Stochastic Processes around Central Dogma

Transcription

1 Stochastic Processes around Central Dogma Hao Ge Beijing International Center for Mathematical Research Biodynamic Optical Imaging Center Peking University, China

2 BICMR: Beijing International Center for Mathematical Research

3 BIOPIC: Biodynamic Optical Imaging Center

4 Introduction

5 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

6 Nobel prize in Chemistry 03: Models for chemical reactions QM: Quantum mechanics MM: Molecular mechanics (Newtonian mechanics) Langevin dynamics: one way to add temperature

7 Example: Perrin s experiment (D diffusion) Single particle trajectories Multiple particle (final positions) r What s the probability distribution function here? <r >= 4Dt J. Perrin (870-94) Nobel Prize in 96 Ann. Chim. Phys. VIII 8, 5-4 (909) Annalen der Physik 7, A.Einstein ( ) Nobel Prize in 9

8 Example: Single-molecule experiments Single Ion channel Single-molecule enzyme kinetics E. Neher and B. Sakman Nobel Prize in 99 Lu, et al. Science (998)

9 Example3: Single-cell dynamics (in vivo) Create a strain of E. coli with two identical copies of a promoter to which either the Green Fluorescent Protein or the Red Fluorescent Protein is attached. An experiment that shows that gene expression is noisy

10 Noisy or not?

11 Single-cell dynamics (in vivo) Eldar, A. and Elowitz, M. Nature (00) Choi, et al. Science (008)

12 Simplest model of central dogma and its applications

13 Central Dogma Copy numbers in a single cell Bacteria Eukaryotic cells DNA or ~ mrna A few Protein

14 Simplest Chemical-masterequation model for central dogma DNA k mrna k Protein g g Var k mrna g mrna Cov Var k g Var Protein k k g k k g Protein g g g g mrna, Protein g k mrnavarprotein g g k g g k

15 Copy numbers of mrna and protein in single cells is uncorrelated Elongation factor tufa Histogram of correlation coefficients for 80 genes g k g g k g g g Correlation coefficient ρ = 0.0 ± 0.03 Taniguchi et al. Science 39, 533 (00)

16 Translational burst Stochastic production of protein Cai, et al. Science 3, 600 (006) The duration time of the burst (the life time of a single mrna) is quite small compared to the separating time between adjacent bursts. g

17 Occurances Burst size Occurrence An average of 4. mol Exponential distributed B Poisson distribution (process) G( n) 0 0 g e Number of Molecules per Burst g Cai et al. Nature, 440, (006) Yu et al., Science, 3, 600 (006) n t kt kt n e dt q ( ) n! q Number of Expression Events in One Cell-cycle k q k g mean q k b q g

18 Simplified Chemical master equation(cme) dp( n, t) dt n k G( j) P( n j, t) qp( n, t) d( n ) P( n, t) np( n, t). j d d d G () G () G(3) P ss ( n) n b ( b) an ( a n) ( a) n! Negative Binomial distribution a k d, b k d Burst frequency and burst size

19 # of proteins Occurrence Burst frequency and size derived from single cell tracking Gamma distribution p x a x b a e x / b a a: burst frequency; b: burst size x, Protein copy number per cell Time Paulsson & Ehrenberg, PRL 84, 5447 (000) Cai et al. Nature, 440, (006); Friedman et al. PRL 97, 6830 (006)

20 95% of proteins exhibit Gamma distributions a= 0.46 b =.3 All are unimodal (single peak) a = 5.9 b =.3 a = b =6 a = 4.4 b =03 Taniguchi et al Science 39, 533 (00)

21 Bimodal distributions in biology: phenotypic states Ferrell, J. and Machleder, E. Science (998) Choi, et al., Science (008) To, T. and Maheshri, N. Nature (00)

22 Stochastic models incorporating different gene states

23 Regulation of gene expression

24 Transcriptional burst under induced condition Golding et al. Cell (005)

25 DNA topology and transcriptional burst Levens and Larson: Cell (04) (preview) Shasha et al. Cell (04)

26 Supercoiling accumulation and gyrase activity Shasha et al. Cell (04)

27 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+ g g ( n )g g g Copy number of mrna k ( n )g g n+

28 Poisson distribution with a spike at zero When α,β<<k,γ, then.,! ) ( ) ( ) ( ; (0) (0) (0) n n k e n p n p n p e p p p n k k g g g Poisson distribution with a spike (bimodal)

29 Transcriptional burst under induced condition Shasha et al. Cell (04)

30 Two-state model with positive feedback k Max large g

31 Mean-field deterministic model with positive feedback gx x g dx dt g( x) gx x n Max flux Stable Off-state Unstable threshold Stable On-state Sigmoidal influx 0.9 Bifurcation diagram Bifurcation diagram for simple example ON state 0.6 x * 0.5 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, 3, 44 Gupta, et al., Cell, 46, 633 (0)

33 Recall Langevin dynamics and Kramers rate formula Chemical reaction activated by diffusional fluctuations k ǂ g Uǂ a kbt is large e g m P. Langevin (87-946) d x dt m f f f du dx x dx dt t 0; t k BT ; t f s 0,s t. f t U U x m x x ǂ x m x xǂ,x around x. ǂ S ǂ k B a ln ǂ g a ah k T B,x around x A ; H.A. Kramers (894-95)

34 Rate formula linking physics and chemistry Diffusion in the space of reaction coordinates Barrier crossing Jumping process between conformational states of chemical species Ex. A B A k+ B Optical trap Coarse graining McCann et al. Nature, 40, 785 (999)

35 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 j M r ( X ) P( X, t) The trajectory of CME is the analog of the dynamics along the reaction coordinates. What is the analog of coarse-grained conformational states in biology? j Max Delbruck(906-98) Nobel Prize in 969

36 Bimodal distributions in biology: phenotypic states They are all at nonequilibrium Ferrell, J. and Machleder, E. Science (998) steady states! Choi, et al., Science (008) To, T. and Maheshri, N. Nature (00)

37 There are two major sources of fluctuations: another level of complexity in gene regulation? Two-state model with positive feedback k Max large g 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 * When stochastic gene-state switching dominates k, f, h g Continuous Mean-field limit Bifurcation diagram Bifurcation diagram for simple example OFF state dx dt g( x) Rescaled dynamics ON state gx (B) dx dt dx dt k f, h, g k Max hx k Max f k Max large g Fluctuating-rate model gx x gx n Max /K eq Ge, H., Qian, H. and Xie, X.S., arxiv (03)

39 Nonequilibrium landscape function emerges dx dt g( x) gx Dynamics in the mean field limit model Landscape function x ss 0 p0 e x ~ Stable (OFF) 0.73 Unstable Stable (ON) d0 dx x f hx k k gx gx Max Max 0.75 analog to energy function at equilibrium case x x Ge, H., Qian, H. and Xie, X.S., arxiv (03)

40 Rate formulae Gene-state switching is extremely slow Gene-state switching is relatively slow Barrier crossing Transition rates linearly depend on gene-state switching rates Log mean transition time Barrier Max x 0 4 Max= Max=7500 Max=0000 Max=5000 Fluctuating-rate model f k k e 0 Ge, H., Qian, H. and Xie, X.S., arxiv (03) ǂ 0

41 The dynamics is bursty Eldar, A. and Elowitz, M. Nature (00) Cai, et al. Science (006) Burst size b f k gx Max g off f If x d 0, x dx b x ǂ k k 0 e xǂ x b off Walczak,et al.,pnas (005);Choi, et al.,jmb(00);ge,h.,qian,h.and Xie, X.S.,arxiv (03)

42 Ge, H., Qian, H. and Xie, X.S., arxiv (03) When protein copy-number fluctuation dominates Reduced CME x d g Max ln dx g x x x ss p e x

43 Ge, H., Qian, H. and Xie, X.S., arxiv (03) Barrier-crossing rate formula k k e ǂ A is more stable B is more stable Possible reverse of relative stability

44 Warren, P.B. and ten Wolde, P.R., PRL (004); Consistent with previous simulation results mutually repressing operons Mehta, et al., Phys. Biol. (008)

45 Ge, H., Qian, H. and Xie, X.S., arxiv (03) Even local fluctuations are different in the two limiting cases ~ p i '' * * x e i x e i x x x ~ /, i, x x x x

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

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

48 Some general property of the landscape function

49 Emergent landscape in the bistable case Unstable fixed point of deterministic models Maximum: the barrier x Local minimum Mean-field dynamics always goes downhill! p ss x e x Global minimum Stable fixed points of deterministic models

50 Maxwell construction Steady States x* Global minimum abruptly transferred. f 0 (x,q ) q * q x Ge and Qian: PRL (009), JRSI (0)

51 Dynamics of bistable systems alternative attractor Inter-attractorial dynamics 3: abrupt transition via barrier-crossing : relaxation process Intra-attractorial dynamics : fluctuating in local attractor, waiting The uphill dynamics is the rare event, related to phenotype switching, punctuated transition in evolution, et al.

52 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 and Qian: PRL (009), JRSI (0)

53 Multistability and different time scales Conformation(Phenotype) 3 k 3 k -3 k - k k - Conformation(Phenotype) k Conformation(Phenotype)

54 local landscapes and Kramers rate formula V ij i k ij k ji j V ji k ij Constructed locally T ij k ji T ji Phenotypic subspace k k ij ji k k 0 ij 0 ji e e V ij V ji,, Driving force: ij k log k ij ji V ji V ij log p p ss j ss i 3 3 0

55 Multistability: local-global conflictions Ge, H. and Qian, H.: Chaos (0) 3 k - k The emergent Markovian jumping process being nonequilibrium is equivalent to the discontinuity of the local landscapes (time symmetry breaking). Global landscape: from stationary distribution Just cut and glue on the local landscapes (having non-derivative points).

56 Summary Stochastic process has a glorious history in physics, chemistry and biology, and due to the advance of singlemolecule techniques, stochastic modeling and computation will become more and more useful and popular. Two simplified models are proposed. Rate formulae associated with different landscape functions are proposed, which emphasizes noises with multiple origins having distinct characteristics, suggesting further complexity within gene regulation. Nonequilibrium is essential to cell biology.

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

58 Thanks for your attention!