^{1,a)}, Albert Y. Lau

^{2}and Benoît Roux

^{2,3}

### Abstract

Despite the significant advances in free-energy computations for biomolecules, there exists no general method to evaluate the free-energy difference between two conformations of a macromolecule that differ significantly from each other. A crucial ingredient of such a method is the ability to find a path between different conformations that allows an efficient computation of the free energy. In this paper, we introduce a method called “deactivated morphing,” in which one conformation is morphed into another after the internal interactions are completely turned off. An important feature of this method is the (shameless) use of nonphysical paths, which makes the method robustly applicable to conformational changes of arbitrary complexity.

S.P. thanks Vijay Pande for guidance during the early stages of the work and Chris Jarzynski for stimulating discussions on rotational entropy. This research was supported by the U.S. Department of Energy under Contract No. DE-AC02-06CH11357.

I. INTRODUCTION

II. DEACTIVATED MORPHING

A. Conformation ensembles

B. Restraining

C. Deactivating

D. Morphing

E. BAR and the overlap of ensembles

F. Breakdown of conformationalfree energy

III. ALANINE DIPEPTIDE

IV. ALANINE DECAMER-HELIX AND HAIRPIN

V. POSSIBLE IMPROVEMENTS

VI. CONCLUSIONS

### Key Topics

- Free energy
- 81.0
- Proteins
- 42.0
- Molecular conformation
- 21.0
- Macromolecular conformation
- 14.0
- Conformational dynamics
- 11.0

## Figures

DM. An ensemble around conformation A is gradually transformed into another ensemble around conformation B through restraining, deactivating, and morphing. Dashed circles represent position restraints. Often, intermediate states are required in some or all of the steps.

DM. An ensemble around conformation A is gradually transformed into another ensemble around conformation B through restraining, deactivating, and morphing. Dashed circles represent position restraints. Often, intermediate states are required in some or all of the steps.

Four conformations of AlaD selected from the map. The four circles roughly correspond to the boundaries of the basins within RMSD from the four reference conformations. The free-energy map was constructed from an umbrella sampling simulation using 144 windows spaced every in and . For each window, we collected data from a 750 ps run after a 250 ps equilibration period.

Four conformations of AlaD selected from the map. The four circles roughly correspond to the boundaries of the basins within RMSD from the four reference conformations. The free-energy map was constructed from an umbrella sampling simulation using 144 windows spaced every in and . For each window, we collected data from a 750 ps run after a 250 ps equilibration period.

Simulation box containing an AlaD molecule and 275 water molecules.

Simulation box containing an AlaD molecule and 275 water molecules.

Overlap of states in the restraining procedure for AlaD. Each panel contains 15 distributions of from the reference conformation, A, B, C, and D, for 15 different restraining states, (from left to right). The spring constants are , 600, 360, 216, 129.6, 77.76, 46.66, 27.99, 16.80, 10.08, 6.05, 3.63, 2.18, 1.31, and .

Overlap of states in the restraining procedure for AlaD. Each panel contains 15 distributions of from the reference conformation, A, B, C, and D, for 15 different restraining states, (from left to right). The spring constants are , 600, 360, 216, 129.6, 77.76, 46.66, 27.99, 16.80, 10.08, 6.05, 3.63, 2.18, 1.31, and .

Outcome of DM applied to the four conformations of AlaD. The numbers next to arrows denote the free-energy differences (in kcal/mol) associated with the transitions. The numbers next to dashed arrows are the total free-energy differences between the conformation ensembles. The numbers in parentheses are the simulation time (in nanosecond) used to compute the free-energy differences. The free-energy difference between Q and M, denoted by , cancels out exactly.

Outcome of DM applied to the four conformations of AlaD. The numbers next to arrows denote the free-energy differences (in kcal/mol) associated with the transitions. The numbers next to dashed arrows are the total free-energy differences between the conformation ensembles. The numbers in parentheses are the simulation time (in nanosecond) used to compute the free-energy differences. The free-energy difference between Q and M, denoted by , cancels out exactly.

Two conformations of Ala10: helix and hairpin.

Two conformations of Ala10: helix and hairpin.

Simulation box containing an Ala10 molecule and 676 water molecules.

Simulation box containing an Ala10 molecule and 676 water molecules.

Overlap of states in the restraining procedure for Ala10. Each panel contains 30 distributions of from the reference conformation, helix and hairpin, for 30 different restraining states, (from left to right). The spring constants are , 700, 490, 343, 240.1, 168.07, 117.65, 82.35, 57.65, 40.35, 28.25, 19.77, 13.84, 9.69, 6.78, 4.75, 3.32, 2.33, 1.63, 1.14, 0.80, 0.56, 0.39, 0.27, 0.19, 0.13, 0.09, 0.07, 0.05, and .

Overlap of states in the restraining procedure for Ala10. Each panel contains 30 distributions of from the reference conformation, helix and hairpin, for 30 different restraining states, (from left to right). The spring constants are , 700, 490, 343, 240.1, 168.07, 117.65, 82.35, 57.65, 40.35, 28.25, 19.77, 13.84, 9.69, 6.78, 4.75, 3.32, 2.33, 1.63, 1.14, 0.80, 0.56, 0.39, 0.27, 0.19, 0.13, 0.09, 0.07, 0.05, and .

Morphing between helix and hairpin. The free-energy profile over the course of morphing is shown as a function of the parameter in Eq. (11). Of 51 conformations used, six are shown. Meshes represent molecular surfaces, and red tubes are traces of backbone.

Morphing between helix and hairpin. The free-energy profile over the course of morphing is shown as a function of the parameter in Eq. (11). Of 51 conformations used, six are shown. Meshes represent molecular surfaces, and red tubes are traces of backbone.

Outcome of DM applied to the helix (a) and hairpin (b) conformations of Ala10. The numbers next to arrows denote the free-energy differences (in kcal/mol) associated with the transitions. The number next to the dashed arrow is the total free-energy difference between the two conformation ensembles. The numbers in parentheses are the simulation time (in nanosecond) used to compute the free-energy differences. The free-energy difference between Q and M, denoted by , cancels out exactly.

Outcome of DM applied to the helix (a) and hairpin (b) conformations of Ala10. The numbers next to arrows denote the free-energy differences (in kcal/mol) associated with the transitions. The number next to the dashed arrow is the total free-energy difference between the two conformation ensembles. The numbers in parentheses are the simulation time (in nanosecond) used to compute the free-energy differences. The free-energy difference between Q and M, denoted by , cancels out exactly.

Dependence of the free-energy difference on the RMSD cutoff.

Dependence of the free-energy difference on the RMSD cutoff.

## Tables

Protein energy and deactivation free energy (in kcal/mol). The numbers in parentheses are the relative values, with conformation A as a reference.

Protein energy and deactivation free energy (in kcal/mol). The numbers in parentheses are the relative values, with conformation A as a reference.

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