The pace of evolution across fitness valleys.

How fast does a population evolve from one fitness peak to another? We study the dynamics of evolving, asexually reproducing populations in which a certain number of mutations jointly confer a fitness advantage. We consider the time until a population has evolved from one fitness peak to another one with a higher fitness. The order of mutations can either be fixed or random. If the order of mutations is fixed, then the population follows a metaphorical ridge, a single path. If the order of mutations is arbitrary, then there are many ways to evolve to the higher fitness state. We address the time required for fixation in such scenarios and study how it is affected by the order of mutations, the population size, the fitness values and the mutation rate.

[1]  J. Crutchfield,et al.  Metastable evolutionary dynamics: Crossing fitness barriers or escaping via neutral paths? , 1999, Bulletin of mathematical biology.

[2]  M. Eigen,et al.  The Hypercycle: A principle of natural self-organization , 2009 .

[3]  William H. Press,et al.  Numerical Recipes 3rd Edition: The Art of Scientific Computing , 2007 .

[4]  Nordling Co.  A New Theory on the Cancer-inducing Mechanism , 1953 .

[5]  P. Armitage,et al.  The age distribution of cancer and a multi-stage theory of carcinogenesis , 1954, British Journal of Cancer.

[6]  M. Nowak,et al.  Dynamics of cancer progression , 2004, Nature Reviews Cancer.

[7]  K. Kinzler,et al.  Cancer genes and the pathways they control , 2004, Nature Medicine.

[8]  Claus O Wilke,et al.  Quasispecies theory in the context of population genetics , 2005, BMC Evolutionary Biology.

[9]  R. Watson,et al.  PERSPECTIVE: SIGN EPISTASIS AND GENETIC COSTRAINT ON EVOLUTIONARY TRAJECTORIES , 2005, Evolution; international journal of organic evolution.

[10]  R. Punnett,et al.  The Genetical Theory of Natural Selection , 1930, Nature.

[11]  W. Godwin.  Article in Press , 2000 .

[12]  M. Nowak,et al.  Evolutionary dynamics of cancer , 2008 .

[13]  Peter Schuster,et al.  A principle of natural self-organization , 1977, Naturwissenschaften.

[14]  H. Nürnberg.  The Hypercycle. A Principle of Natural Self Organization. , 1981 .

[15]  P. Moran,et al.  The statistical processes of evolutionary theory. , 1963 .

[16]  J. C. FISHER,et al.  Multiple-Mutation Theory of Carcinogenesis , 1958, Nature.

[17]  R. Lenski,et al.  The fate of competing beneficial mutations in an asexual population , 2004, Genetica.

[18]  M. Nowak,et al.  Stochastic Tunnels in Evolutionary Dynamics , 2004, Genetics.

[19]  J. Gillespie.  Some Properties of Finite Populations Experiencing Strong Selection and Weak Mutation , 1983, The American Naturalist.

[20]  Martin A. Nowak,et al.  Genetic Progression and the Waiting Time to Cancer , 2007, PLoS Comput. Biol..

[21]  M. DePristo,et al.  Mutational reversions during adaptive protein evolution. , 2007, Molecular biology and evolution.

[22]  N. Goel,et al.  Stochastic models in biology , 1975 .

[23]  Joachim Krug,et al.  Deterministic and Stochastic Regimes of Asexual Evolution on Rugged Fitness Landscapes , 2006, Genetics.

[24]  Martin A. Nowak,et al.  Comparative lesion sequencing provides insights into tumor evolution , 2008, Proceedings of the National Academy of Sciences.

[25]  T. Antal,et al.  Fixation of Strategies for an Evolutionary Game in Finite Populations , 2005, Bulletin of mathematical biology.

[26]  Nigel F. Delaney,et al.  Darwinian Evolution Can Follow Only Very Few Mutational Paths to Fitter Proteins , 2006, Science.

[27]  D. Weinreich,et al.  The Rank Ordering of Genotypic Fitness Values Predicts Genetic Constraint on Natural Selection on Landscapes Lacking Sign Epistasis , 2005, Genetics.

[28]  John H.Gillespie.  Population Genetics: A Concise Guide , 1997 .

[29]  Claus O. Wilke,et al.  The Speed of Adaptation in Large Asexual Populations , 2004, Genetics.

[30]  G. Fox,et al.  Equally Parsimonious Pathways Through an RNA Sequence Space Are Not Equally Likely , 1997, Journal of Molecular Evolution.

[31]  J. Krug,et al.  Clonal interference in large populations , 2007, Proceedings of the National Academy of Sciences.

[32]  C. Maley.  The Evolutionary Dynamics of Cancer , 2000 .

[33]  M. Eigen,et al.  What is a quasispecies? , 2006, Current topics in microbiology and immunology.

[34]  W. Ewens.  Mathematical Population Genetics , 1980 .

[35]  M. Kimura,et al.  An introduction to population genetics theory , 1971 .

[36]  Martin A Nowak,et al.  Evolutionary dynamics of tumor suppressor gene inactivation. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[37]  E. Hill.  Journal of Theoretical Biology , 1961, Nature.

[38]  L. Chao,et al.  RAPID EVOLUTIONARY ESCAPE BY LARGE POPULATIONS FROM LOCAL FITNESS PEAKS IS LIKELY IN NATURE , 2005, Evolution; international journal of organic evolution.

[39]  G. Parmigiani,et al.  The Consensus Coding Sequences of Human Breast and Colorectal Cancers , 2006, Science.

[40]  C. Nordling.  A New Theory on the Cancer-inducing Mechanism , 1953, British Journal of Cancer.

[41]  M. Nowak,et al.  The fastest evolutionary trajectory. , 2007, Journal of theoretical biology.

[42]  M. Gerstung,et al.  Waiting Time Models of Cancer Progression , 2008, 0807.3638.

[43]  K. Atwood,et al.  Periodic selection in Escherichia coli. , 1951, Proceedings of the National Academy of Sciences of the United States of America.

[44]  R. Lenski,et al.  Parallel changes in gene expression after 20,000 generations of evolution in Escherichia coli , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[45]  G. Parmigiani,et al.  Core Signaling Pathways in Human Pancreatic Cancers Revealed by Global Genomic Analyses , 2008, Science.

[46]  H. Muller.  Some Genetic Aspects of Sex , 1932, The American Naturalist.

[47]  L. Chao,et al.  Evolution of transposable elements: an IS10 insertion increases fitness in Escherichia coli. , 1985, Molecular biology and evolution.

[48]  R. Axelrod,et al.  Evolutionary Dynamics , 2004 .

[49]  A. Knudson.  Mutation and cancer: statistical study of retinoblastoma. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[50]  M. Eigen,et al.  The molecular quasi-species , 2007 .

[51]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[52]  A. Sparks,et al.  The Genomic Landscapes of Human Breast and Colorectal Cancers , 2007, Science.

[53]  D. J. Kiviet,et al.  Empirical fitness landscapes reveal accessible evolutionary paths , 2007, Nature.

[54]  M. Eigen,et al.  The hypercycle. A principle of natural self-organization. Part A: Emergence of the hypercycle. , 1977, Die Naturwissenschaften.