Scaling Rules for Vibrational Energy Transport in Globular Proteins

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Scaling Rules for Vibrational Energy Transport in Globular Proteins Sebastian Buchenberg, David M. Leitner, and Gerhard Stock J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b02514 • Publication Date (Web): 09 Dec 2015 Downloaded from http://pubs.acs.org on December 11, 2015

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To compare the effects of various excitation methods, we also performed nonequilibrium simulations using T-jump excitation (see Fig. S1 of the Supporting Information(SI)). To follow the vibrational energy flow after photoexcitation of the protein at time t = 0, we consider the time P evolution of the average kinetic energy of the ith residue, Ei (t) = m Ei,m (t), where the sum runs over the kinetic energies of all atoms m of residue i. The time-dependent expectation value of this kinetic energy is calculated via an ensemble average over N = 250 nonequilibrium trajectories,

Introduction Experimental and computational studies of energy flow in proteins have long provided a detailed picture of anisotropic transport, which may be associated with the functional dynamics of the system. 1–6 On the experimental side, various groups have employed time-resolved vibrational spectroscopy to monitor energy flow in biomolecules, which initially can be triggered by temperature jump (T-jump), or by photoexitation of a chromophore or a photoswitch. 7–13 Adopting various computational approaches, biomolecular energy flow has been studied in numerous systems in great detail, 14–22 including contributions from through-bond and throughspace transport mechanisms, 3,22 funneling of energy to solvent via heme side chains, 14 and the role of embedded water molecules 18 in energy transport. However, to date no quantitative and general rules have been identified by which energy transport pathways in a globular protein can be predicted. With this end in mind, in this work we present scaling rules for energy transport along the backbone and between polar and nonpolar tertiary contacts of folded proteins. The scaling rules have been identified from the results of extensive nonequilibrium molecular dynamics (MD) simulations of vibrational energy flow in a small protein, which are used to parameterize a master equation. In this way, we obtain energy transfer rates between residues, the residues and solvent, and the initially excited moiety, which are found to closely reproduce the results of the all-atom simulations. Employing a diffusion description of the energy transport along the backbone as well as a harmonic model of the contact-mediated transport, we derive general scaling rules for the energy flow. As a simple model system, we adopt the villin headpiece subdomain (HP36), which has long served as a prototype system for the study of protein dynamics and folding. 23 Moreover, first energy transport experiments have been recently reported for HP36, using either T-jump excitation or a azobenzene photoswitch covalently bound to the side chain of residue Cys-16. 24,25 The protein consists of three alpha helices connected by two turns (Fig. 1a) and –despite its small size– shows many of the typical properties of larger proteins such as a compact core and tertiary contacts.

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b Residue

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Ei (t) =

N 1 X (k) E (t). N k=1 i

(1)

In the discussion below, we are mainly interested in the average increase of the kinetic energy reaching residue i after excitation of the system, ∆Ei (t) = Ei (t) − Eieq , where Eieq is the equilibrium energy of residue i before the excitation. For brevity, we will refer to ∆Ei (t) simply as the residue energy. Figure 2 shows the resulting time evolution of some representative residue energies, following initial excitation of residue Cys-16. As expected, we find in panels (a) and (b) that the neighboring residues of Cys-16 receive energy in a sequential manner. Starting by construction at zero at t = 0, the energies of residues 15, 14, 13 and 12, and 17, 18, 19 and 20 rise sequentially and exhibit maxima of decreasing height. The situation is complicated, however, by tertiary contacts between residues, which may serve as shortcuts for energy flow. For example, we find in panel (e) that energy arrives earlier at residue 4 than residue 5, which lies closer in sequence, due to a contact between residues 4 and 15. We note that the transient oscillatory behavior of the residue energies is not noise but reflects the initial photoexcitation of the system. As shown in Fig. S2, the oscillations are mostly caused by high-frequency torsional motion around the N=N double bond of the azobenzene photoswitch. Backbone transport To interpret the simulation results and to develop a physical model of the energy transport, we adopt the following master equation,

35 30 25 20 15 10 5

dEi (t) = dt

X

kji Ej (t) − kij Ei (t)

j

(2)

∗ −ki,S Ei (t) + kS,i ES (t) + kIVR E16 (t)δi,16 ,

where kij represents the energy transport rate from residue i to residue j and δi,16 = 1 if i = 16 and zero otherwise. The rate kS,i accounts for the “cooling” of the photoexcited protein in the solvent with kinetic energy ES (t) (see below). The initial vibrational energy excitation of the system due to cis-trans photoisomerization ∗ of the photoswitch is modeled by a “virtual” high-energy state E16 of residue Cys-16, that decays with the rate kIVR = 0.57 ps (see the Methods Section of the SI). We note that the ansatz of master equation (2) implies that intra-residue energy flow is significantly faster than inter-residue energy transport (see Fig. 2f). By fitting the master equation to the results of the nonequilibrium MD simulations, all rates of the master equation can in principle be determined. Due to the large number of transition rates (∝ M 2 with M = 36 being the number of residues of HP36), however, this fit is ill-defined and likely to yield nonphysical results. To reduce the number of fitted parameters, we first note that in thermal equilibrium each degree of freedom contains the same amount of kinetic energy; hence we have Eieq /fi = Ejeq /fj , where fi denotes the number of degrees of freedom of residue i. Since at equilibrium dE(t)/dt = 0, we obtain the detailed balance condition,

5 10 15 20 25 30 35 Residue

Figure 1: (a) Structure of a photoswitchable mutant of HP36 with the azobenzene photoswitch attached at Cys-16, indicating in orange and green the side-chains of residues that mediate shortcuts of the energy transport, and in blue residues that represent bottlenecks to energy flow. (b) Contact map of HP36 as defined via the minimal distance ∆xij between two residues i and j, using red for ∆xij < 2.8 ˚ A, green for ∆xij < 3.4 ˚ A, blue for ∆xij < 4 ˚ A, and gray for ∆xij < 4.6 ˚ A. Shown are results obtained from the MD equilibrium structure of HP36 in vacuo (upper triangle) and in water (lower triangle). MD simulations As detailed in the Supporting Information (SI), we performed all-atom MD simulations in explicit water solvent to model the energy transport in a photoswitchable mutant of HP36 initiated by cis-trans photoisomerization. All simulations were performed with the GROMACS program suite, 26 using the GROMOS96 force field 53a6 for HP36, 27 the azobenzene parameters of Ref., 28 and the single point charge water model. 29 We obtained 250 statistically independent structures from a 100 ns NVT equilibrium trajectory at 100 K, which were cooled down to 10 K in order to improve the signal to noise ratio of the subsequent NVE nonequilibrium runs. The photoexcitation of the azobenzene photoswitch was modeled by a previously developed potential-energy surface switching approach. 30

fj kij = fi kji ,

(3)

which relates the back-rate to the forward-rate. To further reduce the number of fitted rates, we next introduce two simple models that describe the energy transport along the protein backbone (“backbone transport”) and via inter-residue contacts (“contact transport”). To model backbone transport, we note that the master equation (2) can be derived from an energy transport diffusion equation E˙ = D d2 E/dx2 by discretizing the x-derivatives in terms of residues of

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The Journal of Physical Chemistry Letters rate C kij

DC = 2 i hδrij

s

fj , fi

10 higher in vacuo than in solvent, reflecting the fact that the major part of the kinetic energy of the solvated protein dissipates rapidly into the solvent. We obtained for the isolated system a backbone diffusion constant DB = 1.42 nm2 ps−1 , which is very similar to the value obtained for the solvated system (1.25 nm2 ps−1 ), and indicates that the solvent has little effect on transport along the backbone. For the simulation conditions applied, the dehydrated protein showed only minor structural changes compared to the solvated system. Hence the contact maps of the two cases are quite similar (Fig. 1b) and the key residues for the contact transport are the same. The resulting contact rates of solvated and isolated system show similar trends, but nonetheless may differ by up to a factor 10 (Table S2), which reflects the difference in solvent accessibility of the contacts. When we p plot the 2in vacuo contact transport rates C as a function of fj /fi /hδrij i (Fig. S6), we again find the kij linear dependence of the transport rate predicted by Eq. (6). The energy diffusion constant obtained for the isolated system, DC = 2.7 10−2 nm2 ps−1 , is about two orders of magnitude larger than the corresponding constant for the solvated system. This effect 2 i of is partly compensated by the generally higher variance hδrij the contact length, which is caused by the lack of friction due to missing solvent water. Put together, the good agreement of MD and master-equation results demonstrate that the scaling rules for backbone and contact transport in Eqs. (5) and (6) hold for solvated as well for isolated protein systems.

(6)

where the proportionality factor DC may be regarded as an effective diffusion constant of the process. Interestingly, the above scaling rule predicts that the rate of contact energy transport is inversely 2 i. Hence proportional to the variance of the contact distance hδrij a rigid contact mediates the vibrational motion instantaneously, while a very floppy contact represents a slow pathway of energy flow. In spite of their similar appearance, the scaling rule of contact transport in Eq. (6) therefore is different from the scaling rule of backbone transport in Eq. (5), which states that the backbone transport rate is inversely proportional to the squared mean of the distance between the atoms of two neighboring residues. Our model, supported by the results of the nonequilibrium simulations, describes energy flow along the backbone by vibrational dynamics of atoms within the residues that form the backbone structure, whereas energy flows between side-chain contacts due to the damped oscillatory dynamics of the side-chains themselves. We note that a relation similar to Eq. (6) was found for the mean first passage times of a harmonic network model. 35,36

2.5 2 -1

1.5

c

kij [ps ]

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1

Concluding remarks To summarize, we have derived two scaling rules that describe the transport of vibrational energy along the backbone and via side-chain contacts. Requiring only the calculation of mean and variance of relatively few atomic distances, the approach holds the potential to predict the pathways and timescales of vibrational energy flow in large proteins. The scaling rules proposed above were validated by a direct comparison to classical nonequilibrium MD simulations. Since the underlying assumptions (diffusion model for backbone transport, harmonic model for contact transport) are expected to be generally fulfilled in proteins, the scaling rules may also describe energy transport in other biomolecules. For example, a recent nonequilibrium MD study of the energy flow along a peptide 310 helix 17 revealed a backbone transport time of 0.4 ps which is in perfect agreement with the corresponding result of 0.38 ps obtained from scaling rule (5). Nonetheless, the application of the theory to other (not necessarily biomolecular) polymers is limited to cases for which the a master equation description holds (see discussion above) and may also require some modification of the theory. For example, when we consider larger proteins where only the outer part of the residues are solvent exposed, one may want to restrict the energy exchange with the solvent to these residues. Moreover, the treatment of different polymers (e.g., DNA) may result in different backbone diffusion coefficients. While quantum effects are neglected in our formulation, recent theoretical considerations suggest that they make only a fairly minor contribution to biomolecular vibrational energy flow. 39 Moreover, the above results on HP36 agree well with related energy transport calculations based on communication networks, 16,22 which approximately account for quantum effects. With the advent of recent time-resolved infrared techniques, 9–13 it will be possible to directly compare the predictions of the present theory to experimental data, such as HP36 when the experiments reach amino acid resolution. As the focus of interest will likely shift to larger biomolecular systems, we expect the scaling rules presented in this study to become particularly useful, providing a much more feasible approach than large-scale all-atom simulation studies or other coarse-grained approaches.

0.5 0 0

5

10

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25

2

√fj/fi/  Figure 3: Energy transport rates for the contact-mediated energy transfer, plotted (in units of nm−2 /1000) as a function of p 2 i. Polar contacts are represented by red crosses, fi /fj /hδrij nonpolar contacts by blue stars. Dashed lines are linear fits to the data. To validate the scaling rule (6), we plot p in Fig. 3 the transport 2 i. Indeed, rates of all contacts in HP36 as a function of fi /fj /hδrij the plot reveals a clear linear dependence for the polar and non-polar tertiary contacts. A fit to the MD results yields the contact diffusion constant DC = 1.1 10−4 nm2 ps−1 and 3.6 10−7 nm2 ps−1 for the polar and non-polar tertiary contacts, respectively, again showing that only polar contacts matter for the biomolecular energy flow in 2 i (which are readily calculated the protein. For given variances hδrij from a short (≈ 1 ns) equilibrium MD simulation), the scaling rule (6) facilitates the calculation of the contact energy transport in a protein from fitting a single constant DC instead of fitting all contact rates individually. The resuling fit is virtually identical to the fit shown in Figs. 2 and S4. We note that –according to the derivation above– the contact diffusion constant DC is directly proportional to 2 i are obtained. The the temperature for which the variances hδrij value of DC quoted above is based on MD simulations at 100 K. If the variances were calculated from a simulation at 300 K, e.g., this value needs to be multiplied with a factor 3.

Acknowledgement We thank Janine Franz and Peter Hamm for numerous instructive and helpful discussions. Support from the Deutsche Forschungsgemeinschaft to GS and the National Science Foundation (NSF CHE-1361776) to DML is gratefully acknowledged.

Effect of the solvent The MD results reported above were performed in explicit water, which effects a “cooling” of the photoexcited protein in the solvent. A fit of the MD results to the master equation (2) yields for the energy transport from the protein to the solvent the transfer time τS,i = 1/kS,i = 5.6 ps, which is in excellent agreement with experimental 37 and previous computational 38 studies. To achieve thermal equilibrium at long times, we furthermore invoke a small back-rate ki,S = 1/(240 ps). To study the effects of the solvent on energy flow in the protein, we have also performed energy transport simulations of HP36 in vacuo. As in Fig. 2 for the solvated system, Fig. S5 shows the time evolution of representative residue energies ∆Ei (t) in the isolated case. While the overall appearance is similar, we note that the energy scale is almost a factor

Supporting Information Available: Details on the MD and the backtracking methods, en-


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Scaling Rules for Vibrational Energy Transport in Globular Proteins

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