Simple quasispecies models for the survival-of-the-flattest effect: The role of space

doi:10.1016/j.jtbi.2007.10.027

Journal of Theoretical Biology

Volume 250, Issue 3, 7 February 2008, Pages 560-568

https://doi.org/10.1016/j.jtbi.2007.10.027 Get rights and content

Abstract

The survival-of-the-flattest effect postulates that under high mutation rates natural selection does not necessarily favor the faster replicators. Under such conditions, genotypes which are robust against deleterious mutational effects may be favored instead, even at the cost of a slower replication. This tantalizing hypothesis has been recently proved using digital organisms, subviral RNA plant pathogens (viroids), and an animal RNA virus. In this work we study a simple theoretical system composed by two competing quasispecies which are located at two widely different fitness landscapes that represent, respectively, a fit and a flat quasispecies. The fit quasispecies is characterized by high replication rate and low mutational robustness, whereas the flat quasispecies is characterized by low replication rate but high mutational robustness. By using a mean field model, in silico simulations with digital replicons and a two-dimensional spatial model given by a stochastic cellular automata (CA), we predict the presence of an absorbing first-order phase transition with critical slowing down between selection for replication speed and selection for mutational robustness, where the surpassing of a critical mutation rate involves the outcompetition of the fit quasispecies by the flat one. Furthermore, it is shown that space, which involves a lower critical mutation rate, broadens the conditions at which the survival-of-the-flattest may occur.

Introduction

The quasispecies theory of molecular evolution, originally developed by Eigen and collaborators (Eigen et al., 1988, Eigen and Schuster, 1979) has become the standard theoretical framework used to model the evolution of RNA viruses (Domingo and Holland, 1997, Domingo, 2002). One of the keystones of theoretical quasispecies, shared by RNA viruses, is that replication fidelity is so low that the number of mutant offspring generated in a population may exceed the number of offspring identical to the parental genotype. This gives rise to highly polymorphic populations in which the frequency of the wildtype and of each mutant genotype not only depends on their replication rates but also on their constant genesis by mutation from genotypes which are close in genotypic space. In a constant environment and in the absence of other external perturbations, this distribution of genotypes is known as the quasispecies (Eigen et al., 1988).

The existence of a population structure in quasispecies strongly affects the way selection acts, because the evolutionary success of individual genomes does not depend anymore on their own replication rate but also on the average growth rate of the quasispecies they belong to. Fast replicating genomes that produce low-fitness offspring can be outcompeted by slow replicating genomes provided the latter inhabits a region of sequence space characterized by high neutrality and connectivity (Schuster and Swetina, 1988, van Nimwegen et al., 1999, Wilke, 2001b). This phenomenon has been dubbed as the quasispecies effect (van Nimwegen et al., 1999, Wilke, 2001a) or more recently as the survival-of-the-flattest (Wilke et al., 2001) in clear reference to Darwin's survival-of-the-fittest concept. Indeed, authors who have casted doubts about the relevance of the quasispecies model to real viruses based their criticism in the fact that the quasispecies effect was never observed in vivo (Holmes and Moya, 2002, Jenkins et al., 2001). However, two recent experiments give strong support to the validity of the quasispecies effect for real viral populations. In the first experiment, two viroids (small circular RNAs that infect plants and do not encode for any protein) populations were allowed to compete at increasing mutation rates (Codoñer et al., 2006). At low mutation rate the faster but genetically homogeneous replicator outcompeted the slower but highly polymorphic one, as expected from the survival-of-the-fittest effect. However, the result of the competition was reversed at high mutation rate and the slower replicator won the competition by taking advantage of its larger mutational robustness. In the second study (Sanjuán et al., 2007), two populations of Vesicular stomatitis virus (VSV) that differed in their replication rates and robustness competed at increasing concentrations of chemical mutagens. Below a certain concentration of mutagens, the fittest VSV outcompeted the flattest one. Above this critical concentration, the competition result was reverted and the flattest VSV systematically displaced the fittest one.

One peculiarity of viral infections that is usually ignored by most quasispecies models is the existence of a spatial population structure. Only a few studies have analyzed the error–threshold transition in a spatial context (Altemeyer and McCaskill, 2001, Toyabe and Sano, 2005). It is well known that spatial dynamics can deeply change the outcome of competition even under the absence of selection (Solé and Bascompte, 2006). Within infected hosts, viruses do not behave as a single well-stirred population but as a collection of subpopulations that colonize and reproduce on different compartments constituted by different tissues and organs. For example, it has been extensively shown that after infection, the Human immunodeficiency virus type 1 (HIV-1) is able to establish well-differentiated populations which are organ-specific and show limited gene flow among them (Bordería et al., 2007, Sanjuán et al., 2004). In these cases, organ-specificity appears not as a consequence of founder events but as a consequence of differences in the adaptive constraints imposed by heterogeneous cell types and the existence of fitness tradeoffs across organs. In the case of plant viruses, spatial structure appears at two different levels. First, it has been shown that different leaves can be infected with different viral subpopulations, which, for the case of perennial plants, may further differentiate into different branches and sub-branches, as it has been shown, for example, for Plum pox virus (PPV) (Jridi et al., 2006). In this case, population structure likely arises as a consequence of the strong bottlenecks associated with the systemic movement of viruses from source to sink leaves (Hall et al., 2001, Li and Roossinck, 2004, Sacristán et al., 2003). Second, within a given infected leaf, that for the sake of simplicity can be considered as a two-dimensional space, populations initiated at different infectious foci do not overlap after confluence but exclude each other, generating a patched distribution of genotypes (Dietrich and Maiss, 2003) (see Fig. 1).

In the present report we analyze the dynamics of two competing quasispecies by using a simple mean field model, in silico simulations with digital replicons, and a stochastic cellular automata (CA) model, which allows to simulate the competition process explicitly considering the potential effect of physical space. In all these approaches, the fit quasispecies has a fast replication rate but is surrounded by genotypes which are strongly deleterious and thus mutations always have a strong negative effect on the average population fitness. On the contrary, the flat one has a lower replication rate but is located at a neutral and highly connected region of sequence space, and thus mutations exert a mild impact on viral fitness. In particular, we are interested in studying the effect of increasing mutation rate on the outcome of the competition process, paying special attention to the role of spatial structure. In short, our results suggest the existence of a critical mutation rate at which an absorbing first-order phase transition with slowing down between selection for fast replication and selection for mutational robustness takes place. Beyond the critical mutation rate, the flat quasispecies outcompetes the fit one. In the vicinity of this transition both quasispecies can coexist during an extremely long time. Moreover, space is shown to broaden the conditions at which the survival-of-the-flattest can be observed. The results of the in silico simulations are in complete agreement with the mean field analysis.

Section snippets

Quasispecies mean field model

As a first approach to the analysis of the competition dynamics between two quasispecies located at two different fitness landscapes, a mean field model has been used. The model considers a perfectly mixed system with a fit quasispecies, x, and a flat one, y. Under the assumption $\sum_{i} (x_{i} + y_{i}) = 1$ , the model is defined by the following set of ODEs: $\frac{d x_{0}}{d t} = f_{x}^{(0)} {Qx}_{0} - x_{0} Φ (x, y) - ε x_{0},$ $\frac{d x_{1}}{d t} = f_{x}^{(0)} (1 - Q) x_{0} + f_{x}^{(1)} x_{1} - x_{1} Φ (x, y) - ε x_{1},$ $\frac{d y_{0}}{d t} = f_{y}^{(0)} {Qy}_{0} + f_{y}^{(1)} y_{1} (1 - Q) - y_{0} Φ (x, y) - ε y_{0},$ $\frac{d y_{1}}{d t} = f_{y}^{(1)} {Qy}_{1} + f_{y}^{(0)} y_{0} (1 - Q) - y_{1} Φ (x, y) - ε y_{1},$ where

In silico bit string model

Next, we extended the previous mean field model by considering two competing populations of quasispecies in the form of binary digital replicators, thus considering the whole mutant spectrum of the quasispecies, within a multidimensional sequence space. This computational approach has been previously used to study viral dynamics and evolution (Codoñer et al., 2006, Solé et al., 1999, Solé et al., 2006). Let us define two populations of digital replicons, given by a fit quasispecies, $S_{i}^{F}$ , and a

Bit string spatial model

We finally develop a spatially extended stochastic model including the whole quasispecies structure of both competing quasispecies. We define a $L \times L$ state space $Ω (L) \in Z^{2}$ , with zero-flux boundary conditions (simulating, for example, the bounded system of plant leaves). Each quasispecies (the fit, $S_{i}^{F} = (s_{i 1}^{F}, \dots, s_{i ν}^{F})$ , and the flat, $S_{i}^{f} = (s_{i 1}^{f}, \dots, s_{i ν}^{f})$ ) have a potential population composed of $| H^{ν} |$ (with $H$ $= 2$ and $ν = 16$ ) binary strings which define the states of the automaton and correspond to the whole

Discussion

In the present work we have explored the dynamics of competition between two quasispecies located at two widely different fitness landscapes that correspond, respectively, to a high-fitness and low-robustness (the fittest landscape) and to a low-fitness and high-robustness (the flattest landscape) scenarios. The effect of different mutation rates on the outcome of the competition between these two quasispecies has been explored. A mean field model, in silico simulations with binary digital

Acknowledgments

This work has been supported by an E.U. PACE Project grant to J. Sardanyés within the 6th Framework Program under Contract FP6-002035 (Programmable Artificial Cell Evolution) and by the Santa Fe Institute. Work in València was supported by Grant BFU2006-14819-C02-01/BMC from the Spanish MEC-FEDER and by the EMBO Young Investigator Program.

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Simple quasispecies models for the survival-of-the-flattest effect: The role of space

Abstract

Introduction

Section snippets

Quasispecies mean field model

In silico bit string model

Bit string spatial model

Discussion

Acknowledgments

Virology

Bull. Math. Biol.

J. Theor. Biol.

Physica D

Bull. Math. Biol.

Error threshold for spatially resolved evolution in the quasispecies model

Phys. Rev. Lett.

Selection promotes organ compartmentalization in HIV-1: evidence from gag and pol genes

Evolution

Fluorescent labeling reveals spatial separation of potyvirus populations in mixed infected “Nicotiana benthamiana” plants

J. Gen. Virol.

Quasispecies theory in virology

J. Virol.

RNA virus mutations and fitness for survival

Annu. Rev. Microbiol.

The Hypercycle. A Principle of Natural Self-Organization

Molecular quasispecies

J. Phys. Chem.

In vitro selection of variants of Human immunodeficiency virus type 1 resistant to 3’-azido-3’deoxythymidine and 2’,3’-dideoxyinosine

J. Virol.