Fundamental Properties of the Evolution of Mutational Robustness Lee Altenberg ∗ Abstract proposed it as a consequence of stabilizing selection, and Conrad (cf. [6, 7]) who proposed it could result Evolution on neutral networks of genotypes from higher-order epistatic mutations which smooth the has been found in models to concentrate on adaptive landscape at other sites and consequently en- genotypes with high mutational robustness, to hance evolvability. Another mechanism proposed is that a degree determined by the topology of the natural selection for adaptations robust to environmen- network. Here analysis is generalized beyond tal variation entails robustness to mutation as a generic neutral networks to arbitrary selection and side-effect [8]. parent-offspring transmission. In this larger An altogether different mechanism for the evolution realm, geometric features determine mutational of mutational robustness was proposed by Bornberg- robustness: the alignment of fitness with the Bauer and Chan [9] and van Nimwegen et al. [10, 11], orthogonalized eigenvectors of the mutation which is that evolution along neutral networks|sets matrix weighted by their eigenvalues. \House of mutationally connected genotypes with equivalent of cards" mutation is found to preclude the evo- fitnesses—would \concentrate at highly connected parts lution of mutational robustness. Genetic load of the network, resulting in phenotypes that are rel- is shown to increase with increasing mutation atively robust against mutations"[11], and \Indepen- in arbitrary single and multiple locus fitness dent of functional fitness, topology per se can lead landscapes. The rate of decrease in population to concentration of evolutionary population at some fitness can never grow as mutation rates get sequences."[9]. higher, showing that \error catastrophes" for Numerous citations of these papers repeat the finding genotype frequencies never cause precipitous that \populations will evolve toward highly connected losses of population fitness. The \inclusive regions of the genome space" (e.g. [12], [13]). However, inheritance" approach taken here naturally ex- neither the original papers nor subsequent studies (to tends these results to a new concept of dispersal my knowledge) provide analytic proofs of this observa- robustness. tion. With few exceptions [14] progress has been limited in identifying exactly which properties of neutral net- Keywords: genetic load | spectral gap | lethal works determine their evolved mutational robustness. mutagenesis | epigenetic mutation | dispersal load As is shown by the following example, more is going on than simply a \tendency to evolve toward highly con- Based on theoretical considerations, Kimura [1] pre- nected parts of the network". arXiv:1508.07866v1 [q-bio.PE] 31 Aug 2015 dicted that the majority of evolutionary changes in the Figure 1 compares the equilibrium population distri- genome in mammals should consist of neutral muta- bution for two neutral networks that have 63 equally tions. Since this time, research has been focused on fit genotypes, and one inviable genotype off the net- understanding the extent and nature of neutral genetic work. The only difference between the networks is their variation in organisms. One approach is to attempt mutational topology. One network is the set of all 64 to derive them from molecular first principles [2, 3]. nucleotide triplets, while the other is the set of copy- The idea that neutrality may not merely be a derived number variants, from 1 to 64 copies. Two properties of consequence of molecular interactions, but actually an the population equilibrium distributions on the neutral evolved property shaped by evolutionary dynamics, had networks stand out: first, the equilibrium frequency of a its first manifestation in Fisher's theory for the evolu- genotype is determined not solely by how many neutral tion of dominance [4], followed by Waddington [5] who neighbors it has, but also by its position within the net- ∗The Konrad Lorenz Institute for Evolution and Cognition work. In the copy-number network, 62 of the genotypes Research, Martinstrasse 12, Klosterneuburg, A3400 Austria, are identical in having only neutral neighbors. Yet the [email protected] stationary distribution increases 20-fold over these 62 1 Fundamental Properties of the Evolution of Mutational Robustness| Lee Altenberg 2 EQUILIBRIUM DISTRIBUTION treatment has the potential to apply widely to diverse STEPWISE COPY-NUMBER MUTATION NETWORK mechanisms of \inclusive inheritance" [21]. 0.025 ··· With arbitrary fitnesses, one can no longer character- MUTATION NETWORK OF A ize mutation neighborhoods simply by the fraction of 0.020 NUCLEOTIDE TRIPLET LOAD = 0.0138 mutations that are neutral, since now the distribution 0.015 of fitness effects of mutation (DFE) includes advanta- 0.010 geous or deleterious mutations, as well as neutral and lethal. The more general statistic is the expected fitness 0.005 • LOAD = 0.0003 of offspring. Averaged over the population, one obtains an aggregate population mutational robustness | the 10 20 30 40 50 60 degree to which the population maintains its fitness in Neutral genotypes Lethal genotype the face of mutation pressure. A complete absence of mutational robustness occurs when the genetic load is the mutation rate, corresponding to the Haldane-Muller Figure 1: Equilibrium distributions and genetic loads principle [22, 23]. Complete mutational robustness, on for two neutral networks having 63 neutral genotypes the other hand, would mean that the population suffers and one lethal genotype (circled), under two mutation no genetic load as the mutation rate increases. A given topologies: Mutation of a nucleotide triplet, and step- adaptive landscape will fall somewhere between these wise copy-number mutation. two extrema. The main results found here are that the population mutational robustness at a mutation-selection balance genotypes as they get more mutationally distant from is determined by abstract spectral properties: the align- the lethal genotype [15]. The same phenomenon is seen ment between the fitnesses and the eigenvectors of the in the trinucleotide network but to a lesser degree. Sec- mutation matrix, weighted by their eigenvalues. ondly, dramatic differences are seen between the two This spectral analysis provide a lens through which neutral networks in the size of the genetic loads they to examine models of mutation, dispersal, an inheri- maintain. The genetic load of the trinucleotide network tance generally. Models of mutation and selection are is 44 times the genetic load of the copy-number network. usually constructed without appreciation of how the as- The differences between the evolutionary outcomes sumptions manifest in the eigenvalues and eigenvectors on these two mutational graphs can only be the result of the mutation matrix (or variation operator in the of their different topologies, but what properties of their case of continuous variation). We will see, for exam- topologies? Numerous results from the field of spectral ple, that the widely encountered [24] \house-of-cards" graph theory are applicable to this question; here no re- mutation model [25] is incapable of supporting muta- view is attempted. While graph theory has been widely tional robustness. The results here provide direction used in models of mutation, the results typically impose for analyzing a wide variety of models for their capacity the assumptions that mutations occur only once during to support the evolution of mutational robustness, and replication, mutation is symmetric and occurs at a sin- enable comparisons to be made between different kinds gle rate, and fitness is either zero or a single other value of mutation processes, including nucleotide base muta- (all assumptions in [9, 11]). tion, epigenetic mark mutation, and gene copy number An alternative approach is taken here, which has variation. The comparisons extend to the spectral prop- proven valuable in previous work [16, 17, 18, 19, 20]. erties of dispersal matrices, thereby unifying the results The specific problem|evolution on neutral networks in for genetic robustness and genetic load with those for nucleotide sequence space|is embedded into the larger \dispersal robustness" and dispersal load. space of problems, which includes arbitrary mutation patterns and arbitrary selection values. The generality forces one to seek the appropriate fundamental mathe- 1 The Setting matical properties. It also expands the applicability of any results. Muta- The object of interest is the state of the population tion is treated generally enough to apply to non-genetic when it has converged in frequencies to an equilibrium information transmission, the principal example being under the following asexual, haploid evolutionary dy- organismal location, where the analog of mutation is namics, where the population is large enough to be dispersal. The results here thus automatically apply to treated as infinite and has discrete non-overlapping gen- dispersal load, and in the process define a new concept erations (semelparity). The only event that changes the of dispersal robustness. Moreover, the generality of the genotypes during reproduction is mutation; there is no Fundamental Properties of the Evolution of Mutational Robustness| Lee Altenberg 3 recombination. From (1), equilibrium frequencies must satisfy wbz^i = Pn n j=1 Mijwjz^j, or in vector form, 1 X z (t + 1) = M (µ) w z (t); (1) i w(t) ij j j > j=1 M(µ) D z^ = (w z^) z^ = wb z^; (4) where where > n is the number of possible haploid genotypes, e is the vector of ones, e = 1 1 1 its trans- pose, ··· zi(t) is the frequency of haploid genotype i in the pop- Pn D := diag[w ] is the diagonal matrix of fitness coeffi- ulation at time t, zi(t) 0, i=1 zi(t) = 1, i ≥ cients, wj is the fitness of genotype j, wi 0, ≥ w = De is the vector of fitness coefficients, and Mij(µ) is the probability, when the mutation rate is Pn > > µ [0; 1], that parent of genotype j has offspring wb := i=1 wiz^i = e Dz^ = w z^ is the mean fitness of 2 Pn of genotype i, Mij 0, Mij = 1 j, and the population.