Nonequilibrium stochastic Processes at Single-molecule and Single-cell levels

Transcription

1 Nonequilibrium stochastic Processes at Single-molecule and Single-cell levels Hao Ge ( 葛颢 ) haoge@pu.edu.cn 1 Beijing International Center for Mathematical Research Biodynamic Optical Imaging Center Peing University, China

2 BICMR: Beijing International Center for Mathematical Research

3 BIOPIC: Biodynamic Optical Imaging Center

4 Summary of Ge group Stochastic theory of nonequilibrium statistical mechanics JSP06,08 PRE09,10,13,14 JCP1; JSTAT15 Stochastic modeling of biophysical systems JPCB08,13; JPA1 Phys. Rep. 1 Science13;Cell14;MSB15 Stochastic Biophysics (Biomath) Nonequilibrium landscape theory and rate formulas PRL09,15 JRSI11; Chaos1 Statistical machine learning of singlecell data CR15

5 Which ind of physical/chemical processes can be described by stochastic processes? Mesoscopic scale (time and space) Single-molecule and single-cell (subcellular) dynamics Trajectory perspective

6 Single-molecule experiments Single Ion channel Single-molecule enzyme inetics E. Neher and B. Saman Nobel Prize in 1991 Lu, et al. Science (1998)

7 Single-cell dynamics (in vivo) Eldar, A. and Elowitz, M. Nature (010) Choi, et al. Science (008)

8 Stochastic theory of nonequilibrium statistical mechanics

9 The fundamental equation in nonequilibrium thermodynamics Second law of thermodynamics Clausius inequality ds Q T rewrite epr ds Q T 0 Entropy production Rudolf Clausius ( ) More general Carl Ecart ( ) P.W. Bridgman ( ) Nobel Prize in 1946 ds epr d d i e S S d i S J X 0 Lars Onsager ( ) Nobel Prize in 1968 Ilya Prigogine ( ) Nobel Prize in 1977

10 Mathematical theory of nonequilibrium steady state Time-independent(stationary) Marov process Min Qian (197-) Recipient of Hua Loo-Keng Mathematics Prize ( 华罗庚数学奖 ) in 013 Ge, H.: Stochastic Theory of Nonequilibrium Statistical Physics (review). Advances in Mathematics(China) 43, (014)

11 Master equation model for the single-molecule system Consider a motor protein with N different conformations R 1,R,,R N. ij is the first-order or pseudo-first-order rate constants for the reaction R i R j. No matter starting from any initial distribution, it will finally approach its stationary distribution satisfying N j1 ss ss p p 0 j ji dp i ( t) dt i ij p Self-assembly or self-organization j j ji p i ij p eq j ji p eq i Detailed balance (equilibrium state) ij

12 NESS thermodynamic force and entropy production rate NESS thermodynamic force NESS flux J ss ij p ss i ij p ss j A ji ss ij B T log p p ss i ss j ij ji NESS entropy production rate T epr ness i j J ss ij A ss ij 0

13 Time-dependent case dpi dt t t p j j ji p i ij t Quasi-stationary distribution N j1 ss ss p t t p t t 0 j ji i ij If { ij (t)} satisfys the detailed balance condition for ss fixing t p t t p t t ss j 0 Boltzmann s law ji Free energy i ij Ei ( t)/ T ss eq p i ( t) pi ( t) E j ( t) / T e j E j ( t)/ T F( t) BT log e j e

14 Mathematical equivalence of Jarzynsi and Hatano-Sasa equalities Jarzynsi equality: local equilibrium e Same theorem for time-dependent Marov process W / T F / T p(0) p eq (0) e W Ge, H. and Qian, M., JMP (007); Ge, H. and Jiang, D.Q., JSP (008); Using Feynman-Kac formula of the time-dependent case eq F p( 0) p (0). Hatano-Sasa equality: without local equilibrium e Q ex / T p(0) p ss (0) 1 S 0 ss p0 p Q ex / T. Are these inequalities already nown in the Second Law of classic thermodynamics? Do they only hold for the whole transition process between two steady states? The traditional Clausius inequality can be in a differential form.

15 Decomposition of mesoscopic thermodynamic forces i j A ij t B T log p p i j t t ij ji t t A ss ij Entropy production Houseeeping heat t A T ij t e p t J t A t 0 i j ss Q h( t) Jij ( t) Aij ( t) 0 i j f d ( t) Jij ( t) Aij ( t) i j Free energy dissipation 0 T p t Q t f t Ge, H., PRE (009); e h A d ss ij t B ij T log All the results have also been proved for multidimensional diffusion process. p p ss i ss j ij t t t t Ge, H. and Qian, H., PRE (010) (013) ij ji

16 Two origins of nonequilibrium t 0 e p for any time t In the absence of external energy input and at steady state. Q h t 0 for any time t In the absence of external energy input f d t 0 for any time t At steady state Ge, H., PRE (009); Ge, H. and Qian, H., PRE (010) (013)

17 Decomposition of entropy production rate f T d 0, e p Q f d h Q 0, h 0. Ge, H., PRE (009); Ge, H. and Qian, H., PRE (010) (013) ds dt e p Q T tot ds dt ds dt Q T ex Q T tot e p 0 Q tot Qh f 0 The new Clausius inequality is stronger than the traditional one. T d

18 Mathematical problems left Mathematical proof for some newly recent developed finite-time fluctuation theorems of diffusion process with time-dependent diffusion coefficients; Mathematical proof for the large deviation principle of sample entropy production of diffusion process on R n ; Stochastic theory of nonequilibrium statistical mechanics of second-order stochastic process;

19 A first step towards the stochastic theory of nonequilibrium statistical mechanics for second-order stochastic system: Time-reversibility and anomalous behavior m d X dt F. X,X t

20 Two different definitions of entropy production rates EPR B 1 B D T x lim 0 1 T F log F P P t s : 0 r s x D m tt s v log s T : 0 s t T dxdv t P F F x, v Spinney, R.E., and Ford, I.J., PRL (01); Lee, H.K., Kwon, C., and Par, H., PRL (013) EPR B D x x v x D m v log t dxdv t Kim, K.H. and Qian, H., PRL (004) EPR 1 0, EPR 0 correspond to time-reversibility and Maxwell-Boltzmann distribution for thermodynamic equilibrium respectively thermodynamic equilibrium Ge, H.: PRE (014)

21 When the external force is only dependent on position F x,v xv Gx D B x T x x Ge, H.: PRE (014) EPR EPR 1 thermodynamic equilibrium 0 T G x x T ; x U x Thermal equilibrium Mechanical equilibrium Flow of inetic energy along the spatial coordinate d dt E inetic t inetic x J W x,t Qx,t x x (measurable) Heat flux J q x J inetic x E inetic t xv x EPR 0 J x J 0 q Thermodynamic equivalence between mesoscopic and macroscopic scales

22 The entropy production rate in the small-noise limit Decomposition of entropy production rate EPR EPR spatial EPR spatial EPR over T x x J ˆ over x t ˆ dx t Celani, et al.: PRL (01); Ge, H.: PRE (014) Entropy production rate of the overdamped-limit over n 6 B xt x xt x ˆ t x dx Anomalous contribution of EPR Hence the overdamped approximation only eeps the dynamics rather than the second law of thermodynamics.

23 Local reciprocal relation between linear coefficients L xx L qx T J J over x q x x L xq x ˆ t L x xx L qx X p X L T x x p qq L xq L n qq X q X 8 6 Reciprocal relation between Soret effect (thermal diffusion) and Dufour effect B q B T x x 3 ˆ t x Ge, H.: JSTAT (015) Come from the second moment of velocity along each dimension. Always hold, even for far-from-equilibrium system.

24 Two-state model of central dogma without feedbac Help to uncover the mechanism of transcriptional burst

25 DNA transcription

26 Regulation of gene expression Induced condition Highly expressed, i.e. low repression

27 Transcriptional burst under induced condition Golding et al. Cell (005)

28 DNA topology and transcriptional burst Anchored DNA segment High-throughput in-vitro experiments Levens and Larson: Cell (014) (preview) Shasha et al. Cell (014)

29 Supercoiling accumulation and gyrase activity Gene OFF Gene ON mrna ø 1 Shasha et al. Cell (014)

30 Gene states Two-state model without feedbac Gene ON Gene OFF Gene OFF Gene ON mrna ø Chemical master equation The mean-field deterministic model has only 1 one stable 1 fixed point! n 0 1 n n+1 ( n 1) Copy number of mrna 1 ( n 1) n+1

31 Poisson distribution with a spie at zero When α,β<< 1,γ, then 1.,! ) ( ) ( ) ( ; (0) (0) (0) n n e n p n p n p e p p p n Poisson distribution with a spie (bimodal) Duty cycle ratio

32 Two-state model of central dogma with positive feedbac A rate formula for stochastic phenotype transition in an intermediate region of gene-state switching

33 Central Dogma Copy numbers in a single cell Bacteria Euaryotic cells DNA 1 or ~ mrna A few Protein Not enough attention has been paid to this fact.

34 Regulation of gene expression An example of gene circuit with positive feedbac: Lac operon

35 Bimodal distributions in biology: multiple phenotypic states Ferrell, J. and Machleder, E. Science (1998) Choi, et al., Science (008) To, T. and Maheshri, N. Science (010)

36 Two-state model with positive feedbac n max large 1

37 Mean-field deterministic model with positive feedbac gx x dx dt g( x) x x n n max Sigmoidal influx flux Stable Off-state Unstable threshold Stable On-state Bifurcation diagram Bifurcation diagram for simple example g * * ( x ) x x * ON state Influx g(x) Outflux γx OFF state Flux-balance plot x /K eq

38 Interconversion of different phenotypic states How to quantify the transition rates between different phenotypic states, provided their existence? Choi, et al., Science (008) Gupta, et al., Cell (011)

39 Recall Langevin dynamics and Kramers rate formula Chemical reaction activated by diffusional fluctuations ǂ Uǂ a BT islarge e γ = η m P. Langevin ( ) d x dt m f f f du dx x dx dt f t 0; t BT; t f s 0, s t. t U U 1 1 x m x x a,x around ǂ x m x xǂ,x around x. ǂ κ = ω γ q = 1 q A = π BT ω a h + = κ BT h = κ BT q e h q A a ΔG e B T ΔU B T x A ; H.A. Kramers ( )

40 From single chemical reaction to biochemical networs (biology) Single chemical reaction Physical state of atoms Conformational state The state of system Emergent state at a higher level Molecular copy number Phenotypic state Single cell: biochemical networ Chemical master equation (CME) dp dt M M X, t r X PX, t r X PX, t j1 j j j j1 j Max Delbruc( ) Nobel Prize in 1969

41 Two-state model with positive feedbac large The analytical results introduced here can be applied to any self-regulating module of a single gene, while the methodology is valid for a much more general context. n max 1

42 Three time scales and three different scenarios ( i) : cell cycle ( ) ( ii) : gene - state switching ( f, hn( n 1) ) ( iii) : synthesis rate of protein ( 1 ) When stochastic gene-state switching is extremely rapid Ao, et al. (004); ( i ) ( iii) ( ii) Huang, et al. (010) When stochastic gene-state switching is extremely slow Qian, et al. (009); ( ii) ( i ) ( iii) Wolynes, et al. (005) When stochastic gene-state switching is relatively slow Wolynes, et al. (005); ( i ) ( ii) ( iii) Ge, et al. (015) When the time scales of (ii) and (iii) are comparable Assaf, et al. (011); ( i ) ( ii) ( iii) Li, et al. (014)

43 (A) x * A single-molecule fluctuatingrate model is derived, 1 f, h Continuous Mean-field limit Bifurcation diagram Bifurcation diagram for simple example OFF state dx dt g( x) Rescaled dynamics ON state x (B) dx dt hx 1 f, h, Fluctuating-rate model dx dt n 1 max n max x f x x n n max 1/K eq Ge, H., Qian, H. and Xie, X.S., PRL (015)

44 Stochastic dynamics of fluctuating-rate model dx dt n 1 max x hx f dx dt n max x Ge, H., Qian, H. and Xie, X.S., PRL (015)

45 Nonequilibrium landscape function emerges As gene-state switching is much faster than the cell cycle Landscape function x ss 0 p0 e x ~ dx dt g( x) x Dynamics in the mean field limit model Stable (OFF) Unstable Stable (ON) d 0 dx x n 1 max f hx x x n max analog to energy function at equilibrium case Quantify the relative stability x of stable fixed points Φ 0 = Φ 0 f x Ge, H., Qian, H. and Xie, X.S., PRL (015)

46 Dynamics of bistable systems alternative attractor Inter-attractorial dynamics 3: abrupt transition via barrier-crossing 1: relaxation process Intra-attractorial dynamics : fluctuating in local attractor, waiting The uphill dynamics is the rare event, related to phenotype switching, punctuated transition in evolution, et al.

47 probabil ity Three time scales Fixed finite molecule numbers n y chemical master equation c y B Stochastic A Deterministic (a) n x (b) fast nonlinear differential equations c x discrete stochastic model among attractors emergent slow stochastic dynamics and landscape (d) A Stochastic B A B (c) appropriate reaction coordinate Ge and Qian: PRL (009), JRSI (011)

48 Rate formulae associated with the landscape function Gene-state switching is relatively slow Gene-state switching is extremely slow Barrier crossing n max e 0 ǂ 0 n max n max n max n max Ge, H., Qian, H. and Xie, X.S., PRL (015) linearly depend on gene-state switching rates Wolynes, et al. PNAS (005)

49 A recent example: HIV therapy (activator + noise enhancer) Gene ON off on 0 e Gene OFF ~ 0 off 0 on off Activator: increasing on, lower the barrier Noise enhancer: Decreasing both on and off, further lower the barrier 0 Weinberger group, Science (014) Significantly increasing the transition rate

50 Rigorous analysis: quasi potential in LDP Local: The Donser-Varadhan large deviation theory for Marov process + Global: The Freidlin-Wentzell large deviation theory for random perturbed dynamic system LDT of Fluctuating-rate model (Switching ODE) Two-scale LDT of Switching(Coupled) Diffusion See Chapter 7 in Freidlin and Wentzell: Random Perturbations of Dynamical Systems ( nd Ed). Springer 1984

51 Compared to previous rate formulae for bursty dynamics Eldar, A. and Elowitz, M. Nature (010) Cai, et al. Science (006) Burst size b 1 f x Max 1 off f If x d 0 1, x dx b x ǂ 0 e xǂ x b off Walcza,et al.,pnas (005);Choi, et al.,jmb (010);Ge,H.,Qian,H.and Xie, X.S.,PRL (015)

52 Voice on Cell

53 Multistability and different time scales Conformation(Phenotype) Conformation(Phenotype) 1 1 Conformation(Phenotype)

54 Local landscapes and Kramers rate formula V ij i ij ji j V ji ij Constructed locally 1 T ij ji 1 T ji Phenotypic subspace ij ji 0 ij 0 ji e e V ij V ji, 1, 1 Driving force: ij 1 log ij ji V ji V ij 1 log p p ss j ss i

55 Multistability: local-global conflictions Ge, H. and Qian, H.: Chaos (01) The emergent Marovian jumping process being nonequilibrium is equivalent to the discontinuity of the local landscapes (time symmetry breaing). Global landscape: from stationary distribution Just cut and glue on the local landscapes (having non-derivative points).

56 Summary Stochastic processes become more and more popular to model the nonequilibrium mesoscopic biophysical dynamics, especially in single-cell biology. Stochastic theory of nonequilibrium statistical mechanics is rigorously studied and further developed. Recently we are interested in the second-order stochastic system with temperature gradient. Stochastic model of central dogma at single-cell level helps to uncover the mechanism of transcriptional burst, and we propose a single-molecule fluctuating-rate model as well as an associated saddle-crossing rate formula for the phenotype transition in an intermediate scenario.

57 Acnowledgement Prof. Hong Qian University of Washington Prof. Sunney Xiaoliang Xie Harvard University Peing University Fundings: NSFC, MOE of PRC

58 Thans for your attention!