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Outline Foundational Models Generalized Birth-Death-Mutation Processes A Mathematical Framework for Density Dependent Population Genetics Todd L. Parsons Department of Biology University of Pennsylvania [email protected] Applied Mathematics and Statistics Seminar University of California, Santa Cruz April 10, 2011 Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Foundational Models Generalized Birth-Death-Mutation Processes Applications Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE G OLDEN A GE OF B IOMATHEMATICS Georgii Frantsevich Gause (1910–1986) The Struggle For Existence Williams & Wilkins Co., Baltimore, 1934 Alfred James Lotka (1880–1949) Vito Volterra (1860–1940) Elements of physical biology Williams & Wilkins Co., Baltimore, 1925 Fluctuations in the abundance of a species considered mathematically Nature, (1926) 118: 558–60. Outline Foundational Models Generalized Birth-Death-Mutation Processes T HE G AUSE -L OTKA -V OLTERRA M ODEL ◮ Deterministic model of dynamics of interacting populations; the density of species i, Yi (t), satisfies K X Ẏi (t) = bi + aij Yj (t) Yi (t). j=1 ◮ Population kept finite by density dependent regulation. ◮ Empirical studies with micro-organisms show excellent fits. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes T HE G AUSE -L OTKA -V OLTERRA M ODEL ◮ Deterministic model of dynamics of interacting populations; the density of species i, Yi (t), satisfies K X Ẏi (t) = bi + aij Yj (t) Yi (t). j=1 ◮ Population kept finite by density dependent regulation. ◮ Empirical studies with micro-organisms show excellent fits. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes T HE G AUSE -L OTKA -V OLTERRA M ODEL ◮ Deterministic model of dynamics of interacting populations; the density of species i, Yi (t), satisfies K X Ẏi (t) = bi + aij Yj (t) Yi (t). j=1 ◮ Population kept finite by density dependent regulation. ◮ Empirical studies with micro-organisms show excellent fits. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications K OLMOGOROV ’ S E QUATIONS Kolmogorov proposed considering the class of dissipative dynamical systems such that Ẏi = Mi (Y(t))Yi (t). ◮ Species i competes (cooperates) with species j if ∂Mi ∂xj ≤ 0 (≥ 0) ◮ Rich enough to include many ecological models and be topologically closed. However, arbtitrary dynamics are possible under these axioms! (Smale, 1975) ◮ Andrey Nikolaevich Kolmogorov (1903–1987) Sulla teoria di Volterra della lotta per l’esistenza Giorn. Instituto Ital. Attuari. (1936) 7: 74–80 Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications K OLMOGOROV ’ S E QUATIONS Kolmogorov proposed considering the class of dissipative dynamical systems such that Ẏi = Mi (Y(t))Yi (t). ◮ Species i competes (cooperates) with species j if ∂Mi ∂xj ≤ 0 (≥ 0) ◮ Rich enough to include many ecological models and be topologically closed. However, arbtitrary dynamics are possible under these axioms! (Smale, 1975) ◮ Andrey Nikolaevich Kolmogorov (1903–1987) Sulla teoria di Volterra della lotta per l’esistenza Giorn. Instituto Ital. Attuari. (1936) 7: 74–80 Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications K OLMOGOROV ’ S E QUATIONS Kolmogorov proposed considering the class of dissipative dynamical systems such that Ẏi = Mi (Y(t))Yi (t). ◮ Species i competes (cooperates) with species j if ∂Mi ∂xj ≤ 0 (≥ 0) ◮ Rich enough to include many ecological models and be topologically closed. However, arbtitrary dynamics are possible under these axioms! (Smale, 1975) ◮ Andrey Nikolaevich Kolmogorov (1903–1987) Sulla teoria di Volterra della lotta per l’esistenza Giorn. Instituto Ital. Attuari. (1936) 7: 74–80 Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications K OLMOGOROV ’ S E QUATIONS Kolmogorov proposed considering the class of dissipative dynamical systems such that Ẏi = Mi (Y(t))Yi (t). ◮ Species i competes (cooperates) with species j if ∂Mi ∂xj ≤ 0 (≥ 0) ◮ Rich enough to include many ecological models and be topologically closed. However, arbtitrary dynamics are possible under these axioms! (Smale, 1975) ◮ Andrey Nikolaevich Kolmogorov (1903–1987) Sulla teoria di Volterra della lotta per l’esistenza Giorn. Instituto Ital. Attuari. (1936) 7: 74–80 Outline Foundational Models Generalized Birth-Death-Mutation Processes C LASSICAL P OPULATION G ENETICS Sewall Wright (1889–1988) Evolution in Mendelian populations Genetics 16, 1931, 97–159 Sir Ronald Aylmer Fisher (1890–1962) The Genetical Theory of Natural Selection Clarendon Press, Oxford, 1930 Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE W RIGHT-F ISHER M ODEL • • • • • • • • • • • • 1 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 3 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 4 .. . ◮ ◮ ◮ Assume a population with exactly N individuals of a possible K types, with discrete generations. Each generation “looks back” to the previous and “picks” a parent. Offspring have the same type as their parent. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE W RIGHT-F ISHER M ODEL •O _@ • • gOOO• @@ OOO _@@@ O ◦ ◦ ◦ ◦ •O •O • ~? O ~~ ◦ ◦ •O 1 ◦ •O _@ • _@ • @@ @@ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • ◦ 2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 3 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 4 .. . ◮ ◮ ◮ Assume a population with exactly N individuals of a possible K types, with discrete generations. Each generation “looks back” to the previous and “picks” a parent. Offspring have the same type as their parent. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE W RIGHT-F ISHER M ODEL •O gO • _@ • _@@OO•O @ OOO @@ • • • • •O •O • ~? O ~~ • • •O 1 • •O _@ • _@ • @@ @@ • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • 2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 3 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 4 .. . ◮ ◮ ◮ Assume a population with exactly N individuals of a possible K types, with discrete generations. Each generation “looks back” to the previous and “picks” a parent. Offspring have the same type as their parent. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE W RIGHT-F ISHER M ODEL •O gO • _@ • •O _@@OO•O @ OOO @@ • ? •O •O •O _@ • @@ ~ ~~ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ •O ◦ •O _@ • _@ • @@ @@ •O _@ • _@ • @@ @@ ◦ ◦ ◦ • •O ~? O ~~ •O •O _@ • @@ ◦ ◦ ◦ ◦ ◦ ◦ •O ◦ ◦ • ◦ ◦ 1 2 3 4 .. . ◮ ◮ ◮ Assume a population with exactly N individuals of a possible K types, with discrete generations. Each generation “looks back” to the previous and “picks” a parent. Offspring have the same type as their parent. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE W RIGHT-F ISHER M ODEL •O gO • _@ • •O _@@OO•O @ OOO @@ • ? •O •O •O _@ • @@ ~ ~~ • • • • • •O ◦ ◦ ◦ ◦ ◦ •O • •O ~? O ~~ •O •O _@ • @@ • • • 2 • •O _@ • _@ • @@ @@ •O _@ • _@ • @@ @@ • • • ◦ ◦ ◦ 4 ◦ ◦ • ◦ ◦ 1 3 .. . ◮ ◮ ◮ Assume a population with exactly N individuals of a possible K types, with discrete generations. Each generation “looks back” to the previous and “picks” a parent. Offspring have the same type as their parent. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE W RIGHT-F ISHER M ODEL •O gO • _@ • •O _@@OO•O @ OOO @@ • ? •O •O •O _@ • @@ ~ ~~ •O • ? •O _@ • •O ~ @@ ~~ ◦ ◦ ◦ ◦ ◦ •O •O •O ◦ •O _@ • _@ • @@ @@ •O _@ • _@ • @@ @@ •O _@ • •O @@ ◦ ◦ ◦ • • •O ~? O ~~ •O •O _@ • @@ •O _@ • •O @@ ◦ ◦ ◦ 1 2 3 4 .. . ◮ ◮ ◮ Assume a population with exactly N individuals of a possible K types, with discrete generations. Each generation “looks back” to the previous and “picks” a parent. Offspring have the same type as their parent. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE W RIGHT-F ISHER M ODEL •O gO • _@ • •O _@@OO•O @ OOO @@ • ? •O •O •O _@ • @@ ~ ~~ •O • ? •O _@ • •O ~ @@ ~ ~ • • • • • •O •O •O • •O _@ • _@ • @@ @@ •O _@ • _@ • @@ @@ •O _@ • •O @@ • • • • • •O ~? O ~~ •O •O _@ • @@ •O _@ • •O @@ • • • 1 2 3 4 .. . ◮ ◮ ◮ Assume a population with exactly N individuals of a possible K types, with discrete generations. Each generation “looks back” to the previous and “picks” a parent. Offspring have the same type as their parent. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications G ENERALIZATIONS •O gO • _@ • •O _@@OO•O @ OOO @@ • ? •O •O •O _@ • @@ ~ ~~ •O • ? •O _@ • •O ~ @@ ~~ • • • • • •O •O •O • •O _@ • _@ • @@ @@ •O _@ • _@ • @@ @@ •O _@ • •O @@ • • • • • •O ~? O ~~ •O •O _@ • @@ •O _@ • •O @@ • • • 1 2 3 4 ◮ When all types have the same expected number of offspring, we have the neutral model. ◮ When type i has 1 + si offspring on average, it is selected with coefficient si We can also include mutation, by allowing individuals to randomly switching from type i to type j, with probability µij per generation. ◮ Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications G ENERALIZATIONS •O gO • _@ • •O _@@OO•O @ OOO @@ • ? •O •O •O _@ • @@ ~ ~~ •O • ? •O _@ • •O ~ @@ ~~ • • • • • •O •O •O • •O _@ • _@ • @@ @@ •O _@ • _@ • @@ @@ •O _@ • •O @@ • • • • • •O ~? O ~~ •O •O _@ • @@ •O _@ • •O @@ • • • 1 2 3 4 ◮ When all types have the same expected number of offspring, we have the neutral model. ◮ When type i has 1 + si offspring on average, it is selected with coefficient si We can also include mutation, by allowing individuals to randomly switching from type i to type j, with probability µij per generation. ◮ Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications G ENERALIZATIONS •O gO • _@ • •O _@@OO•O @ OOO @@ • ? •O •O •O _@ • @@ ~ ~~ •O • ? •O _@ • •O ~ @@ ~~ • • • • • •O •O •O • •O _@ • _@ • @@ @@ •O _@ • _@ • @@ @@ •O _@ • •O @@ • • • • • •O ~? O ~~ •O •O _@ • @@ •O _@ • •O @@ • • • 1 2 3 4 ◮ When all types have the same expected number of offspring, we have the neutral model. ◮ When type i has 1 + si offspring on average, it is selected with coefficient si We can also include mutation, by allowing individuals to randomly switching from type i to type j, with probability µij per generation. ◮ Outline Foundational Models Unfortunately, it is exceedingly difficult to obtain analytical results regarding the Wright-Fisher model. This was resolved in 1955 when Kimura proposed an asymptotic approximation to the Wright-Fisher model for large values of N. Generalized Birth-Death-Mutation Processes Applications Motoo Kimura (1924–1994) Solution of a process of random genetic drift with a continuous model PNAS (1955), 41: 144–150 Outline Foundational Models Generalized Birth-Death-Mutation Processes K IMURA’ S D IFFUSION E QUATION Let XiN (t) be the number of individuals of type i at time t and let PN i (t) = 1 N X (Nt). N i Suppose θij = 2Nµij and σi = 2Nsi are constant. As N → ∞ is large, D PN 2 (t) −→ P(t), a diffusion process with probability density f (t, p) satisfying K 1X ∂ ∂f =− ∂t 2 ∂pi i=1 " + pi σi − − K X j=1 K X j=1 θij pk + K X θji pj j=1 ! # " # K X K 2 X ∂ 1 pi δij − pj f . σj pj f + 2 ∂pi ∂pj i=1 j=1 Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications A PPLICATIONS OF K IMURA’ S D IFFUSION , K = 2, With this, Kimura obtained expressions for the fixation probability of the mutant: suppose µ12 = µ21 = 0 and let T = inf {t > 0 : P(t) ∈ {0, 1}} 1 − e−σp 1 − e−σ = p + σp(1 − p) + O(σ 2 ) P{P(T) = 1|P(0) = p} = If the mutation rate is nonzero, the diffusion approaches a steady-state, for which the frequency spectrum of the two alleles is: lim P {P(t) = p} = pθ12−1 (1 − p)θ21 −1 eσp t→∞ Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications A N U NBRIDGEABLE C HASM ? Ecology ◮ Density dependence ◮ Resource limitation ◮ ◮ ◮ ◮ ◮ Multi-species Competition, mutualism, Allee effects Continuous Deterministic ODEs, PDEs Population Genetics ◮ Fixed population sizes ◮ Sampling arguments ◮ Single species Discrete ◮ ◮ ◮ Stochastic Branching processes, Markov chains, diffusions Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications A N U NBRIDGEABLE C HASM ? Ecology ◮ Density dependence ◮ Resource limitation ◮ ◮ ◮ ◮ ◮ Multi-species Competition, mutualism, Allee effects Continuous Deterministic ODEs, PDEs Population Genetics ◮ Fixed population sizes ◮ Sampling arguments ◮ Single species Discrete ◮ ◮ ◮ Stochastic Branching processes, Markov chains, diffusions Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications A N U NBRIDGEABLE C HASM ? Ecology ◮ Density dependence ◮ Resource limitation ◮ ◮ ◮ ◮ ◮ Multi-species Competition, mutualism, Allee effects Continuous Deterministic ODEs, PDEs Population Genetics ◮ Fixed population sizes ◮ Sampling arguments ◮ Single species Discrete ◮ ◮ ◮ Stochastic Branching processes, Markov chains, diffusions Outline Foundational Models F OOLS RUSH IN . . . Generalized Birth-Death-Mutation Processes Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications G ENERALIZED B IRTH -D EATH -M UTATION P ROCESSES XiN (t) = the number of individuals of type i at time t Event XN (t) → XN (t) + l Rate PK N N N X (t) i=1 βi,n ( N )Xi (t) N δi ( X N(t) )XiN (t) XiN (t) → XiN (t) − 1 ◮ Produce a clutch of n offspring, ni of type i, at per capita rate N N X (t) βi,n ( N ). N ◮ ◮ Per capita mortality rate δi ( X N(t) ), and increasing mortality due to competition. Density dependence arises via a “system size” N, which we’ll see is proportional to the equilibrium carrying capacity. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications G ENERALIZED B IRTH -D EATH -M UTATION P ROCESSES XiN (t) = the number of individuals of type i at time t Event XN (t) → XN (t) + l Rate PK N N N X (t) i=1 βi,n ( N )Xi (t) N δi ( X N(t) )XiN (t) XiN (t) → XiN (t) − 1 ◮ Produce a clutch of n offspring, ni of type i, at per capita rate N N X (t) βi,n ( N ). N ◮ ◮ Per capita mortality rate δi ( X N(t) ), and increasing mortality due to competition. Density dependence arises via a “system size” N, which we’ll see is proportional to the equilibrium carrying capacity. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications G ENERALIZED B IRTH -D EATH -M UTATION P ROCESSES XiN (t) = the number of individuals of type i at time t Event XN (t) → XN (t) + l Rate PK N N N X (t) i=1 βi,n ( N )Xi (t) N δi ( X N(t) )XiN (t) XiN (t) → XiN (t) − 1 ◮ Produce a clutch of n offspring, ni of type i, at per capita rate N N X (t) βi,n ( N ). N ◮ ◮ Per capita mortality rate δi ( X N(t) ), and increasing mortality due to competition. Density dependence arises via a “system size” N, which we’ll see is proportional to the equilibrium carrying capacity. Outline Foundational Models Generalized Birth-Death-Mutation Processes A DDITIONAL A SSUMPTIONS ◮ Weak mutation: ◮ ◮ ◮ ◮ ◮ N nj βi,n (x) < ∞. N There exists βin (x) such that limN→∞ N βi,ne (x) − βin (x) < ∞ i uniformly on compact sets. For i 6= j limN→∞ N P n∈NK 0 There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞ uniformly on compact sets. P Finite mean clutch size: β̄i (x) := ∞ n=1 nβi,n (x) converges uniformly on compact sets. P∞ FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges uniformly on compact sets. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes A DDITIONAL A SSUMPTIONS ◮ Weak mutation: ◮ ◮ ◮ ◮ ◮ N nj βi,n (x) < ∞. N There exists βin (x) such that limN→∞ N βi,ne (x) − βin (x) < ∞ i uniformly on compact sets. For i 6= j limN→∞ N P n∈NK 0 There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞ uniformly on compact sets. P Finite mean clutch size: β̄i (x) := ∞ n=1 nβi,n (x) converges uniformly on compact sets. P∞ FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges uniformly on compact sets. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes A DDITIONAL A SSUMPTIONS ◮ Weak mutation: ◮ ◮ ◮ ◮ ◮ N nj βi,n (x) < ∞. N There exists βin (x) such that limN→∞ N βi,ne (x) − βin (x) < ∞ i uniformly on compact sets. For i 6= j limN→∞ N P n∈NK 0 There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞ uniformly on compact sets. P Finite mean clutch size: β̄i (x) := ∞ n=1 nβi,n (x) converges uniformly on compact sets. P∞ FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges uniformly on compact sets. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes A DDITIONAL A SSUMPTIONS ◮ Weak mutation: ◮ ◮ ◮ ◮ ◮ N nj βi,n (x) < ∞. N There exists βin (x) such that limN→∞ N βi,ne (x) − βin (x) < ∞ i uniformly on compact sets. For i 6= j limN→∞ N P n∈NK 0 There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞ uniformly on compact sets. P Finite mean clutch size: β̄i (x) := ∞ n=1 nβi,n (x) converges uniformly on compact sets. P∞ FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges uniformly on compact sets. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes A DDITIONAL A SSUMPTIONS ◮ Weak mutation: ◮ ◮ ◮ ◮ ◮ N nj βi,n (x) < ∞. N There exists βin (x) such that limN→∞ N βi,ne (x) − βin (x) < ∞ i uniformly on compact sets. For i 6= j limN→∞ N P n∈NK 0 There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞ uniformly on compact sets. P Finite mean clutch size: β̄i (x) := ∞ n=1 nβi,n (x) converges uniformly on compact sets. P∞ FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges uniformly on compact sets. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes A DDITIONAL A SSUMPTIONS ◮ Weak mutation: ◮ ◮ ◮ ◮ ◮ N nj βi,n (x) < ∞. N There exists βin (x) such that limN→∞ N βi,ne (x) − βin (x) < ∞ i uniformly on compact sets. For i 6= j limN→∞ N P n∈NK 0 There exists δi (x) such that limN→∞ N δiN (x) − δi (x) < ∞ uniformly on compact sets. P Finite mean clutch size: β̄i (x) := ∞ n=1 nβi,n (x) converges uniformly on compact sets. P∞ FInite variance in clutch size: β̂i (x) := n=1 n2 βi,n (x) converges uniformly on compact sets. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications L AW OF L ARGE N UMBERS Let YN (t) = 1 N N X (t), and let Y(t, x) be the solution of Ẏi (t) = F(Y(t)) := β̄i (Y(t)) − δi (Y(t)) Yi (t) with Y(0) = x0 . Theorem If YN (0) → x0 as N → ∞ and r < 1, there exists a constant Cr,x0 < 1 such that ( ) N −r P sup Y (t) − Y(t, x0 ) > N < N−r t≤Cr,x0 I’ll call a generalized birth-death-mutation process density-dependent if this dynamical system is dissipative. Then, the Law of Large Numbers is of Kolmogorov type. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications L AW OF L ARGE N UMBERS Let YN (t) = 1 N N X (t), and let Y(t, x) be the solution of Ẏi (t) = F(Y(t)) := β̄i (Y(t)) − δi (Y(t)) Yi (t) with Y(0) = x0 . Theorem If YN (0) → x0 as N → ∞ and r < 1, there exists a constant Cr,x0 < 1 such that ( ) N −r P sup Y (t) − Y(t, x0 ) > N < N−r t≤Cr,x0 I’ll call a generalized birth-death-mutation process density-dependent if this dynamical system is dissipative. Then, the Law of Large Numbers is of Kolmogorov type. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRATEGY ◮ ◮ We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics are effectively deterministic, at least for t = O (ln N). In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value. Thus, we are left with two tasks: ◮ ◮ ◮ ◮ understanding the dynamics when XiN (t) ≪ N, and understanding the dynamics in the vicinity of the attractor. For the former, the picture is fairly complete: XiN (t) ≪ N is a branching process. For the latter, I’ll discuss some special cases. Remember, as Smale as shown, anything is possible! Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRATEGY ◮ ◮ We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics are effectively deterministic, at least for t = O (ln N). In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value. Thus, we are left with two tasks: ◮ ◮ ◮ ◮ understanding the dynamics when XiN (t) ≪ N, and understanding the dynamics in the vicinity of the attractor. For the former, the picture is fairly complete: XiN (t) ≪ N is a branching process. For the latter, I’ll discuss some special cases. Remember, as Smale as shown, anything is possible! Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRATEGY ◮ ◮ We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics are effectively deterministic, at least for t = O (ln N). In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value. Thus, we are left with two tasks: ◮ ◮ ◮ ◮ understanding the dynamics when XiN (t) ≪ N, and understanding the dynamics in the vicinity of the attractor. For the former, the picture is fairly complete: XiN (t) ≪ N is a branching process. For the latter, I’ll discuss some special cases. Remember, as Smale as shown, anything is possible! Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRATEGY ◮ ◮ We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics are effectively deterministic, at least for t = O (ln N). In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value. Thus, we are left with two tasks: ◮ ◮ ◮ ◮ understanding the dynamics when XiN (t) ≪ N, and understanding the dynamics in the vicinity of the attractor. For the former, the picture is fairly complete: XiN (t) ≪ N is a branching process. For the latter, I’ll discuss some special cases. Remember, as Smale as shown, anything is possible! Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRATEGY ◮ ◮ We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics are effectively deterministic, at least for t = O (ln N). In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value. Thus, we are left with two tasks: ◮ ◮ ◮ ◮ understanding the dynamics when XiN (t) ≪ N, and understanding the dynamics in the vicinity of the attractor. For the former, the picture is fairly complete: XiN (t) ≪ N is a branching process. For the latter, I’ll discuss some special cases. Remember, as Smale as shown, anything is possible! Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRATEGY ◮ ◮ We’ve seen that once XiN (t) > εN for some ε > 0, it’s dynamics are effectively deterministic, at least for t = O (ln N). In time O (ln N), Y(t, x0 ) is within O N1 of it’s equilibrium value. Thus, we are left with two tasks: ◮ ◮ ◮ ◮ understanding the dynamics when XiN (t) ≪ N, and understanding the dynamics in the vicinity of the attractor. For the former, the picture is fairly complete: XiN (t) ≪ N is a branching process. For the latter, I’ll discuss some special cases. Remember, as Smale as shown, anything is possible! Outline Foundational Models Generalized Birth-Death-Mutation Processes T HE T HREE R EGIONS OF I NTEREST Phase 2: Logistic Phase 3: Diffusion Type 1 (high birth) 200 400 600 Type 2 (low birth) 0 Number 800 Phase 1: Birth-Death 0 5 10 Time (generations) 15 Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes I NVASION D YNAMICS : Xi (0) ≪ N First assume no mutation. Fix ε > 0 and let Kx,ε (t) := {x ∈ RK+ : kx − Y(t, x0 )k < ε}, Tε,N := sup{t : XiN (t) < εN}, +ε βi,n (t) := −ε (t) := βi,n δi+ε (t) := sup βi,n (x) + ε, x∈Kx0 ,ε (t) x∈Kx0 ,ε (t) inf βi,n (x) − ε, sup δi,n (x) + ε, x∈Kx0 ,ε (t) and δi−ε (t) := inf x∈Kx0 ,ε (t) δi,n (x) − ε. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Then we have: Theorem −ε There exist branching processes Xi−ε (t), Xi (t) and Xi+ε (t) with rates βi,n (t) +ε +ε −ε and δi (t), βi,n (Y(t, x0 )) and δi (Y(t, x0 )), and βi,n (t) and δi (t) such that Xi−ε (t) ≤ Xi (t), XiN (t) ≤ Xi+ε (t) for all t < Tε,N and N suffiiciently large. Moreover, limε→0 Xi−ε (t) = limε→0 Xi−ε (t) = Xi (t). More generally, get a branching process with immigration. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Then we have: Theorem −ε There exist branching processes Xi−ε (t), Xi (t) and Xi+ε (t) with rates βi,n (t) +ε +ε −ε and δi (t), βi,n (Y(t, x0 )) and δi (Y(t, x0 )), and βi,n (t) and δi (t) such that Xi−ε (t) ≤ Xi (t), XiN (t) ≤ Xi+ε (t) for all t < Tε,N and N suffiiciently large. Moreover, limε→0 Xi−ε (t) = limε→0 Xi−ε (t) = Xi (t). More generally, get a branching process with immigration. Outline Foundational Models Generalized Birth-Death-Mutation Processes Corollary lim P XN (t) = 0 = q(x0 , t) := P{Xi (t) = 0} N→∞ Moreover, lim P {Xi (t) > εN} = 1 − q(x0 , +∞). N→∞ and Rs e− 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du δi (Y(s, x0 )) ds ≤ q(x, t) R t − R s β̄ (Y(u,x ))−δ (Y(u,x )) du 0 0 i 1+ 0 e 0 i δi (Y(s, x0 )) ds R t − R s β̄ (Y(u,x ))−δ (Y(u,x )) du β̄ (Y(s,x0 ))−β̂(Y(s,x0 )) 0 0 i δi (Y(s, x0 )) + i e 0 i ds, 2 0 , ≤ R t − R s β̄ (Y(u,x ))−δ (Y(u,x )) du β̄ (Y(s,x0 ))−β̂(Y(s,x0 )) 0 0 i 1+ 0 e 0 i δi (Y(s, x0 )) + i ds 2 Rt 0 and β̄i (x) = β̂i (x) if and only if βi,n (x) ≡ 0 for n > 1. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes I MPLICATIONS I: F ECUNDITY VARIANCE ◮ ◮ ◮ Rt The process Xi (t) has mean E[Xi (t)] = e 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du and variance Var(X(t)) = R t 2 R t β̄(Y(u,x ))−δ(Y(u,x )) du 0 0 s e β̂(Y(s, x )) + δ(Y(s, x )) ds. 0 0 0 β̂(x) ≥ β̄(x), with equality if and only if βi,n (x) ≡ 0 for n > 1. The probability of extinction for a birth and death process with birth and death rates β̄(Y(t, x0 )) and death rate δ(Y(t, x0 )) is exactly Rt ◮ Rs e− 0 β̄(Y(u,x0 ))−δ(Y(u,x0 )) du δ(Y(s, x0 )) ds R t − R s β̄(Y(u,x ))−δ(Y(u,x )) du 0 0 1+ 0e 0 δ(Y(s, x0 )) ds 0 Thus, among all processes with the same mean clutch size and the same death rate, the one with minimal variance has the highest invasion probability! Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes I MPLICATIONS I: F ECUNDITY VARIANCE ◮ ◮ ◮ Rt The process Xi (t) has mean E[Xi (t)] = e 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du and variance Var(X(t)) = R t 2 R t β̄(Y(u,x ))−δ(Y(u,x )) du 0 0 s e β̂(Y(s, x )) + δ(Y(s, x )) ds. 0 0 0 β̂(x) ≥ β̄(x), with equality if and only if βi,n (x) ≡ 0 for n > 1. The probability of extinction for a birth and death process with birth and death rates β̄(Y(t, x0 )) and death rate δ(Y(t, x0 )) is exactly Rt ◮ Rs e− 0 β̄(Y(u,x0 ))−δ(Y(u,x0 )) du δ(Y(s, x0 )) ds R t − R s β̄(Y(u,x ))−δ(Y(u,x )) du 0 0 1+ 0e 0 δ(Y(s, x0 )) ds 0 Thus, among all processes with the same mean clutch size and the same death rate, the one with minimal variance has the highest invasion probability! Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes I MPLICATIONS I: F ECUNDITY VARIANCE ◮ ◮ ◮ Rt The process Xi (t) has mean E[Xi (t)] = e 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du and variance Var(X(t)) = R t 2 R t β̄(Y(u,x ))−δ(Y(u,x )) du 0 0 s e β̂(Y(s, x )) + δ(Y(s, x )) ds. 0 0 0 β̂(x) ≥ β̄(x), with equality if and only if βi,n (x) ≡ 0 for n > 1. The probability of extinction for a birth and death process with birth and death rates β̄(Y(t, x0 )) and death rate δ(Y(t, x0 )) is exactly Rt ◮ Rs e− 0 β̄(Y(u,x0 ))−δ(Y(u,x0 )) du δ(Y(s, x0 )) ds R t − R s β̄(Y(u,x ))−δ(Y(u,x )) du 0 0 1+ 0e 0 δ(Y(s, x0 )) ds 0 Thus, among all processes with the same mean clutch size and the same death rate, the one with minimal variance has the highest invasion probability! Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes I MPLICATIONS I: F ECUNDITY VARIANCE ◮ ◮ ◮ Rt The process Xi (t) has mean E[Xi (t)] = e 0 β̄i (Y(u,x0 ))−δi (Y(u,x0 )) du and variance Var(X(t)) = R t 2 R t β̄(Y(u,x ))−δ(Y(u,x )) du 0 0 s e β̂(Y(s, x )) + δ(Y(s, x )) ds. 0 0 0 β̂(x) ≥ β̄(x), with equality if and only if βi,n (x) ≡ 0 for n > 1. The probability of extinction for a birth and death process with birth and death rates β̄(Y(t, x0 )) and death rate δ(Y(t, x0 )) is exactly Rt ◮ Rs e− 0 β̄(Y(u,x0 ))−δ(Y(u,x0 )) du δ(Y(s, x0 )) ds R t − R s β̄(Y(u,x ))−δ(Y(u,x )) du 0 0 1+ 0e 0 δ(Y(s, x0 )) ds 0 Thus, among all processes with the same mean clutch size and the same death rate, the one with minimal variance has the highest invasion probability! Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -N EUTRALITY Today, I’ll focus on two types of long-term (deterministic) dynamics: ◮ Strong selection: the dynamics have a unique globally stable axial fixed point lim Y(t, x0 ) = x⋆ t→∞ ◮ ◮ wtih x⋆ = x⋆i ei . In this case, invasion ⇒ fixation; from previous results, the probability that type i fixes is 1 − qi (x0 , +∞). Quasi-neutrality: (i) the ω-limit set for Y, Ω is an embedded submanifold of RK+ , and (ii) DF(x⋆ ) has no eigenvalue with positive real part. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -N EUTRALITY Today, I’ll focus on two types of long-term (deterministic) dynamics: ◮ Strong selection: the dynamics have a unique globally stable axial fixed point lim Y(t, x0 ) = x⋆ t→∞ ◮ ◮ wtih x⋆ = x⋆i ei . In this case, invasion ⇒ fixation; from previous results, the probability that type i fixes is 1 − qi (x0 , +∞). Quasi-neutrality: (i) the ω-limit set for Y, Ω is an embedded submanifold of RK+ , and (ii) DF(x⋆ ) has no eigenvalue with positive real part. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -N EUTRALITY Today, I’ll focus on two types of long-term (deterministic) dynamics: ◮ Strong selection: the dynamics have a unique globally stable axial fixed point lim Y(t, x0 ) = x⋆ t→∞ ◮ ◮ wtih x⋆ = x⋆i ei . In this case, invasion ⇒ fixation; from previous results, the probability that type i fixes is 1 − qi (x0 , +∞). Quasi-neutrality: (i) the ω-limit set for Y, Ω is an embedded submanifold of RK+ , and (ii) DF(x⋆ ) has no eigenvalue with positive real part. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -N EUTRALITY Today, I’ll focus on two types of long-term (deterministic) dynamics: ◮ Strong selection: the dynamics have a unique globally stable axial fixed point lim Y(t, x0 ) = x⋆ t→∞ ◮ ◮ wtih x⋆ = x⋆i ei . In this case, invasion ⇒ fixation; from previous results, the probability that type i fixes is 1 − qi (x0 , +∞). Quasi-neutrality: (i) the ω-limit set for Y, Ω is an embedded submanifold of RK+ , and (ii) DF(x⋆ ) has no eigenvalue with positive real part. Outline Foundational Models Generalized Birth-Death-Mutation Processes P ROJECTION . . . Let ZN (t) = 1 N N X (Nt). Theorem There exists a generalized Lyapunov function V(x) such that (i) V(x⋆ ) = 0 if and only if x⋆ ∈ Ω, (ii) there exists λ > 0 such that V̇(x) := ∇V(x) · F(x) < −λV(x), and lim V(ZN (t)) = 0 N→∞ in distribution for all t > 0. i.e. the rescaled process is eventually confined to Ω. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes P ROJECTION . . . Let ZN (t) = 1 N N X (Nt). Theorem There exists a generalized Lyapunov function V(x) such that (i) V(x⋆ ) = 0 if and only if x⋆ ∈ Ω, (ii) there exists λ > 0 such that V̇(x) := ∇V(x) · F(x) < −λV(x), and lim V(ZN (t)) = 0 N→∞ in distribution for all t > 0. i.e. the rescaled process is eventually confined to Ω. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications . . . AND D IFFUSION Let π(x) := lim Y(t, x). t→∞ Theorem Let π0 = π(x0 ). Then, Π(t) = weak − lim π(ZN (t)) = weak − lim ZN (t) N→∞ N→∞ is a diffusion process on Ω with Π(0+) = π 0 . The proof also shows how to construct the Kolmogorov equation for the diffusion, but it’s a mess in general. . . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications . . . AND D IFFUSION Let π(x) := lim Y(t, x). t→∞ Theorem Let π0 = π(x0 ). Then, Π(t) = weak − lim π(ZN (t)) = weak − lim ZN (t) N→∞ N→∞ is a diffusion process on Ω with Π(0+) = π 0 . The proof also shows how to construct the Kolmogorov equation for the diffusion, but it’s a mess in general. . . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE G ENERALIZED M ORAN M ODEL Joint work with Christopher Quince University of Glasgow Joshua B. Plotkin University of Pennsylvania ◮ Parsons, T. L., Quince, C., and Plotkin, J. B. (2010) Some Consequences of demographic stochasticity in population genetics. Genetics. 185 (4): 1345–1354. ◮ Parsons, T. L., Quince, C., and Plotkin, J. B. (2008) Expected times to absorption and fixation for neutral and quasi-neutral haploid populations with density dependence. Theor. Pop. Biol.: 74 (4), 302–310. ◮ Parsons, T. L. and Quince, C. (2007b) Fixation in haploid populations exhibiting density dependence II: The quasi-neutral case. Theor. Pop. Biol. 72 (4): 468–479. ◮ Parsons, T. L. and Quince, C. (2007a) Fixation in haploid populations exhibiting density dependence I: The non-neutral case. Theor. Pop. Biol. 72 (1): 121–135. Outline Foundational Models Generalized Birth-Death-Mutation Processes Consider the Generalized Birth-Death-Mutation Process with P (1 − j6=i µij )βi xi if n = ei , βi,n (x) = µji βj xj if n = ej , 0 otherwise and δi (x) = δi ◮ ◮ ◮ 1+ PK j=1 xj (t) N ! xi Reproduction by binary fission, with per-capita birth rate βi . Intrinsic death rate δi , and increasing mortality due to competition, θ Additionally, with probability µij , µij = Nij , a type i individual gives birth to an individual of type j 6= i. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Consider the Generalized Birth-Death-Mutation Process with P (1 − j6=i µij )βi xi if n = ei , βi,n (x) = µji βj xj if n = ej , 0 otherwise and δi (x) = δi ◮ ◮ ◮ 1+ PK j=1 xj (t) N ! xi Reproduction by binary fission, with per-capita birth rate βi . Intrinsic death rate δi , and increasing mortality due to competition, θ Additionally, with probability µij , µij = Nij , a type i individual gives birth to an individual of type j 6= i. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Consider the Generalized Birth-Death-Mutation Process with P (1 − j6=i µij )βi xi if n = ei , βi,n (x) = µji βj xj if n = ej , 0 otherwise and δi (x) = δi ◮ ◮ ◮ 1+ PK j=1 xj (t) N ! xi Reproduction by binary fission, with per-capita birth rate βi . Intrinsic death rate δi , and increasing mortality due to competition, θ Additionally, with probability µij , µij = Nij , a type i individual gives birth to an individual of type j 6= i. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Consider the Generalized Birth-Death-Mutation Process with P (1 − j6=i µij )βi xi if n = ei , βi,n (x) = µji βj xj if n = ej , 0 otherwise and δi (x) = δi ◮ ◮ ◮ 1+ PK j=1 xj (t) N ! xi Reproduction by binary fission, with per-capita birth rate βi . Intrinsic death rate δi , and increasing mortality due to competition, θ Additionally, with probability µij , µij = Nij , a type i individual gives birth to an individual of type j 6= i. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes The Law of Large Numbers is given by Ẏi (t) = βi Yi (t) − δi Yi (t) 1 + K X j=1 Applications Yj (t) . This dynamical system has an unstable equilibrium at 0 and stable equilibria at all points x⋆ such that β Strong Selection x⋆i = βδii − 1, x⋆j = 0 if δjj < βδii for all j 6= i (here, x⋆i N is the carrying capacity of type i PK ⋆ βi 1 1 Quasi-Neutrality i=1 xi = α − 1 if δi ≡ α for i = 1, . . . , K i.e. to order O (N), all types have the same carrying capacity. Outline Foundational Models Generalized Birth-Death-Mutation Processes The Law of Large Numbers is given by Ẏi (t) = βi Yi (t) − δi Yi (t) 1 + K X j=1 Applications Yj (t) . This dynamical system has an unstable equilibrium at 0 and stable equilibria at all points x⋆ such that β Strong Selection x⋆i = βδii − 1, x⋆j = 0 if δjj < βδii for all j 6= i (here, x⋆i N is the carrying capacity of type i PK ⋆ βi 1 1 Quasi-Neutrality i=1 xi = α − 1 if δi ≡ α for i = 1, . . . , K i.e. to order O (N), all types have the same carrying capacity. Outline Foundational Models Generalized Birth-Death-Mutation Processes The Law of Large Numbers is given by Ẏi (t) = βi Yi (t) − δi Yi (t) 1 + K X j=1 Applications Yj (t) . This dynamical system has an unstable equilibrium at 0 and stable equilibria at all points x⋆ such that β Strong Selection x⋆i = βδii − 1, x⋆j = 0 if δjj < βδii for all j 6= i (here, x⋆i N is the carrying capacity of type i PK ⋆ βi 1 1 Quasi-Neutrality i=1 xi = α − 1 if δi ≡ α for i = 1, . . . , K i.e. to order O (N), all types have the same carrying capacity. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRONG S ELECTION β1 δ1 > βj δj for all j 6= 1. ◮ Without loss of generality, ◮ Assume no mutation or migration: the Law of Large Numbers tells us that if X1N (t) = εN at some t, then if N is sufficiently large, type 1 will almost surely exclude all other types. ◮ i.e. a selective sweep occurs. To find the fixation probability, it suffices to consider the probability that, starting from X1N (0) << N, type 1 reaches εN before going extinct. From our general result, we have ◮ ◮ lim P N→∞ n N N X1 (t) = 0|Xi (0) = X0 o −X 1 = 1 + Rs PK R t − 0 β1 −δ1 1+ j=2 Yj (u) du PK Yj (s) ds e δ1 1 + 0 j=2 0 . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRONG S ELECTION β1 δ1 > βj δj for all j 6= 1. ◮ Without loss of generality, ◮ Assume no mutation or migration: the Law of Large Numbers tells us that if X1N (t) = εN at some t, then if N is sufficiently large, type 1 will almost surely exclude all other types. ◮ i.e. a selective sweep occurs. To find the fixation probability, it suffices to consider the probability that, starting from X1N (0) << N, type 1 reaches εN before going extinct. From our general result, we have ◮ ◮ lim P N→∞ n N N X1 (t) = 0|Xi (0) = X0 o −X 1 = 1 + Rs PK R t − 0 β1 −δ1 1+ j=2 Yj (u) du PK Yj (s) ds e δ1 1 + 0 j=2 0 . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRONG S ELECTION β1 δ1 > βj δj for all j 6= 1. ◮ Without loss of generality, ◮ Assume no mutation or migration: the Law of Large Numbers tells us that if X1N (t) = εN at some t, then if N is sufficiently large, type 1 will almost surely exclude all other types. ◮ i.e. a selective sweep occurs. To find the fixation probability, it suffices to consider the probability that, starting from X1N (0) << N, type 1 reaches εN before going extinct. From our general result, we have ◮ ◮ lim P N→∞ n N N X1 (t) = 0|Xi (0) = X0 o −X 1 = 1 + Rs PK R t − 0 β1 −δ1 1+ j=2 Yj (u) du PK Yj (s) ds e δ1 1 + 0 j=2 0 . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRONG S ELECTION β1 δ1 > βj δj for all j 6= 1. ◮ Without loss of generality, ◮ Assume no mutation or migration: the Law of Large Numbers tells us that if X1N (t) = εN at some t, then if N is sufficiently large, type 1 will almost surely exclude all other types. ◮ i.e. a selective sweep occurs. To find the fixation probability, it suffices to consider the probability that, starting from X1N (0) << N, type 1 reaches εN before going extinct. From our general result, we have ◮ ◮ lim P N→∞ n N N X1 (t) = 0|Xi (0) = X0 o −X 1 = 1 + Rs PK R t − 0 β1 −δ1 1+ j=2 Yj (u) du PK Yj (s) ds e δ1 1 + 0 j=2 0 . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S TRONG S ELECTION β1 δ1 > βj δj for all j 6= 1. ◮ Without loss of generality, ◮ Assume no mutation or migration: the Law of Large Numbers tells us that if X1N (t) = εN at some t, then if N is sufficiently large, type 1 will almost surely exclude all other types. ◮ i.e. a selective sweep occurs. To find the fixation probability, it suffices to consider the probability that, starting from X1N (0) << N, type 1 reaches εN before going extinct. From our general result, we have ◮ ◮ lim P N→∞ n N N X1 (t) = 0|Xi (0) = X0 o −X 1 = 1 + Rs PK R t − 0 β1 −δ1 1+ j=2 Yj (u) du PK Yj (s) ds e δ1 1 + 0 j=2 0 . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S PECIAL CASE : K = 2 ◮ Can explicitly evaluate Y2 (t): Y2 (t) = ◮ The extinction probability of type 1 satisfies lim P N→∞ ◮ β2 − δ 2 . 2 −(β2 −δ2 )t e δ2 + βY22−δ (0) n N N X1 (t) = 0|Xi (0) = X0 o −X 1 = 1 + Rs PK − β −δ 1+ Y (u) du R∞ PK 1 0 1 j=2 j Y (s) ds e δ1 1 + 0 j=2 j Alternatively, if x2 is the initial frequency of type 2, and x⋆2 = βδ22 − 1 is its (asymptotic) carrying capacity in the absence of type 1, a change of variable gives −X 0 1 . 1 + β1 −δ1 β1 δ2 −δ1 β2 R x⋆ 2 (β2 −δ2 (1+ξ)) β (β −δ2 ) ξ β2 −δ2 dξ x2 (β −δ (1+x )) 2 2 x 2 2 2 2 0 . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S PECIAL CASE : K = 2 ◮ Can explicitly evaluate Y2 (t): Y2 (t) = ◮ The extinction probability of type 1 satisfies lim P N→∞ ◮ β2 − δ 2 . 2 −(β2 −δ2 )t e δ2 + βY22−δ (0) n N N X1 (t) = 0|Xi (0) = X0 o −X 1 = 1 + Rs PK − β −δ 1+ Y (u) du R∞ PK 1 0 1 j=2 j Y (s) ds e δ1 1 + 0 j=2 j Alternatively, if x2 is the initial frequency of type 2, and x⋆2 = βδ22 − 1 is its (asymptotic) carrying capacity in the absence of type 1, a change of variable gives −X 0 1 . 1 + β1 −δ1 β1 δ2 −δ1 β2 R x⋆ 2 (β2 −δ2 (1+ξ)) β (β −δ2 ) ξ β2 −δ2 dξ x2 (β −δ (1+x )) 2 2 x 2 2 2 2 0 . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S PECIAL CASE : K = 2 ◮ Can explicitly evaluate Y2 (t): Y2 (t) = ◮ The extinction probability of type 1 satisfies lim P N→∞ ◮ β2 − δ 2 . 2 −(β2 −δ2 )t e δ2 + βY22−δ (0) n N N X1 (t) = 0|Xi (0) = X0 o −X 1 = 1 + Rs PK − β −δ 1+ Y (u) du R∞ PK 1 0 1 j=2 j Y (s) ds e δ1 1 + 0 j=2 j Alternatively, if x2 is the initial frequency of type 2, and x⋆2 = βδ22 − 1 is its (asymptotic) carrying capacity in the absence of type 1, a change of variable gives −X 0 1 . 1 + β1 −δ1 β1 δ2 −δ1 β2 R x⋆ 2 (β2 −δ2 (1+ξ)) β (β −δ2 ) ξ β2 −δ2 dξ x2 (β −δ (1+x )) 2 2 x 2 2 2 2 0 . Outline Foundational Models Generalized Birth-Death-Mutation Processes K = 2, ctd. ◮ In particular, taking x2 → x⋆2 and writing the relative fitness of type 2 as ν = βδ22 / βδ11 , so ν1 = 1 + s, we get where p = ◮ X0 N 1+2 and Ne = 1−ν ⋆ x 1+ν 2 −X0 ≈ e−2Ne sp , Nx⋆ 2 . Thus, when type 1 invades a type 2 population at equilibrium, we have approximately Kimura’s expresson. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes K = 2, ctd. ◮ In particular, taking x2 → x⋆2 and writing the relative fitness of type 2 as ν = βδ22 / βδ11 , so ν1 = 1 + s, we get where p = ◮ X0 N 1+2 and Ne = 1−ν ⋆ x 1+ν 2 −X0 ≈ e−2Ne sp , Nx⋆ 2 . Thus, when type 1 invades a type 2 population at equilibrium, we have approximately Kimura’s expresson. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications F REQUENCY D EPENDENT Ne ◮ ◮ If invasion occurs away from equilibrium, for example, after a bottleneck, there is no analogue for the selection coefficient s. In general, we have a frequency dependent effective population size: Ne (x2 ) = N β1 δ2 −δ1 β2 β1 −δ1 R x⋆ 2 (β2 −δ2 (1+ξ)) β2 (β2 −δ2 ) ξ β2 −δ2 dξ x x2 (β −δ (1+x )) 2 ◮ 2 2 2 This leads to r vs. K selection: fixation probability of beneficial alleles are increased in growing populations and decreased in shrinking populations (Fisher 1930). Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications F REQUENCY D EPENDENT Ne ◮ ◮ If invasion occurs away from equilibrium, for example, after a bottleneck, there is no analogue for the selection coefficient s. In general, we have a frequency dependent effective population size: Ne (x2 ) = N β1 δ2 −δ1 β2 β1 −δ1 R x⋆ 2 (β2 −δ2 (1+ξ)) β2 (β2 −δ2 ) ξ β2 −δ2 dξ x x2 (β −δ (1+x )) 2 ◮ 2 2 2 This leads to r vs. K selection: fixation probability of beneficial alleles are increased in growing populations and decreased in shrinking populations (Fisher 1930). Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications F REQUENCY D EPENDENT Ne ◮ ◮ If invasion occurs away from equilibrium, for example, after a bottleneck, there is no analogue for the selection coefficient s. In general, we have a frequency dependent effective population size: Ne (x2 ) = N β1 δ2 −δ1 β2 β1 −δ1 R x⋆ 2 (β2 −δ2 (1+ξ)) β2 (β2 −δ2 ) ξ β2 −δ2 dξ x x2 (β −δ (1+x )) 2 ◮ 2 2 2 This leads to r vs. K selection: fixation probability of beneficial alleles are increased in growing populations and decreased in shrinking populations (Fisher 1930). Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications B ENEFICIAL M UTATIONS 1 2 u = 0.2 u = 0.4 u = 0.6 0.8 2 ψ(z ) Fixation probability 1.5 0.6 1 β1 = 1, δ 1 = 9/10 0.4 β1 = 1, δ 1 = 7/10 β1 = 10/9, δ 1 = 1 0.2 0 β1 = 10/7, δ 1 = 1 0.5 0 0.05 Frequency of mutant allele p 0.1 0 0 0.2 0.4 z2 0.6 0.8 1 Outline Foundational Models Generalized Birth-Death-Mutation Processes D ELETERIOUS M UTATIONS 0.1 β2 varied δ 2 varied Fixed population size N = 93.14 Fixed population size N = 100 0.001 Relative deviation Mutant fixation probability 0.01 0.0001 1e-05 0.2 0 1 0.95 1e-06 0.9 1 0.95 1.05 1.05 Relative fitness ν Deleterious mutations have much higher fixation probabilities than would be predicted under the equivalent Wright-Fisher/Moran model. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -N EUTRALITY Now consider the case 1 βi ≡ δi α for i = 1, . . . , K ◮ Trade-off: an increased birth rate is compensated by an increased death rate. ◮ In deterministic limit, all types have the same carrying capacity, N⋆ = α1 − 1 N, ◮ ◮ ...and they coexist stably. If the population size were fixed, quasi-neutral types would be competitively neutral in the sense of Kimura. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -N EUTRALITY Now consider the case 1 βi ≡ δi α for i = 1, . . . , K ◮ Trade-off: an increased birth rate is compensated by an increased death rate. ◮ In deterministic limit, all types have the same carrying capacity, N⋆ = α1 − 1 N, ◮ ◮ ...and they coexist stably. If the population size were fixed, quasi-neutral types would be competitively neutral in the sense of Kimura. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -N EUTRALITY Now consider the case 1 βi ≡ δi α for i = 1, . . . , K ◮ Trade-off: an increased birth rate is compensated by an increased death rate. ◮ In deterministic limit, all types have the same carrying capacity, N⋆ = α1 − 1 N, ◮ ◮ ...and they coexist stably. If the population size were fixed, quasi-neutral types would be competitively neutral in the sense of Kimura. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -N EUTRALITY Now consider the case 1 βi ≡ δi α for i = 1, . . . , K ◮ Trade-off: an increased birth rate is compensated by an increased death rate. ◮ In deterministic limit, all types have the same carrying capacity, N⋆ = α1 − 1 N, ◮ ◮ ...and they coexist stably. If the population size were fixed, quasi-neutral types would be competitively neutral in the sense of Kimura. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -N EUTRALITY Now consider the case 1 βi ≡ δi α for i = 1, . . . , K ◮ Trade-off: an increased birth rate is compensated by an increased death rate. ◮ In deterministic limit, all types have the same carrying capacity, N⋆ = α1 − 1 N, ◮ ◮ ...and they coexist stably. If the population size were fixed, quasi-neutral types would be competitively neutral in the sense of Kimura. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications S AMPLE PATHS FOR THE Q UASI -N EUTRAL P ROCESS K=2 2 y2 1 0 0 1 2 y1 stochastic process — deterministic limit — Two sample paths, along with the deterministic trajectories. β2 /β1 = 2. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications K OLMOGOROV E QUATION FOR P(t) Let P(t) = limN→∞ PK 1 N ZN (t). i=1 Zi (t) P(t) is a diffusion process with Kolmogorov equation K−1 ∂f 1X ∂ =− ∂t 2 ∂pi i=1 " − + PK K X j=1 βi p i k=1 βk p k θij βi pi + 2 K X k=1 K X for θij = Nµij . ! # (βk − βi )βk pk f " K−1 K−1 1 X X ∂2 + βi p i 2 ∂pi ∂pj i=1 j=1 θji βj pj j=1 βj p j δij − PK k=1 βk p k ! # f , Unlike Kimura’s diffusion, we have drift terms even when all types have the same expected lifetime reproductive success! Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications K OLMOGOROV E QUATION FOR P(t) Let P(t) = limN→∞ PK 1 N ZN (t). i=1 Zi (t) P(t) is a diffusion process with Kolmogorov equation K−1 ∂f 1X ∂ =− ∂t 2 ∂pi i=1 " − + PK K X j=1 βi p i k=1 βk p k θij βi pi + 2 K X k=1 K X for θij = Nµij . ! # (βk − βi )βk pk f " K−1 K−1 1 X X ∂2 + βi p i 2 ∂pi ∂pj i=1 j=1 θji βj pj j=1 βj p j δij − PK k=1 βk p k ! # f , Unlike Kimura’s diffusion, we have drift terms even when all types have the same expected lifetime reproductive success! Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications I NTERPRETING THE D RIFT T ERMS ◮ ◮ Recall that the Law of Large Numbers was given by Ẏ = F(Y(t)), P where Fi (x) = βi xi − δi xi 1 + Kj=1 xj . The Jacobian, DF, evaluated at a point α1 − 1 p on the carrying simplex has eigenvalues 0 (multiplicity K − 1) and λ(p) = − K X βi p i i=1 with eigenvector tangent to the stable manifold. ◮ The drift terms are b(p) = grad(aij (p)) (ln |λ(p)|) Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications I NTERPRETING THE D RIFT T ERMS ◮ ◮ Recall that the Law of Large Numbers was given by Ẏ = F(Y(t)), P where Fi (x) = βi xi − δi xi 1 + Kj=1 xj . The Jacobian, DF, evaluated at a point α1 − 1 p on the carrying simplex has eigenvalues 0 (multiplicity K − 1) and λ(p) = − K X βi p i i=1 with eigenvector tangent to the stable manifold. ◮ The drift terms are b(p) = grad(aij (p)) (ln |λ(p)|) Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications I NTERPRETING THE D RIFT T ERMS ◮ ◮ Recall that the Law of Large Numbers was given by Ẏ = F(Y(t)), P where Fi (x) = βi xi − δi xi 1 + Kj=1 xj . The Jacobian, DF, evaluated at a point α1 − 1 p on the carrying simplex has eigenvalues 0 (multiplicity K − 1) and λ(p) = − K X βi p i i=1 with eigenvector tangent to the stable manifold. ◮ The drift terms are b(p) = grad(aij (p)) (ln |λ(p)|) Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications I NTERPRETING THE D RIFT T ERMS ◮ The drift is in the direction transverse the carrying simplex in which this eigenvalue increases most rapidly, with respect to the Riemannian metric induced by the diffusion terms (aij )(p). ◮ This provides a stochastic second order effect to Tilman’s R⋆ -theory of competitive exclusion in ecology: the species with greatest equilibrium density (and thus can survive at the lowest resource density) will exclude all others. Here, we see that when two species can both exist at this lowest resource density, the type that can minimize the size of stochastic fluctuations from this minimum level has a competitive advantage. ◮ Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications I NTERPRETING THE D RIFT T ERMS ◮ The drift is in the direction transverse the carrying simplex in which this eigenvalue increases most rapidly, with respect to the Riemannian metric induced by the diffusion terms (aij )(p). ◮ This provides a stochastic second order effect to Tilman’s R⋆ -theory of competitive exclusion in ecology: the species with greatest equilibrium density (and thus can survive at the lowest resource density) will exclude all others. Here, we see that when two species can both exist at this lowest resource density, the type that can minimize the size of stochastic fluctuations from this minimum level has a competitive advantage. ◮ Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications I NTERPRETING THE D RIFT T ERMS ◮ The drift is in the direction transverse the carrying simplex in which this eigenvalue increases most rapidly, with respect to the Riemannian metric induced by the diffusion terms (aij )(p). ◮ This provides a stochastic second order effect to Tilman’s R⋆ -theory of competitive exclusion in ecology: the species with greatest equilibrium density (and thus can survive at the lowest resource density) will exclude all others. Here, we see that when two species can both exist at this lowest resource density, the type that can minimize the size of stochastic fluctuations from this minimum level has a competitive advantage. ◮ Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications F IXATION P ROBABILITY, K = 2, Suppose µ12 = µ21 = 0, and let T = inf {t > 0 : P(t) ∈ {0, 1}} . P {P(T) = 1|P(0) = p} = p + β1 − β2 p(1 − p), β1 + β2 This is (formally) equivalent to selection favouring a lower birth rate: 2 identical to Kimura’s approximation for σ = ββ11 −β +β2 . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications F IXATION P ROBABILITY, K = 2, Suppose µ12 = µ21 = 0, and let T = inf {t > 0 : P(t) ∈ {0, 1}} . P {P(T) = 1|P(0) = p} = p + β1 − β2 p(1 − p), β1 + β2 This is (formally) equivalent to selection favouring a lower birth rate: 2 identical to Kimura’s approximation for σ = ββ11 −β +β2 . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -S TATIONARY D ISTRIBUTION , K = 2 Now suppose µ12 > 0, µ21 > 0. β lim P {P(t) = p} = p t→∞ θ12 β1 −1 2 β θ21 β2 −1 (1 − p) 1 e θ12 β1 + θ21 β2 (β1 −β2 )p , Again, this resembles Kimura’s expression, with β2 θ21 θ12 β1 (β1 − β2 ). θ̃21 = θ21 − 1 σ= + θ̃12 = θ12 − 1 β2 β1 β1 β2 Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -S TATIONARY D ISTRIBUTION , K = 2 Now suppose µ12 > 0, µ21 > 0. β lim P {P(t) = p} = p t→∞ θ12 β1 −1 2 β θ21 β2 −1 (1 − p) 1 e θ12 β1 + θ21 β2 (β1 −β2 )p , Again, this resembles Kimura’s expression, with β2 θ21 θ12 β1 (β1 − β2 ). θ̃21 = θ21 − 1 σ= + θ̃12 = θ12 − 1 β2 β1 β1 β2 Outline Foundational Models Generalized Birth-Death-Mutation Processes F IXATION P ROBABILITY, K = 2 1 Analytic approximation Numerical Probability of mutant fixing 0.8 0.6 0.4 0.2 0 0 200 400 600 Initial total population size Fixation favours high birth rate at low density (N = 1000, β1 = 1, β2 = 2). 800 Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Q UASI -S TATIONARY F REQUENCY S PECTRUM , K = 2 Quasi-neutral birth-death model WF model with selection (γ=-0.6) Probability density 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 Frequency of mutant (type 2) The frequency spectrum is almost identical to the Wright-Fisher frequency spectrum with selection. The type with low birth rate appears to be advantageous (β1 = 2 and β1 = 1). Foundational Models Generalized Birth-Death-Mutation Processes ◮ However, had selection been inferred from a competition assay, only the initial birth-death phase would have been seen. ◮ Yields ŝ = (β1 − β2 )(1 − α) = 0.5. i.e. would have predicted type with high birth rate favoured. ◮ Number of individuals Outline Time (1000’s of generations) Two standard methods of inference yield contradictory results. Applications Foundational Models Generalized Birth-Death-Mutation Processes ◮ However, had selection been inferred from a competition assay, only the initial birth-death phase would have been seen. ◮ Yields ŝ = (β1 − β2 )(1 − α) = 0.5. i.e. would have predicted type with high birth rate favoured. ◮ Number of individuals Outline Time (1000’s of generations) Two standard methods of inference yield contradictory results. Applications Foundational Models Generalized Birth-Death-Mutation Processes ◮ However, had selection been inferred from a competition assay, only the initial birth-death phase would have been seen. ◮ Yields ŝ = (β1 − β2 )(1 − α) = 0.5. i.e. would have predicted type with high birth rate favoured. ◮ Number of individuals Outline Time (1000’s of generations) Two standard methods of inference yield contradictory results. Applications Foundational Models Generalized Birth-Death-Mutation Processes ◮ However, had selection been inferred from a competition assay, only the initial birth-death phase would have been seen. ◮ Yields ŝ = (β1 − β2 )(1 − α) = 0.5. i.e. would have predicted type with high birth rate favoured. ◮ Number of individuals Outline Time (1000’s of generations) Two standard methods of inference yield contradictory results. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes E MPIRICAL S UPPORT ? Phase 2: Logistic Phase 3: Diffusion Type 1 (high birth) 400 600 Type 2 (low birth) 0 200 Number 800 Phase 1: Birth-Death 0 5 10 Time (generations) (Hansen & Hubbell, Science 207, 1980, 1491–1493) 15 Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications E PISTATICALLY I NTERACTING N EUTRAL M UTATIONS Joint work with: Jeremy Draghi University of Pennsylvania Günter Wagner Yale University Joshua Plotkin University of Pennsylvania ◮ Draghi J. A., Parsons, T. L. and Plotkin, J. B. Epistasis increases the rate of conditionally neutral substitution in an adapting population. Genetics 187 (4): 1139–1152. ◮ Draghi J. A., Parsons, T. L., Wagner, G. P., and Plotkin, J. B. (2010) Mutational robustness can facilitate adaptation. Nature 463 (7279): 353–355. (Just a few words to avoid any spoliers. . . ) Outline Foundational Models ◮ ◮ ◮ ◮ ◮ Generalized Birth-Death-Mutation Processes Applications Consider a population that has reached a local fitness peak, i.e. all subsequent mutations are neutral or deleterious. Some neutral mutations, however, may interact to facilitate subsequent beneficial mutations. Suppose neutral mutations happen at rate µn = θNn , and of them, a fraction pne act epistatically. Further, individuals with an epistatic mutation experience beneficial mutations at rate µb = θNb . For simplicity, assume all other mutations are deleterious. What is the waiting time until the first beneficial mutation appears? Outline Foundational Models ◮ ◮ ◮ ◮ ◮ Generalized Birth-Death-Mutation Processes Applications Consider a population that has reached a local fitness peak, i.e. all subsequent mutations are neutral or deleterious. Some neutral mutations, however, may interact to facilitate subsequent beneficial mutations. Suppose neutral mutations happen at rate µn = θNn , and of them, a fraction pne act epistatically. Further, individuals with an epistatic mutation experience beneficial mutations at rate µb = θNb . For simplicity, assume all other mutations are deleterious. What is the waiting time until the first beneficial mutation appears? Outline Foundational Models ◮ ◮ ◮ ◮ ◮ Generalized Birth-Death-Mutation Processes Applications Consider a population that has reached a local fitness peak, i.e. all subsequent mutations are neutral or deleterious. Some neutral mutations, however, may interact to facilitate subsequent beneficial mutations. Suppose neutral mutations happen at rate µn = θNn , and of them, a fraction pne act epistatically. Further, individuals with an epistatic mutation experience beneficial mutations at rate µb = θNb . For simplicity, assume all other mutations are deleterious. What is the waiting time until the first beneficial mutation appears? Outline Foundational Models ◮ ◮ ◮ ◮ ◮ Generalized Birth-Death-Mutation Processes Applications Consider a population that has reached a local fitness peak, i.e. all subsequent mutations are neutral or deleterious. Some neutral mutations, however, may interact to facilitate subsequent beneficial mutations. Suppose neutral mutations happen at rate µn = θNn , and of them, a fraction pne act epistatically. Further, individuals with an epistatic mutation experience beneficial mutations at rate µb = θNb . For simplicity, assume all other mutations are deleterious. What is the waiting time until the first beneficial mutation appears? Outline Foundational Models ◮ ◮ ◮ ◮ ◮ Generalized Birth-Death-Mutation Processes Applications Consider a population that has reached a local fitness peak, i.e. all subsequent mutations are neutral or deleterious. Some neutral mutations, however, may interact to facilitate subsequent beneficial mutations. Suppose neutral mutations happen at rate µn = θNn , and of them, a fraction pne act epistatically. Further, individuals with an epistatic mutation experience beneficial mutations at rate µb = θNb . For simplicity, assume all other mutations are deleterious. What is the waiting time until the first beneficial mutation appears? Outline Foundational Models Generalized Birth-Death-Mutation Processes M UTATIONAL C LASSES We can group the individuals into three classes, A’s, which cannot access beneficial mutations, B’s that can, and C’s, which have beneficial mutations: µn pne µn (1−pne ) 5 A o µn (1−pne ) / µb B _ _ _ _ _ _/ C U µn pne All other mutations are lethal and no replacement occurs. Applications Outline Foundational Models ◮ ◮ Generalized Birth-Death-Mutation Processes For simplicity, assume that the ecological dynamics of the A’s, have reached an equilibrium, x⋆A . Starting from a single individual, the number of B’s, XB (t) grow as a critical Markov branching process with immigration: Transition XB → XB + n XB → XB + 1 XB → XB − 1 ◮ Applications Rate βB,n (x∗ A )XB θn pne x⋆ A δB (x∗ A )XB h θ Rt i b The waiting time until the first C is E e− N 0 XB (s) ds . Outline Foundational Models ◮ ◮ Generalized Birth-Death-Mutation Processes For simplicity, assume that the ecological dynamics of the A’s, have reached an equilibrium, x⋆A . Starting from a single individual, the number of B’s, XB (t) grow as a critical Markov branching process with immigration: Transition XB → XB + n XB → XB + 1 XB → XB − 1 ◮ Applications Rate βB,n (x∗ A )XB θn pne x⋆ A δB (x∗ A )XB h θ Rt i b The waiting time until the first C is E e− N 0 XB (s) ds . Outline Foundational Models ◮ ◮ Generalized Birth-Death-Mutation Processes For simplicity, assume that the ecological dynamics of the A’s, have reached an equilibrium, x⋆A . Starting from a single individual, the number of B’s, XB (t) grow as a critical Markov branching process with immigration: Transition XB → XB + n XB → XB + 1 XB → XB − 1 ◮ Applications Rate βB,n (x∗ A )XB θn pne x⋆ A δB (x∗ A )XB h θ Rt i b The waiting time until the first C is E e− N 0 XB (s) ds . Outline Foundational Models ◮ ◮ ◮ ◮ Applications Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process with Immigration. Rt P⌊t/δ⌋ X (s) ds ≈ n=0 XB (nδ)δ. 0 B Taking δ = √1 , N √ 2 and replace t with Nt, get Ee−θb t Pn P √ Nt XN k=0 k Nt2 . X k Lastly, use known results for the GWI: k=0 where Y has n2 → Y, q θn (1−pne ) −θ Y θb Laplace Transform E e b = sech2 γ 2 , where γ= ◮ Generalized Birth-Death-Mutation Processes P∞ ⋆ n=2 n(n−1)βBl (xA ) P∞ ⋆ )+δ(x⋆ ) . β (x Bl n=2 A A For the implications, go to Jeremy Draghi’s talk in EEB on Wednesday! Outline Foundational Models ◮ ◮ ◮ ◮ Applications Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process with Immigration. Rt P⌊t/δ⌋ X (s) ds ≈ n=0 XB (nδ)δ. 0 B Taking δ = √1 , N √ 2 and replace t with Nt, get Ee−θb t Pn P √ Nt XN k=0 k Nt2 . X k Lastly, use known results for the GWI: k=0 where Y has n2 → Y, q θn (1−pne ) −θ Y θb Laplace Transform E e b = sech2 γ 2 , where γ= ◮ Generalized Birth-Death-Mutation Processes P∞ ⋆ n=2 n(n−1)βBl (xA ) P∞ ⋆ )+δ(x⋆ ) . β (x Bl n=2 A A For the implications, go to Jeremy Draghi’s talk in EEB on Wednesday! Outline Foundational Models ◮ ◮ ◮ ◮ Applications Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process with Immigration. Rt P⌊t/δ⌋ X (s) ds ≈ n=0 XB (nδ)δ. 0 B Taking δ = √1 , N √ 2 and replace t with Nt, get Ee−θb t Pn P √ Nt XN k=0 k Nt2 . X k where Y has Lastly, use known results for the GWI: k=0 n2 → Y, q θn (1−pne ) −θ Y θb Laplace Transform E e b = sech2 γ 2 , where γ= ◮ Generalized Birth-Death-Mutation Processes P∞ ⋆ n=2 n(n−1)βBl (xA ) P∞ ⋆ )+δ(x⋆ ) . β (x Bl n=2 A A For the implications, go to Jeremy Draghi’s talk in EEB on Wednesday! Outline Foundational Models ◮ ◮ ◮ ◮ Applications Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process with Immigration. Rt P⌊t/δ⌋ X (s) ds ≈ n=0 XB (nδ)δ. 0 B Taking δ = √1 , N √ 2 and replace t with Nt, get Ee−θb t Pn P √ Nt XN k=0 k Nt2 . X k Lastly, use known results for the GWI: k=0 where Y has n2 → Y, q θn (1−pne ) −θ Y θb Laplace Transform E e b = sech2 γ 2 , where γ= ◮ Generalized Birth-Death-Mutation Processes P∞ ⋆ n=2 n(n−1)βBl (xA ) P∞ ⋆ )+δ(x⋆ ) . β (x Bl n=2 A A For the implications, go to Jeremy Draghi’s talk in EEB on Wednesday! Outline Foundational Models ◮ ◮ ◮ ◮ Applications Now, for any δ > 0, XB (nδ) is a critical Galton-Watson process with Immigration. Rt P⌊t/δ⌋ X (s) ds ≈ n=0 XB (nδ)δ. 0 B Taking δ = √1 , N √ 2 and replace t with Nt, get Ee−θb t Pn P √ Nt XN k=0 k Nt2 . X k Lastly, use known results for the GWI: k=0 where Y has n2 → Y, q θn (1−pne ) −θ Y θb Laplace Transform E e b = sech2 γ 2 , where γ= ◮ Generalized Birth-Death-Mutation Processes P∞ ⋆ n=2 n(n−1)βBl (xA ) P∞ ⋆ )+δ(x⋆ ) . β (x Bl n=2 A A For the implications, go to Jeremy Draghi’s talk in EEB on Wednesday! Outline Foundational Models Generalized Birth-Death-Mutation Processes E VOLUTION IN M ULTI - STRAIN E PIDEMICS Joint work with Amaury Lambert Université Paris 06 In preparation! Sylvain Gandon CNRS Montpellier Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications A T WO S TRAIN SIR M ODEL WITH D EMOGRAPHY β1 I1 N λN / ? I1 S δ ?? ??α1 δ ?? ? ? >> β I >> N2 2 >> > α2 I2 δ Transition SN → SN + 1 SN → SN − 1 R δ (SN , IiN ) → (SN − 1, IiN + 1) IiN → IiN − 1 Rate λN δSN βi SN IN i N (δ + αi )IiN Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications L AW OF L ARGE N UMBERS Let XN (t) = 1 N S (t) N YiN (t) = 1 N I (t), N i and suppose that as N → ∞, XN (0) → x and YiN (0) → yi . Then, XN and YiN converge almost surely to deterministic limits X(t) and Yi (t) satisfying Ẋ(t) = Ẏi (t) = λ − (β1 Y1 (t) + β2 Y2 (t) + δ) X(t) (βi X(t) − (δ + αi )) Yi (t) with X(0) = x and Y(0) = yi . Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE B ASIC R EPRODUCTION N UMBER ◮ This deterministic dynamical system has equilibria at the points (x⋆ , y⋆1 , y⋆2 ), ◮ ◮ ◮ λ δ λ δ δ + α1 δ + α2 λ . , 0, 0 , , − , 0 , and , 0, − δ β1 δ + α1 β1 β2 δ + α2 β2 The first is stable if βi δ+αi < λδ , i = 1, 2. β1 β2 β1 The second (third) is stable if λδ < δ+α and δ+α (resp. > < δ+α 2 1 1 ). this leads us to define the basic reproduction numbers, βi . Ri0 = δ+α i Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE B ASIC R EPRODUCTION N UMBER ◮ This deterministic dynamical system has equilibria at the points (x⋆ , y⋆1 , y⋆2 ), ◮ ◮ ◮ λ δ λ δ δ + α1 δ + α2 λ . , 0, 0 , , − , 0 , and , 0, − δ β1 δ + α1 β1 β2 δ + α2 β2 The first is stable if βi δ+αi < λδ , i = 1, 2. β1 β2 β1 The second (third) is stable if λδ < δ+α and δ+α (resp. > < δ+α 2 1 1 ). this leads us to define the basic reproduction numbers, βi . Ri0 = δ+α i Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE B ASIC R EPRODUCTION N UMBER ◮ This deterministic dynamical system has equilibria at the points (x⋆ , y⋆1 , y⋆2 ), ◮ ◮ ◮ λ δ λ δ δ + α1 δ + α2 λ . , 0, 0 , , − , 0 , and , 0, − δ β1 δ + α1 β1 β2 δ + α2 β2 The first is stable if βi δ+αi < λδ , i = 1, 2. β1 β2 β1 The second (third) is stable if λδ < δ+α and δ+α (resp. > < δ+α 2 1 1 ). this leads us to define the basic reproduction numbers, βi Ri0 = δ+α . i Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HE B ASIC R EPRODUCTION N UMBER ◮ This deterministic dynamical system has equilibria at the points (x⋆ , y⋆1 , y⋆2 ), ◮ ◮ ◮ λ δ λ δ δ + α1 δ + α2 λ . , 0, 0 , , − , 0 , and , 0, − δ β1 δ + α1 β1 β2 δ + α2 β2 The first is stable if βi δ+αi < λδ , i = 1, 2. β1 β2 β1 The second (third) is stable if λδ < δ+α and δ+α (resp. > < δ+α 2 1 1 ). this leads us to define the basic reproduction numbers, βi . Ri0 = δ+α i Outline Foundational Models Generalized Birth-Death-Mutation Processes I NVASION OF A N EW S TRAIN Consider a single individual infected with strain 1 entering a population where strain 2 is endemic. Then λ 2 (X(t), Y1 (t), Y2 (t)) = (x⋆ , 0, y⋆2 ) = δ+α β2 , 0, δ+α2 − the invasion probability reduces to δ β2 for all t, and 1 1+ Rt 0 e− Rs β̄ (x⋆ ,0,y⋆ )−δ1 (x⋆ ,0,y⋆ ) du 2 2 0 1 δ1 (x⋆ , 0, y⋆ 2 ) ds = 1− δ1 (x⋆ ,0,y⋆ ) 2 β̄1 (x⋆ ,0,y⋆ )−δ1 (x⋆ ,0,y⋆ ) 2 2 As t → ∞, this tends to 1 − R20 R10 1 ⋆ ⋆ ⋆ ⋆ e−(β̄1 (x ,0,y2 )−δ1 (x ,0,y2 ))t − 1 if R10 > R20 , and 0 otherwise. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes I NVASION OF A N EW S TRAIN Consider a single individual infected with strain 1 entering a population where strain 2 is endemic. Then λ 2 (X(t), Y1 (t), Y2 (t)) = (x⋆ , 0, y⋆2 ) = δ+α β2 , 0, δ+α2 − the invasion probability reduces to δ β2 for all t, and 1 1+ Rt 0 e− Rs β̄ (x⋆ ,0,y⋆ )−δ1 (x⋆ ,0,y⋆ ) du 2 2 0 1 δ1 (x⋆ , 0, y⋆ 2 ) ds = 1− δ1 (x⋆ ,0,y⋆ ) 2 β̄1 (x⋆ ,0,y⋆ )−δ1 (x⋆ ,0,y⋆ ) 2 2 As t → ∞, this tends to 1 − R20 R10 1 ⋆ ⋆ ⋆ ⋆ e−(β̄1 (x ,0,y2 )−δ1 (x ,0,y2 ))t − 1 if R10 > R20 , and 0 otherwise. Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes T HE Q UASI -N EUTRAL E PIDEMIC The two strains are quasi-neutral if both have the same R0 : β1 β2 = , δ + α2 δ + α1 in which case Ω = {(x∗ , y∗1 , y∗2 ) ∈ R3+ : β1 y∗1 + β2 y∗2 = λ − δ}. x∗ Applications Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Again, we have a Kolmogorov equation for the frequency of the first strain: ∂f d 1 d2 [a(p)f (p, t)] , = − [b(p)f (p, t)] + ∂p dp 2 dp2 where ne (P) = b(p) = x∗ λ x∗ −δ β1 P + β2 (1 − P) β1 β2 β1 p + β2 (1 − p) p(1 − p)(β2 − β1 ) ne (p) β12 p + β22 (1 − p) 2 β1 (1 − p) + β2 p + β1 β2 (β1 p + β2 (1 − p)) β12 p + β22 (1 − p) and a(p) = 2x∗ β1 β2 (β1 p + β2 (1 − p))3 p(1 − p) ne (p) β12 p + β22 (1 − p) 2 There is a drift towards the type with lower contact rate. Under our assumption of equal R0 , this is equivalent to a drift favouring the type with longer infective period, which may be achieved by reducing either the excess mortality caused by the pathogen, or by a reduced recovery rate. ! , Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications Again, we have a Kolmogorov equation for the frequency of the first strain: ∂f d 1 d2 [a(p)f (p, t)] , = − [b(p)f (p, t)] + ∂p dp 2 dp2 where ne (P) = b(p) = x∗ λ x∗ −δ β1 P + β2 (1 − P) β1 β2 β1 p + β2 (1 − p) p(1 − p)(β2 − β1 ) ne (p) β12 p + β22 (1 − p) 2 β1 (1 − p) + β2 p + β1 β2 (β1 p + β2 (1 − p)) β12 p + β22 (1 − p) and a(p) = 2x∗ β1 β2 (β1 p + β2 (1 − p))3 p(1 − p) ne (p) β12 p + β22 (1 − p) 2 There is a drift towards the type with lower contact rate. Under our assumption of equal R0 , this is equivalent to a drift favouring the type with longer infective period, which may be achieved by reducing either the excess mortality caused by the pathogen, or by a reduced recovery rate. ! , Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications T HANKS . . . ◮ . . . to you all for listening! ◮ . . . to Marc Mangel and Qi Gong for being such a gracious hosts! . . . to the Burroughs Wellcome Fund for financial support. ◮ ◮ . . . to Warren Ewens, and to all the members of the Plotkin lab, for many stimulating discussions. Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE ◮ These results invite a comparison to John Gillespie’s results on fecundity variance polymorphism. ◮ The model is similar to the Wright-Fisher model, where each individual’s offspring number is a type-dependent i.i.d. random variable with mean 1 + µNi and variance σi2 . ◮ Each generation is composed of N individuals sampled uniformly from the offspring of the previous generation. ◮ When σ12 − σ22 = O (1), the proportion of type 1 admits a diffusion approximation with forward equation ∂f d p(1 − p) (µ1 − µ2 ) + (σ12 − σ22 ) f =− ∂t dp 1 d2 + p(1 − p) σ22 p + σ12 (1 − p) f . 2 2 dp Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE ◮ These results invite a comparison to John Gillespie’s results on fecundity variance polymorphism. ◮ The model is similar to the Wright-Fisher model, where each individual’s offspring number is a type-dependent i.i.d. random variable with mean 1 + µNi and variance σi2 . ◮ Each generation is composed of N individuals sampled uniformly from the offspring of the previous generation. ◮ When σ12 − σ22 = O (1), the proportion of type 1 admits a diffusion approximation with forward equation d ∂f p(1 − p) (µ1 − µ2 ) + (σ12 − σ22 ) f =− ∂t dp 1 d2 + p(1 − p) σ22 p + σ12 (1 − p) f . 2 2 dp Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE ◮ These results invite a comparison to John Gillespie’s results on fecundity variance polymorphism. ◮ The model is similar to the Wright-Fisher model, where each individual’s offspring number is a type-dependent i.i.d. random variable with mean 1 + µNi and variance σi2 . ◮ Each generation is composed of N individuals sampled uniformly from the offspring of the previous generation. ◮ When σ12 − σ22 = O (1), the proportion of type 1 admits a diffusion approximation with forward equation ∂f d p(1 − p) (µ1 − µ2 ) + (σ12 − σ22 ) f =− ∂t dp 1 d2 + p(1 − p) σ22 p + σ12 (1 − p) f . 2 2 dp Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE ◮ These results invite a comparison to John Gillespie’s results on fecundity variance polymorphism. ◮ The model is similar to the Wright-Fisher model, where each individual’s offspring number is a type-dependent i.i.d. random variable with mean 1 + µNi and variance σi2 . ◮ Each generation is composed of N individuals sampled uniformly from the offspring of the previous generation. ◮ When σ12 − σ22 = O (1), the proportion of type 1 admits a diffusion approximation with forward equation ∂f d p(1 − p) (µ1 − µ2 ) + (σ12 − σ22 ) f =− ∂t dp 1 d2 + p(1 − p) σ22 p + σ12 (1 − p) f . 2 2 dp Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE ◮ ◮ Despite superficial similarities, these results are qualitatively different. To see this, consider the lifetime reproductive output of an individual of type i born at time t = 0, which is O ∼ Poisson(βi L) ◮ ◮ where L is a non-stationary exponential process with rate ! PK N j=1 Xj (t) . δi 1 + N The Law of Large Numbers tells us that to O N1 , δi , L ∼ Exponential α We then have, to O N1 , Var(O) = E [Var(O)|L] + Var (E [O|L]) = α2 β 2 αβi + 2 i = 2. δi δi Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE ◮ ◮ Despite superficial similarities, these results are qualitatively different. To see this, consider the lifetime reproductive output of an individual of type i born at time t = 0, which is O ∼ Poisson(βi L) ◮ ◮ where L is a non-stationary exponential process with rate ! PK N j=1 Xj (t) . δi 1 + N The Law of Large Numbers tells us that to O N1 , δi , L ∼ Exponential α We then have, to O N1 , Var(O) = E [Var(O)|L] + Var (E [O|L]) = α2 β 2 αβi + 2 i = 2. δi δi Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications D ENSITY D EPENDENCE VS . F ECUNDITY VARIANCE ◮ ◮ Despite superficial similarities, these results are qualitatively different. To see this, consider the lifetime reproductive output of an individual of type i born at time t = 0, which is O ∼ Poisson(βi L) ◮ ◮ where L is a non-stationary exponential process with rate ! PK N j=1 Xj (t) . δi 1 + N The Law of Large Numbers tells us that to O N1 , δi , L ∼ Exponential α We then have, to O N1 , Var(O) = E [Var(O)|L] + Var (E [O|L]) = α2 β 2 αβi + 2 i = 2. δi δi Outline Foundational Models Generalized Birth-Death-Mutation Processes Applications F IXATION P ROBABILITY FOR THE M UTANT WITH B OTTLENECK , K = 2 0.5 0.4 0.3 0.1 0.2 Fixation probability 0.6 If we impose bottlenecks by sampling a small number of individuals at regular intervals to start a new population (β1 = 2, β2 = 0.4), 5 10 20 50 100 200 500 1000 2000 Bottleneck frequency (generations) Fixation probabilities vary substantially with the interval length. Outline Foundational Models Generalized Birth-Death-Mutation Processes C OMPETITION A SSAYS Measuring adaptation 1:1 mixture Transfer sample to solid media Evolved population Competition Ancestral population (Buckling et al., Nature 457, 2009, 824 – 829) Estimate frequency of each population Applications