^{1}and David J. Wales

^{1}

### Abstract

We present the results of several benchmarks comparing the relative efficiency of different coordinate systems in optimizing polypeptide geometries. Cartesian, natural internal, and primitive internal coordinates are employed in quasi-Newton minimization for a variety of biomolecules. The peptides and proteins used in these benchmarks range in size from 16 to 999 residues. They vary in complexity from polyalanine helices to a -barrel enzyme. We find that the relative performance of the different coordinate systems depends on the parameters of the optimization method, the starting point for the optimization, and the size of the system studied. In general, internal coordinates were found to be advantageous for small peptides. For larger structures, Cartesians appear to be more efficient for empirical potentials where the energy and gradient can be evaluated relatively quickly compared to the cost of the coordinate transformations.

I. INTRODUCTION

II. METHODS

A. Internal coordinate systems

B. Coordinate transformation algorithm

C. L-BFGS optimization algorithm

D. Test polypeptides

III. RESULTS

A. Effects of L-BFGS memory

B. Effects of transformation parameters

C. Polyalanine helices

D. Effect of starting points for optimization

IV. CONCLUSIONS

## Figures

(Color) Structures for the three polypeptides (GB1, EnH, and CypA) used in the benchmarks. (a) A conformation from the native ensemble for each system, optimized to a convergence tolerance of rms force units. (b) Two of the 50 starting points used in the L-BFGS optimization benchmarks. The conformations were obtained by stepping off from randomly selected transition states parallel to the Hessian eigenvector corresponding to the unique negative Hessian eigenvalue.

(Color) Structures for the three polypeptides (GB1, EnH, and CypA) used in the benchmarks. (a) A conformation from the native ensemble for each system, optimized to a convergence tolerance of rms force units. (b) Two of the 50 starting points used in the L-BFGS optimization benchmarks. The conformations were obtained by stepping off from randomly selected transition states parallel to the Hessian eigenvector corresponding to the unique negative Hessian eigenvalue.

(Color online) Efficiency of Cartesian (C), primitive internal (P), and natural internal (N) coordinate systems in optimizing peptide structures. Fifty different starting structures were minimized for each of three polypeptides (GB1, EnH, and CypA). The average number of L-BFGS steps and the average total CPU time is plotted as a function of the number of previous steps used to build up the inverse Hessian approximation . The results for primitive and natural internals are indistinguishable in some cases. The rms force convergence criterion for all minimizations was .

(Color online) Efficiency of Cartesian (C), primitive internal (P), and natural internal (N) coordinate systems in optimizing peptide structures. Fifty different starting structures were minimized for each of three polypeptides (GB1, EnH, and CypA). The average number of L-BFGS steps and the average total CPU time is plotted as a function of the number of previous steps used to build up the inverse Hessian approximation . The results for primitive and natural internals are indistinguishable in some cases. The rms force convergence criterion for all minimizations was .

(Color online) Effect of parameters on GB1 energy minimization with primitive (P) and natural (N) internal coordinates. The average CPU time required to minimize 50 starting structures to an rms force of units is plotted as a function of the spectral shift used to make the matrix positive definite and of the convergence tolerance for the iterative back transformation.

(Color online) Effect of parameters on GB1 energy minimization with primitive (P) and natural (N) internal coordinates. The average CPU time required to minimize 50 starting structures to an rms force of units is plotted as a function of the spectral shift used to make the matrix positive definite and of the convergence tolerance for the iterative back transformation.

(Color online) Comparison of Cartesian (C), primitive internal (P), and natural internal (N) coordinate systems for optimizing polyalanine helices of different sizes. The L-BFGS memory parameter is for Cartesian coordinates and for internal coordinates. A spectral shift parameter of was used in this calculation. The rms force convergence criterion was .

(Color online) Comparison of Cartesian (C), primitive internal (P), and natural internal (N) coordinate systems for optimizing polyalanine helices of different sizes. The L-BFGS memory parameter is for Cartesian coordinates and for internal coordinates. A spectral shift parameter of was used in this calculation. The rms force convergence criterion was .

(Color online) Structures of 100 residue polyalanine helices after minimization with (a) natural internal and (b) Cartesian coordinates. Structures were minimized to an rms force tolerance of .

(Color online) Structures of 100 residue polyalanine helices after minimization with (a) natural internal and (b) Cartesian coordinates. Structures were minimized to an rms force tolerance of .

(Color online) Comparison of Cartesian (C), primitive internal (P), natural internal (N), and mixed internal (M) coordinate systems for optimizing polyalanine helices that have been previously minimized and perturbed. The number of gradient calls and CPU time are averaged over 50 different perturbed structures for each helix size. Parameters are the same as in Fig. 4. For the mixed coordinate system, -matrix coordinates were used for the backbone atoms only, with natural coordinates used for the side chains. The rms force convergence criterion was .

(Color online) Comparison of Cartesian (C), primitive internal (P), natural internal (N), and mixed internal (M) coordinate systems for optimizing polyalanine helices that have been previously minimized and perturbed. The number of gradient calls and CPU time are averaged over 50 different perturbed structures for each helix size. Parameters are the same as in Fig. 4. For the mixed coordinate system, -matrix coordinates were used for the backbone atoms only, with natural coordinates used for the side chains. The rms force convergence criterion was .

(Color online) Efficiency of Cartesian (C), primitive internal (P), and natural internal (N) coordinate systems in optimizing GB1 structures close to a local minimum. Fifty different starting structures were obtained by perturbing an optimized hairpin configuration. The average number of L-BFGS steps and the average total CPU time (in seconds) is plotted as a function of the number of previous steps used to build up the inverse Hessian approximation . The rms force convergence criterion was .

(Color online) Efficiency of Cartesian (C), primitive internal (P), and natural internal (N) coordinate systems in optimizing GB1 structures close to a local minimum. Fifty different starting structures were obtained by perturbing an optimized hairpin configuration. The average number of L-BFGS steps and the average total CPU time (in seconds) is plotted as a function of the number of previous steps used to build up the inverse Hessian approximation . The rms force convergence criterion was .

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