Introduction 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. 1994 1983 1998 2 2 2 2001 a priori Theory NMR interaction tensors and coordinate transformations P 1 1 \documentclass[12pt]{minimal}
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\vec{\nu}_z=\vec{\nu}_{\rm A-B}\\
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\begin{document}$$D=\left(\vec{e}_x,\;\vec{e}_y,\;\vec{e}_z\right)
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P^{\prime}_{\alpha\beta}=D_{\alpha\alpha^{\prime}}D_{\beta\beta^{\prime}}P_{\alpha^{\prime}\beta^{\prime}}
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E_{\rm pseudo}=\frac{k}{2}\sum_{\alpha\beta}\sum_i{\left(P_{\alpha\beta}^{{\rm {theo}}_{i}}-P_{\alpha\beta}^{\exp_{i}}\right)}^2$$\end{document} k zz 3 1994 P theo 3 5 \documentclass[12pt]{minimal}
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P_{\alpha\beta}^{\langle\,\rangle t}=\langle D_{\alpha\alpha^{\prime}}D_{\beta\beta^{\prime}}P_{\alpha^{\prime}\beta^{\prime}}\rangle_t $$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$${D_{\alpha\alpha^{\prime}}D_{\beta\beta^{\prime}}}$$\end{document} 1991 6 \documentclass[12pt]{minimal}
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\langle D_{\alpha\alpha^{\prime}}D_{\beta\beta^{\prime}}P_{\alpha^{\prime}\beta^{\prime}}\rangle_t=\frac{1}{N}
\int_0^{t^{\prime}}\hbox{e}^{t^{\prime}/\tau}D_{\alpha\alpha^{\prime}}(t)D_{\beta\beta^{\prime}}(t)\,\hbox{d}t^{\prime}P_{\alpha^{\prime}\beta^{\prime}}$$\end{document} N t 6 S S n n S n 7 \documentclass[12pt]{minimal}
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{\bf S}_0={\bf P}\\
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{\bf S}_{n+1}=\frac{1}{N_{n+1}}\left(N_n{\bf S}_n\hbox{e}^{-\Delta
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\begin{aligned}
&F_{x_{\gamma_j}}^{\rm A}=k\sum\limits_{\alpha\beta}^{3}
(P_{\alpha\beta}^{\rm theo_A}-P_{\alpha\beta}^{\rm exp_A})\frac{\partial}{\partial x_{\gamma_j}}P_{\alpha\beta}^{\rm theo_A}\\
&\frac{\partial}{\partial x_{\gamma_j}} P_{\alpha\beta}^{\rm theo_A}=
\left(D_{\beta\beta^{\prime}}\frac{\partial}{\partial x_{\gamma_j}}D_{\alpha\alpha^{\prime}}
+D_{\alpha\alpha^{\prime}}\frac{\partial}{\partial x_{\gamma_j}}D_{\beta\beta^{\prime}}\right)P_{\alpha\beta}^{\rm theo_A}
\end{aligned}
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\partial=\frac{\partial}{\partial x_{\alpha_j}}\quad\alpha=\{1,2,3\}\quad \hbox{and}\,
j=\{\hbox{A,B,C}\}
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\vec{\nu}_{\rm A-B}\times\partial\vec{\nu}_{\rm B-C}\\
\partial\vec{\nu}_x=\partial\vec{\nu}_y\times\vec{\nu}_{\rm A-B}
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\vec{e}=\frac{\vec{\nu}}{{\left(\vec{\nu}\vec{\nu}\right)}^{1/2}}\\
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\begin{document}$${\partial D_{\alpha\beta}=\partial e_{\alpha\beta}}$$\end{document} 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. 8 \documentclass[12pt]{minimal}
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\begin{aligned}
&f=\hbox{e}^{\frac{(P^{\rm theo}-P^{\rm exp})}{\Delta P}}\\
&c(P^{\rm theo}-P^{\rm exp})=\frac{f-f^{-1}}{f+f^{-1}}
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\begin{document}$${P^{\rm theo}-P^{\rm exp} < \Delta P}$$\end{document} k P Order parameter calculation D 1964 12 \documentclass[12pt]{minimal}
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S_{ij}=1/2\langle 3\cos\Theta_i\cos\Theta_j-\delta_{ij}\rangle
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P_{\alpha\beta}^{{\rm A}_t}=D_{\alpha\alpha^{\prime}}^{\rm PAS_A}D_{\beta\beta^{\prime}}^{\rm PAS_A}P_{\alpha^{\prime}\beta^{\prime}}^{\langle\,\rangle t}\\
W_{\alpha\beta}^{\rm A}=\frac{1}{2}\langle 3D_{z\alpha}^{\rm PAS_A}
D_{z\beta}^{\rm PAS_A}-\delta_{\alpha\beta}\rangle_t \end{array}
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S_{kl}^{\rm inertia}\frac{1}{2}\left\langle 3D_{zk}^{\rm PAS}
D_{zl}^{\rm PAS}-\delta_{kl}\right\rangle_t\quad k,l=\{a,b,c\}
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{\bf S}=\left(\begin{array}{lll}
-\frac{1}{2}S-\xi&0&0\\
0&-\frac{1}{2}S+\xi&0\\
0&0&S
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{\bf S}=\left(\begin{array}{lll}
-0.5&0&0\\
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0&0&1
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\begin{document}$${f=1-\hbox{e}^{-t/\rho}}$$\end{document} 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 C Q 17 \documentclass[12pt]{minimal}
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\begin{document}$$C_{\alpha\beta}^Q=\frac{e^2}{h}V_{\alpha\beta}Q
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\nu_{Q}^{\pm}=\pm\frac{3}{4}D_{z\alpha}D_{z\beta}C_{\alpha\beta}^{Q}
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\begin{document}$${\eta_{Q}=0.06\;(\eta_{Q}=
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\begin{document}$${C_{33}^{Q} > C_{11}^{Q} > C_{22}^{Q}}$$\end{document} 1998 3 3 2 3 Program implementation 2003 2002 2001 2001 http://www.cosmos-software.de 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 2 2005 2 2 B 0 2 2 2 2005 Fig. 2 2 2 2 a b c  2 4 2 P 3 2 6 2 4 2005 Table 1 Deuterium quadrupolar coupling tensors Group zz Q yy Q xx Q Source 2 2 193 −102.29 −90.71 1974 3 2 175 −87.5 −87.5 1974 3 58.33 −29.17 −29.17 3 2 a a Table 2 General parameters for the MD simulations with orientational constraints Parameter Value Target temperature 293 K MD time step 0.5 fs Coupling time η to the heat bath 0.05 ps P 11 1 kHz 6 7 200 ps Time constant ρ for the exponential rise of pseudo-forces 200 ps Total MD duration 1 ns Fig. 3 2 B 0  Fig. 4 2 2005  3 C xy C xz C yz 6 3 2 2 2 2 2 B 0 3 Table 3 2 Site a Constrained MD splitting Δν (kHz) b 2 + c 93.0 2 + c 40.5 2 + c 40.5 a B 0 b 2005 c 2 2 20 \documentclass[12pt]{minimal}
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\Delta\nu^{\rm A}=W_{\alpha\alpha}^{\rm A}C_{\alpha\alpha}^{Q}
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$$\end{document} 21 3 C Q 2 2 1 W A 20 W A 4 Table 4 Simulated order parameters for pyrene in POPC Site W xx W yy W zz W xy W xz W yz 2 +0.021 −0.342 +0.321 ≤0.03 2 +0.181 −0.310 +0.129 ≤0.07 2 +0.165 −0.294 +0.129 ≤0.03  S bb S cc S aa S ab S ac S bc a +0.048 −0.275 +0.228 ≤0.006 a 2 2 S W A S W A 2 S 4 W 2 4 S W S S aa a S cc S bb 16 S c \documentclass[12pt]{minimal}
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\begin{document}$${\langle\cos \Theta_{c}^{2}\rangle}$$\end{document} 2005 S zz S cc Cholesterol 2 1984 1999 2 5 Fig. 5 2  2 1992 3 5 2 1999 5 2 B 0 5 2 Table 5 2 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 3 −33.0 −33.8  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  5 2 6 2 2 5 Fig. 6 2 B 0 5 5 3  a 6 16 1999 a a 1999 1984 Table 6 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  a a z 7 a Fig. 7 a a z  PGLa 1999 2 2005 2 3 3 2 3- 1 α β 2005 1999 3 + B 0 2 7 Table 7 2 2005 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 8 a c b Fig. 8 2 3 2 a b α  9 a b c a b c a 2 2005 a α b α 8 10 b b 2005 Fig. 9 2 2005 a b c  Fig. 10 b b  2 8 a 10 8 S 15 10 Table 8 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 10 7 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. 2