Models of Switching in Biophysical Contexts

Transcription

1 Models of Switching in Biophysical Contexts Martin R. Evans SUPA, School of Physics and Astronomy, University of Edinburgh, U.K. March 7, 2011 Collaborators: Paolo Visco (MSC, Paris) Rosalind J. Allen (Edinburgh) Satya N. Majumdar (LPTMS, Paris)

2 Plan Plan I Thoughts on biophysical modelling II Switching between states III Microscopic dynamics: model linear feedback switch IV Macroscopic dynamics: population growth in a catastrophic environment References: P. Visco, R. J. Allen, M. R. Evans, Phys. Rev. Lett. 2008, Phys. Rev. E 2009 P. Visco, R. J. Allen, S. N. Majumdar, M. R. Evans, Biophysical Journal 2010

3 Physics vs Biology Physics vs Biology Physics Unifying Principles Effective Theories; minimal models Mathematical proof e.g. Free energy minimisation Biology System details Models with many parameters to fit data Argumentation e.g. Evolutionary pressures

4 What can physicists bring to biophysical modelling? Ideas from (statistical) physics many particle behaviour non equilibrium phenomena fluctuations and stochastic effects idea of scales Model building savoir faire minimal models exact solutions; good approximations

5 What can physicists bring to biophysical modelling? Ideas from (statistical) physics many particle behaviour non equilibrium phenomena fluctuations and stochastic effects idea of scales Model building savoir faire minimal models exact solutions; good approximations What physicists shouldn t bring Arrogance and ignorance

6 Plan Plan I Thoughts on biophysical modelling II Switching between states III Microscopic dynamics: model linear feedback switch IV Macroscopic dynamics: population growth in a catastrophic environment References: P. Visco, R. J. Allen, M. R. Evans, Phys. Rev. Lett. 2008, Phys. Rev. E 2009 P. Visco, R. J. Allen, S. N. Majumdar, M. R. Evans, Biophysical Journal 2010

7 II Switching: Basic Biology for Physicists Gene Stretch of DNA 1000 base pairs long Transcription by RNAp mrna Translation amino acids production of proteins... Phenotype Regulation regulatory sites at ends of gene known as operators gene switched off/on by binding of repressors/enhancers known as Transcription factors genes can produce transcription factors for themselves or other genes genetic network

8 Heterogeneity Populations of bacteria are often heterogeneous even if environmentally and genetically identical Happens when bacteria frequently switches between different states: Multistable genetic switches Stochastic oscillation between different states It represents a strategy against environmental changes and stresses Examples: Bacterial persistence Phase variation (e.g.: fimbriae)

9 Example: Bacterial persistence Non persistent state Persistent state Vulnerable to antibiotics Resists against antibiotics Fast growth Very slow growth Balaban, Merrin, Chait, Kowalik, Leibler, Science, 2004

10 Examples of population strategies Bet Hedging small fraction of population in unfit persistor state which can survive catastrophes e.g. antibiotics Once and for all Population splits into groups with long lived phenotypes i.e. bistability Defence against immune response small fraction of population in fit state since too successful a population would evoke an immune response

11 Models of genetic switches Existing models for bistable gene regulatory networks Mutually repressing genes Positive feedback loop Typically: bistable systems effective potential But this is not how the most common bacterial phase variation systems work! state state 1 state 2

12 Example: Uropathogenic E.Coli Urinary tract Attached, fimbriated Detached, fimbriated Detached, non fimbriated Attached vulnerable to immune system Detached vulnerable to flushing

13 Example: Uropathogenic E. Coli Non fimbriated state Fimbriated state M. R. Evans Models of Switching in Biophysical Contexts

14 The fim switch DNA inversion switch fim system controls production of fimbriae short piece of DNA can be inserted in two orientations in one orientation fimbrial genes transcribed and fimbriae produced ( on state ) inversion of DNA element mediated by recombinase enzymes FimE recombinase which flips the switch on to off is produced more strongly in the on state orientational control

15 III Model of stochastic linear feedback switch Reaction network k R 1 1 k S 2 on Son + R 1 k S on + R on 3 1 S off + R 1 k 4 S on k4 S off R 1 variations switching reactions Reaction rates k 1 k 2 k on R 1 decay R 1 production 3 R 1 mediated switching (only on to off) k 4 spontaneous switching (both directions)

16 III Model of stochastic linear feedback switch Reaction network k R 1 1 k S 2 on Son + R 1 k S on + R on 3 1 S off + R 1 k 4 S on k4 S off R 1 variations switching reactions In the on state: dn dt = k 2 k 1 n

17 III Model of stochastic linear feedback switch Reaction network k R 1 1 k S 2 on Son + R 1 k S on + R on 3 1 S off + R 1 k 4 S on k4 S off R 1 variations switching reactions In the on state: n(t) = k 2 k 1 [1 exp( k 1 t)] n k 2 k 1 t

18 III Model of stochastic linear feedback switch Reaction network k R 1 1 k S 2 on Son + R 1 k S on + R on 3 1 S off + R 1 k 4 S on k4 S off R 1 variations switching reactions In the off state: dn dt = k 1 n n k 2 k 1 t

19 III Model of stochastic linear feedback switch Reaction network k R 1 1 k S 2 on Son + R 1 k S on + R on 3 1 S off + R 1 k 4 S on k4 S off R 1 variations switching reactions In the off state: n(t) = n 0 exp( k 1 t) n k 2 k 1 t

20 Scales k 2 k 1 0 n τ R τ on Three timescales τ off t τ R relaxation time of n: 1/k 1 τ on on to off switching time: 1/( n on k3 on + k on τ off off to on switching time: 1/k4 off Two n scales n on asymptotic value of n in the on state: k 2 /k 1 n off asymptotic value of n in the off state: 0 4 )

21 Some examples τ R τ S τ R τ S τ R τ S

22 Statistical description Define p s (n, t) probability that there are n enzymes and the switch is in position s {on, off} at time t. Master equation dp s (n) dt = k 1 [(n + 1)p s (n + 1) np s (n)] + k s 2 [p s(n 1) p s (n)] + n[k 1 s 3 p 1 s (n) k s 3 p s(n)] + k 4 [p 1 s (n) p s (n)] Removal of R 1 Production of R 1 (if the switch is on) R 1 mediated switching spontaneous switching

23 Steady state: two coupled equations (n + 1)k 1 p on (n + 1) + k 2 p on (n 1) + k 4 p off (n) (n + 1)k 1 p off (n + 1) + nk on 3 p on(n) + k 4 p on (n) = (nk 1 + k 2 + nk on 3 + k 4)p on (n) = (nk 1 + k 4 )p off (n) Exact solution p on (n) = a 0 (u 1 u 0 ) n n! (η) n (ζ) n 1F 1 (η + n, ζ + n, u 0 ) p off (n) = κδ n,0 + k 2 p on (n 1) p on (n) k 1 n where u 1, u 0, η, ζ are combinations of the reaction rates and κ, a 0 are normalising constants

24 Test against simulations Plot of p(n) = p on (n) + p off (n) Perfect agreement

25 Flipping time distribution Flipping time distribution F on (t)dt probability that the switch flips at time t t + dt Compare with mean first passage times and persistence distributions of stochastic processes We would like to see peak around typical time to be in on-state F(t) m t Can the model achieve this? require df(t) dt > 0 t=0

26 Measurement ensemble F (T ) depends on the initial condition of n i Two choices: Switch change ensemble SCE Steady state ensemble SSE Initial distribution W (n i ) defines the ensemble

27 Relation between ensembles on switch position T T 1 T 2 off Probability that t is an interval T : Prob(T )dt = t time T F SCE (T )dt 0 T F SCE (T )dt Prob(T 2 T )dt = θ(t T 2)dT T (uniform) F SSE (T 2 ) = T 2 Prob(T 2 T )Prob(T )dt = T 2 0 F SCE (T )dt TF SCE (T )dt General relation from renewal theory: df SSE dt = FSCE (T ) T SCE

28 Peak in the distribution never a peak in SSE in SCE require k 2 k on 3 (k 4) 2 k on 3 (k 1 + 2k 4 ) n W (k on 3 )2 n 2 W > 0 k k 2 = k 2 = k3 on Visco, Allen, Evans, PRL (2008);

29 IV Population dynamics in changing environments Consider whether switching rate to a less fit state is advantageous for the population Previous studies Thattai and Van Oudenaarden, Genetics environments, 2 phenotypes, Poissonian environmental changes Kussell and Leibler, Science 2005 many environments and phenotypes, different phentotypes have preferred environment Random switching between phenotypes good strategy when environmental changes unpredictable

30 General scenario Single environment Population of bacteria, say, with two possibles states for individuals: Fit state has fast growth Unfit (persistor) state has slow growth but withstands catastrophes Catastrophes occur stochastically, coupled to growth of population Question: what is best strategy of population to maximise growth?

31 Model Deterministic growth Two subpopulations n A and n B. Exponential growth rates γ A > γ B Individuals switch states with rates k A, k B dn A dt dn B dt = γ A n A + k B n B k A n A, = γ B n B + k A n A k B n B. Stochastic catastrophes Catastrophe rate β(n A, n B ) β is the environmental response function When a catastrophe occurs n A n A < n A, with probability density ν(n A n A ). ν is the catastrophe strength distribution

32 Fitness Biological definition: instantaneous growth rate of population Here f is fraction of population in fit state dn dt Deterministic growth: n A f = n A + n B = γ A n A + γ B n B = (γ B + γf )n df dt = v(f ) = γ(f + f )(f f ), where γ = γ A γ B and f ± are the roots of ( f 2 1 k ) A + k B f k B γ γ = 0.

33 Typical trajectory log(n A ) log(n B ) f + f(t) t Piecewise Deterministic Markov Processes - used extensively in context of queueing theory

34 Catastrophe rate Threshold sigmoid function Catastrophes triggered when a threshold is reached Our choice: β λ (f ) = ξ 2 ( 1 + ) f f λ2 + (f f ) 2 parameters ξ plateau value f threshold value βλ(f) 1 λ = 0.01 λ = 0.1 λ = 1 λ = 10 λ sharpness of the transition f

35 Catastrophe strength n A n A = u n A, where 0 < u < 1 is a random number sampled from: P(u) = (α + 1)u α α > 1 n ν(n n) α = 0.5 α = 0 α = 1 α = 10 1 < α < 0 strong catastrophes α > 0 weak catastrophes n /n To each jump n A n A corresponds a jump f f Catastrophe strengh distribution µ(f f ), where µ(f f ) = Θ(f f ) d m(f ) df m(f ) with m(f ) = ( f ) 1+α 1 f

36 Exact Solution for Stationary State Constant flux condition v(f ) 0 f f f f + β(f )µ(f f ) Recall df dt = v(f ) µ(f f ) = d m(f ) df m(f ) ( ) 1+α f m(f ) = 1 f p(f )v(f ) = f+ f df df p(f )β(f )µ(f f ) f 0

37 Exact Solution for Stationary State Constant flux condition v(f ) 0 f f f f + β(f )µ(f f ) Recall df dt = v(f ) µ(f f ) = d m(f ) df m(f ) ( ) 1+α f m(f ) = 1 f p(f )v(f ) = f+ f df p(f )β(f ) m(f ) f df 0 d df m(f )

38 Exact Solution for Stationary State Constant flux condition v(f ) 0 f f f f + β(f )µ(f f ) Recall df dt = v(f ) µ(f f ) = d m(f ) df m(f ) ( ) 1+α f m(f ) = 1 f p(f )v(f ) m(f ) = f+ f df p(f )β(f ) m(f )

39 Exact Solution for Stationary State Constant flux condition v(f ) 0 f f f f + β(f )µ(f f ) Recall df dt = v(f ) µ(f f ) = d m(f ) df m(f ) ( ) 1+α f m(f ) = 1 f d p(f )v(f ) p(f )β(f ) = df m(f ) m(f )

40 Exact Solution for Stationary State Constant flux condition v(f ) 0 f f f f + β(f )µ(f f ) Recall df dt = v(f ) µ(f f ) = d m(f ) df m(f ) ( ) 1+α f m(f ) = 1 f d p(f )v(f ) p(f )β(f ) = df m(f ) m(f ) p(f ) = C m(f ) ( ) β(f ) v(f ) exp v(f )

41 Examples 1.5 (a) (b) p(f/f+) (c) (d) p(f/f+) f /f f /f +

42 Optimal strategies We characterise the population strategy by the value of k A, which is the control parameter for the population balance. We define Optimal Strategies as the values of k A which maximise the average fitness f in the stationary state. Two optimal strategies emerge: f λ = 10 λ = 1 λ = 0.1 λ = k A

43 Optimal strategies cont. Two possible optimal strategies 1 k A = 0 (no switching to unfit state) 2 k A ka where ka yields f + = f (saturation fitness = response threshold) f(t) 1 no switching switching t t

44 Conclusions for population dynamics in changing environments New kind of environments Catastrophic Responsive Switching can be a good strategy Threshold mechanism (different from bet hedging) Two main strategies: no switching: grow faster oblivious to catastrophes switching: grow slower but try not to get caught Outlook: Generalise to saturating populations

45 Summary Plan I Thoughts on biophysical modelling II Switching between states III Microscopic dynamics: model linear feedback switch IV Macroscopic dynamics: population growth in a catastrophic environment References: P. Visco, R. J. Allen, M.R. Evans, Phys. Rev. Lett 2008, Phys. Rev. E 2009 P. Visco, R. J. Allen, S. N. Majumdar, M.R. Evans, Biophysical Journal 2010