Error Thresholds and their Relation to Optimal Mutation Rates Gabriela Ochoa Inman Harvey and Hilary Buxton Centre for the Study of Evolution Centre for Computational Neuroscience and Rob otics COGS The University of Sussex Falmer Brighton BN QH UK fgabroinmanhhilarybgcogssusxacuk Abstract The error threshold a notion from molecular evolution is the critical mutation rate b eyond which structures obtained by the evolutionary pro cess are destroyed more frequently than selection can repro duce them We argue that this notion is closely related to the more familiar notion of optimal mutation rates in Evolutionary Algorithms EAs This corresp ondence has b een intuitively p erceived b efore However no previous study to our knowledge has b een aimed at explicitly testing the hyp othesis of such a relationshi p Here wepropose a metho dology for doing so Results on a restricted range of tness land scap es suggest that these two notions are indeed correlated There is not however a critically precise optimal mutation rate but rather a range of values pro ducing similar nearoptimal p erformance When recombi nation is used b oth error thresholds and optimal mutation ranges are lower than in the asexual case This knowledge mayhave b oth theoret ical relevance in understanding EA b ehavior and practical implicati ons for setting optimal values of evolutionary parameters Intro duction The error threshold a notion from molecular evolution is the critical mu tation rate b eyond which structures obtained by the evolutionary pro cess are destroyed more frequently than selection can repro duce them With mutation rates ab ove this critical value an optimal solution would not b e stable in the p opulation ie the probability that the p opulation loses these structures is no longer negligible On the other hand an optimum mutation rate a more fa miliar notion within the EAs community isthemutation value whichsolves a sp ecied search or optimization problem with optimal eciency that is with the least numb er of generations or function evaluations The notion of error threshold seems to b e intuitively related to the idea of an optimal balance b etween exploitation and exploration in genetic search In this sense we argue that optimal mutation rates are related to error thresholds The aim of this pap er is to test this hyp othesis using an empirical approach together with knowledge from molecular evolution theory Optimal parameter settings have b een the sub ject of numerous studies within the EA community and particular emphasis has b een placed on nding optimal mutation rates There is however no conclusive agreement on what is b est most p eople use what has worked well in previously rep orted cases It is very dicult to formulate a priori general principles ab out parameter settings in view of the variety of problem typ es enco dings and p erformance cri teria p ossible in dierent applications Our hyp othesis that optimal mutation rates are correlated to the notion of error thresholds promises practical rele vance and useful guidelines in nding optimal parameter settings thus enhancing evolutionary search In the remainder of the pap er we summarize the knowledge from molecular evolution relevant to our argument the notions of quasispecies and error thresh olds we discuss the relation b etween error thresholds and optimal mutation rates and we describ e the tness landscap e used for our exp eriments the Royal Staircase functions Thereafter we describ e the empirical metho dology used to test the hyp othesis under studywe present the exp erimental results obtained and we discuss the insight gained Quasisp ecies And Error Thresholds The concept of a quasisp ecies was develop ed in the context of p olynucleotide replication and in particular studies of early RNA evolution A pro tein space or more generally a sequence space can b e mo delled as the space of all p ossible sequences of length drawn from a nite alphab et of size A Each sequence has a tness value which sp ecies its replication rate or exp ected numb er of ospring p er unit time The tnesses of all A p ossible sequences dene a tness landscap e When A a binary alphab et the tness land scap e is equivalent to sp ecifying tness valuesateachvertex of a dimensional hyp ercub e with some mathematical imagination and some caution this can b e pictured as spread out over a geographical landscap e where tness is analogous to height and the dynamics of evolution of a p opulation corresp onds to movement of the p opulation over such a landscap e Given an innite p opulation and a sp ecied mutation rate governing errors in asexual replication one can determine the stationary sequence distribution reached after any transients from some original distribution have died away Unless the mutation rate is to o large or dierences in tnesses to o small the p opulation will typically cluster around the ttest sequences forming a concentrated cloud the average Hamming distance b etween twomemb ers of such a distribution drawn at random will b e relatively small Such a clustered distribution is called a quasisp ecies As the mutation rate is increased the lo cal distribution widens and ultimately loses its hold on the lo cal optimum This can b e seen at its clearest in an extreme form of a tness landscap e whichcontains a single p eak of tness all other sequences having a tness of With an innite p opulation there is a phase transition at a particular error rate p the mutation rate at eachofthe lo ci in a sequence In this critical error rate the error threshold is determined analytically Equation and it is dened as the rate ab ovewhich the prop ortion of the innite p opulation on the p eak drops to chance levels ln p In equation represents the selectiveadvantage of the master sequence over the rest of the p opulation and the chromosome length In the simplest case is the ratio of the master sequence repro duction rate tness to the average repro duction rate of the rest Error Thresholds In Finite Populations In the calculations of an error threshold for innite asexually replicating p opulations are extended to nite p opulations we shall call the critical rate p M for a p opulation of size M Finite p opulations lose grip on the solitary spike of sup erior tness easily b ecause of the added hazard of natural uctuations in this case In we derived a reformulation of the Nowak and Schuster analytical expression This new expression equation explicitly approximates the extent of the reduction in the error threshold as wemove from innite to nite p opulations The expression strictly should b e an innite series in which successive terms get smaller here we are ignoring all after the rst few p p ln ln p p p M M M Error Thresholds and Optimal Mutation Rates The notion of error threshold seems to b e intuitively related to the idea of an optimal balance b etween exploitation and exploration in genetic search Too low amutation rate implies to o little exploration in the limit of zero mutation suc cessive generations of selection removeallvariety from the p opulation and once the p opulation has converged to a single p ointingenotyp e space all further ex ploration ceases On the other hand clearlymutation rates can b e to o excessive in the limit where mutation places a randomly chosen allele at every lo cus on an ospring genotyp e then the evolutionary pro cess has degenerated into random search with no exploitation of the information acquired in preceding generations Any optimal mutation rate must lie b etween these two extremes but its precise p osition will dep end on several factors including in particular structure of the tness landscap e It can however b e hyp othesized that where evolution pro ceeds through a successiveaccumulation of information then a mutation rate close to the error threshold is an optimal mutation rate for the landscap e under study since this should maximise the search done through mutation sub ject to the constraint of not losing information already gained The main purp ose of our pap er is to empirically test this hyp othesis section Some biological evidence supp orts the relationship b etween error thresholds and optimal mutation rates Eigen and Schuster havepointed out that viruses which are very eciently evolving entities live within and close to the error thresholds given by the known rates of nucleotide mutations This corresp on dence has also b een noticed b efore in the GA community Hesser and Manner devised a heuristic formula for optimal setting of mutation rates inspired by previous work on error thresholds Kauman p also suggest a relationship b etween these two notions Royal Staircase Fitness Functions van Nimwegen and Crutcheld prop osed the Royal Staircase functions for analyzing ep o chal evolutionary search This class of functions are related to the previous Royal Road functions In the authors justify their particular choice of tness function b oth in terms of biological motivations and in terms of articial evolution issues In short many biological systems and articial evolu tion problems have highly degenerate genotyp etophenotyp e maps that is the mapping from genetic sp ecication to tness is a manytoone function Conse quently the numb er of dierent tness values that genotyp es can takeismuch smaller than the numb er of dierentgenotyp es Moreover due to its high dimen sionality it is p ossible for the genotyp e to break into networks of connected sets of equaltness genotyp e that can reacheach other via elementary