12,551 research outputs found
Robust Moment Closure Method for the Chemical Master Equation
The Chemical Master Equation (CME) is used to stochastically model
biochemical reaction networks, under the Markovian assumption. The low-order
statistical moments induced by the CME are often the key quantities that one is
interested in. However, in most cases, the moments equation is not closed; in
the sense that the first moments depend on the higher order moments, for
any positive integer . In this paper, we develop a moment closure technique
in which the higher order moments are approximated by an affine function of the
lower order moments. We refer to such functions as the affine Moment Closure
Functions (MCF) and prove that they are optimal in the worst-case context, in
which no a priori information on the probability distribution is available.
Furthermore, we cast the problem of finding the optimal affine MCF as a linear
program, which is tractable. We utilize the affine MCFs to derive a finite
dimensional linear system that approximates the low-order moments. We quantify
the approximation error in terms of the induced norm of some
linear system. Our results can be effectively used to approximate the low-order
moments and characterize the noise properties of the biochemical network under
study
Stochastic focusing coupled with negative feedback enables robust regulation in biochemical reaction networks
Nature presents multiple intriguing examples of processes which proceed at
high precision and regularity. This remarkable stability is frequently counter
to modelers' experience with the inherent stochasticity of chemical reactions
in the regime of low copy numbers. Moreover, the effects of noise and
nonlinearities can lead to "counter-intuitive" behavior, as demonstrated for a
basic enzymatic reaction scheme that can display stochastic focusing (SF).
Under the assumption of rapid signal fluctuations, SF has been shown to convert
a graded response into a threshold mechanism, thus attenuating the detrimental
effects of signal noise. However, when the rapid fluctuation assumption is
violated, this gain in sensitivity is generally obtained at the cost of very
large product variance, and this unpredictable behavior may be one possible
explanation of why, more than a decade after its introduction, SF has still not
been observed in real biochemical systems.
In this work we explore the noise properties of a simple enzymatic reaction
mechanism with a small and fluctuating number of active enzymes that behaves as
a high-gain, noisy amplifier due to SF caused by slow enzyme fluctuations. We
then show that the inclusion of a plausible negative feedback mechanism turns
the system from a noisy signal detector to a strong homeostatic mechanism by
exchanging high gain with strong attenuation in output noise and robustness to
parameter variations. Moreover, we observe that the discrepancy between
deterministic and stochastic descriptions of stochastically focused systems in
the evolution of the means almost completely disappears, despite very low
molecule counts and the additional nonlinearity due to feedback.
The reaction mechanism considered here can provide a possible resolution to
the apparent conflict between intrinsic noise and high precision in critical
intracellular processes
Markovian Dynamics on Complex Reaction Networks
Complex networks, comprised of individual elements that interact with each
other through reaction channels, are ubiquitous across many scientific and
engineering disciplines. Examples include biochemical, pharmacokinetic,
epidemiological, ecological, social, neural, and multi-agent networks. A common
approach to modeling such networks is by a master equation that governs the
dynamic evolution of the joint probability mass function of the underling
population process and naturally leads to Markovian dynamics for such process.
Due however to the nonlinear nature of most reactions, the computation and
analysis of the resulting stochastic population dynamics is a difficult task.
This review article provides a coherent and comprehensive coverage of recently
developed approaches and methods to tackle this problem. After reviewing a
general framework for modeling Markovian reaction networks and giving specific
examples, the authors present numerical and computational techniques capable of
evaluating or approximating the solution of the master equation, discuss a
recently developed approach for studying the stationary behavior of Markovian
reaction networks using a potential energy landscape perspective, and provide
an introduction to the emerging theory of thermodynamic analysis of such
networks. Three representative problems of opinion formation, transcription
regulation, and neural network dynamics are used as illustrative examples.Comment: 52 pages, 11 figures, for freely available MATLAB software, see
http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.htm
Model reduction for stochastic CaMKII reaction kinetics in synapses by graph-constrained correlation dynamics.
A stochastic reaction network model of Ca(2+) dynamics in synapses (Pepke et al PLoS Comput. Biol. 6 e1000675) is expressed and simulated using rule-based reaction modeling notation in dynamical grammars and in MCell. The model tracks the response of calmodulin and CaMKII to calcium influx in synapses. Data from numerically intensive simulations is used to train a reduced model that, out of sample, correctly predicts the evolution of interaction parameters characterizing the instantaneous probability distribution over molecular states in the much larger fine-scale models. The novel model reduction method, 'graph-constrained correlation dynamics', requires a graph of plausible state variables and interactions as input. It parametrically optimizes a set of constant coefficients appearing in differential equations governing the time-varying interaction parameters that determine all correlations between variables in the reduced model at any time slice
The auxiliary region method: A hybrid method for coupling PDE- and Brownian-based dynamics for reaction-diffusion systems
Reaction-diffusion systems are used to represent many biological and physical
phenomena. They model the random motion of particles (diffusion) and
interactions between them (reactions). Such systems can be modelled at multiple
scales with varying degrees of accuracy and computational efficiency. When
representing genuinely multiscale phenomena, fine-scale models can be
prohibitively expensive, whereas coarser models, although cheaper, often lack
sufficient detail to accurately represent the phenomenon at hand. Spatial
hybrid methods couple two or more of these representations in order to improve
efficiency without compromising accuracy.
In this paper, we present a novel spatial hybrid method, which we call the
auxiliary region method (ARM), which couples PDE and Brownian-based
representations of reaction-diffusion systems. Numerical PDE solutions on one
side of an interface are coupled to Brownian-based dynamics on the other side
using compartment-based "auxiliary regions". We demonstrate that the hybrid
method is able to simulate reaction-diffusion dynamics for a number of
different test problems with high accuracy. Further, we undertake error
analysis on the ARM which demonstrates that it is robust to changes in the free
parameters in the model, where previous coupling algorithms are not. In
particular, we envisage that the method will be applicable for a wide range of
spatial multi-scales problems including, filopodial dynamics, intracellular
signalling, embryogenesis and travelling wave phenomena.Comment: 29 pages, 14 figures, 2 table
Circular Stochastic Fluctuations in SIS Epidemics with Heterogeneous Contacts Among Sub-populations
The conceptual difference between equilibrium and non-equilibrium steady
state (NESS) is well established in physics and chemistry. This distinction,
however, is not widely appreciated in dynamical descriptions of biological
populations in terms of differential equations in which fixed point, steady
state, and equilibrium are all synonymous. We study NESS in a stochastic SIS
(susceptible-infectious-susceptible) system with heterogeneous individuals in
their contact behavior represented in terms of subgroups. In the infinite
population limit, the stochastic dynamics yields a system of deterministic
evolution equations for population densities; and for very large but finite
system a diffusion process is obtained. We report the emergence of a circular
dynamics in the diffusion process, with an intrinsic frequency, near the
endemic steady state. The endemic steady state is represented by a stable node
in the deterministic dynamics; As a NESS phenomenon, the circular motion is
caused by the intrinsic heterogeneity within the subgroups, leading to a broken
symmetry and time irreversibility.Comment: 29 pages, 5 figure
Moment Closure - A Brief Review
Moment closure methods appear in myriad scientific disciplines in the
modelling of complex systems. The goal is to achieve a closed form of a large,
usually even infinite, set of coupled differential (or difference) equations.
Each equation describes the evolution of one "moment", a suitable
coarse-grained quantity computable from the full state space. If the system is
too large for analytical and/or numerical methods, then one aims to reduce it
by finding a moment closure relation expressing "higher-order moments" in terms
of "lower-order moments". In this brief review, we focus on highlighting how
moment closure methods occur in different contexts. We also conjecture via a
geometric explanation why it has been difficult to rigorously justify many
moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in
mathematics, physics, chemistry and quantitative biolog
Uncoupled Analysis of Stochastic Reaction Networks in Fluctuating Environments
The dynamics of stochastic reaction networks within cells are inevitably
modulated by factors considered extrinsic to the network such as for instance
the fluctuations in ribsome copy numbers for a gene regulatory network. While
several recent studies demonstrate the importance of accounting for such
extrinsic components, the resulting models are typically hard to analyze. In
this work we develop a general mathematical framework that allows to uncouple
the network from its dynamic environment by incorporating only the
environment's effect onto the network into a new model. More technically, we
show how such fluctuating extrinsic components (e.g., chemical species) can be
marginalized in order to obtain this decoupled model. We derive its
corresponding process- and master equations and show how stochastic simulations
can be performed. Using several case studies, we demonstrate the significance
of the approach. For instance, we exemplarily formulate and solve a marginal
master equation describing the protein translation and degradation in a
fluctuating environment.Comment: 7 pages, 4 figures, Appendix attached as SI.pdf, under submissio
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