title: Collective Fluctuations in the Dynamics of Adaptation and Other Traveling Waves.
abstract: The dynamics of adaptation are difficult to predict because it is highly stochastic even in large populations. The uncertainty emerges from random genetic drift arising in a vanguard of particularly fit individuals of the population. Several approaches have been developed to analyze the crucial role of genetic drift on the expected dynamics of adaptation, including the mean fitness of the entire population, or the fate of newly arising beneficial deleterious mutations. However, little is known about how genetic drift causes fluctuations to emerge on the population level, where it becomes palpable as variations in the adaptation speed and the fitness distribution. Yet these phenomena control the decay of genetic diversity and variability in evolution experiments and are key to a truly predictive understanding of evolutionary processes. Here, we show that correlations induced by these emergent fluctuations can be computed at any arbitrary order by a suitable choice of a dynamical constraint. The resulting linear equations exhibit fluctuation-induced terms that amplify short-distance correlations and suppress long-distance ones. These terms, which are in general not small, control the decay of genetic diversity and, for wave-tip dominated ("pulled") waves, lead to anticorrelations between the tip of the wave and the lagging bulk of the population. While it is natural to consider the process of adaptation as a branching random walk in fitness space subject to a constraint (due to finite resources), we show that other traveling wave phenomena in ecology and evolution likewise fall into this class of constrained branching random walks. Our methods, therefore, provide a systematic approach toward analyzing fluctuations in a wide range of population biological processes, such as adaptation, genetic meltdown, species invasions, or epidemics.
authors.affiliation: Biophysics and Evolutionary Dynamics Group, Departments of Physics and Integrative Biology, University of California, Berkeley, California 94720-3220 ohallats@berkeley.edu.
authors.forename: Oskar
authors.initials: O
authors.lastname: Hallatschek
authors.affiliation: Biophysics and Evolutionary Dynamics Group, Max Planck Institute for Dynamics and Self-Organization, 33077 Göttingen, Germany.
authors.forename: Lukas
authors.initials: L
authors.lastname: Geyrhofer
mesh_terms: D000220:Adaptation, Biological / Q000235:genetics*; D019143:Evolution, Molecular; D040961:Genetic Drift; D005828:Genetics, Population; D008957:Models, Genetic; D009154:Mutation; D011156:Population Density
keywords: Fisher waves; asexual adaptation; exact approaches; higher-order correlation functions; stochastic modeling; traveling-wave models; tuned models
references.citation: Theor Popul Biol. 2008 Feb;73(1):24-46
references.pmid: 18023832
references.citation: Philos Trans R Soc Lond B Biol Sci. 2009 Jun 12;364(1523):1483-9
references.pmid: 19414463
references.citation: Genetics. 2013 Feb;193(2):565-85
references.pmid: 23222656
references.citation: Genetics. 2007 Jul;176(3):1759-98
references.pmid: 17483432
references.citation: Proc Natl Acad Sci U S A. 2011 Feb 1;108(5):1783-7
references.pmid: 21187435
references.citation: Philos Trans R Soc Lond B Biol Sci. 2005 Jul 29;360(1459):1411-25
references.pmid: 16048784
references.citation: Proc Natl Acad Sci U S A. 2012 Mar 27;109(13):4950-5
references.pmid: 22371564
references.citation: Science. 2007 Aug 10;317(5839):813-5
references.pmid: 17690297
references.citation: J Mol Microbiol Biotechnol. 2011;21(1-2):20-35
references.pmid: 22248540
references.citation: J R Soc Interface. 2014 May 07;11(96):20140172
references.pmid: 24806706
references.citation: Genetics. 2010 Feb;184(2):467-81
references.pmid: 19948891
references.citation: Genetics. 2012 Aug;191(4):1309-19
references.pmid: 22661327
references.citation: Curr Biol. 2007 Mar 6;17(5):385-94
references.pmid: 17331728
references.citation: Theor Popul Biol. 2013 May;85:86-102
references.pmid: 23337315
references.citation: Genetics. 2014 Apr;196(4):1167-83
references.pmid: 24429280
references.citation: Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Jun;61(6 Pt B):6789-801
references.pmid: 11088374
references.citation: Science. 2006 Mar 17;311(5767):1615-7
references.pmid: 16543462
references.citation: PLoS Genet. 2014 Mar 06;10(3):e1004182
references.pmid: 24603313
references.citation: Phys Rev Lett. 1996 Jun 3;76(23):4440-4443
references.pmid: 10061290
references.citation: Genetics. 2011 Dec;189(4):1361-75
references.pmid: 21926305
references.citation: Theor Popul Biol. 1984 Apr;25(2):138-93
references.pmid: 6729751
references.citation: Genetica. 1998;102-103(1-6):127-44
references.pmid: 9720276
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Oct;76(4 Pt 1):041104
references.pmid: 17994933
references.citation: Trends Ecol Evol. 2012 Oct;27(10):547-60
references.pmid: 22819306
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2006 May;73(5 Pt 2):056126
references.pmid: 16803017
references.citation: Proc Natl Acad Sci U S A. 2013 Sep 24;110(39):15836-41
references.pmid: 24019480
references.citation: Nature. 2015 Mar 12;519(7542):181-6
references.pmid: 25731169
references.citation: Phys Rev Lett. 2009 Sep 4;103(10):108103
references.pmid: 19792344
references.citation: Genet Res. 2007 Dec;89(5-6):491-503
references.pmid: 18976539
references.citation: Theor Popul Biol. 2008 Feb;73(1):158-70
references.pmid: 17963807
references.citation: Genetics. 2012 Aug;191(4):1283-93
references.pmid: 22649084
references.citation: Proc Natl Acad Sci U S A. 2003 Jan 21;100(2):587-92
references.pmid: 12525686
references.citation: Phys Rev Lett. 2005 Apr 22;94(15):158302
references.pmid: 15904196
references.citation: Proc Natl Acad Sci U S A. 1978 Jan;75(1):423-6
references.pmid: 16592484
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Dec;72(6 Pt 2):066126
references.pmid: 16486029
references.citation: Proc Natl Acad Sci U S A. 2014 Jan 21;111(3):1037-42
references.pmid: 24395776
references.citation: Proc Natl Acad Sci U S A. 2013 Jan 8;110(2):437-42
references.pmid: 23269838
references.citation: Rep Prog Phys. 2014 Oct;77(10):102602
references.pmid: 25303141
references.citation: Proc Natl Acad Sci U S A. 1965 Sep;54(3):731-6
references.pmid: 5217452
references.citation: Genet Res. 1975 Dec;26(3):221-35
references.pmid: 1225762
references.citation: Annu Rev Ecol Evol Syst. 2014 Nov 1;45:179-201
references.pmid: 26740803
references.citation: Proc Natl Acad Sci U S A. 2007 Dec 11;104(50):19926-30
references.pmid: 18056799
references.citation: Phys Rev Lett. 2014 Apr 11;112(14):148103
references.pmid: 24766021