Macromolecular Evolution: Dynamical Ordering in Sequence Space

Virus Evolution on Fitness Landscapes

The landscape paradigm is revisited in the light of evolution in simple systems. A brief overview of different classes of fitness landscapes is followed by a more detailed discussion of the RNA model, which is currently the only evolutionary model that allows for a comprehensive molecular analysis of a fitness landscape. Neutral networks of genotypes are indispensable for the success of evolution. Important insights into the evolutionary mechanism are gained by considering the topology of sequence and shape spaces. The dynamic concept of molecular quasispecies is viewed in the light of the landscape paradigm. The distribution of fitness values in state space is mirrored by the population structures of mutant distributions. Two classes of thresholds for replication error or mutations are important: (i) the-conventional-genotypic error threshold, which separates ordered replication from random drift on neutral networks, and (ii) a phenotypic error threshold above which the molecular phenotype is lost. Empirical landscapes are reviewed and finally, the implications of the landscape concept for virus evolution are discussed.

Degeneracy and genetic assimilation in RNA evolution

BACKGROUND

The neutral theory of Motoo Kimura stipulates that evolution is mostly driven by neutral mutations. However adaptive pressure eventually leads to changes in phenotype that involve non-neutral mutations. The relation between neutrality and adaptation has been studied in the context of RNA before and here we further study transitional mutations in the context of degenerate (plastic) RNA sequences and genetic assimilation. We propose quasineutral mutations, i.e. mutations which preserve an element of the phenotype set, as minimal mutations and study their properties. We also propose a general probabilistic interpretation of genetic assimilation and specialize it to the Boltzmann ensemble of RNA sequences.

RESULTS

We show that degenerate sequences i.e. sequences with more than one structure at the MFE level have the highest evolvability among all sequences and are central to evolutionary innovation. Degenerate sequences also tend to cluster together in the sequence space. The selective pressure in an evolutionary simulation causes the population to move towards regions with more degenerate sequences, i.e. regions at the intersection of different neutral networks, and this causes the number of such sequences to increase well beyond the average percentage of degenerate sequences in the sequence space. We also observe that evolution by quasineutral mutations tends to conserve the number of base pairs in structures and thereby maintains structural integrity even in the presence of pressure to the contrary.

CONCLUSIONS

We conclude that degenerate RNA sequences play a major role in evolutionary adaptation.

Evolutionary dynamics of a polymorphic self-replicator population with a finite population size and hyper mutation rate

Self-replicating biomolecules, subject to experimental evolution, exhibit hyper mutation rates where the genotypes of most offspring have at least a one point mutation. Thus, we formulated the evolutionary dynamics of an asexual self-replicator population with a finite population size and hyper mutation rate, based on the probability density of fitnesses (fitness distribution) for the evolving population. As a case study, we used a Kauffman's "NK fitness landscape". We deduced recurrence relations for the first three cumulants of the fitness distribution and compared them with the results of computer simulations. We found that the evolutionary dynamics is classified in terms of two modes of selection: the "radical mode" and the "gentle mode". In the radical mode, only a small number of genotypes with the highest or near highest fitness values can leave offspring. In the gentle mode, genotypes with moderate fitness values can leave offspring. We clarified how the evolutionary equilibrium and climbing rate depend on given parameters such as gradient and ruggedness of the landscape, mutation rate and population size, in terms of the two modes of selection. Roughly, the radical mode conducts the fast climbing but attains to the stationary states with low fitness, while the gentle mode conducts the slow climbing but attains to the stationary states with high fitness.

Extracting characteristic properties of fitness landscape from in vitro molecular evolution: a case study on infectivity of fd phage to E.coli

We have developed a methodology for extracting characteristic properties of a fitness landscape of interest by analyzing fitness data on an in vitro molecular evolution. The in vitro evolution is required to be conducted as the following "adaptive walk": a single parent sequence generates N mutant sequences as its offsprings, and the fittest individual among the N offsprings will become a new parent in the next generation. N is the library size of mutants to be screened in a single generation. Our theory of the adaptive walk on the "NK landscape" suggests the following: the adaptive walker starting from a random sequence climbs the landscape easily in an early stage, and then reaches a stationary phase in which the mutation-selection-random drift balance sets in. The stationary fitness value is nearly proportional to square root of ln N. Our analysis is performed from the following points: (1) stationary fitness values, (2) time series of fitness in the transitional state, (3) mutant's fitness distribution, and (4) the strength of selection pressure. Applying our methodology, we analyzed experimental data on the in vitro evolution of a random polypeptide (139 amino acids) toward acquiring infectivity (= ability to infect) of fd phage. As a result, we estimated that k is about 27 in this system, indicating that an arbitrary residue in a sequence is affected from other 23% residues. In this article, we demonstrated that the experimental data is consistent with our theoretical equations quantitatively, and that our methodology for extracting characteristic properties of a fitness landscape may be effective.

Experimental rugged fitness landscape in protein sequence space

The fitness landscape in sequence space determines the process of biomolecular evolution. To plot the fitness landscape of protein function, we carried out in vitro molecular evolution beginning with a defective fd phage carrying a random polypeptide of 139 amino acids in place of the g3p minor coat protein D2 domain, which is essential for phage infection. After 20 cycles of random substitution at sites 12–130 of the initial random polypeptide and selection for infectivity, the selected phage showed a 1.7×10⁴-fold increase in infectivity, defined as the number of infected cells per ml of phage suspension. Fitness was defined as the logarithm of infectivity, and we analyzed (1) the dependence of stationary fitness on library size, which increased gradually, and (2) the time course of changes in fitness in transitional phases, based on an original theory regarding the evolutionary dynamics in Kauffman's n-k fitness landscape model. In the landscape model, single mutations at single sites among n sites affect the contribution of k other sites to fitness. Based on the results of these analyses, k was estimated to be 18–24. According to the estimated parameters, the landscape was plotted as a smooth surface up to a relative fitness of 0.4 of the global peak, whereas the landscape had a highly rugged surface with many local peaks above this relative fitness value. Based on the landscapes of these two different surfaces, it appears possible for adaptive walks with only random substitutions to climb with relative ease up to the middle region of the fitness landscape from any primordial or random sequence, whereas an enormous range of sequence diversity is required to climb further up the rugged surface above the middle region.

Surveying a local fitness landscape of a protein with epistatic sites for the study of directed evolution

We present a method for analysis of a fitness landscape of a biopolymer with significantly epistatic sites. The analysis is based on a quasi-additive fitness model. The fitness model is constructed with additive terms conducted by "site-fitness" and epistatic terms conducted by "pair-fitness," where the site-fitness is a fitness contribution from an independent residue and the pair-fitness is a fitness contribution from a pair of epistatic residues. As a case study, we analyzed the sequence-fitness data for 45 clones of thermostable prolyl endopeptidase mutants. They were generated by a mutation scrambling method, which can accumulate advantageous mutations. The fitness contributions from 14 single-point mutations including E67Q and Q656R were identified by the analysis. As a result, we found that the fitness model with a significant epistatic term by a pair of the 67th site and 656th site was in good agreement with the experimental data and that the explored landscape in the binary 14-dimensional sequence space is still a mountainous landscape with twin peaks. The validity was supported by the analysis of mutant fitness distributions derived from another mutation scrambling experiment and by (3D) structural data.

Theory of evolutionary molecular engineering through simultaneous accumulation of advantageous mutations

We examined the effectiveness of an "adaptive leap" strategy using the "mutation scrambling" method as an efficient optimization technique (Uchiyama, 2000;J. Biochem.128, 441-447) for cases where mutational (rough) additivity holds in fitness. The mutation scrambling method is composed of the following three processes: (1) preliminary selection of several advantageous single-point mutations introduced in a wild-type sequence; (2) preparation of various multiple-point mutants incorporating the advantageous mutant residue or wild-type residue at each of the selected sites, by scrambling the mutant residues and wild-type residues (this process is called mutation scrambling); and (3) selection of the fittest through screening of the mutant pool. The fitness distribution in the mutant pool is controlled by the mixing ratio of the mutant residues to the wild-type residues. We focused on the mutant fitness distribution and obtained the optimal mixing ratio which efficiently generates superior multiple-point mutants with high fitnesses. As a result, we found that the optimal ratio lies between 7/3 and 9/1 in realistic cases. Particularly, this strategy works well in cases where the number of component mutations is large and the size of the population to be screened is small. Analysis of the mutant fitness distributions with various mixing ratios is also useful to explore local fitness landscapes.

Analysis of a local fitness landscape with a model of the rough Mt. Fuji-type landscape: application to prolyl endopeptidase and thermolysin

A method of analysis of a local fitness landscape for a current biopolymer is presented. Based on the assumption of additivity of mutational effects in the biopolymer, we assigned a site-fitness to each residue at each site. The assigned values of site-fitnesses were obtained by the least-squares method to minimize discrepancies between experimental fitnesses and theoretical ones. As test cases, we analyzed a section of a local landscape for the thermostability of prolyl endopeptidase and that for the enzymatic activity of thermolysin. These sections were proved to be of the rough Mt. Fuji-type with straight theta values of larger than 1.0, where straight theta is defined as the ratio of the "mean slope" to the "degree of roughness" on the fitness surface. Furthermore, we theoretically explained discrepancies between the fitnesses of multiple mutants and those predicted based on strict additivity of the component mutations by using a model of the rough Mt. Fuji-type landscape. According to this model, the discrepancies depend on the local landscape property (such as the straight theta value) and the location of the wild type on the landscape and the mean change in fitness by the component mutations. Our results suggest that this model may provide a good approximation of real sections of local landscapes for current biopolymers phenomenologically.

Directed evolution of enzyme catalysts

Directed enzyme evolution has emerged in the past few years as a powerful alternative to rational approaches for engineering biocatalysts. Prerequisites for successful directed evolution are functional expression in a suitable microbial host, a rapid screen for the desired feature(s) and a well-thought-out working strategy for navigating protein landscapes. The rapidly growing body of literature on enzyme evolution in vitro includes techniques for creating and searching combinatorial enzyme libraries, as well as several successful examples of different evolutionary strategies being used.

Genotypes with phenotypes: Adventures in an RNA toy world

Evolution has created the complexity of the animate world and deciphering the language of evolution is the key towards understanding nature. The dynamics of evolution is simplified by considering it as a superposition of three less sophisticated processes: population dynamics, population support dynamics, and genotype-phenotype mapping. Evolution of molecules in laboratory assays provides a sufficiently simple system for the quantitative analysis of the three phenomena. Coarse-grained notions of structures like RNA secondary structures are used as model phenotypes. They provide an excellent tool for a comprehensive analysis of the entire complex of molecular evolution. The mapping from RNA genotypes into secondary structures is highly redundant. In order to find at least one sequence for every common structures one need only search a (relatively) small part of sequence space. The existence of selectively neutral phenotypes plays an important role for the the success and the efficiency of evolutionary optimization. Molecular evolution found a highly promising technological application in the design of biomolecules with predefined properties.

A computer model of evolutionary optimization

Molecular evolution is viewed as a typical combinatorial optimization problem. We analyse a chemical reaction model which considers RNA replication including correct copying and point mutations together with hydrolytic degradation and the dilution flux of a flow reactor. The corresponding stochastic reaction network is implemented on a computer in order to investigate some basic features of evolutionary optimization dynamics. Characteristic features of real molecular systems are mimicked by folding binary sequences into unknotted two-dimensional structures. Selective values are derived from these molecular 'phenotypes' by an evaluation procedure which assigns numerical values to different elements of the secondary structure. The fitness function obtained thereby contains nontrivial long-range interactions which are typical for real systems. The fitness landscape also reveals quite involved and bizarre local topologies which we consider also representative of polynucleotide replication in actually occurring systems. Optimization operates on an ensemble of sequences via mutation and natural selection. The strategy observed in the simulation experiments is fairly general and resembles closely a heuristic widely applied in operations research areas. Despite the relative smallness of the system--we study 2000 molecules of chain length v = 70 in a typical simulation experiment--features typical for the evolution of real populations are observed as there are error thresholds for replication, evolutionary steps and quasistationary sequence distributions. The relative importance of selectively neutral or almost neutral variants is discussed quantitatively. Four characteristic ensemble properties, entropy of the distribution, ensemble correlation, mean Hamming distance and diversity of the population, are computed and checked for their sensitivity in recording major optimization events during the simulation.