Proc. Natl. Acad. Sci. USA Vol. 96, pp. 9716–9720, August 1999 Evolution Neutral evolution of mutational robustness ERIK VAN NIMWEGEN*†,JAMES P. CRUTCHFIELD*‡, AND MARTIJN HUYNEN†§ *Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501; §Biocomputing Group, European Molecular Biology Laboratory, Meyerhofstrasse 1, 69012 Heidelberg, Germany; and †Bioinformatics Group, University of Utrecht, Padualaan 8, NL-3584-CH Utrecht, The Netherlands Communicated by Hans Frauenfelder, Los Alamos National Laboratory, Los Alamos, NM, June 16, 1999 (received for review March 18, 1999) ABSTRACT We introduce and analyze a general model of observed in biological RNA structures (15) and in simulations a population evolving over a network of selectively neutral of the evolution of RNA secondary structure (16). However, genotypes. We show that the population’s limit distribution on an analytical understanding of the phenomenon, the underly- the neutral network is solely determined by the network ing mechanisms, and their dependence on evolutionary pa- topology and given by the principal eigenvector of the net- rameters—such as mutation rate, population size, selection work’s adjacency matrix. Moreover, the average number of advantage, and the topology of the neutral network—has up neutral mutant neighbors per individual is given by the matrix to now not been available. spectral radius. These results quantify the extent to which Here, we develop a model for the evolution of populations populations evolve mutational robustness—the insensitivity on neutral networks and show analytically that, for biologically of the phenotype to mutations—and thus reduce genetic load. relevant population sizes and mutation rates, a population’s Because the average neutrality is independent of evolutionary distribution over a neutral network is determined solely by the parameters—such as mutation rate, population size, and network’s topology. Consequently, one can infer important selective advantage—one can infer global statistics of neutral structural information about neutral networks from data on network topology by using simple population data available evolving populations, even without specific knowledge of the from in vitro or in vivo evolution. Populations evolving on evolutionary parameters. Simulations of the evolution of a neutral networks of RNA secondary structures show excellent population of RNA sequences, evolving on a neutral network agreement with our theoretical predictions. defined with respect to secondary structure, confirm our theoretical predictions and illustrate their application to in- Kimura’s (1) contention that a majority of genotypic change in ferring network topology. evolution is selectively neutral has gained renewed attention Modeling Neutrality with the recent analysis of evolutionary optimization methods (2, 3) and the discovery of neutral networks in genotype- We assume that genotype space contains a neutral network of phenotype models for RNA secondary structure (4–6) and high, but equal fitness, genotypes on which the majority of a protein structure (7). It was found that collections of mutually population is concentrated and that the neighboring parts of neutral genotypes, which are connected via single mutational genotype space consist of genotypes with markedly lower steps, form extended networks that permeate large regions of fitness. The genotype space consists of all sequences of length genotype space. Intuitively, a large degeneracy in genotype- L over a finite alphabet Ꮽ of A symbols. The neutral network phenotype maps, when combined with the high connectivity of on which the population moves can be most naturally regarded (high-dimensional) genotype spaces, readily leads to such as a graph G embedded in this genotype space. The vertex set extended neutral networks. This intuition is now supported by of G consists of all genotypes that are on the neutral network; recent theoretical results (8, 9). its size is denoted by ͉G͉. Two vertices are connected by an edge In evolution of ribozymes in vitro, mutations responsible for if and only if they differ by a single point mutation. an increase in fitness are only a small minority of the total We will investigate the dynamics of a population evolving on number of accepted mutations (10). This fact indicates that, this neutral network and analyze the dependence of several even in adaptive evolution, the majority of point mutations is population statistics on the topology of the graph G. With neutral. The fact that only a minority of loci is conserved in these results, we will then show how measuring various pop- sequences evolved from a single ancestor similarly indicates a ulation statistics enables one to infer the structural properties high degeneracy in ribozymal genotype-phenotype maps (11). of G. Neutrality is also implicated in experiments where RNA For the evolutionary process, we assume a discrete- sequences evolve a given structure starting from a range of generation selection-mutation dynamics with constant popu- different initial genotypes (12). More generally, neutrality in lation size M. Individuals on the neutral network G have a RNA and protein genotype-phenotype maps is indicated by fitness of ␴. Individuals outside the neutral network have the observation that their structures are much better con- fitnesses that are considerably smaller than ␴. With the ap- served during evolution than their sequences (13, 14). proximations we use, the exact fitness values for genotypes off Given the presence of neutral networks that preserve struc- G turn out to be immaterial. Each generation, M individuals ture or function in sequence space, one asks, how does an are selected with replacement and with probability propor- evolving population distribute itself over a neutral network? tional to fitness and then mutated with probability ␮. These Can we detect and analyze structural properties of neutral individuals form the next generation. networks from data on biological or in vitro populations? To This dynamical system is a discrete-time version of Eigen’s what extent does a population evolve toward highly connected molecular-evolution model (17). Our analysis can be translated parts of the network, resulting in sequences that are relatively directly to the continuous-time equations for the Eigen model. insensitive to mutations? Such mutational robustness has been The results remain essentially unchanged. Although our analysis can be extended to more complicated The publication costs of this article were defrayed in part by page charge mutation schemes, we will assume that only single point payment. This article must therefore be hereby marked ‘‘advertisement’’ in accordance with 18 U.S.C. §1734 solely to indicate this fact. ‡To whom reprint requests should be addressed. E-mail: chaos@ PNAS is available online at www.pnas.org. santafe.edu. 9716 Downloaded by guest on September 23, 2021 Evolution: van Nimwegen et al. Proc. Natl. Acad. Sci. USA 96 (1999) 9717 mutations can occur at each reproduction of an individual. to remain on the neutral network G. If asymptotically a ␮ ͗␯͘ With probability , one of the L symbols is chosen with fraction Ps of the population is located at genotype s, then Ϫ ␯ uniform probability and is mutated to one of the A 1 other is simply the average of s over the asymptotic distribution on ͗␯͘ϭ ⌺ ␯ ͞ ͗␯͘ symbols. Thus, with a mutation, a genotype s moves with the network sʦG sPs P.AsEq.3 shows, the average Ϫ ͗ ͘ϭ⌺ ͞ uniform probability to one of the L(A 1) neighboring points is simply related to the population neutrality d sʦGdsPs P. in genotype space. Moreover, using Eq. 2, we can directly relate the population For the results presented below to hold, it is not necessary neutrality ͗d͘ to the average fitness ͗ f͘: that all genotypes in G have exactly the same fitness. As in any ␴ Ϫ ͗ ͘ model of neutral evolution (1, 18), it is sufficient to assume that f ͗d͘ ϭ L͑A Ϫ 1͒ͫ1 Ϫ ͬ. [4] the fitness differentials between distinct genotypes in G are ␮␴ smaller than the reciprocal 1͞M of the population size. Addi- tionally, we assume that the fitness differentials between Despite the fact that neither the details of the topology of G genotypes in G and genotypes outside G are much larger than nor the fitness values of the genotypes lying off the neutral 1͞M. These assumptions break down when there is a contin- network are given, one can relate the population neutrality ͗d͘ uum of fitness differentials between genotypes or in the case of the individuals on the neutral network directly to the of very small population size, which readily allows the spread- average fitness ͗ f͘ in the population. It may seem surprising ing of mildly deleterious mutations (19). that such a simple relation is possible at all. Because the population consists partly of sequences off the neutral net- Infinite-Population Solution work, one expects that the average fitness is determined in part by the fitnesses of these sequences. However, under the The first step is to solve for the asymptotic distribution of the assumption that back mutations from low-fitness genotypes off population over the neutral network G in the limit of very large the neutral network onto G are negligible, the fitnesses of population size. sequences outside G influence only the total proportion P of Once the (infinite) population has come to equilibrium, individuals on the network but not the average fitness in the there will be a constant proportion P of the population located population. ͗ ͘ on the network G and a constant average fitness f in the Eq. 4 shows that the population neutrality ͗d͘ can be population. Under selection, the proportion of individuals on inferred from the average fitness and other parameters—such ␴ ͗͞ ͘ the neutral network increases from P to P f . Under muta- as mutation rate. However, as we will now show, the popula- ͗␯͘ tion, a proportion of these individuals remains on the tion neutrality ͗d͘ can also be obtained independently from Ϫ͗␯͘ network, whereas a proportion 1 falls off the neutral knowledge of the topology of G alone.