Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations

Maximum-likelihood estimates of the parameters of stochastic differential equations are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed-form expression for the transitional probability density function of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This article provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox–Ingersoll–Ross and Ornstein–Uhlenbeck equations respectively.

Modeling prostate cancer: a perspective on transgenic mouse models

Despite considerable success in treatment of early stage localized prostate cancer (PC), acute inadequacy of late stage PC treatment and its inherent heterogeneity poses a formidable challenge. Clearly, an improved understanding of PC genesis and progression along with the development of new targeted therapies are warranted. Animal models, especially, transgenic immunocompetent mouse models, have proven to be the best ally in this respect. A series of models have been developed by modulation of expression of genes implicated in cancer-genesis and progression; mainly, modulation of expression of oncogenes, steroid hormone receptors, growth factors and their receptors, cell cycle and apoptosis regulators, and tumor suppressor genes have been used. Such models have contributed significantly to our understanding of the molecular and pathological aspects of PC initiation and progression. In particular, the transgenic mouse models based on multiple genetic alterations can more accurately address the inherent complexity of PC, not only in revealing the mechanisms of tumorigenesis and progression but also for clinically relevant evaluation of new therapies. Further, with advances in conditional knockout technologies, otherwise embryonically lethal gene changes can be incorporated leading to the development of new generation transgenics, thus adding significantly to our existing knowledge base. Different models and their relevance to PC research are discussed.

Semantic modeling method for configurable enterprise information systems

Business practices vary from one company to another and business practices often need to be changed due to changes of business environments. To satisfy different business practices, enterprise systems need to be customized. To keep up with ongoing business practice changes, enterprise systems need to be adapted. Because of rigidity and complexity, the customization and adaption of enterprise systems often takes excessive time with potential failures and budget shortfall. Moreover, enterprise systems often drag business behind because they cannot be rapidly adapted to support business practice changes. Extensive literature has addressed this issue by identifying success or failure factors, implementation approaches, and project management strategies. Those efforts were aimed at learning lessons from post implementation experiences to help future projects. This research looks into this issue from a different angle. It attempts to address this issue by delivering a systematic method for developing flexible enterprise systems which can be easily tailored for different business practices or rapidly adapted when business practices change. First, this research examines the role of system models in the context of enterprise system development; and the relationship of system models with software programs in the contexts of computer aided software engineering (CASE), model driven architecture (MDA) and workflow management system (WfMS). Then, by applying the analogical reasoning method, this research initiates a concept of model driven enterprise systems. The novelty of model driven enterprise systems is that it extracts system models from software programs and makes system models able to stay independent of software programs. In the paradigm of model driven enterprise systems, system models act as instructors to guide and control the behavior of software programs. Software programs function by interpreting instructions in system models. This mechanism exposes the opportunity to tailor such a system by changing system models. To make this true, system models should be represented in a language which can be easily understood by human beings and can also be effectively interpreted by computers. In this research, various semantic representations are investigated to support model driven enterprise systems. The significance of this research is 1) the transplantation of the successful structure for flexibility in modern machines and WfMS to enterprise systems; and 2) the advancement of MDA by extending the role of system models from guiding system development to controlling system behaviors. This research contributes to the area relevant to enterprise systems from three perspectives: 1) a new paradigm of enterprise systems, in which enterprise systems consist of two essential elements: system models and software programs. These two elements are loosely coupled and can exist independently; 2) semantic representations, which can effectively represent business entities, entity relationships, business logic and information processing logic in a semantic manner. Semantic representations are the key enabling techniques of model driven enterprise systems; and 3) a brand new role of system models; traditionally the role of system models is to guide developers to write system source code. This research promotes the role of system models to control the behaviors of enterprise.

Generalized binomial Tau-leap method for biochemical kinetics incorporating both delay and intrinsic noise

The delay stochastic simulation algorithm (DSSA) by Barrio et al. [Plos Comput. Biol.2, 117–E (2006)] was developed to simulate delayed processes in cell biology in the presence of intrinsic noise, that is, when there are small-to-moderate numbers of certain key molecules present in a chemical reaction system. These delayed processes can faithfully represent complex interactions and mechanisms that imply a number of spatiotemporal processes often not explicitly modeled such as transcription and translation, basic in the modeling of cell signaling pathways. However, for systems with widely varying reaction rate constants or large numbers of molecules, the simulation time steps of both the stochastic simulation algorithm (SSA) and the DSSA can become very small causing considerable computational overheads. In order to overcome the limit of small step sizes, various τ-leap strategies have been suggested for improving computational performance of the SSA. In this paper, we present a binomial τ- DSSA method that extends the τ-leap idea to the delay setting and avoids drawing insufficient numbers of reactions, a common shortcoming of existing binomial τ-leap methods that becomes evident when dealing with complex chemical interactions. The resulting inaccuracies are most evident in the delayed case, even when considering reaction products as potential reactants within the same time step in which they are produced. Moreover, we extend the framework to account for multicellular systems with different degrees of intercellular communication. We apply these ideas to two important genetic regulatory models, namely, the hes1 gene, implicated as a molecular clock, and a Her1/Her 7 model for coupled oscillating cells.

Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Stochastic simulation of chemical reactions in spatially complex media

Discrete stochastic simulations, via techniques such as the Stochastic Simulation Algorithm (SSA) are a powerful tool for understanding the dynamics of chemical kinetics when there are low numbers of certain molecular species. However, an important constraint is the assumption of well-mixedness and homogeneity. In this paper, we show how to use Monte Carlo simulations to estimate an anomalous diffusion parameter that encapsulates the crowdedness of the spatial environment. We then use this parameter to replace the rate constants of bimolecular reactions by a time-dependent power law to produce an SSA valid in cases where anomalous diffusion occurs or the system is not well-mixed (ASSA). Simulations then show that ASSA can successfully predict the temporal dynamics of chemical kinetics in a spatially constrained environment.

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.