Nonequilibrium landscape theory of chemical reaction systems and its applications: From stochastic thermodynamics to single-cell biology

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1 Nonequilibrium landscape theory of chemical reaction systems and its applications: From stochastic thermodynamics to single-cell biology Hao Ge ( 葛颢 ) haoge@pu.edu.cn 1 Beijing International Center for Mathematical Research 2 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 JSP06,08,17 PRE09,10,13,14,16 JCP12; JSTAT15 Stochastic theory of nonequilibrium statistical mechanics Nonequilibrium landscape theory and rate formulas PRL09,15 JRSI11; Chaos12 Theory Stochastic Biophysics (Biomath) Providing analytical tools Providing scientific problems JPCB08,13,16; JPA12; SPA17 Phys. Rep. 12 Science13;Cell14;MSB15 Stochastic modeling of biophysical systems Statistical machine learning of single-cell data Applications

5 Interactions among math, physics, chemistry and biology

6 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

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

8 Single-cell dynamics (in vivo) Eldar, A. and Elowitz, M. Nature (2010) Choi, et al. Science (2008)

9 Nonequilibrium thermodynamics of chemical reaction systems A self-consistent theory based on only the internal inetics

10 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

11 Biochemical cycle inetics 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)

12 Mathematical theory of nonequilibrium steady state Time-independent(stationary) Marov process Min Qian (1927-) Recipient of Hua Loo-Keng Mathematics Prize ( 华罗庚数学奖 ) in 2013 Ge, H.: Stochastic Theory of Nonequilibrium Statistical Physics (review). Advances in Mathematics(China) 43, (2014)

13 Scientific questions we want to address Eldar, A. and Elowitz, M. Nature (2010) Only internal dynamics/inetics can be observed and identified in many real experiments; Is there any nonequilibrium thermodynamic theory can be built only based on internal dynamics/inetics? What is the use of it? Our answer: yes! We only have partial answers so far.

14 Master equation model for the single-molecule system Consider a motor protein with N different conformations R 1,R 2,,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

15 Time reversibility Time-reversed process: still satisfy the master equation dpi dt t p j ji piij j ji p ss i ij ss p j ji ji p ss j ji p ss i ij i0i1 i1i 2 i2i3 in i0 For each cycle, 1, c i0, i1,, in i i 0 n i i n n1 i i n1 n2 i 1i0 Kolmogorov cyclic condition

16 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

17 Energy transduction efficiency at NESS A mechanical system coupled fully reversibly to a chemical reactions, with a constant force resisting the mechanical movement driven by the chemical gradient. Energy Energy Input (Chemical) Input (Mechanical) ness h d or ness h d Energy Output (Mechanical) Energy Output (Chemical) Energy output Energy input ness h d Energy output Energy output 1 h ness d ness T epr all dissipated! Ge, H. and Qian, H.: PRE (2013)

18 Time-dependent case dpi dt t t p j j ji p i ij t If { ij (t)} satisfies the detailed balance condition for ss ss fixed t p t t p t t Boltzmann s law Free energy ss ss p t t p t t 0 N What does the Second Law j ji i ij j1 Quasi-stationary distribution loo lie here? j 0 ji 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

19 Decomposition of mesoscopic thermodynamic forces A ij t B T log p p i j t t i ij ji t t A ss ij Entropy production t A ij All the results here have also been t j T epr( t) J ( t) A ( t) proved for multidimensional ij i j diffusion process. Houseeeping heat Free energy dissipation T epr A ss ij ss Q h ( t) Jij ( t) Aij ( t) i j f d ( t) Jij ( t) Aij ( t) i j Ge, H., PRE (2009); t t t Q h f d t B T log p p ss i ss j t t ij t t Esposito, M. et al. PRE (2007) Ge, H. and Qian, H., PRE (2010) (2013) ij ji

20 Two origins of nonequilibrium t 0 epr 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 (2009); Ge, H. and Qian, H., PRE (2010) (2013)

21 Extended Clausius inequality f d 0, Q h 0, Ge, H., PRE (2009); Ge, H. and Qian, H., PRE (2010) (2013) S B i T p i epr log p i f d Q h ds dt 0. epr Q T tot ds dt ds dt Q T ex Q T tot epr 0 Q tot Qh f 0 The new Clausius inequality is stronger than the traditional one. T d

22 A generalization of free energy and its balance equation Ge, H., PRE (2009); Generalized free energy H B T i p i log Ge, H. and Qian, H., PRE (2010) (2013) p p i ss i dh - dt W d f d E in E dis 0. Dissipative wor in Jarzynsi equality Vanish in time-independent systems E in E dis Q T h 0 epr 0 Generalized free energy input Generalized free energy dissipation

23 General chemical reaction system at macroscopic level Forward flux R +l x ν l1 + X 1 + ν l2 + X ν ln + X N ν l1 X 1 + ν l2 X ν ln X N Bacward flux R l x l = 1,2,, M Stoichiometric matrix S N M = {s ij } s ij =ν ji Flux vector J M 1 = {j l x } j l x =R +l x R l x x N 1 : concentration vector Rate equation dx dt = S J x SJ x =0 Steady state

24 Entropy production rate M epr x = l=1 R +l x R l x log R +l x R l x 0 epr x = 0 R +l x =R l x, l Equilibrium state: detailed balance Equilibrium state is always steady state Wegscheider-Lewis cycle condition: For any M-dimensional vector ξ = ξ 1, ξ 2,, ξ M satisfying Sξ = 0 have σ M l=1 ξ l log R +l x R l x = 0, x. Right-null-space

25 Chemical reaction system at the mesoscopic level Copy number of all the chemical species n = n 1, n 2,, n N n = x V Denote ν l = ν l1, ν l2,, ν ln, ν li = ν li ν li + n n + ν l with rate r +l n n n ν l with rate r l n Chemical master equation Assuming Marovian dynamics dp V n, t dt M = ሾp V n ν l, t r +l n ν l l=1 + p V n + ν l, t r l n + ν l

26 Mesoscopic nonequilibrium thermodynamics dh (meso) dt = E in E dis. H (meso) = n p V n, t log p V n, t p V ss n M E in = l=1 n p V n, t r +l n p V n + ν l, t r l n + ν l log p V ss n r +l n p V ss n + ν l r l n + ν l M E dis = l=1 n p V n, t r +l n p V n + ν l, t r l n + ν l log p V n, t r +l n p V n + ν l, t r l n + ν l Ge, H., PRE (2009); Ge, H. and Qian, H., PRE (2010) (2013)

27 Law of large number: from mesoscopic to macroscopic In the limit of V tending to infinity, with the same initial state r +l xv /V R +l x r l xv /V R l x n V t V x t, for any t. Kurtz. T. JCP (1972) M Large deviation principle and rate function 1 V log p V ss xv φ ss x R +l x 1 e ν l x φ ss x + R l x 1 e ν l x φ ss x = 0 Shwartz, A. and Weiss, A. Large deviation for performance analysis. (1995) l=1 Hu, G. Z. Physi (1982)

28 Emergent macroscopic nonequilibrium thermodynamics H (meso) V φ ss x dh (meso) dt dφ ss x dt = E in E dis. V +. = cmf x epr x 0. A generalization of macroscopic free energy Lyapunov property M E in V cmf x = l=1 R +l x R l x log R +l x R l x eν l x φ ss x 0 Free energy input E dis V M epr x = l=1 R +l x R l x log R +l x R l x 0 Entropy production rate Ge, H. and Qian, H., PRE (2016); JSP (2016)

29 Decomposition of macroscopic thermodynamic forces = log R +l x R l x log R +l x R l x + ν l x φ ss x + ν l x φ ss x log R +l x R l x + ν l x φ ss x =log R +l x R l x if mass-action law is applied and complex balance is satisfied. Generally this decomposition can not be achieved if one only considers the macroscopic deterministic dynamics. Ge, H. and Qian, H., Chem. Phys. (2016); PRE (2016); JSP (2016)

30 Strong and wea detail balance conditions epr x = 0 if and only ifr +l x =R l x Strong detailed balance condition (equilibrium) cmf x = 0 if and only if log R +l x R l x Wea detailed balance condition = ν l x φ ss x, l Wea detailed balance condition for each x is equivalent to Wegscheider-Lewis cycle condition. dφ ss x dt = 0 if and only if the systems is at steady state. Ge, H. and Qian, H., PRE (2016); JSP (2016)

31 General fluctuation-dissipation theorem for chemical reactions ν l1 + X 1 + ν l2 + X ν ln + X N ν l1 X 1 + ν l2 X ν ln X N Rate equation Ξ ij = 2 φ ss x, x i x j dx dt R +l x R l x = SJ x F x M A ij = l=1 l = 1,2,, M F x =0 Stable fixed point R +l x + R l x ν li ν lj Bij = F i x, x j Variance matrix Fluctuation matrix Dissipation matrix ΞAΞ = ΞB BΞ Ge, H. and Qian, H., PRE (2016); JSP (2016)

32 More is different The elementary entities of science X obey the laws of science Y. But this hierarchy does not imply that science X is just applied Y. At each stage entirely new laws, concepts and generalizations are necessary, requiring inspiration and creativity to just as great a degree as in the previous one. P.W. Anderson: Science 177, (1972)

33 Two-state model of central dogma with positive feedbac A rate formula for stochastic phenotype transition in an intermediate region of gene-state switching Ge, H., Qian, H. and Xie, X.S., Phys. Rev. Lett. (2015)

34 Central Dogma Copy numbers in a single cell Bacteria Euaryotic cells DNA 1 or 2 ~2 mrna A few Protein Not enough attention has been paid to this fact.

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

36 Bimodal distributions in biology: multiple phenotypic states Ferrell, J. and Machleder, E. Science (1998) Choi, et al., Science (2008) To, T. and Maheshri, N. Science (2010)

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

38 Mean-field deterministic model with positive feedbac g(x) x γ dx dt = g x γx x = n n max Sigmoidal influx flux Stable Off-state Unstable threshold Stable On-state Bifurcation diagram Bifurcation diagram for simple example g x = γx ON state x * 0.5 Influx g(x) Outflux γx OFF state Flux-balance plot x /K eq

39 Interconversion of different phenotypic states How to quantify the transition rates between different phenotypic states, provided their existence? Choi, et al., Science (2008) Gupta, et al., Cell (2011)

40 Recall Langevin dynamics and Kramers rate formula Chemical reaction activated by diffusional fluctuations ǂ 2 Uǂ a BT is large e γ = η m P. Langevin ( ) 2 d x dt m 2 f f f du dx x dx dt f t 0; 2t 2 BT; t f s 0, s t. t U U x m x x a,x around ǂ2 2 x m x xǂ,x around x. ǂ κ = ω γ q = 1 q A = 2π BT ω a h + = κ BT h = κ BT q e h q A a ΔG e B T ΔU B T 2 x A ; H.A. Kramers ( )

41 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) M M X, t r X PX, t r X PX, t dp dt j1 j j j j1 j Max Delbruc( ) Nobel Prize in 1969

42 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

43 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. (2004); ( i ) ( iii ) ( ii ) Huang, et al. (2010); When stochastic gene-state switching is extremely slow Qian, et al. (2009); ( ii ) ( i ) (iii ) Wolynes, et al. (2005); When stochastic gene-state switching is relatively slow Wolynes, et al. (2005); ( i ) ( ii ) (iii ) Ge, et al. (2015) When the time scales of (ii) and (iii) are comparable Assaf, et al. (2011); ( i ) ( ii )(iii ) Li, et al. (2014)

44 A single-molecule fluctuatingrate model is derived (A) x * , f, തh γ Continuous Mean-field limit Bifurcation diagram Bifurcation diagram for simple example Rescaled dynamics (B) ON state 1 f, തh, γ Fluctuating-rate model (Piecewise deterministic Marov process) dx dt = 1 n max γx തhx 2 dx dt = g x γx x = f n n max OFF state dx dt = 2 n max γx /K eq Ge, H., Qian, H. and Xie, X.S., PRL (2015)

45 Stochastic dynamics of fluctuating-rate model dx dt = 1 n max γx തhx 2 f dx dt = 2 n max γx Ge, H., Qian, H. and Xie, X.S., PRL (2015)

46 Nonequilibrium landscape function emerges As gene-state switching is much faster than the cell cycle Landscape function p ss x e x Φ dx = g x γx dt Dynamics in the mean field limit model Stable (OFF) Unstable Stable (ON) d dx x n 1 max 2 f hx 2 x x n max analog to energy function at equilibrium case Quantify the relative stability x of stable fixed points Φ = Φ f x Ge, H., Qian, H. and Xie, X.S., PRL (2015)

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, H. and Qian, H.: PRL (2009), JRSI (2011)

48 Rate formulae associated with the landscape function Gene-state switching is relatively slow Gene-state switching is extremely slow Barrier crossing n max 0 e Φ n max n max n max n max Ge, H., Qian, H. and Xie, X.S., PRL (2015) linearly depend on gene-state switching rates Wolynes, et al. PNAS (2005)

49 A recent example: HIV therapy (activator + noise enhancer) Gene ON off on Gene OFF 0 e Φ ǂ ~ on off off Activator: increasing on, lower the barrier Noise enhancer: Decreasing both on and off, further lower the barrier Weinberger group, Science (2014) 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 (2 nd Ed). Springer 1984

51 Compared to previous rate formulae for bursty dynamics Eldar, A. and Elowitz, M. Nature (2010) Cai, et al. Science (2006) If d dx x Burst size 1, x b x ǂ b 1 f n 1 max x off f 0 e x ǂ x b off Walcza,et al.,pnas (2005);Choi, et al.,jmb (2010);Ge,H.,Qian,H.and Xie, X.S.,PRL (2015)

52 Voice on Cell

53 Summary Stochastic processes become more and more popular to model the nonequilibrium mesoscopic biophysical dynamics. Stochastic theory of nonequilibrium transient states is proposed and has been applied to general chemical reaction systems. We propose a single-molecule fluctuating-rate model for central dogma at single-cell level, and derive an associated saddle-crossing rate formula for the phenotype transition in an intermediate scenario.

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

55 Thans for your attention!