Communication pubs.acs.org/JACS
Dynamic Descriptions of Highly Flexible Molecules from NMR Dipolar Couplings: Physical Basis and Limitations Nicola Salvi, Loïc Salmon, and Martin Blackledge* Institut de Biologie Structurale (IBS), CEA, CNRS, University Grenoble Alpes, Grenoble 38044, France S Supporting Information *
Planck’s constant. The term within angular parentheses in eq 1 depends on the alignment properties of the protein, and on the conformational properties of the internuclear vector. This average can be expressed in terms of the orientational probability P(Ω):
ABSTRACT: Biomolecules that control physiological function by changing their conformation play key roles in biology and remain poorly characterized. NMR dipolar couplings (DCs) depend intrinsically on both molecular shape and structural fluctuations, thereby providing the enticing prospect of tracking these conformational changes at atomic detail. Although this dual dependence has until now severely complicated analysis of DCs from highly dynamic systems, general approaches have recently been proposed that simplify interpretation of experimental DCs, by entirely eliminating molecular alignment from the analysis. Using simple and intuitive simulation of target ensembles, we investigate the impact of such approaches on the resulting descriptions of the conformational energy landscape. We find that ensemble descriptions of highly flexible systems derived from DCs without explicit consideration of the alignment properties of the constituent conformations can be compromised and inaccurate, despite exhibiting high correlation with experimental measurement.
3cos2 θ − 1 2
3cos2 θ − 1 2
(2)
=
3 2
∑
⟨cos βk cos βl ⟩⟨cos αk cos αl⟩ −
k , l ∈ (x , y , z)
1 2 (3)
where βk and αk describe respectively the orientation of the magnetic field and of the internuclear vector within the molecular frame. The first average refers to the probability of the axes of the molecular frame to be oriented along the magnetic field, and takes the form of a time-independent, symmetric and traceless second rank tensor S (the Saupe tensor) relative to which structural and dynamic information can be quantified, leading to
arge-scale protein dynamics lie at the heart of innumerable biological processes, including signal transduction, allostery and remodelling of functional protein complexes.1 The ability of proteins to morphologically adapt to their environment often relies on the presence of significant levels of intrinsic disorder, providing flexible linkers and extreme levels of conformational freedom to achieve this plasticity. Understanding the free energy landscapes of proteins with such high levels of flexibility remains one of the major challenges of contemporary structural biology.2 Solution state nuclear magnetic resonance (NMR) is currently the most powerful experimental tool to achieve this aim.3−6 In particular, dipolar couplings (DCs), measured under conditions of partial alignment, are remarkably sensitive to the conformational properties of different regions of the protein, and, despite significant orientational degeneracy, can help delineate the contours of conformational space populated by biomolecules in solution. The dipolar interaction between two spins 1/2 I and S is dependent on the anisotropic averaging properties of θIS, the instantaneous angle between the I−S bond vector and the static magnetic field:
DIS = KIS
∑
Skl⟨cos αk cos αl⟩
k , l ∈ (x , y , z)
(4)
This formalism not only provides a convenient mathematical framework for DC averaging but also allows for a physical representation of molecular dynamics in which the reorientation properties of the system relative to the magnetic field are treated as a second-rank tensor and local motions are described within a frame associated with this tensor. For steric alignment, the tensor can be estimated on the basis of molecular shape.8 Approaches to structural or dynamic analysis of DCs in folded proteins have normally estimated components of the common alignment tensor a priori, or more commonly, simultaneously to vector orientation properties.9−21 Methods that do not rely on prediction of the physical basis of alignment, exploiting for example singular value decomposition to determine numerical values of Skl, can accurately estimate substate populations from commonly accessible DC data.22−25 In the case of time-dependent molecular shape, the situation is more complex. If the system samples N molecular shapes, the
(1)
with KIS = −γIγSμ0h/8πrIS3, rIS the internuclear distance, γ the gyromagnetic ratio, μ0 the permeability of free space, and h © 2017 American Chemical Society
∫ P(Ω)P2(cos θ)dΩ
In the case of a time-independent molecular shape, the standard approach is to deconvolute this complex average into components reporting on molecular alignment,7 and vector orientation relative to this coordinate system:
L
DIS = KIS⟨P2cos(θIS)⟩
=
Received: February 14, 2017 Published: March 14, 2017 5011
DOI: 10.1021/jacs.7b01566 J. Am. Chem. Soc. 2017, 139, 5011−5014
Communication
Journal of the American Chemical Society measured DC must account for the heterogeneous orientational probability Pn(Ω) of each molecular shape n, with population pn: N
DIS = KIS ∑ pn
∫ Pn(Ω)P2(cos θn)dΩ
n=1
(5)
By introducing the individual alignment probabilities Snkl for each molecular shape, this can be reframed as
Figure 1. Representation of the simulation system. (A) 10 000 conformers sample accessible orientations of the two domains: (red) N-terminal domain, all superposed, and (blue) linker (25 amino acids) and C-terminal domain. (B, C) DCs are predicted for each conformer on the basis of shape (solid and dashed lines show principal alignment directions for two conformers). Target DCs are averaged over subensembles of conformers.
N
DIS = KIS ∑ pn { n=1
Skln⟨cos αk , ncos αl , n⟩}
∑ k , l ∈ (x , y , z)
(6)
If the different conformers present significantly different shapes or alignment properties, the determination of common alignment properties over the entire ensemble is inappropriate. Although it is possible to predict alignment on the basis of molecular shape, approaches that have been used to interpret DCs from intrinsically disordered or multidomain proteins,26−29 similar procedures are less straightforward, and more computationally demanding, when alignment is not steric.30,31 For these reasons, the recent suggestion that the alignment properties of the molecule can be circumvented by treating eq 1 directly, for example as the target function for restrained molecular dynamics refinement, provides a potentially attractive solution to the analysis of high dynamic proteins by NMR.32−36 By-passing the use of an alignment tensor, the double sum in eq 6 is avoided and eq 1 reframed into a single average over vector orientations in an explicit multiconformational ensemble relative to a common (magnetic field) axis b:
Synthetic target data sets (Dexp ij ) were generated by averaging DCs from subensembles of N = 2 or 4 randomly selected conformations, following eq 8. Ensembles of N = 2, 4, 8, or 16 conformations were then selected from this pool (after removal of target conformers) to minimize over M target DCs: M
χ2 =
N
∑ {KISa ∑ pn P2(cosθm,n) − Dmexp}2 m=1
n=1
(9)
For simplicity, each conformer was equally weighted, so that p = 1/N. Very close agreement was indeed found (Figure 2) by
N
DIS = KISa ∑ pn P2(cos θn) n=1
(7)
a is a common scaling factor reporting on the overall level of alignment. The physical restraints imposed on the effective alignment tensor when using this approach to study proteins of fixed molecular shape were recently discussed.37,38 Further developments invoke multiple directions bj, to allow simultaneous analysis of multiple alignment conditions without explicitly calculating physical interactions with the media.39 Accurate and efficient calculation of the physical basis of aligning interactions is one of the key challenges associated with analysis of DCs measured from partially aligned flexible proteins. Alleviating the need to simulate alignment of the protein under any alignment conditions would therefore greatly simplify analytical procedures. Here we test the physical basis of this approach and investigate whether such a description can indeed provide a meaningful description of conformational disorder in dynamic systems. 10 000 conformations of a model protein comprising two rigid domains connected by a flexible linker (25 amino acids in length) were generated (Figure 1), and the restricted molecular alignment and resultant nonaveraged DCs predicted on the basis of the shape of each conformer. For simplicity, we assume DC averaging is affected only by molecular reorientation and overall shape modulation, and not by local motion:
Figure 2. Analysis using a single alignment medium. (A) Typical correlation between target (Dexp ij x-axis) and DCs from ensembles of 16 conformers. (B, C) Typical correlation between target (x-axis) and shape-predicted DCs for two equally valid solutions. (D) Correlation of shape-predicted DCs for solutions shown in panels B and C.
selecting subsets of conformers from the available pool. The significant orientational degeneracy of DCs40,41 precludes straightforward analysis of the validity of these ensembles in terms of domain orientations within the ensembles. However, their accuracy in terms of angular sampling of the dipolarcoupled vectors can be precisely gauged, by comparing the
N
DIS = KIS ∑ pn n=1
∑ k , l ∈ (x , y , z)
Skln cos αk , ncos αl , n (8) 5012
DOI: 10.1021/jacs.7b01566 J. Am. Chem. Soc. 2017, 139, 5011−5014
Communication
Journal of the American Chemical Society
with both sets of target DCs, again show very poor agreement with the orientational properties of the target ensemble (Figure 3 and S4, Supporting Information). Repeat solutions again disagree with each other.
sterically predicted DCs with those calculated for the target ensemble. If the selection correctly describes the physical properties of the system, the ensemble should reproduce sterically predicted DCs averaged according to eq 8. Although the average over eq 7 agrees effectively perfectly with the target calculated over eq 8, DCs predicted on the basis of shape of members of the ensembles disagree with the target and, equally strikingly, repeat solutions disagree with each other (Figure 2). This is true for all tested ensemble sizes and randomly selected targets (Supporting Information). This result reveals that ensembles that are equivalent in terms of χ2 can comprise conformers with very different orientational properties, indicating that the angular information contained in the ensembles is different to the target and incorrect. Notably, fixing b to an arbitrary direction always produces (different) ensembles in excellent agreement with the target (Figures S1 and S2, Supporting Information). This calculation demonstrates that the true physical average (eq 6 or 8) cannot be simplified by considering an average over eq 7 that ignores the orientational probability of each conformer. It is interesting to examine the reason why effectively exact agreement with DCs can be achieved despite incorrect angular sampling. Equation 6 or 8 can be expressed in matrix format:
Figure 3. Analysis using two alignment media. (A, B) Target DCs (Dexp ij , blue) predicted from shape (A) and electrostatic charge (B) by ensembles of 16 conformers calculated using eq 7 (fit to both sets simultaneously). Red: Typical correlation between target and predicted DCs selected using eq 1. (C) Correlation of shape-predicted DCs for two equivalent solutions.
The possible impact of using eq 7 to describe conformational disorder is graphically illustrated in Figure 4. DCs were predicted from a third (positively charged) alignment medium, and averaged over two conformers constituting a simple target ensemble. Subensembles of four conformers were selected from the remaining pool using the three DC data sets, averaging DCs using either eq 8 or eq 7 to match the target. Figure 4 (and S5, S6) show that in the presence of three alignment conditions, eq 8 can reasonably reproduce the relative domain orientations of the target, aided by asymmetric surface charge distributions and overall shape that encode physical interactions with all three alignment media. Practical application of such an approach is limited by orientational degeneracy, the actual nature of the dynamic equilibrium and available degrees of freedom, as well as identification of different alignment conditions and accuracy of DC prediction. Nevertheless, these hypothetical examples
N
DIS = KIS ∑ XISSn n=1
(10)
where the column vectors Sn describe the alignment properties of the N different molecular forms, and XIS the local conformational properties of the internuclear vector. Minimisation of eq 9 is equivalent to identifying a basis set of vector orientations, or average spherical harmonics X′ that solve the expression: DIS = KISa X′S′
(11)
using a single axially symmetric tensor S′, common to all the conformers in the selected ensemble. The axial symmetry of S′ is a direct consequence of describing conformational sampling relative to the magnetic field vector b uniquely in terms of the polar angle θ (eq 7). The reason why the ensembles agree so well with the target DCs appears to derive from the rotational averaging properties of spherical harmonics, so that, assuming sufficient degrees of conformational freedom are available, X′ can always be found that solves eq 11 in the presence of a single axially symmetric tensor S′. As far as we can tell, the associated conformational sampling of X′ is not related to the conformational origin of DIS in a simple way, because it fails to account for the physical nature of the interaction between the solute and the alignment medium. Clearly, if this approach cannot accurately reproduce an ensemble of rigid structures, it is unlikely to succeed when applied to the significantly higher conformational freedom available to an ensemble of pliable structures present in ensemble restrained MD. In principle, DCs measured in one alignment medium are not sufficient to unambiguously determine the orientation of bond vectors or chiral elements. This orientational degeneracy can be raised by measuring DCs in the presence of differently, for example electrostatically, aligning media. For this reason, we have also predicted DCs due to alignment of the 10 000 conformers using an electrostatic alignment.30,31 Target DCs were again predicted from a subensemble of 4 structures, and used to select ensembles of N = 16 conformations from this pool using eq 9. The resulting ensembles, although agreeing
Figure 4. Ensemble selection of 4 conformers reproducing DCs from 3 alignment media and a target ensemble of 2 structures (red: a and b) for a two-domain (I/II) protein. All ensembles are superimposed on domain I (orange). Ensembles selected using eq 8 (A, blue), or (7) (B, green). (C) Reproduction of DCs for (A) blue, and (B) green. Linkers contained no DCs and are not shown for clarity. See Supporting Information for orthogonal view. 5013
DOI: 10.1021/jacs.7b01566 J. Am. Chem. Soc. 2017, 139, 5011−5014
Communication
Journal of the American Chemical Society
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again show that although DCs are equally well reproduced, eq 7 does not reproduce the nature of the target. In view of these results, it is evident that averaging procedures that ignore the dependence of angular order on the shape of each conformation sampled over the time and ensemble average cannot correctly reproduce the conformational properties of the ensemble, despite mathematically reproducing the DCs. This apparently paradoxical observation has its origin in the physically incorrect interpretation of eq 1 inherent to approaches that do not evoke distinct alignment of each molecular shape. Although this equation is general, precise averaging procedures need to be respected in order to reproduce the physical behavior of the considered system. If this is not the case, numerically acceptable solutions can be found, but these do not provide any information about the physically encoded motional modes. This underlines the importance of rigorously considering the impact of specific alignment of each conformation on the measured average DC.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/jacs.7b01566. Simulation details and further results (PDF) All pdb files and simulated RDCs, target RDCs, resulting selected ensembles and effective alignment tensors (ZIP)
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AUTHOR INFORMATION
Corresponding Author
*
[email protected] ORCID
Martin Blackledge: 0000-0003-0935-721X Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Swiss National Science Foundation Early Postdoc Mobility Fellowship P2ELP2_148858 (NS) and FRISBI (ANR-10INSB-05-02) and GRAL (ANR-10-LABX-49-01).
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DOI: 10.1021/jacs.7b01566 J. Am. Chem. Soc. 2017, 139, 5011−5014
