^{1,a)}, Godfrey S. Beddard

^{1}, Emanuele Paci

^{2}and David R. Glowacki

^{3,a)}

### Abstract

Molecular dynamics (MD) methods are increasingly widespread, but simulation of rare events in complex molecular systems remains a challenge. We recently introduced the boxed molecular dynamics (BXD) method, which accelerates rare events, and simultaneously provides both kinetic and thermodynamic information. We illustrate how the BXD method may be used to obtain high-resolution kinetic data from explicit MD simulations, spanning picoseconds to microseconds. The method is applied to investigate the loop formation dynamics and kinetics of cyclisation for a range of polypeptides, and recovers a power law dependence of the instantaneous rate coefficient over six orders of magnitude in time, in good agreement with experimental observations. Analysis of our BXD results shows that this power law behaviour arises when there is a broad and nearly uniform spectrum of reaction rate coefficients. For the systems investigated in this work, where the free energy surfaces have relatively small barriers, the kinetics is very sensitive to the initial conditions: strongly non-equilibrium conditions give rise to power law kinetics, while equilibrium initial conditions result in a rate coefficient with only a weak dependence on time. These results suggest that BXD may offer us a powerful and general algorithm for describing kinetics and thermodynamics in chemical and biochemical systems.

D.R.G. is supported by EPSRC Programme Grant No. EP/G00224X. The computational resources for the group of DS have been provided by EPSRC Grant Nos. EP/I014500/1 and EP/J001481/1. We would like to thank Martin Volk and Martin Gruebele for useful comments.

I. INTRODUCTION

II. THEORY OF THE BXD METHOD

III. HIGH RESOLUTION KINETICS

IV. RESULTS AND DISCUSSION

A. BXD numerical experiment

B. Rate coefficient distribution

V. CONCLUSIONS

### Key Topics

- Peptides
- 33.0
- Reaction rate constants
- 22.0
- Free energy
- 20.0
- Molecular dynamics
- 20.0
- Eigenvalues
- 12.0

##### C12

## Figures

Schematic of how BXD works with a reaction coordinate, ρ, split into m boxes.

Schematic of how BXD works with a reaction coordinate, ρ, split into m boxes.

Schematic of the procedure for deriving arbitrarily high-resolution kinetic data from BXD simulations. The procedure relies on introducing a new box c between boxes a and b, and then expressing the new rate coefficients in terms of k ab and k ba .

Schematic of the procedure for deriving arbitrarily high-resolution kinetic data from BXD simulations. The procedure relies on introducing a new box c between boxes a and b, and then expressing the new rate coefficients in terms of k ab and k ba .

log10(k inst ) as a function of log10(1<![CDATA[/]] >t) for four different peptides. The left-hand panels show the results of BXD calculations (dotted red line) compared with a 1<![CDATA[/]] >t ^{α} power law, with α = 1 (black line) and α = 0.93 (blue dashed line). The red squares indicate the timescales over which the BXD results are reliable, as described in the text. The right-hand panels show each peptide's free energy (or potential of mean force) as function of its extension. The position of the “absorbing boundary” where the loop formation is assumed to be irreversible (i.e., the border between the boxes 0 and 1) is indicated by the black arrow.

log10(k inst ) as a function of log10(1<![CDATA[/]] >t) for four different peptides. The left-hand panels show the results of BXD calculations (dotted red line) compared with a 1<![CDATA[/]] >t ^{α} power law, with α = 1 (black line) and α = 0.93 (blue dashed line). The red squares indicate the timescales over which the BXD results are reliable, as described in the text. The right-hand panels show each peptide's free energy (or potential of mean force) as function of its extension. The position of the “absorbing boundary” where the loop formation is assumed to be irreversible (i.e., the border between the boxes 0 and 1) is indicated by the black arrow.

BXD results obtained from 10-ALA. Panel A shows the free energy as a function of extension coordinate. Panel B shows log10(k inst ) as a function of log10(1<![CDATA[/]] >t) using the strongly non-equilibrium initial conditions in Eq. (8) , and how increasing the number of boxes with Eqs. (7) extends the timescale over which power law kinetics are observed. Panel C shows the same results as B, but with a 225 × 225 kinetic matrix. It also shows the overlap between the BXD results and unbiased MD results as described in the text. Panel D shows log10(k inst ) as a function of log10(1<![CDATA[/]] >t) using equilibrium initial conditions. The final two panels show the eigenvalues of the kinetic matrix M weighted with their coefficients F(k) in Eq. (11) for (E) non-equilibrium and (F) equilibrium initial conditions.

BXD results obtained from 10-ALA. Panel A shows the free energy as a function of extension coordinate. Panel B shows log10(k inst ) as a function of log10(1<![CDATA[/]] >t) using the strongly non-equilibrium initial conditions in Eq. (8) , and how increasing the number of boxes with Eqs. (7) extends the timescale over which power law kinetics are observed. Panel C shows the same results as B, but with a 225 × 225 kinetic matrix. It also shows the overlap between the BXD results and unbiased MD results as described in the text. Panel D shows log10(k inst ) as a function of log10(1<![CDATA[/]] >t) using equilibrium initial conditions. The final two panels show the eigenvalues of the kinetic matrix M weighted with their coefficients F(k) in Eq. (11) for (E) non-equilibrium and (F) equilibrium initial conditions.

Routes selected randomly on a landscape. At time t 2 the hikers who took the route t 3 are still walking but the hikers on the short route t 1 have already returned. Therefore at the moment t 2 the instantaneous rate constant of return is equal to 1<![CDATA[/]] >t 2. If the start and end-point of all routes are close then there is a large number of short routes, which is not the case when start and end point are distant from each other.

Routes selected randomly on a landscape. At time t 2 the hikers who took the route t 3 are still walking but the hikers on the short route t 1 have already returned. Therefore at the moment t 2 the instantaneous rate constant of return is equal to 1<![CDATA[/]] >t 2. If the start and end-point of all routes are close then there is a large number of short routes, which is not the case when start and end point are distant from each other.

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