^{1}and Markus Meuwly

^{1}

### Abstract

The dynamics of processes relevant to chemistry and biophysics on rough free energy landscapes is investigated using a recently developed algorithm to solve the Smoluchowski equation. Two different processes are considered: ligand rebinding in MbCO and protein folding. For the rebinding dynamics of carbon monoxide (CO) to native myoglobin (Mb) from locations around the active site, the two-dimensional free energysurface (FES) is constructed using extensive molecular dynamics simulations. The surface describes the minima in the state (bound MbCO), CO in the distal pocket and in the Xe4 pocket, and the transitions between these states and allows to study the diffusion of CO in detail. For the folding dynamics of protein G, a previously determined two-dimensional FES was available. To follow the diffusive dynamics on these rough free energysurfaces, the Smoluchowski equation is solved using the recently developed hierarchical discrete approximation method. From the relaxation of the initial nonequilibrium distribution, experimentally accessible quantities such as the rebinding time for CO or the folding time for protein G can be calculated. It is found that the free energy barrier for CO in the Xe4 pocket and in the distal pocket ( state) closer to the heme iron is which is considerably larger than the inner barrier which separates the bound state and the state. For the folding of protein G, a barrier of between the unfolded and the folded state is consistent with folding times of the order of milliseconds.

The authors acknowledge financial support from the Schweizerischer Nationalfonds through a Förderungsprofessur. We thank Professor C. L. Brooks III for providing us with the free energy surface of folding protein G.

I. INTRODUCTION

II. COMPUTATIONAL METHODS

A. Molecular dynamics simulations

B. Solution of the Smoluchowski equation

III. RESULTS

A. Rebinding in MbCO

B. The folding time of protein G

IV. CONCLUSION

### Key Topics

- Free energy
- 23.0
- Protein folding
- 21.0
- Diffusion
- 17.0
- Proteins
- 17.0
- Molecular dynamics
- 15.0

## Figures

(a) Coordinate system for CO rebinding in Mb. The center of mass of the CO molecule is at position with a distance to the center of mass of the four pyrrole nitrogen atoms which is at (0,0,0). The projection of defines the angle . The vector , which is used in the definition of , is given by the direction of the projection of and the length of . Note that . (b) is the free energy surface from the present simulations (see also Fig. 2) with the position of the Fe atom, the state, and the Xe4 pocket labeled. (c) shows the projection of the position of the CO center of mass onto the heme plane drawn together with a wire frame of the heme unit from previous MD simulations of unbound CO (Ref. 20). The projection in (c) is rotated to illustrate that representations (b) and (c) cover a very similar phase space. The earlier simulations were too short to sample the second state Xe4(2) (see Fig. 2).

(a) Coordinate system for CO rebinding in Mb. The center of mass of the CO molecule is at position with a distance to the center of mass of the four pyrrole nitrogen atoms which is at (0,0,0). The projection of defines the angle . The vector , which is used in the definition of , is given by the direction of the projection of and the length of . Note that . (b) is the free energy surface from the present simulations (see also Fig. 2) with the position of the Fe atom, the state, and the Xe4 pocket labeled. (c) shows the projection of the position of the CO center of mass onto the heme plane drawn together with a wire frame of the heme unit from previous MD simulations of unbound CO (Ref. 20). The projection in (c) is rotated to illustrate that representations (b) and (c) cover a very similar phase space. The earlier simulations were too short to sample the second state Xe4(2) (see Fig. 2).

(Color) (a) Contour map of FES for CO rebinding to native myoglobin. The state is the bound state, whereas DP (distal pocket, state), Xe4(1), and Xe4(2) correspond to unbound CO. The ruggedness of the FES can be clearly seen. Contour lines are given in kcal/mol. (b) Minimum energy path along the yellow arrow in panel (a). (c) Probability distribution of CO for starting from (centered at locations “X”). At , the bound state is already populated, whereas much of the population still resides in the Xe4 pocket. Red contours are low probability, blue contours are high probability, and green contour is intermediate. (d) Active site region of MbCO with CO in the state (1), at the transition between DP and Xe4 (2), in Xe4(1) (3), and in Xe4(2) (4). Residues that form the top and bottom of the Xe4 cavity are colored blue and gold, respectively.

(Color) (a) Contour map of FES for CO rebinding to native myoglobin. The state is the bound state, whereas DP (distal pocket, state), Xe4(1), and Xe4(2) correspond to unbound CO. The ruggedness of the FES can be clearly seen. Contour lines are given in kcal/mol. (b) Minimum energy path along the yellow arrow in panel (a). (c) Probability distribution of CO for starting from (centered at locations “X”). At , the bound state is already populated, whereas much of the population still resides in the Xe4 pocket. Red contours are low probability, blue contours are high probability, and green contour is intermediate. (d) Active site region of MbCO with CO in the state (1), at the transition between DP and Xe4 (2), in Xe4(1) (3), and in Xe4(2) (4). Residues that form the top and bottom of the Xe4 cavity are colored blue and gold, respectively.

Free energy surface as a function of (number of native contacts) and (the radius of gyration in Å). (a) original FES; (b) FES with a modified barrier between the unfolded and the folded state. Contours are separated by .

Free energy surface as a function of (number of native contacts) and (the radius of gyration in Å). (a) original FES; (b) FES with a modified barrier between the unfolded and the folded state. Contours are separated by .

(Color online) Folding time (ps) for protein G depending on the transition barrier and the diffusion coefficient : the green curve (solid black line) corresponds to , the blue line (dot dashed black) corresponds to the -dependent diffusion coefficient, the red line (solid grey) corresponds to , and the black line to . The circles are calculated folding times and the continuous lines are the fit. Dashed lines mark the experimentally observed folding times: exponential phase with a time scale of and the slower, denaturant-dependent time constant (Ref. 32).

(Color online) Folding time (ps) for protein G depending on the transition barrier and the diffusion coefficient : the green curve (solid black line) corresponds to , the blue line (dot dashed black) corresponds to the -dependent diffusion coefficient, the red line (solid grey) corresponds to , and the black line to . The circles are calculated folding times and the continuous lines are the fit. Dashed lines mark the experimentally observed folding times: exponential phase with a time scale of and the slower, denaturant-dependent time constant (Ref. 32).

## Tables

The positions and energies of the minima in the unbound state on the FES for , the corresponding center-of-mass distance (Å), and the values of the forward and backward free energy barriers (in kcal/mol) between the states indicated.

The positions and energies of the minima in the unbound state on the FES for , the corresponding center-of-mass distance (Å), and the values of the forward and backward free energy barriers (in kcal/mol) between the states indicated.

The rebinding times (ns) for CO depending on the initial distribution or for different values of (kcal/mol).

The rebinding times (ns) for CO depending on the initial distribution or for different values of (kcal/mol).

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