Stochastic Processes at Single-molecule and Single-cell levels

Transcription

1 Stochastic Processes at Single-molecule and Single-cell levels Hao Ge 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 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

5 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 ( ) Nobel Prize in 926 Ann. Chim. Phys. VIII 8, 5-4 (909) Annalen der Physi 7, (905) A.Einstein ( ) Nobel Prize in 92

6 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)

7 Example3: Single-cell dynamics (in vivo) Eldar, A. and Elowitz, M. Nature (200) Choi, et al. Science (2008)

8 The simplest three-state example: single-molecule enzyme inetics

9 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.

10 From concentration to probability Pseudo-first order reaction constants 0 0 = [ ] [ ] S = 3 3 P The evolution of probability distribution Reactant E [P] S P EP - 0 [S] -2 Product 2 ES [S] and [P] held constant

11 Gillespie algorithm (Exact simulation) Generate two random numbers for each step: r and r 2. Purely Analytical

12 Simulated turnover traces of a single molecule S P ν (t) Min, et al. Nano Lett, (2005) Ge, H.: J. Phys. Chem. B (2008) ν + ( t) : S P ν ( t) : P S the number of occurrences of forward and bacward cycles up to time t

13 Steady-state cycle fluxes and nonequilibrium steady state J ss = ν ( 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 inetics J ss + ν + ( t) lim = t t + [ S] VS KmS [ S] [ P] + K K ss = ms mp ; J ν ( t) = lim = t t + [ P] VP KmP [ S] [ P] + K K 0 ss 23[ S] J + Chemical potential difference: µ = BT ln = ln( ) 0 BT ss [ P] J 2 3 ms J mp ss. = 0 µ = 0

14 Waiting cycle times S E ES EP E ES EP E P 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.

15 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. (205); Ge, et al. (205)

16 Nonequilibrium steady state Ge, H., et al. Phys. Rep. (202) B P S T log = = µ µ µ ) ( ) ( < = = < = 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 µ µ µ = = = ) ) ( ( ) ) ( ( ) ( = T t W B e Second law in terms of equality Free energy conservation 0 T log epr B = Entropy production: Free energy dissipation Free energy input µ S µ P µ = 0. m Equilibriu = µ Traditional Second law ( ) ( ) µ ν = t t W

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

18 Central Dogma Copy numbers in a single cell Bacteria Euaryotic cells DNA or 2 ~2 mrna A few Protein

19 Transcriptional burst under induced condition Golding et al. Cell (2005)

20 Regulation of gene expression All are not the mechanism of transcriptional burst under induced condition

21 DNA topology and transcriptional burst Anchored DNA segment High-throughput in-vitro experiments Levens and Larson: Cell (204) (preview) Shasha et al. Cell (204)

22 Supercoiling accumulation and gyrase activity β Gene OFF Gene ON mrna ø α γ Shasha et al. Cell (204)

23 Gene states Two-state model without feedbac Gene ON Gene OFF β Gene OFF Gene ON mrna ø α Chemical master equation The mean-field deterministic model has only one stable fixed point! 0 n 0 n n+ γ 2γ ( n +)γ β α β α β α β α γ 2γ Copy number of mrna ( n +)γ γ n+

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

25 Transcriptional burst under induced condition Shasha et al. Cell (204)

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

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

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

29 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)

30 Two-state model with positive feedbac n = max large γ

31 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 0.9 Bifurcation diagram g * * ( x ) = γx x * ON state Influx g(x) Outflux γx OFF state Flux-balance plot x /K eq

32 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)

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

34 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 ν ) P( X ν, t) r ( X ) P( X, t) j= j j j j= j Max Delbruc(906-98) Nobel Prize in 969

35 Three time scales and three different scenarios ( i) : cell cycle ( γ ) ( ii) : gene - state switching ( f, hn( n ) ( iii) : synthesis rate of protein ( ) ) 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)

36 Ubiquitous bursty dynamics and relatively slow gene-state switching Eldar, A. and Elowitz, M. Nature (200) Cai, et al. Science (2006) Golding et al. Cell (2005) Suter, et al. Science (20)

37 Start from the full Chemical Master Equation description n = max γ 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.

38 (A) x * A single-molecule fluctuatingrate model is derived, f, h >> γ Continuous Mean-field limit Bifurcation diagram OFF state dx dt = g( x) Rescaled dynamics ON state γx (B) dx dt 2 hx >> f, h, γ Fluctuating-rate model dx dt = = n max n 2 max γx f x = γx n n max /K eq Ge, H., Qian, H. and Xie, X.S., PRL (205)

39 Nonequilibrium landscape function emerges dx dt = g( x) γx Dynamics in the mean field limit model Landscape function Φ ( x) ( x) e ss 0 p0 ~ Φ Stable (OFF) 0.73 Unstable Stable (ON) d Φ 0 dx ( x) = n max 2 f hx + 2 γx γx n max analog to energy function at equilibrium case x x Ge, H., Qian, H. and Xie, X.S., PRL (205)

40 Rate formulae Gene-state switching is relatively slow Gene-state switching is extremely slow Barrier crossing n max e 0 Φ ǂ 0 n max n max n max n max Ge, H., Qian, H. and Xie, X.S., PRL (205) linearly depend on gene-state switching rates Wolynes, et al. PNAS (2005)

41 A recent example: HIV therapy (activator + noise enhancer) 0 Φ e 0 Φ = Φ~ 0 off 0 on off Activator: increasing on, lower the barrier Noise enhancer: Decreasing both on and off, further lower the barrier Weinberger group, Science (204) Significantly increasing the transition rate

42 Compared to previous rate formulae for bursty dynamics Eldar, A. and Elowitz, M. Nature (200) Cai, et al. Science (2006) Burst size b f γx Max γ off f If ( x) dφ 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.,PRL (205)

43 If assuming the extremely rapid gene-state switching Reduced CME d Φ dx ( x) g( x) = n max ln γx ( ) Φ ( x) x e ss p Ge, H., Qian, H. and Xie, X.S., PRL (205)

44 Barrier-crossing rate formula e Φ ǂ A is more stable B is more stable Possible reverse of relative stability Ge, H., Qian, H. and Xie, X.S., PRL (205)

45 Even local fluctuations can be different ~ Φ p i '' * * Φ ( x ) Φ ( x) e e i ( x )( x x ) i / 2 = ~ Φ , i, x x x x Ge, H., Qian, H. and Xie, X.S., PRL (205)

46 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 * = v( x) w( x) x = [E ] dt Φ CME Landscape functions associated with rate formulae w( x) w dx Φ = 2 v( x) w + = ln DA ( x) v( x) ( x) v( 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)

47 Landscape perspective for cellular dynamics Uninduced E. coli Induced Pluripotent State Waddington, The Strategy of Genes, 957 Fibroblasts Blood Pancreas Liver Differentiated State

48 Any essential difference between these two levels? Physics e β Chemistry(single reaction) Given a priori E b Equilibrium A B Φ e ǂ Emergent from dynamics Phenotype 2 Nonequilibrium Phenotype Chemistry (reaction system) Biology

49 Multistability and different time scales Conformation(Phenotype) Conformation(Phenotype) Conformation(Phenotype) 2

50 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 ω V ij ω V ji, ω >>, ω >> Driving force: µ ω ij = log ω ij ji V ji V ij log ω p p ss j ss i µ µ 2 + µ > 0

51 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).

52 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 propose a rate formula between phenotypic states at an intermediate region.

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

54 Thans for your attention!