Supplemental Material for Global Langevin model of multidimensional biomolecular dynamics

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1 Supplemental Material for Global Langevin model of multidimensional biomolecular dynamics Norbert Schaudinnus, Benjamin Lickert, Mithun Biswas and Gerhard Stock Biomolecular Dynamics, Institute of Physics, Albert Ludwigs University, 74 Freiburg, Germany (Dated: September, 6) The first two authors contributed equally.

2 I. ALANINE DIPEPTIDE (ALAD) A. MD simulation details We conducted MD simulations of alanine dipeptide (Ac-Ala-NHCH 3, AlaD) at 3K, using the AMBER ff99sb force field in the GROMACS program suite. The structures were immersed in a dodecahedral box of TIP3P 3 water molecules such that there was at least Å of separation between the solute and the edge of the box. Periodic boundary conditions were used and the long range electrostatics was treated with the particle mesh Ewald method 4 with a Fourier spacing of. Å and sixth order interpolation to compute potential and forces between the grid points. For both the van der Waals and Coulomb interactions the cut-off was set to Å. All bonds were constrained using the LINCS algorithm. 5 The integration time step was fs. and coordinates were saved every steps during the simulations. A high frequency for saving the frames was necessary to provide sufficient data for Langevin modeling. First the water molecules were energy minimized using the steepest decent method. To equilibrate the system at the target temperature and to find an optimum box size, the system was simulated for ns in a NPT ensemble. Pressure was kept constant with a Berendsen barostat 6 having coupling constant τ P = ps and a compressibility of β = bar. For temperature control, the Bussi-Donadio-Parinello thermostat 7 with a coupling time τ T = ps was used. The last 5 ns of the NPT run was used to calculate the mean size of the water box. Finally, the production run for the MD simulations was performed in the NVT ensemble. B. Definition of reaction coordinate The backbone dihedral angles φ, ψ are often used to describe conformational dynamics of peptides. The Ramachandran plot of AlaD in Fig. Sa reveals four distinct metastable states: Left- and right-handed helical conformation α L and α R, the β sheet and the polyproline II state P II. While the most frequent conformational transition occurs between β sheet and P II, the α L state is very sparsely visited. In the course of our 5 ns MD simulation we observe only 3 transitions. Hence, in order to establish a simple one-dimensional model we restrained our system to the framed area in Fig. Sa, by only considering parts of the trajectory that stay in that region. Since in this area the dynamics of the coordinates φ and ψ are in good approximation uncorrelated, we focused on the essential coordinate ψ. In order to achieve a path-connected coordinate space, we shifted our system cyclically along the φ direction by φ = 5. Along the ψ direction, we excluded the very few transitions across the highest free energy barrier at ψ = to construct a simple linear representation of the cyclic coordinate. For convenience, we introduced the shifted coordinate x = ψ 7.

3 An excerpt of the resulting MD time series x(t) is shown in Fig. Sb. ψ - β PII α R α L a 8 4 x - b φ t [ns] FIG. S: (a) Ramachandran plot of alanine dipeptide. We focus on the marked area and impose cyclic shifts φ = 5 and ψ = 7 to achieve a path-connected collective coordinate. (b) The resulting time series resolves the transition α R β/p II along the shifted coordinate x = ψ 7. C. Calculation of first passage time and transition path time The free energy of AlaD projected on x = ψ 7 shows two minima centered at x 83 and x 85, respectively, which correspond to the states α R and β/p II, see Fig. Sa. To define the extension of the states, we chose a radius of r = 5, i.e. x i ± r define the borders of the states. Based on the definition of the states (centers and radii), we follow the trajectory after exiting one state till it enters the other state which directly yields the transition path time τ tp. To calculate the first passage time τ FP, we first assign all frames along the trajectory without a state definition to the last state visited. Subsequently, for each frame in the initial state, the first passage time τ FP denotes the number of steps it takes to reach the other state. 3

4 D. Properties of the noise model Using various time steps δt for the propagation of the dle, Fig. S4 shows (a) the autocorrelation function and (b) mean and standard deviation of the noise (for calculation, seeref. 8 ). Whilealltimestepsproducezeromeanandunitstandarddeviation,weneedδt fs to have the autocorrelation function decay to (almost) zero in the first time step (which reflects delta-correlated white noise). Note that the behavior of the noise autocorrelation function for longer times does not affect the dle simulations. noise autocorrelation a 4fs fs fs 4fs noise mean/sd 3 - b 4fs fs fs 4fs timestep ψ FIG. S: Properties of the noise model of AlaD, obtained for various time steps δt of the dle simulations. Shown are (a) autocorrelation function and (b) mean and standard deviation of the noise. 4

5 II. REPRESENTATION OF MULTIDIMENSIONAL FIELDS As explained in the main text, we obtain from the dle various fields, e.g., the friction ˆΓ(x)andthenoiseamplitude ˆK(x). Duetofinitestatistics, thesefieldestimatesaretypically noisy, in particular for multivariate observables. Since the construction of a global model requires sufficiently smooth and well sampled fields, in the following we propose two simple algorithms that smoothen and extrapolate a given field. As an illustrative example, we consider a scalar field, b(x,x ), depending on two variables. By using some suitable grid (i,j) the field is histogrammed into bins, which gives for each bin n(i,j) raw data points with an average value b org (i,j). Smoothing of this field is achieved by employing a weighted average over all neighboring bins that contain at least a single entry, b(i,j) = n tot i+ j+ k=i l=j n(k,l)b org (k,l), () where n tot = i+ j+ k=i l=j n(k,l) denotes the total number of data points. The procedure may be repeated several times in order to achieve sufficiently smooth fields. We applied the proposed algorithm to two-dimensional projections of the dle field estimates for the three-dimensional model of heptaalanine (Ala 7 ). As a representative example, Fig. S3 shows the friction fields Γ (x,x ) and Γ (x,x ) before and after smoothing. Note that the shape and the value range are preserved by the algorithm. By comparing the value ranges, the figure also reveals that the off-diagonal part of the friction matrix Γ is much smaller that the diagonal part. In a second step, one may improve the representation of sparsely sampled regions by replacing the field values of empty bins by the average trend of the surrounding bins. That is, given some empty bin with n(i,j) =, we scan its neighborhood for nonempty bins b(k,l) which satisfy k i and l j. In order to estimate the value of b(i,j), we make first-order Taylor expansions for the neighboring points. E.g., starting at point (i, j) and extrapolating in the direction of the first index, this approximation reads b(i,j)+(b(i,j) b(i,j)). Averaging over all possible directions, the estimate for b(i,j) can be written as b(i,j) = A i+ j+ k=i l=j b(k,l)+b ijkl, b ijkl = b(i+k i, j+l j) δ ijkl, () where δ ijkl δ(n(k,l))δ(n(i+k i,j+l j)) assures that the bins n(k,l) and n(i+k i,j+l j) are not empty and A = i+ j+ k=i l=j δ ijkl denotes the number of contributing terms. Subsequent to the extrapolation, another smoothening step is applied as described above. 5

6 a x b x x c x d x. x x - x FIG. S 3: Effect of the smoothening algorithm demonstrated for friction field Γ of the threedimensional model of Ala7. The diagonal element Γ (x, x ) is shown (a) from a binning of the dle estimates and (b) after applying the smoothening algorithm to it. Similarly, the off-diagonal element Γ (x, x ) is shown (c) before and (d) after smoothening. 6

7 III. HEPTAALANINE (ALA 7 ) A. Definition of reaction coordinates As for AlaD, the dihedral angles ψ i reveal the essential dynamical information of Ala 7. So, we again chose shifted coordinates x i = ψ i 7 (i =,,3) of the three inner residues as reaction coordinates. For purposes of presentation (see Fig. 4a in the main text), we rotated our data by α = 45 clockwise around the axis v = (,,) T. This rotation is described by x = R(α) x with 3 sin ( α)+cos(α) 3 sin ( α)+ 3 sin(α) 3 sin ( α) 3 sin(α) R(α) = 3 sin ( α) 3 sin(α) 3 sin ( α)+cos(α) 3 sin ( α)+ 3 sin(α). (3) 3 sin ( α)+ 3 sin(α) 3 sin ( α) 3 sin(α) 3 sin ( α)+cos(α) B. EVB model parameters The EVB model of the free energy of Ala 7 [Eq. (6) in the main text] is described by the energy matrix V(x,x,x 3 ) with diagonal elements (in units of k B T) V (x,x,x 3 ) =.43(x +43.) +.445(x +3.6) +.44(x ) V (x,x,x 3 ) =.48(x 4.59) +.94(x 38.8) +.99(x ) V 33 (x,x,x 3 ) =.5(x +9.6) +.49(x 64.65) +.97(x ) V 44 (x,x,x 3 ) =.6(x +5.3) +.55(x 6.9) +.4(x ) V 55 (x,x,x 3 ) =.73(x +68.9) +.659(x 8.4) +.646(x ) V 66 (x,x,x 3 ) =.6(x.3) +.6(x 97.) +.6(x ) +7.4 V 77 (x,x,x 3 ) =.95(x 33.4) +.5(x ) +.46(x ) V 88 (x,x,x 3 ) =.56(x 79.3) +.9(x +84.5) +.54(x ) and the off-diagonal part V coup =

8 C. Properties of the noise model Using various time steps δt for the propagation of the dle, Fig. S?? shows (a) the autocorrelation function and (b) mean and standard deviation of the noise. The minimum of δt = ps is determined by the write-out time of the MD simulation, as a maximum we choose δt = ps. In all cases, the noise is found to be delta-correlated and shows zero mean and unit standard deviation noise autocorrelation ps ps 4 ps ps a timestep 4 noise mean/stdev ps ps 4 ps ps b x FIG. S4: Properties of the noise model of Ala 7, obtained for various time steps δt of the dle simulations. Shown are (a) autocorrelation function and (b) mean and standard deviation of the noise. 8

9 D. Calculation of transition rates To describe the conformational dynamics of the three-dimensional model of Ala 7, we calculate the transition rates k i j between the eight states of the system. As written in the main text, the eight states can be labeled as abc with a,b,c = α,β in line with the structures of the three inner residues of Ala 7. In the rotated space (see above), the eight states are centered at x (ααα) = (,, ) T, x (βαα) = (45,, 4) T, x (αβα) = ( 4,45, ) T, x (ββα) = (,3, 6) T, x (αββ) = ( 6,,3) T, x (βββ) = (8,8,8) T, x (ααβ) = (, 4,45) T and x (βαβ) = (3, 6, ) T. Approximating the shape of the states by a sphere, we furthermore choose a radius r = 5 to define the region of each state. By knowing centers and widths of these states, a state trajectory n(t) was calculated that assigns a state n to every MD frame. Based on this definition, we counted the number of transitions N i j between state i (start) and state j (ending) for all states which occur within a certain lag time τ lag. We chose τ lag = ps, which is also the time step of the MD and dle (and hence the most natural choice). The transition probability i j for the transition p i j is then p i j = N i j / k N i k, (5) with the summation over k being performed over all eight ending states (so, i = k is included too). Then, the transition rates k i j can be calculated by k i j = p i j /τ lag. (6) Errors are estimated by cutting the trajectory into pieces and calculating the standard deviation of the rates for the different pieces. 9

E. Modifications of the model As explained in the main text, we constructed a modified model of Ala 7, in which the four barriers are increased over which transitions of the central angle occur, i.e., aαb aβb where a,b {α,β}.

S5b shows the outcome of a mle simulation starting in the lower part of the free energy landscape. Due to the increase of the barriers, the dynamics is restricted to the lower part.

10 E. Modifications of the model As explained in the main text, we constructed a modified model of Ala 7, in which the four barriers are increased over which transitions of the central angle occur, i.e., aαb aβb where a,b {α,β}. Fig. S5a shows the resulting free energy landscape of the modified EVB model. As a further illustration, Fig. S5b shows the outcome of a mle simulation starting in the lower part of the free energy landscape. Due to the increase of the barriers, the dynamics is restricted to the lower part. x x a x x b FIG. S5: (a) Free energy landscape of a modified EVB model of Ala 7. (b) Due to the high barriers between the upper and lower part of the energy landscape, a mle mle simulation starting in the lower part remains restricted to the lower part. F. Calculation of mean first passage times To calculate mean first passage times τ MFP of Ala 7 listed in Table I of the main text, we need to generalize the simple method for two states (see Sec. IC) to the case of several states. As a simple example, we consider three states a,b,c. To calculate the mean first passage time from a c, we consider an example trajectory with total time T taken to reach from a to c and is written as: T = t a, +t b, +t a, ; where t m,i is the lifetime in state m during the i-th visit. Since a trajectory can start in state a in any one of the multiple visits, we obtain a distribution of passage times from a c. The mean first passage time τ MFP is the probability of starting a trajectory during the i-th visit times the averaged first passage time from this visit to c. In the example above, we find t a, τ MFP = ( t a, t a, +t a, +t t a, b, +t a, )+ ( t a, t a, +t a, ) (7)

11 V. Hornak, R. Abel, A. Okur, B. Strockbine, A. Roitberg, and C. Simmerling, Comparison of multiple Amber force fields and development of improved protein backbone parameters, Proteins 65, 7 (6). B. Hess, C. Kutzner, D. van der Spoel, and E. Lindahl, Gromacs 4: Algorithms for highly efficient, load-balanced, and scalable molecular simulation, Journal of Chemical Theory and Computation 4, 435 (8). 3 W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. Klein, Comparison of simple potential functions for simulating liquid water, J. Chem. Phys. 79, 96 (983). 4 T. Darden, D. York, and L. Petersen, Particle mesh Ewald: An N log(n) method for Ewald sums in large systems, J. Chem. Phys. 98, 89 (993). 5 B. Hess, H. Bekker, H. J. C. Berendsen, and J. G. E. M. Fraaije, LINCS: A linear constraint solver for molecular simulations, J. Comp. Chem. 8, 463 (997). 6 H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. Dinola, and J. R. Haak, Molecular dynamics with coupling to an external bath, J. Chem. Phys. 8, 3684 (984). 7 G. Bussi, D. Donadio, and M. Parrinello, Canonical sampling through velocity rescaling, J. Chem. Phys. 6, 4 (7). 8 N. Schaudinnus, A. J. Rzepiela, R. Hegger, and G. Stock, Data driven Langevin modeling of biomolecular dynamics, J. Chem. Phys. 38, 46 (3).

Molecular modeling. A fragment sequence of 24 residues encompassing the region of interest of WT-

SUPPLEMENTARY DATA Molecular dynamics Molecular modeling. A fragment sequence of 24 residues encompassing the region of interest of WT- KISS1R, i.e. the last intracellular domain (Figure S1a), has been