Solvent Friction Effects Propagate over the Entire Protein Molecule

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Solvent Friction Effects Propagate over the Entire Protein Molecule through Low-Frequency Collective Modes Kei Moritsugu,*,† Akinori Kidera,† and Jeremy C. Smith‡ †

Graduate School of Medical Life Science, Yokohama City University, 1-7-29 Suehiro-cho, Tsurumi-ku, Yokohama 230-0045, Japan Center for Molecular Biophysics, University of Tennessee/Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States

‡

ABSTRACT: Protein solvation dynamics has been investigated using atom-dependent Langevin friction coefficients derived directly from molecular dynamics (MD) simulations. To determine the effect of solvation on the atomic friction coefficients, solution and vacuum MD simulations were performed for lysozyme and staphylococcal nuclease and analyzed by Langevin mode analysis. The coefficients thus derived are roughly correlated with the atomic solvent-accessible surface area (ASA), as expected from the fact that friction occurs as the result of collisions with solvent molecules. However, a considerable number of atoms with higher friction coefficients are found inside the core region. Hence, the influence of solvent friction propagates into the protein core. The internal coefficients have large contributions from the low-frequency modes, yielding a simple picture of the surface-to-core long-range damping via solvation governed by collective low-frequency modes. To make use of these findings in implicit-solvent modeling, we compare the all-atom friction results with those obtained using Langevin dynamics (LD) with two empirical representations: the constant-friction and the ASA-dependent (Pastor− Karplus) friction models. The constant-friction model overestimates the core and underestimates the surface damping whereas the ASA-dependent friction model, which damps protein atoms only on the solvent-accessible surface, reproduces well the friction coefficients for both the surface and core regions observed in the explicit-solvent MD simulations. Therefore, in LD simulation, the solvent friction coefficients should be imposed only on the protein surface.

1. INTRODUCTION Molecular dynamics (MD) simulation has been extensively used to study dynamical properties of biomolecules1 and sample their accessible potential energy landscapes.2 However, although computer power has increased massively over the past decades, MD simulation in which solvent atoms are explicitly treated is restricted to timescales too short for many biologically relevant motions of proteins. To overcome this difficulty, implicit-solvent MD simulation is applied, thus saving the computational cost of calculating solute−solvent and solvent−solvent interactions, which usually requires most of the computational time when simulating fully solvated systems.3,4 Solvation effects can be considered as modifying effective potentials of mean force and dynamical effects on these potentials.5 Implicit solvation has been examined extensively from an energetic point of view: the electrostatic interactions can be calculated using, for example, the Poisson−Boltzmann6 or generalized Born equations,7−9 and the nonpolar hydrophobic interactions have been estimated via the atomic solventaccessible surface areas (ASA).10,11 For the purpose of introducing the dynamical contribution of solvation in implicit solvent MD studies, the Langevin equation has been used, in which a friction term and a random force corresponding to the dissipation-fluctuation theorem are added to Newtonian dynamics.13−20 The collisions of solvent molecules with solute atoms are represented phenomenologically via the friction © 2014 American Chemical Society

term. Conventionally, the friction coefficient has been determined “heuristically” from a hydrodynamic equation, such as Stokes’ law,21 either as a constant value for all atoms (the “constant model”)13−18 or with a value proportional to the ASA12,19 using the Pastor−Karplus model22 (the “ASA model”). However, how protein solvation affects relative effective frictional coefficients of atoms on the surface and in the core dynamics is not understood. For example, it is known that the constant friction model with the friction coefficients evaluated according to the Stokes’ law (∼60−80 ps−1) leads to an excess damping of protein dynamics relative to that in explicit solvent.14,17 Therefore, the constant friction model with ∼0.5−10 ps−1 is usually used for Langevin dynamics (LD) simulations of biomolecules as a thermostat. However, the reason for the overestimation has not been explicitly examined. Furthermore, recent neutron scattering work on the effect of surface hydration water has shown that the effect of surface hydration increases the volume of localized single-well diffusion in such a way that the hydration effect propagates from the protein surface into the dry core.23 How the associated friction coefficients are modified is not known. Here, we obtain an understanding of the solvent frictional contribution to protein dynamics by the quantification of atomReceived: April 23, 2014 Revised: June 5, 2014 Published: June 25, 2014 8559

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dependent Langevin friction coefficients. To do this, we first perform full-scale MD simulations in explicit solvent to evaluate the atom-dependent friction, and then examine which friction model best reproduces the results of the explicit solvent simulations when employed using coarser-level, implicit-solvent LD simulations. To quantify the atom friction coefficients, the decay of the velocity autocorrelation function (VACF) of the associated atom must be characterized. This is, however, not straightforward, since damping of the atomic VACF depends not only on friction but also on vibrations with various ranges of frequencies (see Theory and Methods section for detail).24 To overcome this difficulty, the MD simulation results are analyzed assuming that the VACF decays as a damped harmonic oscillator based on Langevin equation.20 In Langevin mode analysis, the potential energy is approximated as harmonic, allowing normal-mode analysis and thus a Langevin frequency and friction coefficient to be derived for each normal mode (i.e., each so-called “Langevin mode”) by fitting the VACF of the associated normal mode to that of the onedimensional Langevin equation.25−27 In our previous studies, protein dynamics was described by calculating the vibrational frequencies and frictions of normal modes which were directly derived from MD trajectories, and this allowed the temperature and hydration dependence of the protein vibrational dynamics to be characterized.26,27 In the present study, we address quantification of the atom-dependent Langevin friction coefficients using the normal-mode friction coefficients thus derived. To test whether protein structural dependence of friction dynamics exists, the analysis was performed on simulations of two proteins: lysozyme and staphylococcal nuclease (SNase), the former containing mostly helices and the latter sheets. The Langevin friction coefficients thus derived are analyzed to understand the dynamical solvation effects on the protein molecule. We find propagation of solvent damping effects from the surface to the core via low-frequency collective modes. However, we also find that, in implicit solvent simulations with Langevin dynamics, solvent friction coefficients should be imposed only on the protein surface atoms.

set 2233 was used for the potential function. The vacuum simulations were performed in the microcanonical ensemble with the average kinetic energy corresponding to the system temperature of 300 K, followed by heating to the target temperature of 300 K during 30 ps and equilibration for 100 ps with velocity scaling in the NVE ensemble. In the vacuum simulation, the electrostatic interaction was treated using the generalized Born implementation of CHARMM (GBSW),8,9 and the apolar solvation term using the surface tension parameter set to a value of 0.005 kcal/mol/Å2.10,11 The atomic solvent-accessible surface area was calculated using a probe radius of 1.4 Å.22 For the solution simulations, electrostatic interactions were calculated with a dielectric constant of 1 using the particle-mesh Ewald method.34,35 The systems were energy minimized with 1000 steps of the conjugate gradient method. Then, each simulation system was uniformly heated to 300 K during 30 ps and equilibrated for 100 ps with velocity scaling in the NVE ensemble. Subsequent simulations were carried out at constant temperature (300 K) and pressure (1 atm) conditions (the NPT ensemble), for 500 ps equilibrium run. Finally, the production runs were carried out in the microcanonical ensemble with the average kinetic energy corresponding to the system temperature of 300 K. The lengths of all the MD production runs were 1 ns. Although short by modern standards, this simulation length is more than adequate for determining the friction coefficients. Both the atomic coordinates and velocities were saved every 5 fs for analysis. Normal-mode analyses were performed using the CHARMM program with the vacuum model and potential function. For these, the average coordinates of the 1 ns MD production run trajectories were used as the starting structure. The structures were energy minimized with 1000 steps of the steepest descent method, followed by adopted basis Newton−Raphson minimization32 until a rms energy gradient was reached of 10 ps−1), probably due to anharmonic motions, another source of error. To avoid these problems, we adopted the following approximation

j

where Hvac and Hwat are the second-derivative (Hessian) matrices for the vacuum and solution simulations, respectively. γvac,i originates from damping by intraprotein interactions via anharmonicity, whereas γwat,i arises from both surrounding solvent molecules and intraprotein interactions. The atomic friction coefficient arising from solvent effects, which is renamed as Δγatom,i, is then assumed to be quantified as the difference of the two, i.e., Δγatom,i = γwat,i − γvac,i. Now we consider how to calculate the friction coefficients from the MD trajectories. The derivation of Langevin mode analysis allows the friction, γi, to be calculated using the VACF.20,24,26 γi = −β

∑ uin2Δγmode,n n

= −∑ H vac, ijrj − miγvac, iri̇ + R vac, i(t )

j

(6)

After a set of normal-mode friction coefficients Δγmode,n (or the difference between in vacuum and in solvent, Δγmode,n = γwat,n − γvac,n) is obtained, the atom friction, Δγatom,i, can be determined using the diagonal friction approximation for both as follows

Here, the friction γi was quantified directly using both the vacuum and solution MD simulations. The two simulations were analyzed based on the model of Langevin mode,20 assuming that the potential energy is harmonic around the equilibrium position as follows mi

for ω > 250 cm−1

Δγatom, i = γ

(10)

where γ is a constant friction equal to ∑iΔγatom,i/N with N being the total number of non-hydrogen atoms in the protein. The values of γ for the two proteins derived were similar, 2.3 ps−1 for lysozyme and 2.6 ps−1 for SNase. The friction coefficients for non-hydrogen atoms in the ASA model are given by

(5)

where vn̅ is the mass-weighted normal-mode velocity, and ϖn = (ωn2 − γn2/4)1/2 is the effective frequency for underdamped modes (γn < 2ωn) and ϖ0,n = (γn2/4 − ωn2)1/2 holds for overdamped modes (γn ≥ 2ωn). γn (and ωn as well) was then calculated by directly fitting eq 5 to the VACF of mode n calculated from the MD trajectories over the range t = 0−5 ps.26 The integral of the square deviation between the MD and model VACF, i.e., ∫ 50 dt |ψMD(t) − ψmodel(t)|2, was calculated for both the underdamped/overdamped models (eq 5) as an evaluation function, allowing determination of whether the associated dynamics is underdamped or overdamped.26 In the high-frequency region (with the normal-mode frequency, ωNMA ≡ ω > 250 cm−1), since the vibrational contribution is small, the VACF is approximately modeled only by the damping term as

Δγatom, i = κSi

(11)

2

where Si (Å ) is the solvent-accessible surface area of atom i and κ (ps−1/Å2) is calculated as ∑i(Δγatom,i/Si)/N. The calculated values of κ for the two proteins were again similar: 0.67 for lysozyme and 0.74 for SNase. These united-atom models of friction, neglecting hydrogen atoms, have been commonly used,31 and use of an all-atom model, assigning friction also to hydrogen atoms, did not change the vibrational density of states, g(ω), of the united-atom model (results not shown). 2.4. Vibrational Density of States. The atom friction coefficients derived were examined by comparing the vibrational density of states, g(ω), from the associated implicit solvent LD with g(ω) from the explicit solvent MD. For this 8561

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comparison the spectra of low-frequency modes ( 100 cm−1, in contrast, the distributions of both the vacuum and solution frictions seem similar to each other, indicating a smaller contribution from solvation dynamics.23 As a result, γwat,mode has a small frequency dependence. Friction coefficients on the atom basis, Δγatom, were then calculated using the normal-mode frictions (see Theory and Methods for details). The averages of Δγatom over the nonhydrogen atoms are similar for lysozyme (2.3 ps−1) and SNase (2.6 ps−1), implying transferability of these frictional parameters between different protein structural classes. Since the averages of Δγatom for hydrogen atoms are much smaller (1.2 ps−1 for lysozyme and 1.4 ps−1 for SNase) than those for the non8562

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shifted, with the friction monotonically increasing with frequency; see also Figure 1, which may correspond to the situation in which the intraprotein interactions are uniformly stiffened. In contrast, the frequency dependence of the ASA model friction is similar to that of the solution MD simulation (more or less constant independent of the frequency; see also Figure 1), with a large Δγmode in the low-frequency region. The difference between the two models is also visible in the averaged atom-dependent friction coefficient, ⟨Δγatom⟩. Figure 5

Figure 3. Vibrational density of states, g(ω), from vacuum MD (gray dashed curve), solution MD (gray curve), and Langevin dynamics together with the constant-friction model (black dashed curve) and the ASA-friction model (black curve). (a) Lysozyme and (b) SNase.

surface atoms (Δγatom ≥ 10 ps−1), whereas in the constant model (Δγatom = 2.3 or 2.6 ps−1 for lysozyme and SNase, respectively) the atomic frictions on the surface are not sufficiently high to damp the low-frequency modes, and the friction on the core atoms with small-amplitude motions is overestimated. This is clearly evidenced by the values of ⟨γmode⟩, i.e., the average of γmode over the normal modes with a bin width of 50 cm−1 (Figure 4). This averaging procedure smooths the fluctuating values of γmode yielding the result that the constant-friction model produces almost the same frequency dependence of ⟨γmode⟩ as does the vacuum simulation, albeit

Figure 5. Non-hydrogen atom frictions as a function of ASA, averaged over bins with a width of 5 Å2, ⟨Δγatom⟩. Solution MD (gray curve) and Langevin dynamics together with the constant-friction model (black dashed curve) and the ASA-friction model (black curve). (a) Lysozyme and (b) SNase.

shows that the LD simulation with the ASA model correctly mimics the results of the solution MD, yielding atomic frictional coefficients proportional to the ASA. In contrast, the LD simulation with the constant model has values of ⟨Δγatom⟩ with a much smaller ASA dependence, implying that both the core and the surface regions have similar frictional properties. These results indicate that the ASA model is far superior to the constant model as a representation of solvent damping in LD simulation. To strengthen the usefulness of the ASA model, Δγatom was again calculated from the LD simulation with the ASA model. Figure 6 shows that the surface-to-core propagation of the solvation effects through the low-frequency modes, which is observed in the solution MD simulations (see Figure 2), is indeed recovered by the ASA model (see eq 11 or the relation between ASA and the friction coefficient). Note that this surface-to-core propagation leads to the consequence that the input friction for the LD simulation results in extremely different output friction or Δγatom (Figure 6).

4. CONCLUSION The present study targeted a phenomenological understanding of protein solvation dynamics through the evaluation of atomdependent Langevin friction coefficients. To do this, a multiscale scheme was adopted, applying Langevin mode analysis to both vacuum and solution MD trajectories of lysozyme and SNase, thus yielding normal-mode friction coefficients both in vacuum and in solution. The solvationinduced atomic friction coefficients, derived as the difference of the two, were found to be roughly correlated with the atomic ASA, but also to have high values in the protein core. This

Figure 4. Normal-mode friction coefficients as a function of frequency, averaged over bins with a width of 50 cm−1, ⟨γmode⟩. Vacuum MD (gray dashed curve), solution MD (gray curve), and Langevin dynamics together with the constant-friction model (black dashed curve) and the ASA-friction model (black curve). (a) Lysozyme and (b) SNase. 8563

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(4) Chen, J.; Brooks, C. L., III; Khandogin, J. Recent Advances in Implicit Solvent-based Methods for Biomolecular Simulations. Curr. Opin. Struct. Biol. 2008, 18, 140−148. (5) Smith, J. C.; Merzel, F.; Bondar, A. N.; Tournier, A.; Fischer, S. Structure, Dynamics and Reactions of Protein Hydration Water. Philos. Trans. R. Soc. London B 2004, 359, 1181−1190. (6) Smart, J. L.; Marrone, T. J.; McCammon, J. A. Conformational Sampling with Poisson-Boltzmann Forces and a Stochastic Dynamics/ Monte Carlo Method: Application to Alanine Dipeptide. J. Comput. Chem. 1997, 18, 1750−1759. (7) Still, W. C.; Tempczyk, A.; Hawley, R. C.; Hendrickson, T. Semianalytical Treatment of Solvation for Molecular Mechanics and Dynamics. J. Am. Chem. Soc. 1990, 112, 6127−6129. (8) Im, W.; Lee, M. S.; Brooks, C. L., III Generalized Born Model with a Simple Smoothing Function. J. Comput. Chem. 2003, 24, 1691− 1702. (9) Im, W.; Feig, M.; Brooks, C. L., III An Implicit Membrane Generalized Born Theory for the Study of Structure, Stability, and Interactions of Membrane Proteins. Biophys. J. 2003, 85, 2900−2918. (10) Simonson, T.; Brunger, A. T. Solvation Free Energies Estimated from Macroscopic Continuum Theory: An Accuracy Assessment. J. Phys. Chem. 1994, 98, 4683−4694. (11) Sitkoff, D.; Sharp, K. A.; Honig, B. Accurate Calculation of Hydration Free Energies Using Macroscopic Continuum Models. J. Phys. Chem. 1994, 98, 1978−1988. (12) Shen, M. Y.; Freed, K. F. Long Time Dynamics of Metenkephalin: Comparison of Explicit and Implicit Solvent Models. Biophys. J. 2002, 82, 1791−1808. (13) Smith, J. C.; Cusack, S.; Tidor, B.; Karplus, M. Inelastic Neutron Scattering Analysis of Low-frequency Motions in Proteins: Harmonic and Damped Harmonic Models of BPTI. J. Chem. Phys. 1990, 93, 2974−2991. (14) Loncharich, R. J.; Brooks, B. R.; Pastor, R. W. Langevin Dynamics of Peptides: the Frictional Dependence of Isomerization Rates of N-acetylalanyl-N′-methylamide. Biopolymers 1992, 32, 523− 535. (15) Li, X.; Hassan, S. A.; Mehler, E. L. Long Dynamics Simulations of Proteins Using Atomistic Force Fields and a Continuum Representation of Solvent Effects: Calculation of Structural and Dynamic Properties. Proteins 2005, 60, 464−484. (16) Khalili, M.; Liwo, A.; Jagielska, A.; Scheraga, H. A. Molecular Dynamics with the united-Residue Model of Polypeptide Chains. II. Langevin and Berendsen-bath Dynamics and Tests on Model Alphahelical Systems. J. Phys. Chem. B 2005, 109, 13798−13810. (17) Hamelberg, D.; Shen, T.; McCammon, J. A. Insight into the Role of Hydration on Protein Dynamics. J. Chem. Phys. 2006, 125, 094905. (18) Jagielska, A.; Scheraga, H. A. Influence of Temperature, Friction, and Random Forces on Folding of the B-domain of Staphylococcal Protein A: All-atom Molecular Dynamics in Implicit Solvent. J. Comput. Chem. 2007, 28, 1068−1082. (19) Kent, A.; Jha, A. K.; Fitzgerald, J. E.; Freed, K. F. Benchmarking Implicit Solvent Folding Simulations of the Amyloid Beta(10−35) Fragment. J. Phys. Chem. B 2008, 112, 6175−6186. (20) Lamm, G.; Szabo, A. Langevin Modes of Macromolecules. J. Chem. Phys. 1986, 85, 7334−7348. (21) Chandrasekhar, S. Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys. 1943, 15, 1−89. (22) Pastor, R. W.; Karplus, M. Parametrizatlon of the Friction Constant for Stochastic Simulations of Polymers. J. Phys. Chem. 1988, 92, 2636−2641. (23) Hong, L.; Cheng, X.; Glass, D. C.; Smith, J. C. Surface Hydration Amplifies Single-well Protein Atom Diffusion Propagating into the Macromolecular Core. Phys. Rev. Lett. 2012, 108, 238102. (24) Kitao, A.; Hirata, F.; Go̅, N. The effects of Solvent on the Conformation and the Collective Motions of Protein: Normal Mode Analysis and Molecular Dynamics Simulations of Melittin in Water and in Vacuum. Chem. Phys. 1991, 158, 447−472.

Figure 6. Non-hydrogen atom friction coefficients, Δγatom, calculated from Langevin dynamics together with the ASA-friction model plotted as a function of the input atom friction coefficients in Langevin dynamics simulation, γatom,LDin. (a) Lysozyme and (b) SNase.

propagation of solvent damping effects is due to low-frequency collective modes, which were found to undergo considerable damping by the surrounding water molecules. Two simple models of the Langevin friction coefficients, the constant-friction and ASA-dependent friction models, were examined. It was found that LD with the ASA model mimics the phenomenological collisions between the surface atoms and the surrounding solvent, resulting in the propagation of dynamic damping effects into the protein core through lowfrequency collective modes. In contrast, the constant model underestimates the damping at the protein surface. Therefore, in Langevin dynamics simulation solvent friction coefficients should be imposed only on the protein surface.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +81-45-508-7233. Fax: +81-45-508-7367. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS K.M. and A.K. acknowledge support by the MEXT grand challenge program using next-generation supercomputing. K.M. was supported by MEXT Grant-in-Aid for Young Scientists, 25840060, and A.K. by MEXT, Grant-in-Aid for Scientific Research, 23247027. J.C.S. acknowledges funds from the U.S. Department of Energy via a Laboratory-Directed Research and Development grant. The computations were partly performed on the RIKEN Integrated Cluster of Clusters (RICC).

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