Nonequilibrium Response in a Model for Sensory Adapta8on

Transcription

1 The 7 th KIAS Conference on Sta8s8cal Physics, 4-7 July 2016 Nonequilibrium Sta8s8cal Physics of Complex Systems Nonequilibrium Response in a Model for Sensory Adapta8on Shouwen Wang and Lei- Han Tang Beijing Computa0onal Science Research Center Hong Kong Bap0st University In collabora0on with Kyogo Kawaguchi (HMS) and Shin-ichi Sasa (Kyoto) Supported in part by the RGC/HKSAR

2 RICHARD FEYNMAN: everything that is living can be understood in terms of the jiggling and wiggling of atoms Transmembrane G protein coupled receptor (GPCR) + interleukin- 8 Courtesy of Yang Wang and H. Liu

3 Swimming bacteria (Howard Berg)

4 E. coli chemotaxis pathway Run Tumble SL Porter, GH Wadhams, JP Armitage, Nature Reviews Microbiology 9, (2011)

5 Sensory adapta0on Yuhai Tu, Annu Rev Biophys 42: (2013) ligand concentra0on receptor ac0vity

6 Sensory adapta0on Yuhai Tu, Annu Rev Biophys 42: (2013) ligand concentra0on receptor ac0vity

7 Adapta0on: nonequilibrium methyla0on dynamics Scheme: s m spectator a Suppose (a,m) follows equilibrium relaxa0onal dynamics under a poten0al F(a,m) γ a a = a F + η a γ m m = m F + η m Contradic0on! m activates a 2 F m a > 0 a represses m 2 F a m < 0

8 A discrete state Markov model of receptor dynamics Lan, Sartori, Neumann, Sourjik, Tu Nature Physics 2012, 8:422 Energy cost for adapta0on Fluctua0ons, noise

9 Discrete Markov Model of Adapta0on Lan et al., Nature Physics 2012, 8:422 Scheme: m spectator s a Fast receptor ac0va0on/deac0va0on at a given m (< 1 second): a = 1, active ω 1 (m,s) Slow methyla0on dynamics (> 1 min): αω m ω m a = 1 ω 0 (m,s) a = 0, inactive ω 0 ω 1 ( m,s) m,s ( ) = e ΔE(m,s) m = 0 m = 1 m = m 0 a = 0 ω m αω m ΔE(m,s) = e m ( m 1 m) + ln 1+ s / k i 1+ s / k a α = nonequilibrium parameter

10 Phase diagram perfect adapta0on α c α EQ α Adapta0on is possible only when α < α c = 1 < α EQ = exp( e m / 2) Lan et al., Nature Physics 8, 422 )2012); Sartori and Tu, PRL 115, 118 (2015)

11 Phase diagram perfect adapta0on α c α EQ α Landscape: NESS distribu0on P(m) can be computed exactly in the limit of fast ac0vity switching α = α EQ no adapta0on α = α c V eff (m) = ln P(m) α < α c adapta0on Driving changes the steady- state distribu0on completely

12 Kapitza s Pendulum Pyotr Kapitza ( ) russian physicist

13 The Energy-speed-accuracy (ESA) trade-off relation Lan et al., Nature Physics 2012, 8:422 adapta0on accuracy: ε = a a 0 driven by energy Claim: dissipated power methyla0on frequency

14 Got us onto an expedi0on of Rate of energy dissipa0on Shouwen Wang 3 rd yr graduate student Harada- Sasa equality Viola0on of the fluctua0on- response rela0on in 0me- scale separated systems S- W Wang, Y Lan and L- H Tang, JSTAT P07025 (2015) S- W Wang, K Kawaguchi, S- i Sasa, and L- H Tang, arxiv:

15 Energy dissipa0on rate Power needed to keep the system in the NESS: W = ( J AB J BA ) AB forward prob current ln J AB J BA backward prob current in units of k B T Exactly: W = ω m ( 1 α + boundary)ln α EQ α Increases with decreasing α boundary = αp(0,0) + αp(1,m 0 ) P(1,0) P(0,m 0 )

16 Modified ESA tradeoff NESS a = 0 a = 1 ε = a a 0 α m d { } m d = min m*,m 0 m * Driven down by both α and methyla0on range m * total dissipated power W ω m ln α EQ α ω m ln ε m d ε 0 No fundamental link between accuracy of adapta0on and power

17 Power the transient response?

18 Dissipa0on decomposed into a and m channels Drive transient response m- dynamics dissipa0on Drive adapta0on? a- dynamics dissipa0on

19 The Harada- Sasa equality Energy dissipa0on J x = γ x 2 + dω 2π C x ω ( ) 2TR x '( ω ) Viola0on of the fluctua0on- response rela0on (FDT) x(t) = observable (methyla0on level quasi- con0nuous) γ = fric0on coefficient associated with x (?) T. Harada and S.- i. Sasa, Phys Rev Leq 95, (2005)

20 Response and correla0on func0ons for methyla0on speed ε = τ a τ m τ m 1 τ a 1 FDT sa0sfied at both low and high frequency ends Plateau behavior in the intermediate frequency range As 0mescale separa0on increases, window size increases while viola0on strength decreases

21 A General Markov Framework to Treat Time Scale Separated Systems

22 Markov States: n = p k p, q: coarse- grained states (slow) k, l: microstates within each coarse- grained state (fast) Transi0on rate matrix: M pk q l = ε 1 δ pq M q (1) kl + M pk q l fast dynamics coupling

23 Transi8on rate matrix spectrum Master equa0on: dp n dt = M nm P m (t) m Right and les eigenvectors: Mx j = λ j x j, y j M = λ j y j Time scale separated system M pk q l = ε 1 δ pq M q (1) Degenerate kl + M pk q l perturba0on Block diagonalized! M pq = k,l M (1) pk q l P ss ( l q) Emergent dynamics for coarse- grained variables Landscape!

24 Correla8on and response func8ons Perturba0on introduced through modified transi0on rates: w(m n) = w(m n)exp h( Q n Q m ) / T original model transi0on rates observable Q(t) defined on a trajectory n t : Q(t) = Q nt Velocity response: R Q (t τ ) = δ Q(t) δ h(τ ) Velocity correla0on: C Q (t τ ) = Q(t) Q(τ )

25 Correla8on and response func8ons in frequency domain Right and les eigenvectors: Mx j = λ j x j, y j M = λ j y j j = 1,2,, N, in ascending order of the real part ( ) ( ) 2 N Velocity response: R (ω ) = 2α Q jβ j λ j 1 1 i ω λ j j=2 1+ ω λ j N Velocity correla0on: C (ω ) = α 1 Q jϕ j 1 1+ ( ω λ j ) 2 j=2 Important proper0es: α j,β j,ϕ j are linear weighted averages of Q. a) Equilibrium with detailed balance λ j β j = Tϕ j C Q = 2TR Q ' FRR or FDT! b) General sum rule ( ) N α j λ j β j Tϕ j = 0 j=2

26 Applied to 8me- scale separated system dissipa0ve cycle General result on the FRR viola0on spectrum: C Q (ω ) 2TR Q '(ω ) = 2 K j=2 Vanishes if the effec0ve dynamics sa0sfies detailed balance α j T ϕ j λ j β j ( ) 2 + εv s 1+ ω λ j Harada- Sasa: dissipa0on on intermediate 0me scales, Hidden Entropy Produc8on! Split the sum into slow and fast modes N = + j=2 K j=2 N j=k+1 Only slow modes make significant contribu0ons for ω < τ f 1 1 j K

27 Summary: 0mescale separated driven system Biomolecular devices NESS energy dissipa0on Nonequilibrium noise, Harada- Sasa equality Possible detailed balance for coarse- grained dynamics viola0on of the FRR (effec0ve) energy landscape design, control

28 Thank you for your aqen0on!