models defined by systems of ordinary differential equations (ODEs). This problem is important because many processes in different fields of science are modelled by systems of ODEs. Various estimation methods based on smoothing have been suggested to bypass numerical integration of the ODE system. In this paper, we do not propose another method based on smoothing but show how some of the existing ones can be brought together under one unifying framework. The framework is based on generalized Tikhonov regularization and extremum estimation. We define an approximation of the ODE solution by viewing the system of ODEs as an operator equation and exploiting the connection with regularization theory. Combining the introduced regularized solution with an extremum criterion function provides a general framework for estimating parameters in ODEs, which can handle partially observed systems. If the extremum criterion function is the negative log‐likelihood, then suitable regularized solutions yield estimators that are consistent and asymptotically efficient. The well‐known generalized profiling procedure fits into the proposed framework. Copyright © 2016 John Wiley & Sons, Ltd.

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones.

Chapter 11 - Applications of Systems of Ordinary Differential Equations

A perturbation-based estimate algorithm for parameters of coupled ordinary differential equations, applications from chemical reactions to metabolic dynamics

In this paper we study the periodic problem with a period equal to 1 for a two-dimensional system of second-order ordinary differential equations, in which the main nonlinear part is generated by a polynomial in one complex variable. It is proven that if the convex hull of the roots of the generating polynomial does not contain numbers that are multiples of 2 i*pi, then there is an a priori estimate for solutions to the periodic problem. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, the solvability of the periodic problem for any perturbation from a given class is proven. The system of equations under consideration does not reduce to a similar system of first-order equations with the main positive homogeneous nonlinear part. For systems of first-order equations, the periodic problem was studied in the works of V.A. Pliss, M.A. Krasnoselskii and their followers using methods of a priori estimation and calculation of the mapping degree of vector fields. It is known that an a priori estimate of solutions to boundary value problems for systems of nonlinear ordinary second order differential equations is fraught with difficulties associated with an estimate of the first-order derivative of the solution when the solution itself is bounded. In this paper, using the example of a periodic problem for the considered system of second-order equations, it is established that the a priori estimate is deducible if we combine methods for studying similar systems of first-order equations and methods for qualitative research of singularly perturbed systems of equations. The results obtained can be further generalized for multidimensional systems of second-order equations, applying the idea of the directing function method.

Nonlinear Model Predictive Control for an Exothermic Reaction in an Adiabatic CSTR

In this chapter, we provide a method for solving systems of linear ordinary differential equations by using techniques associated with the calculation of eigenvalues, eigenvectors and generalized eigenvectors of matrices. We learn in calculus how to solve differential equations and the system of differential equations. Here, we firstly show how to represent a system of differential equations in a matrix formulation. Then, using the Jordan canonical form and, whenever possible, the diagonal canonical form of matrices, we will describe a process aimed at solving systems of linear differential equations in a very efficient way.

We develop a Bayesian approach to estimate the parameters of ordinary differential equations (ODE) from the observed noisy data. Our method does not need to solve ODE directly. We replace the ODE constraint with a probability expression and combine it with the nonparametric data fitting procedure into a joint likelihood framework. One advantage of the proposed method is that for some ODE systems, one can obtain closed form conditional posterior distributions for all variables which substantially reduce the computational cost and facilitate the convergence process. An efficient Riemann manifold based hybrid Monte Carlo scheme is implemented to generate samples for variables whose conditional posterior distribution cannot be written in terms of closed form. Our approach can be applied to situations where the state variables are only partially observed. The usefulness of the proposed method is demonstrated through applications to both simulated and real data.

The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.