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Register a new userThe linear-noise approximation and the chemical master equation exactly agree up to second-order moments for a class of chemical systems
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Ramon Grima
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It is well known that the linear-noise approximation (LNA) exactly agrees with the chemical master equation, up to second-order moments, for chemical systems composed of zero and first-order reactions. Here we show that this is also a property of the LNA for a subset of chemical systems with second-order reactions. This agreement is independent of the number of interacting molecules.
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Coagulation and fragmentation (CF) is a fundamental process by which particles attach to each other to form clusters while existing clusters break up into smaller ones. It is a ubiquitous process that plays a key role in many physical and biological phenomena. CF is typically a stochastic process that often occurs in confined spaces with a limited number of available particles. In this study, we use the discrete Chemical Master Equation (dCME) to describe the CF process. Using the newly developed Accurate Chemical Master Equation (ACME) method, we calculate the time-dependent behavior of the CF system. We investigate the effects of a number important factors that influence the overall behavior of the system, including the dimensionality, the ratio of attachment to detachment rates among clusters, and the initial conditions. By comparing CF in one and three dimensions we conclude that systems in higher dimensions are more likely to form large clusters. We also demonstrate how the ratio of the attachment to detachment rates affect the dynamics and the steady-state of the system. Finally, we demonstrate the relationship between the formation of large clusters and the initial condition.
This is a short review of two common approximations in stochastic chemical and biochemical kinetics. It will appear as Chapter 6 in the book Quantitative Biology: Theory, Computational Methods and Examples of Models edited by Brian Munsky, Lev Tsimring and Bill Hlavacek (to be published in late 2017/2018 by MIT Press). All chapter references in this article refer to chapters in the aforementioned book.
We give a first principles derivation of the stochastic partial differential equations that describe the chemical reactions of the Gray-Scott model (GS): $U+2V {stackrel {lambda}{rightarrow}} 3 V;$ and $V {stackrel {mu}{rightarrow}} P$, $U {stackrel { u}{rightarrow}} Q$, with a constant feed rate for $U$. We find that the conservation of probability ensured by the chemical master equation leads to a modification of the usual differential equations for the GS model which now involves two composite fields and also intrinsic noise terms. One of the composites is $psi_1 = phi_v^2$, where $ < phi_v >_{eta} = v$ is the concentration of the species $V$ and the averaging is over the internal noise $eta_{u,v,psi_1}$. The second composite field is the product of three fields $ chi = lambda phi_u phi_v^2$ and requires a noise source to ensure probability conservation. A third composite $psi_2 = phi_{u} phi_{v}$ can be also be identified from the noise-induced reactions. The Hamiltonian that governs the time evolution of the many-body wave function, associated with the master equation, has a broken U(1) symmetry related to particle number conservation. By expanding around the (broken symmetry) zero energy solution of the Hamiltonian (by performing a Doi shift) one obtains from our path integral formulation the usual reaction diffusion equation, at the classical level. The Langevin equations that are derived from the chemical master equation have multiplicative noise sources for the density fields $phi_u, phi_v, chi$ that induce higher order processes such as $n rightarrow n$ scattering for $n > 3$. The amplitude of the noise acting on $ phi_v$ is itself stochastic in nature.
The stochastic nature of chemical reactions involving randomly fluctuating population sizes has lead to a growing research interest in discrete-state stochastic models and their analysis. A widely-used approach is the description of the temporal evolution of the system in terms of a chemical master equation (CME). In this paper we study two approaches for approximating the underlying probability distributions of the CME. The first approach is based on an integration of the statistical moments and the reconstruction of the distribution based on the maximum entropy principle. The second approach relies on an analytical approximation of the probability distribution of the CME using the system size expansion, considering higher-order terms than the linear noise approximation. We consider gene expression networks with unimodal and multimodal protein distributions to compare the accuracy of the two approaches. We find that both methods provide accurate approximations to the distributions of the CME while having different benefits and limitations in applications.
We use the Percus-Yevick approach in the chemical-potential route to evaluate the equation of state of hard hyperspheres in five dimensions. The evaluation requires the derivation of an analytical expression for the contact value of the pair distribution function between particles of the bulk fluid and a solute particle with arbitrary size. The equation of state is compared with those obtained from the conventional virial and compressibility thermodynamic routes and the associated virial coefficients are computed. The pressure calculated from all routes is exact up to third density order, but it deviates with respect to simulation data as density increases, the compressibility and the chemical-potential routes exhibiting smaller deviations than the virial route. Accurate linear interpolations between the compressibility route and either the chemical-potential route or the virial one are constructed.
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