scholarly journals CLUE: exact maximal reduction of kinetic models by constrained lumping of differential equations

2021 ◽  
Author(s):  
Alexey Ovchinnikov ◽  
Isabel Pérez Verona ◽  
Gleb Pogudin ◽  
Mirco Tribastone
Keyword(s):  
Differential Equations ◽  
Kinetic Models ◽  
Maximal Reduction

scholarly journals CLUE: Exact maximal reduction of kinetic models by constrained lumping of differential equations

2021 ◽  
Author(s):  
Alexey Ovchinnikov ◽  
Isabel Pérez Verona ◽  
Gleb Pogudin ◽  
Mirco Tribastone
Keyword(s):  
Differential Equations ◽  
Model Reduction ◽  
Nonlinear Differential Equations ◽  
Linear Mapping ◽  
Supplementary Information ◽  
Biological Processes ◽  
Linear Combinations ◽  
Parameter Estimations ◽  
Found Model ◽  
Maximal Reduction

Abstract Motivation Detailed mechanistic models of biological processes can pose significant challenges for analysis and parameter estimations due to the large number of equations used to track the dynamics of all distinct configurations in which each involved biochemical species can be found. Model reduction can help tame such complexity by providing a lower-dimensional model in which each macro-variable can be directly related to the original variables. Results We present CLUE, an algorithm for exact model reduction of systems of polynomial differential equations by constrained linear lumping. It computes the smallest dimensional reduction as a linear mapping of the state space such that the reduced model preserves the dynamics of user-specified linear combinations of the original variables. Even though CLUE works with nonlinear differential equations, it is based on linear algebra tools, which makes it applicable to high-dimensional models. Using case studies from the literature, we show how CLUE can substantially lower model dimensionality and help extract biologically intelligible insights from the reduction. Availability An implementation of the algorithm and relevant resources to replicate the experiments herein reported are freely available for download at https://github.com/pogudingleb/CLUE. Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals Diffusive and Anti-Diffusive Behavior for Kinetic Models of Opinion Dynamics

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1024 ◽  
Author(s):  
Mirosław Lachowicz ◽  
Henryk Leszczyński ◽  
Elżbieta Puźniakowska–Gałuch
Keyword(s):  
Differential Equations ◽  
Kinetic Models ◽  
Essential Role ◽  
Opinion Dynamics ◽  
Individual State ◽  
Mesoscopic Scale ◽  
Macroscopic Scale ◽  
Rigorous Result ◽  
Numerical Examples ◽  
Political Sciences

In the present paper, we study a class of nonlinear integro-differential equations of a kinetic type describing the dynamics of opinion for two types of societies: conformist ( σ = 1 ) and anti-conformist ( σ = - 1 ). The essential role is played by the symmetric nature of interactions. The class may be related to the mesoscopic scale of description. This means that we are going to statistically describe an individual state of an agent of the system. We show that the corresponding equations result at the macroscopic scale in two different pictures: anti-diffusive ( σ = 1 ) and diffusive ( σ = - 1 ). We provide a rigorous result on the convergence. The result captures the macroscopic behavior resulting from the mesoscopic one. In numerical examples, we observe both unipolar and bipolar behavior known in political sciences.

FROM MICROSCOPIC TO MACROSCOPIC DESCRIPTION FOR GENERALIZED KINETIC MODELS

2002 ◽  
Vol 12 (07) ◽  
pp. 985-1005 ◽  
Author(s):  
MIROSŁAW LACHOWICZ
Keyword(s):  
Differential Equations ◽  
Kinetic Models ◽  
General Class ◽  
Hydrodynamic Limit ◽  
Bilinear Systems ◽  
Macroscopic Description ◽  
Research Perspectives ◽  
Diffusive Limit

In this paper a review of some results and research perspectives for the general class of bilinear systems of Boltzmann-like integro-differential equations (generalized kinetic models) describing the dynamics of individuals undergoing kinetic (stochastic) interactions is presented. Some macroscopic limits (the diffusive limit and the hydrodynamic limit) are discussed.

scholarly journals Coupling kinetic models and advection-diffusion equations. 1. Framework development and application to sucrose translocation and metabolism in sugarcane

2021 ◽  
Author(s):  
Lafras Uys ◽  
Jan-Hendrik S Hofmeyr ◽  
Johann M Rohwer
Keyword(s):  
Differential Equations ◽  
Kinetic Models ◽  
Numerical Solutions ◽  
Spatial Information ◽  
Initial Conditions ◽  
Reaction Rates ◽  
Diffusion Reaction ◽  
Sucrose Accumulation ◽  
Fluid Medium ◽  
Advection Diffusion

Abstract The sugarcane stalk, besides being the main structural component of the plant, is also the major storage organ for carbohydrates. Previous studies have modelled the sucrose accumulation pathway in the internodal storage parenchyma of sugarcane using kinetic models cast as systems of ordinary differential equations. To address the shortcomings of these models, which did not include subcellular compartmentation or spatial information, the present study extends the original models within an advection-diffusion-reaction framework, requiring the use of partial differential equations to model sucrose metabolism coupled to phloem translocation.We propose a kinetic model of a coupled reaction network where species can be involved in chemical reactions and/or be transported over long distances in a fluid medium by advection or diffusion. Darcy’s law is used to model fluid flow and allows a simplified, phenomenological approach to be applied to translocation in the phloem. Similarly, generic reversible Hill equations are used to model biochemical reaction rates. Numerical solutions to this formulation are demonstrated with time-course analysis of a simplified model of sucrose accumulation. The model shows sucrose accumulation in the vacuoles of stalk parenchyma cells, and is moreover able to demonstrate the up-regulation of photosynthesis in response to a change in sink demand. The model presented is able to capture the spatio-temporal evolution of the system from a set of initial conditions by combining phloem flow, diffusion, transport of metabolites between compartments and biochemical enzyme-catalysed reactions in a rigorous, quantitative framework that can form the basis for future modelling and experimental design.

scholarly journals CHAOTIC OSCILLATIONS IN SIMPLEST CHEMICAL REACTION

2018 ◽  
Vol 61 (4-5) ◽  
pp. 133
Author(s):  
Nikolay I. Kol'tsov
Keyword(s):  
Kinetic Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Dynamic Model ◽  
Lyapunov Exponents ◽  
Chemical Reactions ◽  
Chemical Reaction ◽  
Kinetic Models ◽  
Intermediate Stage ◽  
Chaotic Oscillations

It is known that chaotic oscillations for chemical reactions can be described by non-stationary kinetic models consisting of three ordinary differential equations.  Rossler established the first examples of chemical reactions, including the two-route five-stage reaction of the Villamovski-Rossler, with three intermediate substances, containing three autocatalytic on intermediates stages, the dynamic model of which describes chaotic oscillations. In given article presents a simple one-route four-stages reaction A+E=D involving two autocatalytic and one linear on intermediate stage, the non-stationary kinetic model of which describes chaotic oscillations. The non-stationary kinetic model under the assumption of quasistationarity with respect to the main substances within the framework of the law of acting masses is a system of three ordinary differential equations. The presence of chaos is confirmed by numerical calculations of the kinetic model and Lyapunov exponentials. The Lyapunov exponents satisfy the condition L1+L2+L3<0, which proves the existence of chaotic oscillations.Forcitation:Kol'tsov N.I. Chaotic oscillations in simplest chemical reaction. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 2018. V. 61. N 4-5. P. 133-135

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