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.

High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations

The pioneering work of Runge and Kutta a hundred years ago has ultimately led to suites of sophisticated numerical methods suitable for solving complex systems of deterministic ordinary differential equations. However, in many modelling situations, the appropriate representation is a stochastic differential equation and here numerical methods are much less sophisticated. In this paper a very general class of stochastic Runge-Kutta methods is presented and much more efficient classes of explicit methods than previous extant methods are constructed. In particular, a method of strong order 2 with a deterministic component based on the classical Runge-Kutta method is constructed and some numerical results are presented to demonstrate the efficacy of this approach.

Numerical solutions of stochastic differential equations – implementation and stability issues

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively.

A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively. (C) 2000 Elsevier Science B.V. All rights reserved.

High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

In many modeling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary differential equation (SODE). However, unlike the deterministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by K. Burrage and P.M. Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But K. Burrage and P.M. Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or Ito) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known stochastic Runge-Kutta methods can suffer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if an s-stage SRK contains Stratonovich integrals up to order p then the strong order of the SRK cannot exceed min{(p + 1)/2, (s - 1)/2), p greater than or equal to 2, s greater than or equal to 3 or 1 if p = 1.

Transcriptome analysis of neutrophils after endurance exercise reveals novel signaling mechanisms in the immune response to physiological stress

Neutrophils serve as an intriguing model for the study of innate immune cellular activity induced by physiological stress. We measured changes in the transcriptome of circulating neutrophils following an experimental exercise trial (EXTRI) consisting of 1 h of intense cycling immediately followed by 1 h of intense running. Blood samples were taken at baseline, 3 h, 48 h, and 96 h post-EXTRI from eight healthy, endurance-trained, male subjects. RNA was extracted from isolated neutrophils. Differential gene expression was evaluated using Illumina microarrays and validated with quantitative PCR. Gene set enrichment analysis identified enriched molecular signatures chosen from the Molecular Signatures Database. Blood concentrations of muscle damage indexes, neutrophils, interleukin (IL)-6 and IL-10 were increased (P < 0.05) 3 h post-EXTRI. Upregulated groups of functionally related genes 3 h post-EXTRI included gene sets associated with the recognition of tissue damage, the IL-1 receptor, and Toll-like receptor (TLR) pathways (familywise error rate, P value < 0.05). The core enrichment for these pathways included TLRs, low-affinity immunoglobulin receptors, S100 calcium binding protein A12, and negative regulators of innate immunity, e.g., IL-1 receptor antagonist, and IL-1 receptor associated kinase-3. Plasma myoglobin changes correlated with neutrophil TLR4 gene expression (r = 0.74; P < 0.05). Neutrophils had returned to their nonactivated state 48 h post-EXTRI, indicating that their initial proinflammatory response was transient and rapidly counterregulated. This study provides novel insight into the signaling mechanisms underlying the neutrophil responses to endurance exercise, suggesting that their transcriptional activity was particularly induced by damage-associated molecule patterns, hypothetically originating from the leakage of muscle components into the circulation.

The measurement of adenosine and estrogen receptor expression in rat brains following ovariectomy using quantitative PCR analysis

In our laboratory we have developed a quantitative-polymerase chain reaction (Q-PCR) strategy to examine the differential expression of adenosine receptor (ADOR), A(1), A(2A), A(2B) and A(3), and estrogen receptors (ER) alpha and beta. Brain and uterine mRNA were first used to optimise specific amplification conditions prior to SYBR Green I real time analysis of receptor subtype expression. SYBR Green I provided a convenient and sensitive means of examining specific PCR amplification product in real time, and allowed the generation of standard curves from which relative receptor abundance could be determined. Real time Q-PCR analysis was then performed, to examine changes in receptor expression levels in brains of adult female Wistar rats 3-month post ovariectomy. Comparison with sham-operated age-matched control rats demonstrated both comparative and absolute-copy number changes in receptor levels. Evaluation of both analytical methods investigated 18S rRNA as an internal reference for comparative gene expression analysis in the brain. The results of this study revealed preferential repression of ADORA(2A) (>4-fold down) and consistent (>2-fold) down-regulation of ADORA(1), ADORA(3), and ER-beta, following ovariectomy. No change was found in ADORA(2B) or ER-alpha. Analysis of absolute copy number in this study revealed a correlation between receptor expression in response to ovariectomy, and relative receptor subtype abundance in the brain.

A transcription factor map as revealed by a genome-wide gene expression analysis of whole-blood mRNA transcriptome in multiple sclerosis

Background Several lines of evidence suggests that transcription factors are involved in the pathogenesis of Multiple Sclerosis (MS) but a complete mapping the whole network has been elusive. One of the reasons is that there are several clinical subtypes of MS and transcription factors which may be involved in one subtype may not be in others. We investigated the possibility that this network could be mapped using microarray technologies and modern bioinformatics methods on a dataset from whole blood in 99 untreated MS patients (36 Relapse Remitting MS, 43 Primary Progressive MS, and 20 Secondary Progressive MS) and 45 age-matched healthy controls, Methodology/Principal Findings We have used two different analytical methodologies: a differential expression analysis and a differential co-expression analysis, which have converged on a significant number of regulatory motifs that seem to be statistically overrepresented in genes which are either differentially expressed (or differentially co-expressed) in cases and controls (e.g. V$KROX_Q6, p-value < 3.31E-6; V$CREBP1_Q2, p-value < 9.93E-6, V$YY1_02, p-value < 1.65E-5). Conclusions/significance: Our analysis uncovered a network of transcription factors that potentially dysregulate several genes in MS or one or more of its disease subtypes. Analysing the published literature we have found that these transcription factors are involved in the early T-lymphocyte specification and commitment as well as in oligodendrocytes dedifferentiation and development. The most significant transcription factors motifs were for the Early Growth response EGR/KROX family, ATF2, YY1 (Yin and Yang 1), E2F-1/DP-1 and E2F-4/DP-2 heterodimers, SOX5, and CREB and ATF families.

Gene expression analysis of diagnostic biopsies predicts pathological response to neoadjuvant chemoradiotherapy of esophageal cancer

OBJECTIVE: This study explored gene expression differences in predicting response to chemoradiotherapy in esophageal cancer. PURPOSE:: A major pathological response to neoadjuvant chemoradiation is observed in about 40% of esophageal cancer patients and is associated with favorable outcomes. However, patients with tumors of similar histology, differentiation, and stage can have vastly different responses to the same neoadjuvant therapy. This dichotomy may be due to differences in the molecular genetic environment of the tumor cells. BACKGROUND DATA: Diagnostic biopsies were obtained from a training cohort of esophageal cancer patients (13), and extracted RNA was hybridized to genome expression microarrays. The resulting gene expression data was verified by qRT-PCR. In a larger, independent validation cohort (27), we examined differential gene expression by qRT-PCR. The ability of differentially-regulated genes to predict response to therapy was assessed in a multivariate leave-one-out cross-validation model. RESULTS: Although 411 genes were differentially expressed between normal and tumor tissue, only 103 genes were altered between responder and non-responder tumor; and 67 genes differentially expressed >2-fold. These included genes previously reported in esophageal cancer and a number of novel genes. In the validation cohort, 8 of 12 selected genes were significantly different between the response groups. In the predictive model, 5 of 8 genes could predict response to therapy with 95% accuracy in a subset (74%) of patients. CONCLUSIONS: This study has identified a gene microarray pattern and a set of genes associated with response to neoadjuvant chemoradiation in esophageal cancer. The potential of these genes as biomarkers of response to treatment warrants further investigation. Copyright © 2009 by Lippincott Williams & Wilkins.

Differential pathologic variables and outcomes across the spectrum of adenocarcinoma of the esophagogastric junction

Background Adenocarcinoma of the esophagogastric junction (AEG) as described by Siewert et al. is classified as one entity in the latest (7th Edition) American Joint Cancer Committee/International Union Against Cancer (AJCC/UICC) manual, compared with the previous mix of esophageal and gastric staging systems. The origin of AEG tumors, esophageal or gastric, and their biology remain controversial, particularly for AEG type II (cardia) tumors. Methods We adapted a large prospective database (n = 520: 180 type I, 182 type II, 158 type III) to compare AEG tumors under the new TNM system Pathological variables associated with prognosis were compared (pT, pN, stage, differentiation, R status, lymphovascular invasion, perineural involvement, number of positive nodes, percent of positive nodes, and tumor length), as well as overall survival. Results Compared with AEG type I tumors, type II and type III tumors had significantly (p\0.05) more advanced pN stages, greater number and percentage of positive nodes, poorer differentiation, more radial margin involvement, and more perineural invasion. In AEG type I, 14/180 patients (8%) had[6 involved nodes (pN3), compared with 16 and 30% of patients classified type II and III, respectively. Median survival was significantly (p = 0.03) improved for type I patients (38 months) compared with those with tumors classified as type II (28 months) and type III (24 months). In multivariate analysis node positivity and pN staging but not AEG site had an impact on survival. Conclusions In this series AEG type I is associated with more favorable pathologic features and improved outcomes compared with AEG type II and III. This may reflect earlier diagnosis, but an alternative possibility, that type I may be a unique paradigm with more favorable biology, requires further study. © Société Internationale de Chirurgie 2010.

Metabolomic analysis of siderophore cheater mutants reveals metabolic costs of expression in uropathogenic escherichia coli

Bacterial siderophores are a group of chemically diverse, virulence-associated secondary metabolites whose expression exerts metabolic costs. A combined bacterial genetic and metabolomic approach revealed differential metabolomic impacts associated with biosynthesis of different siderophore structural families. Despite myriad genetic differences, the metabolome of a cheater mutant lacking a single set of siderophore biosynthetic genes more closely approximate that of a nonpathogenic K12 strain than its isogenic, uropathogen parent strain. Siderophore types associated with greater metabolomic perturbations are less common among human isolates, suggesting that metabolic costs influence success in a human population. Although different siderophores share a common iron acquisition function, our analysis shows how a metabolomic approach can distinguish their relative metabolic impacts in E.coli.

Genome-wide DNA methylation analysis of human brain tissue from schizophrenia patients

Recent studies suggest that genetic and environmental factors do not account for all the schizophrenia risk and epigenetics also plays a role in disease susceptibility. DNA methylation is a heritable epigenetic modification that can regulate gene expression. Genome-Wide DNA methylation analysis was performed on post-mortem human brain tissue from 24 patients with schizophrenia and 24 unaffected controls. DNA methylation was assessed at over 485 000 CpG sites using the Illumina Infinium Human Methylation450 Bead Chip. After adjusting for age and post-mortem interval (PMI), 4 641 probes corresponding to 2 929 unique genes were found to be differentially methylated. Of those genes, 1 291 were located in a CpG island and 817 were in a promoter region. These include NOS1, AKT1, DTNBP1, DNMT1, PPP3CC and SOX10 which have previously been associated with schizophrenia. More than 100 of these genes overlap with a previous DNA methylation study of peripheral blood from schizophrenia patients in which 27 000 CpG sites were analysed. Unsupervised clustering analysis of the top 3 000 most variable probes revealed two distinct groups with significantly more people with schizophrenia in cluster one compared to controls (p = 1.74x10-4). The first cluster was composed of 88% of patients with schizophrenia and only 12% controls while the second cluster was composed of 27% of patients with schizophrenia and 73% controls. These results strongly suggest that differential DNA methylation is important in schizophrenia etiology and add support for the use of DNA methylation profiles as a future prognostic indicator of schizophrenia.