Nonequilibrium stochastic Processes at Single-molecule and Single-cell levels Hao Ge ( 葛颢 ) haoge@pku.edu.cn Beijing International Center for Mathematical Research 2 Biodynamic Optical Imaging Center Peking University, China http://bicmr.pku.edu.cn/~gehao/
BICMR: Beijing International Center for Mathematical Research
BIOPIC: Biodynamic Optical Imaging Center
Summary of Ge group JSP06,08,7 PRE09,0,3,4,6 JCP2; JSTAT5 Stochastic theory of nonequilibrium statistical mechanics Nonequilibrium landscape theory and rate formulas PRL09,5 JRSI; Chaos2 Theory Stochastic Biophysics (Biomath) Providing analytical tools Providing scientific problems JPCB08,3,6; JPA2; SPA7 Phys. Rep. 2 Science3;Cell4;MSB5 Stochastic modeling of biophysical systems Statistical machine learning of single-cell data Applications
Interactions among math, physics, chemistry and biology
Which kind of physical/chemical processes can be described by stochastic processes? Mesoscopic scale (time and space) Single-molecule and single-cell (subcellular) dynamics Trajectory perspective
Single-molecule experiments Single Ion channel Single-molecule enzyme kinetics E. Neher and B. Sakman Nobel Prize in 99 Lu, et al. Science (998)
Single-cell dynamics (in vivo) Eldar, A. and Elowitz, M. Nature (200) Choi, et al. Science (2008)
The simplest three-state example: single-molecule enzyme kinetics Ge, H.: J. Phys. Chem. B (2008); J. Phys. A (202) Ge, H., et al. Phys. Rep. (202) Ge, et al. Stoc. Proc. Appl. In press
Reversible Michaelis-Menten enzyme kinetics Two reversible Michaelis-Menten reactions Ge, H.: J. Phys. Chem. B (2008) Ge, H., Qian, M. and Qian, H.: Phys. Rep. (202) Kinetic scheme of a simple reversible enzyme. From the perspective of a single enzyme molecule, the reaction is unimolecular and cyclic.
Simulated turnover traces of a single molecule S P (t) Min, et al. Nano Lett, (2005) Ge, H.: J. Phys. Chem. B (2008) ( t) : S P ( t) : P S the number of occurrences of forward and backward cycles up to time t
Steady-state cycle fluxes and nonequilibrium steady state J ss ( t) lim t t [ S] VS V KmS [ S] K ms [ P] KmP [ P] K P mp J ss J ss. Ge, H.: JPCB (2008) Ge, H., Qian, M. and Qian,H.: Phys. Rep. (202) Michaelis-Menten kinetics J ss ( t) lim t t [ S] VS KmS [ S] [ P] K K ms mp ; J ss ( t) lim t t [ P] VP KmP [ S] [ P] K K 0 ss k k2k3[ S] J Chemical potential difference: kbt ln k ln( ) 0 BT ss k k k [ P] J 2 3 ms J mp ss. 0 0
Waiting cycle times S k k 2 k 3 k k 2 k 3-3 -2-0 2 3 E ES EP E ES EP E P k - k -2 k -3 k - k -2 k -3 The kinetic scheme for computing the waiting cycle times, which also serves for molecular motors. T J ss J ss ; T J ss ; T J ss.
Generalized Haldane equality P E ( T t T T ) PE ( T t T T ). Superposed distributions! Carter, N. J.; Cross, R. A. Nature (2005) Waiting cycle time T is independent of whether the enzyme completes a forward cycle or a backward cycle, although the probability weight of these two cycles might be rather different. Average time course of forward and backward steps Qian, H. and Xie, X.S.: PRE (2006) Ge, H.: J. Phys. Chem. B (2008); J. Phys. A (202) Jia, et al. Ann. Appl. Prob. (206): general Markovian jumping process Ge, et al. Stoc. Proc. Appl. In press: diffusion on a circle
Nonequilibrium steady state Ge, H., et al. Phys. Rep. (202) 3 2 3 2 B P S k k k k k k T log k ) ( ) ( T T t T P T T t T P E E Ge, H. J. Phys. Chem. B (2008); J. Phys. A (202) Generalized Haldane Equality Fluctuation theorem of fluctuating chemical work T k n E E B e n t W P n t W P ) ) ( ( ) ) ( ( ) ( T k t W B e Second law in terms of equality Free energy conservation 0 t W epr t Entropy production: Free energy dissipation Free energy input S P 0. m Equilibriu Traditional Second law t W t
Steady-state Gallavotti-Cohen type fluctuation theorems, J t t J t t t J t t t J t J t J has the large deviation principle with rate function I. log P t J,J x,x I x,x,t I t t 2 2 x, x I x, x x 2 2 x2 J t J has the large deviation principle with rate function I 2. t log P t I J t x I2x, t x I xx 2 2 Jia, et al. Ann. Appl. Prob. (206): Ge, et al. Stoc. Proc. Appl. In press
Quasi-time-reversal invariance of d B.M. and 3d Bessel process translation The distribution is the same as a random flipped 3d Bessel Process P E ( T t T T ) PE ( T t T T ) For diffusion on a d circle Ge, et al. Stoc. Proc. Appl. In press
Two-state model of central dogma without feedback Help to uncover the mechanism of transcriptional burst Chong, S.S., Chen, C.Y., Ge, H. and Xie, X.S. Cell (204)
DNA transcription
Regulation of gene expression Induced condition Highly expressed, i.e. low repression
Transcriptional burst under induced condition Golding et al. Cell (2005)
DNA topology and transcriptional burst Anchored DNA segment Levens and Larson: Cell (204) (preview) High-throughput in-vitro experiments Chong, S.S., Chen, C.Y., Ge, H. and Xie, X.S. Cell (204)
Supercoiling accumulation and gyrase activity Gene OFF Gene ON mrna ø k Chong, S.S., Chen, C.Y., Ge, H. and Xie, X.S. Cell (204)
Gene states Two-state model without feedback Gene ON Gene OFF Gene OFF Gene ON mrna ø Chemical master equation The mean-field deterministic model has only k one stable k fixed kpoint! 0 n 0 n n+ 2 ( n ) 2 Copy number of mrna k ( n ) n+
Poisson distribution with a spike at zero When α,β<<k,γ, then.,! ) ( ) ( ) ( ; (0) (0) (0) 2 2 n n k e n p n p n p e p p p n k k Poisson distribution with a spike (bimodal) Duty cycle ratio Chong, S.S., Chen, C.Y., Ge, H. and Xie, X.S. Cell (204)
Two-state model of central dogma with positive feedback 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. (205)
Central Dogma Copy numbers in a single cell Bacteria Eukaryotic cells DNA or 2 ~2 mrna A few - 0 3 Protein - 0 4-0 6 Not enough attention has been paid to this fact.
Regulation of gene expression An example of gene circuit with positive feedback: Lac operon
Bimodal distributions in biology: multiple phenotypic states Ferrell, J. and Machleder, E. Science (998) Choi, et al., Science (2008) To, T. and Maheshri, N. Science (200)
Two-state model with positive feedback n max large k
Mean-field deterministic model with positive feedback g(x) x γ dx dt = g x γx x = n n max Sigmoidal influx flux Stable Off-state Unstable threshold Stable On-state 0.9 Bifurcation diagram Bifurcation diagram for simple example g x = γx 0.8 0.7 0.6 ON state x * 0.5 Influx g(x) Outflux γx 0.4 0.3 0.2 OFF state Flux-balance plot x 0. 0 0 5 0 5 20 25 30 /K eq
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 (20)
Recall Langevin dynamics and Kramers rate formula Chemical reaction activated by diffusional fluctuations k ǂ 2 Uǂ a kbt is large e γ = η m P. Langevin (872-946) 2 d x dt m 2 f f f du dx x dx dt f t 0; 2t 2 kbt; t f s 0, s t. t U U 2 2 2 x m x x a,x around ǂ2 2 x m x xǂ,x around x. ǂ κ = ω γ q = q A = 2πk BT ω a h k + = κ k BT h = κ k BT q e h q A a ΔG e k B T ΔU k B T 2 x A ; H.A. Kramers (894-952)
From single chemical reaction to biochemical networks (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 network Chemical master equation (CME) M M X, t r X PX, t r X PX, t dp dt j j j j j j Max Delbruck(906-98) Nobel Prize in 969
Two-state model with positive feedback 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 k
Three time scales and three different scenarios ( i) : cell cycle ( ) ( ii) : gene - state switching ( f, hn( n ) ( iii) : synthesis rate of protein( k ) ) When stochastic gene-state switching is extremely rapid Ao, et al. (2004); ( i ) ( iii ) ( ii ) Huang, et al. (200); 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. (205) When the time scales of (ii) and (iii) are comparable Assaf, et al. (20); ( i ) ( ii )(iii ) Li, et al. (204)
A single-molecule fluctuatingrate model is derived (A) x * 0.9 0.8 0.7 0.6 0.5 0.4 k, f, തh γ Continuous Mean-field limit Bifurcation diagram Bifurcation diagram for simple example Rescaled dynamics (B) ON state k f, തh, γ Fluctuating-rate model (Piecewise deterministic Markov process) dx dt = k n max γx തhx 2 dx dt = g x γx x = f n n max 0.3 0.2 0. OFF state dx dt = k 2 n max γx 0 0 5 0 5 20 25 30 /K eq Ge, H., Qian, H. and Xie, X.S., PRL (205)
Stochastic dynamics of fluctuating-rate model dx dt = k n max γx തhx 2 f dx dt = k 2 n max γx Ge, H., Qian, H. and Xie, X.S., PRL (205)
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 6 4 2 0-2 Stable (OFF) 0.73 0.725 Unstable Stable (ON) d dx x k n max 2 f hx k2 x x n max analog to energy function at equilibrium case -4 0.7-6 Quantify the relative stability 0.705 0.7 0.695-8 0.69 0.0 0.005 0.0 0.05 0.02 x of stable fixed points -0 0.72 0.75 Φ = Φ f -2 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x Ge, H., Qian, H. and Xie, X.S., PRL (205)
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 (20)
Rate formulae associated with the landscape function Gene-state switching is relatively slow Gene-state switching is extremely slow Barrier crossing n max k k 0 e Φ n max n max n max n max Ge, H., Qian, H. and Xie, X.S., PRL (205) k linearly depend on gene-state switching rates Wolynes, et al. PNAS (2005)
A recent example: HIV therapy (activator + noise enhancer) Gene ON k off k on Gene OFF k k 0 e Φ ǂ ~ kon k off koff Activator: increasing k on, lower the barrier Noise enhancer: Decreasing both k on and k off, further lower the barrier Weinberger group, Science (204) Significantly increasing the transition rate
Rigorous analysis: quasi potential in LDP Local: The Donsker-Varadhan large deviation theory for Markov 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 984
Compared to previous rate formulae for bursty dynamics Eldar, A. and Elowitz, M. Nature (200) Cai, et al. Science (2006) If d dx x Burst size, x x b ǂ b f k n max x off f k k 0 e x ǂ x b off Walczak,et al.,pnas (2005);Choi, et al.,jmb (200);Ge,H.,Qian,H.and Xie, X.S.,PRL (205)
Voice on Cell
Summary Stochastic processes become more and more popular to model the nonequilibrium mesoscopic biophysical dynamics. Stochastic theory of nonequilibrium statistical mechanics is rigorously studied and further developed, including an unexpected equality for the cycle symmetry. 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.
Acknowledgement Prof. Hong Qian University of Washington Prof. Sunney Xiaoliang Xie Harvard University Peking University Fundings: NSFC, MOE of PRC
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