Introduction 2005 1989 2006 S LS 2 1982a b 1989 2006 2002 2002 2004 2005 2001 2001 c 1997 2004 2001 2006 2005 2005 In this study we inspect critically the concept of a common alignment frame for a protein structure on the basis of experimental data. We find that a variation in molecular alignment due to the ps–ns fluctuations can contribute significantly to the RDC-probed order parameter in the case of a small globular protein. Theory Protein motions captured by RDCs residual 1997 molecular perspective 1 \documentclass[12pt]{minimal}
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\begin{document}$$S_\Upomega^2 $$\end{document} S τ 2 −6 2001 1997 S τ 2 Dynamics modulated alignment 2006 2004 Signature of dynamics modulated alignment 2005 2001 \documentclass[12pt]{minimal}
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\begin{document}$$\vartheta^{2}=2/3\sum {\vartheta_{ij}^2}. $$\end{document} ij 1999 1 \documentclass[12pt]{minimal}
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\begin{document}$$ RDC=\left(\frac{\mu_o}{4\pi} \right)\frac{\gamma_A \gamma_B h}{2\pi^{2}r_{AB}^3}\sum_{ij=\{x,y,z\}} {\vartheta_{ij} c_i c_j}, $$\end{document} c i c j A B r AB h o S LS 2 2 α 2006 \documentclass[12pt]{minimal}
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\begin{document}$$\left\langle {\vartheta_{\rm NH}^2} \right\rangle_o =0.878 \pm 0.002$$\end{document} α \documentclass[12pt]{minimal}
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\begin{document}$$\left\langle {\vartheta_{{\rm C}^{\alpha}{\rm CO}}^2} \right\rangle_o =0.955\pm 0.001$$\end{document} S LS 2 2003 1995 α 2005 2006 2003 2 α \documentclass[12pt]{minimal}
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\begin{document}$$ \delta =\frac{\left\langle {\vartheta_{{\rm C}^{\alpha}{\rm CO}}^2} \right\rangle -\left\langle {\vartheta_{\rm NH}^2} \right\rangle}{\left\langle {\vartheta_{{\rm C}^{\alpha}{\rm CO}}^2} \right\rangle}=1-\frac{\left\langle {\vartheta_{\rm NH}^2} \right\rangle}{\left\langle {\vartheta_{{\rm C}^{\alpha}{\rm CO}}^2} \right\rangle} $$\end{document} 2004a b 1998 2003 2003 1999 2 Amount of intrinsic bond fluctuations \documentclass[12pt]{minimal}
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\begin{document}$$\left\langle {\vartheta_{{\rm C}^{\alpha}{\rm CO}}^2} \right\rangle_o $$\end{document} o o o Local alignment frames 2001 2 2006 Methods 1998 2004 2003 2006 2000 2004 2 2004a 2003 2004b 1998 2003 1 1998 2003 Table 1 2 2 −6 Protein Medium \documentclass[12pt]{minimal}
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\begin{document}$${\delta \pm \Updelta \delta}$$\end{document} T (K) a CTAB 2.073 2.349 0.117 ± 0.006 302 PEG 0.614 0.750 0.181 ± 0.009 304 PAG+ 1.079 1.242 0.131 ± 0.007 298 PAG− 1.113 1.320 0.158 ± 0.006 298 Pf1 1.377 1.731 0.205 ± 0.005 302 b Bicelles 0.695 0.788 0.118 ± 0.005 304 CTAB 2.269 2.685 0.155 ± 0.003 304 c Pf1 4.451 4.728 0.059 ± 0.001 318 Experimental temperature is shown for an easy comparison with δ variation a NH \documentclass[12pt]{minimal}
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\begin{document}$${\varepsilon_{{\rm C}^{\alpha}{\rm CO}}}=0.10\,\hbox{Hz};$$\end{document} b NH c 1999 1 hybrid model 5 2 Table 2 \documentclass[12pt]{minimal}
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\begin{document}$$\Upomega$$\end{document} Protein Medium \documentclass[12pt]{minimal}
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\begin{document}$${\Upomega_{{\rm C}^{\alpha}{\rm CO}}}$$\end{document} direct SVD GB1 Negative 31.7 ± 0.2 31.1 ± 0.2 0.172 ± 0.007 0.109 ± 0.008 Neutral 11.82 ± 0.04 7.38 ± 0.02 0.114 ± 0.001 0.116 ± 0.008 Positive 40.36 ± 0.05 36.23 ± 0.03 0.385 ± 0.004 0.208 ± 0.002 GB3 Negative 12.60 ± 0.10 9.54 ± 0.06 0.082 ± 0.003 0.097 ± 0.009 Neutral 11.82 ± 0.09 7.75 ± 0.05 0.096 ± 0.003 0.100 ± 0.009 Positive 23.0 ± 0.4 20.6 ± 0.4 0.119 ± 0.010 0.107 ± 0.007 UBI Negative 13.21 ± 0.04 9.87 ± 0.02 0.090 ± 0.001 0.106 ± 0.003 Neutral 16.95 ± 0.09 13.16 ± 0.08 0.080 ± 0.002 0.085 ± 0.005 Positive 15.84 ± 0.06 12.86 ± 0.05 0.092 ± 0.002 0.104 ± 0.007 Results and discussion Analysis of experimental data 1 2003 1999 α \documentclass[12pt]{minimal}
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\begin{document}$$\delta > \delta_{o}$$\end{document} α r NH 1992 \documentclass[12pt]{minimal}
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\begin{document}$$r_{{\rm C}^{\alpha}{\rm CO}} =1.53\,\hbox{\AA}$$\end{document} 1991 1972 r NH 1998 α α S LS 2 Simple model of wobble Y 00 3 \documentclass[12pt]{minimal}
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\begin{document}$$ \vartheta^{2}=\frac{4\pi}{5}\sum_{m=-2}^2 {\left\langle {Y_{0m} Y_{0m}^\ast} \right\rangle \approx \frac{1}{4}\left({3\left\langle {c^{2}} \right\rangle_{\theta,\Upomega} -1} \right)^{2}} $$\end{document} \documentclass[12pt]{minimal}
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\begin{document}$$ \vartheta^{2}\approx \frac{1}{4}\left[ {3\cos^{2}\left({\theta_o +\Upomega} \right)-1} \right]^{2} $$\end{document} o \documentclass[12pt]{minimal}
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\begin{document}$$(\delta, \Upomega)$$\end{document} 2 4 Analysis of simulation data The sources of the medium-dependent variation in δ found from the experimental data can be many. Fortunately, they can be assessed via computer simulations. We consider the variation in the protein shape and charge distribution due to the ps–ns dynamics. These fluctuations give rise to a family of conformations whose members do not all align in exactly the same way. Simultaneous variation in the shape and net electric dipole moment may lead to a substantial alignment fluctuation. However, it is also conceivable that dynamics of the nematogen contributes to the protein alignment dispersion. Other microscopic heterogeneity in obstructing nematogens may also add to the variation of protein alignments but we are unequipped to examine these effects. Neither are the possible effects of hydration fully explored in this study. α c i c j 1 α \documentclass[12pt]{minimal}
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\begin{document}$$\left\langle {c^{2}} \right\rangle_{\theta,\Upomega} $$\end{document} 2 2 1 2 Non-uniform bond direction sampling 2000 f i \documentclass[12pt]{minimal}
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\begin{document}$$f_{\rm z}=f_{\rm y}=f_{\rm x} = 1/3,$$\end{document} 3 Table 3 The fraction of vectors oriented along the three principal axis calculated for the MD simulated ensembles of GB1, GB3 and ubiquitin Protein Bond f z f y f x GB1 NH 0.54 0.26 0.20 α 0.45 0.29 0.26 GB3 NH 0.55 0.26 0.19 α 0.45 0.29 0.26 UBI NH 0.49 0.31 0.21 α 0.37 0.34 0.29 α α 2 4 α α Table 4 SVD SVD α Protein Medium SVD SVD GB1 Negative 0.109 0.113 ± 0.008 Neutral 0.116 0.116 ± 0.007 Positive 0.208 0.130 ± 0.008 GB3 Negative 0.097 0.116 ± 0.006 Neutral 0.100 0.106 ± 0.006 Positive 0.107 0.113 ± 0.006 UBI Negative 0.106 0.109 ± 0.003 Neutral 0.085 0.018 ± 0.008 Positive 0.104 0.092 ± 0.007 Table 5 Hybrid model Protein Medium δ hybrid GB3 CTAB 0.117 0.133 ± 0.005 Pf1 0.205 0.265 ± 0.004 UBI Bicelles 0.118 0.093 ± 0.008 CTAB 0.155 0.139 ± 0.006 hybrid Hydration shell Since the MD simulated conformational ensembles seemed to be less susceptible to DMA than NMR-based ensembles, it was plausible that PALES failed to impose a realistic dispersion of alignments. We hypothesised that the addition of a hydration shell to the molecules would make them more globular and therefore more susceptible to alignment dispersion due to steric interactions. To test whether this hypothesis could account for the observed lack of dispersion, a hydration layer needed to be added to the molecules. This was done with a rough model of the hydration shell in the form of increased atomic radii in PALES. Atomic radii were increased by 1, 2, 3, and 4 Å, but no increase in DMA was observed neither for GB1 nor for ubiquitin. The lack of a hydration shell does not therefore seem to be the cause for the difference in DMA between simulations and experiments. Hybrid model 5 3 Fig. 3 \documentclass[12pt]{minimal}
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\begin{document}$$\Upomega$$\end{document}  Competition of interactions 4 Fig. 4 a b c d  It is more intricate to rationalize the reduced order parameters in a charged media as both molecular shape and charge distribution fluctuations are present. It may not be possible to simplify the dynamics of a complex set of interactions to provide lucid understanding to the causes of DMA. To begin with, GB1, GB3, ubiquitin and calerythrin carry under experimental conditions a net charge of −1.22, −3.09, +1.39 and −8.45, respectively. In the repulsive electrostatic potential the alignment is, to a crude approximation, governed by the direction and size of the net electric dipole moment in relation to the molecular shape tensor. 4 1 2003 In general a small globular protein that carries many charges and yet only a small net electric dipole moment is the most vulnerable to molecular alignment fluctuations. The shape as well as the charge distribution is perturbed by dynamics of long side-chains at the molecular surface. For large proteins the effects are expected to have less impact on the overall molecular alignment. Conclusions α 2000 1999 2001 2001 Here we were able to consider only alignment effects due to protein dynamics that occur in the ps–ns range owing to the limitations of molecular dynamics simulations. Other states accessible via slower motions are of course also expected to contribute to the variation in molecular alignment depending on their population, shapes and charge distributions. We also expect that dynamics of nematogens and other variation in the media contribute as well to the bond vector dispersion underlying the observed RDC signals. 2005 c S 2 2