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 Time Population size Bigger 1944(!)

so not a resistant genotype generic effect various hypotheses around 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: switch to persisters a burden Time Population size rarely persisters frequently persisters intermittent adverse conditions: switch can be advantageous Time Population size rarely persisters frequently persisters

deterministic rate equation model

• ptimal 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?

• riginal observation: life insurance against extinction (not for growth race)

fitness meaningless! instead: mean time to extinction (MTE) Aim

1

general method to treat extinction: rare, but important large fluctuation

2

use and quantitative effect of persisters? (also: adverse conditions)

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−αn+βm, ˙ m = αn−βm

m n N FM F0

fixed points (FP) saddle F0 at n = 0 = m (extinction) stable node FM at

nM = N(1−1/B), mM = (α/β)nM

population relaxes → FM, established does not tell us anything about extinction

Extinction of Population with Persisters Model & Method

Stochastic System: Quasi-Stable Distribution (QSD)

master equation for prob. distrib.

dPn,m dt = ˆ HPn,m

single stationary eigenstate δn,0;m,0

• thers decay FM no longer stable

Pn,m ≡ 0 πn,m m n N F0 FM

quasi-stable distribution (QSD) πn,m, decay time τ ∼ exp(N) (≫ others) slowly leaks to extinction probability P0,0(t)

τ = 1/π1,0 = mean time to extinction (MTE)

Extinction of Population with Persisters Model & Method

WKB Approximation and HAMILTONian System

ansatz for N ≫ 1

Kubo ’73, Dykman et al. ’94, Elgart & Kamenev ’04

πn,m = exp[−NS(x,y)]

with

x = n N , y = m N continuous

leading order in 1/N

H(x,y,px,py) := (epx −1)Bx(1−x)+

• e−px −1
• x

+

• e−px+py −1
• αx+
• epx−py −1
• βy = 0,

px = ∂S ∂x, py = ∂S ∂y

identify:

∂H/∂t ≡ 0 HAMILTON-JACOBI for conserved “energy” E ≡ 0

coordinates x,y, action S, conjugate momenta px,py, HAMILTONian H HAMILTON’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

FM (px = 0 = py)

• F∅ =

x = 0 = y, px = −lnB = py

y x 1 F∅ t → +∞ FM t → −∞ E = 0

meaning?

• cf. talks: Gabrielli, Meerson

action along instanton

S =

F∅

FM

(px dx+py dy−H dt) ˆ = entropic barrier against extinction

MTE τ ≃ exp(NS) HAMILTON path of minimal action

most likely path to extinction

Extinction of Population with Persisters Constant Environment

Regime and Assumptions

normal persister

α β B = 1+δ 1

assumptions close to bifurcation

δ := B−1 ≪ 1

slow switching α, β ≪ δ ≪ 1 slowness

ε := β δ ≪ 1,

persister/normals

Γ := α β

Extinction of Population with Persisters Constant Environment

Multi-Scale Ansatz

rescale: x = δ ·X, px = δ ·PX ... and t = T/δ

dX dT = X(2PX −X +1)−ε(ΓX −Y), dY dT = ε(ΓX −Y), dPX dT = −PX(PX −2X +1)+εΓ(PX −PY), dPY dT = −ε(PX −PY), X, PX — fast T

and

Y, PY — slow T′ := εT (formally separate) ε ≪ 1 as perturbation: X = X0(T)+εX1(T,T′)+..., Y = Y0(T′)+εY1(T′)+... system of PDEs for ε-orders

normals ∼ ε0: 1d-system, solved persisters ∼ ε1 resolve only slow time scale: driving by normals ≃ step functions

Extinction of Population with Persisters Constant Environment

Theory and Numerical Solution

Theory — Numerical

1 1 FM F∅

Normals X Persisters Y

−1 −1 FM F∅ PX PY 1 −1 FM F∅

Normals X

PX 1 −1 FM F∅

Persisters Y

PY Γ = 1, ε = 0.1 0.0 0.5 1.0 1.5 2.0 2.5 1 2 3

s0 = 1 2 +Γ

Switching rate ratio Γ Action s

ε = 0.1 ε = 0.2

numerically: iterative algorithm

Chernykh & Stepanov ’01, Elgart & Kamenev ’04

Extinction of Population with Persisters Constant Environment

Mean Time to Extinction

τ ≃ exp

• Nδ 2

1 2 +Γ

• exponential increase, but also larger population size. . .

compare with normals-only MTE compensate by same carrying capacity K = Nδ(1+Γ):

τ τ1d = exp

• KδΓ

2(1+Γ)

• time-scale separation delayed extinction “maximal” action (rectangle)

Extinction of Population with Persisters Catastrophe

Catastrophe Model

model for time tc = Tc/δ = T′

c/(εδ):

persisters unaffected, normals’ B → 0 1d-system (normals only) solved

Assaf et al. ’09

0.0 0.2 0.4 0.6 0.8 1.0 ∆P0,0 tc

Time ≪ τ Extinction Probability P0,0

MTE too crude: dominated by systems surviving catastrophe instead: extinction probability increase (EPIC) ∆P0,0 maths the same:

∆P0,0 ≃ exp(−NS)

Extinction of Population with Persisters Catastrophe

WKB Theory: Re-Trace Steps. . .

instanton: before catastrophe HAMILTONian Hc (Tc fixes start / end points) after (matched segments)

[before/after also changed by cat. if #species > 1]

regime and rescaling as before:

dX dT = −X δ +XPX −ε(ΓX −Y), dY dT = ε(ΓX −Y), dPX dT = PX δ − P2

X

2 +εΓ(PX −PY), dPY dT = −ε(PX −PY)

normals exp. decay (rate 1/δ ≫ 1) — always strong catastrophe normals ∼ ε0: effective 1d-system again

Assaf et al. ’09

persisters ∼ ε1: driven by X0, PX0 again

• nly difference: driving has “step” at start / end of catastrophe

Extinction of Population with Persisters Catastrophe

Instantons: Theory and Numerical Solution

short cat. Tc = 0.2 Theory — Numerical long cat. Tc = 10

1 1 FM F∅

Normals X Persisters Y

−1 −1 FM F∅ PX PY 1 −1 FM F∅

Normals X

PX 1 −1 FM F∅

Persisters Y

PY Γ = 1, δ = 0.1, ε = 0.1 1 1 FM F∅

Normals X Persisters Y

−1 −1 FM F∅ PX PY 1 −1 FM F∅

Normals X

PX 1 −1 FM F∅

Persisters Y

PY Γ = 1, δ = 0.1, ε = 0.1

normals nearly extinct after catastrophe, persisters survive much longer theory improves for longer Tc

Extinction of Population with Persisters Catastrophe

Action: Theory and Numerical Solution

1 2 3 4 5 0.0 0.5 1.0 1.5

s0,c =

• 1

1+eTc/δ +Γe−T′

c

• Catastrophe length Tc

Action s

Γ = 1, δ = 0.1, ε = 0.1

10−1 100 101 102 10−2 10−1 100

short T′

c ≪ 1 normals die on very fast scale t ∼ 1

— persisters cannot resolve any change long T′

c 1 persisters mimic X, PX time shifts,

measured on slow scale T′ ∼ 1 of switching back

Extinction of Population with Persisters Catastrophe

Extinction Probability Increase (EPIC)

∆P0,0 ≃ exp

• −Nδ 2
• 1

1+etc +Γe−βtc

• compare with normals-only EPIC

same carrying capacity K = Nδ(1+Γ), still exponentially reduced:

∆P0,0 ∆P1d

0,0

= exp −KδΓ 1+Γ

• e−βtc −

1 1+etc

• ptimal benefit

∆P0,0 ∆P1d

0,0

≃ exp

• − KδΓ

1+Γ

• for tc ≫ 1 ≫ βtc,

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

Remarks and Summary

Summary

Methodology preaching to the choir: rare fluctuations very important WKB method path to extinction and MTE / EPIC works w/ “arbitrarily” complicated models (num. ODEs) Bacterial Populations persisters exponentially increase MTE / reduce EPIC fundamental mechanism: time-scale separation due to slow switching explains existence of persisters in a natural and robust way

Thank You!

Partially funded by the Minerva foundation.