Accelerating Steered Molecular Dynamics: Toward Smaller Velocities

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Accelerating steered molecular dynamics: towards smaller velocities in forced unfolding simulations Christian Mücksch, and Herbert M. Urbassek J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.5b01024 • Publication Date (Web): 09 Feb 2016 Downloaded from http://pubs.acs.org on February 17, 2016

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Journal of Chemical Theory and Computation

Accelerating steered molecular dynamics: towards smaller velocities in forced unfolding simulations Christian Mücksch and Herbert M. Urbassek∗ Fachbereich Physik und Forschungszentrum OPTIMAS, University of Kaiserslautern, Erwin-Schrödinger-Straße, D-67663 Kaiserslautern, Germany E-mail: [email protected]

Abstract The simulation of forced unfolding experiments, in which proteins are pulled apart, is conventionally done using steered molecular dynamics. We present here a hybrid scheme in which accelerated molecular dynamics is used together with steered molecular dynamics. We show that the new scheme changes the force-distance curves mainly in the region around the force maximum and thus demonstrate that the improved equilibration of the protein-solvent system brought about by using accelerated molecular dynamics makes the simulation more comparable to experimental data.

∗

To whom correspondence should be addressed

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Introduction

The technique of force spectroscopy using an atomic force microscope (AFM) has various applications such as the determination of mechanical properties of biomolecules. Molecular dynamics (MD) simulations can support these studies and provide insight into structural rearrangements and the underlying energy landscape during unfolding events. However molecular dynamics simulations of AFM experiments, also known as steered molecular dynamics (SMD), have the problem that even the slowest pulling velocities are several orders of magnitude larger than in experiment due to limitations in computational resources. 1 The forces needed to stretch proteins depend very much on the applied pulling velocity. 2 In AFM experiments thermal fluctuations are sufficient to cross energetic barriers, 3 whereas in the case of most molecular dynamics simulations friction of the solvent and the protein’s elastic response will dominate, therefore resulting in much higher forces. Since the turn of the century several techniques have been presented of how to improve the application of MD to slow processes. These considerations started with relatively clear and transparent setups such as the diffusion of single adatoms on a surface. 4–7 A breakthrough in the application of such methods to soft matter such as protein molecules was given by Hamelberg et al. 8 By adding a bias potential to the dihedral potential, 8 and furthermore to the total potential, 9 the natural dynamics are altered in order to escape potential energy minima; this technique has been termed (dual) accelerated molecular dynamics (aMD). Since the energy landscape is altered at transition state regions a boost factor for calculating the reweighting time is not easily available. 10 Hamelberg et al. 8 note that their ‘approach accurately and efficiently explores conformational space with improved sampling and converges to the correct canonical probability distribution’ and can hence be used for sampling proteins in equilibrium situations. In this study we want to investigate in how far acceleration can be used for a driven biological system that is forced far from equilibrium. Based on an idea by Kim and Falk 11 presented for frictional sliding processes of dry solids we use acceleration to lower the ef2


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fective pulling velocity. However, while these authors proposed an algorithm in which the system is simulated in several replicas at different slider positions in parallel – thus including the technique of parallel-replica dynamics 5 – we follow the time evolution sequentially in our hybrid scheme; this is necessary due to the big computational effort accompanying simulations of large biological systems. We will employ the technique of displacement controlled pulling (DCP), 12 in which the molecule is stretched during one timestep and then the system is equilibrated in its new position. We perform the equilibration step using aMD and denote this hybrid technique as accelerated displacement controlled pulling (aDCP). The use of DCP allows us to clearly separate the steps of pulling – pushing the molecule into a state far from equilibrium – and the subsequent relaxation to equilibrium which is supported (accelerated) by aMD. The main idea behind our novel approach is that while an external force used in SMD lowers the energy barriers, accelerated MD raises the potential energies near the minima so that barrier crossing events occur even faster (see Fig. 1).

V(r)

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MD SMD accelerated SMD r

Figure 1: Schematics of the dependence of the protein potential energy V (r) along a reaction coordinate r. As a result of steered molecular dynamics energy barriers between different states are lowered while accelerated MD raises the potential energy near the minima. In our simulations we use the protein spectrin as a model system since it has been studied extensively, both by AFM experiments 13 and by MD. 14,15 These proteins, associated with red blood cells, show α-helical linkers that play a vital role in the elastic properties of erythrocytes 3



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which are affected in diseases such as hereditary spherocytosis and elliptocytosis. 1 Note that we are not interested here in the biophysics of this particular molecule but simply take it as a well-studied example.

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Methods

All molecular dynamics simulations were carried out by NAMD 2.9 16 together with the CHARMM27 force field 17 and the TIP3P 18 water model. The crystal structure of repeat units 8 and 9 of human erythroid β-spectrin was obtained from the protein data bank (PDB-ID 1S35 19 ); the molecule was aligned along the z axis. It was then solvated in a A in x, y, z direction plus a NaCl concentration water box with an extension of 40 × 40 × 420 ˚ of 0.1 M resulting in a total of 70451 atoms. After energy minimization the model system was equilibrated for 2 ns at a temperature of 300 K and constant pressure of 1 atm with the protein’s backbone atoms restrained. Following the equilibration SMD simulations were carried out with N-terminal amino acid fixed and the C-terminal subjected to the pulling force. Simulation times ranged between 8 ns and 300 ns using different pulling velocities between 0.1 m/s and 10 m/s in z direction with a spring constant set to 0.01 N/m as used in the according experiments. 13 A time step of 2.0 fs enabled by the SHAKE 20 algorithm was employed to ensure rigid hydrogen atoms while van-der-Waals interactions were cut off at 12 ˚ A. The pulling code of NAMD was extended to implement DCP, therefore enabling a manual update of the cantilever positions according to the intended velocity. With DCP it is now possible to control the amount of time the system spends at a fixed cantilever position instead of moving the cantilever at a constant velocity. In SMD the pulling velocity, vSMD , simply corresponds to the path length ∆s covered by the cantilever in a given time, ∆tSMD . However, in aMD the period of time which the cantilever stays at a given position is effectively prolongated, ∆taMD ≫ ∆tSMD , due to the accelerated dynamics, and hence the corresponding

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pulling velocity is shortened:

vaMD =

∆s ∆s ≪ vSMD = . ∆taMD ∆tSMD

(1)

Here, the implementation of dual accelerated MD 8,9,21 in NAMD 22 was used, which we employed in a similar manner for studying protein adsorption. 23 While the pulling velocity is usually presented in units of m/s, for DCP simulations we write it as X ˚ A/Y ps; this short-term notation means that a displacement of X ˚ A is followed by a relaxation of Y ps. The force distance curves obtained using classical constant velocity SMD, DCP and aDCP are compared at three pulling velocities of 10 m/s, 2 m/s and 0.5 m/s. Note, that it is not straightforward to determine the actual pulling velocity due to the bias in the dynamics, especially for rather complicated systems like biomolecules. 10 To estimate the accelerated time we fit our SMD data together with experimental data 13 to a model introduced by Heymann and Grubm¨ uller 24 which approximates the dependence of the unfolding force Funfold on the pulling velocity v of the cantilever by

Funfold (v) = γv +

kB T v ln . L k0 ∆L

(2)

Here, the friction of the water environment with friction coefficient γ dominates for fast pulling velocities (v > 1 m/s) whereas a logarithmic dependence of the unfolding force on the velocity applies for slow pulling velocities driven by the thermal energy kB T . ∆L is the scatter width of the unfolding lengths L and k0 denotes the temperature-dependent rate of spontaneous dissociation in the absence of external forces. From Law et al. 13 we assume k0 = 4 · 10−3 s−1 and match ∆L to meet the experimental unfolding force as provided for the puling velocity of 10−6 m/s in the same paper. Snapshots were rendered using VMD 25 and Tachyon. 26

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Results and Discussion

The force-distance curves presented in Fig. 2 have been obtained using a number of different techniques: 1. constant-velocity SMD; this is the conventionally used MD technique to model molecule pulling; 2. DCP using conventional MD; 3. aDCP, which is our newly proposed method; 4. accelerated constant-velocity SMD for comparison. These techniques have been used for different pull velocities and displacement steps in order to assess their performance. In addition, the effect of statistics has been addressed. The results will be discussed in the following. The general pattern of all force-distance curves follow is similar in that forces rise very A pulling distance up to the maximum force and then decline. steeply during the first 50 ˚ The force maximum is caused by rupture events in either the central α-helical linker or in the terminal helices. Especially for the largest velocity (Fig. 2a) a force plateau can be observed which occurs due to the strong viscous forces governing at the high speeds used in these simulations. With lower pulling velocities a steeper decline at large distances can be observed which is due to reduced viscous forces. The force maxima are listed in Table 1; it shows the plausible trend that the lowest forces show up for the slowest pulling velocities. These features are the same for all techniques used here. Table 1: Comparison of unfolding forces obtained from the different force profiles in Fig. 2.

constant-velocity SMD displacement controlled pulling displacement controlled pulling aMD

10 m/s

2 m/s

0.5 m/s

539.947 pN 510.219 pN 468.13 pN

391.538 pN 384.675 pN 259.729 pN

274.734 pN 239.408 pN 175.655 pN

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Figure 2: Force-distance curve for pulling velocities of (a) 10 m/s, (b)(d) 2 m/s, and (c) 0.5 m/s. In (b) the the step size of the cantilever used in DCP was varied. Note the change in ordinate scale. Graph (d) shows the variations obtained for two different configurations (C1 and C2) during equilibration.

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Both conventional SMD and DCP work without acceleration techniques. In all cases, we see that DCP decreases the force needed to stretch the protein; the only exception is provided by the 2 ˚ A/100 ps pulling in Fig. 2b. The improved performance of DCP can be explained by noting that – while at a fixed cantilever position – the protein finds more ways to adapt to the applied force and equilibrate. The use of acceleration techniques has an even larger impact on the force-distance curves. Here, the amount of force needed to stretch the protein is reduced by a significant amount. The reason hereto lies in the accelerated equilibration to the external pulling force during which also conformational changes are promoted. Although the dynamics of the unfolding process are affected by introducing acceleration the shape of the force-distance curves remains qualitatively quite similar to the un-accelerated dynamics. In particular, the initial rise in the force profile as well as the decline after the maximum pulling force seem mainly unaltered. This is due to the fact that in the beginning stages of pulling – whether accelerated or not – the same central α-helical linker or terminal helices will unfold, while in the final stages the force is dominated by viscous drag forces exerted by the water molecules. In Fig. 2b, we explore an alternative route of accelerating SMD simulations, where we use acceleration during the constant-velocity SMD for the exemplary case of 2 m/s pulling velocity. This means that at every simulation timestep the protein is stretched and has to adapt to the new position. We see that the use of acceleration in this simulation strongly lowers the force profile as compared to classical SMD, and makes it close to the aDCP profile. However, in particular the force maximum is still higher than in aDCP due to the less effective equilibration to a constantly changing cantilever position, making this technique slightly inferior to the aDCP favored by us. But the influence of the actual pulling has also for the aDCP method a big relevance. Simply due to the fact that the system can spend more time for relaxation to the external cantilever force when the velocity is reduced a smaller force is measured.

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Furthemore, we explore in Fig. 2d the effect of statistical deviations, i.e., that the initial configuration of the protein at the start of the pull can have consequences for the outcome of the pulling process. To this end, we perform one pull simulation also on a second configuration that has simply been obtained by taking another molecule configuration during the equilibration of the molecule to demonstrate the effect of starting conditions. Fig. 2d shows that the conclusions drawn above on the performance of the various methods studied here hold for both initial configurations studied; in particular the strong decrease of the maximum force when acceleration techniques are used. We note that for simulational studies of the biophysics of the unfolding processes, many more configurations usually have to be taken into account in order to reliably assess statistical fluctuations which may play an important role in the complex response of the protein to the mechanical force. 15 Fig. 3 compares snapshots taken at the exact same cantilever position using constantvelocity SMD and aDCP, respectively. It shows that using aDCP the pulling process has advanced much farther. This figure demonstrates that the use of aDCP has a stronger influence on the protein conformation than would have been guessed from the similarity of the force-distance curves. This can be explained by looking at the dynamics of the unfolding process shown in Fig. 4. We clearly see that the whole process of unfolding is speeded up. The top of the simulation box for 0.5 m/s pulling velocity is reached about 20 ns earlier for aDCP. Thus the use of aDCP results in a higher computational efficiency in that it reduces the amount of computational time needed for the simulation. Qualitatively, our results resemble those found in AFM experiments, compare Fig. 2 of Ref. 13 where the same spring constant was used as in our simulations. However, the rise in our force-distance profiles is steeper and our measured unfolding forces obtained with aDCP are still a factor of 2–4 larger than in experiment. This is mainly due to the still remaining difference in pulling velocity. Previous MD simulations of spectrin pulling used classical SMD techniques. 15 These authors obtained somewhat higher maximum forces than the ones obtained in our simulations

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Figure 3: Comparison between constant-velocity SMD and aDCP. Here, the cantilever is at A and therefore both snapshots are taken at the same the exact same position of 321.856 ˚ simulation time of 53.4 ns. α-helices are shown in purple, coil structure in silver and turns in cyan.

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Figure 4: Force dynamics curve for 0.5 m/s pulling velocity. The rising forces after the decline are due to interactions with the replica when the protein is already fully elongated to the top of the simulation box.

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(Table 1) using SMD. This is caused by their use of a considerably harder spring, while our spring was adapted to the experimental AFM tip data provided in Ref. 13 Their hard spring also influences the shape of their force-distance curves rendering them non-comparable to our work. 600 500 600

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Figure 5: Unfolding forces, obtained by SMD simulation, as a function of the pulling velocity v. The fit curve, Eq. (2), is discussed in the text. The inset shows the velocity at logarithmic scale where it is more obvious to see that the fit curve passes through the experimental data as measured by Law et al. 13

Our SMD data cover a velocity range of 0.1–10 m/s, but are still far above the experimental velocities of 10−6 m/s. 13 We use the theoretical dependence of the unfolding force on the velocity as provided by Eq. (2) to interpolate between our simulation results and the experimental data, see Fig. 5. The parameters used in the fit are L = 2.5 ˚ A and γ = 26.1 pN s/m; these values are in the range of typical SMD simulations. 24 For ∆L the fit requires the relatively large value of ∆L = 1.12 · 10−5 m. Note that due to missing statistics this fit can only be taken to obtain a rough estimate of the effective pulling speeds corresponding to our aDCP simulations. With the unfolding forces shown in Table 1, Fig. 5 allows us to b vfitted = 6.23 m/s, vaDCP = 2 m/s = b vfitted = 0.25 m/s and estimate vaDCP = 10 m/s = b vfitted = 2.12 · 10−3 m/s. These values give us an idea about the decrease vaDCP = 0.5 m/s =

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in effective pulling velocity – and the concomitant increase in real time – achieved by the use of aMD. Due to the logarithmic velocity dependence, Eq. (2), the effect of aMD is most pronounced at slow pulling velocities.

4

Conclusions

We explored the application of acceleration techniques for unfolding simulations of biophysical relevance. To this end we propose the hybrid technique of accelerated displacement controlled pulling (aDCP), and compare it to classical constant-velocity SMD simulations. Acceleration results in a faster relaxation and adaptation to the applied pulling force. It brings the simulation closer to the results obtained at experimental pulling speeds since the effective pulling velocities obtained by our hybrid method are considerably smaller than those that can be obtained nowadays by using classical SMD simulations. The principle of this approach is that while SMD lowers the barriers in the free-energy landscape between the folded and the unfolded state, accelerated MD raises the potential energies near the minima. This results in a higher barrier crossing rate than by SMD itself. One of the main objectives of such unfolding simulations would be to reach experimental pulling velocities. Besides the comparison of the experimental and simulational force-distance curves, the simulations may allow to identify structural features during the unfolding event that are not accessible to experiment. The details of the dynamics during the pulling are affected; however, the force-distance curves follow a similar trend as in the classical simulations. This means that the underlying unfolding process is not much altered but the barriers in the protein’s energy landscape are lowered. An important result of using acceleration is a speedup of the simulation, due to the faster relaxation of the protein to the external cantilever force. For a quantitative study of the biophysics of the spectrin molecule and for detailed comparison with AFM experiments, 13 considerably more simulation runs would be needed in

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order to obtain statistically reliable data. For qualitative purposes the present similarities between experiment and simulation found in the force-distance profiles is reassuring. Anyhow, usage of aDCP does not relieve the practitioner of the necessity to use as long relaxation times as possible, and thus of the smallest pulling speeds affordable. Future simulations using aDCP could try to reach longer relaxation times, making a comparison with experimental data more likely. Also the influence of different spring constants should be studied.

Acknowledgement We acknowledge financial support by Deutsche Forschungsgemeinschaft within projects Ur 32/26-1. Furthermore we appreciate the computational resources provided by the compute cluster ‘Elwetritsch’ of the University of Kaiserslautern. We acknowledge discussions with Luis Sandoval.

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Accelerating Steered Molecular Dynamics: Toward Smaller Velocities

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