Doi-Peliti path integral methods for stochastic systems with partial exclusion

Doi-Peliti methods are developed for stochastic models with finite maximum occupation numbers per site. We provide a generalized framework for the different Fock spaces reported in the literature. Paragrassmannian techniques are then utilized to construct path integral formulations of factorial moments. We show that for many models of interest, a Magnus expansion is required to construct a suitable action, meaning actions containing a finite number of terms are not always feasible. However, for such systems, perturbative techniques are still viable, and for some examples, including carrying capacity population dynamics, and diffusion with partial exclusion, the expansions are exactly summable

PDE models of adder mechanisms in cellular proliferation

Cell division is a process that involves many biochemical steps and complex biophysical mechanisms. To simplify the understanding of what triggers cell division, three basic models that subsume more microscopic cellular processes associated with cell division have been proposed. Cells can divide based on the time elapsed since their birth, their size, and/or the volume added since their birth-the timer, sizer, and adder models, respectively. Here, we propose unified adder-sizer models and investigate some of the properties of different adder processes arising in cellular proliferation. Although the adder-sizer model provides a direct way to model cell population structure, we illustrate how it is mathematically related to the well-known model in which cell division depends on age and size. Existence and uniqueness of weak solutions to our 2+1-dimensional PDE model are proved, leading to the convergence of the discretized numerical solutions and allowing us to numerically compute the dynamics of cell population densities. We then generalize our PDE model to incorporate recent experimental findings of a system exhibiting mother-daughter correlations in cellular growth rates. Numerical experiments illustrating possible average cell volume blowup and the dynamical behavior of cell populations with mother-daughter correlated growth rates are carried out. Finally, motivated by new experimental findings, we extend our adder model cases where the controlling variable is the added size between DNA replication initiation points in the cell cycle

Reaction Diffusion Systems and Extensions of Quantum Stochastic Processes

Reaction diffusion systems describe the behaviour of dynamic, interacting, particulate systems. Quantum stochastic processes generalise Brownian motion and Poisson processes, having operator valued It\^{o} calculus machinery. Here it is shown that the three standard noises of quantum stochastic processes can be extended to model reaction diffusion systems, the methods being exemplified with spatial birth-death processes. The usual approach for these systems are master equations, or Doi-Peliti path integration techniques. The machinery described here provide efficient analyses for many systems of interest, and offer an alternative set of tools to investigate such problems.Comment: 29 page

Time series path integral expansions for stochastic processes

A form of time series path integral expansion is provided that enables both analytic and numerical temporal effect calculations for a range of stochastic processes. All methods rely on finding a suitable reproducing kernel associated with an underlying representative algebra to perform the expansion. Birth–death processes can be analysed with these techniques, using either standard Doi-Peliti coherent states, or the su(1 , 1) Lie algebra. These result in simplest expansions for processes with linear or quadratic rates, respectively. The techniques are also adapted to diffusion processes. The resulting series differ from those found in standard Dyson time series field theory techniques

Computational Cancer Biology: An Evolutionary Perspective

ISSN:1553-734XISSN:1553-735

The relative timing of mutations in a breast cancer genome.

Many tumors have highly rearranged genomes, but a major unknown is the relative importance and timing of genome rearrangements compared to sequence-level mutation. Chromosome instability might arise early, be a late event contributing little to cancer development, or happen as a single catastrophic event. Another unknown is which of the point mutations and rearrangements are selected. To address these questions we show, using the breast cancer cell line HCC1187 as a model, that we can reconstruct the likely history of a breast cancer genome. We assembled probably the most complete map to date of a cancer genome, by combining molecular cytogenetic analysis with sequence data. In particular, we assigned most sequence-level mutations to individual chromosomes by sequencing of flow sorted chromosomes. The parent of origin of each chromosome was assigned from SNP arrays. We were then able to classify most of the mutations as earlier or later according to whether they occurred before or after a landmark event in the evolution of the genome, endoreduplication (duplication of its entire genome). Genome rearrangements and sequence-level mutations were fairly evenly divided earlier and later, suggesting that genetic instability was relatively constant throughout the life of this tumor, and chromosome instability was not a late event. Mutations that caused chromosome instability would be in the earlier set. Strikingly, the great majority of inactivating mutations and in-frame gene fusions happened earlier. The non-random timing of some of the mutations may be evidence that they were selected

A Path Integral Approach to Age Dependent Branching Processes

Age dependent population dynamics are frequently modeled with generalizations of the classic McKendrick-von Foerster equation. These are deterministic systems, and a stochastic generalization was recently reported in [1,2]. Here we develop a fully stochastic theory for age-structured populations via quantum field theoretical Doi-Peliti techniques. This results in a path integral formulation where birth and death events correspond to cubic and quadratic interaction terms. This formalism allows us to efficiently recapitulate the results in [1,2], exemplifying the utility of Doi-Peliti methods. Furthermore, we find that the path integral formulation for age-structured moments has an exact perturbative expansion that explicitly relates to the hereditary structure between correlated individuals. These methods are then generalized with a binary fission model of cell division

A hierarchical kinetic theory of birth, death, and fission in age-structured interacting populations

We study mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we develop a complete kinetic framework for age-structured interacting populations undergoing birth, death and fission processes in spatially dependent environments. We define the full probability density for the population-size age chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behavior. Our approach utilizes mixed-type, multidimensional probability distributions similar to those employed in the study of gas kinetics and with terms that satisfy BBGKY-like equation hierarchies

Heterarchy of Transcription Factors Driving Basal and Luminal Cell Phenotypes in Human Urothelium

Cell differentiation is effected by complex networks of transcription factors that co-ordinate re-organisation of the chromatin landscape. The hierarchies of these relationships can be difficult to dissect. During in vitro differentiation of normal human uro-epithelial cells, formaldehyde-assisted isolation of regulatory elements (FAIRE-seq) and RNA-seq were used to identify alterations in chromatin accessibility and gene expression changes following activation of the nuclear receptor PPARG as a differentiation-initiating event. Regions of chromatin identified by FAIRE-seq as having altered accessibility during differentiation were found to be enriched with sequence-specific binding motifs for transcription factors predicted to be involved in driving basal and differentiated urothelial cell phenotypes, including FOXA1, P63, GRHL2, CTCF and GATA3. In addition, co-occurrence of GATA3 motifs was observed within sub-sets of differentiation-specific peaks containing P63 or FOXA1 after induction of differentiation. Changes in abundance of GRHL2, GATA3, and P63 were observed in immunoblots of chromatin-enriched extracts. Transient siRNA knockdown of P63 revealed that P63 favoured a basal-like phenotype by inhibiting differentiation and promoting expression of basal marker genes. GATA3 siRNA prevented differentiation-associated downregulation of P63 protein and transcript, and demonstrated positive feedback of GATA3 on PPARG transcript, but showed no effect on FOXA1 transcript or protein expression. This approach indicates that as a transcriptionally-regulated programme, urothelial differentiation operates as a heterarchy wherein GATA3 is able to co-operate with FOXA1 to drive expression of luminal marker genes, but that P63 has potential to transrepress expression of the same genes