Stochastic Processes around Central Dogma

Transcription

1 Stochastic Processes around Central Dogma Hao Ge Beijing International Center for Mathematical Research 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 Single-molecule experiments Single Ion channel Single-molecule enzyme inetics E. Neher and B. Saman Nobel Prize in 99 Lu, et al. Science (998)

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

7 Marov processes Diffusion process (Langevin dynamics, stochastic differential equation): The motion of molecule in solution The motion of each atom or atom group within a single molecule (in solution) Master equation (jumping process) The jumping between different conformational states of a single molecule The copy-number fluctuations of chemical species

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

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

10 Simplest model for central dogma DNA mrna Protein γ γ mrna = γ Var ( mrna) = γ ρ = Cov Var Var Protein = γ γ ( Protein) = + γ γ γ + γ ( mrna, Protein) γ = ( mrna) Var( Protein) γ + γ + γ + γ

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

12 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. γ >>

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

14 Simplified Chemical master equation(cme) dp( n, t) n = G( j) P( n j, t) qp( n, t) + d dt j= d d d [( n + ) P( n +, t) np( n, )]. t G () G () G(3) P ss ( n) n b ( + b) = a+ n Γ( a + n) Γ( a) n! Negative Binomial distribution d a =, b = d Burst frequency and burst size

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

16 95% of proteins exhibit Gamma distributions a= 0.46 b =.3 All are unimodal (single pea) a = 5.9 b =.3 a = b =6 a = 4.4 b =03 Taniguchi et al Science 39, 533 (00)

17 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)

18 Two-state model without ON feedbac OFF β OFF ON mrna ø α 0 n 0 n When α,β<<,γ, then p( 0 ) = β e α + β γ γ α β + ; p( n ) = e α + β α + β γ γ n! n n+ γ γ ( n +)γ β α β α β α β α The mean-field deterministic model has only one stable fixed point! γ γ ( n +)γ n+,n. Poisson distribution with a spie (bimodal)

19 Transcriptional burst under induced condition Shasha et al. Cell (04) in press β/α Wild Type Gyrase Gyrase SGS Inhibition Overexpression Insertion Plasmid-borne System

20 Mean-field deterministic model with positive feedbac g( x) x γ dx dt = g( x) γx Sigmoidal influx flux Stable Off-state Unstable threshold Stable On-state 0.9 Bifurcation diagram ON state 0.6 x * 0.5 Influx g(n) Outflux γn OFF state Flux-balance plot n /K eq

21 Two-state model with positive feedbac A minimal model Max = large γ Chemical master equation

22 Interconversion of different phenotypic states How to quantify the transition rates between different phenotypic states, provided their existence? Choi, et al., Science, 3, 44 (008) Gupta, et al., Cell, 46, 633 (0)

23 Langevin dynamics and Kramers rate formula Chemical reaction activated by diffusional fluctuations + ǂ ω ω γ π Uǂ a BT ( η is large) = e γ = η m P. Langevin (87-946) d x dt m f f f ( x) du dx = η + dx dt ( t) = 0; ( t) = η BT ; ( t) f ( s) = 0,s t. f ( t) U U ( x) mω ( x x ) ǂ ( x) mω ( x xǂ ),x around x. ǂ S ǂ = B a ln ω ǂ κ = γ a ωah π T B,x around x A ; H.A. Kramers (894-95)

24 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, 40, 785 (999)

25 From single chemical reaction to biochemical networs (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. j Max Delbruc(906-98) Nobel Prize in 969 What is the analog of coarse-grained conformational states in biology?

26 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)

27 Two-state model with positive feedbac A minimal model Max = large γ The analytical results introduced Chemical master here equation can be applied to any self-regulating module of a single gene, while the methodology is valid for a much more general context. There are two major sources of fluctuations.

28 (A) x * When stochastic gene-state switching dominates, f, h >> γ Continuous Mean-field limit Bifurcation diagram OFF state dx dt = g( x) Rescaled dynamics ON state γx (B) dx dt dx dt >>, f h, γ Max = hx Max = f Max = large γ Fluctuating-rate model γx x = γx n Max /K eq Ge, H., Qian, H. and Xie, X.S., arxiv (03)

29 Nonequilibrium landscape function emerges dx dt = g( x) γx Dynamics in the mean field limit model ~ Φ dφ ( x) Landscape function 0 p 0 Φ0 ( x ) ( x) e 0 - Stable (OFF) Unstable Stable (ON) dx = f hx + γx γx Max Max x x Ge, H., Qian, H. and Xie, X.S., arxiv (03)

30 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 e 0 Φ Ge, H., Qian, H. and Xie, X.S., arxiv (03) ǂ 0

31 The dynamics is bursty Eldar, A. and Elowitz, M. Nature (00) Cai, et al. Science (006) 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 (005); Choi, et al., JMB (00); Ge, H., Qian, H. and Xie, X.S., arxiv3.6776(03)

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

33 Local fluctuations are different in the two limiting cases ~ Φ p i '' * * Φ ( x ) Φ ( x) e e i ( x )( x x ) i / = ~ Φ 4 3.5, i, x x.5.5 x x Ge, H., Qian, H. and Xie, X.S., arxiv (03)

34 Barrier-crossing rate formula e Φ ǂ A is more stable B is more stable Possible reverse of relative stability

35 Consistent with previous simulation results mutually repressing operons Mehta, et al., Phys. Biol. (008) Warren, P.B. and ten Wolde, P.R., PRL (004); JPCB (005)

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

37 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 Nonequilibrium Phenotype Chemistry (reaction system) Biology

38 Nonequilibrium is essential to cell biology. 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.

39 Acnowledgement Prof. Min Qian Peing University Prof. Hong Qian University of Washington Prof. Sunney Xiaoliang Xie Harvard University Peing University

40 Thans for your attention!

41 Multistability: nonequilibrium steady states at different time scales Conformation(Phenotype) Conformation(Phenotype) Conformation(Phenotype)

42 Multistability: 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 µ > 0

43 Multistability: local-global conflictions and nonequilibrium steady state Ge, H. and Qian, H.: Chaos (0) 3 - 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).