Metastable evolutionary dynamics: crossing fitness barriers or escaping via neutral paths?

We analytically study the dynamics of evolving populations that exhibit metastability on the level of phenotype or fitness. In constant selective environments, such metastable behavior is caused by two qualitatively different mechanisms. On the one hand, populations may become pinned at a local fitness optimum, being separated from higher-fitness genotypes by a fitness barrier of low-fitness genotypes. On the other hand, the population may only be metastable on the level of phenotype or fitness while, at the same time, diffusing over neutral networks of selectively neutral genotypes. Metastability occurs in this case because the population is separated from higher-fitness genotypes by an entropy barrier: the population must explore large portions of these neutral networks before it discovers a rare connection to fitter phenotypes. We derive analytical expressions for the barrier crossing times in both the fitness barrier and entropy barrier regime. In contrast with 'landscape' evolutionary models, we show that the waiting times to reach higher fitness depend strongly on the width of a fitness barrier and much less on its height. The analysis further shows that crossing entropy barriers is faster by orders of magnitude than fitness barrier crossing. Thus, when populations are trapped in a metastable phenotypic state, they are most likely to escape by crossing an entropy barrier, along a neutral path in genotype space. If no such escape route along a neutral path exists, a population is most likely to cross a fitness barrier where the barrier is narrowest, rather than where the barrier is shallowest.

Dynamics of one-pass germinal center models: implications for affinity maturation.

During an immune response, the affinity of antibodies that react with the antigen that triggered the response increases with time, a phenomenon known as affinity maturation. The molecular basis of affinity maturation has been partially elucidated. It involves the somatic mutation of immunoglobulin V-region genes within antigen-stimulated germinal center B cells and the subsequent selection of high affinity variants. This mutation and selection process is extremely efficient and produces large numbers of high affinity variants. Studies of the architecture of germinal centers suggested that B cells divide in the dark zone of the germinal center, then migrate to the light zone, where they undergo selection based on their interaction with antigen-loaded follicular dendritic cells, after which they exit the germinal center through the mantle zone. Kepler and Perelson questioned this architecturally driven view of the germinal center reaction. They, as well as others, argued that the large number of point mutations observed in germinal center B cell V-region genes, frequently 5 to 10 and sometimes higher, would most likely render cells incapable of binding the antigen, if no selection step was interposed between rounds of mutations. To clarify this issue, we address the question of whether a mechanism in which mutants are generated and then selected in one pass, with no post-selection amplification, can account for the observed efficiency of affinity maturation. We analyse a set of one-pass models of the germinal center reaction, with decaying antigen, and mutation occurring at transcription or at replication. We show that under all the scenarios, the proportion of high affinity cells in the output of a germinal center varies logarithmically with their selection probability. For biologically realistic parameters, the efficiency of this process is in clear disagreement with the experimental data. Furthermore, we discuss a set of, possibly counterintuitive, more general features of one-pass selection models that follow from our analysis. We believe that these results may also provide useful intuitions in other cases where a population is subjected to selection mediated by a selective force that decays over time.

Neutral evolution of mutational robustness.

We introduce and analyze a general model of a population evolving over a network of selectively neutral genotypes. We show that the population's limit distribution on the neutral network is solely determined by the network topology and given by the principal eigenvector of the network's adjacency matrix. Moreover, the average number of neutral mutant neighbors per individual is given by the matrix spectral radius. These results quantify the extent to which populations evolve mutational robustness-the insensitivity of the phenotype to mutations-and thus reduce genetic load. Because the average neutrality is independent of evolutionary parameters-such as mutation rate, population size, and selective advantage-one can infer global statistics of neutral network topology by using simple population data available from in vitro or in vivo evolution. Populations evolving on neutral networks of RNA secondary structures show excellent agreement with our theoretical predictions.

The frequency distribution of gene family sizes in complete genomes.

We compare the frequency distribution of gene family sizes in the complete genomes of six bacteria (Escherichia coli, Haemophilus influenzae, Helicobacter pylori, Mycoplasma genitalium, Mycoplasma pneumoniae, and Synechocystis sp. PCC6803), two Archaea (Methanococcus jannaschii and Methanobacterium thermoautotrophicum), one eukaryote (Saccharomyces cerevisiae), the vaccinia virus, and the bacteriophage T4. The sizes of the gene families versus their frequencies show power-law distributions that tend to become flatter (have a larger exponent) as the number of genes in the genome increases. Power-law distributions generally occur as the limit distribution of a multiplicative stochastic process with a boundary constraint. We discuss various models that can account for a multiplicative process determining the sizes of gene families in the genome. In particular, we argue that, in order to explain the observed distributions, gene families have to behave in a coherent fashion within the genome; i.e., the probabilities of duplications of genes within a gene family are not independent of each other. Likewise, the probabilities of deletions of genes within a gene family are not independent of each other.