Biology subjects

Durrett, R.

Publications and source records attributed to Durrett, R..

5 recordsLinked to original sources

bioRxiv2018

A heterogeneous spatial model in which savanna and forest coexist in a stable equilibrium

In work with a variety of co-authors, Staver and Levin have argued that savanna and forest coexist as alternative stable states with discontinuous changes in density of trees at the boundary. Here we formulate a nonhomogeneous spatial model of the competition between forest and savanna. We prove that coexistence occurs for a time that is exponential in the size of the system, and that after an initial transient, boundaries between the alternative equilibria remain stable.

ecology↗

bioRxiv2018

Genotype patterns in growing solid tumors

Over the past decade, the theory of tumor evolution has largely focused on the selective sweeps model. According to this theory, tumors evolve by a succession of clonal expansions that are initiated by driver mutations. In a 2015 analysis of colon cancer data, Sottoriva et al [34] proposed an alternative theory of tumor evolution, the so-called Big Bang model, in which one or more driver mutations are acquired by the founder gland, and the evolutionary dynamics within the expanding population are predominantly neutral. In this paper we will describe a simple mathematical model that reproduces qualitative features of the observed paatterns of genetic variability and makes quantitative predictions.

cancer biology↗

bioRxiv2018

Extrapolating Weak Selection in Evolutionary Games

This work is inspired by a 2013 paper from Arne Traulsens lab at the Max Plank Institute for Evolutionary Biology [10]. They studied the small mutation limit of evolutionary games. It has been shown that for 2 x 2 games the ranking of the strategies does not change as strength of selection is increased [11]. The point of the 2013 paper is that when there are three or more strategies the ordering can change as selection is increased. Wu et al [10] did numerical computations for fixed N. Here, we will instead let the strength of selection {beta} = c/N and let N [->] {infty} to obtain formulas for the invadability probabilities{phi} ij that determine the rankings. These formulas, which are integrals on [0, 1], are intractable calculus problems but can be easily evaluated numerically. Here, we concentrate on simple formulas for the ranking order when c is small or c is large. The results we have obtained for five concrete examples lead us to doubt the accuracy of insights that are derived in the small mutation limit for games with three strategies.

evolutionary biology↗

bioRxiv2018

Genealogies in Growing Solid Tumors

Over the past two decades, the theory of tumor evolution has largely focused on the selective sweeps model. According to this theory, tumors evolve by a succession of clonal expansions that are initiated by driver mutations that have a fitness advantage over the resident types. A 2015 study of colon cancer [44] has suggested an alternative theory of tumor evolution, the so-called Big Bang model, in which all of the necessary driver mutations are acquired before expansion began, and the evolutionary dynamics within the expanding population are predominantly neutral. In this paper, we will describe a simple mathematical model inspired by work of Hallatschek and Nelson [25] that makes quantitative predictions about spatial patterns of genetic variability. While this model has some success in matching observed patterns in two dimensions, it fails miserably in three dimensions. Despite this failure, we think the model analyzed here will be a useful first step in building an accurate model.

cancer biology↗

bioRxiv2018

A new look at multi-stage models of cancer incidence

Multi-stage models have long been used in combination with SEER data to make inferences about the mechanisms underlying cancer initiation. The main method for studying these models mathematically has been the computation of generating functions by solving hyperbolic partial differential equations. Here, we analyze these models using a probabilistic approach similar to the one Durrett and Moseley [7] used to study branching process models of cancer. This more intuitive approach leads to simpler formulas and new insights into the behavior of these models. Unfortunately, the examples we consider suggest that fitting multi-stage models has very little power to make inferences about the number of stages unless parameters are constrained to take on realistic values.

cancer biology↗