Crossing a fitness valley as a metastable transition in a stochastic population model. (English) Zbl 1433.92033

Bovier, Anton; Coquille, Loren; Smadi, Charline

Ann. Appl. Probab. 29, No. 6, 3541-3589 (2019).

Summary: We consider a stochastic model of population dynamics where each individual is characterised by a trait in \(\{0,1,\ldots,L\}\) and has a natural reproduction rate, a logistic death rate due to age or competition and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape exhibits a valley: mutant individuals with negative fitness have to be created in order for the population to reach a trait with positive fitness. We focus on the limit of large population and rare mutations at several speeds. In particular, when the mutation rate is low enough, metastability occurs: the exit time of the valley is an exponentially distributed random variable.

MSC:

92D25	Population dynamics (general)
92D40	Ecology
92D15	Problems related to evolution
60J85	Applications of branching processes

Keywords:

eco-evolution; birth and death process with immigration; selective sweep; coupling; competitive Lotka-Volterra system with mutations

References:

[1]	Abu Awad, D. and Billiard, S. (2017). The double edged sword: The demographic consequences of the evolution of self-fertilization. Evolution 71 1178-1190.
[2]	Alexander, H. K. (2013). Conditional distributions and waiting times in multitype branching processes. Adv. in Appl. Probab. 45 692-718. · Zbl 1276.92081 · doi:10.1239/aap/1377868535
[3]	Baar, M., Bovier, A. and Champagnat, N. (2017). From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step. Ann. Appl. Probab. 27 1093-1170. · Zbl 1371.92094 · doi:10.1214/16-AAP1227
[4]	Baar, M., Coquille, L., Mayer, H., Hölzel, M., Rogava, M., Tüting, T. and Bovier, A. (2016). A stochastic model for immunotherapy of cancer. Sci. Rep. 6 24169.
[5]	Billiard, S. and Smadi, C. (2017). The interplay of two mutations in a population of varying size: A stochastic eco-evolutionary model for clonal interference. Stochastic Process. Appl. 127 701-748. · Zbl 1378.92055 · doi:10.1016/j.spa.2016.06.024
[6]	Bolker, B. and Pacala, S. W. (1997). Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol. 52 179-197. · Zbl 0890.92020 · doi:10.1006/tpbi.1997.1331
[7]	Bolker, B. M. and Pacala, S. W. (1999). Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal. Amer. Nat. 153 575-602.
[8]	Bovier, A., Coquille, L. and Neukirch, R. (2018). The recovery of a recessive allele in a Mendelian diploid model. J. Math. Biol. 77 971-1033. · Zbl 1415.92123 · doi:10.1007/s00285-018-1240-z
[9]	Bovier, A. and den Hollander, F. (2015). Metastability: A Potential-Theoretic Approach. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 351. Springer, Cham. · Zbl 1339.60002
[10]	Bovier, A. and Wang, S.-D. (2013). Trait substitution trees on two time scales analysis. Markov Process. Related Fields 19 607-642. · Zbl 1301.92060
[11]	Brink-Spalink, R. and Smadi, C. (2017). Genealogies of two linked neutral loci after a selective sweep in a large population of stochastically varying size. Adv. in Appl. Probab. 49 279-326. · Zbl 1429.92115 · doi:10.1017/apr.2016.88
[12]	Britton, T. and Pardoux, E. (2017). Stochastic epidemics in a homogeneous community. In preparation. · Zbl 1447.92402
[13]	Carter, A. J. and Wagner, G. P. (2002). Evolution of functionally conserved enhancers can be accelerated in large populations: A population-genetic model. Proc. R. Soc. Lond., B Biol. Sci. 269 953-960.
[14]	Champagnat, N. (2006). A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stochastic Process. Appl. 116 1127-1160. · Zbl 1100.60055 · doi:10.1016/j.spa.2006.01.004
[15]	Champagnat, N., Ferrière, R. and Ben Arous, G. (2001). The canonical equation of adaptive dynamics: A mathematical view. Selection 2 73-83.
[16]	Champagnat, N., Ferrière, R. and Méléard, S. (2008). From individual stochastic processes to macroscopic models in adaptive evolution. Stoch. Models 24 2-44. · Zbl 1157.60339 · doi:10.1080/15326340802437710
[17]	Champagnat, N. and Méléard, S. (2011). Polymorphic evolution sequence and evolutionary branching. Probab. Theory Related Fields 151 45-94. · Zbl 1225.92040
[18]	Chazottes, J.-R., Collet, P. and Méléard, S. (2016). Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes. Probab. Theory Related Fields 164 285-332. · Zbl 1335.92073 · doi:10.1007/s00440-014-0612-6
[19]	Chazottes, J.-R., Collet, P. and Méléard, S. (2017). On time scales and quasi-stationary distributions for multitype birth-and-death processes. arXiv preprint arXiv:1702.05369. · Zbl 1434.60237 · doi:10.1214/18-AIHP948
[20]	Coron, C., Costa, M., Leman, H. and Smadi, C. (2018). A stochastic model for speciation by mating preferences. J. Math. Biol. 76 1421-1463. · Zbl 1390.60282 · doi:10.1007/s00285-017-1175-9
[21]	Coron, C., Méléard, S., Porcher, E. and Robert, A. (2013). Quantifying the mutational meltdown in diploid populations. Amer. Nat. 181 623-636.
[22]	Cowperthwaite, M. C., Bull, J. J. and Meyers, L. A. (2006). From bad to good: Fitness reversals and the ascent of deleterious mutations. PLoS Comput. Biol. 2 e141.
[23]	DePristo, M. A., Hartl, D. L. and Weinreich, D. M. (2007). Mutational reversions during adaptive protein evolution. Mol. Biol. Evol. 24 1608-1610.
[24]	Dieckmann, U. and Law, R. (1996). The dynamical theory of coevolution: A derivation from stochastic ecological processes. J. Math. Biol. 34 579-612. · Zbl 0845.92013 · doi:10.1007/BF02409751
[25]	Dieckmann, U. and Law, R. (2000). Moment approximations of individual-based models. In The Geometry of Ecological Interactions: Simplifying Spatial Complexity (U. Dieckmann, R. Law and J. A. J. Metz, eds.) 252-270. Cambridge University Press, Cambridge.
[26]	Durrett, R. and Mayberry, J. (2011). Traveling waves of selective sweeps. Ann. Appl. Probab. 21 699-744. · Zbl 1219.92037 · doi:10.1214/10-AAP721
[27]	Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. · Zbl 0592.60049
[28]	Fournier, N. and Méléard, S. (2004). A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 1880-1919. · Zbl 1060.92055 · doi:10.1214/105051604000000882
[29]	Geritz, S. A., Metz, J. A., Kisdi, É. and Meszéna, G. (1997). Dynamics of adaptation and evolutionary branching. Phys. Rev. Lett. 78 2024.
[30]	Giachetti, C. and Holland, J. J. (1988). Altered replicase specificity is responsible for resistance to defective interfering particle interference of an Sdi-mutant of vesicular stomatitis virus. J. Virol. 62 3614-3621.
[31]	Gillespie, J. H. (1984). Molecular evolution over the mutational landscape. Evolution 38 1116-1129.
[32]	Gokhale, C. S., Iwasa, Y., Nowak, M. A. and Traulsen, A. (2009). The pace of evolution across fitness valleys. J. Theoret. Biol. 259 613-620. · Zbl 1402.92311 · doi:10.1016/j.jtbi.2009.04.011
[33]	Haeno, H., Maruvka, Y. E., Iwasa, Y. and Michor, F. (2013). Stochastic tunneling of two mutations in a population of cancer cells. PLoS ONE 8 e65724.
[34]	Iwasa, Y., Michor, F. and Nowak, M. A. (2004). Evolutionary dynamics of invasion and escape. J. Theoret. Biol. 226 205-214. · Zbl 1439.92132 · doi:10.1016/j.jtbi.2003.08.014
[35]	Iwasa, Y., Michor, F. and Nowak, M. A. (2004). Stochastic tunnels in evolutionary dynamics. Genetics 166 1571-1579.
[36]	Leman, H. (2016). Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence. Stoch. Partial Differ. Equ. Anal. Comput. 4 791-826. · Zbl 1387.60117 · doi:10.1007/s40072-016-0077-y
[37]	Lenski, R. E., Ofria, C., Pennock, R. T. and Adami, C. (2003). The evolutionary origin of complex features. Nature 423 139-144.
[38]	Maisnier-Patin, S., Berg, O. G., Liljas, L. and Andersson, D. I. (2002). Compensatory adaptation to the deleterious effect of antibiotic resistance in Salmonella typhimurium. Mol. Microbiol. 46 355-366.
[39]	Metz, J. A., Geritz, S. A., Meszéna, G., Jacobs, F. J. and Van Heerwaarden, J. S. (1995). Adaptive dynamics: A geometrical study of the consequences of nearly faithful reproduction. WP-95-099. · Zbl 0972.92024
[40]	Moore, F. B.-G. and Tonsor, S. J. (1994). A simulation of Wright’s shifting-balance process: Migration and the three phases. Evolution 48 69-80.
[41]	O’Hara, P. J., Nichol, S. T., Horodyski, F. M. and Holland, J. J. (1984). Vesicular stomatitis virus defective interfering particles can contain extensive genomic sequence rearrangements and base substitutions. Cell 36 915-924.
[42]	Sagitov, S. and Serra, M. C. (2009). Multitype Bienaymé-Galton-Watson processes escaping extinction. Adv. in Appl. Probab. 41 225-246. · Zbl 1161.60031 · doi:10.1239/aap/1240319583
[43]	Schrag, S. J., Perrot, V. and Levin, B. R. (1997). Adaptation to the fitness costs of antibiotic resistance in Escherichia coli. Proc. R. Soc. Lond., B Biol. Sci. 264 1287-1291.
[44]	Serra, M. C. (2006). On the waiting time to escape. J. Appl. Probab. 43 296-302. · Zbl 1097.60069 · doi:10.1239/jap/1143936262
[45]	Serra, M. C. and Haccou, P. (2007). Dynamics of escape mutants. Theor. Popul. Biol. 72 167-178. · Zbl 1123.92027 · doi:10.1016/j.tpb.2007.01.005
[46]	Smadi, C. (2017). The effect of recurrent mutations on genetic diversity in a large population of varying size. Acta Appl. Math. 149 11-51. · Zbl 1373.92087 · doi:10.1007/s10440-016-0086-x
[47]	Tran, V. C. (2008). Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM Probab. Stat. 12 345-386. · Zbl 1187.92071 · doi:10.1051/ps:2007052
[48]	van der Hofstad, R. (2016). Random Graphs and Complex Networks. Cambridge Univ. Press, Cambridge. · Zbl 1361.05002
[49]	Wade, M. J. and Goodnight, C. J. (1991). Wright’s shifting balance theory: An experimental study. Science 253 1015-1018.
[50]	Weinreich, D. M. and Chao, L. (2005). Rapid evolutionary escape by large populations from local fitness peaks is likely in nature. Evolution 59 1175-1182.
[51]	Weissman, D. B., Desai, M. M., Fisher, D. S. and Feldman, M. W. (2009). The rate at which asexual populations cross fitness valleys. Theor. Popul. Biol. 75 286-300. · Zbl 1213.92051 · doi:10.1016/j.tpb.2009.02.006
[52]	Wright, S. (1965). Factor interaction and linkage in evolution. Proc. R. Soc. Lond., B Biol. Sci. 162 80-104.