# Segmenting Proteins into Tripeptides to Enhance Conformational Sampling with Monte Carlo Methods

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Implemented Move Classes and Parameter Settings

- -
- The simplest class of trial moves, largely applied to sample the conformation of chain-like molecules, consists of perturbing a randomly selected bond torsion and then propagating the motion toward the end of the chain. Such moves, usually called pivot moves, are named here OneTorsion moves. They are illustrated in Figure 1a.
- -
- The second move class is named ConRot, since it is inspired by the concerted rotations proposed by Dodd et al. [17]. It has been implemented using the proposed tripeptide-based model as follows: an amino-acid residue is randomly selected and one of its bond torsions ($\varphi $ or $\psi $) is randomly perturbed; the backbone conformation of the next three residues (the next tripeptide) is modified by inverse kinematics in order to maintain fixed ends (see Section 3.2 for details). The move class is illustrated in Figure 1b.
- -
- The third move class, called OneParticle moves, corresponds to the simplest move class involving tripeptide-based particle perturbations, as described in Section 3.2, and illustrated in Figure 1c.
- -
- The last move class, called Hinge moves, corresponds to the rigid-body block moves described in Section 3.2, and illustrated in Figure 1d. The number of consecutive particles affected by the move is randomly sampled at each iteration between 3 and 10 (i.e., moves involve between 9 and 30 residues).

#### 2.2. Test Systems

#### 2.3. Computational Performance

^{®}Xeon

^{®}E5-1650 processor at 3.2 GHz. These results show that, for a similar MC acceptance rate, generating a given number of samples ($2\times {10}^{6}$ in this case) requires more computational resources using the OneTorsion move class compared to the other move classes, which apply fixed-end motions. The reason is that, although the OneTorsion move class does not require solving inverse kinematics, a significant amount of computing time is needed for propagating atom motions along the chain by forward kinematics and to recompute interaction energies between atom pairs. In other words, the computing time needed by the ConRot, OneParticle and Hinge move classes to solve inverse kinematics is largely compensated since only the positions of a small number of atoms need to be updated after each move, which also decreases the cost of energy recalculation. Nevertheless, computing time is probably not the most important indicator of the performance of the different move classes. Indeed, the numbers in Table 2 have to be analyzed together with other data related to the quality of the sampling strategy in terms of energy and distance distributions, as discussed below.

#### 2.4. Distribution of Sampled States

#### 2.5. Exploration Efficiency Analysis

#### 2.5.1. Time Dependent RMSD Function

#### 2.5.2. Autocorrelation Time

#### 2.6. Additional Results for Ubiquitin

## 3. Materials and Methods

#### 3.1. Protein Model

#### 3.1.1. Mechanistic Model

#### 3.1.2. Decomposition into Tripeptides

#### 3.2. Devising Move Classes

#### 3.2.1. Perturbing Particles

#### 3.2.2. Solving Inverse Kinematics for a Tripeptide

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Woolfson, M.M. An Introduction to X-ray Crystallography; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Kay, L.E. NMR studies of protein structure and dynamics. J. Magn. Reson.
**2005**, 173, 193–207. [Google Scholar] [CrossRef] [PubMed] - Lange, O.F.; Lakomek, N.A.; Farès, C.; Schröder, G.F.; Walter, K.F.A.; Becker, S.; Meiler, J.; Grubmüller, H.; Griesinger, C.; de Groot, B.L. Recognition Dynamics Up to Microseconds Revealed from an RDC-Derived Ubiquitin Ensemble in Solution. Science
**2008**, 320, 1471–1475. [Google Scholar] [CrossRef] [PubMed] - Fenwick, R.B.; van den Bedem, H.; Fraser, J.S.; Wright, P.E. Integrated description of protein dynamics from room-temperature X-ray crystallography and NMR. Proc. Natl. Acad. Sci. USA
**2014**, 111, E445–E454. [Google Scholar] [CrossRef] [PubMed] - Leach, A.R. Molecular Modelling: Principles and Applications; Pearson Education: Harlow, UK, 2001. [Google Scholar]
- Frenkel, D.; Smit, B. Understanding Molecular Simulations: From Algorihtms to Applications; Academic Press: London, UK, 2002. [Google Scholar]
- Wales, D. Energy Landscapes: Applications to Clusters, Biomolecules and Glasses; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Donald, B.R. Algorithms in Structural Molecular Biology; The MIT Press: Cambridge, MA, USA, 2011. [Google Scholar]
- Al-Bluwi, I.; Siméon, T.; Cortés, J. Motion Planning Algorithms for Molecular Simulations: A Survey. Comput. Sci. Rev.
**2012**, 6, 125–143. [Google Scholar] [CrossRef] - Gipson, B.; Hsu, D.; Kavraki, L.; Latombe, J.C. Computational models of protein kinematics and dynamics: Beyond simulation. Ann. Rev. Anal. Chem.
**2012**, 5, 273–291. [Google Scholar] [CrossRef] [PubMed] - Shehu, A.; Plaku, E. A Survey of Computational Treatments of Biomolecules by Robotics-Inspired Methods Modeling Equilibrium Structure and Dynamic. J. Artif. Intell. Res.
**2016**, 57, 509–572. [Google Scholar] - Cortés, J.; Al-Bluwi, I. A Robotics Approach To Enhance Conformational Sampling Of Proteins. In Proceedings of the ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Chicago, IL, USA, 12–15 August 2012; Volume 4, pp. 1177–1186. [Google Scholar]
- Parsons, D.; Canny, J. Geometric Problems in Molecular Biology and Robotics. Proc. Int. Conf. Intell. Syst. Mol. Biol.
**1994**, 2, 322–330. [Google Scholar] [PubMed] - Siciliano, B.; Khatib, O. (Eds.) Springer Handbook of Robotics; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Metropolis, N.; Rosenbluth, A.; Rosenbluth, M.; Teller, A.; Teller, E. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys.
**1953**, 21, 1087–1092. [Google Scholar] [CrossRef] - Gō, N.; Scheraga, H. Ring Closure and Local Conformational Deformations of Chain Molecules. Macromolecules
**1970**, 3, 178–187. [Google Scholar] [CrossRef] - Dodd, L.R.; Boone, T.D.; Theodorou, D.N. A Concerted Rotation Algorithm for Atomistic Monte Carlo Simulation of Polymer Melts and Glasses. Mol. Phys.
**1993**, 78, 961–996. [Google Scholar] [CrossRef] - Leontidis, E.; de Pablo, J.J.; Laso, M.; Suter, U.W. A Critical Evaluation of Novel Algorithms for the Off-Lattice Monte Carlo Simulation of Condensed Polymer Phases. Adv. Polym. Sci.
**1994**, 116, 283–318. [Google Scholar] - Wu, N.G.; Deem, M.W. Efficient Monte Carlo Methods for Cyclic Peptides. Mol. Phys.
**1999**, 94, 559–580. [Google Scholar] [CrossRef] - Betancourt, M.R. Efficient Monte Carlo Trial Moves for Polypeptide Simulations. J. Chem. Phys.
**2005**, 123, 174905. [Google Scholar] [CrossRef] [PubMed] - Davis, I.W.; Arendall, W.B., III; Richardson, D.C.; Richardson, J.S. The Backrub Motion: How Protein Backbone Shrugs When a Sidechain Dances. Structure
**2006**, 14, 265–274. [Google Scholar] [CrossRef] [PubMed] - Bottaro, S.; Boomsma, W.; Johansson, K.E.; Andreetta, C.; Hamelryck, T.; Ferkinghoff-Borg, J. Subtle Monte Carlo Updates in Dense Molecular Systems. J. Chem. Theory Comput.
**2012**, 8, 695–702. [Google Scholar] [CrossRef] [PubMed] - Vitalis, A.; Pappu, R.V. Methods for Monte Carlo Simulations of Biomacromolecules. Annu. Rep. Comput. Chem.
**2009**, 5, 49–76. [Google Scholar] [PubMed] - Kollman, P.; Dixon, R.; Cornell, W.; Fox, T.; Chipot, C.; Pohorille, A. The development/application of a “minimalist” organic/biochemical molecular mechanic force field using a combination of ab initio calculations and experimental data. In Computer Simulation of Biomolecular Systems; van Gunsteren, W., Weiner, P., Wilkinson, A., Eds.; Springer: Dordrecht, The Netherlands, 1997; Volume 3, pp. 83–96. [Google Scholar]
- Bondi, A. Van der Waals Volumes and Radii. J. Phys. Chem.
**1964**, 68, 441–451. [Google Scholar] [CrossRef] - Kurochkina, N.; Guha, U. SH3 domains: Modules of protein–protein interactions. Biophys. Rev.
**2013**, 5, 29–39. [Google Scholar] [CrossRef] [PubMed] - Brocca, S.; Samalíkovà, M.; Uversky, V.N.; Lotti, M.; Vanoni, M.; Alberghina, L.; Grandori, R. Order propensity of an intrinsically disordered protein, the cyclin-dependent-kinase inhibitor Sic1. Proteins
**2009**, 76, 731–746. [Google Scholar] [CrossRef] [PubMed] - Bernadó, P.; Blanchard, L.; Timmins, P.; Marion, D.; Ruigrok, R.; Blackledge, M. A structural model for unfolded proteins from residual dipolar couplings and small-angle X-ray scattering. Proc. Natl. Acad. Sci. USA
**2005**, 102, 17002–17005. [Google Scholar] [CrossRef] [PubMed] - Krivov, G.G.; Shapovalov, M.V.; Dunbrack, R.L., Jr. Improved Prediction of Protein Side-chain Conformations with SCWRL4. Proteins Struct. Funct. Bioinf.
**2009**, 77, 778–795. [Google Scholar] [CrossRef] [PubMed] - Ulmschneider, J.P.; Jorgensen, W.L. Monte Carlo backbone sampling for polypeptides with variable bond angles and dihedral angles using concerted rotations and a Gaussian bias. J. Chem. Phys.
**2003**, 118, 4261–4271. [Google Scholar] [CrossRef] - Pickart, C.M.; Eddins, M.J. Ubiquitin: Structures, functions, mechanisms. Biochim. Biophys. Acta (BBA) Mol. Cell Res.
**2004**, 1695, 55–72. [Google Scholar] [CrossRef] [PubMed] - Scott, R.; Scheraga, H. Conformational Analysis of Macromolecules. II. The Rotational Isomeric States of the Normal Hydrocarbons. J. Chem. Phys.
**1966**, 44, 3054–3069. [Google Scholar] [CrossRef] - Ulmschneider, J.P.; Jorgensen, W.L. Polypeptide Folding Using Monte Carlo Sampling, Concerted Rotation, and Continuum Solvation. J. Am. Chem. Soc.
**2004**, 126, 1849–1857. [Google Scholar] [CrossRef] [PubMed] - Craig, J.J. Introduction to Robotics; Addison-Wesley: Reading, MA, USA, 1989. [Google Scholar]
- Nilmeier, J.; Hua, L.; Coutsias, E.A.; Jacobson, M.P. Assessing Protein Loop Flexibility by Hierarchical Monte Carlo Sampling. J. Chem. Theory Comput.
**2011**, 7, 1564–1574. [Google Scholar] [CrossRef] [PubMed] - Nilmeier, J.; Jacobson, M.P. Multiscale Monte Carlo Sampling of Protein Sidechains: Application to Binding Pocket Flexibility. J. Chem. Theory Comput.
**2008**, 4, 835–846. [Google Scholar] [CrossRef] [PubMed] - Nilmeier, J.; Jacobson, M.P. Monte Carlo Sampling with Hierarchical Move Sets: POSH Monte Carlo. J. Chem. Theory Comput.
**2009**, 5, 1968–1984. [Google Scholar] [CrossRef] [PubMed] - Wu, N.G.; Deem, M.W. Analytical Rebridging Monte Carlo: Application to cis/trans Isomerization in Proline-Containing, Cyclic Peptides. J. Chem. Phys.
**1999**, 111, 6625–6632. [Google Scholar] [CrossRef] - Canutescu, A.A.; Dunbrack, R.L., Jr. Cyclic Coordinate Descent: A Robotics Algorithm for Protein Loop Closure. Protein Sci.
**2003**, 12, 963–972. [Google Scholar] [CrossRef] [PubMed] - Renaud, M. A simplified inverse kinematic model calculation method for all 6R type manipulators. In Current Advances in Mechanical Design and Production VII; Hassan, M.F., Megahed, S.M., Eds.; Pergamon: New York, NY, USA, 2000; pp. 57–66. [Google Scholar]
- Renaud, M. Calcul des Modèles Géométriques Inverses des Robots Manipulateurs 6R; Rapport LAAS 06332; LAAS: Toulouse, France, 2006. [Google Scholar]
- Lee, H.Y.; Liang, C.G. A New Vector Theory for the Analysis of Spatial Mechanisms. Mech. Mach. Theory
**1988**, 23, 209–217. [Google Scholar] [CrossRef] - Lee, H.Y.; Liang, C.G. Displacement Analysis of the General Spatial 7-Link 7R Mechanisms. Mech. Mach. Theory
**1988**, 23, 219–226. [Google Scholar] - Manocha, D.; Canny, J.F. Efficient Inverse Kinematics for General 6R Manipulators. IEEE Trans. Robot. Autom.
**1994**, 10, 648–657. [Google Scholar] [CrossRef] - Golub, G.H.; Van Loan, C.F. Matrix Computations, 3rd ed.; Johns Hopkins University Press: Baltimore, MD, USA, 1996. [Google Scholar]
- Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Demmel, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; et al. LAPACK Users’ Guide, 3rd ed.; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1999. [Google Scholar]
- Cortés, J.; Siméon, T.; Remaud-Siméon, M.; Tran, V. Geometric Algorithms for the Conformational Analysis of Long Protein Loops. J. Comput. Chem.
**2004**, 25, 956–967. [Google Scholar] [CrossRef] [PubMed] - Cortés, J.; Carrión, S.; Curco, D.; Renaud, M.; Aleman, C. Relaxation of Amorphous Multichain Polymer Systems using Inverse Kinematics. Polymer
**2010**, 51, 4008–4014. [Google Scholar] [CrossRef] - Dinner, A.R. Local deformations of polymers with nonplanar rigid main-chain internal coordinates. J. Comput. Chem.
**2000**, 21, 1132–1144. [Google Scholar] [CrossRef] - Coutsias, E.A.; Seok, C.; Jacobson, M.P.; Dill, K.A. A Kinematic View of Loop Closure. J. Comput. Chem.
**2004**, 25, 510–528. [Google Scholar] [CrossRef] [PubMed] - Mezei, M. Efficient Monte Carlo Sampling of Long Molecular Chains Using Local Moves, Tested on a Solvated Lipid Bilayer. J. Chem. Phys.
**2003**, 118, 3874–3879. [Google Scholar] [CrossRef] - Okamoto, Y. Generalized-ensemble algorithms: Enhanced sampling techniques for Monte Carlo and molecular dynamics simulations. J. Mol. Graph. Model.
**2004**, 22, 425–439. [Google Scholar] [CrossRef] [PubMed] - Carr, J.M.; Wales, D.J. Global optimization and folding pathways of selected alpha-helical proteins. J. Chem. Phys.
**2005**, 123, 234901. [Google Scholar] [CrossRef] [PubMed] - Cortés, J.; Le, D.; Iehl, R.; Siméon, T. Simulating ligand-induced conformational changes in proteins using a mechanical disassembly method. Phys. Chem. Chem. Phys.
**2010**, 12, 8268–8276. [Google Scholar] [CrossRef] [PubMed] - Mandell, D.J.; Kortemme, T. Backbone flexibility in computational protein design. Curr. Opin. Biotechnol.
**2009**, 20, 420–428. [Google Scholar] [CrossRef] [PubMed]

Sample Availability: Not Available. |

**Figure 1.**Illustration of the implemented move classes. (

**a**) pivot move, which is the most frequently used move class within MC methods applied to chain-like molecules; (

**b**) concerted rotation, as implemented using the proposed tripeptide-based model; (

**c**) local move produced by the perturbation of the particle associated with one tripeptide; and (

**d**) hinge move produced by a correlated perturbation of a set of particles. The protein backbone trace before and after the move are drawn in black and blue, respectively. Tripeptides that change their conformation are represented with dashed lines.

**Figure 3.**Projection of sampled states on distance vs. energy plots for the SH3 domain and the Sic1 protein.

**Figure 4.**Plots of the function $\mathtt{rmsd}\left(\tau \right)$ for each move class during MC runs for the SH3 domain (

**left**) and the Sic1 protein (

**right**).

**Figure 5.**Plots of the average autocorrelation function over the 20 central dihedral angles of 14-alanine for the ConRot, OneParticle, and Mixed move classes.

**Figure 6.**Autocorrelation times ${\tau}_{int}$ (in MC steps) of the 20 central dihedral angles of 14-alanine for the ConRot, OneParticle, and Mixed move classes.

**Figure 8.**Ubiquitin C${}_{\alpha}$ root mean squared fluctuations (RMSF) from $15\times {10}^{6}$ steps of MC simulations using ConRot, OneParticle, and Mixed move classes.

**Figure 9.**Evolution of the cumulative C${}_{\alpha}$ root mean squared fluctuations (RMSF) of ubiquitin with the number of MC steps using ConRot, OneParticle, and Mixed move classes.

**Figure 10.**Geometric model of the protein backbone: (

**a**) in a general case; (

**b**) in the particular case of a rigid peptide bond. In the general case, the mDH parameters involved in the homogeneous transformation matrices between consecutive atom groups are obtained directly from bond lengths and bond angles. In the case of a rigid peptide bond, this requires some (simple) geometric operations. Note that the origin of a frame attached to an atom group does not necessarily correspond to an atom position.

**Figure 11.**Illustration of the protein subdivision approach. Fragments of three amino-acid residues are treated as kinematic chains, similar to robotic manipulators.

OneTorsion | ConRot | OneParticle | Hinge | ||
---|---|---|---|---|---|

${\mathit{\delta}}_{\mathit{b}}$ | ${\mathit{\delta}}_{\mathit{b}}$ | ${\mathit{\delta}}_{\mathit{pt}}$ | ${\mathit{\delta}}_{\mathit{pr}}$ | ${\mathit{\delta}}_{\mathit{pr}}$ | |

SH3 domain | 0.01 rad | 0.025 rad | 0.05 Å | 0.003 rad | 0.01 rad |

Sic1 protein | 0.02 rad | 0.025 rad | 0.05 Å | 0.003 rad | 0.02 rad |

Move Class | Acc. Rate | # Iterations | T${}_{\mathbf{CPU}}$ | |
---|---|---|---|---|

SH3 domain | OneTorsion | 0.68 | $3.28\times {10}^{6}$ | 63 h |

ConRot | 0.56 | $3.55\times {10}^{6}$ | 51 h | |

OneParticle | 0.42 | $4.06\times {10}^{6}$ | 56 h | |

Hinge | 0.59 | $3.46\times {10}^{6}$ | 57 h | |

Mixed | 0.56 | $3.58\times {10}^{6}$ | 57 h | |

Sic1 protein | OneTorsion | 0.56 | $3.54\times {10}^{6}$ | 89 h |

ConRot | 0.65 | $3.25\times {10}^{6}$ | 63 h | |

OneParticle | 0.52 | $3.66\times {10}^{6}$ | 69 h | |

Hinge | 0.53 | $3.60\times {10}^{6}$ | 75 h | |

Mixed | 0.57 | $3.49\times {10}^{6}$ | 74 h |

**Table 3.**Autocorrelation times ${\tau}_{int}$ (in MC steps) of the ConRot, OneTorsion, and Mixed move classes.

Move Class | Average | Min | Median | Max |
---|---|---|---|---|

ConRot | 6016 | 512 | 5315 | 14,070 |

OneParticle | 3426 | 343 | 1215 | 11,281 |

Mixed | 1518 | 157 | 985 | 3706 |

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**MDPI and ACS Style**

Denarie, L.; Al-Bluwi, I.; Vaisset, M.; Siméon, T.; Cortés, J.
Segmenting Proteins into Tripeptides to Enhance Conformational Sampling with Monte Carlo Methods. *Molecules* **2018**, *23*, 373.
https://doi.org/10.3390/molecules23020373

**AMA Style**

Denarie L, Al-Bluwi I, Vaisset M, Siméon T, Cortés J.
Segmenting Proteins into Tripeptides to Enhance Conformational Sampling with Monte Carlo Methods. *Molecules*. 2018; 23(2):373.
https://doi.org/10.3390/molecules23020373

**Chicago/Turabian Style**

Denarie, Laurent, Ibrahim Al-Bluwi, Marc Vaisset, Thierry Siméon, and Juan Cortés.
2018. "Segmenting Proteins into Tripeptides to Enhance Conformational Sampling with Monte Carlo Methods" *Molecules* 23, no. 2: 373.
https://doi.org/10.3390/molecules23020373