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In living cells, the diffusion of molecules is fundamental for the proper functioning of many intracellular processes and for their correct response to external or internal signaling. In this paper, we consider the diffusion of a pivotal protein involved in bacterial chemotaxis, and show how to model and simulate the formation of concentration gradients of this protein along the bacterial cell. For this purpose, we exploit the framework of P systems to represent the partition of the bacterial volume into separate but communicating virtual volumes, and make use of communication rules to mimic the diffusive events. We simulate this system by means of the multivolume stochastic simulation algorithm τ-DPP, and present the effects of different diffusion coefficients and of a varying number of virtual volumes over the response of the chemotactic pathway.

Gene expression and its regulation is a nonequilibrium stochastic process. Different molecules are involved in several biochemical steps in this process with low copies. It is observed that the stochasticity in biochemical processes is mainly due to the low copy number of the molecules present in the system. Several studies also show that the nonequilibrium biochemical processes require energy cost. But cellular system has developed itself through natural evolution by minimizing energy cost for optimum output. Here, we study the role of stochasticity qualitatively in a network of two genes using stochastic simulation method and approximately measure the energy consumption for the gene expression process. We find that the noise in gene expression process reduces the energy cost of protein synthesis. Therefore, we argued that the stochasticity in gene expression may be a choice of cellular system for protein synthesis with minimum energy cost.

This paper addresses the complex job shop scheduling problem with the consideration of non-identical job sizes. By simultaneously considering practical constraints of sequence dependent setup times, incompatible job families and job dependent batch processing time, we formulate this problem into a simulation optimization problem based on the disjunctive graph representation. In order to find scheduling policies that minimise the expectation of mean weighted tardiness, we propose a genetic programming based hyper heuristic to generate efficient dispatching rules. And then, based on the nested partition framework together with the optimal computing budget allocation technique, a hybrid rule selection algorithm is proposed for searching machine group specified rule combinations. Numerical results show that the proposed algorithms outperform benchmark algorithms in both solution quality and robustness.

In this paper we discuss a noisy mean field model for the genetic toggle switch. We show that this model approximates very well the characteristics of the system, observed using the exact Gillespie stochastic simulation algorithm. Also, we show that the system can be made exponentially stable depending on reaction parameters.

The Lotka–Volterra model of cyclic competitions among three species with noise was investigated by numerical simulations. Our results indicate that the multiplicative noise is responsible for the alternately sinusoidal competitive oscillations of species, the optimizations of response of the system via stochastic resonance, the spatial patterns and the temporal oscillations of the spatial correlation coefficients.

The laser intensity Langevin equations for a two-mode ring laser subject to both pump noise and quantum noise have been obtained by a phase-locking method. By means of stochastic simulation, the mode competition between two modes for different combinations of the strength of the pump noise, the coupling constant and the difference of pump parameters have been investigated. The results show the following. (i) The pump noise dramatically affects the mode competition. (ii) The mode competition is quite sensitive to changing of the coupling constant near the line center. (iii) The mode competition gets stronger and stronger with increasing absolute value of the difference of pump parameters.

A master equation for a scheme of chemical Chua system has been simulated stochastically. Two pairs of reaction simulations, one pair corresponding to deterministic chaos and the other pair to limit cycle, are carried out. The initial conditions differ with only one molecule for the former, while one hundred molecules for the latter. The rapid separation of time traces is shown only in the former case. This marked difference of dynamic behavior between the two cases might reveal the intrinsic random nature of deterministic chemical chaos, i.e. initial-condition sensitivity based on random collision events.

Intrinsic fluctuation of Chua system is studied with master equation method. Our results have shown that the intrinsic noise indeed exerted considerable influence on Chua system. In contrast to that of deterministic equation the patterns of time evolution and attractor have been greatly altered under the influence of intrinsic noise.

In this paper two stochastic kinetic models of a faulty ubiquitin–proteasome are presented. In the first model, mutant variants of the α-synuclein, induced by environmental stress factors, inhibit the ubiquitin–proteasome pathway that is in charge to start the degradation of misfolded proteins. In the second model mutant variants of parkin proteins are not able to transfer the ubiquitin molecules on the faulty proteins, so that they are not targeted and recognized as toxic products. The models are specified in BlenX and simulated with Beta Workbench (Beta WB) simulator. BlenX is a new programming language that has been recently designed by the CoSBi team of researchers to describe components and interactions among components of a biochemical system. It implements a process calculus in which the communication and synchronization between computational processes abstract the biochemical interaction and reactions between molecules, proteins and functional complexes. The aim of this study is twofold: (i) to move the current focus on the structural studies of the components of the protein ubiquitination and degradation to the dynamics of such components in a typical systems biology perspective; (ii) to provide first stochastic models in a new language suitable to systemic approach to the modelling of biochemical kinetics. This tool makes it possible for biologists and medical researchers a high-level modeling of molecular mechanisms underlying the pathogenesis of the disease. Consequently, the suitable abstraction provided by the language makes possible the elucidation of important and still elusive aspects of the involved biochemical interactions. The agreement of the simulations of our models with the available qualitative experimental observations and previous theoretical studies supports the confidence in the model validity.

We develop some simple simulation algorithms for CIR and Wishart processes. We investigate rigorously the square of a matrix valued Ornstein–Uhlenbeck process, the main idea being to split the generator and to reduce the problem to the simulation of the square of a matrix valued Ornstein–Uhlenbeck process to be added to a deterministic process. In this way, we provide a weak second-order scheme that requires only the simulation of i.i.d. Gaussian r.v.'s and simple matrix manipulations.

The flashing Brownian ratchet is a stochastic process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, the latter being a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. The result is directed motion. In the presence of a static homogeneous force that acts in the direction opposite to that of the directed motion, there is a reduction (or even a reversal) of the directed motion effect. Such a process may be called a tilted flashing Brownian ratchet. We show how one can study this process numerically, using a random walk approximation or, equivalently, using numerical solution of the Fokker–Planck equation. Stochastic simulation is another viable method.

This paper proposes a new decision support approach for applying stochastic simulation to the multiplicative analytic hierarchy process (AHP) in order to deal with issues concerning the scale parameter. The paper suggests a new approach that captures the influence from the scale parameter by making use of probability distributions. Herein, the uncertainty both with regard to the scale and the inherent randomness from the parameter is captured by probabilistic input and output distributions. Provided that each alternative and criteria under consideration are independent it is assumed that the embedded uncertainty from the progression factors remains the same. The result is then an interval estimate for each alternative’s final scores. This can lead to overlapping intervals of scores which may be interpreted as possible rank reversals. Thus, the decision support approach makes it possible to calculate the probability of overlapping for any given set of pairwise comparisons.

Understanding seismic energy input and dissipation mechanism is necessary for energy-based seismic design of complex underground structures. Due to the intrinsic uncertainty of ground motion, stochastic methods are usually needed. In this paper, we study the seismic energy input and dissipation mechanism in an underground structure using the probability density evolution method (PDEM). It is found that the cumulative hysteretic energy dissipation ($E_h)$ of the underground structure is reduced by 55% compared with the above-ground structure due to soil constraint. The columns, walls and slabs at the bottom of the underground structure have a high demand for $E_h$, which are vulnerable spots of the structure.

p53 kinetics plays a key role in regulating cell fate. Based on the p53 gene regulatory network composed by the core regulatory factors ATM, Mdm2, Wip1, and PIDD, the effect of the delays in the process of transcription and translation of Mdm2 and Wip1 on the dynamics of p53 is studied theoretically and numerically. The results show that these two time delays can affect the stability of the positive equilibrium. With the increase of delays, the dynamics of p53 presents an oscillating state. Further, we also study the effects of PIDD and chemotherapeutic drug etoposide on the kinetics of p53. The model indicates that (i) PIDD low-level expression does not significantly affect p53 oscillatory behavior, but high-level expression could induce two-phase kinetics of p53; (ii) Too high and too low concentration of etoposide is not conducive to p53 oscillation. These results are in good agreement with experimental findings. Finally, we consider the influence of internal noise on the system through Binomial $\tau$-leap algorithm. Stochastic simulations reveal that high-intensity noise completely destroys p53 dynamics in the deterministic model, whereas low-intensity noise does not alter p53 dynamics. Interestingly, for the stable focus, the internal noise with appropriate intensity can induce quasi-limit cycle oscillations of the system. Our work may provide the useful insights for the development of anticancer therapy.

Stochastic effect in cellular systems has been an important topic in systems biology. Stochastic modeling and simulation methods are important tools to study stochastic effect. Given the low efficiency of stochastic simulation algorithms, the hybrid method, which combines an ordinary differential equation (ODE) system with a stochastic chemically reacting system, shows its unique advantages in the modeling and simulation of biochemical systems. The efficiency of the hybrid method is usually limited by reactions in the stochastic subsystem, which are modeled and simulated using Gillespie’s framework and frequently interrupt the integration of the ODE subsystem. In this paper, we develop an efficient implementation approach for the hybrid method coupled with traditional ODE solvers. We also compare the efficiency of the hybrid methods with three widely used ODE solvers RADAU5, DASSL, and DLSODAR. Numerical experiments with three biochemical models are presented. A detailed discussion is presented for the performances of three ODE solvers.

Importance Sampling is a variance reduction technique possessing the potential of zero-variance estimators in its optimal case. It has been successfully applied in a variety of settings ranging from Monte Carlo methods for static models to simulations of complex dynamical systems governed by stochastic processes. We demonstrate the applicability of Importance Sampling to the simulation of coupled molecular reactions constituting biological or genetic networks. This fills a gap between great efforts spent on enhanced trajectory generation and the largely neglected issue of reduced variance among trajectories in the context of biological and genetic networks.