ISSN 00063509, Biophysics, 2014, Vol. 59, No. 1, pp. 28–34. © Pleiades Publishing, Inc., 2014. Original Russian Text © A.A. Koshevoy, E.O. Stepanov, Yu.B. Porozov, 2014, published in Biofizika, 2014, Vol. 59, No. 1, pp. 37–44.

MOLECULAR BIOPHYSICS

Method of Prediction and Optimization of Conformational Motion of Proteins Based on Mass Transportation Principle A. A. Koshevoya, E. O. Stepanovb, c, and Yu. B. Porozova a

ITMO University, St. Petersburg, 197101 Russia St. Petersburg Department of Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, 191023 Russia c St. Petersburg State University, Peterhof, St. Petersburg, 198504 Russia email: [email protected]

b

Received April 15, 2013; in final form, November 29, 2013

Abstract—The paper highlights approaches to fast prediction of protein conformational mobility. A new mathematical model based on the transportation principle is proposed. We describe an algorithm and soft ware developed for a construction of the possible trajectories of the largescale conformational motions of proteins (i.e. movements that occur within relatively large time intervals of the order of milliseconds). The modeling showed that the proposed method provides adequate, in terms of current knowledge of the biology and physics of proteins, results and allows simulation of largescale conformational transitions for less time. Keywords: conformational motion of protein, transportation principle, optimization, coarsegrained methods DOI: 10.1134/S0006350914010035

INTRODUCTION

intensiveness of computations, not allowing execution of modeling of the mobility of large molecules of dura tion up to several millisecond and longer. Some meth ods possess inferior accuracy of modeling and may give results in which errors are present, such as violations of the laws of protein physics, high energies, inadmissible values of covalent and torsion angles and lengths of covalent bonds in the constructed conformational transitions. However the principal shortcoming of the majority of modern methods of modeling the motions of protein molecules consists in that they, regardless of the concrete numerical algorithm used (let us name in the quality of an example the probabilistic roadmaps [1–3], stochastic roadmap simulation [4, 5], rapidly exploring random trees [6, 7], simple gradient descent [8, 9] and the like), as a rule, are based on analysis of the internal energy of a molecule, do not take (and cannot take) into account the external forces acting on the molecule in the process of motion. Therefore such methods can give adequate prediction about molecule motions in cases when external forces can be neglected. For prediction of largescale motions (on intervals of the order of milliseconds) of a protein mol ecule, the external forces appear definitive (in the pro cess of such motion a molecule may pass, for example, from the nucleus to the membrane of a cell, being sub jected to the action of a large number of different fac tors), yet predicting them does not appear possible. For overcoming this shortcoming in modeling of

At the present time the paradigm “protein structure determines its function” undergoes substantial changes. It has been shown that there is quite an inti mate connection between the functional activity of a protein and its conformational dynamics in micro and millisecond ranges. The cause of a dependency of protein functions on its conformational mobility hides in the character of conformational motion and its attendant events. During a transition there is a change in the characteristics of intermolecular contacts—on the surface of a macromolecule the properties, charac teristics and the very availability of active centers undergo significant changes. The work of the system responsible for molecular recognition and the phe nomenon known in the literature as “induced fit” is conditioned by intramolecular mobility. Mobile loops of proteins, owing to which the functions of vaccines are realized, keep in motion. Modeling of protein mobility turns out to be exceptionally important for the needs of theoretical metabolomics, during study ing signal pathways and development of new medici nal means. With its aid it becomes possible to predict side effects of new active substances. The presently developed methods of modeling the conformational mobility of proteins have a number of limitations and shortcomings. A shortcoming tradi tionally noted in the literature presents as high labor 28

PREDICTION OF PROTEIN SLOW MOTION BASED ON MASS TRANSPORT

29

Table 1. Methods of modeling the motions of proteins and the corresponding time intervals Methods

Time scale

Molecular dynamics

From pico to nanoseconds [4]. With the use of specialized computational techniques and software, up to 100 μs Milliseconds Milliseconds Milliseconds

Stochastic roadmap simulation Rapidly exploring random trees Analysis of normal modes

largescale molecular motions, developed in recent years are approaches intentionally departing from exact calculations of the internal energy of a molecule and analyzing its spatial structure [10–25]. Regret fully, such methods often predictably give results poorly consistent with the available knowledge on the physics of molecular motions. By the present time a series of methods have been developed for modeling the conformational mobility of protein molecules, but each of the proposed meth ods possesses certain shortcomings. With the aid of methods of molecular dynamics (MD) [26–29] it is possible to model processes of duration up to hundreds of nanoseconds (Table 1). Thereto related are small scale (<1 Å) and mediumscale (1–4 Å) motions of the molecule. At that with increasing quantity of atoms and duration of modeled dynamics there is a sharp increase in the time required for conduction a simula tion. MD protocols are not suitable for modeling con formational motions that may last hundreds of micro seconds and longer and the interconformational mean square deviation of which constitutes 5–30 Å [30–32], though with the use of specialized computational techniques and software the simulation time can be increased to 100 μs [33]. Besides that, some MD pro tocols may close the modeling into repeating cycles and do not allow overcoming energy barriers between conformations. This imposes limitations on the appli cability of the given group of methods by a narrow class of modeling tasks. Another approach, based on quantum mechanics, possesses greater accuracy for modeling on short time intervals (nanoseconds) but is totally inapplicable for modeling motions in micro and millisecond intervals of time at the expense of extremely prolonged time of computation, and what is most important, at the expense of the absence of ade quate information about external impacts on the mol ecule on a prolonged time interval. Therefore for modeling of the mobility of large molecules of proteins on large segments of time in the main use is made of methods that propose their own models of a protein molecule and its mobility, execut ing in calculations sufficiently rough approximations of real conformational motions. Planning of the tra jectory of the change by a protein of its conformation appears an exceptionally important task, inasmuch as exactly such “slow” motions are responsible for the functional activity of proteins [31–33]. It should be BIOPHYSICS

Vol. 59

No. 1

2014

noted that during modeling of protein mobility in micro and millisecond ranges, investigators are not interested in lowamplitude oscillations and vibrations of separate atoms, which are necessarily taken into account by methods of MD, inasmuch as such oscilla tions are “not seen” in the regarded time scale. Pro ceeding from this, it is possible to speak about expedi ence of application of coarse models of motion. The methods of modeling conformational motion intended for prediction of motion on micro and mil lisecond intervals of time, as a rule are based on solving a problem of planning the motion of an object with a large number of degrees of freedom in an unchanging medium. In computational structural biology the methods of solving the task of motion planning have found broad application in predicting the possible tra jectory of the motion of molecules. Investigations conducted with the use of the given methods have made a significant contribution into understanding the molecular kinetics of biologically significant mac romolecules in such applications thereof as investiga tion of energy landscapes, protein folding, process of ligand binding [34–38]. In the present paper, pro posed is a method of modeling the conformational motion of a protein molecule that can be related to the regarded class. The aim of the given work is to present of a new method of prediction of largescale conformational motions of a protein molecule, not having the indi cated shortcomings. The proposed method is based on solving a problem of minimization of a special func tional, resembling the classical Monge–Kantorovich mass transfer problem [39], but with constraints set by the physics of proteins. Similar functionals are actively used, in particular, with aims of interpolation in prob lems of image processing. In the present paper we present a description of the mathematical model of a protein molecule used in the work, of the proposed algorithm of finding the trajectory of transition of the protein molecule from one conformation into another, and also perform qualitative comparison of interconformational trajectories obtained by various methods.

30

KOSHEVOY et al.

MATHEMATICAL MODEL OF CONFORMATIONAL MOTION. GEOMETRY OF PROTEIN MOLECULE In protein molecules they distinguish the main chain, consisting of atoms consecutively connected by i i+1 covalent chemical bonds: Ni, C α , Ci, Ni + 1, C α , Ci + 1, …, and side chains, unique for various amino acids. The flexibility of the chain of protein molecule stipulates its conformational mobility. The variability of protein molecule geometry is determined by the possibility of varying the torsion (dihedral) angles ϕ and ψ. Torsion angle ω between covalent bonds i i+1 i i+1 changes insignificantly, C α –C and N –C α assuming values close to 180° or 0° . The lengths of covalent bonds between atoms of the protein molecule and values of valence angles between neighboring bonds may change only insignificantly, remaining in a limited range of admissible values. We will call a conformation of a protein molecule a vector x = (x1, …, xn) of Cartesian coordinates of the main chain atoms. We take as preset the initial conformation of the molecule x0 and the final confor mation x1. For conformation x = (x1, …, xn) we denote Δx i : = x i + 1 – x i , r i : = Δx i , i = 1, ..., n – 1, α i : = ∠( – Δx i , Δx i + 1 ) , i = 1, ..., n – 2. In what follow we will assume in accordance with the physicochemical properties of covalent bonds that: 0

1

r i = r i , i = 1, ..., n – 1, 0

1

α i = α i ∈ [ 0...π ], i = 1, ..., n – 2, where ri – lengths of covalent bonds, while αi – planar (valence) angles. Let us call conformation x admissible if 0

1

r i = r i = r i , i = 1, ..., n – 1, 0

1

α i = α i = α i , i = 1, ..., n – 2.

(1)

Let us call an admissible motion of a protein mole cule a Lipschitz function γ: [0, 1] → ⺢3n such that γ(t)—admissible conformation of a protein molecule for any t ∈ [0, 1]. In accordance with (1) this signifies that upon such motion the bond lengths and valence angles do not undergo changes. Confront conformation x with the vector of torsion angles (ψ1, … ψn–3). Torsion angle ψi ∈ [–π, π] – the angle between two planes in which there lie atoms with coordinates xi, xi + 1, xi + 2 and xi + 1, xi + 2, xi + 3 respec tively. The sign of a torsion angle is determined by the direction of rotation of plane xi + 1, xi + 2, xi + 3 about bond [xi + 1, xi + 2].

By the protein conformation (coordinates of mole cule atoms) one can calculate the vector of torsion angles, the values of planar (valence) angles, the lengths of covalent bonds. Conversely, by torsion angles, planar (valence) angles and bonds lengths, cal culation of the molecule coordinates is impossible, inasmuch as two conformations differing only by par allel translation or rotation have identical values of listed parameters. In other words, the listed parame ters define the conformation of a molecule to an accu racy of parallel translation or rotation. Let there be set conformations x0, x1. It is necessary to find a possible motion, transition, rendering one conformation into the other to an accuracy of possible shift (parallel translation) or rotation of any of these conformations. For solving this problem we propose the following mathematical model: motion γ is found as a solution of the problem of minimization among all admissible motions between x0 and T(x1) of functional m

F ( γ ): =

∑m l , p j j

j=1

where lj – path passed by jth atom of the main chain during motion γ, mj – mass of jth atom, p – parame ter (usually we choose p = 1 or p = 2), T runs through possible solidbody motions (shifts and turns) of the molecule. For numerical modeling an admissible motion γ: [0, 1] → ⺢3n is presented in the form of a sequence of a set number M + 2 of intermediate conformations M+1

⎧ ⎛ i ⎞⎫ , of which the initial and final confor ⎨ γ ⎝ ⎠ ⎬ ⎩ M + 1 ⎭i = 0 mations are set (such a sequence we will call approxi mate motion). Therewith the initial conformation coincides with x0, while the final conformation differs from x1 by a solidbody motion (shift and turn) not fixed beforehand. Numerical modeling of motion is actualized by means of applying a method of conjugate gradients [40] to a functional F˜ , set on approximate motions by formula M+1

m

F˜ ( ˜γ ): =

∑

p

m j l j , where l j : =

j=1

∑

k k–1 x˜ j – x˜ j ,

(2)

k=1

k M+1 where ˜γ – approximate motion ˜γ = { x˜ } k = 0 : = M+1

⎧ ⎛ i ⎞⎫ , and presenting as a natural approxi ⎨ γ ⎝ ⎠ ⎬ ⎩ M + 1 ⎭i = 0 mation of functional F. It is actualized in two steps: at the first step, actualized is selection of a “reasonable” initial motion, and then at the second step, realized is the conjugate gradients method proper with the cho sen initial motion. During optimization both at the first and at the second step, only the vectors of torsion BIOPHYSICS

Vol. 59

No. 1

2014

PREDICTION OF PROTEIN SLOW MOTION BASED ON MASS TRANSPORT

31

Table 2. Protein structures used for comparison of algorithms Protein

Length, aa

Calcitonin

32

CyanovirinN Calmodulin Gammadelta resolvase

101 149 183

Transition

RMSD, Å

1BYV : 1 → 1BYV : 7 1BZB : 1 → 1BZB : 10 1J4V : 1 → 1L5B : 1 1PRW : 1 → 1OSA : 1 1GDT : 1 → 2RSL : 1

7.77 10.70 6.41 16.25 2.11

Table 3. Results of modeling of conformational mobility of proteins

Parameters of geometry

Quantity of valence angles beyond allowance Quantity of valence bonds the lengths of which reside beyond allowance Quantity of angles phi and psi in border range Quantity of angles phi and psi in forbidden range

Improved algorithm Linear interpolation of linear interpolation Model of Cartesian of Cartesian coordinates of mixed coordinates and torsion angles elastic networks (Movie Maker) [19], with energy minimization [24], M ± m M±m of intermediate conformations (server Morph) [42], M ± m 41.088 ± 1.149

16.957 ± 1.225

56.088 ± 0.736

9.341 ± 0.619

60.137 ± 1.149

8.189 ± 0.910

59.471 ± 0.711

3.395 ± 0.360

4.114 ± 0.265

3.112 ± 0.266

17.955 ± 0.905

4.667 ± 0.392

1.522 ± 0.158

0.995 ± 0.130

15.213 ± 1.191

1.959 ± 0.272

angles are changed, and also shifts and rotations of the molecule are selected. It should be noted that even for an approximate functional, in this way found is not the point of global minimum, but only approaching is the point of one of local minima. However, from the point of view of modeling the motions corresponding to local minima of functional F are no less interesting than motions corresponding to global minima. The proposed mathematical model and algorithm have been realized in a program package Protein Motion Prediction Framework (PMPF) [41]. RESULTS AND DISCUSSION For verification of the proposed method of model ing we have conducted comparative analysis of the results of work of PMPF and the results obtained with the aid of the following algorithms: Mixed Elastic Network Model [12]; algorithm of linear interpolation of Cartesian coordinates [25]; improved algorithm of linear interpolation of Cartesian coordinates and tor sion angles with energy minimization of intermediate conformations [42]. The indicated methods were used for constructing trajectories of conformational transi tions of the following proteins: eel calcitonin (two pairs of conformations), cyanovirinN, calmodulin, gammadelta resolvase. To each method inputted were BIOPHYSICS

Vol. 59

Model PMPF, M±m

No. 1

2014

two conformations of protein molecule (initial and final) in PDB files (Table 2). Evaluation of the quality of the geometry of inter mediate state constituting the constructed trajectories of conformational motions was executed with the aid of PROCHECK software [43]. For initial, final and every intermediate conformation we determined the percentage of covalent bonds and valence angles the values of which lie beyond allowance, and the percent age of amino acid residues whose values of torsion angles ϕ and ψ lie in border and forbidden regions of the Ramachandran map (Table 3). We calculated mean values M and their standard errors m by the con structed trajectories of motions of all test proteins. Modeling of the conformational mobility of eel calcitonin (PDB ID: 1BYV, transition from conforma tion 1 to conformation 7) has shown that the molecule geometry in the process of transitions built by PATH ENM and MovieMaker algorithms is strongly dis torted. The quantity of valence angles beyond allow ance reaches 50%, of covalent bond lengths beyond allowance – 60 and 70% for algorithms PATHENM and MovieMaker respectively. The MolMovdb algorithm has shown acceptable results, nonetheless being substantially inferior to the proposed algorithm PMPF in the quantity of valence angles beyond allowance.

32

KOSHEVOY et al. 1

10

1 10

Model of conformational transition from conformation 1 (from 1BZB) into conformation 10 (from 1BZB). Numer als denote the first and last atoms of the main chain of cor responding conformations.

By the values of torsion angles ϕ and ψ in border and forbidden ranges the proposed algorithm and MovieMaker algorithm have similar indices, while MolMovdb algorithm— somewhat better. Analysis of conformational transition of eel calci tonin (PDB ID: 1BZB, transition from conforma tion 1 to conformation 10) has shown a similar to the preceding picture of change in the values of stere ochemical indices. The proposed method PMPF and algorithm MolMovdb show similar results, therewith the pro posed method wins in percentages of admissible valence angles and bond lengths, somewhat losing to the latter in the quantity of torsion angles residing in the allowed range, which may be explained by tran sient and local loss of secondary structure by the chain. The results of modeling and the predicted trajectory of conformational motion are presented in the figure. The protein is represented by atoms of the main chain without side amino acid chains. It is shown that the trajectory of the variable part of the molecule looks sufficiently smooth. At the same time the modeling

has not led to significant conformational rearrange ments in the central part of the molecule. Modeling of the conformational transition for a molecule of cyanovirinN (PDB ID 1J4V1L5B) and subsequent analysis of the obtained data have shown that in the number of valence angles beyond allowance the proposed algorithm and MolMovdb algorithm show close results, at some regions of the trajectories insignificantly yielding in indices, on some others win ning over each other. Algorithms PATHENM and MovieMaker significantly lose to the indicated meth ods both in the number of valence angles and in bond length beyond allowance. At the same time the num ber of torsion angles ϕ and ψ residing in border and forbidden regions did not exceed several percent for all compared methods except algorithm PATHENM. Analysis of constructed transitions for a molecule of calmodulin (PDB ID 1PRW, 1OSA) has shown that the MolMovdb algorithm significantly loses to the proposed algorithm of PMPF in the number of valence angles and in lengths of covalent bond residing beyond allowance. For the transition constructed by MolMovdb, the number of valence angles beyond allowance constituted 40%, while the number of bon lengths beyond allowance for some intermediate exceeded 30%. For the transitions constructed by the proposed algorithm of PMPF, the values of the given indices do not exceed 20 and 5% respectively. Calculation of stereochemical indices of the trajec tory obtained by MovieMaker algorithm gave a small quantity of values of torsion angles ϕ and ψ in border and forbidden ranges, not exceeding several percent. At the same time the number of the number of valence angles and lengths of bonds beyond allowance for transition of the molecule constructed upon the use of MovieMaker reached 80%. The PATHENM algorithm shows the worst results in the number of torsion angles ϕ and ψ in bor der and forbidden ranges among all regarded methods. In the number of valence angles and lengths of bonds beyond allowance PATHENM somewhat wins over the MovieMaker algorithm, but loses to other methods. The quantity of valence angles of bond length beyond allowance in intermediate conformations for transition of a molecule of protein gammadelta resolvase (PDB ID 1GDT2RSL) constructed with the aid of PMPF algorithm did not exceed that for ini tial and final conformation, while algorithm Mol Movdb slightly improved the values of given indices for intermediate conformations. For the constructed transitions of protein molecule all abovelisted methods have shown close results in the quantity of torsion angles ϕ and ψ in border and forbidden ranges. The number of torsion angles in the border range for intermediate conformations of transitions did not exceed 7%, in the forbidden range—3%. BIOPHYSICS

Vol. 59

No. 1

2014

PREDICTION OF PROTEIN SLOW MOTION BASED ON MASS TRANSPORT CONCLUSIONS Modeling of the conformational mobility of mac romolecule presents an open problem of structural computational biology. Solution thereof cannot be obtained by methods of molecular dynamics based on minimization of molecule energy: conformational transitions that may proceed over a duration of tens of milliseconds are practically impossible to model by such methods on the strength of the absence of infor mation about external influences on the molecule, which on the large intervals of time appear definitive. In the work, proposed is a new method of predic tion of conformation motion of protein macromole cules. In the basis of the method there lies a new entity—cost of admissible motion, resembling a Monge–Kantorovich functional (more exactly— Kantorovich–Wasserstein metrics) emerging in a problem of optimal mass transfer, and which is mini mized in the process of the work of the algorithm. In this way, the algorithm tries to solve the mass transfer problem in an optimal manner, but with account taken of restraints imposed by the physics of protein. In the work, performed is comparison of the pro posed method with a series of existing algorithms of modeling conformation motion on a set of five pairs of conformations of various proteins. Comparative anal ysis has been performed by the main stereochemical indices of intermediate conformations of the obtained transitions. The results of analysis have shown that the proposed algorithm in a series of indices substantially exceeds the majority of compared methods, notice ably outstripping them also by the computation time. The proposed method has been realized in a pro gram package PMPF, which presents as a flexible soft ware complex for conduction of investigations in the field of prediction of the conformational mobility of protein molecules. The developed software may also be used for investigations in adjoining fields of bioin formatics: prediction of protein structure, modeling of processes of folding and docking. Sophistication of the developed method and soft ware for its realization may proceed by the way of including into calculation the flexibility of side chains of amino acids. Very substantial progress is seen as construction of a complex method combining in itself local minimization, heuristics and evolutionary algo rithms or random adaptive search for finding a global minimum of the cost of admissible motion with taking account of (and circumventing) possible mutual inter sections of the chain. Later on, possible, mass transfer will be replaced with charge transfer, aboriginal con straints will be added on some turns etc. A prospective direction of practical usage of the results of work presents as application of the developed method to problems of prediction of protein structure, modeling of processes of folding and dynamic docking of protein molecules. BIOPHYSICS

Vol. 59

No. 1

2014

33

The PMPF software is freely accessible upon request. ACKNOWLEDGMENTS This work was partially financially supported by Government of Russian Federation, Grant 074U01 and State Contract no. 14.514.11.4068. REFERENCES 1. N. M. Amato, K. A. Dill, and G. Song, J. Comput. Biol. 10, 239 (2003). 2. G. Song and N. M. Amato, in Proc. ACM Int. Conf. on Computational Biology (RECOMB) (2001), pp. 287– 296. 3. L. Kavraki, P. Svestka, J. C. Latombe, et al., IEEE Trans. Robot. Automat. 12 (4), 566 (1996). 4. M. S. Apaydin, A. P. Singh, D. L. Brutlag, et al., in IEEE International Conference on Robotics and Automa tion, Ed. by A. Singh (IEEE Press, New York, 2001), pp. 932–939. 5. M. S. Apaydin, D. L. Brutlag, C. Guestrin, et al., J. Comput. Biol. 10 (3–4), 257 (2003). 6. S. M. LaValle and J. J. Kuffner, in Algorithmic and Com putational Robotics: New Directions, Ed. by B. Donald, K. Lynch, D. Rus (A.K. Peters, Wellesley, Massachu setts, 2001), pp. 293–308. 7. B. Raveh, A. Enosh, O. SchuelerFurman, et al., PLoS Comput. Biol. 5 (2), e1000 295 (2009). 8. R. Elber and D. Shalloway, J. Chem Phys. 112, 5539 (2000). 9. A. Horovitz, A. Amir, O. Danziger, et al., Proc. Natl. Acad. Sci. USA 99, 14095 (2002). 10. Z. Yang, P. Majek, and I. Bahar, PLOS Comput. Biol. 5, e1000360 (2009). 11. M. K. Kim, R. L. Jernigan, and G. S. Chirikjian, Bio phys. J. 83, 1620 (2002). 12. W. Zheng, B. R. Brooks, and G. Hummer, Proteins 69 (1), 43 (2007). 13. W. Zheng and S. Doniach, Proc. Natl. Acad. Sci. USA 100, 13253 (2003). 14. M. M. Tirion, Phys. Rev. Lett. 77, 1905 (1996). 15. H. W. T. Vlijmen and M. Karplus, J. Phys. Chem. 103, 3009 (1999). 16. D. A. Case, in Rigidity Theory and Applications, Ed. by M. Thorpe, P. Duxbury (Springer, Fundamental Mate rials Research, 2002), pp. 329–344. 17. B. R. Brooks, D. Janesic, and M. Karplus, J. Comp. Chem. 16, 1522 (1995). 18. B. R. Brooks, D. Janesic, and M. Karplus, J. Comp. Chem. 16, 1543 (1995). 19. B. R. Brooks, D. Janesic, and M. Karplus, J. Comp. Chem. 16, 1554 (1995). 20. K. Noonan, D. O’Brien, and J. Snoeyink, Intern. J. Robotics Res. 24, 971 (2005). 21. D. Manocha and J. F. Canny, IEEE Transactions on Robotics and Automation 10, 648 (1994). 22. W. J. Wedemeyer and H. A. Scheraga, J. Comp. Chem. 20, 819 (1999).

34

KOSHEVOY et al.

23. E. A. Coutsias, C. Seok, M. P. Jacobson, et al., J. Comp. Chem. 25, 510 (2004). 24. R. Maiti, G. H. van Domselaar, and D. S. Wishart, Nucl. Acids Res. 32, W590 (2004). 25. R. Maiti, G. H. van Domselaar, and D. S. Wishart, Nucl. Acids Res. 33, W358 (2005). 26. I. Bahar, T. R. Lezon, L. W. Yang, et al., Annu. Rev. Biophys. 39, 23 (2010). 27. K. V. Shaitan, N. K. Balabaev, A. S. Lemak, et al., Biofizika 42, 47 (1997). 28. B. J. Alder and T. E. Wainwright, J. Chem. Phys. 31, 459 (1959). 29. J. A. McCammon, B. R. Gelin, and M. Karplus, Nature 267, 585 (1977). 30. M. Karplus and J. Kuriyan, Proc. Natl. Acad. Sci. USA 102, 6679 (2005). 31. K. A. HenzlerWildman, M. Lei, V. Thai, et al., Nature 450, 913 (2007). 32. V. A. Feher and J. Cavanagh, Nature 400, 289 (1999). 33. D. E. Shaw, P. Maragakis, K. LindorffLarsen, et al., Science 330 (6002), 341 (2010). 34. M. S. Apaydin, A. P. Singh, D. L. Brutlag, et al., in IEEE International Conference on Robotics and Automa

35. 36. 37. 38. 39.

40. 41. 42. 43.

tion, Ed. by A. Singh (IEEE Press, New York, 2001), pp. 932–939. N. M. Amato, K. A. Dill, and G. Song, J. Comput. Biol. 10, 239 (2003). S. Thomas, G. Song, and N. M. Amato, Phys. Biol. 2, S148 (2005). J. Cortes, T. Simeon, V. Ruiz de Angulo, et al., Bioin formatics 21, i116 (2005). J. Cortes, L. Jaillet, and T. Simeon, in IEEE Interna tional Conference on Robotics and Automation (IEEE Press, New York, 2007), pp. 3301–3306. L. C. Evans, in Partial Differential Equations and MongeKantorovich Mass Transfer. Current Develop ments in Mathematics (Cambridge, MA Int. Press, Bos ton, MA, 1999), pp. 65–126. J. Nocedal and S. J. Wright, Numerical Optimization (Springer Verlag, New York, 1999). Yu. B. Porozov, A. A. Koshevoy, and E. O. Stepanov, Certificate of State Registration for Computer Software No. 2011619647. Registered 21.12.2011. S. Flores, N. Echols, D. Milburn, et al., Nucl. Acids Res. 34, D296 (2006). R. A. Laskowski, M. W. MacArthur, D. S. Moss, et al., J. App. Cryst. 26, 283 (1993).

BIOPHYSICS

Vol. 59

No. 1

2014