Range Expansions & Spatial Population Genetics

Transcription

1 Range Expansions & Spatial Population Genetics 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 Gene Surfing and Survival of the Luckiest The fate of mutations in a spreading population Oskar Hallatschek Kirill Korolev Severine Atis, Bryan Weinstein, Andrew Murray vs.

3 Fisher Waves and Population Dynamics ct () population of species at time t in region dc() t dt births - deaths + saturation + migration 1798 T.R. Matthus dc() t dt ac(), t a P.F. Verhulst dc() t dt 2 ac() t bc () t c* stable population size: c a/ b biomass of yeast you are here J. Maynard Smith, Evolutionary Genetics B B B B B B B B time (hours) B 1937 R.A. Fisher c c( r, t) crt (, ) t 2 2 D crt (, ) acrt (, ) bc( rt, )

4 Population Waves In One Dimension (R. A. Fisher, Kolmogorov-Petrovsky-Piscounov, 1937) t 2 cxt (, ) D cxt (, ) acxt (, )[1 ( b/ a)( cxt (, )]; letcxt (, ) f( x vt) 2 x c(x,t) c(x,t) c(x,t) a/ b a/ b a/ b at e v 2Dt Schematic time development of a wavefront solution of Fisher s equation on the infinite line. (J.D. Murray, Mathematical Biology) Interface velocity v = 2 Da Interface width = D / a

5 Genetic drift for a neutral mutation, (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) u( p, t) t 2 [ DG u( p, t)] 2 p D ( p) p(1 p)/(4 N) G pt () pt ()(1 pt ())/2 N () t t ( t) ( t') ( t t') (Ito calculus)

6 Genetic drift for a neutral mutation, (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( p, t) [ DG u( p, t)] 2 t p D ( p) p(1 p)/(4 N) G Finite populations go to fixation for long times 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!

7 Successful Surfing (1d) c s (x), steady state population density (O. Hallatschek) c x, comoving frame

8 c s Successful Surfing (1d) (x), steady state population density c x, comoving frame

9 c s Successful Surfing (1d) (x), steady state population density c x, comoving frame

10 c s Successful Surfing (1d) (x), steady state population density c x, comoving frame

11 c s Successful Surfing (1d) (x), steady state population density c x, comoving frame

12 Often however c s (x), steady state population density (O. Hallatschek) c x, comoving frame

13 c s Often however (x), steady state population density c x, comoving frame

14 c s Often however (x), steady state population density c x, comoving frame

15 c s Often however (x), steady state population density c x, comoving frame

16 c s Often however (x), steady state population density surfing is not achieved! c x, comoving frame

17 Gene Surfing in nonmotile E. coli (thanks to Tom Shimizu, Berg Lab, for strains) HCB1553/pVS133 Background DH5 Amp(R) Ori pbr HCB1550/pVS130 Background DH5 Amp(R) Ori pbr PTrc99A: yfp A206K Ptrc PTrc99A: ecfp A206K Ptrc mixture, 1550/1553

18 Gene Surfing in nonmotile E. coli (thanks to Tom Shimizu, Berg Lab, for strains) HCB1553/pVS133 Background DH5 Amp(R) Ori pbr HCB1550/pVS130 Background DH5 Amp(R) Ori pbr PTrc99A: yfp A206K Ptrc PTrc99A: ecfp A206K Ptrc mixture, 1550/1553 Cyan Red 1mm

19 Genetic demixing in non-motile E. coli (thanks to Tom Shimizu, Berg Lab, for strains) HCB1553/pVS133 Background DH5 Amp(R) Ori pbr HCB1550/pVS130 Background DH5 Amp(R) Ori pbr PTrc99A: yfp A206K Ptrc PTrc99A: ecfp A206K Ptrc a r = doubling time of red cells a g = doubling time of green cells = (1+s)a r s = selective advantage (s=0) mixture, 1550/1553 Cyan Red

20 Genetic demixing time effective population size for range expansions ΔR fix = 1mm t ( p) 2 N [(1 p)ln(1 p) pln p] fix eff g (2ln 2) N 2.83N eff g eff g v Da g cm / sec; 1560sec (2.83 N ) v R N 22 eff g fix eff

21 What would happen if we could replay the tape of life? R 0 R 0 R 0 Ecoli. range expansions: can infer the radius R of the homeland from data at the boundary: Nsec R0v/2D W Chiral range expansion (Kirill Korolev) 0

22 Gene surfing in the dilute limit: survival of the luckiest 95%-5% mixture, founder population ~ %-2% mixture, founder population ~ 5000

23 Gene surfing and simple models of tumor growth Exponential phase succeeded by linear growth 1 : Growth confined to tumor edge 1,2 Rt () vt Growth may stop (due to mechanical stress 3 or nutrient shielding 4 ), creating a "treadmilling" effect Prevascular tumors can grow to radius R 0 ~ mm. R 0 /a ~ 20, a = cell diameter necrotic core (volume loss) quiescent region (cells not dividing) thin growing region cells Hemispherical yeast colonies 1 A Bru et al. Biophys. J (2003) 2 T Roose et al. SIAM Rev. 49(2) 179 (2007) 3 F Montel et al. PRL (2011) Confocal images of nutrient shielding in S. cerevisiae RED: protein with growth-independent expression GREEN: ribosomal protein indicating growth 4 Lavrentovich, Koschwanez,drn, PRE 87, (2013)

24 Selection and driver mutations driver mutations form logarithmic spirals: Mean field theory () r s(2 s)ln( r/ r0 ) s selective advantage Max Lavrentovich, ( Domany-Kinzel models Friday talk) *Cancer often results from mutations in normal cells, leading to abnormal growth. or they die out! *A series of driver mutations (in oncogenes, tumor suppressor genes, etc.) can lead to benign or malignant cell clusters. What is the survival probability of a single frontier cell with a driver mutation?

25 Exact asymptotics in two dimensions (cell size a is important ) Successful escape from genetic drift to a deterministic logarithmic spiral determined by dimensionless radial selective advantage s R / a ( a cell size) 0 R0 initial radius of tumor 1, strong selection 1, weak selection () t noisy logarithmic spiral described by a stochastic differential equation d () t sa 2 () t (), t R() t R0 vt dt R() t g 2 2 () t a / g R (), t () t (') t 2 ( t t')

26 lim P ( s, R, a,,, v, t) ( R / a, s R / a ) t Survival probability of a driver mutation in two dimensions: surv 0 g 0 t surv ( x R / a, ) 0 0 s R / a 0 LARGE survival probability at s =0 Much larger than a/2πr 0 (treadmilling circle result) inflation at frontier helps neutral passenger mutations

27 We obtain similar results for driver mutations on spherical colonies Surface of an inflating sphere P fix (s) vs. Large survival probability at s =0! compare a treadmilling sphere... lim P ( s) a / 4 R 8 10 s 0 fix when R 10 a, inflation amplifies 27 fixation prob. by a factor of

28 Spatial population genetics and fluid flow Life first evolved in a liquid environment! Cyanobacterium Synechococcus Effect of compressible fluid flow on the Fisher Equation What is the effect on fluid motion on spatial population genetics? R. Benzi, M. Jensen P. Perlecar, S. Pigolotti F. Toschi Severine Atis, Bryan Weinstein & Andrew Murray

29 Buoyant population dyanamics in Silico (P. Perlekar and F. Toschi) Reynolds number Schmidt number Doubling time/eddy turnover time τ 2 /τ eddy / TN & WI 11/04/11

30 Compressible population genetics with two interacting species Compressible turbulent flow (Re ~10 5) 2 2 ( u) / ( u ) 0.17 i j time Survival of the luckiest Wanted: simple repeatable experiments that explore how fluid flows affect spatial population genetics. High Reynolds numbers might be hard to achieve in the laboratory, but low Reynolds numbers can also be biologically relevant Can we impose a flow with pumps and syringes, such that the doubling time τ 2 << τ eddy, where τ eddy is a eddy turnover time?

31 Actually, microorganisms grown on a liquid but highly viscous substrates create their own flows (without pumps and syringes!) Hard Agar Epoch of genetic demixing stretched out. Liquid Media Yeast on a 1% hard agar YPD plate (viscosity η = ) Yeast on a liquid but highly viscous YPD media with 3% cellulose (η 600 Pa-s) (the viscosity of water is η 10-3 Pa-s; our viscosities are times larger) The colony metabolism (after only two generation times) generates fluid flows that dilate the colony radially.

32 Deformations of features inside colony in liquid regimes are consistent with flow-induced dilation!

33 PIV (particle imaging velocimetry) measurements quantify the self-induced flow under the colony Air Liquid Like an anti-helmholtz coil!

34 What is the origin of the flow beneath colonies growing on liquid substrates? = dextrose wikipedia

35 Flow simulations Isobars and isoclines in the absence of flow (η = ) A thresholdless baroclinic instability generates a ring of vorticity beneath the colony.

36 Liquid-like fingering instabilities (moderate substrate viscosity η 450 Pa-s) Happy 4 th of July (and Bastille Day!) Dynamics of a finger: Rayleigh-Plateau type instability with growth?

37 Solid-like colony fragmentation (low substrate viscosity η 85 Pa-s) Colony takes over plate in one day!

38 Microbes on highly viscous liquids generate buoyant flows that alter colony morphology and evolutionary dynamics