Markov models for accumulating mutations

Abstract

Abstract. We introduce and analyze a waiting time model for the accumulation of genetic changes. The continuous time conjunctive Bayesian network is defined by a partially ordered set of mutations and by the rate of fixation of each mutation. The partial order encodes constraints on the order in which mutations can fixate in the population, shedding light on the mutational pathways underlying the evolutionary process. We study a censored version of the model and derive equations for an EM algorithm to perform maximum likelihood estimation of the model parameters. We also show how to select the maximum likelihood poset. The model is applied to genetic data from different cancers and from drug resistant HIV samples, indicating implications for diagnosis and treatment.

Lectures on Algebraic Statistics

“…We consider families of discrete random variables that arise from randomly censoring exponential random variables. This example describes a special case of a censored continuous time conjunctive Bayesian network; see [15] for more details and derivations. A random variable T is exponentially distributed with rate parameter λ > 0 if it has the (Lebesgue) density function…”

Simultaneous Inference of Cancer Pathways and Tumor Progression from Cross-Sectional Mutation Data

“…A number of methods for inferring temporal progression of mutations from cross-sectional data have been introduced (Desper et al, 2000(Desper et al, , 1999Beerenwinkel et al, 2005a,b;Rahnenführer et al, 2005;Tofigh et al, 2011;Hjelm et al, 2006;Gerstung et al, 2009;Beerenwinkel and Sullivant, 2009;Beerenwinkel et al, 2006Beerenwinkel et al, , 2007Gerstung et al, 2011;Sakoparnig and Beerenwinkel, 2012;Shahrabi Farahani and Lagergren, 2013) (see section 1.1). These methods consider models of increasing complexity for cancer progression: trees, mixtures of trees, and Bayesian network models with different constraints.…”

Learning Monotonic Genotype-Phenotype Maps

“…For the fixed optimal model structure shown in Figure 3, the model on posets , Beerenwinkel and Sullivant, 2009, Gerstung et al, 2009, tree posets or mixtures of trees (Beerenwinkel et al, 2005), and general Bayesian networks (Deforche et al, 2006). It can also be regarded as a model for regressing viral resistance phenotype on genotype.…”

“…Several statistical models have been proposed for this purpose, including Bayesian networks (Klingler and Brutlag, 1994, Deforche, Silander, Camacho, Grossman, Soares, Laethem, Kantor, Moreau, and Va n d a m m e , 2006, Poon, Lewis, Pond, and Frost, 2007) and dependency networks (Carlson, Brumme, Rousseau, Brumme, Matthews, Kadie, Mullins, Walker, Harrigan, Goulder, and Heckerman, 2008). Order constraints represent a specific type of dependency and a specialized Bayesian network model, called conjuctive Bayesian network (CBN), has been proposed that uses a partial order to represent these constraints (Beerenwinkel, Eriksson, and Sturmfels, 2006, Beerenwinkel and Sullivant, 2009, Gerstung, Baudis, Moch, and Beerenwinkel, 2009.…”