Exact, time-independent estimation of clone size distributions in normal and mutated cells

Biological tools such as genetic lineage tracing, three dimensional confocal microscopy and next generation DNA sequencing are providing new ways to quantify the distribution of clones of normal and mutated cells. Population-wide clone size distributions in vivo are complicated by multiple cell types, and overlapping birth and death processes. This has led to the increased need for mathematically informed models to understand their biological significance. Standard approaches usually require knowledge of clonal age. We show that modelling on clone size independent of time is an alternative method that offers certain analytical advantages; it can help parameterize these models, and obtain distributions for counts of mutated or proliferating cells, for example. When applied to a general birth-death process common in epithelial progenitors this takes the form of a gamblers ruin problem, the solution of which relates to counting Motzkin lattice paths. Applying this approach to mutational processes, an alternative, exact, formulation of the classic Luria Delbruck problem emerges. This approach can be extended beyond neutral models of mutant clonal evolution, and also describe some distributions relating to sub-clones within a tumour. The approaches above are generally applicable to any Markovian branching process where the dynamics of different "coloured" daughter branches are of interest

Solving the chemical master equation using sliding windows

<p>Abstract</p> <p>Background</p> <p>The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species.</p> <p>Results</p> <p>In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy.</p> <p>Conclusions</p> <p>The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori.</p

A Simple Stochastic Model with Environmental Transmission Explains Multi-Year Periodicity in Outbreaks of Avian Flu

Avian influenza virus reveals persistent and recurrent outbreaks in North American wild waterfowl, and exhibits major outbreaks at 2–8 years intervals in duck populations. The standard susceptible-infected- recovered (SIR) framework, which includes seasonal migration and reproduction, but lacks environmental transmission, is unable to reproduce the multi-periodic patterns of avian influenza epidemics. In this paper, we argue that a fully stochastic theory based on environmental transmission provides a simple, plausible explanation for the phenomenon of multi-year periodic outbreaks of avian flu. Our theory predicts complex fluctuations with a dominant period of 2 to 8 years which essentially depends on the intensity of environmental transmission. A wavelet analysis of the observed data supports this prediction. Furthermore, using master equations and van Kampen system-size expansion techniques, we provide an analytical expression for the spectrum of stochastic fluctuations, revealing how the outbreak period varies with the environmental transmission

Dynamical density functional theory for orientable colloids including inertia and hydrodynamic interactions

Over the last few decades, classical density-functional theory (DFT) and its dynamic extensions (DDFTs) have become powerful tools in the study of colloidal fluids. Recently, previous DDFTs for spherically-symmetric particles have been generalised to take into account both inertia and hydrodynamic interactions, two effects which strongly influence non-equilibrium properties. The present work further generalises this framework to systems of anisotropic particles. Starting from the Liouville equation and utilising Zwanzig's projection-operator techniques, we derive the kinetic equation for the Brownian particle distribution function, and by averaging over all but one particle, a DDFT equation is obtained. Whilst this equation has some similarities with DDFTs for spherically-symmetric colloids, it involves a translational-rotational coupling which affects the diffusivity of the (asymmetric) particles. We further show that, in the overdamped (high friction) limit, the DDFT is considerably simplified and is in agreement with a previous DDFT for colloids with arbitrary shape particles.Comment: dynamical density functional theory ; colloidal fluids ; arbitrary-shape particles ; orientable colloid

Recent Developments in Particle Tracking Diagnostics for Turbulence Research

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