Projection Scheme Wei WANG, Siqin CAO, Fu Kit SHEONG and Xuhui - - PowerPoint PPT Presentation

Extend the applicability of Markov State Model by the

Projection Scheme

Wei WANG, Siqin CAO, Fu Kit SHEONG and Xuhui HUANG* Department of Chemistry The Hong Kong University of Science and Technology

Motivation: why Markov state model

Motivation: why Markov state model

Conformations Microstates Macrostates

p(nτ) = [T(τ)]n p(0)

Motivation: why extend MSM

Xuhui Huang … Vijay S. Pande, Pacific Symposium on Biocomputing 15:228-239(2010)

Motivation: why extend MSM

• MSM is “Markov” State Model
• The Markovian lagtime is required to build a

Markov model

Daniel A. Silvaa, Dahlia R. Weissb, Fátima Pardo A., Lin-Tai Da, Michael Levitt, Dong Wang, PNAS (2014), 111, 7665-7670

: 1.

• 2. unique

3. 4.

The projection scheme

{T|PT} ATv = V PT vPT = D−1

N ADnV

P = D−1

N ADnAT

Theory 1: evaluate the macrostate model

• A good macrostate model should:
• 1. not mix microstates
• 2. be able to recover the correct kinetics

Yab = φPK(a)Tv0(b) p φPK(a)TvPK(a) p φ0(b)Tv0(b)

Theory 1: evaluate the macrostate model

Theory 1: evaluate the macrostate model

Theory 1: evaluate the macrostate model

Theory 2: recover the long time kinetics

M0 = Vdiag{λM}V−1 Mg = Vdiag{λT}V−1 T0 Lumping Backward Projection

Theory 2: recover the long time kinetics

• The kinetics recovery requires eigenvector matching
• The kinetics recovery can predict long time kinetics

Theory 2: recover the long time kinetics

• Recovering scheme 1: general case

ATp(mτM) = P(mτM) = ⇒ Mg = Vdiag{λT}V−1

Theory 2: recover the long time kinetics

Chapman-Kolmogorov test recovered Macrostate transitions Mg

Theory 2: recover the long time kinetics

• Recovering scheme 2: a time dilation

α = ln λT ln λM = const. ⇐ ⇒ P(αt) = ATP(t)

Dyson US, https://www.youtube.com/watch?v=j8Gi4bMtNNw

Theory 2: recover the long time kinetics

• Recovering scheme 2: a time dilation

α = ln λT ln λM = const. ⇐ ⇒ P(αt) = ATP(t)

Theory 2: recover the long time kinetics

Chapman-Kolmogorov test

• riginal Macrostate transitions M0

Theory 2: recover the long time kinetics

Chapman-Kolmogorov test recovered Macrostate transitions Mg

Conclusions

• A lumping evaluation:
• By comparing eigenvectors of microstate and

backward projection of macrostate

• A kinetics recovery:
• By matching the long time kinetics
• And non-Markovian macrostate transition can be used

to build the Markov State Model Yab = φPK(a)Tv0(b) p φPK(a)TvPK(a) p φ0(b)Tv0(b) Mg = Vdiag{λT}V−1

Acknowledgement

• Prof. HUMMER, Gerhard
• Prof. YAO, Yuan
• Dr. Peter, CHEUNG

Funding: Hong Kong Research Grant Council National Science Foundation of China

• Mr. JIANG, Hanlun
• Dr. PARDO, Fatima
• Mr. LIU, Song

Thank you! and Happy New Year

Appendix: why implied time scale changes

VS. B A Barrier hinders transition from A to B B A Large space hinders transition from A to B

Appendix: eigenvector is eigenmode

δp(t − 0) = Tp(0) − p(0) δp(2t − 0) = (T + 1)δp(0) δp(nt − 0) = (1 + T + T 2 + ... + T n)δp(0)

• 1. The transition is at eigenvectors:
• 2. The transition is not at eigenvectors:

∴ Transition mode maintains if it is eigenvector

δp(0) ∝ v(1) δp(nt − 0) = 1 − λ(1)n 1 − λ(1) δp(0) ∝ δp(0) δp(nt 0) = X

i

ai 1 λ(i)n 1 λ(i) v(i)6 /δp(0) δp(0) = X

i

aiv(i)

via Mathematical induction

Appendix: the variational principle

• The recovery of the kinetics is consistent with the

variational principle at long lagtime: Φ(a)TM0(mτ)V(a) ≤ Φ(a)TMg(mτ)V(a) @* X

1≤i≤n

y2

aiλT(i)m ≤ λT(a)m

1 A

Chapman-Kolmogorov test

Theory 2: recover the long time kinetics

Macrostate transitions at τ=4x80