Stochastic delay differential equations for genetic regulatory networks

Time delay is an important aspect in the modelling of genetic regulation due to slow biochemical reactions such as gene transcription and translation, and protein diffusion between the cytosol and nucleus. In this paper we introduce a general mathematical formalism via stochastic delay differential equations for describing time delays in genetic regulatory networks. Based on recent developments with the delay stochastic simulation algorithm, the delay chemical masterequation and the delay reaction rate equation are developed for describing biological reactions with time delay, which leads to stochastic delay differential equations derived from the Langevin approach. Two simple genetic regulatory networks are used to study the impact of' intrinsic noise on the system dynamics where there are delays. (c) 2006 Elsevier B.V. All rights reserved.

Oscillatory regulation of hes1: Discrete stochastic delay modelling and simulation

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.

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.

A multi-scaled approach for simulating chemical reaction systems

In this paper we give an overview of some very recent work, as well as presenting a new approach, on the stochastic simulation of multi-scaled 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 present a new approach which couples the three regimes mentioned above. We then apply this approach to a biologically inspired problem involving the expression and activity of LacZ and LacY proteins in E. coli, and conclude with a discussion on the significance of this work.

A genetic estimation algorithm for parameters of stochastic ordinary differential equations

A generic method for the estimation of parameters for Stochastic Ordinary Differential Equations (SODEs) is introduced and developed. This algorithm, called the GePERs method, utilises a genetic optimisation algorithm to minimise a stochastic objective function based on the Kolmogorov-Smirnov statistic. Numerical simulations are utilised to form the KS statistic. Further, the examination of some of the factors that improve the precision of the estimates is conducted. This method is used to estimate parameters of diffusion equations and jump-diffusion equations. It is also applied to the problem of model selection for the Queensland electricity market. (C) 2003 Elsevier B.V. All rights reserved.

Robust algorithms for solving stochastic partial differential equations

A robust semi-implicit central partial difference algorithm for the numerical solution of coupled stochastic parabolic partial differential equations (PDEs) is described. This can be used for calculating correlation functions of systems of interacting stochastic fields. Such field equations can arise in the description of Hamiltonian and open systems in the physics of nonlinear processes, and may include multiplicative noise sources. The algorithm can be used for studying the properties of nonlinear quantum or classical field theories. The general approach is outlined and applied to a specific example, namely the quantum statistical fluctuations of ultra-short optical pulses in chi((2)) parametric waveguides. This example uses a non-diagonal coherent state representation, and correctly predicts the sub-shot noise level spectral fluctuations observed in homodyne detection measurements. It is expected that the methods used wilt be applicable for higher-order correlation functions and other physical problems as well. A stochastic differencing technique for reducing sampling errors is also introduced. This involves solving nonlinear stochastic parabolic PDEs in combination with a reference process, which uses the Wigner representation in the example presented here. A computer implementation on MIMD parallel architectures is discussed. (C) 1997 Academic Press.

The stochastic design of force-minimized compact magnets for high-field magnetic resonance imaging applications

New designs for force-minimized compact high-field clinical MRI magnets are described. The design method is a modified simulated annealing (SA) procedure which includes Maxwell forces in the error function to be minimized. This permits an automated force reduction in the magnet designs while controlling the overall dimensions of the system. As SA optimization requires many iterations to achieve a final design, it is important that each iteration in the procedure is rapid. We have therefore developed a rapid force calculation algorithm. Novel designs for short 3- and 4-T clinical MRI systems are presented in which force reduction has been invoked. The final designs provide large homogeneous regions and reduced stray fields in remarkable short magnets. A shielded 4-T design that is approximately 30% shorter than current designs is presented. This novel magnet generates a full 50-cm diameter homogeneous region.

Two-link approximation schemes for loss networks with linear structure and trunk reservation

Loss networks have long been used to model various types of telecommunication network, including circuit-switched networks. Such networks often use admission controls, such as trunk reservation, to optimize revenue or stabilize the behaviour of the network. Unfortunately, an exact analysis of such networks is not usually possible, and reduced-load approximations such as the Erlang Fixed Point (EFP) approximation have been widely used. The performance of these approximations is typically very good for networks without controls, under several regimes. There is evidence, however, that in networks with controls, these approximations will in general perform less well. We propose an extension to the EFP approximation that gives marked improvement for a simple ring-shaped network with trunk reservation. It is based on the idea of considering pairs of links together, thus making greater allowance for dependencies between neighbouring links than does the EFP approximation, which only considers links in isolation.

A simulated annealing algorithm for finding consensus sequences

Motivation: A consensus sequence for a family of related sequences is, as the name suggests, a sequence that captures the features common to most members of the family. Consensus sequences are important in various DNA sequencing applications and are a convenient way to characterize a family of molecules. Results: This paper describes a new algorithm for finding a consensus sequence, using the popular optimization method known as simulated annealing. Unlike the conventional approach of finding a consensus sequence by first forming a multiple sequence alignment, this algorithm searches for a sequence that minimises the sum of pairwise distances to each of the input sequences. The resulting consensus sequence can then be used to induce a multiple sequence alignment. The time required by the algorithm scales linearly with the number of input sequences and quadratically with the length of the consensus sequence. We present results demonstrating the high quality of the consensus sequences and alignments produced by the new algorithm. For comparison, we also present similar results obtained using ClustalW. The new algorithm outperforms ClustalW in many cases.

A monolithic smoothing-gap algorithm for contact-impact based on the signed distance function

A new algorithm has been developed for smoothing the surfaces in finite element formulations of contact-impact. A key feature of this method is that the smoothing is done implicitly by constructing smooth signed distance functions for the bodies. These functions are then employed for the computation of the gap and other variables needed for implementation of contact-impact. The smoothed signed distance functions are constructed by a moving least-squares approximation with a polynomial basis. Results show that when nodes are placed on a surface, the surface can be reproduced with an error of about one per cent or less with either a quadratic or a linear basis. With a quadratic basis, the method exactly reproduces a circle or a sphere even for coarse meshes. Results are presented for contact problems involving the contact of circular bodies. Copyright (C) 2002 John Wiley Sons, Ltd.

The tree cut and merge algorithm for estimation of network reliability

This article presents Monte Carlo techniques for estimating network reliability. For highly reliable networks, techniques based on graph evolution models provide very good performance. However, they are known to have significant simulation cost. An existing hybrid scheme (based on partitioning the time space) is available to speed up the simulations; however, there are difficulties with optimizing the important parameter associated with this scheme. To overcome these difficulties, a new hybrid scheme (based on partitioning the edge set) is proposed in this article. The proposed scheme shows orders of magnitude improvement of performance over the existing techniques in certain classes of network. It also provides reliability bounds with little overhead.

A reduced load approximation accounting for link interactions in a loss network

This paper is concerned with evaluating the performance of loss networks. Accurate determination of loss network performance can assist in the design and dimen- sioning of telecommunications networks. However, exact determination can be difficult and generally cannot be done in reasonable time. For these reasons there is much interest in developing fast and accurate approximations. We develop a reduced load approximation that improves on the famous Erlang fixed point approximation (EFPA) in a variety of circumstances. We illustrate our results with reference to a range of networks for which the EFPA may be expected to perform badly.

Binomial leap methods for simulating stochastic chemical kinetics

This paper discusses efficient simulation methods for stochastic chemical kinetics. Based on the tau-leap and midpoint tau-leap methods of Gillespie [D. T. Gillespie, J. Chem. Phys. 115, 1716 (2001)], binomial random variables are used in these leap methods rather than Poisson random variables. The motivation for this approach is to improve the efficiency of the Poisson leap methods by using larger stepsizes. Unlike Poisson random variables whose range of sample values is from zero to infinity, binomial random variables have a finite range of sample values. This probabilistic property has been used to restrict possible reaction numbers and to avoid negative molecular numbers in stochastic simulations when larger stepsize is used. In this approach a binomial random variable is defined for a single reaction channel in order to keep the reaction number of this channel below the numbers of molecules that undergo this reaction channel. A sampling technique is also designed for the total reaction number of a reactant species that undergoes two or more reaction channels. Samples for the total reaction number are not greater than the molecular number of this species. In addition, probability properties of the binomial random variables provide stepsize conditions for restricting reaction numbers in a chosen time interval. These stepsize conditions are important properties of robust leap control strategies. Numerical results indicate that the proposed binomial leap methods can be applied to a wide range of chemical reaction systems with very good accuracy and significant improvement on efficiency over existing approaches. (C) 2004 American Institute of Physics.

QRT: A Qr-based tridiagonalization algorithm for nonsymmetric matrices

The stable similarity reduction of a nonsymmetric square matrix to tridiagonal form has been a long-standing problem in numerical linear algebra. The biorthogonal Lanczos process is in principle a candidate method for this task, but in practice it is confined to sparse matrices and is restarted periodically because roundoff errors affect its three-term recurrence scheme and degrade the biorthogonality after a few steps. This adds to its vulnerability to serious breakdowns or near-breakdowns, the handling of which involves recovery strategies such as the look-ahead technique, which needs a careful implementation to produce a block-tridiagonal form with unpredictable block sizes. Other candidate methods, geared generally towards full matrices, rely on elementary similarity transformations that are prone to numerical instabilities. Such concomitant difficulties have hampered finding a satisfactory solution to the problem for either sparse or full matrices. This study focuses primarily on full matrices. After outlining earlier tridiagonalization algorithms from within a general framework, we present a new elimination technique combining orthogonal similarity transformations that are stable. We also discuss heuristics to circumvent breakdowns. Applications of this study include eigenvalue calculation and the approximation of matrix functions.

GANN: Genetic algorithm neural networks for the detection of conserved combinations of features in DNA

Background: The multitude of motif detection algorithms developed to date have largely focused on the detection of patterns in primary sequence. Since sequence-dependent DNA structure and flexibility may also play a role in protein-DNA interactions, the simultaneous exploration of sequence-and structure-based hypotheses about the composition of binding sites and the ordering of features in a regulatory region should be considered as well. The consideration of structural features requires the development of new detection tools that can deal with data types other than primary sequence. Results: GANN ( available at http://bioinformatics.org.au/gann) is a machine learning tool for the detection of conserved features in DNA. The software suite contains programs to extract different regions of genomic DNA from flat files and convert these sequences to indices that reflect sequence and structural composition or the presence of specific protein binding sites. The machine learning component allows the classification of different types of sequences based on subsamples of these indices, and can identify the best combinations of indices and machine learning architecture for sequence discrimination. Another key feature of GANN is the replicated splitting of data into training and test sets, and the implementation of negative controls. In validation experiments, GANN successfully merged important sequence and structural features to yield good predictive models for synthetic and real regulatory regions. Conclusion: GANN is a flexible tool that can search through large sets of sequence and structural feature combinations to identify those that best characterize a set of sequences.