Semi-analytical solutions of the Schnakenberg model of a reaction-diffusion cell with feedback

Noufaey, K.S. Al

doi:10.1016/j.rinp.2018.03.017

Cited by 15 publications

(15 citation statements)

References 14 publications

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“…To test the multi-dimensional PDE simulation, and to verify the file interface system, we implemented the well-studied two species reaction system commonly known as the Schnakenberg system (Li et al, 2018;Schnakenberg, 1979) and shown in equations ( 5)-( 7). When these reactions are modelled with mass action kinetics they are shown to display temporal oscillations and diffusion driven spatial patterning for distinct, defined parameter regimes (Al Noufaey, 2018;. These phenomena were reproduced here using ChemChaste.…”

Section: Spatial Simulation Accuracy In Chemchaste: Fisher-kpp Equationmentioning

confidence: 78%

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ChemChaste: Simulating spatially inhomogenous biochemical reaction-diffusion systems for modelling cell-environment feedbacks

Johnson

Fletcher

Soyer

2021

Preprint

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MotivationSpatial organisation plays an important role in the function of many biological systems, from cell fate specification in animal development to multi-step metabolic conversions in microbial communities. The study of such systems benefits from the use of spatially explicit computational models that combine a discrete description of cells with a continuum description of one or more chemicals diffusing within a surrounding bulk medium. These models allow the in silico testing and refinement of mechanistic hypotheses. However, most existing models of this type do not account for concurrent bulk and intracellular biochemical reactions and their possible coupling.ResultsHere, we describe ChemChaste, an extension for the open-source C++ computational biology library Chaste. ChemChaste enables the spatial simulation of both multicellular and bulk biochemistry by expanding on Chaste’s existing capabilities. In particular, ChemChaste enables: (i) simulation of an arbitrary number of spatially diffusing chemicals; (ii) spatially heterogeneous chemical diffusion coefficients; and (iii) inclusion of both bulk and intracellular biochemical reactions and their coupling. ChemChaste also introduces a file-based interface that allows users to define the parameters relating to these functional features without the need to interact directly with Chaste’s core C++ code. We describe ChemChaste and demonstrate its functionality using a selection of chemical and biochemical exemplars, with a focus on demonstrating increased ability in modelling bulk chemical reactions and their coupling with intracellular reactions.Availability and implementationChemChaste is a free, open-source C++ library, available via GitHub at https://github.com/OSS-Lab/ChemChaste under the BSD license.ContactO.Soyer@warwick.ac.uk or a.g.fletcher@sheffield.ac.uk

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Section: Spatial Simulation Accuracy In Chemchaste: Fisher-kpp Equationmentioning

confidence: 78%

“…Two parameter sets were considered: one for temporal oscillations; and one for diffusion-driven patterning (Al Noufaey, 2018). Temporal oscillations are present when the homogeneous system, equations ( 8)-( 9), display limit cycle behaviour.…”

Section: Spatial Simulation Accuracy In Chemchaste: Fisher-kpp Equationmentioning

confidence: 99%

ChemChaste: Simulating spatially inhomogenous biochemical reaction-diffusion systems for modelling cell-environment feedbacks

Johnson

Fletcher

Soyer

2021

Preprint

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“…Finally, semi-analytical solutions of Schnakenberg’s equation, including limit cycles, can be viewed in reference to Noufaey [ 32 ], where Equation (5) is, or is not, verified to obtain patterns.…”

Section: Necessary Conditions For Generating Proto-tissues Structuresmentioning

confidence: 99%

Prebiotic Aggregates (Tissues) Emerging from Reaction–Diffusion: Formation Time, Configuration Entropy and Optimal Spatial Dimension

Flores

2022

Entropy

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For the formation of a proto-tissue, rather than a protocell, the use of reactant dynamics in a finite spatial region is considered. The framework is established on the basic concepts of replication, diversity, and heredity. Heredity, in the sense of the continuity of information and alike traits, is characterized by the number of equivalent patterns conferring viability against selection processes. In the case of structural parameters and the diffusion coefficient of ribonucleic acid, the formation time ranges between a few years to some decades, depending on the spatial dimension (fractional or not). As long as equivalent patterns exist, the configuration entropy of proto-tissues can be defined and used as a practical tool. Consequently, the maximal diversity and weak fluctuations, for which proto-tissues can develop, occur at the spatial dimension 2.5.

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“…where ϕ = ϕðx, tÞ and ψ = ψðx, tÞ represent the concentration and α and β are positive arbitrary constants and represent the concentration of A and B. D 1 and D 2 are the diffusion coefficients of the chemicals Φ and Ψ, for detail study (see [8][9][10][11]).…”

Section: Introductionmentioning

confidence: 99%

“…In such circumstances, fractional order derivatives are more reliable to describe these real word problems, (see [12][13][14][15][16][17][18][19]). In this regard, the proposed model has been studied by various researchers from both analytical and numerical points of view (see [11,20,21]).…”

Section: Introductionmentioning

confidence: 99%

Numerical Investigation of Chemical Schnakenberg Mathematical Model

Khan

Ali

Hamadneh

et al. 2021

Journal of Nanomaterials

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Schnakenberg model is known as one of the influential model used in several biological processes. The proposed model is an autocatalytic reaction in nature that arises in various biological models. In such kind of reactions, the rate of reaction speeds up as the reaction proceeds. It is because when a product itself acts as a catalyst. In fact, model endows fractional derivatives that got great advancement in the investigation of mathematical modeling with memory effect. Therefore, in the present paper, the authors develop a scheme for the solution of fractional order Schnakenberg model. The proposed model describes an auto chemical reaction with possible oscillatory behavior which may have several applications in biological and biochemical processes. In this work, the authors generalized the concept of integer order Schnakenberg model to fractional order Schnakenberg model. We provided the approximate solution for the underlying generalized nonlinear Schnakenberg model in the sense of Caputo differential operator via Laplace Adomian decomposition method (LADM). Furthermore, we established the general scheme for the considered model in the form of infinite series by the aforementioned technique. The consequent results obtained by the proposed technique ensure that LADM is an effective and accurate techniques to handle nonlinear partial differential equations as compared to the other available numerical techniques. Finally, the obtained numerical solution is visualized graphically by MATLAB to describe the dynamics of desired solution.

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Semi-analytical solutions of the Schnakenberg model of a reaction-diffusion cell with feedback

Cited by 15 publications

References 14 publications

ChemChaste: Simulating spatially inhomogenous biochemical reaction-diffusion systems for modelling cell-environment feedbacks

ChemChaste: Simulating spatially inhomogenous biochemical reaction-diffusion systems for modelling cell-environment feedbacks

Prebiotic Aggregates (Tissues) Emerging from Reaction–Diffusion: Formation Time, Configuration Entropy and Optimal Spatial Dimension

Numerical Investigation of Chemical Schnakenberg Mathematical Model

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