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    Nonlinear Differential Equations of Chemically Reacting Systems - Semantic Scholar

    The sheer number allows us to describe this inherently stochastic process deterministically. It can be represented as a set of two uni-directional reactions f orward and b ackward :. The rate of the second reaction the backward process is in analogy proportional to the square of the concentration of NOBr. We will now integrate this system of ordinary differential equations numerically as an initial value problem IVP using the odeint solver provided by scipy :.

    The expected signature of this function is:.

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    • Writing the rhs function by hand for larger reaction systems quickly becomes tedious. Ideally we would like to construct it from a symbolic representation having a symbolic representation of the problem opens up many possibilities as we will soon see. It represents the cumulative effect of external stimuli on the chemical species and has been used to discover an input amplification phenomenon in the MAPK pathway [4] , and to study the activation of cell membrane receptor such as the epidermal growth factor and the erythropoietin receptors [5] , [6].

      We focus on networks of biochemical reactions subject to molecular diffusion and spatiotemporal stimuli. We aim at obtaining exact formulae for the time-integrals of species concentrations. An analytic approach can reveal structural properties of the model under consideration, as opposed to simulation-based studies where it is unclear if predictions are rather a consequence of the particular parameter values examined.

      In the case of diffusionless systems, the work in [7] provided exact expressions for the norm of a class of signalling cascades.

      Oscillation theorems for second order nonlinear forced differential equations

      However, similar results for reaction-diffusion systems remain elusive, owing to the fact that the vast majority of nonlinear reaction-diffusion systems are analytically intractable. A complete solution to this problem, for any reaction-diffusion network, may require analytic solutions of the reaction-diffusion partial differential equation PDE.

      We have previously identified a class of nonlinear networks in which the time-integrals of some species can be computed as a series [8]. Here we build on these results and show that in this class the time-integrals satisfy a linear inhomogeneous differential equation. Solving the derived equation leads to analytic expressions for the time-integrals without knowing the solution of the nonlinear PDE. We further provide a graphical characterization of the class of networks in terms of the Species-Reaction graph [9].

      This provides a simple test to determine if a given network belongs to the derived class and to explore other network topologies that are amenable to our analysis. We consider networks composed of species interacting through reactions: 1 where is the reactant or product for the reaction.

      The numbers and denote the stoichiometric coefficients of the corresponding species. The nonlinear vector function contains the reaction rates, whereas the matrices with , and describe the stoichiometry, diffusion coefficients, and which species are subject to external stimuli. We focus on the response of the reaction-diffusion network to an initial spatial perturbation and a transient spatiotemporal stimulus such that and.

      For simplicity here we will focus on a 1D domain with the same boundary conditions for all species. Once the effect of the stimulus has vanished, we assume that the network reaches a unique homogeneous equilibrium. One way of quantifying the network response is by means of the time-integral: 3 which is finite provided that the equilibrium is exponentially stable. We relabel and partition the species and reaction rate vectors as follows:.

      The affine reaction rates contain a combination of zeroth and first order terms of the form , with a vector of constant production rates and is a matrix of first order kinetic constants. The nonlinear rates typically model saturable binding kinetics such as Michaelis-Menten or Hill kinetics [10] , together with linear dissociation note that in our notation reversible reactions are taken as a single rate. If we can find a labelling for the reaction rates so that the stoichiometric matrix has a block-triangular form 4 with , and. C1 the species in do not diffuse, and.

      C2 the number of species in is equal to the number of nonlinear reactions i. The solution of 5 must satisfy boundary conditions consistent with those of the reaction-diffusion PDE. Equation 5 is an inhomogeneous linear differential equation with constant coefficients, and therefore depending on the spatial profile of the stimuli and initial condition , it may be possible to obtain a closed-form solution for the time-integral. In the general case when a closed-form solution is not available, equation 5 can be solved by projecting the solution on an orthonormal basis for the spatial domain.

      To this end, we write and , where is a complete orthonormal basis of. We choose the basis as orthonormal eigenfunctions of the Laplacian, i.


      The time-integrals are then with coefficients 6 and. The derived series is exact and we can use it to compute the time-integrals of without knowing the solution of the nonlinear PDE. Most importantly, the series coefficients are explicitly given in terms of the geometry and boundary conditions comprised in the eigenvalues , the initial condition and the equilibrium comprised in the function , and the total concentration supplied to and consumed from the network comprised in the integral of.

      Note that these coefficients can also be obtained by linearizing the PDE in 2 , but such an approach provides no guarantee of the exactness of the solution. Linearized solutions neglect the nonlinear terms in the PDE, and therefore they are valid only for small perturbations around the equilibrium.

      In our case, conditions C1 and C2 guarantee that the derived time-integral is exact, defining a class of nonlinear networks for which the time-integral can be computed analytically for small or large perturbations. Conditions C1 and C2 are structural hence independent of the functional form of the nonlinearities and can be interpreted in terms of a graph.

      We use the Species-Reaction graph Fig. The graph is bipartite—so that reaction nodes only link to species nodes, and vice versa —and is defined as follows: an S-node is connected to an R-node if depends on i. As shown in Fig. With these definitions, conditions C1 and C2 amount to:. The conditions amount to the graph having a possibly disjoint subgraph containing every red R-node with all their adjacent red S-nodes not linked with any black R-nodes; in this subgraph, the number of red R-nodes and red S-nodes must be the same.

      A Networks with two nonlinear reactions satisfying the conditions. The red subgraphs are marked with dashed boxes. B Networks that do not satisfy the conditions. C A generic complex-formation mechanism satisfying the conditions. A spatially-fixed molecule binds a diffusible ligand to form a complex. Species and are synthesized at a constant rate and linearly degraded.

      External stimuli of ligand can be modeled via a spatiotemporal influx. The Species-Reaction graph is shown in the inset. D Genetic regulation via protein sequestration [12] is an instance of the mechanism in C.

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      A nondiffusive inhibitor sequesters a transcriptional activator to form an inactive complex , causing the downregulation of gene expression. As illustrated by the examples in Fig. Under these conditions we can use our formula to compute the time-integrals of all the species outside the subgraph.