Stochastic models for regulatory networks of the genetic toggle switch

Bistability arises within a wide range of biological systems from the A phage switch in bacteria to cellular signal transduction pathways in mammalian cells. Changes in regulatory mechanisms may result in genetic switching in a bistable system. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. Although deterministic models have been used for studying the existence of bistability properties under various system conditions, these models cannot realize cell-to-cell fluctuations in genetic switching. However, there is a lag in the development of stochastic models for studying the impact of noise in bistable systems because of the lack of detailed knowledge of biochemical reactions, kinetic rates, and molecular numbers. in this work, we develop a previously undescribed general technique for developing quantitative stochastic models for large-scale genetic regulatory networks by introducing Poisson random variables into deterministic models described by ordinary differential equations. Two stochastic models have been proposed for the genetic toggle switch interfaced with either the SOS signaling pathway or a quorum-sensing signaling pathway, and we have successfully realized experimental results showing bimodal population distributions. Because the introduced stochastic models are based on widely used ordinary differential equation models, the success of this work suggests that this approach is a very promising one for studying noise in large-scale genetic regulatory networks.

Respiratory gene clusters of Metallosphaera sedula - differential expression and transcriptional organization

Metallosphaera sedula is a thermoacidophilic Crenarchaeon which is capable of leaching metals from sulfidic ores. The authors have investigated the presence and expression of genes encoding respiratory complexes in this organism when grown heterotrophically or chemolithotrophically on either sulfur or pyrite. The presence of three gene clusters, encoding two terminal oxidase complexes, the quinol oxidase SoxABCD and the SoxM oxidase supercomplex, and a gene cluster encoding a high-potential cytochrome b and components of a bc(1) complex analogue (cbsBA-soxL2N gene cluster) was established. Expression studies showed that the soxM gene was expressed to high levels during heterotrophic growth of M. sedula on yeast extract, while the soxABCD mRNA was most abundant in cells grown on sulfur. Reduced-minus-oxidized difference spectra of cell membranes showed cytochrome-related peaks that correspond to published spectra of Sulfolobus-type terminal oxidase complexes. In pyrite-grown cells, expression levels of the two monitored oxidase gene clusters were reduced by a factor of 10-12 relative to maximal expression levels, although spectra of membranes clearly contained oxidase-associated haems, suggesting the presence of additional gene clusters encoding terminal oxidases in M. sedula. Pyrite- and sulfur-grown cells contained high levels of the cbsA transcript, which encodes a membrane-bound cytochrome b with a possible role in iron oxidation or chemolithotrophy. The cbsA gene is not co-transcribed with the soxL2N genes, and therefore does not appear to be an integral part of this bc(1) complex analogue. The data show for the first time the differential expression of the Sulfolobus-type terminal oxidase gene clusters in a Crenarchaeon in response to changing growth modes.

Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage lambda

Bistability and switching are two important aspects of the genetic regulatory network of phage. Positive and negative feedbacks are key regulatory mechanisms in this network. By the introduction of threshold values, the developmental pathway of A phage is divided into different stages. If the protein level reaches a threshold value, positive or negative feedback will be effective and regulate the process of development. Using this regulatory mechanism, we present a quantitative model to realize bistability and switching of phage based on experimental data. This model gives descriptions of decisive mechanisms for different pathways in induction. A stochastic model is also introduced for describing statistical properties of switching in induction. A stochastic degradation rate is used to represent intrinsic noise in induction for switching the system from the lysogenic pathway to the lysis pathway. The approach in this paper represents an attempt to describe the regulatory mechanism in genetic regulatory network under the influence of intrinsic noise in the framework of continuous models. (C) 2003 Elsevier Ltd. All rights reserved.

The prion protein gene: Identifying regulatory signals using marsupial sequence

The function of the prion protein gene (PRNP) and its normal product PrPC is elusive. We used comparative genomics as a strategy to understand the normal function of PRNP. As the reliability of comparisons increases with the number of species and increased evolutionary distance, we isolated and sequenced a 66.5 kb BAC containing the PRNP gene from a distantly related mammal, the model Australian marsupial Macropus eugenii (tammar wallaby). Marsupials are separated from eutherians such as human and mouse by roughly 180 million years of independent evolution. We found that tammar PRNP, like human PRNP, has two exons. Prion proteins encoded by the tammar wallaby and a distantly related marsupial, Monodelphis domestica (Brazilian opossum) PRNP contain proximal PrP repeats with a distinct, marsupial-specific composition and a variable number. Comparisons of tammar wallaby PRNP with PRNPs from human, mouse, bovine and ovine allowed us to identify non-coding gene regions conserved across the marsupial-eutherian evolutionary distance, which are candidates for regulatory regions. In the PRNP 3' UTR we found a conserved signal for nuclear-specific polyadenylation and the putative cytoplasmic polyadenylation element (CPE), indicating that post-transcriptional control of PRNP mRNA activity is important. Phylogenetic footprinting revealed conserved potential binding sites for the MZF-1 transcription factor in both upstream promoter and intron/intron 1, and for the MEF2, MyTI, Oct-1 and NFAT transcription factors in the intron(s). The presence of a conserved NFAT-binding site and CPE indicates involvement of PrPC in signal transduction and synaptic plasticity. (c) 2004 Elsevier B.V. All rights reserved.

Network topology and the evolution of dynamics in an artificial genetic regulatory network model created by whole genome duplication and divergence

Topological measures of large-scale complex networks are applied to a specific artificial regulatory network model created through a whole genome duplication and divergence mechanism. This class of networks share topological features with natural transcriptional regulatory networks. Specifically, these networks display scale-free and small-world topology and possess subgraph distributions similar to those of natural networks. Thus, the topologies inherent in natural networks may be in part due to their method of creation rather than being exclusively shaped by subsequent evolution under selection. The evolvability of the dynamics of these networks is also examined by evolving networks in simulation to obtain three simple types of output dynamics. The networks obtained from this process show a wide variety of topologies and numbers of genes indicating that it is relatively easy to evolve these classes of dynamics in this model. (c) 2006 Elsevier Ireland Ltd. All rights reserved.

Cluster analysis of the p53 genetic regulatory network: topology and biology

We describe a network module detection approach which combines a rapid and robust clustering algorithm with an objective measure of the coherence of the modules identified. The approach is applied to the network of genetic regulatory interactions surrounding the tumor suppressor gene p53. This algorithm identifies ten clusters in the p53 network, which are visually coherent and biologically plausible.

Implicit stochastic Runge-Kutta methods for stochastic differential equations

In this paper we construct implicit stochastic Runge-Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods.

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the best choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy. (C) 2004 Elsevier B.V. All rights reserved.

Numerical methods for strong solutions of stochastic differential equations: an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Stochastic dynamic programming for election timing: A game theory approach

In this paper, we consider dynamic programming for the election timing in the majoritarian parliamentary system such as in Australia, where the government has a constitutional right to call an early election. This right can give the government an advantage to remain in power for as long as possible by calling an election, when its popularity is high. On the other hand, the opposition's natural objective is to gain power, and it will apply controls termed as "boosts" to reduce the chance of the government being re-elected by introducing policy and economic responses. In this paper, we explore equilibrium solutions to the government, and the opposition strategies in a political game using stochastic dynamic programming. Results are given in terms of the expected remaining life in power, call and boost probabilities at each time at any level of popularity.