Population Genetics on Land

Transcription

1 Population Genetics on Land Oskar Hallatschek & drn, (E. coli) In 500 generations. Large mammals expand over ~10 4 km Bacteria (in a Petri dish) expand ~ 1 cm K. Korolev et al., Reviews of Modern Physics 82, 1691 (2010) K. Foster, J. Xavier, K. Korolev, drn (genetic demixing in P. Aeruginosa)

2 Genetic fluctuations in a neutral population, (M. Kimura) Populus 5.3 N = 20 a A u(p,t)= probability allele A has frequency p at time t. Finite populations go to fixation for long times (using, e.g., Fisher-Wright population sampling) u( f, t) t 2 [ Du G ( f, t)] 2 f D ( f) f(1 f)/(4 N) G df ()/ t dt f (1 f )/2 N() t () t (') t 2 ( tt') (use Ito calculus...)

3 Survival of the Luckiest (M. Kimura) Populus 5.3 N = 20 u(p,t)= probability allele A has frequency p at time t. Finite populations go to fixation for long times(using, e.g., Fisher-Wright population sampling) 2 u( f, t) [ Du G ( f, t)] 2 t f D ( f) f(1 f)/(4 N) G Finite populations go to fixation for long times. t fix = 1/D G ~ Nτ g Probability of fixation of a single neutral mutation in a population of size N is just 1/N But N is small in the vicinity of an expanding population front!

4 Linear Inoculations: Genetic demixing results from number fluctuations at the frontier View from the top: razor blade innoculation View of a dust mote on a Petri dish as the wave advances. t 1 Side view as the population wave advances: t 2 > t 1 Fisher genetic waves

5 What would happen if we could replay the tape of life? N R v D sec 0 /2 W Chiral range expansion!

6 But life probably evolved first in a liquid environment ~2-3 billion years ago, water covered most of the earth Fossilized, oxygen-producing cyanobacteria have been dated at ~2 billion years ago. Cyanobacterium Synechococcus ralf wagner.de/blaualgen englisch.htm Oxygenic cyanobacteria transformed the atmosphere via photosynthesis Spatial growth and evolutionary competition took place in liquid environments at both high low Reynolds numbers These photosynthetic organisms may control their height to resist down welling currents and stay close to the ocean surface. Bloom of cyanobacteria in Lake Atitlán, Guatemala NASA Earth observatory

7 Life at High Reynolds Number (population genetics on land and at sea) Competition and cooperation on land and at sea Range expansions and genetic competitions on solid surfaces Cyanobacteria (& plankton) in the ocean Effect of compressible turbulent advection on the Fisher Equation --Quasilocalization on velocity sinks --Greatly reduced carrying capacity Effect on turbulence on population genetics --Population bottlenecks and altered fixation times R. Benzi M. Jensen P. Perlecar S. Pigolotti F. Toschi P. Perlekar, R. Benzi, F. Toschi and drn, Population Dynamics at High Reynolds Numbers, Phys. Rev. Lett. 105, (2010) S. Pigolotti, R. Benzi, M. Jensen and drn. Population Genetics in Compressible Flows Phys. Rev. Lett. 108, (2012). S. Pigolotti et al. Growth, competition and cooperation in spatial population genetics, Theoretical Population Biology 84, 72 (2013).

8 Navier-Stokes Equations and Reynolds Numbers urt (, ) = velocity field (incompressible) 2 tu( u) u (1/ 0) p u Navier & u 0, p( r, t) = pressure density; 0 viscosity Stokes ~1841 Kolmogorov Energy Cascade ~1941 Reynolds Number Re Lu / Reynolds_number Re ~ Re ~ BP Gulf oil spill

9 Turbulent advection & diffusion of a passive scalar Plume like emission in a pipe Shan & Dimotakis JFM 566, 47 (2006) Re j = 1000 v jet /v flow = 10 Volcanic emission in Iceland A 150 km wide view of the ash cloud from the 2010 eruption of the Eyjafjallajökull volcano 2 cxt (, ) [ uxtcxt (, ) (, )] D cxt (, ) t D diffusion constant, u( x, t) 0 Peclet number Pe ul / D 1 Multiply by 2 crt (, ) & average over space and ti me: When advected by an incompressible turbulent velocity field in a confined volume, cxt (, ) is mixed to uniformity in a few large eddy turnover times. d 2 2 c ( r, t) 2 D c( r, t) dt But. the behavior of floater particles is different!!

10 Contrast: compressible passive scalar advection in two dimensions Limit μ 0 described by compressible passive scalar with a conserved probability distribution P(r,t) t 2 crt (, ) [ urtcrt (, ) (, )] D crt (, ) Multiply by 2c & average over space and realizations: d dt c ( r, t) c ( r, t)[ u( r, t)] 2 D c( r, t) Uniformly mixed steady state inevitable for incompressible fluids (also true for μ > 0) Peaks in P(r,t) correlated with sinks in u(r,t) ( u( r, t) 0) ( u( r, t) 0) Nontrivial steady state fluctuations result Lagrangian tracer particles on a surface flow Boffetta et al., PRL93, (2004)

11 Connection with compressible passive scalar advection in 2d Limit μ 0 described by compressible passive scalar with a conserved probability distribution P(r,t) t 2 crt (, ) [ urtcrt (, ) (, )] D crt (, ) Multiply by 2P & average over space and time: d dt c ( r, t) c ( r, t)[ u( r, t)] 2 D c( r, t) Uniformly mixed steady state inevitable for incompressible fluids (also true for μ > 0) Peaks in P(r,t) correlated with sinks in u(r,t) ( u( r, t) 0) ( u( r, t) 0) Nontrivial steady state fluctuations result Floater particles at the surface of a turbulent flow W. Goldburg et al. New Journal of Physics 6, 53 (2004)

12 Steady incompressible flow ux( x, y) Fsin(2 y/ L) u ( x, y) Fsin(2 x/ L), u 0 y time Compressible population genetics with two interacting species Compressible turbulent flow (Re ~10 5) 2 2 ( u) / ( iuj) 0.17

13 Striated plankton populations in oceanic flows Phytoplankton blooms at high Reynolds number in the Norwegian Sea and near Iceland 110 km bin/viewrecord?5278.see also, Tel. et al. Phys. Rep. 413, 91 (2005). periments/ice/channel_islands/ mixing layer m. Phytoplankton (see also zooplankton & bacterioplankton) A. P. Martin, Prog. Oceanography 57, 125 (2003) Re LU / Large eddy turnover time 50 days Small eddy turnover time 5 minutes Plankton doubling time hours

14 PNAS 107, (2010) Chlorophyll map Velocity field from altimetry diatoms (green) Prochlorococcus (red) Synechococcus (dark blue) nanoeukaryotes (yellow) Phaeocystis (magenta) coccolithophorids (cyan). Dominant species types

15 Compressible advection of microorganism density c(x,t) 2 cxt (, ) [ u( xtcxt, ) (, )] D cxt (, ) cxt (, )[1 cxt (, )] t u( x, t) 0; u zuz 0; uz ~10 m/ day; z ~ 20m u 1/ day 1/ cell divison time! u( x, t) effective 2d compressible turbulent velocity field; growth rate... Advection by an effectively compressible two dimensional velocity field results for organisms with a buoyancy that keeps them close to the ocean surface. accumulation points Uop.whoi.edu/projects/projects.htm c.f. john crimaldi s talk

16 Intermittent microorganism fluctuations in two dimensions (Prasad Perlecar, F. Toschi, R. Benzi, ) 2 cxt (, ) [ uxtcxt (, ) (, )] D cxt (, ) cxt (, )[1 cxt (, )] t Generate 2d compressible velocity field by assuming microorganisms populate a fixed slice of a simulation of 3d incompressible turbulence with Re 10 5 degree of compressibility 2 2 ( u) / ( iuj) 0.17 urt (, ) See 2d movie

17 Life in silico + / TN & WI 11/04/11 P. Perlekar and F. Toschi

18 Quasi localization in two dimensions Bacterial localization length : 2 D D=0.05 1/4 One point closure predicts a localization length ξ ~ D 1/2 D=0.01 Synechococcus: ~1/3 can swim at velocity ~25um/sec. Increases D eff by factor of D= Z( r, t) d r c( r, t) carrying capacity Average carrying capacity <Z> 1 as μ <Z> suffers an 80% reduction as μ 0.

19 Stochastic Fluid Mechanics of Two Interacting Species See S. Pigolotti, R. Benzi, M. Jensen, F. Toschi, drn, Growth, competition and cooperation in spatial population genetics,, Theoretical Population Biology 84, 72 (2013). time Gillespie algorithm [ uxtc (, ) A ] [ uxtc (, ) B ] A AA B BB (1 ) AB A (1 ) BA B A and control the B strength of interactions (mutualism, competitive exclusion, etc.)

20 Mutualists on various substrates Melanie Mueller, Andrew Murray & drn Abundant nutrients (incuding Leu & Trp) Mutualism unimportant Strong genetic demixing Abundant nutrients (but Leu & Trp are missing) Obligate mutualism genetic demixing suppressed

21 Turbulence induced localization Front speed Front thickness / TN N & WI WI 11/04/11 Federico Toschi P. Perlekar and F. Toschi

22 Effect of compressible turbulence on population genetics with Lagrangian inertial particles (1d) (R. Benzi, M. Jensen, S. Pigolotti, drn)

23 Long-lived persistor states can have enormous fixation times.. u 0 0 ux ( ) usin(2 x/ L) -u 0 Start with well mixed initial conditions and a single sink R R vs. G G vs. R G vs. G R even number of genetic boundaries odd number of genetic boundaries

24 Long-lived persistor states can have enormous fixation times.. Start with well mixed initial conditions and a single sink in a converging flow R R vs. G G vs. R G vs. G R Selective Advantage Meter S = 0.0 S = 0.3 See also: A. Brandenburg and T. Multamaki, How long can left and right handed life forms coexist?, Int. J. of Astrobiology 3, 209 (2004)