Stochastic chemical kinetics and the quasi-steady-state assumption : application to the stochastic simulation algorithm and chemical master equation

Recently the application of the quasi-steady-state approximation (QSSA) to the stochastic simulation algorithm (SSA) was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)] and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and, in particular, to the finite state projection algorithm [Munsky and Khammash, J. Chem. Phys. 124, 044104 (2006)], in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the (deterministic) total QSSA (tQSSA) and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis–Menten enzyme kinetics, double phosphorylation, the Goldbeter–Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.

Stochastic analysis of the VEGF receptor response curve

Recently, an analysis of the response curve of the vascular endothelial growth factor (VEGF) receptor and its application to cancer therapy was described in [T. Alarcón, and K. Page, J. R. Soc. Lond. Interface 4, 283–304 (2007)]. The analysis is significantly extended here by demonstrating that an alternative computational strategy, namely the Krylov FSP algorithm for the direct solution of the chemical master equation, is feasible for the study of the receptor model. The new method allows us to further investigate the hypothesis of symmetry in the stochastic fluctuations of the response. Also, by augmenting the original model with a single reversible reaction we formulate a plausible mechanism capable of realizing a bimodal response, which is reported experimentally but which is not exhibited by the original model. The significance of these findings for mechanisms of tumour resistance to antiangiogenic therapy is discussed.

Accelerated leap methods for simulating discrete stochastic chemical kinetics

Biologists are increasingly conscious of the critical role that noise plays in cellular functions such as genetic regulation, often in connection with fluctuations in small numbers of key regulatory molecules. This has inspired the development of models that capture this fundamentally discrete and stochastic nature of cellular biology - most notably the Gillespie stochastic simulation algorithm (SSA). The SSA simulates a temporally homogeneous, discrete-state, continuous-time Markov process, and of course the corresponding probabilities and numbers of each molecular species must all remain positive. While accurately serving this purpose, the SSA can be computationally inefficient due to very small time stepping so faster approximations such as the Poisson and Binomial τ-leap methods have been suggested. This work places these leap methods in the context of numerical methods for the solution of stochastic differential equations (SDEs) driven by Poisson noise. This allows analogues of Euler-Maruyuma, Milstein and even higher order methods to be developed through the Itô-Taylor expansions as well as similar derivative-free Runge-Kutta approaches. Numerical results demonstrate that these novel methods compare favourably with existing techniques for simulating biochemical reactions by more accurately capturing crucial properties such as the mean and variance than existing methods.

Parallel implementation of stochastic simulation for large scale cellular processes

Experimental and theoretical studies have shown the importance of stochastic processes in genetic regulatory networks and cellular processes. Cellular networks and genetic circuits often involve small numbers of key proteins such as transcriptional factors and signaling proteins. In recent years stochastic models have been used successfully for studying noise in biological pathways, and stochastic modelling of biological systems has become a very important research field in computational biology. One of the challenge problems in this field is the reduction of the huge computing time in stochastic simulations. Based on the system of the mitogen-activated protein kinase cascade that is activated by epidermal growth factor, this work give a parallel implementation by using OpenMP and parallelism across the simulation. Special attention is paid to the independence of the generated random numbers in parallel computing, that is a key criterion for the success of stochastic simulations. Numerical results indicate that parallel computers can be used as an efficient tool for simulating the dynamics of large-scale genetic regulatory networks and cellular processes

Supplement : Efficient weak second-order stochastic Runge-Kutta methods for non-commutative Stratonvich stochastic differential equations

This paper gives a modification of a class of stochastic Runge–Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.

Evolving genetic regulatory networks performing as stochastic switches

Recent studies have shown that small genetic regulatory networks (GRNs) can be evolved in silico displaying certain dynamics in the underlying mathematical model. It is expected that evolutionary approaches can help to gain a better understanding of biological design principles and assist in the engineering of genetic networks. To take the stochastic nature of GRNs into account, our evolutionary approach models GRNs as biochemical reaction networks based on simple enzyme kinetics and simulates them by using Gillespie’s stochastic simulation algorithm (SSA). We have already demonstrated the relevance of considering intrinsic stochasticity by evolving GRNs that show oscillatory dynamics in the SSA but not in the ODE regime. Here, we present and discuss first results in the evolution of GRNs performing as stochastic switches.

Automated recognition of drunk driving on highways from video sequences

A new method for the detection of abnormal vehicle trajectories is proposed. It couples optical flow extraction of vehicle velocities with a neural network classifier. Abnormal trajectories are indicative of drunk or sleepy drivers. A single feature of the vehicle, eg., a tail light, is isolated and the optical flow computed only around this feature rather than at each pixel in the image.

Evaluation of novel Streptococcus pyogenes vaccine candidates incorporating multiple conserved sequences from the C-repeat region of the M-protein

A major challenge for Streptococcus pyogenes vaccine development is the identification of epitopes that confer protection from infection by multiple S. pyogenes M-types. Here we have identified and characterised the distribution of common variant sequences from individual repeat units of the C-repeat region (CRR) of M-proteins representing 77 different M-types. Three polyvalent fusion vaccine candidates (SV1, SV2 and SV3) incorporating the most common variants were subsequently expressed and purified, and demonstrated to be alpha-helical by Circular Dichroism (CD), a secondary conformational characteristic of the CRR in the M-protein. Antibodies raised against each of these constructs recognise M-proteins that vary in their CRR, and bind to the surface of multiple S. pyogenes isolates. Antibodies raised against SV1, containing five variant sequences, also kill heterologous S. pyogenes isolates in in vitro bactericidal assays. Further structural characterisation of this construct demonstrated the conformation of SV1 was stable at different pHs, and thermal unfolding of SV1 a reversible process. Our findings demonstrate that linkage of multiple variant sequences into a single recombinant construct overcomes the need to embed the variant sequences in foreign helix promoting flanking sequences for conformational stability, and demonstrates the viability of the polyvalent candidates as global S. pyogenes vaccine candidates.

Combined mitochondrial and nuclear DNA sequences resolve the interrelations of the major Australasian marsupial radiations

Australasian marsupials include three major radiations, the insectivorous/carnivorous Dasyuromorphia, the omnivorous bandicoots (Peramelemorphia), and the largely herbivorous diprotodontians. Morphologists have generally considered the bandicoots and diprotodontians to be closely related, most prominently because they are both syndactylous (with the 2nd and 3rd pedal digits being fused). Molecular studies have been unable to confirm or reject this Syndactyla hypothesis. Here we present new mitochondrial (mt) genomes from a spiny bandicoot (Echymipera rufescens) and two dasyurids, a fat-tailed dunnart (Sminthopsis crassicaudata) and a northern quoll (Dasyurus hallucatus). By comparing trees derived from pairwise base-frequency differences between taxa with standard (absolute, uncorrected) distance trees, we infer that composition bias among mt protein-coding and RNA sequences is sufficient to mislead tree reconstruction. This can explain incongruence between trees obtained from mt and nuclear data sets. However, after excluding major sources of compositional heterogeneity, both the “reduced-bias” mt and nuclear data sets clearly favor a bandicoot plus dasyuromorphian association, as well as a grouping of kangaroos and possums (Phalangeriformes) among diprotodontians. Notably, alternatives to these groupings could only be confidently rejected by combining the mt and nuclear data. Elsewhere on the tree, Dromiciops appears to be sister to the monophyletic Australasian marsupials, whereas the placement of the marsupial mole (Notoryctes) remains problematic. More generally, we contend that it is desirable to combine mt genome and nuclear sequences for inferring vertebrate phylogeny, but as separately modeled process partitions. This strategy depends on detecting and excluding (or accounting for) major sources of nonhistorical signal, such as from compositional nonstationarity.

Modeling ion channel dynamics through reflected stochastic differential equations

Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks. © 2012 American Physical Society.

Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise

In this paper we extend the ideas of Brugnano, Iavernaro and Trigiante in their development of HBVM($s,r$) methods to construct symplectic Runge-Kutta methods for all values of $s$ and $r$ with $s\geq r$. However, these methods do not see the dramatic performance improvement that HBVMs can attain. Nevertheless, in the case of additive stochastic Hamiltonian problems an extension of these ideas, which requires the simulation of an independent Wiener process at each stage of a Runge-Kutta method, leads to methods that have very favourable properties. These ideas are illustrated by some simple numerical tests for the modified midpoint rule.