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.

Broadening of transgenic adenocarcinoma of the mouse prostate (TRAMP) model to represent late stage androgen depletion independent cancer

BACKGROUND The transgenic adenocarcinoma of the mouse prostate (TRAMP) model closely mimics PC-progression as it occurs in humans. However, the timing of disease incidence and progression (especially late stage) makes it logistically difficult to conduct experiments synchronously and economically. The development and characterization of androgen depletion independent (ADI) TRAMP sublines are reported. METHODS Sublines were derived from androgen-sensitive TRAMP-C1 and TRAMP-C2 cell lines by androgen deprivation in vitro and in vivo. Epithelial origin (cytokeratin) and expression of late stage biomarkers (E-cadherin and KAI-1) were evaluated using immunohistochemistry. Androgen receptor (AR) status was assessed through quantitative real time PCR, Western blotting, and immunohistochemistry. Coexpression of AR and E-cadherin was also evaluated. Clonogenicity and invasive potential were measured by soft agar and matrigel invasion assays. Proliferation/survival of sublines in response to androgen was assessed by WST-1 assay. In vivo growth of subcutaneous tumors was assessed in castrated and sham-castrated C57BL/6 mice. RESULTS The sublines were epithelial and displayed ADI in vitro and in vivo. Compared to the parental lines, these showed (1) significantly faster growth rates in vitro and in vivo independent of androgen depletion, (2) greater tumorigenic, and invasive potential in vitro. All showed substantial downregulation in expression levels of tumor suppressor, E-cadherin, and metastatis suppressor, KAI-1. Interestingly, the percentage of cells expressing AR with downregulated E-cadherin was higher in ADI cells, suggesting a possible interaction between the two pathways. CONCLUSIONS The TRAMP model now encompasses ADI sublines potentially representing different phenotypes with increased tumorigenicity and invasiveness.

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.

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.