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Rate constants: 
from Sinai billiards 
to protein folding 
Boris Fačkovec 
Wales group 
Department of Chemistry 
University of Cambridge 
Alumni festival 
26th September 2014 
Discretised relaxation path sampling (DRPS) 1/4 bf269@cam.ac.uk
Rare events 
• large separation of smallest and largest time scales 
kAB / exp(−E) 
Discretised relaxation path sampling (DRPS) 2 
/4 bf269@cam.ac.uk 
molecular vibration 
(~1 fs) 
folding 
(~1 ms) 
A 
B 
J 
I 
H 
G 
E 
F 
C 
D 
A C 
B D 
J H E 
I G F
Transition state theory 
eq.flux 
eq.population 
• cell-to-cell rate 
constant in DRPS is 
based on relaxation: 
position 
kTST 
AB = 
potential 
probability 
density 
position 
relaxation 
Discretised relaxation path sampling (DRPS) 3/4 bf269@cam.ac.uk
From Sinai billiards to folding 
30 
20 
10372 J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 M. Muthukumar 
Native 
10 
polymer reversal polymer translocation peptide dynamics 
FIG. 1. Polymer escape in transition. 
Discretised relaxation path sampling (DRPS) 4/4 bf269@cam.ac.uk 
where the unnecessary constant terms are ignored. kB is the 
FIG. 3. Plot of Fm in units of kBT against f for ! 1!!0.69!! 2!. Curves 
represent: solid line, N$%/kBT!0; long-dashed line, N$%/kBT!1; 
dashed line, N$%/kBT!1. 
20 30 40 50 
Stationary Point 
0 10 20 30 40 0 
Stationary Point 
Potential Energy Difference (kcal/mol) 
B 
Native 
Unfolded 
biomolecular folding 
10 
20 
30 
Energy Difference (kcal/mol) 
A B 
Sinai billiards 
-11.40 -11.48 
-12.54 
-11.50 
-11.04 
4 
3 
2 
1 
-10.77 
Cluster rearrangement
QUESTIONS?

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Rate constants: from Sinai billiards to protein folding

  • 1. Rate constants: from Sinai billiards to protein folding Boris Fačkovec Wales group Department of Chemistry University of Cambridge Alumni festival 26th September 2014 Discretised relaxation path sampling (DRPS) 1/4 bf269@cam.ac.uk
  • 2. Rare events • large separation of smallest and largest time scales kAB / exp(−E) Discretised relaxation path sampling (DRPS) 2 /4 bf269@cam.ac.uk molecular vibration (~1 fs) folding (~1 ms) A B J I H G E F C D A C B D J H E I G F
  • 3. Transition state theory eq.flux eq.population • cell-to-cell rate constant in DRPS is based on relaxation: position kTST AB = potential probability density position relaxation Discretised relaxation path sampling (DRPS) 3/4 bf269@cam.ac.uk
  • 4. From Sinai billiards to folding 30 20 10372 J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 M. Muthukumar Native 10 polymer reversal polymer translocation peptide dynamics FIG. 1. Polymer escape in transition. Discretised relaxation path sampling (DRPS) 4/4 bf269@cam.ac.uk where the unnecessary constant terms are ignored. kB is the FIG. 3. Plot of Fm in units of kBT against f for ! 1!!0.69!! 2!. Curves represent: solid line, N$%/kBT!0; long-dashed line, N$%/kBT!1; dashed line, N$%/kBT!1. 20 30 40 50 Stationary Point 0 10 20 30 40 0 Stationary Point Potential Energy Difference (kcal/mol) B Native Unfolded biomolecular folding 10 20 30 Energy Difference (kcal/mol) A B Sinai billiards -11.40 -11.48 -12.54 -11.50 -11.04 4 3 2 1 -10.77 Cluster rearrangement