15,351 research outputs found
Jump-Diffusion Approximation of Stochastic Reaction Dynamics: Error bounds and Algorithms
Biochemical reactions can happen on different time scales and also the
abundance of species in these reactions can be very different from each other.
Classical approaches, such as deterministic or stochastic approach, fail to
account for or to exploit this multi-scale nature, respectively. In this paper,
we propose a jump-diffusion approximation for multi-scale Markov jump processes
that couples the two modeling approaches. An error bound of the proposed
approximation is derived and used to partition the reactions into fast and slow
sets, where the fast set is simulated by a stochastic differential equation and
the slow set is modeled by a discrete chain. The error bound leads to a very
efficient dynamic partitioning algorithm which has been implemented for several
multi-scale reaction systems. The gain in computational efficiency is
illustrated by a realistically sized model of a signal transduction cascade
coupled to a gene expression dynamics.Comment: 32 pages, 7 figure
Effective simulation techniques for biological systems
In this paper we give an overview of some very recent work on the stochastic simulation of systems involving chemical reactions. In many biological systems (such as genetic regulation and cellular dynamics) there is a mix between small numbers of key regulatory proteins, and medium and large numbers of molecules. In addition, it is important to be able to follow the trajectories of individual molecules by taking proper account of the randomness inherent in such a system. We describe different types of simulation techniques (including the stochastic simulation algorithm, Poisson Runge-Kutta methods and the Balanced Euler method) for treating simulations in the three different reaction regimes: slow, medium and fast. We then review some recent techniques on the treatment of coupled slow and fast reactions for stochastic chemical kinetics and discuss how novel computing implementations can enhance the performance of these simulations
Modeling and simulating chemical reactions
Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain large-scale limit of a sequence of finer-scale probabilistic models. In studying this hierarchy of models, students can be exposed to a range of modern ideas in applied and computational mathematics. This article introduces some of the basic concepts in an accessible manner and points to some challenges that currently occupy researchers in this area. Short, downloadable MATLAB codes are listed and described
Boolean network model predicts cell cycle sequence of fission yeast
A Boolean network model of the cell-cycle regulatory network of fission yeast
(Schizosaccharomyces Pombe) is constructed solely on the basis of the known
biochemical interaction topology. Simulating the model in the computer,
faithfully reproduces the known sequence of regulatory activity patterns along
the cell cycle of the living cell. Contrary to existing differential equation
models, no parameters enter the model except the structure of the regulatory
circuitry. The dynamical properties of the model indicate that the biological
dynamical sequence is robustly implemented in the regulatory network, with the
biological stationary state G1 corresponding to the dominant attractor in state
space, and with the biological regulatory sequence being a strongly attractive
trajectory. Comparing the fission yeast cell-cycle model to a similar model of
the corresponding network in S. cerevisiae, a remarkable difference in
circuitry, as well as dynamics is observed. While the latter operates in a
strongly damped mode, driven by external excitation, the S. pombe network
represents an auto-excited system with external damping.Comment: 10 pages, 3 figure
Transition from stochastic to deterministic behavior in calcium oscillations
Simulation and modeling is becoming more and more important when studying complex biochemical systems. Most often, ordinary differential equations are employed for this purpose. However, these are only applicable when the numbers of participating molecules in the biochemical systems are large enough to be treated as concentrations. For smaller systems, stochastic simulations on discrete particle basis are more accurate. Unfortunately, there are no general rules for determining which method should be employed for exactly which problem to get the most realistic result. Therefore, we study the transition from stochastic to deterministic behavior in a widely studied system, namely the signal transduction via calcium, especially calcium oscillations. We observe that the transition occurs within a range of particle numbers, which roughly corresponds to the number of receptors and channels in the cell, and depends heavily on the attractive properties of the phase space of the respective systems dynamics. We conclude that the attractive properties of a system, expressed, e.g., by the divergence of the system, are a good measure for determining which simulation algorithm is appropriate in terms of speed and realism
The Nondeterministic Waiting Time Algorithm: A Review
We present briefly the Nondeterministic Waiting Time algorithm. Our technique
for the simulation of biochemical reaction networks has the ability to mimic
the Gillespie Algorithm for some networks and solutions to ordinary
differential equations for other networks, depending on the rules of the
system, the kinetic rates and numbers of molecules. We provide a full
description of the algorithm as well as specifics on its implementation. Some
results for two well-known models are reported. We have used the algorithm to
explore Fas-mediated apoptosis models in cancerous and HIV-1 infected T cells
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