Stochastic Processes at Single-molecule and Single-cell levels Hao Ge haoge@pu.edu.cn Beijing International Center for Mathematical Research 2 Biodynamic Optical Imaging Center Peing University, China http://bicmr.pu.edu.cn/personal/gehao/
BICMR: Beijing International Center for Mathematical Research
BIOPIC: Biodynamic Optical Imaging Center
Which ind of physical/chemical processes can be described by stochastic processes? Mesoscopic scale (time and space) Single-molecule and single-cell (subcellular) dynamics Marovian v.s. Non-Marovian
Example: Perrin s experiment (2D Brownian motion) Single particle trajectories Multiple particle (final positions) r What s the probability distribution function here? <r 2 >= 4Dt J. Perrin (870-942) Nobel Prize in 926 Ann. Chim. Phys. VIII 8, 5-4 (909) Annalen der Physi 7, 549 560 (905) A.Einstein (879-955) Nobel Prize in 92
Example2: Single-molecule experiments Single Ion channel Single-molecule enzyme inetics E. Neher and B. Saman Nobel Prize in 99 Lu, et al. Science (998)
Example3: 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 inetics
Reversible Michaelis-Menten enzyme inetics 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.
From concentration to probability Pseudo-first order reaction constants 0 0 [ ] [ ] S 3 3 P The evolution of probability distribution Reactant E 3 0-3 [P] S P EP - 0 [S] -2 Product 2 ES [S] and [P] held constant
Gillespie algorithm (Exact simulation) Generate two random numbers for each step: r and r 2. Purely Analytical
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 bacward 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] K [ P] K P mp mp J ss J ss. Ge, H.: JPCB (2008) Ge, H., Qian, M. and Qian,H.: Phys. Rep. (202) Michaelis-Menten inetics J ss [ S] VS ( t) KmS lim ; t t [ S] [ P] K K ms mp J ss [ P] VP ( t) KmP lim. t t [ S] [ P] K K 0 ss 23[ S] J Chemical potential difference: BT ln ln( ) 0 BT ss [ P] J 2 3 ms J mp ss 0 0
Waiting cycle times S 2 3 2 3-3 -2-0 2 3 E ES EP E ES EP E P - -2-3 - -2-3 The inetic 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! Average time course of forward and bacward steps Carter, N. J.; Cross, R. A. Nature (2005) Waiting cycle time T is independent of whether the enzyme completes a forward cycle or a bacward cycle, although the probability weight of these two cycles might be rather different. Ge, H.: J. Phys. Chem. B (2008); J. Phys. A (202) Jia, et al. (204); Ge, et al. (204)
Nonequilibrium steady state Ge, H., et al. Phys. Rep. (202) 3 2 3 2 B P S T log ) ( ) ( 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 wor T n E E B e n t W P n t W P ) ) ( ( ) ) ( ( ) ( T t W B e Second law in terms of equality Free energy conservation 0 T log epr 3 2 3 2 B Entropy production: Free energy dissipation Free energy input S P 0. m Equilibriu Traditional Second law t W t
Two-state model of central dogma with positive feedbac Noise with different origins in stochastic cellular phenotype switching
Two-state model with positive feedbac x max large
Bimodal distributions in biology: phenotypic states Ferrell, J. and Machleder, E. Science (998) Choi, et al., Science (2008) To, T. and Maheshri, N. Nature (200)
Mean-field deterministic model with positive feedbac gx x dx dt g( x) x x n x max Sigmoidal influx flux Stable Off-state Unstable threshold Stable On-state 0.9 Bifurcation diagram Bifurcation diagram for simple example 0.8 0.7 ON state Influx g(x) Outflux γx x * 0.6 0.5 0.4 0.3 0.2 g OFF state * * ( x ) x 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 ǂ 2 Uǂ a BT is large e m P. Langevin (872-946) 2 d x dt m 2 f f f du dx x dx dt t 0; 2t 2 BT ; t f s 0,s t. f t U U 2 x m x x ǂ2 2 x m x xǂ,x around x. ǂ S ǂ 2 2 B a ln ǂ a 2 ah 2 T B,x around x A ; H.A. Kramers (894-952)
Rate formula lining physics and chemistry Diffusion in the space of reaction coordinates Barrier crossing Jumping process between conformational states of chemical species Ex. A B A + B Optical trap Coarse graining McCann et al. Nature (999)
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) Max Delbruc(906-98) Nobel Prize in 969
There are two major sources of fluctuations: another level of complexity in gene regulation? Two-state model with positive feedbac 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.
Three time scales and three different scenarios ( i ): cell cycle( ) ( ii) : gene - state switching( f ( iii) : synthesis rate of TF (,hn( n ) )) When the TF copy-number fluctuation dominates ( ( i i ) ) ( iii) ( ii) ( ii) When stochastic gene-state switching dominates ( ii) ( i ) ( iii) Ao, et al. (2004); Huang, et al. (200) Qian, et al. (2009); Wolynes, et al. (2005) ( iii) Wolynes, et al. (2005) More relevant to Lac operon (e.g. Xie group, 2006-200)
(A) x * 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. When stochastic gene-state switching dominates, f, h Continuous Mean-field limit Bifurcation diagram Bifurcation diagram for simple example OFF state dx dt g( x) Rescaled dynamics ON state x 0 0 5 0 5 20 25 30 (B) dx dt 2 hx f, h, Fluctuating-rate model dx dt x max x 2 max x f x x n x m ax /K eq Ge, H., Qian, H. and Xie, X.S., arxiv32.6776 (203)
Nonequilibrium landscape function emerges dx dt g( x) x Dynamics in the mean field limit model Landscape function x ss 0 p0 e x ~ 0 6 4 2 0-2 Stable (OFF) 0.73 Unstable Stable (ON) d0 dx x x max 2 f hx 2 x x x max 0.725 analog to energy function at equilibrium case -4-6 -8-0 0.72 0.75 0.7 0.705 0.7 0.695 0.69 0.0 0.005 0.0 0.05 0.02 x -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., arxiv32.6776 (203)
Rate formulae Gene-state switching is extremely slow Gene-state switching is relatively slow Barrier crossing e 0 ǂ 0 Transition rates linearly depend on gene-state switching rates Ge, H., Qian, H. and Xie, X.S., arxiv32.6776 (203)
The dynamics is bursty Eldar, A. and Elowitz, M. Nature (200) Cai, et al. Science (2006) Burst size b f x Max off f If x d 0, x dx b x ǂ 0 e xǂ x b off Walcza,et al.,pnas (2005);Choi, et al.,jmb (200);Ge,H.,Qian,H.and Xie, X.S.,arxiv32.6776 (203)
When protein copy-number fluctuation dominates Reduced CME d dx x x max ln x g x x ss p e x Ge, H., Qian, H. and Xie, X.S., arxiv32.6776 (203)
Barrier-crossing rate formula e ǂ A is more stable B is more stable Possible reverse of relative stability Ge, H., Qian, H. and Xie, X.S., arxiv32.6776 (203)
Even local fluctuations are different in the two limiting cases ~ 0 6 4 p i '' * * x e i x e i x x x ~ 4 3.5 2 / 2, i,2 2 3 0-2 -4-6 -8-0 0.73 0.725 0.72 0.75 0.7 0.705 0.7 0.695 0.69 0.0 0.005 0.0 0.05 0.02-0.5 x 0.0 0.005 0.0 0.05 0.02-2 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x 2.5 2.5 0.5 0-0.5-0.5055-0.506-0.5065-0.507-0.5075-0.508-0.5085-0.509-0.5095 x - 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., arxiv32.6776 (203)
Kramers formula for the diffusion approximation of CME K v 0 v v 2 E E* 0E* E* 2E* 3E* (N-)E* NE* The mean-field dynamics goes downhill along both functions, but the predicted relative P w 0 w w stability of phenotypic states can 2 be opposite! dx * vx wx x [E ] dt CME Landscape functions associated with rate formulae wx wx dx 2 vx wx ln DA v v x x dx Walcza, et al., PNAS (2005); Ao, P., Genet Genomics (2009); Wang, et al. Biophys. J. (200); Zhou, et al., JRSI (202); Vellela, M. and Qian, H., JRSI (2009); Ge, H. and Qian, H., JRSI (20)
Landscape perspective for cellular dynamics Uninduced E. coli Induced Pluripotent State Waddington, The Strategy of Genes, 957 Fibroblasts Blood Pancreas Liver Differentiated State
Any essential difference between these two levels? Physics e Chemistry(single reaction) Given a priori E b Equilibrium A B e ǂ Emergent from dynamics Phenotype 2 Nonequilibrium Phenotype Chemistry (reaction system) Biology
Multistability and different time scales Conformation(Phenotype) 3 3-3 -2 2 - Conformation(Phenotype) Conformation(Phenotype) 2
local landscapes and Kramers rate formula V ij i ij ji j V ji ij Constructed locally T ij ji T ji Phenotypic subspace ij ji 0 ij 0 ji e e V ij V ji,, Driving force: ij log ij ji V ji V ij log p p ss j ss i 2 23 3 0
Multistability: local-global conflictions Ge, H. and Qian, H.: Chaos (202) 3-2 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).
Summary Due to the advance of single-molecule techniques, stochastic modeling and computation will become more and more useful and popular. Three-state model of single-molecule inetics at nonequilibrium steady state exhibits rich cyclic dynamics and an unexpected equality emerges, followed by fluctuation theorems. Two-state model of central dogma at single-cell level helps to uncover the mechanism of transcriptional burst, and reveals an additional level of complexity within gene regulation.
Acnowledgement Prof. Min Qian Peing University Prof. Hong Qian University of Washington Prof. Sunney Xiaoliang Xie Harvard University Peing University
Thans for your attention!