Stochastic Transcription Elongation via Rule Based Modelling

Transcription

1 Stochastic Transcription Elongation via Rule Based Modelling Masahiro HAMANO SASB, Saint Malo

2 Purpose of This Talk 2

3 mechano-chemical TE as Rule Based Modelling 3

4 Table of Contents 4

5 Table of Contents 5

6 Transcription Elongation (TE) direction of transcription unzipped DNA nascent RNA 5 RNA exit channel 3 active site PPi NTP release entry secondary channel 6

7 Translocations of TEC ( pre post back forward 7

8 Brownian Ratchet Mechanism of Elongation TEC (n,fwdtrack) TEC (n,pre) TEC (n,backtrack) k 1 k 1 NTP TEC k 2 (n,post) k 2 TEC (n,post) NTP PPi TEC (n+1,pre) TEC (n,post) NTP k 3 TEC k (n+1,pre) PPi k 4 3 k 4 PPi TEC (n+1,pre) Sequence-dependent kinetic model for transcription elongation by RNA polymerase, J Mol Biol. (2004) 8

9 Table of Contents 9

10 Stochastic Process Calculi and their Markov Semantics synchronization blocking (inhibition) composable but not enough descriptrive! splitting of customers 10

11 Stochastic Process Calculi Provide Discrete/Stochastic Modelling in Biology 11

12 Single-nucleotide-level description 12

13 action of RNA polymerase. scent RNA 5 RNA exit channel Nucleotide and TEC s Muti-Partite Active Sites direction of transcription unzipped DNA Fig. 1. Structure of TEC active site 3 PPi NTP release entry secondary channel alized by Transcription Elongation Complex ith the template DNA nucleotide and the nascent N RNA e is mechano-chemical cycle of NTP binding: C to pre-translocation and TEC must move returning to post-trans location. Stochastic seen in transcription regulation, are due to longation, and moreover that in the elongais often disrupted by transcriptional N pauses, location of RNA l along DNA. r N(l,r) W W m n riment (such as magnetic and optical tweezoretical biochemical models have been prolongation in terms of individual biochemical ur [28,2,11,22,4,19]. Such theoretical mod- P site A site PPi how RNA couples chemical catalysis energy The starting of this present paper is that tures a fundamental notion explaining the tability of TEC along various stages of the work is able to represent biochemical internd is able to capture the stability of multiple interaction during mechano-chemical moveur rule based description, systems of master 2 window frame W for TEC m-p act sts as Agents W with counters (n,m) (n, m) P site A site PPi post-translocation: W [N(r), ] schematically N OH pre-translocation: W [ N(r 1 ),N(l 1,r) ] schematically N N OH loading: W [N(r), NTP] schematically N OH NTP OH in the subscript of N designates the 3 OH for the next nu m<0 m=0 m> 0 n 3 5 }{{} W 13

14 The ratchet pathway as Rules I. Elong Z N W 3 n [, ] W 2 n [, ] W n+1 [ N(r 1 ),N(l 1,r) ] W n [ N(r 1 ),N(l 1,r) ] W n [N(r), ] W n+1 [N(r), NTP] W PPi n+1 [ N(r 1 ),N(l 1,r) ] W 1 n W 2 n [ N(r 1 ),N(l 1,r 1 ) ] [ N(r 1 ),N(l 1,r 1 ) ]. In these rules, the left downward-successive rules correspond to the back-track pathway of (1). The middle upward-successive rules correspond to the forward-track pathway of (1). The other rules are for the main pathway (that is the horizontal one) of (1). 14

15 The ratchet pathway as Rules II: To coaser/coarsets grained rules Elong Z N Elong N Elong) 15

16 est grained rules Elong Z N. The ratchet pathway as Rules III ules of Elong Z N ).... Wn 3 [, ] W 2 n [, ] Elong N W n [, ] W n [N(r), )] W n+1 [ N(r 1 ),N(l 1,r) ] W 3 n [, ] W n [ N(r 1 ),N(l 1,r) ] W n [N(r), ] W n+1 [N(r), NTP] W PPi n+1 [ N(r 1 ),N(l 1,r) ] W n [ N(r 1 ),N(l 1,r) ] [ W n N(r 1 ),N(l 1,r 1 ) ] W 1 n W 2 n [ N(r 1 ),N(l 1,r 1 ) ] [ N(r 1 ),N(l 1,r 1 ) ]. In these W rules, 2 the [, left ] downward-successive rules correspond n to the back-track pathway of (1). The middle upward-successive rules correspond to the forward-track pathway of (1). The other rules are for the main pathway (that is the horizontal one) of (1). W n [ N(r 1 ),N(l 1,r) ] W n [N(r), ] W n+ 16

17 [ ] The ratchet pathway as Rules IV: Finally, coarsest grained rules Elong) W [, ] W [N(r), ] W [ N(r 1 ),N(l 1,r) ] W [N(r), NTP] W [ N(r 1 ),N(l 1,r 1 ) ] W PPi [ N(r 1 ),N(l 1,r) ] 17

18 [ ] [ The ratchet pathway as Rules V: Conversely, more complicated pathway by augmenting sites on W TEC (n,pre) NTP NTP k i1 k i1 TEC (n,pre) k i2 TEC (n,post) NTP NTP TEC k (n,post) i2 Kinetic modeling of transcription elongation, In RNA polymerases as molecular motors in the Royal Society of Chemistry (2009) (Agent) (Rules ) W = P site A site E site W n [ N(r 1 ),N(l 1,r), NTP ] W n [N(r), NTP, ] [ W n N(r 1 ),N(l 1,r), ] W n [N(r),, ] 18

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20 Table of Contents 20

21 Markov Process and Stationary Distribution pt(i,j) Kolmogorov fwd equation) p t = p t Q stationary distribution π p t = π for all t detailed balance, πq =0 π(j) k q(j, k) = k π(k) q(k, j) π(j) q(j, k) =π(k) q(k, j) 21

22 Wegscheider Condition Is Chemist's Algorithm to Check Reversibility ϕ (i,j) φ q(i, j) log q(j, i) =0 (i,j) φ q(i, j) q(j, i) =1 ϕ 22

23 Chemical Master Equation (CME) : Kolmogorov Eq Arisen from Chemical Reactions CME d dt p t(x) = k {λ k (x (ν k ν k )) p t (x (ν k ν k )) λ k (x) p t (x)} Though analytically unsolvable CME Behaviour of average yields a continuous model < x i (t) >= x x i p t (x) reaction rate equation d dt < x i(t) >= = k (ν ik ν ik ) < λ k (x(t)) > (ν ik ν ik )λ k (x(t)) k 23

24 ngation. State Transition for cally described am of Fig 3 2, W n β 3 α 3 W n+1 bability is idenant of the cor- Wn β α W 1 e pathways (1) α n+1 2 Wn 0 β 4 b = k α 4 W b n+1 0 1, α 1 = β 1 k 3, and β 2 = a Wn the vertical ard unloading of 1 rows represent dt w[0] merization. The arrows from front to deep represent ack reversely. We assume d that depolymerization occurs n = owing to the biological stability of TEC. d Fig. ( 3. local state transition n = aw n [ 1] + α 4 w [0] n 1 + β for 4w [0] Elong Z ) n+1 + α N 2w n [ ] + bw n [1] d dt w[ ] d dt w[1] n = dt w[ (n 1)] (α 4 + β 2 + a + b + β 4 )w n [0] ( ) ( α 3 w [ ] n 1 + β 3w [ ] n+1 + β 2w n [0] ) + α 1 w [1] n 1 ( ( β 1 w [ ] ) n+1 + aw[0] n (α 3 + β 3 + α 2 + β 1 )w n [ ] ) (b + α 1 )w [1] aw [ (n 1)] n n = bw n [ (n 2)] d ( ) dt w[ j] n = aw n [ (j+1)] + bw n [ (j 1)] Wn+1 1 struction, the forward-track pathway is omitted in the rest of the paper her than Elong Z N )wherez is the negative integers. 7 (Rules of Elong Z N ) n [ ] W n N(r 1 ),N(l 1,r) W (a + b)w n [ j] for 1 j<n 1 n [N N W 3 n W 2 n 24

25 W n State Transition for... W 1 n W n β 2 α 2 W 0 n b β 3 α 3... W 1 W n+1 n+1 α 1 Elong N W n+1 β 4 α 4 W 0 β n+1 1 a Wn 1 Wn+1 1 w [ ] n = 1 j n 1 w [ j] n and d dt w[ ] n = 1 j n 1 Z d dt w[ j] n = bw n [0] aw [ 1] n 25

26 Finally, State Transition for Elong) w [ ] b a w [ ] β 2 α 1 α 2 β 1 w [0] b a w [1] d dt w[0] = aw [ 1] + α 2 w [ ] + bw [1] (β 2 + a + b)w [0] d dt w[ ] = β 2 w [0] + α 1 w [1] (α 2 + β 1 )w [ ] d dt w[1] = β 1 w [ ] + aw [0] (b + α 1 )w [1] 26

27 Equilibrium Distribution of w [ ] b a w [ ] β 2 α 1 α 2 β 1 w [0] (a/b)(α 1 /β 1 )(α 2 /β 2 )=1 b a Elong) w [1] w [0] = bβ 1 τ w [1] = aβ 1 τ w [ ] = aα 1 τ τ =1/(bβ 1 + aβ 1 + aα 1 ) 27

28 Quasi-steady state approximation Elong Z N Elong N Elong) 28

29 Elong Z N Elong N Quasi-Steady State approximation { } w [j] n (t) w [j] w n (t) where w n = j {1,0,, } w [j] n d dt w n(t) = θ + w n 1 (θ + + θ )w n + θ w n+1 in which θ + = α 4 bβ 1 + α 3 aα 1 + α 1 aβ 1 and θ = β 4 bβ 1 + β 3 aα 1 + β 1 aα 1 29

30 Table of Contents 30

31 Boltzmann distribution w [j] n = π(w j n)= 1 Z n exp ( E(W j n) k B T ) with Z n = j exp ( E(W j n) k B T ) 31

32 K eq = a The main reaction pathway is the following b =exp(e(w n) E(W part k B T This means that the equilibrium con 0, and Q is given by the equilibrium constant K eq,thuswehave eq. That is TEC (n,fwdtrack) E(W i 1 ition rates atec and (n,backtrack) b at the equilibrium via Boltzmann distribution. tion pathway as Michaelis-Menten kinetics:) ntal pathway of (1) follows the kinetic W n βlaw 3 ofαmichaelis-menten 3 W (as the last two reactions) in the presence of a competitive n+1 inrst reaction) [11,2]. In our... β α 1 2 Wagent 1 based... framework, W 1 the enzyme n difference of reaction. Note that (1 xp( E(W n) 0 E(Wn) 1 n ) E(W )=exp( n) i ) (15) k B T k k B TNTP PPi 1 TEC k 2 (n,pre) TEC he equilibrium constant Kk eq (n,post) TEC 1 is solely determined k (n,post) NTP 2 by the free energy ion. Note that (15) is equivalent to the detailed balance condition between the transition rates a and rapid n+1 convergence of the stationar β 4 α 4 W n+1 0 β 1 directly derivable only under the probabilistic α 2 assumption of the e of the stationary distribution without Wn 0 employing the two tradis of the rapid equilibrium (or more generally b a Wn 1 quasi-steady-state) W f the mass balance for the total concentration [21,12]. n+1 1 d p Z 0 be respectively the probabilities w n [1], w [ ] n+1 and w[0] n. Then atrix Q for Kolmogorov backward ) equation p (t) = Qp(t) for iven by Q =. Direct calculation Qp(t) = 0 TEC (n+1,pre) (The main reaction pathway a The whole horizontal pathway of enzyme kinetics (as the last two hibitor (as the first reaction) [11,2 kinetics is more directly derivable tional assumptions of the rapid eq assumption nor of the mass balanc Let p 1, p and p 0 be respectiv ( (q 1 +q 01 ) q 1 q 10 q 1 q 1 0 q 01 0 q 10 the transition matrix Q for Kolm p =(p i ) i=1,,0 is given p i W by Q i = n onary distribution in which p (with p 1 and p 0 ) is given by q q q ( ( leads to the stationary distribution 32

33 The main reaction pathway as Michaelis-Menten kinetics ( ) M-M kinetics is derivable from the stationary distribution of Kolmogorov p (t) = Qp(t) q 1 = k 2 and q 1 = k 2 [NTP] p = 1 q 1 (1+ q 01 )+1 q 1 q 10 Since the probabi = [NTP] k 2 k 2 (1 + q 01 q 10 )+[NTP] v/v max = p, Michaelis-Menten v = v max [NTP] k 2 k 2 (1 + q 01 q 10 )+[NTP] q 33

34 More generally combining M-M and Boltzmann ( ) Steady-state elongation rate v = v max[ntp] K n +[NTP] with K n = k 2 k 2 log j w [j] n w [0] n j exp ((E(W 0 n) E(W j n))/k B T ) 34

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36 p(x,t) j(x, t) = D p x + F γ p(x, t) diffusion (random motion) where diffusion coef. D and external force F (x, t) applied to RNAp with γ drag coef. p t = j x = D 2 p x + 2 x ( U x p) (Note: F = U x drift (directed motion) with energy U) 36

37 0=D dp dx + 1 du γ dx p p(x) exp ( U(x) Dγ ) U(x) 37