All-atom molecular dynamics simulations using orientational constraints from anisotropic NMR samples
Abstract
Orientational constraints obtained from solid state NMR experiments on anisotropic samples are used here in molecular dynamics (MD) simulations for determining the structure and dynamics of several different membrane-bound molecules. The new MD technique is based on the inclusion of orientation dependent pseudo-forces in the COSMOS-NMR force field. These forces drive molecular rotations and re-orientations in the simulation, such that the motional time-averages of the tensorial NMR properties approach the experimentally measured parameters. The orientational-constraint-driven MD simulations are universally applicable to all NMR interaction tensors, such as chemical shifts, dipolar couplings and quadrupolar interactions. The strategy does not depend on the initial choice of coordinates, and is in principle suitable for any flexible molecule. To test the method on three systems of increasing complexity, we used as constraints some deuterium quadrupolar couplings from the literature on pyrene, cholesterol and an antimicrobial peptide embedded in oriented lipid bilayers. The MD simulations were able to reproduce the NMR parameters within experimental error. The alignment of the three membrane-bound molecules and some aspects of their conformation were thus derived from the NMR data, in good agreement with previous analyses. Furthermore, the new approach yielded for the first time the distribution of segmental orientations with respect to the membrane and the order parameter tensors of all three systems.
Keywords
Molecular dynamics simulations Orientational NMR constraints ^{2}H-NMR Oriented samples Cholesterol Pyrene PGLa peptide Order parameters Force field calculationsIntroduction
Using anisotropic media, such as partially oriented bicelles or macroscopically oriented membranes, NMR investigations can reveal a wealth of information about molecular properties, namely conformation, orientation and dynamics. In many solid state NMR studies of membrane-active peptides and transmembrane proteins, the samples are conveniently prepared with macroscopically oriented bilayers to obtain structural information. The NMR data analysis relies on a uniform alignment of all molecules with respect to the static magnetic field, as it makes use of the orientation dependence of the chemical shift, quadrupolar coupling or dipolar coupling interactions. In contrast to single crystal studies, where the molecules are immobilized in a unique conformation, in the case of lipid membranes and liquid crystalline systems one has to consider a wide distribution of molecular orientations and anisotropic motions. Here, we apply for the first time a new MD strategy to deduce such structural and dynamics information on three representative compounds in biomembranes with increasing complexity: (i) pyrene, (ii) cholesterol, and (iii) the antimicrobial peptide PGLa.
A convenient nucleus for solid state NMR investigations of oriented membranes is deuterium for several reasons (for reviews see e.g., Vold 1994; Davis 1983; Ulrich and Grage 1998). First, the influence of ^{2}H substitutions on the molecular structure and dynamics is negligible. More importantly, in most biologically relevant cases the quadrupolar interaction tensor of a carbon-bound deuterium is to a good approximation aligned parallel to the ^{2}H–C bond and directly reflects the local molecular orientation and dynamics of the labelled segment. Using ^{2}H-NMR, the alignment and dynamic behaviour of lipids, small organic guest molecules and membrane-active peptides has been studied in numerous examples. Most previous strategies to extract the molecular orientation from NMR data, however, have been restricted to molecules with a rigid conformation, e.g. peptides assuming an ideal α-helix with fixed backbone torsion angles. In these studies, several individual labels are usually placed into strategic positions on the rigid molecular part. The measured quadrupolar splittings are then compared in a least squares RMSD analysis with the predicted values upon systematically varying the molecular alignment. The best-fit molecular orientation is determined from a 3D error plot as the one yielding minimal differences between the observed and calculated NMR parameters. However, this RMSD (root mean square deviation) analysis can only give preliminary results for flexible molecules such as peptides or proteins with a high degree of internal mobility.
A proper way to account for motional averaging of the NMR parameters caused by molecular mobility is to run all-atom MD simulations. When applied to biomembranes, however, such simulations pose two closely connected problems: (i) large size of the system and (ii) long simulation times. In particular, water surrounding the lipid bilayer has to be included into the MD simulations, and time-spans have to be covered approaching the NMR time scale. Moreover, the volume of interest has to be surrounded by multiple copies of the central cell to avoid boundary effects. Nevertheless, such simulations have been demonstrated to be feasible for moderately sized membrane segments hosting medium size molecules (see e.g. Tieleman et al. 2001). In the present contribution, we propose an alternative strategy, in which the oriented medium is not explicitly considered, but instead the ordering membrane environment is replaced in the MD simulations by pseudo-forces derived from the measured NMR parameters. This way sufficiently long MD runs become possible and allow to calculate the motionally averaged parameters observed in the NMR experiments. As a result, the full information about molecular orientation, order, segmental motions and even aspects of the molecular conformation could be revealed.
There is one remarkable advantage in the calculations involving orientational constraints. Since an experimentally observed NMR value usually represents a time average over the motions of the molecules, knowledge of the geometry and strength of these molecular motions is a prerequisite for properly interpreting the NMR data in terms of structural parameters. The order parameter, which is typically used to describe the effect of molecular motion on the NMR spectra, however, is often not known a priori. In the present approach, the molecular motions are intrinsically included and the order parameters are obtained as part of the simulation result.
Theory
NMR interaction tensors and coordinate transformations
Atom A is selected as the position of the nucleus of interest (for instance ^{2}H), whose property P_{A} is to be calculated. In case there are several possible ways of selecting atoms B and C, they will be defined such that the valence of bond B–C is a maximum (the valence can be estimated from the bond length, see O’Keefe and Brese 1991). This selection ensures that the y-axis will be perpendicular to a π-system if it exists, and the x-axis will be positioned within the π-plane.
For most deuterium quadrupolar couplings, as well as vicinal dipolar couplings like ^{15}N–^{1}H and ^{13}C–^{1}H the tensor P_{A} will be to a good approximation diagonal in this coordinate system, hence we can easily assign the principal values of P_{A} to the three axes. In cases where the chemical shift tensor is to be analysed, the same molecular coordinate system can be used as in those calculations based on the bond polarization theory (BPT, see Sternberg 1988). Therefore, all these types of NMR interactions can be treated within the same formalism.
In static NMR experiments the tensor components are observed in a coordinate system of reference whose z-axis is aligned with B_{0}. In this frame the measured frequency or splitting is given by the zz-component of the tensor P. However, in the approach presented here we can incorporate further tensor components in the MD simulations, which are accessible for example by measuring the sample at further tilt angles.
Constraints and calculation of molecular properties
Calculation of pseudo-forces
Because of the time dependence of the transformation matrices, the derivatives were calculated continuously during the MD simulation at each time step. The time average is only calculated for the NMR property.
Order parameter calculation
Molecular dynamics simulation
The integration of the equations of motions is based on Verlet’s algorithm (Verlet 1967), and time steps of 0.5 fs were employed to sample all high frequency hydrogen atom vibrations. In constrained MD simulations it is generally necessary to control the temperature during the simulation time. This is accomplished by coupling the molecular system to a heat bath which dissipates the heat generated by the pseudo-forces. All prevailing differences between the constraints and their calculated values are sources of heat. To obtain an NTV assemble (with conserved particle number N, temperature T, and volume V), we introduced a proper thermostating procedure (see Evans and Morriss 1990). The coupling to the thermostat is controlled by a coupling time constant η which should be much larger than the time step. This time span η allows an adjustment of the range of thermal fluctuations in the simulated molecular system.
To prevent too large pseudo-forces at the start of the MD simulation the pseudo-forces were gradually increased towards their final values during the MD simulation. To this aim we introduced time dependent scaling factors \({f=1-\hbox{e}^{-t/\rho}}\), which approach the value 1.0 in an exponential fashion. The time constant was set in most cases to 200 ps, leading to a relatively smooth course of the temperature.
When applying the NMR orientational constraints during an MD run, the resulting pseudo-forces will “heat up” the system and enhance its rotational degrees of freedoms. Because the averaging procedure depends on the molecular re-orientations caused by the NMR constraints, some net rotational motion will prevail up to the end of the simulation. In standard MD simulations any overall molecular rotations and translations are subtracted from the velocities, since these external degrees of freedom are not of interest. In the present orientationally constrained calculations, however, only the net translations of the systems are removed.
Parametrization
In the case of a C–^{2}H bond the deuterium quadrupolar coupling depends only weakly on the molecular surrounding, hence representative values from static solid state NMR measurements of characteristic substances can be used as quadrupolar constants (Table 1). In this selection we have to keep in mind that the quadrupolar coupling constants depend strongly on the hybridization of the carbon, and to some extent also on the polarization of the C–^{2}H bonds by partial charges caused by electronegative groups.
The coordinate system for the local tensors is defined in Fig. 1. The zz-component is aligned with the C–^{2}H bond direction, and the yy-component is perpendicular to any π system. For an aromatic C–^{2}H (sp^{2}) bond an asymmetry parameter of \({\eta_{Q}=0.06\;(\eta_{Q}= (C_{11}^{Q}-C_{22}^{Q})/C_{33}^{Q}}\) with \({C_{33}^{Q} > C_{11}^{Q} > C_{22}^{Q}}\)) was assumed, and the tensor axes were assigned according to calculations of Bailey (1998). The value for the –CD_{3} group is obtained from the C(sp^{3})–^{2}H value scaled by 1/3, assuming rapid rotation around the C–CD_{3} axis.
Program implementation
The routines to apply the proposed orientational constraints are included into the COSMOS-NMR force field, which has been used in a number of previous applications, but which had so far been restricted to distance and chemical shift constraints (see e.g. Sternberg et al. 2003; Witter et al. 2002), and for the force field (Möllhoff and Sternberg 2001; Sternberg et al. 2001). These two types of NMR constraints can be combined with the new orientational constraints as will be demonstrated in the example of PGLa below. The authors provide the backend version of the full COSMOS program, containing all computational procedures without the graphics and modelling interface (GUI) (see http://www.cosmos-software.de). The COSMOS-backend (C++) was compiled for several operation systems including Windows, Unix and Linux.
Applications
The use of NMR orientational constraints is particularly well suited to gain insight into the alignment and dynamics of molecules embedded in biomembranes. In the three examples presented here, we will first demonstrate our new MD approach on pyrene as a simple model compound dissolved in lipid bilayers, then we will apply it to cholesterol as an intrinsic membrane lipid, and finally to the antimicrobial peptide PGLa, which forms an amphiphilic α-helix in membranes.
Pyrene
Deuterium quadrupolar coupling tensors
General parameters for the MD simulations with orientational constraints
In this first simulation we had constrained only the three principal components of the quadrupolar tensor. It turned out that in this case small off-diagonal values did not approach zero but remain in the range of −1.5 to 2 kHz up to the end of the simulation (see Fig. 3). At the beginning of the simulation a high pseudo-energy of 31,000 kJ/mol was encountered, which dropped to 0.8 kJ/mol by the end of the simulation. Because of the large pseudo-energies at the start, we realized that it is preferable to first run a preliminary MD simulation (1–4 ns) with much smaller pseudo-forces, and then to step up the pseudo- forces after some initial averaging has been performed. In the next two cases studies the pseudo-forces were switched on exponentially to avoid long preliminary equilibration periods.
In a second MD simulation of pyrene over 7.8 ns, we also constrained the three off-diagonal tensor components C_{xy}, C_{xz}, and C_{yz} to zero, which are averaged by rotations of the molecule about the membrane normal. The corresponding values calculated from the MD run yielded off-diagonal elements with absolute values smaller than 0.23 kHz. The pseudo-energy increased by a factor of two, as it now contains contributions from both the diagonal and off-diagonal tensor components. In an attempt to compensate for the higher pseudo-energy, we set the memory time to a higher value of τ = 1,000 ps (see Eq. 6). Altogether we obtained smaller fluctuations of the coupling tensor components and much narrower lines in simulated spectra, which can be most likely attributed to the damping effect of the memory function outweighing the higher pseudo-energies.
Calculated quadrupolar splittings in a 7.8 ns constrained MD simulation of pyrene, compared with the experimental values obtained from a ^{2}H-NMR measurement of the deuterated pyrene in oriented POPC membranes
Site | Sign of tensor component^{a} | Constrained MD splitting Δν (kHz) | NMR experimental^{b} splitting Δ ν (kHz) |
---|---|---|---|
^{2}H1 | + | 91.4 (91.39, 91.4)^{c} | 93.0 |
^{2}H2 | + | 39.9 (39.8–40.0)^{c} | 40.5 |
^{2}H3 | + | 39.7 (39.6–39.8)^{c} | 40.5 |
Simulated order parameters for pyrene in POPC
Site | W_{xx} | W_{yy} | W_{zz} | | W_{xy},W_{xz},W_{yz}| |
---|---|---|---|---|
^{2}H1 | +0.021 | −0.342 | +0.321 | ≤0.03 |
^{2}H2 | +0.181 | −0.310 | +0.129 | ≤0.07 |
^{2}H3 | +0.165 | −0.294 | +0.129 | ≤0.03 |
S_{bb} | S_{cc} | S_{aa} | |S_{ab}, S_{ac}, S_{bc}| | |
Molecule^{a} | +0.048 | −0.275 | +0.228 | ≤0.006 |
For a perfectly rigid pyrene molecule the molecular Saupe order tensor S and the segmental order tensors W^{A} are linked by fixed conformation tensors. With our choice of the molecular coordinate system, S and W^{A} are identical for the site ^{2}H1, and for the other two sites the tensors are related by simple geometric expressions. Comparing the molecular order tensor S (last row in Table 4) and the segmental order tensor W (e.g. of the site ^{2}H1, which is expressed in the same coordinate system, first row in Table 4), it is noticed that most of the molecular S tensor components are slightly smaller than the corresponding values of the segmental order tensor W. This effect is caused by bond vibrations and molecular twists occurring during the MD simulation. However, as these contributions to the dynamics (manifested in the difference between segmental and molecular order) are small, the mobility of the pyrene in the membrane is well described by the molecular order tensor S. Further qualitative conclusions on the behaviour of the pyrene molecule in the membrane can thus be drawn from the order tensor. Because S_{aa} constitutes the largest (signed value) component, it is obvious that the corresponding axis a (in the system of inertia), and with it the long molecular axis of pyrene, shows some preference for an orientation in the direction of the membrane normal. Since S_{cc} and S_{bb} are different, the molecular motions are not axially symmetric, but display a preference for a second axis, expressed in a biaxiality parameter ξ = 0.11 (see Eq. 16). This pronounced biaxiality is a consequence of the restricted motions of the flat pyrene molecule within the lipid matrix. A further indication for such a restricted motion is the fact that the smallest component of the order tensor S is found along the c-axis direction perpendicular to the aromatic ring system. This direction has thus the smallest average component \({\langle\cos \Theta_{c}^{2}\rangle}\) along the membrane normal and is most conserved, as expected for the planar shape of the molecule. In their all-atom MD simulation including 128 lipid molecules and water, Hoff et al. (2005) obtained essentially the same results as in this work, with the exception that they found a much higher Saupe order component S_{zz} = 0.42 (in this work denoted with S_{cc}).
Cholesterol
The authors examined six different models of cholesterol structures proposed in the literature using an RMSD analysis to assign all observed lines in the ^{2}H-NMR spectrum. In this analysis the quadrupolar splittings were calculated from the respective rigid molecular models, treating the molecular orientation and the Saupe tensor as free variables. It turned out that the results depended crucially on the selection of the molecular coordinates, and only the data from a neutron diffraction analysis (McMullan et al. 1992) of a cholesterol derivative (20-CH_{3}-methylpregnene-3,20-diol) produced convincing results. This strong dependence on the exact coordinates is regarded as one of the main drawbacks of RMSD analyses of rigid molecular models.
Calculated deuterium quadrupolar splittings from a constrained MD simulation compared with the experimental ^{2}H-NMR data (Marsan et al. 1999)
Site | Constrained MD splitting of methyl-pregnenediol (kHz) | Constrained MD splitting of cholesterol (kHz) | Experimental NMR splitting (kHz) |
---|---|---|---|
H2a | −101.5 | −101.4 | 101.68 |
H2e | −66.7 | −67.7 | 67.86 |
H3_1 | −107.1 | −106.7 | 107.30 |
H4e | −63.2 | −62.3 | 62.68 |
H4a | −94.3 | −93.0 | 94.98 |
H6 | −7.29 | −6.69 | 6.44 |
H7a | −92.9 | −95.4 | 96.12 |
H7e | −91.6 | −91.3 | 91.48 |
C18–CD_{3} | −33.0 | −33.8 | |
C19–CD_{3} | −37.5 | −34.7 | |
H1a | −104.2 | −99.2 | |
H1e | −69.4 | −46.1 | |
H8_1 | −107.0 | −100.3 | |
H9_1 | −112.6 | −103.0 | |
H11a | −106.4 | −102.7 | |
H11e | −50.7 | −74.7 | |
H12a | −46.4 | −13.1 | |
H12e | −103.1 | −101.1 | |
H14_1 | −103.4 | −100.3 | |
H15_1 | −102.1 | −81.8 | |
H15_2 | −93.5 | −88.9 | |
H16_1 | −17.6 | −40.9 | |
H16_2 | −23.1 | −27.6 | |
H17_1 | −85.7 | −92.7 |
Molecular Saupe order tensor derived from the constrained MD simulation, compared with the values derived from the previous RMSD analysis (Marsan et al. 1999)
Tensor component | MD simulation of methylpregnenediol | MD simulation of cholesterol | Static RMSD analysis |
---|---|---|---|
S_{aa} | 0.87 | 0.88 | 0.94 |
S_{bb} | −0.44 | −0.44 | −0.48 |
S_{cc} | −0.43 | −0.44 | −0.46 |
S_{ab} | 0.0 | −0.11 | |
S_{ac} | 0.0 | −0.12 | |
S_{bc} | 0.0 | 0.0 |
PGLa
In our third case study, the new MD approach was applied to a membrane bound peptide. PGLa is a 21-residue cationic peptide (GMASKAGAIAGKIAKVALKAL-NH2) from the magainin family of antibiotics present in frog skin, which folds into an amphiphilic α-helix when bound to lipid bilayers. The mechanism of antimicrobial activity has been addressed in numerous studies and is attributed to the perturbation of bacterial membranes. Solid state NMR has yielded much insight into its structure and dynamic behaviour in model membranes (Bechinger 1999). For our MD simulations we used the constraints from the ^{2}H-NMR investigations of Strandberg et al. (2005), who had labelled four native alanine residues (positions 6, 8, 10 and 14) and two isoleucines (positions 9 and 13) one by one with ^{2}H_{3}-alanine. At a peptide-to-lipid ratio of 1:200 in DMPC, it was found from the quadrupolar splittings of these six CD_{3}-groups that the α-helical PGLa is aligned flat in the plane of the membrane, and the peptide undergoes fast rotational diffusion about the membrane normal at 35°C. From the same data we now derived the full quadrupolar coupling tensors for the six ^{2}H-labelled sites, including the sign of the tensor elements, and used these as constraints for the MD simulations. For the quadrupolar coupling tensors we used the parameters of the CD_{3-}group (Table 1), both for the native Ala substitutions as well as for the nominal constraint along the C_{α}–C_{β} segment of isoleucine.
It is known from previous NMR studies (Strandberg et al. 2005) that membrane-bound PGLa forms an α-helix in the range of the labelled stretch, and further evidence for an α-helical conformation between residues 6 and 21 is provided by (Bechinger 1999). We therefore started the MD simulations with an idealized α-helix, and for all backbone hydrogen bonds we introduced 18 additional distance constraints of 1.86 Å to keep the molecular model helical. At the end of the simulation, the RMSD of the backbone hydrogen bond lengths from their ideal values was only 0.2 Å, thus confirming that that the molecule stayed indeed mostly helical during the MD simulation. It also had to be taken into account that the four lysine side chains and N-terminus are positively charged. Since charged NH_{3}^{+} groups have a strong tendency to form hydrogen bonds, we added a water molecule near each hydrogen atom of a charged group. This way any undesired hydrogen bonds of lysines to the backbone could be prevented, and indeed at the end of the simulation all 15 water molecules were still in contact with the peptide.
Compared to the two small-molecule case studies above, the NMR data on PGLa provide only few constraints on the large and rather flexible molecule, as the pseudo-forces act only on six amino acids out of 21. Therefore, unrealistically large-pseudo forces at the start of the MD simulation have to be avoided. As a first measure, the pseudo-forces were increased exponentially to their final values during the MD simulation with a time constant of ρ = 100 ps, like in the cholesterol simulation. From the NMR studies it was known that PGLa rotates rapidly (on the NMR time scale) about the membrane normal. This motion opens up a second possibility to scale the pseudo-forces properly. We calculated rotational averages of the quadrupolar tensors in every step of the MD. Only three tensor values rotated by 120° around the director axis (in this case the membrane normal oriented parallel to the B_{0} field directions) are necessary to obtain a mean tensor within the limits of a fast rotation. All off-diagonal values of the simulated tensors thus become zero and only the principal values of the tensors are left as constraints. The mean temperature at the end of the simulation was only 11 K above the target temperature of 293 K, indicating that exceedingly high pseudo-forces have been successfully avoided indeed, which would otherwise have lead to additional heat production.
Calculated deuterium quadrupolar splittings from a constrained MD simulation of PGLa, compared with the experimental ^{2}H-NMR data (Strandberg et al. 2005) in DMPC
Site | MD (kHz) | NMR (kHz) |
---|---|---|
Ala3 | −22.2 | |
Ala6 | +15.7 | +15.6 |
Ala8 | +17.1 | +17.2 |
Ala10 | −15.0 | −15.0 |
Ala14 | −26.8 | −26.6 |
Ala17 | +17.1 | |
Ala20 | −25.5 | |
Ile9 | −4.9 | −5.2 |
Ile13 | +26.2 | 26.4 |
Molecular Saupe order tensor from a constrained MD simulation of PGLa
Tensor component | MD |
---|---|
S_{aa} | +0.49 |
S_{bb} | −0.28 |
S_{cc} | −0.46 |
S_{ab} | −0.46 |
S_{ac} | +0.06 |
S_{bc} | −0.08 |
At a first glance the rather regular oscillations of PGLa in Fig. 10, or the rotations of cholesterol (Fig. 7), may not seem to reflect reality. However, we have to keep in mind that our MD simulations are performed with single molecules in vacuum, and the temperature is controlled by a continuously acting NTV “thermostat” adjustment. In nature, heat is transferred by stochastic interactions with other molecules, thereby introducing a stochastic behaviour of molecular rotations and re-orientations. This stochastic aspect is not present in our MD simulations, but nonetheless the amplitudes, velocities and directions of the motions adopt realistic values as the NMR constraints have to be satisfied. The molecule in the simulation cannot stop in one preferred orientation, because then no averaging would be performed anymore.
Conclusions
Solid state NMR is a valuable technique to gain insight into the behaviour of peptides and proteins in oriented media, provided the data can be interpreted in terms of molecular structure and dynamics. In this contribution we developed a new strategy in which all-atom MD simulations and NMR data obtained from oriented samples are combined to obtain such structural and motional information. To this aim, a molecular mechanics force field (in this case COSMOS-NMR) was extended to include pseudo-forces, which drive the molecular dynamics to meet the NMR constraints. They “heat up” molecular rotations or re-orientations, leading to proper averaging of the calculated tensor values such that the calculated tensor values agree with the corresponding experimental observations. The orientational constraints can be further combined with intramolecular constraints such as distances or chemical shifts. This way, similar results can be obtained as in full membrane MD simulations, but without the computational burden of having to perform a detailed simulation of the lipids and surrounding water molecules. Because they are performed in vacuum, the constrained MD simulations can be completed in relatively short simulation times (≤1 ns), still reaching a complete averaging of the NMR observables.
In three case studies, ranging from a small rigid compound to a 21-residue membrane-active peptide, the MD simulations with orientational NMR constraints succeeded to produce a detailed picture of the molecular motions and orientations in oriented membranes. It could be demonstrated that this new method is not limited to rigid molecules and does not depend on the choice of the initial coordinates. Deuterium quadrupolar splittings from ^{2}H-labelled pyrene, cholesterol and PGLa in oriented lipid bilayers have been used as constraints in the present examples, but the general formalism presented will be applicable to all kinds of tensorial NMR properties.
Notes
Acknowledgement
Many helpful discussions with Dr. Stephan Grage are gratefully acknowledged.
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