Measuring the Initial Transient: Reflected Brownian Motion

We analyze the convergence to equilibrium of one-dimensional reflected Brownian motion (RBM) and compute a number of related initial transient formulae. These formulae are of interest as approximations to the initial transient for queueing systems in heavy traffic, and help us to identify settings in which initialization bias is significant. We conclude with a discussion of mean square error for RBM. Our analysis supports the view that initial transient effects for RBM and related models are typically of modest size relative to the intrinsic stochastic variability, unless one chooses an especially poor initialization.Comment: 14 pages, 3 figure

Mathematical Modelling of Cell-Fate Decision in Response to Death Receptor Engagement

Cytokines such as TNF and FASL can trigger death or survival depending on cell lines and cellular conditions. The mechanistic details of how a cell chooses among these cell fates are still unclear. The understanding of these processes is important since they are altered in many diseases, including cancer and AIDS. Using a discrete modelling formalism, we present a mathematical model of cell fate decision recapitulating and integrating the most consistent facts extracted from the literature. This model provides a generic high-level view of the interplays between NFκB pro-survival pathway, RIP1-dependent necrosis, and the apoptosis pathway in response to death receptor-mediated signals. Wild type simulations demonstrate robust segregation of cellular responses to receptor engagement. Model simulations recapitulate documented phenotypes of protein knockdowns and enable the prediction of the effects of novel knockdowns. In silico experiments simulate the outcomes following ligand removal at different stages, and suggest experimental approaches to further validate and specialise the model for particular cell types. We also propose a reduced conceptual model implementing the logic of the decision process. This analysis gives specific predictions regarding cross-talks between the three pathways, as well as the transient role of RIP1 protein in necrosis, and confirms the phenotypes of novel perturbations. Our wild type and mutant simulations provide novel insights to restore apoptosis in defective cells. The model analysis expands our understanding of how cell fate decision is made. Moreover, our current model can be used to assess contradictory or controversial data from the literature. Ultimately, it constitutes a valuable reasoning tool to delineate novel experiments

Oscillation patterns in negative feedback loops

Organisms are equipped with regulatory systems that display a variety of dynamical behaviours ranging from simple stable steady states, to switching and multistability, to oscillations. Earlier work has shown that oscillations in protein concentrations or gene expression levels are related to the presence of at least one negative feedback loop in the regulatory network. Here we study the dynamics of a very general class of negative feedback loops. Our main result is that in these systems the sequence of maxima and minima of the concentrations is uniquely determined by the topology of the loop and the activating/repressing nature of the interaction between pairs of variables. This allows us to devise an algorithm to reconstruct the topology of oscillating negative feedback loops from their time series; this method applies even when some variables are missing from the data set, or if the time series shows transients, like damped oscillations. We illustrate the relevance and the limits of validity of our method with three examples: p53-Mdm2 oscillations, circadian gene expression in cyanobacteria, and cyclic binding of cofactors at the estrogen-sensitive pS2 promoter.Comment: 10 pages, 8 figure

Action semantics of unified modeling language

The Uni ed Modeling Language or UML, as a visual and general purpose modeling language, has been around for more than a decade, gaining increasingly wide application and becoming the de-facto industrial standard for modeling software systems. However, the dynamic semantics of UML behaviours are only described in natural languages. Speci cation in natural languages inevitably involves vagueness, lacks reasonability and discourages mechanical language implementation. Such semi-formality of UML causes wide concern for researchers, including us. The formal semantics of UML demands more readability and extensibility due to its fast evolution and a wider range of users. Therefore we adopt Action Semantics (AS), mainly created by Peter Mosses, to formalize the dynamic semantics of UML, because AS can satisfy these needs advantageously compared to other frameworks. Instead of de ning UML directly, we design an action language, called ALx, and use it as the intermediary between a typical executable UML and its action semantics. ALx is highly heterogeneous, combining the features of Object Oriented Programming Languages, Object Query Languages, Model Description Languages and more complex behaviours like state machines. Adopting AS to formalize such a heterogeneous language is in turn of signi cance in exploring the adequacy and applicability of AS. In order to give assurance of the validity of the action semantics of ALx, a prototype ALx-to-Java translator is implemented, underpinned by our formal semantic description of the action language and using the Model Driven Approach (MDA). We argue that MDA is a feasible way of implementing this source-to-source language translator because the cornerstone of MDA, UML, is adequate to specify the static aspect of programming languages, and MDA provides executable transformation languages to model mapping rules between languages. We also construct a translator using a commonly-used conventional approach, in i which a tool is employed to generate the lexical scanner and the parser, and then other components including the type checker, symbol table constructor, intermediate representation producer and code generator, are coded manually. Then we compare the conventional approach with the MDA. The result shows that MDA has advantages over the conventional method in the aspect of code quality but is inferior to the latter in terms of system performance

Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations

We present a new Monte Carlo method for obtaining solutions of the Boltzmann equation for describing phonon transport in micro and nanoscale devices. The proposed method can resolve arbitrarily small signals (e.g. temperature differences) at small constant cost and thus represents a considerable improvement compared to traditional Monte Carlo methods whose cost increases quadratically with decreasing signal. This is achieved via a control-variate variance reduction formulation in which the stochastic particle description only solves for the deviation from a nearby equilibrium, while the latter is described analytically. We also show that simulating an energy-based Boltzmann equation results in an algorithm that lends itself naturally to exact energy conservation thereby considerably improving the simulation fidelity. Simulations using the proposed method are used to investigate the effect of porosity on the effective thermal conductivity of silicon. We also present simulations of a recently developed thermal conductivity spectroscopy process. The latter simulations demonstrate how the computational gains introduced by the proposed method enable the simulation of otherwise intractable multiscale phenomena

Graph Theory and Networks in Biology

In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss recent work on identifying and modelling the structure of bio-molecular networks, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronization and disease propagation.Comment: 52 pages, 5 figures, Survey Pape

In-depth analysis of the Naming Game dynamics: the homogeneous mixing case

Language emergence and evolution has recently gained growing attention through multi-agent models and mathematical frameworks to study their behavior. Here we investigate further the Naming Game, a model able to account for the emergence of a shared vocabulary of form-meaning associations through social/cultural learning. Due to the simplicity of both the structure of the agents and their interaction rules, the dynamics of this model can be analyzed in great detail using numerical simulations and analytical arguments. This paper first reviews some existing results and then presents a new overall understanding.Comment: 30 pages, 19 figures (few in reduced definition). In press in IJMP