Master Equation For Models Research Articles

Mathematical Modelling of Clonal Stem Cell Dynamics.

Studying cell fate dynamics is complicated by the fact that direct in vivo observation of individual cell fate outcomesis usually not possible and only multicellular data of cell clones can be obtained. In this situation, experimental dataalone is not sufficient to validate biological models because the hypotheses and the datacannot be directly compared and thus standard statistical tests cannot be leveraged. On the other hand, mathematical modelling can bridge the scales between a hypothesis and measured data via quantitative predictions from a mathematical model. Here, we describe how to implement the rules behind a hypothesis (cell fate outcomes) one-to-one as a stochastic model, how to evaluate such a rule-based model mathematicallyvia analytical calculation or stochastic simulations of the model's Master equation, and to predict the outcomes of clonal statistics for respective hypotheses. We also illustrate two approaches to compare these predictions directly with the clonal data to assess the models.

Monte carlo simulation of bacterial disinfection: Nonlinear and time-explicit intensity of transition

Bacteria being disinfected in fluid media are discrete entities and mesoscopic in size; moreover, they are incessantly as well as irregularly in motion and in collision among themselves or with the surrounding solid surfaces. As such, it is highly likely that some of the attributes of the bacterial population, for example, their number concentration, will fluctuate randomly. This is especially the case at the tail-end of disinfection when the population of bacteria is sparse. It might be effectual, therefore, to explore the resultant random fluctuations via a stochastic paradigm. Proposed herein is a Markovian stochastic model for the rate of bacterial disinfection, whose intensity of transition takes into account the contact time of the bacteria with the disinfecting agent to eliminate any given percentage of the bacteria in terms of a nonlinear function of time. The model's master equation has been simulated by resorting to the Monte Carlo method to circumvent the undue complexities in solving it analytically or numerically via conventional numerical techniques. For illustration, the mean, the variance (standard deviation), and the coefficient of variation of the number concentration of bacteria during disinfection have been estimated through Monte Carlo simulation. The results of simulation compare favorably with the available experimental data as well as with those computed from the corresponding deterministic model.