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Rotational Diffusion Depends on Box Size in Molecular Dynamics Simulations Max Linke, Juergen Koefinger, and Gerhard Hummer J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b01090 • Publication Date (Web): 11 May 2018 Downloaded from http://pubs.acs.org on May 12, 2018

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The Journal of Physical Chemistry Letters

Rotational Diffusion Depends on Box Size in Molecular Dynamics Simulations Max Linke,† Jürgen Köfinger,† and Gerhard Hummer∗,† †Max Planck Institute of Biophysics, Max-von-Laue-Str. 3, 60438 Frankfurt am Main, Germany ‡Department of Physics, Goethe University Frankfurt, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany E-mail: [email protected]

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Abstract We show that the rotational dynamics of proteins and nucleic acids determined from molecular dynamics simulations under periodic boundary conditions suffer from significant finite-size effects. We remove the box-size dependence of the rotational diffusion coefficients by adding a hydrodynamic correction kB T /6ηV with kB Boltzmann’s constant, T the absolute temperature, η the solvent shear viscosity, and V the box volume. We show that this correction accounts for the finite-size dependence of the rotational diffusion coefficients of myoglobin and a B-DNA dodecamer in aqueous solution. The resulting hydrodynamic radii are in excellent agreement with experiment.

Graphical TOC Entry

Keywords Rotational Diffusion Tensor, Finite-Size Effects, Quaternion, B-DNA, Hydrodynamics, Rotational Friction

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Hydrodynamics captures the large finite-size effects in the translational self-diffusion coefficients determined from molecular dynamics (MD) simulations of neat fluids 1–4 and lipid membranes. 4,5 The difference between the Stokes friction in an infinite system and in a system under periodic boundary conditions (PBC) determines the hydrodynamic correction. For a particle in a neat fluid, the translational diffusion coefficient from MD simulation is predicted to be underestimated by a correction that depends only on the box size and shape but is independent of the diffusion coefficient of the particle. For a cubic simulation box of length L, addition of 2.837 kB T /6πηL corrects the translational diffusion coefficient, 2,3 where η is the shear viscosity, kB is Boltzmann’s constant, and T is the absolute temperature. This strong dependence on size is a consequence of PBC imposing severe constraints on the hydrodynamic flow, including zero net momentum. Do similarly large finite-size effects afflict rotational diffusion, which plays an important role in macromolecular binding and the theory of nuclear magnetic resonance (NMR) and other spectroscopies? To answer this question, we derive a hydrodynamic correction and test it using MD simulations, from which we extract rotational diffusion tensors using a recently developed quaternion-based algorithm. 6 Our hydrodynamic model treats the molecule of interest as a rigid particle rotating at constant, small angular velocity in an incompressible fluid. In this Stokes regime of hydrodynamics, the angular velocity is given by the torque divided by the rotational friction coefficient ζ r , which is connected to the rotational diffusion coefficient by an Einstein relation, D = kB T /ζ r . The hydrodynamic finite-size correction is the difference between these hydrodynamic diffusion coefficients for an infinite system, with the fluid at rest at infinity, and for a periodic simulation box. The velocity field of the hydrodynamic flow generated by a rotating particle in a periodically replicated box satisfies PBC. Figure 1 compares the flow field around a sphere rotating about the z axis in infinite space and in a cube under PBC. For the latter, symmetry imposes that tangential flows vanish on the four faces at x = ±L/2 and y = ±L/2 that are aligned with the rotation axis, which amounts to a no-slip-like boundary condition. For proteins and

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ω

ω

z infinite

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y x

PBC

Figure 1: Effect of PBC on the Stokes flow around a rotating sphere. (Left) Unrestricted rotational flow in infinite space. (Right) Fluid flow in a cubic box of length L under PBC. The sphere of radius Rh = 0.3L rotates at constant angular velocity ω about the z axis. The velocity field of the surrounding fluid is shown as arrows at the mid-plane, z = 0, and at z = L/2. The unrestricted velocity field is the “rotlet” v(r) = (Rh /r)3 ω × r where ω = (0, 0, ω)T . The vertical blue arrow indicates the axis of rotation ω. For visualization under PBC, we show the simple-cubic lattice sum of rotlets evaluated using Ewald summation, 7 not the full solution of the Stokes equation for no-slip boundary conditions on the surface of the rotating sphere. 8 other macromolecules in a tight box, one may expect significant additional friction from such restrictions on the fluid flow. Zuzovsky et al. 8 generalized Hasimoto’s 1 calculation for the Stokes flow through periodic lattices of stationary spheres to the flow generated by lattices of rotating spheres with no-slip boundary condition. Detailed analytical and numerical calculations for simple cubic (sc), face-centered cubic (fcc), and body-centered cubic (bcc) lattices of rotating spheres 8–10 showed that, to lowest order, the reciprocal of the rotational friction coefficient depends linearly on the volume fraction, 1/ζ r = 1−vmol /V , where vmol = 4πRh 3 /3 is the hydrodynamic volume expressed in terms of the hydrodynamic radius Rh of the rotating sphere. The Wigner-Seitz cells of these three Bravais lattices correspond to cubic (sc), rhombic dodecahedral (fcc), and truncated octahedral (bcc) simulation boxes, respectively. We recover this dependence of the rotational friction on the relative volume of a rotating 4


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sphere from a simple hydrodynamic model of spherical Couette flow between two counterrotating concentric spheres with no-slip boundary conditions. Inspired by Figure 1 and adapting Fushiki’s approach for translational diffusion, 11 we replace the simulation box of volume V by the Wigner-Seitz sphere of equal volume, i.e., with radius R = (3V /4π)1/3 . In the Stokes regime of laminar flow, 12 the friction coefficient ζ r for the rotation of the inner sphere of radius Rh relative to the outer sphere increases by a factor (1 − (Rh /R)3 )−1 over the infinite-system value obtained in the limit R → ∞, which indeed agrees with the detailed calculation 8–10 to lowest order in the volume fraction. Assuming the same box-size dependence for the rotational friction coefficient also for aspherical particles simulated under PBC and the Einstein relation D = kB T /ζ r , we expect that the rotational diffusion coefficient DPBC observed in a simulation depends on box volume as DPBC

vmol D0 = = 1− V

4πRh 3 1− 3V

D0

(1)

where D0 is the infinite-system rotational diffusion coefficient and vmol = 4πRh 3 /3 is the effective volume of the rotating particle. For large, near-spherical macromolecules, we expect the rotational diffusion coefficient to satisfy the Stokes-Einstein relation, D0 = kB T (8πηRh 3 )−1 , which further simplifies the box-size dependence eq 1 to

DPBC

kB T 4πRh 3 kB T = 1− 3 = D0 − 3V 6ηV 8πηRh

(2)

We expect from eq 1 that the apparent rotational diffusion coefficient depends linearly on the inverse box volume V −1 with intercept D0 , and from eq 2 that the slope depends solely on temperature and viscosity. To test the hydrodynamic predictions, we calculated the rotational diffusion tensors for a B-DNA dodecamer and horse-heart myoglobin over a wide range of box volumes. We simulated the Dickerson-Drew dodecamer B-DNA double helix (PDB code 1DUF 13 ) in cubic boxes of volumes 233 nm3 , 365 nm3 , and 580 nm3 in 5 × 1.4 µs, 5 × 1.9 µs, and 10 × 1 µs MD

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trajectories, respectively. The box volumes were chosen so that the minimal distances of the solute to the box edges were 1 nm, 1.5 nm, and 2 nm, respectively. For the B-DNA simulations, we used the GROMACS 2016.3 package, 14 the Amber Parmbsc1 forcefield, 15 TIP3P water, 16 150 mM NaCl, 17 a temperature 18 of 298.15 K, a pressure 19 of 1 bar, and a time step of 2 fs. We simulated horse-heart myoglobin (PDB code 1DWR 20 ) in rhombic dodecahedral boxes of volumes 206 nm3 , 240 nm3 , 293 nm3 , 366 nm3 , and 516 nm3 in 5×1.4 µs, 10×2.3 µs, 5×1.1 µs, 5×2.4 µs, and 5×2.9 µs MD trajectories, respectively. The box volumes were chosen so that the minimal distance of the solute to the box edges was 0.8 nm, 1 nm, 1.25 nm, 1.5 nm, and 2 nm, respectively. The myoglobin simulations used the GROMACS 5.1.4 package, 14 the Amber99sb*-ildn forcefield for the protein, 21–23 the Giammona-Case force field for the CO-bound heme, 24 TIP3P water, 16 150 mM NaCl, 25 a temperature 18 of 293 K, a pressure 19 of 1 bar, and a time step of 2 fs. Other simulation details for the two systems are as in ref 6. From the trajectories, we calculated the rotational diffusion tensors using a least-χ2 fit to the time-dependent quaternion covariances averaged over all trajectories of the same box size (Table S1, and Figures S1 and S2). 6,26 We then used one third of the trace of the diffusion tensor to define an invariant mean rotational diffusion coefficient, DPBC = (DPBC,1 + DPBC,2 + DPBC,3 )/3, for each system and box size, with DPBC,i the eigenvalues of the diffusion tensor. We estimated statistical uncertainties from 100 independent rotational Brownian dynamics simulations of the same durations as the MD simulations, with our best estimate as input diffusion tensor. 6,26 Figure 2 shows the dependence of the mean rotational diffusion coefficients of B-DNA and myoglobin on the inverse box volume. For both systems, we find that the apparent diffusion coefficient DPBC decreases linearly with 1/V , as expected from eq 1. Also shown are single-parameter fits of eq 2. We find that both the absolute values of DPBC and the dependence on box volume can be explained by a single hydrodynamic radius Rh . For myoglobin and the B-DNA, we obtained values of Rh = 2.15(3) nm and Rh = 1.72(3) nm, respectively,

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0.05

B-DNA

B DPBC [ns 1]

A

Myoglobin D0

DPBC [ns 1]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.10

D0

0.09

0.04 0.000 0.004 V 1 [nm 3]

0.000 0.004 V 1 [nm 3]

Figure 2: Mean rotational diffusion coefficients DPBC from MD simulations of (A) myoglobin and (B) B-DNA as a function of the inverse box volume. Results obtained from the quaternion covariances 6 averaged over all trajectories are shown as blue filled circles. Error bars indicate standard errors of the mean (SEM), as estimated 6 from repeated Brownian dynamics simulations using the best-fit model as input. Two-parameter (Rh , D0 ) straight-line fits of eq 1 are shown as blue dashed lines. One-parameter (D0 ) fits of eq 2 are shown as blue solid lines. Individual estimates of the infinite-system value D0 from eq 2 are shown as open gray circles. Global estimates of D0 from eq 2 are shown as horizontal gray dashed lines. with TIP3P water viscosities of 0.326(16) mPas at 293 K and 0.321(16) mPas at 298.15 K, 6,27 respectively. These calculated values of Rh are in excellent agreement with the NMR experimental values of 2.12(2) nm for horse-heart myoglobin 28,29 and 1.71 nm for the DrewDickerson B-DNA dodecamer, 30 respectively. Here we converted second-order (P2 ) rotational correlation times τ2 from NMR into hydrodynamic radii using Rh = (3kB T τ2 /4πη)1/3 , which follows from τ2 = 1/(6D0 ) and D0 = kB T (8πηRh 3 )−1 , with water viscosities of η = 1.0016 mPa s and η = 0.975 mPa s at 20 ◦ C and at 21 ◦ C, respectively. Figure 2 also shows mean rotational diffusion coefficients corrected according to eq 2 7



A

Myoglobin

B 0.14

DPBC,1

0.05

DPBC,2

DPBC [ns 1]

DPBC [ns 1]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.04 DPBC,3

0.12

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B-DNA DPBC,1

0.10 0.08

DPBC,2 DPBC,3

0.06 0.000 0.004 V 1 [nm 3]

0.000 0.004 V 1 [nm 3]

Figure 3: Box-volume dependence of the eigenvalues DPBC,i of the rotational diffusion tensors of (A) myoglobin and (B) B-DNA. Values of DPBC,i are shown as filled circles with SEM as error bars. Two-parameter (Rh , D0 ) straight-line fits of eq 1 are shown as dashed lines. Single-parameter (D0 ) fits of eq 2 are shown as solid lines. by adding kB T /6ηV . For myoglobin, the correction removes the box-size dependence. For B-DNA, the data suggest a residual box size dependence, albeit within the statistical errors. A possible cause for the small deviations is the near-cylindrical shape of B-DNA, which deviates noticeably from the sphere assumed in the hydrodynamic model. To investigate the dependence on molecule shape, we show in Figure 3 the eigenvalues DPBC,i (i = 1, 2, 3) of the rotational diffusion tensors. 6 We find that also the eigenvalues are in good agreement with eq 2, i.e., D0,i ≈ DPBC,i + kB T /6ηV , albeit with larger scatter than the mean rotational diffusion coefficients in Figure 2. The dependence of the rotational dynamics on the box size is also apparent in the orientational correlation functions. Figure 4 shows the correlation functions hhP1 (cos(θ(t)))ii, 8


1.0 0.6 0.4 0.3 0.2 0.1

Myoglobin B-DNA

0

2

4 6 time t [ns]

8

10

increasing volumes

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


P1(cos( (t)))

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Figure 4: Orientational correlation function hhP1 (cos(θ(t)))ii for myoglobin and B-DNA averaged over MD trajectories and orientations of the reference vector for small (dark blue) to large simulation boxes (yellow). where P1 is the Legendre polynomial of order 1, θ(t) is the angle a vector attached to the molecule has traveled in time t, and hh. . .ii indicates a double average over the trajectories and isotropic orientations of the vector in the frame of the molecule. The average P1 correlation function was calculated directly from the variance of the scalar quaternion component. 6 Both for myoglobin and B-DNA, we find that increasing the box size accelerates the rotational dynamics, consistent with the speedup in the rotational diffusion coefficient shown in Figure 2. As further validation, we compare in Figure 5 the prediction of our hydrodynamic model to the simulation results by Takemura and Kitao 31 for the third IgG-binding domain of Protein G (GB3). The hydrodynamic model captures the simulation results without fitting, using only experimental inputs. For the solvent viscosity, we used the bulk water value

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D0Exp

0.054 D [ns 1]

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0.050 0.046 0.042 0.000

0.004

0.008 0.012 V 1 [nm 3]

0.016

Figure 5: Comparison of rotational diffusion coefficients for protein GB3 calculated by Takemura and Kitao 31 (symbols) to the predictions (solid green line) of the hydrodynamic model eq 2 using as input, without fit, the NMR value of D 32,33 (horizontal dashed line) and the bulk-water viscosity at 300 K. The rotational diffusion coefficients were calculated from the simulation data as D = 1/6τ , where τ is the rotational correlation time listed in Table 5 of ref 31. of 8.5 × 10−4 Pa s because the SPC/Eb water model developed by Takemura and Kitao 31 captures the experimental translational diffusion coefficient. For the rotational diffusion coefficient, we used a value measured by NMR 32 and corrected for D2 O and temperature. 33 In Figure S3, we show that earlier simulation results for ubiquitin 34 with three different water models follow our hydrodynamic model with a single hydrodynamic radius independent of water model. Because of their small size, the rotational dynamics of water molecules in the bulk phase at ambient conditions is not noticeably affected by box size for typical system sizes, again consistent with the hydrodynamic model (Figure S4 and Table S2). In summary, we derived a hydrodynamic finite-size correction to the rotational diffusion 10


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coefficient. In the simplest form, eq 2, the infinite-system value is obtained by adding kB T /6ηV to the simulation estimate. Eq 2 predicts an absolute correction independent of the size of the molecule, and a relative correction given by the fraction of the box volume occupied by the rotating molecule, (D0 − DPBC )/D0 = vmol /V . We note that small and highly anisotropic simulation boxes and highly anisotropic molecules might require a more refined hydrodynamic correction. Moreover, the box-size dependence of rotational diffusion could also be affected by long-range electrostatic interactions in the periodic systems, and by thermostats that transfer momentum between distant particles in a non-physical manner violating hydrodynamics. However, we find no indication of such effects as our hydrodynamic model fully explains the observed size dependence within the error. For myoglobin in a box of typical size filled with TIP3P water, the relative correction is about 20 % of the infinitesystem value. Including finite-size corrections thus becomes important when MD simulations are used for quantitative comparisons to experiments sensitive to rotational dynamics, such as NMR or fluorescence. We showed here that MD simulations, after finite-size correction, produce hydrodynamic radii in excellent agreement with experiment.

Acknowledgement We acknowledge financial support by the Max Planck Society.

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Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b00240. Table listing fitted diffusion coefficients for B-DNA and myoglobin for different box volumes; figure showing finite-size effect for ubiquitin with different water models using data from Takemura and Kitao; 34 figures showing the quaternion covariance functions for B-DNA and myoglobin; figures and table of hP1 (cos(θ(t)))i and correlation times of pure TIP3P water for different box volumes.

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