THE SLOW-SCALE SSA. Yang Cao, Dan Gillespie, Linda Petzold

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1 THE SLOW-SCALE SSA Yang Cao, Dan Gillepie, Linda Petzold With thank to: Muruhan Rathinam, Hana El-Samad, John Doyle and E. Haeltine & J. Rawling, J. Chem. Phy 117, 6959 (2002) Chri Rao & Adam Arkin, J. Chem. Phy 118, 4999 (2003) Support: AFOSR/DARPA, MSI/Sandia, DOE, NSF, ICB/ARO

2 MOTIVATION Cellular reaction oten take place on vatly dierent time cale. - Fat and low reaction. Fat and low pecie. - Fat reaction uually le important than low one. - Example: S c 1 S 2 1 c2 c (at) and S S (low). SSA will pend mot o it time imulating the at reaction. I there a way to kip over the at reaction, and directly imulate only the low one?

3 SOME ISSUES Do at and low apply to reaction or to pecie? How can the at/low part o the ytem be eparated? Are there at/low ubytem that obey Markovian mater equation? What doe theory ay a multi-cale SSA hould do?

4 ASSUMPTIONS A well-tirred ytem at contant volume and temperature. N pecie { S K S }. Sytem tate Xt () ( X(), t, X () t ) X () t i 1,, N number o S i molecule at time t. M reaction { R R } = K, 1, K, M. Propenity unction a1, K, am, a ( x) dt probability, given Xt () When 1 = x, that R will ire in next dt. R ire, the ytem tate change rom x to ( ) 1, K,, { i} ν ν ν N N x + ν, ν i the toichiometric matrix Some reaction/pecie are very much ater than the other.

5 - OUR PROPOSED APPROACH - 1 t Partition the reaction: { } ( R ) 1, K, RM = R, R Fat reaction { R R } 1,, RM K. Slow reaction { R R } 1,, RM K. Criterion: a ( x) a ( x), and or mot x. Thi criterion i looe, ubect to later reinement.

6 2 nd Partition the pecie: { } ( S ) 1, K, SN = S, S ( ) Xt () = X(), t X() t Fat pecie { } 1,, S S K SN { X () t X } 1 (), t, XN () t K. Slow pecie { } 1,, S S K SN { X () t X } 1(), t, XN () t K. Criterion: A pecie i at i it population get changed by at leat one at reaction; otherwie, the pecie i low. Some ubtle point: A low pecie cannot get changed by a at reaction, but a at pecie can get changed by a low reaction. a ( x) = a ( x, x ), a ( x) = a ( x, x ). A at pecie population need not be large.

7 3 rd Deine the virtual at proce X ˆ () t. X ˆ () t the at pecie population driven by only the at reaction. X ˆ () t i X () t with all the low reaction witched o. X () t i a phyically real but non-markovian proce. It doe not atiy an ordinary mater equation. X ˆ () t i a phyically ictitiou but Markovian proce. It doe atiy an ordinary mater equation: The ME or the at pecie, driven by only the at reaction, with all low pecie population held contant. The CME or X ˆ () t will be impler than the CME or Xt. ( ) ˆ ˆ Px (, tx, x, t) Pr X( t) = x Xt ( ) = ( x, x). It determine { }

8 4 th Now impoe two condition on X ˆ () t. Condition 1. ˆ () X t mut be table; i.e., ˆ ˆ lim Px (, tx, t) Px (, x) mut exit. t Condition 2. Xˆ () t Xˆ ( ) in a time that i very mall compared to the time between low reaction event. I change in the initial choice o at and low reaction are required to ecure thee condition, o be it. Thee condition hould be atiiable whenever tine i preent. Then the at reaction will be le important than the low one, o kipping over them will make ene. I thee condition cannot be atiied, the at reaction are not le important than the low one, and kipping over them i not a good idea.

9 5 th A new theorem. Theorem: With Xt () = ( x, x), let be a time increment that i large compared to the time or Xˆ () t Xˆ ( ), but mall compared to the time between low reaction. Then the probability that one in [, tt+ ) i approximately a ( x ; x ), where a ( x ; x ) Pˆ ( x, x, x ) a ( x, x ). x R will occur a ( x ; x ) i called the low-cale propenity unction or It the average o It replace a ( x, x ) w.r.t. X ˆ ( ). a ( x, x ) on the time cale o the low reaction. R. For at leat ome ytem, it can be computed. Knowing it and X ˆ ( ) will allow u to approximately imulate the evolution o the ytem one low reaction at a time.

10 Proo o the theorem: x, x X ˆ ( t ), x ( ) ( ) t t t + dt t+ { } ( ˆ ) + Prob R in [ t, t dt ) a X ( t ), x dt. time t+ { } ( ˆ ) + Prob R in [ t, t ) a X ( t ), x dt t = t 1 t+ ˆ ( ), t ( ) a X t x dt { ( temporal average o a ˆ ( ), )} X t x { ( enemble average o a ˆ ( ), )} X x ˆ = Px (, x, x) a ( x, x). x QED

11 Will need to compute a ( x ; x ) a ( Xˆ ( ), x ) : Never require more than the irt two moment o X ˆ ( ). Thoe two moment can oten be adequately approximated in term o the olution o the determinitic RRE. Will need to generate random ample o X ˆ ( ): Thi eldom can be done exactly. I we can etimate ˆ ( ) var Xˆ ( ), we can oten approximate ˆ ( ) There are way o etimating X X and { } a a normal random variable. X ˆ ( ) and var { Xˆ ( )} that don t require a ull computation o Px ˆ(, x, x). Thi i an important problem area, requiring urther work.

12 Initialize: Given 1. In tate 2. Compute The Slow-Scale SSA Xt ( ) = ( x, x), et t t0, ( x, x ) at time t, evaluate M 0 = 1 x x 0, x x 0. a ( x ; x ) or = 1, K, M. a ( x ; x ) = a ( x ; x ). With r 1, r 2 = U (0,1), take 1 1 τ = ln, a0 ( x ; x ) r =1 = mallet integer atiying a ( x ; x ) r a ( x ; x ) 3. Advance ytem to the next low reaction ( 4. Record R at time t + τ ): t t+ τ, xi xi + ν i ( i= 1, K, N), xi xi + ν i ( i= 1, K, N ), Then relax the at variable: x ample o Xˆ ( ). Xt () = ( x, x) i deired. Then return to 1, or ele top.

13 An Example: S S c1 1 S2 c 2 c3 2 S3 (at) (low) ˆ () ˆ X t + X () t = cont. X ˆ ( ) can be calculated exactly. - ˆ c X 1 ( ) i binomial: 2( x1+ x2) cc 1 2( x1+ x2) mean =, variance = 2 c1+ c2 ( c1+ c2) - Xˆ 2( ) = x ˆ 1+ x2 X1( ). ˆ cc( x + x) a3( x3; x1, x2) = c3 X2( ) =. c + c Relaxation time or Xˆ () t Xˆ ( ) i 1( c1+ c2). 1 c1+ c2 Condition or uing the SSA:. c + c c c ( x + x )

14 X 1 S 1 <--> S 2 (at), S 2 --> S 3 (low) (c 1 =1, c 2 =2, c 3 = ) - Exact SSA Run - - Point plotted ater each low reaction. - Over 40 million reaction imulated. - Fat/low reaction: 76,000 (avg) molecular population X X time

15 X 1 S 1 <--> S 2 (at), S 2 --> S 3 (low) (c 1 =1, c 2 =2, c 3 = ) - Slow-Scale SSA Run - - Only low reaction imulated. - Point plotted ater each reaction (low) reaction in all. molecular population X X time

16 2000 X S 1 <--> S 2 (at), S 2 --> S 3 (low) molecular population (c 1 =10, c 2 =40,000, c 3 =2) - Exact SSA Run - - X 1 and X 3 only. - Point plotted ater each low reaction. - Over 23 million reaction imulated. - Fat/low reaction: 39,000 (avg) X time

17 2000 X S 1 <--> S 2 (at), S 2 --> S 3 (low) molecular population (c 1 =10, c 2 =40,000, c 3 =2) - Slow-Scale SSA Run - - X 1 and X 3 only. - Only low reaction imulated. - Point plotted ater each reaction (low) reaction in all. X time

18 8 molecular population S 1 <--> S 2 (at), S 2 --> S 3 (low) (c 1 =10, c 2 =40,000, c 3 =2) - Exact SSA Run - X 2 only - plotted at equally paced time interval o time unit time

19 8 molecular population S 1 <--> S 2 (at), S 2 --> S 3 (low) (c 1 =10, c 2 =40,000, c 3 =2) - Slow-Scale SSA Run - X 2 only - plotted "relaxed" ater each imulated (low) reaction time

20 c 1 S1+ S1 S2 (at) c Another Example: 2 c3 c 4 S1, S2 S3 (low) The decay-dimerizing reaction, but in a very ti regime. ˆ () 2 ˆ X t + X () t = cont. But no imple ormula or X ˆ ( ). 1 2 Can approximate the mean and variance o ˆ ( ) X a 2 ˆ 1 c2 c2 X2 ( ) 2( x1+ 2 x2 ) + 2( x1+ 2 x2 ) + 4( x1+ 2 x2 ) 4 c1 c1 c { } 2 Xˆ 2 ( ) var Xˆ 2( ), 4 c Xˆ ( ) + c 2( x + 2 x ) c ( ) , Xˆ ( ) = ( x + 2 x ) 2 Xˆ ( ) Slow-cale propenitie: a3( x) = c ˆ ˆ 3 X1( ), a4( x) = c4 X2( ). Sample ˆ 2 ( ) ( ) X approximately a Xˆ { ˆ } 2( ),var X2( ) N.

21 X 2 R 1, R 2 : S 1 + S 1 <--> S 2 (at) R 3, R 4 : S 1 --> O, S 2 --> S 3 (low) ( c 1 =1, c 2 =200, c 3 =0.02, c 4 =0.004) - Exact SSA Run - molecular population X 1 - Point plotted ater each low reaction million reaction in all. - 1,742 low reaction. - Fat/low reaction: 14, X time

22 X 2 R 1, R 2 : S 1 + S 1 <--> S 2 (at) R 3, R 4 : S 1 --> O, S 2 --> S 3 (low) ( c 1 =1, c 2 =200, c 3 =0.02, c 4 =0.004) - Slow-Scale SSA Run - molecular population X 1 - Only low reaction imulated. - Point plotted ater each reaction. - 1,741 (low) reaction in all. 200 X time

23 SUMMARY Have produced a clear operational procedure or deining - at and low reaction, - at and low pecie, - a virtual at proce X ˆ () t (which i Markovian). Have elucidated the critical role o the two tine condition: - X ˆ () t mut be table; - Xˆ () t Xˆ ( ) very rapidly on the time cale o the low reaction. Have etablihed The Slow-Scale Propenity Function Theorem: I the time cale o the at and low reaction are well eparated, then on the time cale o the low reaction a ( x, x ) can be approximately replaced with a ( x ; x ) a ( Xˆ ( ), x ). Lead to the Slow-Scale SSA, which kip over all the at reaction.

24 RESULTS SO FAR Showed or two imple ytem that the SSA work well. - It greatly outperorm explicit tau-leaping on very ti ytem. Demontrated that neither the ize nor the importance o luctuation can be inerred rom molecular population level alone. Clariied the ource o the high-requency iltering eect oberved in the E. coli heat hock repone mechanim.

25 FUTURE EFFORTS Apply the SSA to the enzyme-ubtrate reaction, and explore the connection with Michaeli-Menten kinetic. Invetigate the relation between the SSA and implicit tau-leaping. Develop general method or computing the aymptotic propertie o common virtual at procee. See about urther accelerating the SSA by uing regular tau-leaping a a ollow-on procedure.