Extinction of Bacterial Populations: A Change of Paradigm? Ingo - PowerPoint PPT Presentation

Extinction of Bacterial Populations: A Change of Paradigm? Ingo Lohmar (w/ Baruch Meerson) Racah Institute of Physics, The Hebrew University, Jerusalem MPIPKS Dresden — LAFNES’11 — 2011-07-12

Introduction Experiments Bacterial Persistence antibiotic Population size Bigger 1944(!) so not a resistant genotype generic effect various hypotheses around Time single-cell experiments Balaban et al. ’04 isogenetic population, same environment individual bacteria switch stochastically between two phenotypes: “normals” grow fast & susceptible to antibiotics “persisters” grow much slower & hardly susceptible What’s the Use?

Introduction Theory Exponential Growth Stage — Fitness lab conditions (w/o antibiotics): exponential growth focus: outgrow other species — fitness = asymptotic net growth rate good environmental conditions: intermittent adverse conditions: switch to persisters a burden switch can be advantageous rarely persisters rarely persisters frequently persisters frequently persisters Population size Population size Time Time deterministic rate equation model optimal fitness: Kussell & Leibler ’05 time spent as one phenotype ≃ duration of its beneficial environment

Introduction Theory Our Take: Extinction of Established Populations in vivo space / resources limited � bounded growth then natural to consider established populations births / deaths stochastic, ultimately: rare fluctuations � extinction role of persisters? original observation: life insurance against extinction (not for growth race) fitness meaningless! instead: mean time to extinction (MTE) Aim general method to treat extinction: rare, but important large fluctuation 1 use and quantitative effect of persisters? (also: adverse conditions) 2

Extinction of Population with Persisters Model & Method Deterministic Rate Equations (RE) well-mixed two-species system normals n : unit death rate, birth rate B ( 1 − n / N ) ( B > 1 viable) add persisters m : no birth, no death, just switching at rates α , β n = Bn ( 1 − n / N ) − n − α n + β m , ˙ m = α n − β m ˙ m fixed points (FP) F M saddle F 0 at n = 0 = m (extinction) stable node F M at n M = N ( 1 − 1 / B ) , m M = ( α / β ) n M population relaxes → F M , established n F 0 N does not tell us anything about extinction

Extinction of Population with Persisters Model & Method Stochastic System: Quasi-Stable Distribution (QSD) m master equation for prob. distrib. d P n , m π n , m = ˆ HP n , m d t F M single stationary eigenstate δ n , 0; m , 0 others decay � F M no longer stable n F 0 P n , m ≡ 0 N quasi-stable distribution (QSD) π n , m , decay time τ ∼ exp ( N ) ( ≫ others) slowly leaks to extinction probability P 0 , 0 ( t ) τ = 1 / π 1 , 0 = mean time to extinction (MTE)

Extinction of Population with Persisters Model & Method WKB Approximation and H AMILTON ian System ansatz for N ≫ 1 Kubo ’73, Dykman et al. ’94, Elgart & Kamenev ’04 x = n N , y = m π n , m = exp [ − NS ( x , y )] with N continuous leading order in 1 / N p x = ∂ S ∂ x , H ( x , y , p x , p y ) : = ( e p x − 1 ) Bx ( 1 − x )+ � e − p x − 1 � x e − p x + p y − 1 e p x − p y − 1 p y = ∂ S � � � � + α x + β y = 0 , ∂ y identify: ∂ H / ∂ t ≡ 0 � H AMILTON -J ACOBI for conserved “energy” E ≡ 0 coordinates x , y , action S , conjugate momenta p x , p y , H AMILTON ian H H AMILTON ’s eqs. � particular system path through state space

Extinction of Population with Persisters Model & Method Instanton and Action wanted: “instanton”, a heteroclinic E = 0 -orbit quasi-stable RE-FP fluctuational FP � x = 0 = y , F M ( p x = 0 = p y ) F ∅ = � p x = − ln B = p y meaning? cf. talks: Gabrielli, Meerson y action along instanton � F ∅ S = ( p x d x + p y d y − H d t ) t → − ∞ F M F M = entropic barrier against extinction ˆ E = 0 MTE τ ≃ exp ( NS ) x F ∅ t → + ∞ 1 H AMILTON � path of minimal action � most likely path to extinction

Extinction of Population with Persisters Constant Environment Regime and Assumptions assumptions close to bifurcation δ : = B − 1 ≪ 1 B = 1 + δ 1 slow switching α , β ≪ δ ≪ 1 normal ε : = β slowness δ ≪ 1 , α β Γ : = α persister / normals persister β

Extinction of Population with Persisters Constant Environment Multi-Scale Ansatz rescale: x = δ · X , p x = δ · P X ... and t = T / δ d X d Y d T = X ( 2 P X − X + 1 ) − ε ( Γ X − Y ) , d T = ε ( Γ X − Y ) , d P X d P Y d T = − P X ( P X − 2 X + 1 )+ ε Γ ( P X − P Y ) , d T = − ε ( P X − P Y ) , Y , P Y — slow T ′ : = ε T (formally separate) X , P X — fast T and ε ≪ 1 as perturbation: X = X 0 ( T )+ ε X 1 ( T , T ′ )+ ..., Y = Y 0 ( T ′ )+ ε Y 1 ( T ′ )+ ... � system of PDEs for ε -orders normals ∼ ε 0 : 1d-system, solved persisters ∼ ε 1 resolve only slow time scale: driving by normals ≃ step functions

Extinction of Population with Persisters Constant Environment Theory and Numerical Solution Theory — Numerical Γ = 1 , ε = 0 . 1 1 0 F M F M 3 Persisters Y s 0 = 1 2 + Γ P Y Action s 2 F ∅ F ∅ 0 − 1 0 Normals X 1 − 1 P X 0 1 ε = 0 . 1 F M 0 0 F M ε = 0 . 2 0 P X P Y 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 Switching rate ratio Γ F ∅ − 1 F ∅ − 1 0 Normals X 1 0 Persisters Y 1 numerically: iterative algorithm Chernykh & Stepanov ’01, Elgart & Kamenev ’04

Extinction of Population with Persisters Constant Environment Mean Time to Extinction � � 1 �� N δ 2 τ ≃ exp 2 + Γ exponential increase, but also larger population size. . . compare with normals-only MTE compensate by same carrying capacity K = N δ ( 1 + Γ ) : τ � K δ Γ � = exp τ 1d 2 ( 1 + Γ ) time-scale separation � delayed extinction � “maximal” action (rectangle)

Extinction of Population with Persisters Catastrophe Catastrophe Model model for time t c = T c / δ = T ′ c / ( εδ ) : persisters unaffected, normals’ B → 0 1d-system (normals only) solved Assaf et al. ’09 1 . 0 Extinction Probability P 0 , 0 MTE too crude: dominated by t c 0 . 8 systems surviving catastrophe 0 . 6 instead: extinction probability 0 . 4 ∆ P 0 , 0 increase (EPIC) ∆ P 0 , 0 0 . 2 maths the same: 0 . 0 ∆ P 0 , 0 ≃ exp ( − NS ) Time ≪ τ

Extinction of Population with Persisters Catastrophe WKB Theory: Re-Trace Steps. . . H AMILTON ian H c instanton: before catastrophe after (matched segments) ( T c fixes start / end points) [before/after also changed by cat. if # species > 1 ] regime and rescaling as before: d X d T = − X d Y δ + XP X − ε ( Γ X − Y ) , d T = ε ( Γ X − Y ) , δ − P 2 d P X d T = P X d P Y X 2 + ε Γ ( P X − P Y ) , d T = − ε ( P X − P Y ) normals exp. decay (rate 1 / δ ≫ 1 ) — always strong catastrophe normals ∼ ε 0 : effective 1d-system again Assaf et al. ’09 persisters ∼ ε 1 : driven by X 0 , P X 0 again only difference: driving has “step” at start / end of catastrophe

Extinction of Population with Persisters Catastrophe Instantons: Theory and Numerical Solution short cat. T c = 0 . 2 long cat. T c = 10 Theory — Numerical Γ = 1 , δ = 0 . 1 , ε = 0 . 1 Γ = 1 , δ = 0 . 1 , ε = 0 . 1 F M F M 1 0 1 0 F M F M Persisters Y Persisters Y P Y P Y F ∅ F ∅ 0 F ∅ − 1 0 F ∅ − 1 0 Normals X 1 − 1 P X 0 0 Normals X 1 − 1 P X 0 0 0 F M 0 0 F M F M F M P X P Y P X P Y F ∅ F ∅ F ∅ F ∅ − 1 − 1 − 1 − 1 0 Normals X 1 0 Persisters Y 1 0 Normals X 1 0 Persisters Y 1 normals nearly extinct after catastrophe, persisters survive much longer theory improves for longer T c

Extinction of Population with Persisters Catastrophe Action: Theory and Numerical Solution Γ = 1 , δ = 0 . 1 , ε = 0 . 1 1 . 5 10 0 10 − 1 10 − 2 1 . 0 Action s 10 − 1 10 0 10 1 10 2 0 . 5 � � 1 1 + e T c / δ + Γ e − T ′ s 0 , c = c 0 . 0 0 1 2 3 4 5 Catastrophe length T c short T ′ c ≪ 1 normals die on very fast scale t ∼ 1 — persisters cannot resolve any change long T ′ c � 1 persisters mimic X , P X time shifts, measured on slow scale T ′ ∼ 1 of switching back

Extinction of Population with Persisters Catastrophe Extinction Probability Increase (EPIC) � � �� 1 − N δ 2 1 + e t c + Γ e − β t c ∆ P 0 , 0 ≃ exp compare with normals-only EPIC same carrying capacity K = N δ ( 1 + Γ ) , still exponentially reduced: ∆ P 0 , 0 � − K δ Γ � 1 �� e − β t c − = exp ∆ P 1d 1 + e t c 1 + Γ 0 , 0 optimal benefit ∆ P 0 , 0 � − K δ Γ � for t c ≫ 1 ≫ β t c , ≃ exp ∆ P 1d 1 + Γ 0 , 0 compare to constant environment MTE ratio: catastrophe squares benefit

Remarks and Summary Remarks compatible points of view focus on fitness � optimal switching rates survival very different � relative slowness matters neglected effects slow death and growth of persisters — expect: qualitatively similar “internal” competition for resources? � cost of persisters