Evolution on distributive lattices

Abstract

We consider the directed evolution of a population after an intervention that has significantly altered the underlying fitness landscape. We model the space of genotypes as a distributive lattice; the fitness landscape is a real-valued function on that lattice. The risk of escape from intervention, i.e., the probability that the population develops an escape mutant before extinction, is encoded in the risk polynomial. Tools from algebraic combinatorics are applied to compute the risk polynomial in terms of the fitness landscape. In an application to the development of drug resistance in HIV, we study the risk of viral escape from treatment with the protease inhibitors ritonavir and indinavir.

Introduction

The evolutionary fate of a population is determined by the replication dynamics of the ensemble and by the reproductive success of its individuals. We are interested in scenarios where most individuals have a low fitness, eventually leading to extinction, and only a few types of individuals (“escape mutants”) can survive permanently. These situations often arise due to a significant change of the underlying fitness landscape. For example, a virus that has been transmitted to a new host is confronted with a new immune response. Likewise, medical interventions such as radiation therapy, vaccination, or chemotherapy result in altered fitness landscapes for the targeted agents, which may be bacteria, viruses, or cancer cells.

Given a population and such a hostile fitness landscape, the central question is whether the population will survive. In the case of medical interventions we wish to know the probability of successful treatment. Answering this question involves computing the risk of evolutionary escape, i.e., the probability that the population develops an escape mutant before extinction. We present a mathematical framework for computing such probabilities.

Our primary application is the evolution of drug resistance during treatment of HIV infected patients (Clavel and Hance, 2004). We consider therapy with two different protease inhibitors (PIs). These compounds interfere with HIV particle maturation by inhibiting the viral protease enzyme. The effectiveness of PI therapy is limited by the development of drug resistance. Rapid and highly error prone replication of a large virus population generates mutants that resist the selective pressure of drug therapy. PI resistance is caused by mutations in the protease gene that reduce the binding affinity of the drug to the enzyme. These mutations have been shown to accumulate in a stepwise manner (Berkhout, 1999). For most PIs, no single mutation confers a significant level of resistance, but multiple mutations are required for escape from drug pressure. Quantitative predictions of the probability of successful PI treatment would help in finding effective antiretroviral combination therapies. Selecting a drug combination amounts to controlling the viral fitness landscape.

We regard the directed evolution of a population towards an escape state as a fluctuation on a fitness landscape. The space of genotypes is modeled as follows. We start with a finite partially ordered set (poset) $\mathcal{E}$ whose elements are called events. The events are non-reversible mutations with some constraints on their order of occurrence. Such constraints are primarily due to epistatic effects between different loci in a genome (Bonhoeffer et al., 2004). The event constraints define the poset structure: $e_1<e_2$ in $\mathcal{E}$ means that event $e_1$ must occur before event $e_2$ can occur. Each genotype g is represented by a subset of $\mathcal{E}$, namely, the set of all events that occurred to create g. Thus a genotype g is an order ideal in the poset $\mathcal{E}$. The space of genotypes $\mathcal{G}$ is the set of all order ideals in $\mathcal{E}$, which is a distributive lattice (Stanley, 1997, Section 3.4). The order relation on $\mathcal{G}$ is set inclusion and corresponds to the accumulation of mutations. This mathematical formulation is reasonable in the above situations, where a population is exposed to strong selective pressure.

The risk of escape is governed by the structure of $\mathcal{G}$, the fitness function on $\mathcal{G}$, and the population dynamics (such as the mutation rates and population size). Our focus is on the dependency of the risk of escape on the assigned fitness values for each genotype $g\in \mathcal{G}$. This leads us to the risk polynomial, which is shown to be equivalent to a well-known object in algebraic combinatorics. Indeed, one of the objectives of this work is to provide a bridge between algebraic combinatorics and evolutionary biology.

This paper is organized as follows. In Section 2 we formalize our model of a static fitness landscape on the genotype lattice $\mathcal{G}$ derived from an event poset $\mathcal{E}$, and we discuss evolution on the lattice $\mathcal{G}$. In Section 3 we review the multistate branching process studied by Iwasa et al., 2003, Iwasa et al., 2004.

In Section 4 we study the Bayesian networks which arise from identifying the events in $\mathcal{E}$ with binary random variables. These statistical models can be used to infer the genotype space from given data. For conjunctive Bayesian networks we recover the distributive lattice of order ideals in $\mathcal{E}$. Of particular interest is the case where $\mathcal{E}$ is a directed forest: here the Bayesian network is a mutagenetic tree model (Beerenwinkel and Drton, 2005; Beerenwinkel et al., 2005a, Beerenwinkel et al., 2005b). The application of our methods to the development of PI resistance in HIV is presented in Section 5.

The appendix summarizes various representations of the risk polynomial in terms of structures from algebraic combinatorics. Efficient methods for computing the risk polynomial and their implementation are presented.