Linear Reaction Networks Research Articles

Variance of filtered signals: Characterization for linear reaction networks and application to neurotransmission dynamics

Neurotransmission at chemical synapses relies on the calcium-induced fusion of synaptic vesicles with the presynaptic membrane. The distance of the synaptic vesicle to the calcium channels determines the release probability and consequently the postsynaptic signal. Suitable models of the process need to capture both the mean and the variance observed in electrophysiological measurements of the postsynaptic current. In this work, we propose a method to directly compute the exact first- and second-order moments for signals generated by a linear reaction network under convolution with an impulse response function, rendering computationally expensive numerical simulations of the underlying stochastic counting process obsolete. We show that the autocorrelation of the process is central for the calculation of the filtered signal’s second-order moments, and derive a system of PDEs for the cross-correlation functions (including the autocorrelations) of linear reaction networks with time-dependent rates. Finally, we employ our method to efficiently compare different spatial coarse graining approaches for a specific model of synaptic vesicle fusion. Beyond the application to neurotransmission processes, the developed theory can be applied to any linear reaction system that produces a filtered stochastic signal.

Hypocoercivity and Fast Reaction Limit for Linear Reaction Networks with Kinetic Transport

The long time behavior of a model for a first order, weakly reversible chemical reaction network is considered, where the movement of the reacting species is described by kinetic transport. The reactions are triggered by collisions with a nonmoving background with constant temperature, determining the post-reactional equilibrium velocity distributions. Species with different particle masses are considered, with a strong separation between two groups of light and heavy particles. As an approximation, the heavy species are modeled as nonmoving. Under the assumption of at least one moving species, long time convergence is proven by hypocoercivity methods for the cases of positions in a flat torus and in whole space. In the former case the result is exponential convergence to a spatially constant equilibrium, and in the latter it is algebraic decay to zero, at the same rate as solutions of parabolic equations. This is no surprise since it is also shown that the macroscopic (or reaction dominated) behavior is governed by the diffusion equation.

Diffusion-dynamics laws in stochastic reaction networks.

Many biological activities are induced by cellular chemical reactions of diffusing reactants. The dynamics of such systems can be captured by stochastic reaction networks. A recent numerical study has shown that diffusion can significantly enhance the fluctuations in gene regulatory networks. However, the universal relation between diffusion and stochastic system dynamics remains veiled. Within the approximation of reaction-diffusion master equation (RDME), we find general relation that the steady-state distribution in complex balanced networks is diffusion-independent. Here, complex balance is the nonequilibrium generalization of detailed balance. We also find that for a diffusion-included network with a Poisson-like steady-state distribution, the diffusion can be ignored at steady state. We then derive a necessary and sufficient condition for networks holding such steady-state distributions. Moreover, we show that for linear reaction networks the RDME reduces to the chemical master equation, which implies that the stochastic dynamics of networks is unaffected by diffusion at any arbitrary time. Our findings shed light on the fundamental question of when diffusion can be neglected, or (if nonnegligible) its effects on the stochastic dynamics of the reaction network.

The Spatial Chemical Langevin Equation and Reaction Diffusion Master Equations: moments and qualitative solutions.

BackgroundIt has been established that stochastic effects play an important role in spatio-temporal biochemical networks. A popular method of representing such stochastic systems is the Reaction Diffusion Master Equation (RDME). However, simulating sample paths from the RDME can be computationally expensive, particularly at large populations. Here we investigate an uncommon, but much faster alternative: the Spatial Chemical Langevin Equation (SCLE).MethodsWe investigate moment equations and correlation functions analytically, then we compare sample paths and moments of the SCLE to the RDME and associated deterministic solutions. Sample paths are generated computationally by the Next Subvolume method (RDME) and the Euler-Maruyama method (SCLE), while a deterministic solution is obtained with an Euler method. We consider the Gray-Scott model, a well-known pattern generating system, and a predator–prey system with spatially inhomogeneous parameters as sample applications.ResultsFor linear reaction networks, it is well known that the first order moments of all three approaches match, that the RDME and SCLE match to the second moment, and that all approaches diverge at third order moments. For non-linear reaction networks, differential equations governing moments do not form a closed system, but a general moment equation can be compared term wise. All approaches match at the leading order, and the RDME and SCLE match at the second leading order. As expected, the SCLE captures many dynamics of the RDME where deterministic methods fail to represent them. However, areas of the parameter space in the Gray-Scott model exist where either the SCLE and RDME give qualitatively different predictions, or the RDME predicts patterns, while the SCLE does not.ConclusionsThe SCLE provides a fast alternative to existing methods for simulation of spatial stochastic biochemical networks, capturing many aspects of dynamics represented by the RDME. This becomes very useful in search of quantitative parameters yielding desired qualitative solutions. However, there exist parameter sets where both the qualitative and quantitative behaviour of the SCLE can differ when compared to the RDME, so care should be taken in its use for applications demanding greater accuracy.Electronic supplementary materialThe online version of this article (doi:10.1186/s12976-015-0001-6) contains supplementary material, which is available to authorized users.

Identifiability of rate coefficients in linear reaction networks from isothermal transient experimental data

For linear reaction networks, graph-theoretical methods are used to find two types of kinetic transfer functions in the Laplace domain for isothermal uniform open systems under transient regime. Rules are formulated to find the functional form of these transfer functions from properties of the reaction network, viz the number of components and the length of shortest paths. In many cases, some reaction intermediates are not observed. This is typical in heterogeneous catalysis, where the intermediates are bound to the catalytic surface. The mechanistic details are then manifested only through some transfer functions. This often causes the rate coefficients to be unidentifiable. A graph-theoretical rule is derived which allows assessing whether reaction networks are structurally locally identifiable. It is applied to the interaction of propane with a copper oxide/ceria-catalyst, as studied with a temporal analysis of products (TAP) setup.

An equation-free probabilistic steady-state approximation: Dynamic application to the stochastic simulation of biochemical reaction networks

Stochastic chemical kinetics more accurately describes the dynamics of "small" chemical systems, such as biological cells. Many real systems contain dynamical stiffness, which causes the exact stochastic simulation algorithm or other kinetic Monte Carlo methods to spend the majority of their time executing frequently occurring reaction events. Previous methods have successfully applied a type of probabilistic steady-state approximation by deriving an evolution equation, such as the chemical master equation, for the relaxed fast dynamics and using the solution of that equation to determine the slow dynamics. However, because the solution of the chemical master equation is limited to small, carefully selected, or linear reaction networks, an alternate equation-free method would be highly useful. We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. We demonstrate the accuracy and efficiency of the method using multiple interesting examples, including a highly nonlinear protein-protein interaction network. The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.

Fate and transport of species in a linear reaction network with different retardation coefficients

A response function approach is used to generate semi-analytical solutions to model the transport of species within a first-order reaction network. If the retardation coefficients of these species are not the same, product species will either be delayed or displaced forward relative to the reactants. A novel approach is used to simulate linear, kinetic response functions (KRFs) for species in plug flow, then to integrate with the conservative residence time density function ( E-curve) to yield a residence time density for the reacted species. This kinetic E-curve can be used with linear operator methods such as convolution to yield the semi-analytical solution for nonuniform inputs, nonideal mixing, and complex reaction networks. Closed-form analytical solutions are presented for the linear response functions for straight, three-member reaction networks in plug flow. Only irreversible reactions are considered. Several examples with various mixing conditions are shown and compared to other analytical solutions. Although the approach is related to other transfer function models, the kinetic residence time density is a transfer function that is generated directly from the plug-flow response functions and the numerical evaluation of the E-curve. The advantage is that a wide variety of mixing conditions and reaction networks may be considered without the need to generate analytical transfer functions for each species and mixing condition.

Kinetic investigation of complex pseudo mass action systems by use of the pulse method

AbstractWith simultaneous isomerization and cracking of the five hexane isomers on a bifunctional Pt/Al2 O3‐catalyst as a model reaction, which has been shown to follow pseudo mass action kinetics, it is demonstrated how the microcatalytic pulse technique can be applied for the kinetic analysis of complex linear reaction networks. From the experimental data relative rate constants for the individual reaction steps have been obtained by use of the Wei‐Prater‐method. The results are compared with results from continuous operation.