Stochastic Processes in Physics, Chemistry and Biology

Transcription

1 Stochastic Processes in Physics, Chemistry and Biology a personal perspective Hao Ge haoge@pku.edu.cn 1 Beijing International Center for Mathematical Research 2 Biodynamic Optical Imaging Center Peking University, China

2 Which kind of physical/chemical processes can be described by stochastic processes? Mesoscopic scale (time and space) Single-molecule and single-cell (subcellular) dynamics Markovian v.s. Non-Markovian

3 Example1: Perrin s experiment (2D diffusion) Single particle trajectories Multiple particle (final positions) r What s the probability distribution function here? <r 2 >= 4Dt J. Perrin ( ) A.Einstein ( ) Nobel Prize in 1926 Ann. Chim. Phys. VIII 18, (1909) Annalen der Physik 17, (1905) Nobel Prize in 1921

4 Example2: Single-molecule experiments Chalfie, M. Nat. Rev. Mol. Cell Biol. (2009) E. Neher and B. Sakman Nobel Prize in 1991 Lu, et al. Science (1998)

5 Example3: Single-cell dynamics Eldar, A. and Elowitz, M. Nature (2010) Choi, et al. Science (2008)

6 Markov processes Diffusion process (stochastic differential equation): The motion of molecule in solution The motion of each atom or atom group within a single molecule (in solution) Master equation (jumping process) The jumping between different conformational states of a single molecule The copy-number fluctuations of chemical species

7 Two different views of a same stochastic process Probability distribution view Chemical Master Equation Fokker-Planck equation Discrete Markov Chain Diffusion process Gillespie algorithm Stochastic differential equation Trajectory view The trajectory view of stochastic process is indeed beyond the theory of corresponding differential equations.

8 Why the trajectory view is important? Stochastic trajectories are captured in singlemolecule and single-cell tracking experiments; Integral along trajectories (e.g. Feynman- Kac formula) and first passage time of the trajectories are always involved in theoretical analysis; New theory in physics, chemistry and biology emerge from the trajectory perspective.

9 A remark on Brownian motion For physicists and chemists: mdv( t Ornstein-Uhlenbeck process ) = η v( t )dt + Langevin dynamics 2ηk B T db( t ) Mean square displacement x At short time: At long time: ( t) dt' v( t' ) = t 0 kbt t m k T η 2 2 ( x( t) ) 2 B ( x( t) ) 2 t = 2Dt 2k T η D = kbt η Einstein Relation! In probability theory: B dx( t ) = db( t ) Overdamped limit First-order SDE P. Langevin ( )

10 Nobel prize in Chemistry 2013 QM: Quantum mechanics MM: Molecular mechanics (Newtonian mechanics) Langevin dynamics: one way to add temperature

11 Rate formulas and coarse graining at different levels Kramers rate formula for chemical reaction: from physics to chemistry Rate formula of chemical master equation: from chemistry to biology

12 Reaction-rate theory: from physics to chemistry Diffusion in the space of reaction coordinates Barrier crossing Jumping process between conformational states of chemical species Ex. A B A k 1 B Optical trap p A B( t) = k 1 e k 1 t Coarse graining p B k 2 B A( t) = A k 2 e k 2 t McCann et al. Nature, 402, 785 (1999) First-order reaction

13 Arrhenius plot and Kramers rate formula Unimolecule reaction k = Ae β E b Chemical reaction activated by diffusional fluctuations 2 d x dt m 2 ( x) du dx = η + dx dt f ( x) S. Arrhenius ( ) Nobel Prize in 1903 k + ω ω η 2π b a βeb ( η + ) = e H.A. Kramers ( )

14 First-passage-time approach to diffusion controlled reactions K. Schulten, Z. Schulten, and A. Szabo J. Chem. Phys. (1980s) k ( x) du η dx = + dx T 2 d T dx B η 2 ( ) lim x 2ηk B T db( t ) ( x) 1 du ( x) dt ( x) dt x dx η = 0, dx T dx = 1 ( x ) = 0. C T ( x) = η k T B x C x e U k ( y ) U ( z ) B T dy z e k B T dz 1 = = T x B x T B e U k ( x) Even can be applied to bimolecular association reactions k + e ( x) B T U k dx ( x) B T dx ωaωb e 2πη βe b

15 Stochastic reaction rate: exponentially distributed A k+ B P k k + + x 1 e x,η + E. Olivieri and M.E. Vares:Large Deviations and Metastability. Cambridge University Press (2005) Martin V. Day: On the Exponential Exit Law on the Small Parameter Exit Problem. Stochastics 8, (1983) Martin V. Day: Recent Progress on the Small Parameter Exit Problem. Stochastics 20, (1987) Pierre Collet, Servet Martinez, Jaime San Martin: Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems. Springer (2013)

16 From single chemical reaction to biochemical networks (biology) How can we describe the state of a single cell biochemically? Chemical master equation (CME): The state of the system transfers from conformational states to molecular copy-numbers of chemical species P( X, t) = M r ( X v ) P( X v, t) dt j j j t j= 1 j= 1 Law of large number: Chemical Master Equation System size V tends to infinity M r ( X ) P( X, t) j Law of Mass Action (ODE) Max Delbruck( ) Nobel Prize in 1969 Kurtz, T. JCP (1972) analog of the dynamics along the reaction coordinates What is the analog of conformational states in biology?

17 Phenotypic states in biology: analog to conformational states Ferrell, J. and Machleder, E. Science (1998) Choi, et al., Science (2008) To, T. and Maheshri, N. Nature (2010)

18 An example: Phosphorylation cycle ATP ADP K * E E * P Pi K and K * are inactive and active forms of a kinase. E * is the phosphorylated E, a signaling molecule. Usually E is functionally active, i.e., turned-on.

19 Deterministic and stochastic models K E E* P dx dt x = [E * ] = v( x) w( x) v 0 v 1 v 2 0E* 1E* 2E* 3E* (N-1)E* NE* w 0 w 1 w 2

20 Steady distribution for large V We could derive that p( x) =P( N = xv ) E * x vy ( ) exp V dy log wy ( ) = exp Vφ ( x) Emergent Landscape ( ) * * * * d[e ] ϕ ( x) = 0 vx ( ) = wx ( ) = 0 dt Steady states (WKB, LDP)

21 Emergent landscape and deterministic dynamics φ Dynamics in the deterministic model dx dt = v( x) w( x) Stable Unstable Stable Maximum: the barrier dφ( x ) dx = log v( x ) w( x ) Local minimum dφ( x(t)) dt 0 Global minimum (More stable one) p ss V(n) exp x = n V,V +. x ( Vφ( x) ),

22 Kramers-like rate formula for Chemical Master Equation Qian, H. and Ge, H., Molecular & cellular biomechanics (2012) Zhang, X.Y., Ge, H. and Qian, H., Stud. Appl. Math. (2012) H 2 1 H 1 2 The switching rates between attractors: 1 VH VH 2 1 k1 2 = e, k2 1 = e. T T 1 2 The barrier H here may not be the same as the barrier in the previous landscape φ(x) for multistability of high dimensional cases. See Ge, H. and Qian, H.: Chaos (2012) 2 1

23 Life time of phenotypic states Qian, H. and Ge, H., Molecular & cellular biomechanics (2012) switching time in msec x x yrs 30 yrs 10 hrs assumed all the biochemical reaction rates are on the order of millisecond total number of molecule E

24 Protein Copy Numbers in Yeast 1024 Ghaemmaghami, S. et. al. (2003) Global analysis of protein expression in yeast, Nature, 425,

25 Any key difference between these two levels? Physics k e β E b Equilibrium Chemistry Given a priori A B k e VH b Emergent from dynamics Phenotype 2 Chemistry Nonequilibrium Phenotype 1 Biology

26 Landscape perspective for cellular dynamics Uninduced E. coli Induced Pluripotent State Waddington, The Strategy of Genes, 1957 Fibroblasts Blood Pancreas Liver Differentiated State

27 The history of nonequilibrium steady state in chemical biology From Macroscopic to mesoscopic From thermodynamics to statistical mechanics From chemical oscillation to subcellular stochastic kinetics

28 What is life Erwin Schrödinger was the first one trying to apply mathematical/physical arguments to study living matters; He explains that living matter evades the decay to thermodynamical equilibrium (death) by homeostatically maintaining negative entropy in an open system.

29 Nonequilibrium thermodynamics and chemical oscillation Belousov Zhabotinsky reaction Brusselator Ilya Prigogine Nobel Prize in 1977 Entropy production Nonequilibrium steady state Nonequilibrium thermodynamics epr ds = de S + dis = d = is J k X k k 0

30 Nonequilibrium statistical physics Nonequilibrium thermodynamics is not easy to be applied. Needs the idea of stochastic process; Understand where is the energy input and output T.L. Hill Muscle contraction: transduction from Ion pump at cell membrane chemical energy to mechanical energy T.L. Hill, Nature (1982) E. Eisenberg and T.L. Hill, Science (1985)

31 Mathematical theory of nonequilibrium steady state Min Qian (1927-) Recipient of Hua Loo-Keng Mathematics Prize ( 华罗庚数学奖 ) in 2013

32 How to break equilibrium mechanically or chemically? Nonconservative force ( x) du mdv( t) = η v( t) + f dt + 2ηk BT db( t) dx U. Seifert: Rep. Prog. Phys. (2012) Gradient of chemical free energy Ge, H., Qian, M. and Qian, H.: Phys. Rep. (2012)

33 Condition of equilibrium: Detailed balance From wikipedia: At equilibrium, each elementary process (such as collisions or elementary reactions) should be equilibrated by its reverse process. Mechanical dynamics Chemical dynamics Macroscopic dx = F( x) F ( c ) dt Mesoscopic dc = dt x: position (momentum) c: concentration Stochastic differential equation (diffusion) Markovian process on the discrete state space of n x: position (momentum) n=cv: molecular number Unified mathematical theory Qian, M. et al., Time-reversibility since 1970s Call for the thermodynamics/statistical mechanics along stochastic trajectory

34 Stochastic thermodynamics: thermodynamics of stochastic processes First and second law along trajectory Fluctuation theorems Nonequilibrium steady state

35 A quote from J.C. Maxwell The truth of the second law is therefore a statistical, not mathematical, truth, for it depends on the fact the bodies we deal with consist of millions of molecules, and that we never can get hold of single molecules. Hence the second law of thermodynamics is continuously violated, and that to a considerable extent, in any sufficiently small group of molecules belonging to a real body. Nature 17, 278 (1878) J.C. Maxwell ( )

36 Fluctuation theorems Evans, Gallavotti, Jarzynski, Crooks, Szabo, Seifert, Esposito, et al. since 1990s ω ω t : 0, { } For any trajectory = t T T Define a physical function on the trajectory F ( ) ω with probability P Find another process P ~ which is normally time-reversal of the original one dp F ~ dp ( ω) = log ( ω) e F ( ω ) P = 1 or P P ( F( ω) = a) ~ F( ω) = a a ~ = e F( ω) 0 ( ) P The second Law Ge, H. and Qian, M.: JMP, JPA (2007); Ge, H. and Jiang, D.Q. JSP (2008); Ge, H.: JPA (2012)

37 Experimental validation and application Wang, et al.: Phys. Rev. Lett. (2002) Gupta, et al. Nat. Phys. (2011)

38 Questions left ( ) 0 F ω Are these ensemble-averaged inequalities already known in the Second Law of classic thermodynamics? It is exactly our motivation for studying the superstructure of thermodynamics. P

39 Two origins of irreversibility Te p 1. = dh dt dh dt 0, + F 2. in F in 0; 0. Ge, H.: PRE (2009); Ge, H. and Qian, H.: PRE (2010, 2013) e p characterizes total time irreversibility in a Markov process. When system reaches stationary, dh/dt=0. When system is closed (i.e., no active energy drive, detailed balaned) F in =0. Boltzmann: -dh/dt = Te p >0 but F in =0; Prigogine(Brussel school, NESS): F in = Te p >0 but dh/dt =0. dh/dt 0 in driven systems is self-organization.

40 Two Clausius inequalities ds dt = e p h T d ds dt = 1 T dh dt ( h F ) d T in ds dt h T d ( e 0) p ds dt hd Fin dh + 0 T T dt With detailed balance (F in =0), they are equivalent. e p = dh, dt Without detailed balance (F in >0), the new one is stronger than the traditional one. F in 0

41 Anomalous thermodynamics with temperature gradient Underdamped dynamics spatial epr over epr epr = Ξ + Ξ over over Ξ 2 d x m = η ( x) v + G( x) + σ ( x) db( t) 2 dt T σ ( x) σ ( x) = 2η ( x) k T ( x) Q n = + 2 k 6 B Q = Q( x, t) dx over over = Q ( x, t) dx [ T ( x) ] x ˆ ρtdx η( x) T ( x) 2 B Celani, et al.: PRL (2012); Ge, H.: submitted (2013) At steady state ss Q ( x) ss dx = epr k T ( x) B ss, over Q ( x) ss, over k B T ( x) dx = epr ss ss, over Q ( x) Q ( x) ss, over k B T ( x) + dx Ξ Overdamped m small η large dynamics η ( x) dx = G( x) + σ ( x) db(t) Neither Ito nor Stratonovich!

42 Polymer dynamics: inside a single molecule

43 Static model: equilibrium conformations Trajectory of Brownian motion freely-jointed chain freely-rotating chain worm-like chain Self-avoiding random walk Paul Flory Nobel Prize in 1974 Excluded volume 2 d ~ N 3 5 SLE (Stochastic or Schramm Loewner evolution) Related fields medals: Wendelin Werner (2006); Stanislav Smirnov (2010)

44 Dynamic model: Rouse and Zim Gaussian chain ( ) ) ( 2 3 ) ( t f x x x a T k t f x U t x n n n n B n n n + + = + = + η ( ) n n N n B x x a T k U = + = 2 1 ~, ~ N N D G τ ~, ~ N N D G τ Zimm model (1956) Rouse model (1953) include hydrodynamic interactions mediated by solvent dt t s db T k s t s X t t s X B ), ( 2 ), ( ), ( 2 2 η α η + = SPDE Bruno Zim ( ) theta solvent

45 Stochasticity in advanced experimental techniques

46 Fluorescence correlation spectroscopy (FCS) Madge, Elson & Webb. Phys. Rev. Lett. 29, 705 (1972) 4D / 2 c ω = τ Autocorrelation function ) / ( ) ( ) ( ) ( ) ( ) ( c t I I t I t I t I G τ τ τ τ + = + = Autocorrelation τ Gaussian observation of Brownian particles ω

47 Stochastic optical reconstruction microscopy (STORM) Diffraction limit Spatial resolution: ~200 nm Activation Deactivation Photobleach Dark State Fluorescent State Permanent damage Xiaowei Zhuang Since 2006 Spatial resolution: ~20 nm

48 Other important examples Coupled (Switched) diffusion process: molecular motors; Generalized Langevin dynamics and fractional Brownian motion: within a single protein molecule; Onsager-Machulp functional: pathway in protein folding; Facilitated diffusion: target searching on DNA;

49 Take home message Stochastic process has a glorious history in physics, chemistry and biology; Due to the advance of single-molecule techniques, stochastic modeling will become more and more useful and popular; Nonequilibrium is the essence of living entities, and it would help us to understand the difference between in vivo and in vitro experimental observations; A series of problems in applied mathematics arises from stochastic processes in physics, chemistry and biology.

50 If you want to know more about mathematical modeling for biochemical systems, join my course Mathematical Biophysics ( 生物数学物理 ) in spring (2014).

51 Acknowledgement Prof. Min Qian Peking University Prof. Hong Qian University of Washington Prof. Sunney Xiaoliang Xie Harvard University Peking University

52 Thanks for your attention!

53 More experimental examples

54 1D diffusion of protein sliding along DNA Blainey, et al.: 103, 5752 PNAS (2006)

55 Stochasticity in central dogma Stochastic production of protein Occurrence An average of 4.2 mol Number of Molecules per Burst Burst size: exponential distributed B Protein copy-number distribution of cell population a 1 x / b x e p( x) = a b Γ( a) Gamma distribution Xie, et al. Nature (2006); PRL (2006); Science (2006,2010)