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 p ( n τ ) = [ T ( τ )] n p (0) Conformations Microstates Macrostates

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

The projection scheme : { T | P T } 1. A T v = V 2. unique P T v P T = D − 1 3. N AD n V 4. P = D − 1 N AD n A T

Theory 1: evaluate the macrostate model • A good macrostate model should: • 1. not mix microstates • 2. be able to recover the correct kinetics φ P K ( a ) T v 0 ( b ) Y ab = p p φ P K ( a ) T v P K ( a ) φ 0 ( b ) T v 0 ( 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 M g = V diag { λ T } V − 1 T 0 Lumping Backward Projection M 0 = V diag { λ M } V − 1

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 A T p ( m τ M ) = P ( m τ M ) = ⇒ M g = V diag { λ T } V − 1

Theory 2: recover the long time kinetics recovered Macrostate transitions M g Chapman-Kolmogorov test

Theory 2: recover the long time kinetics • Recovering scheme 2: a time dilation α = ln λ T ⇒ P ( α t ) = A T P ( t ) = const . ⇐ ln λ M Dyson US, https://www.youtube.com/watch?v=j8Gi4bMtNNw

Theory 2: recover the long time kinetics • Recovering scheme 2: a time dilation α = ln λ T ⇒ P ( α t ) = A T P ( t ) = const . ⇐ ln λ M

Theory 2: recover the long time kinetics original Macrostate transitions M 0 Chapman-Kolmogorov test

Theory 2: recover the long time kinetics recovered Macrostate transitions M g Chapman-Kolmogorov test

Conclusions • A lumping evaluation: • By comparing eigenvectors of microstate and backward projection of macrostate φ P K ( a ) T v 0 ( b ) Y ab = p p φ P K ( a ) T v P K ( a ) φ 0 ( b ) T v 0 ( b ) • A kinetics recovery: • By matching the long time kinetics M g = V diag { λ T } V − 1 • And non-Markovian macrostate transition can be used to build the Markov State Model

Acknowledgement Prof. HUMMER, Gerhard Mr. JIANG, Hanlun Prof. YAO, Yuan Dr. PARDO, Fatima Dr. Peter, CHEUNG Mr. LIU, Song Funding : Hong Kong Research Grant Council National Science Foundation of China

Thank you! and Happy New Year

Appendix: why implied time scale changes A B Barrier hinders transition from A to B VS. A B Large space hinders transition from A to B

Appendix: eigenvector is eigenmode δ p ( t − 0) = Tp (0) − p (0) δ p (2 t − 0) = ( T + 1) δ p (0) via Mathematical induction δ p ( nt − 0) = (1 + T + T 2 + ... + T n ) δ p (0) 1. The transition is at eigenvectors: δ p (0) ∝ v (1) δ p ( nt − 0) = 1 − λ (1) n 1 − λ (1) δ p (0) ∝ δ p (0) 2. The transition is not at eigenvectors: X δ p (0) = a i v ( i ) i 1 � λ ( i ) n X δ p ( nt � 0) = 1 � λ ( i ) v ( i ) 6 / δ p (0) a i i ∴ Transition mode maintains if it is eigenvector

Appendix: the variational principle • The recovery of the kinetics is consistent with the variational principle at long lagtime: Φ ( a ) T M 0 ( m τ ) V ( a ) ≤ Φ ( a ) T M g ( m τ ) V ( a ) 0 1 ai λ T ( i ) m ≤ λ T ( a ) m X y 2 @ * A 1 ≤ i ≤ n

Theory 2: recover the long time kinetics Macrostate transitions at τ =4x80 Chapman-Kolmogorov test