Predicting Molecular Evolutionary Trajectories in Principle and in Practice


The discrete, digital nature of DNA (deoxyribonucleic acid) defines a large but countable number of mutational trajectories between any two alleles. Even though some set of mutations jointly confer an improvement in an organism's reproductive success, it need not follow that all mutational trajectories to that allele are equivalent in the eyes of evolution by natural selection. To a good approximation, the key question is whether all mutations improve reproductive success in all combinations; if they do not, then some subset (even, possibly all) trajectories will be selectively inaccessible. Mutations that are only conditionally beneficial are said to exhibit sign epistasis, which several recent empirical studies demonstrate is pervasive. Biochemical and biophysical considerations of protein biology are beginning to shed light on the underlying mechanisms of sign epistasis and current theoretical attention is focused on its influence on selectively accessible trajectories in sexual population.

Key concepts:

  • Phenotypic change over evolutionary time must reflect underlying changes in genetic constitution of the population.

  • Under reasonable assumptions, it is possible to enumerate all mutational trajectories a population might follow in the course of evolving from one genetic variant to another.

  • Recent experimental work demonstrates that for several proteins, only a very small fraction of mutational trajectories are accessible to populations evolving under the action of natural selection.

  • The identity of selectively accessible mutational trajectories reflects the underlying biochemistry and biophysics of the protein in question.

Keywords: epistasis; fitness landscapes; molecular evolution; natural selection; population genetics; protein evolution

Figure 1.

Sequence space. Shown here for k=3 variable nucleotides, designated left‐to‐right ‘A’, ‘B’ and ‘C’. Left: We associate a distinct spatial dimension with each mutation (see key), so mutationally adjacent alleles are spatially adjacent. Centre: Representative trajectories from allele abc to allele ABC, named by the order in which mutations appear (reversions designated by lower‐case, simultaneous fixations designated by multiple mutations in a single column). Top‐to‐bottom: a classical trajectory in which mutations fix individually and without reversion, two nonclassical trajectories in which mutations transiently revert, two nonclassical trajectories in which more than one mutation fixes simultaneously, one nonclassical trajectory in which a mutation reverts and two mutations fix simultaneously. Right: The fitness function maps from each allele (indicated by a string of +'s and −'s signalling presence or absence of each mutation) to the reproductive success conferred to the organism (represented by the symbol w). Note that this mapping is sometimes referred as a fitness or adaptive surface or landscape (beginning with Wright, ). However, this usage is mathematically less precise than fitness function and connotations and intuitions based on everyday experience of three‐dimensional surfaces and landscapes can be misleading (e.g. Gavrilets, ).

Figure 2.

Phenotypic decomposition of two β‐lactamase mutations conferring increased drug resistance in terms of catalytic activity and thermodynamic folding stability. M182T exhibits sign epistasis for cefotaxime resistance, because it reduces drug resistance (grey bars) on the wild‐type background but increases it sharply in the presence of G238S. (data from Wang et al., ; Weinreich et al., ). Note that neither catalytic activity nor thermodynamic folding stability exhibit sign epistasis, which instead is the consequence of pleiotropic effects of both mutations on these traits.

Figure 3.

Fisher's geometric model of adaptation for n=2 traits. All combinations of trait values are represented in n‐dimensional space (here, the plane); the optimal value is represented by the point labelled O and an individual labelled X, is displaced from this optimum (as perhaps following an environmental perturbation) by distance d. Fitness is some declining function of d. Mutations may simultaneously influence multiple traits. Left: To be beneficial, a mutation in wild‐type X must yield a phenotype X′ lying within the circle passing through X and centred at O. Thus, the probability that a mutation will be beneficial rises to half as its phenotypic perturbation r drops to zero. Inset: Y represents another organism with fitness equal to that of X. Note, however, that the phenotypically identical mutation exhibits sign epistasis: it is beneficial on X but deleterious on Y. Right: Mutational interactions in TEM‐1 β‐lactamase between catalytic activity and thermodynamic stability in determining cefotaxime resistance (see Figure ), represented qualitatively in Fisher's geometric model.



Ambler RP, Coulson AFW, Frère J‐M et al. (1991) A standard numbering scheme for the Class A β‐lactamases. Biochemical Journal 276: 269–272.

Bloom JD, Silberg JJ, Wilke CO et al. (2005) Thermodynamic prediction of protein neutrality. Proceedings of the National Academy of Sciences of the USA 102: 606–611.

Bridgham JT, Carroll SM and Thornton JW (2007) Evolution of hormone‐receptor complexity by molecular exploitation. Science 312: 97–100.

Brunet E, Rouzine IM and Wilke CO (2008) The stochastic edge in adaptive evolution. Genetics 179: 603–620.

Bustamante CD, Fledel‐Alon A, Williamson S et al. (2005) Natural selection on protein‐coding genes in the human genome. Nature 437: 1153–1157.

Camps M, Herman A, Loh E and Loeb LA (2007) Genetic constraints on protein evolution. Critical Reviews in Biochemistry and Molecular Biology 42: 313–326.

Carter AJR and Wagner GP (2002) Evolution of functionally conserved enhancers can be accelerated in large populations: a population‐genetic model. Proceedings of the Royal Society of London, Series B 269: 953–960.

Coyne JA, Barton NH and Turelli M (2000) Is Wright's shifting balance process important in evolution? Evolution 54: 306–317.

Dean AM and Thornton JW (2007) Mechanistic approaches to the study of evolution: the functional synthesis. Nature Reviews. Genetics 8: 675–688.

DePristo MA, Hartl DL and Weinreich DM (2007) Mutational reversions during adaptive protein evolution. Molecular Biology and Evolution 24: 1608–1610.

DePristo MA, Weinreich DM and Hartl DL (2005) Missense meandering through sequence space: a biophysical perspective on protein evolution. Nature Reviews. Genetics 6: 678–687.

Desai MM and Fisher DS (2007) Beneficial mutation‐selection balance and the effect of linkage on positive selection. Genetics 176: 1759–1798.

Desai M, Fisher D and Murray AW (2007) The speed of evolution and maintenance of variation in asexual populations. Current Biology 17: 385–394.

Fisher RA (1930) The Genetical Theory of Natural Selection, p. 272. Oxford, UK: Clarendon Press.

Gavrilets S (2004) Fitness Landscapes and The Origin of Species, p. 476. Princeton, NJ: Princeton University Press.

Gerrish PJ and Lenski RE (1998) The fate of competing beneficial mutation in an asexual population. Genetica 102/103: 127–144.

Gillespie JH (1984) Molecular evolution over the mutational landscape. Evolution 38: 1116–1129.

Goodnight CJ and Wade MJ (2000) The ongoing synthesis: a reply to Coyne, Barton and Turelli. Evolution 54: 317–324.

Gu X (2007) Evolutionary framework for protein sequence evolution and gene pleiotropy. Genetics 175: 1813–1822.

Haldane JBS (1927) A mathematical theory of natural and artificial selection. Part V. Proceedings of the Cambridge Philosophical Society 23: 838–844.

Hall BG (2002) Predicting evolution by in vitro evolution requires determining evolutionary pathways. Antimicrobial Agents and Chemotherapy 46: 3035–3038.

Kimura M (1962) On the probability of fixation of mutant genes in a population. Genetics 47: 713–719.

Lee Y‐H, D'souza LM and Fox GE (1997) Equally parsimonious pathways through an RNA sequence space are not equally likely. Journal of Molecular Evolution 45: 278–284.

Lozevsky ER, Chookajorn T, Brown KM et al. (2009) Stepwise acquisition of pyrimethamine resistance in the malaria parasite. Proceedings of the National Academy of Sciences of the USA 106: 12025–12030.

Lunzer M, Miller SP, Felsheim R and Dean AM (2005) The biochemical architecture of an ancient adaptive landscape. Science 310: 499–501.

Maisnier‐Patin S, Berg OG, Lijas L and Andersson DI (2002) Compensatory adaptation to the deleterious effect of antibiotic resistance in Salmonella typhimurium. Molecular Microbiology 46: 355–366.

Malcolm BA, Wilson KP, Matthews BW, Kirsch JF and Wilson AC (1990) Ancestral lysozymes reconstructed, neutrality tested, and thermostability linked to hydrocarbon packing. Nature 345: 86–89.

Maynard Smith J (1970) Natural selection and the concept of a protein space. Nature 225: 563–565.

Orr HA (2005) Theories of adaptation: what they do and don't say. Genetica 123: 3–13.

Perno CF, Svicher V and Ceccherini‐Silberstein F (2006) Novel drug resistance mutations in HIV: recognition and clinical relevance. AIDS Reviews 8: 179–190.

Poelwijk F, Kiviet DJ and Tans SJ (2006) Evolutionary potential of a duplicated repressor‐operator pair: simulating pathways using mutational data. PLoS Computational Biology 2: e58.

Raquet X, Lamotte‐Brasseur J and Fonzé E (1994) TEM β‐lactamase mutants hydrolysing third‐generation cephalosporins. Journal of Molecular Biology 2444: 625–639.

Raquet X, Vanhove M, Lamotte‐Brasseur J et al. (1995) Stability of TEM β‐lactamase mutants hydrolyzing third generation cephalosporins. Proteins 23: 63–72.

Reetz MT, Wang L‐W and Bocola M (2006) Directed evolution of enantioselective enzymes: iterative cycles of CASTing for probing protein‐sequence space. Angewadte Chemie 118: 1258–1263.

Rokyta DR, Beisel CJ and Joyce P (2006) Properties of adaptive walks on uncorrelated landscapes under strong selection and weak mutation. Journal of Theoretical Biology 243: 114–120.

Schrag SJ, Perrot V and Levin BR (1997) Adaptation to the fitness cost of antibiotic resistance in E. coli. Proceedings of the Royal Society of London, Series B 264: 1287–1291.

Sideraki V, Huang W, Palzkill T and Gilbert HF (2001) A secondary drug resistance mutation of TEM‐1 β‐lactamase that suppresses misfolding and aggregation. Proceedings of the National Academy of Sciences of the USA 98: 283–288.

Smith NGC and Eyre‐Walker A (2002) Adaptive protein evolution in Drosophila. Nature 415: 1022–1024.

Tomatis PE, Fablane SM, Simona F et al. (2008) Adaptive protein evolution grants organismal fitness by improving catalysis and flexibility. Proceedings of the National Academy of Sciences of the USA 105: 20605–20610.

Wang Z, Minasov G and Shoichet BK (2002) Evolution of an antibiotic resistance enzyme constrained by stability and activity trade‐offs. Journal of Molecular Biology 320: 85–95.

Watson RA (2005) Compositional evolution: the impact of sex, symbiosis and modularity on the gradualist framework of evolution, p. 324. Cambridge, MA: MIT Press.

Watson RA, Weinreich DM and Wakeley J (2006) Effects of Intra‐gene epistasis on the benefit of sexual recombination. Biochemical Society Transactions 34: 560–561.

Weinreich DM and Chao L (2005) Rapid evolutionary escape by large populations from local fitness peaks is likely in nature. Evolution 59: 1175–1182.

Weinreich DM, Delaney NF, DePristo MA and Hartl DL (2006) Darwinian evolution can follow only very mutational paths to fitter proteins. Science 312: 111–114.

Weinreich DM, Watson RA and Chao L (2005) Sign epistasis and genetic constraint on evolutionary trajectories. Evolution 59: 1165–1174.

Whitlock MC, Phillips PC, Moore FB‐G and Tonsor SJ (1995) Multiple fitness peaks and epistasis. Annual Review of Ecology and Systematics 26: 601–629.

Wright S (1932) The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: Jones DF (ed.) Proceedings of the Sixth International Congress of Genetics, pp. 356–366. Menasha, WI: Brooklyn Botanic Garden.

Zhou H‐Z, Wlodek ST and McCammon JA (1998) Conformational gating as a mechanism for enzyme specificity. Proceedings of the National Academy of Sciences of the USA 95: 9280–9283.

Further Reading

Crow JF and Kimura M (1970) An Introduction to Population Genetics Theory, p. 591. New York: Harper & Row.

Gillespie JH (1991) The Causes of Molecular Evolution, p. 336. Oxford, UK: Oxford University Press.

Gillespie JH (2004) Population Genetics: A Concise Guide, p. 214. Baltimore, MD: Johns Hopkins University Press.

Hartl DL and Clark AG (2007) Principles of Population Genetics, p. 565. Sunderland, MA: Sinauer Associates.

Poelwijk F, Kiviet DJ, Weinreich DM and Tans SJ (2007) Empirical fitness landscapes reveal accessible evolutionary paths. Nature 445: 383–386.

Rice SH (2004) Evolutionary Theory: Mathematical and Conceptual Foundations, p. 370. Sunderland, MA: Sinauer Associates.

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Weinreich, Daniel M(Mar 2010) Predicting Molecular Evolutionary Trajectories in Principle and in Practice. In: eLS. John Wiley & Sons Ltd, Chichester. [doi: 10.1002/9780470015902.a0022174]