Non-equilibrium statistical physics, population genetics and evolution Marija Vucelja The Rockefeller University UVa Physics colloquium, 2013
outline • traditional view of population genetics: mutations, recombination and selection • questions of interest D arwin’s finches • why all hasn’t been solved yet? • relations to statistical physics • spin glass (clonal interference) • polymers, path integrals, localization phenomena (phenotype switching) • unusual kind of non-equilibrium statistical physics: • new processes like recombination • effects of discreteness appearing even in the “thermodynamical limit” 2
early ideas on evolution and the standard picture g = { s 1 , ..., s L } genotype = spin configuration s i ∈ { +1 , − 1 } allele = spin state example: eye color Gregor Mendel Alfred Russel Jean-Baptiste Charles Darwin Wallace Lamarck mutation creates variation: only favorable survive fitness landscape F ( g, environment) fitness = energy 3
mutation - spin flip M : g → Mg = { s 1 , . . . , Ms i , . . . , s L } clones = organisms’ with the same genotype successional mutations strong selection & weak mutations present in small populations concurrent mutations strong selection & strong mutations present in large populations we see: clonal competition & multiple mutations clonal competition weak beneficial mutation present in large populations disregarding multiple mutations Desai, Fisher, 2007 4
recombination - no physics analog humans and other organisms with two sets of chromosomes n o s i | ξ i s ( m ) + (1 − ξ i ) s ( f ) R : g ( m ) , g ( f ) → g = i i viruses recombine in the hosts’ cell s n HIV: 2 RNA strains i a r t s A N R 8 - 7 : a z n e u fl n i 5
selection proxy for fitness = the expected reproductive success F ( g, environment) spin interactions X X X F = ¯ F + f i s i + f ij s i s j + f ijk s i s j s k + . . . i i<j i<j<k number of individuals with genotype g n g n g ( t ) = ( F g − ¯ ˙ F ) n g ( t ) + noise N F = 1 X ¯ chemical potential n g F g N g R ¯ = e F g t − F dt = e F g t n g ( t ) Boltzmann statistics time = inverse temperature N N Z 1 ⌧ N ⌧ 2 L population = random sample from genotype space 6
ρ ( F , t ) fitness distribution X ρ ( F, t ) = δ ( F − F g ) Z g h F i ⌘ d Fp ( F, t ) F Fundamental theorem of natural selection d d t h F i = h ( F � h F i ) 2 i variance = selection strength, since only n g R. A. Fisher with F g > F grow R ¯ n g ( t ) = e F g t − F dt 7
Quasi-linkage equilibrium Kimura, 1965 perturbative: weak selection & weak interactions compared to mutation & recombination that act to decorrelate spins (bio: alelles) F int interactions (bio: epistasis) X X X F = ¯ F + f i s i + f ij s i s j + f ijk s i s j s k + . . . i i<j i<j<k F 0 non-interacting (bio: additive) fitness distribution separable solution: distributions for non-interacting ρ ( F, t ) = p ( F 0 , t ) ω ( F int , t ) and interacting part d d t h s i i ⇡ function of h s i i , h s i s j i description with first two moments: h s i s j i ⇡ function of h s i i , h s j i 8
Quasi-linkage equilibrium is allele competition Neher, Shraiman 2009 no mutations recombination rate r selection strength = variance of F = σ 2 perturbation: r/ σ > 1 color = different allele same colors grouped together 2 F = F 0 + F int σ / int 2 σ σ int2 = variance F int variance ratio r/ σ 9 recombination/selection
Clonal Competition Neher, MV, Mezard, B. Shraiman 2009 Quasi- Linkage Clonal Condensation Equilibrium weakly nonlinear strongly interacting perturbative non-perturbative How do large clones form and persist? • fungi Relevant for: • microorganisms • plants • contiguous parts of chromosome facultatively recombining populations like: y e a s t • pathogens such as HIV and influenza • nematodes rapidly adapting populations, hybridization zone, population bottlenecks, HIV, influenza 10
Coalescence rate = good measure of distance in genotype space 0 Y(t) = probability that in a population of N individuals 2 are from the same clone (have a common ancestor) t if two genotypes are identical than they had a common ancestor in the past t − ∆ t t 1 1 N probability of two individuals in the present generation, having the same O ( N − 1 ) ancestor one generation in the past ( O ( N − 1 ) Y ( t, ... ) = actually: O (1) clonal condensation few clones grow to form a significant fraction of the population. 11
Selection t F j F j the most fit F i genotype overtakes the F i whole population Y = 1 Y = 0 ln N ∝ ( F max − F ) t t c t t c t σ σ σ σ “condensation time” time for the fittest clone to over takes the whole population 12
the Random Energy Model and spin glasses 1 2 πσ 2 e − F 2 many spins contribute independently; Central Limit ρ ( F ) = 2 σ 2 Theorem √ statistics of clones is identical to that of the Random Energy Model spin glass: disordered magnet with frustrated interactions and stochastic positions of spins Random Energy Model = a set of configurations and an energy functional over those configurations. ρ ( E ) are i.i.d. random variables drawn from { E i } “sample” = particular realization of such a process 13
Recombination - reshuffles genetic variations and produces novel combinations of existing alleles Ρ H F L Ρ H F L t F F r σ t F j F j F i F i F k Y = 0 Y = 1 ln N ∝ ( F max − F ) t t c t t c t σ σ σ σ 2 3 4 N � 1 X � � n ( g ) = ˙ F ( { g } ) − F ( t ) n ( g ) K ( g | g 0 , g 00 ) n ( g 0 ) n ( g 00 ) − n ( g ) + r 5 g 0 , g 00 Smoluchowski equation 14
Condensation r = 0 Ρ H F L F ρ ( F ) At short t the averages are dominated by vicinity of the peak of t > t c At the dominant contribution shifts to the leading edge of the distribution With time the population shifts to fitter and fitter genotypes and eventually condenses. ! 2 + *X probability of two individuals n i ( t ) h Y t i = being identical P j n j ( t ) spin-glass order parameter i * N + Z ∞ i 0 =1 e ( Fi 0 � r � F ) t d zze 2( F i − F ) t − z P N X h Y t i = 0 i =1 � N − 1 Z ∞ Z Z d F i ρ ( F i ) n 2 i ( t ) e − zn i ( t ) d F j ρ ( F j ) e − zn j ( t ) h Y t i = N d zz 0 15
Effective model N − M M R t R t tj d t 0 F ( t 0 ) = N ( F j − r )( t − t j ) − 0 d t 0 F ( t 0 ) + X e ( F i − r ) t − X e i =1 j =1 forefathers recombinants ! 2 + *X n i ( t ) h Y t i = P j n j ( t ) i averaging: • statistics of fitness • Poisson process - arrival of M recombinants at times t j t c ( r ) 1 = t c 1 − r/r c p 2 ln( N ) r c ≡ σ theory Similar behavior observed in Barton 1983, Franklin and Levontin, 1970 16
Heritability - how does the fitness of recombinants relate to that of the parents. r r c Y = 1 Y = 0 h 2 heritability h 2 t c t parents highly heritable trait offspring trait of low heritability parents offspring 17
without heritability r/ σ = 1 . 0 r/ σ = 1 . 8 r/ σ = 0 . 2 r/ σ = 2 . 4 clone = color same colors grouped increasing recombination rate delays the transition to condensed state 18
high heritability r/ σ = 1 . 0 r/ σ = 0 . 2 r/ σ = 1 . 8 r/ σ = 2 . 4 Large clones cease to exist. t c = σ − 1 p 2 ln( N ) Most population is made out of short lived genotypes. 19
Traveling solutions for additive fitness h 2 = 1 No aging = old genotypes are replaced; dominant genotypes have a finite characteristic age τ j τ 2 j σ 2 n j = e ( A j − r ) τ j − 2 2 ln N − r 2 h Y ( r ) i ⇡ 2 N − 1 + re − r σ − 1 √ 2 σ 2 ( Cohen et al., 2005a; Desai and Fisher, 2007; Hallatschek, 2011; Neher et al., 2010; Rouzine et al., 2003; Tsimring et al., 1996) 20
Future directions: clonal condensation Connection to real world populations: heritability (between 0 and 1), large number of loci. - more complicated models would reveal more structure than this simple “dust/clone” dichotomy 0 < h 2 < 1 • better understanding of a “mixed” phase • adding mutations • REM Sherrington - Kirkpatrick • dynamics population getting homogeneous and diverse log h Y ∞ ( r, h ) i 21
Inference of fitness of the leaves from genealogical trees Dayarian, Shraiman 2012 influenza sequences US and Canada 2010-2011 Drummond and Rambaut, 2007 heuristic approach 22
Phenotype internal set of states of individual organisms set of organisms’ traits example: stripes, color, biochemical or physiological properties, behavior... phenotypic switching phenotype-genotype map environmental fluctuations = external forcing temporal fluctuations drive the system away from equilibrium equilibrium perspective - individual histories (trajectories in the phenotypic space observed in the population ) + environment 23
Phenotypic switching Responsive and stochastic switching Kussell, Leibler 2005 Kussell, Leibler, 2010 24
Persistors different behavior, dividing more slowly antibiotic persistence Balaban et al, 2004 hipA7 - high persistence mutants 25
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