Polar Versus Non-polar Local Ordering at Mobile Sites in Proteins

Article pubs.acs.org/JPCB

Polar Versus Non-polar Local Ordering at Mobile Sites in Proteins: Slowly Relaxing Local Structure Analysis of 15N Relaxation in the Third Immunoglobulin-Binding Domain of Streptococcal Protein G Oren Tchaicheeyan and Eva Meirovitch* The Mina and Everard Goodman Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan 52900 Israel S Supporting Information *

ABSTRACT: We developed recently the slowly relaxing local structure (SRLS) approach for studying restricted motions in proteins by NMR. The spatial restrictions have been described by potentials comprising the traditional L = 2, K = 0, 2 spherical harmonics. However, the latter are associated with non-polar ordering whereas protein-anchored probes experience polar ordering, described by odd-L spherical harmonics. Here we extend the SRLS potential to include the L = 1, K = 0, 1 spherical harmonics and analyze 15N−1H relaxation from the third immunoglobulinbinding domain of streptococcal protein G (GB3) with the polar L = 1 potential (coefficients c10 and c11) or the non-polar L = 2 potential (coefficients c20 and c22). Strong potentials, with ⟨c10⟩ ∼ 60 for L = 1 and ⟨c20⟩ ∼ 20 for L = 2 (in units of kBT), are detected. In the α-helix of GB3 the coefficients of the rhombic terms are c11 ∼ c22 ∼ 0; in the preceding (following) chain segment they are ⟨c11⟩ ∼ 6 for L = 1 and ⟨c22⟩ ∼ 14 for L = 2 (⟨c11⟩ ∼ 3 for L = 1 and ⟨c22⟩ ∼ 7 for L = 2). The local diffusion rate, D2, lies in the 5 × 109−1 × 1011 s−1 range; it is generally larger for L = 1. The main ordering axis deviates moderately from the N−H bond. Corresponding L = 1 and L = 2 potentials and probability density functions are illustrated for residues A26 of the α-helix, Y3 of the β1-strand, and L12 of the β1/β2 loop; they differ considerably. Polar/orientational ordering is shown to be associated with GB3 binding to its cognate Fab fragment. The polarity of the local ordering is clearly an important factor.

1. INTRODUCTION It is widely accepted today that protein function is predicated on both 3D structure1 and dynamics.2 Stochastic approaches for elucidating protein dynamics by NMR, typically applied to experimental relaxation parameters,3−7 emerged as particularly useful tools for studying internal mobility in proteins.8−15 The internal dynamics materialize as restricted motions of protein moieties comprising NMR nuclei (e.g., 15N−1H bonds). The spatial restrictions at the site of the motion of the probe are typically expressed by potentials, which represent structural information. These potentials are associated with ordering tensors, related to preferential arrangement in space of molecular axes, hence to local geometric features. Thus, important kinetic (rate-constant-related), structure-related and geometry-related characteristics can be elucidated by studying motions that take place in anisotropic environments, notably internal protein surroundings.8−15 Symmetry-related properties of local structure and geometry, which have not been considered in substantial detail, are the subject matter of this study. Restricted motions in proteins are also amenable to investigation with molecular dynamics (MD) methods.16−19 MD simulations are typically predictive in nature, forgoing the utilization of informative experimental relaxation parameters.11−15 In quite a few cases MD simulations have not © 2016 American Chemical Society

reproduced satisfactorily order parameters derived independently from the experimental relaxation parameters.16 In principle, one could reproduce with MD simulations the experimental relaxation parameters themselves within the scope of fitting schemes. However, this requires MD-derived time correlation functions (TCFs) (which underlie the experimental relaxation parameters) that are good approximations to the actual TCFs.14,19 Simple TCFs associated with NMR relaxation have been derived in the past from MD trajectories.17 However, research conducted in recent years has shown that the actual TCFs are not simple.11−15 Thus, methods for deriving from MD trajectories TCFs that are good approximations to the actual TCFs have yet to be developed.14 In addition, a large number of MD trajectories have to be generated so as to optimize iteratively the relevant force-field parameters. This scenario is not viable at present; addressing it effectively with MD calculations (in a broad perspective) is an important future prospect.19 Thus, currently stochastic models are still important methods for studying restricted motions in proteins. Let us consider from a general perspective the key element of local ordering. Received: October 20, 2015 Revised: December 31, 2015 Published: January 5, 2016 386

DOI: 10.1021/acs.jpcb.5b10244 J. Phys. Chem. B 2016, 120, 386−395

Article

The Journal of Physical Chemistry B This physical quantity is typically expressed by a potential, U, expanded in the complete basis set of the Wigner rotation matrix elements, DLMK.6,7 For a rigid probe reorienting in a uniaxial medium one has3−7 ∞

u(θ , φ) ≡ U (θ , φ)/kBT = − ∑

with the secondary structure context, from the perspective of ordering polarity. Connection between the polarity of the local ordering and biological function has also been established. Thus, we found that the only chain segment where polar and non-polar ordering differ substantially, and polar ordering has unique characteristics, is the one involved in binding the cognate Fab fragment of GB3. Future prospects include the application of the L = 1, K = 0, 1 potential to additional proteins, inclusion of L = 3 terms in the SRLS potential, and the utilization of potentials containing both even-L and odd-L terms. This study is relevant in a broader scope. Potentials featuring odd-rank spherical harmonics are relevant to 31P relaxation in phospholipid bilayers and liposomes,23 dielectric relaxation of ferroelectric systems,24 and molecules experiencing electricfield-implied polar ordering.25 To our knowledge, in those cases where L = 1 potentials were used, only the axial (K = 0) term was considered. A theoretical summary of SRLS is given in section 2; the complete SRLS formalism is presented in the Supporting Information. Results and discussion are delineated in section 3. Our conclusions appear in section 4.

+L

∑

cKLD0LK (0, θ , φ)

L = 1 K =−L

(1)

cLK

with u(θ,φ) and the coefficients being dimensionless. Given that the director of the medium has been taken uniaxial, only the two polar angles, θ and φ, are needed to describe the molecular orientation with respect to the director. This is a typical simplification, requiring just the DL0K(0, θ, φ) functions, which are proportional to the corresponding spherical harmonics. We have developed in recent year the slowly relaxing local structure (SRLS) approach for studying NMR relaxation in proteins.12−15 SRLS is a (complex) extension of the general theories for treating restricted motions, notably in macroscopically oriented liquid crystals (LCs). For probes with D2h symmetry the (real) SRLS potential has been typically given by the L = 2, K = 0, 2 spherical harmonics;5−7 in some cases, it also comprised the L = 4, K = 0, 2 terms.9 There is, however, a symmetry-related distinction between the local ordering in a LC and the local ordering of, e.g., an N− H bond in a protein. LC media exhibit inversion symmetry. One may consider the ordering of probes dissolved therein to reflect this property. Such ordering is non-polar in nature, often referred to as alignment.20,21 On the other hand, the ordering of a protein-anchored probe lacks inversion symmetry. Such ordering is polar in nature, often referred to as orientation.20,21 To investigate the implications of ordering polarity we enhance in this study the SRLS potential to also include the L = 1, K = 0, 1 spherical harmonics. Subsequently we analyze in parallel 15N relaxation parameters from the third immunoglobulin-binding domain of streptococcal protein G (GB3)22 using the rhombic L = 1, K = 0, 1 potential, or the rhombic L = 2, K = 0, 2 potential (the simple axial L = 1 and L = 2 potentials were found to generate results for motional correlation functions that are similar in nature, with different potential coefficients8). The physical quantities entering the SRLS model include a local potential in terms of which a local ordering tensor is defined, global and local diffusion tensors, and the relative orientation of the model-related and magnetic tensors. In this study we allow the following parameters to vary: the axial and rhombic potential coefficients c10 and c11 for L = 1, or c20 and c22 for L = 2; the rate constant, D2, of the local diffusion tensor, taken isotropic; and the angle βOF−DF, where OF denotes the local ordering frame and DF the magnetic 15N−1H dipolar frame (which points along the N−H bond). Comparison between the best-fit values of c10, c11, D2(L = 1), βOF−DF(L = 1), and c20, c22, D2(L = 2) and βOF−DF(L = 2), is expected to clarify the distinction between alignment and orientation. We propose a scheme for characterizing the local ordering at mobile sites in proteins in terms of the local potential, u, and the associated probability density functions (PDF), exp(−u), represented in both Cartesian and spherical coordinates. This scheme is applied to the representative residues A26 of the αhelix, Y3 of the β1 strand, and L12 of the β1/β2 loop of GB3, for both the L = 1 and L = 2 scenarios. Comparison between corresponding potentials and PDFs reveals effects associated

2. THEORETICAL SUMMARY The slowly relaxing local structure approach8−10 has been applied to NMR relaxation in proteins11−15 as a two-body coupled-rotator approach. In aqueous solution the globally reorienting protein (body 1) represents an unrestricted rotator. The locally reorienting probe (body 2) represents a restricted rotator, as invariably its motion takes place in the presence of an ordering potential. Given that the spatial restraints experienced by the mobile probe are imposed by the mobile protein, the rotational degrees of freedom of the two bodies are statistically dependent, i.e., dynamically coupled.8 In the limit of large time scale separation between the two motions, one may consider these degrees of freedom to be statistically independent, i.e., decoupled.8 This physical picture is substantiated in terms of a Smoluchowski operator. The corresponding Smoluchowski equation is solved. The solution consists of generic TCFs which depend on rank, L, and order, K, of the Wigner rotation matrix elements comprising the ordering potential. These TCFs are subjected to Fourier transformation to yield the corresponding generic spectral densities. The latter are linearly combined into the measurable spectral densities according to the local geometry, given in this case by the relative orientation of the local diffusion/local ordering and magnetic (15N−1H dipolar and 15N chemical shift anisotropy (CSA)) tensors.11−15 The measurable spectral densities underlie the expressions for the experimental relaxation parameters, in this case 15N T1, T2 and 15N−{1H} NOE. For a uniaxial orienting medium (director) the ordering potential to be actually used is given by the real linear combinations of the Wigner rotation matrix elements, DL0K. This requires that cLK = (−)K(cL−K)* (refs 9 and 26), yielding ∞ L u(θ , φ) = −∑ {c0LD00 (0, θ , 0) L L

+

∑ cKL±[D0LK (0, θ , φ) + D0L− K (0, θ , φ)]} K>0

387

(2a)


Article

The Journal of Physical Chemistry B

Figure 1. Experimental 15N R1 (a), R2 (b) 15N−{1H} NOE (c) and R2/R1 (d) data:21 11.7 T, black; 14.1 T, red; 16.4 T, green; and 18.8 T , blue.

for even L, and

u(θ , φ) ⎛ 3⎞ ⎛1⎞ 2 ≅ −c02⎜ ⎟[3(cos θ )2 − 1] − c 22⎜ ⎟(sin θ) cos 2φ ⎝2⎠ ⎝ 2⎠

∞

u(θ , φ) = −

∑ {c0LD00L (0, θ , 0) L

(4)

L

+

∑

cKL±[D0LK (0,

θ , φ) −

D0L− K (0,

The following order parameters are calculated:

θ , φ)]}

K>0

(2b)

L S0L = ⟨D00 (0, θ , 0)⟩,

for odd L. The coefficients c20,c22± = c22 ± c2−2 are themselves irreducible tensor components. The Cartesian tensor with components c2ij = c2ji is diagonal in the local ordering/local diffusion frame, M (i.e., Tr(c2ii) = 0),5 with complete specification given by c20 and c22+. The coefficient c20 multiplies the real function D200(0, θ, 0), which is proportional to the spherical harmonic Y20(θ). The coefficient c22+ multiplies the real function (D202(0, θ, φ) + (D20−2(0, θ, φ)), which is proportional to the real spherical harmonic function Y22(θ, φ) + Y2−2(θ, φ) For simplicity, the plus sign in the designation of c22+ is omitted. Likewise, the coefficients c10, c11+ = c11 + c1−1 and c11− = c11 − c1−1 are associated with the real functions D100(0, θ, 0) and (D101(0, θ, φ) − D10−1(0, θ, φ)), and the imaginary function (D101(0, θ, φ) + D10−1(0, θ, φ)), which are proportional to the real spherical harmonic functions Y10(θ) and Y1−1(θ, φ) − Y11(θ, φ) and the imaginary spherical harmonic function Y11(θ, φ) + Y1−1(θ, φ), respectively. Orientational ordering is properly represented by linear combinations of Y10(θ) and Y1−1(θ, φ) − Y11(θ, φ). For simplicity, the minus sign in the designation of c11+ is omitted. Thus, the expression for the L = 1 potential used in this study is given by u(θ , φ) ≅ − c01 cos θ − c11( 2 ) sin θ cos φ

L = 1, 2

(5a)

1 S11 = ⟨(D01 (0, θ , φ) − D01− 1(0, θ , φ))⟩

(5b)

2 S22 = ⟨(D02 (0, θ , φ) + D02− 2(0, θ , φ))⟩

(5c)

The ensemble averages are defined as

∫ D0LK (0, θ , φ)e−u(θ , φ) sin θ dθ dφ , ∫ e−u(θ , φ) sin θ dθ dφ

K = 0, 1, or 2 (6)

The Saupe order parameters, Sxx, Syy, and Szz, are related to the irreducible order parameters, S20 and S22, as5−7 Sxx = +( 3/2 S22 − S02)/2

(7a)

Syy = −( 3/2 S22 + S02)/2

(7b)

Szz = S02

(7c)

3D potential surfaces are shown in Cartesian coordinates, x = sin θ cos φ|u|, y = sin θ sin φ|u|, and z = cos θ|u|.27 In this representation, the value of the potential is given by the distance from the origin of the coordinate frame to any point on the 3D surface. Positive and negative values are colored red and blue, respectively. 3D surfaces of the potential are also shown in spherical coordinates, θ and φ. In this representation the value of the potential, u(θ, φ), is depicted on the z-axis, and color-coding is used to illustrate the variations. PDFs are shown in both Cartesian coordinates, x = sin θ cos φ|exp(−u)|, y = sin θ sin φ |exp(−u)|, and z = cos θ|exp(−u)|,

(3)

and the expression for L = 2 potential is given by 388


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Figure 2. Best-fit values of the potential coefficient, c10, of the axial term of the L = 1 potential (black), and the potential coefficient, c20, of the axial term of the L = 2 potential (red), as a function of residue number (a). Best-fit values of the potential coefficient, c11, of the rhombic term of the L = 1 potential (black), and the potential coefficient, c22, of the rhombic term of the L = 2 potential (red), as a function of residue number (b).

and spherical coordinates, θ and φ. Given that the PDF is an exponential function, large positive potential values virtually do not contribute to its visual display. Thus, due to complementary and supplementary features, the four representations generate a comprehensive picture of the local spatial restraints.

starting values of the potential coefficients have been taken from the 5−20 range. The starting value of D2 has been taken from the 1.0 × 109 to 5.0 × 1010 s−1 range. The fitting process converged consistently to the global minimum (except for landing in an easily identified local minimum). The reduced χ2, i.e., χ2/df (where df denotes the number of degrees of freedom) was typically satisfactory according to standard criteria. Errors are estimated based on variations in the best-fit parameters implied by excluding data acquired at given magnetic fields (out of five magnetic fields22). Considerations associated with the sensitivity of the analysis to parameter variations (see below) have also been invoked. The 15N CSA was taken −172 ppm, the backbone N−H bond length 1.02 Å, and the 15N−1H dipolar/15N CSA tensor tilt −17°.28 3.2. Global Diffusion. We use D1 = (5.01 ± 0.03) × 107 s−1 as global diffusion rate of GB3. This corresponds to correlation time for isotropic global motion given by τm = 3.33 ns.28 Evidence and considerations justifying the assumption of isotropic global diffusion appear in ref 28. 3.3. SRLS Analysis of the GB3 Data. Figure 2 shows the best-fit values of the potential coefficients c10 and c11 (eq 3) corresponding to the L = 1, K = 0, 1 potential (black), and the potential coefficients c20 and c22 (eq 4) corresponding to the L = 2, K = 0, 2 potential (red). The coefficients c10 and c20 estimate potential strength; the coefficients c11 and c22 estimate potential rhombicity. The axial potential coefficients are, on average, c10 ∼ 60 and c20 ∼ 20 (Figure 2a). These values represent strong potentials. Both c10 and c20 are relatively small in the first part of the α-helix and the α-helix/β3 loops. Interestingly, c10 is relatively large in the β1/β2-loop. The rhombic potential coefficients, c11 and c22 (Figure 2b), are virtually equal to zero for the α-helix (with a few exceptions which might be associated with experimental imperfections). For the chain segment preceding the α-helix

3. RESULTS AND DISCUSSION 3.1. General Considerations. In a previous study we analyzed with SRLS 15N T1, T2 and 15N−{1H} NOE from GB3, acquired in ref 22 at 9.4, 11.7, 14.1, 16.4, and 18.8 T.28 In this study we analyze the same data, shown (excluding the 9.4 T data) in Figure 1, using the rhombic local potential given by eq 3, or the rhombic local potential given by eq 4. That the asymmetry of the local spatial restraints has to be accounted for was shown by us12−15 and others.29−31 Hence, we allow the local potential to be rhombic. The ordering tensor frame (OF) is not known at the start of the data-fitting process. In ref 28, we set the starting orientation of ZOF parallel to the Cαi−1 − Cαi axis, in agreement with refs 29−31 (for example, in ref 31, where the Gaussian axial fluctuations (GAF) model is used to analyze NMR relaxation analysis in proteins, this geometric feature is encoded). Here we set ZOF parallel to the N−H bond. This orientation is encoded in simple (familiar) methods for NMR relaxation analysis, such as model-free (MF).32−34 As shown below, the two different starting values of ZOF mentioned above yield (as expected) practically the same best-fit local potential in the data-fitting process. The local structural restrictions are represented by c10 and c11 or c20 and c22. The local geometry is represented by the angle βOF−DF (for simplicity, the angle αOF−DF is set equal to zero). The local diffusion, taken isotropic, is represented by the rate, D2 [s−1]. Thus, 4 parameters, representing key features of local N−H bond dynamics, are allowed to vary in our calculations. As indicated above, the starting value of βOF−DF is zero. The 389


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Figure 3. Best-fit values of S10 (axial component of the L = 1 ordering tensor; black) and S20 (axial component of the L = 2 ordering tensor; red) as a function of residue number (a). Best-fit values of S11 (rhombic component of the L = 1 ordering tensor; black) and S22 (rhombic component of the L = 2 ordering tensor; red) as a function of residue number (b). Best-fit values of the angle βOF‑DF between the principal axes of the local ordering tensor frame, OF, and the dipolar tensor frame, DF, for L = 1 (black) and L = 2 (red), as a function of residue number (c).

one has, on average, c11 ∼ 6 and c22 ∼ 14. For the chain segment following the α-helix one has, on average, c11 ∼ 3 and c22 ∼ 7. For all of the potential coefficients we estimate the errors at 10% for the larger values and 5% for the smaller values. Many individual values are within the error margin of the average values. The black traces in parts a and b of Figure 3 show the principal values, S10 and S11, of the irreducible L = 1 ordering tensor (eqs 3, 5 and 6). The red traces in parts a and b of Figure 3 show the principal values, S20 and S22, of the irreducible L = 2 ordering tensor (eqs 4, 5, and 6). The orientations of these tensors are given by the angles βOF−DF (Figure 3c). In the well-structured regions of GB3 the order parameters S10 and S20 are, on average, equal to 0.93 (Figure 3a). In the β1/β2 loop (αhelix/b3 loop) S10 and S20 are relatively small whereas S11 and S22 are relatively large with substantial difference (virtually no difference) between the L = 1 and L = 2 cases (Figure 3, parts a and b). Interestingly, S10 and S11 are constant, whereas S20 and S22 vary substantially, in the second half of the β1-strand and in the β1/β2 loop. The angle βOF−DF ranges from +7° and −20° (Figure 3c). This represents moderate-to-considerable deviation of the main ordering axis, ZOF, from the N−H bond. Corresponding angles for the L = 1 and L = 2 scenarios do not differ much except for the second half of the β1-strand and the β1/β2 loop, where they differ substantially. |βOF−DF| = 20° for L = 1, occurring in the β1/β2 loop, is the largest values. The value of |βOF−DF| for L = 1 is larger than the value of |βOF−DF| for L = 2; this trend is exceptional. In Figure 4 we show the irreducible order parameters S10 and 2 S0 (Figure 4a) and S11 and S22 (Figure 4b) averaged over the various secondary structure elements and loops of GB3. S10 is somewhat larger, except for the β1/β2 loop where it is much

Figure 4. Average values of S10 (axial component of the L = 1 ordering tensor; black) and S20 (axial component of the L = 2 ordering tensor; red) for the various secondary structure elements and loops of GB3, as a function of residue number (a). Average values of S11 (rhombic component of the L = 1 ordering tensor; black) and S22 (rhombic component of the L = 2 ordering tensor; red) for the various secondary structure elements and loops of GB3 as a function of residue number (b).

larger, than S20 (Figure 4a). S11 is somewhat smaller, except for the β1 and β2 strands, and the β1/β2 loop, where it is much smaller, than S22 (Figure 4b). In general, corresponding average order parameters exhibit similar trends, except for the N-, and C-termini of the β1/β2 loop, where both the axial and rhombic order parameters exhibit opposite trends. 390


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Figure 5. Best-fit values of the Saupe ordering tensor components (eq 5): Sxx (black for L = 1 and red for L = 2) as a function of residue number (a), Syy (black for L = 1 and red for L = 2) as a function of residue number (b), and Szz (black for L = 1 and red for L = 2) as a function of residue number (c).

The effect of ordering polarity is reflected clearly by the magnitude and sign of the principal values Sxx and Syy of the Saupe ordering tensors for L = 1 (black) and L = 2 (red), shown in Figure 5, parts a and b. The L = 2 tensor components exhibit the familiar pattern of strong non-polar alignment and moderate-to-considerable rhombicity characteristic of secondrank tensors. Thus, in the well-structures regions of the GB3 structure Szz (same as S20) assumes, on average, the large value of 0.93 whereas in loops (β1/β2 and α-helix/β3) Szz assumes smaller values (Figure 5c, red). Sxx lies in the −0.45 to −0.35 range; its absolute value is smaller in the β1/β2 and α-helix/β3 loops (Figure 5a, red). Syy is nearly −0.44 throughout the GB3 backbone (Figures 5b, red). For L = 1 Szz is, on average, also 0.93 in the well-structures regions of the GB3 structure, and smaller in the β1/β2 and αhelix/β3 loops (Figure 5c, black). On the other hand, the average value of Sxx is somewhat below zero (Figure 5a, black), and the average value of Syy is on the order of 0.07 (Figure 5b, black). This pattern, which represents strong polar orientation and moderate ordering rhombicity within the scope of “L = 1 ordering”, is clearly different from the familiar “L = 2 ordering” pattern. Figure 6 shows the local diffusion rate, D2 as a function of residue number. The values of D2 lie in the 1 × 1010−5 × 1011 s−1 range for L = 1 (black) and in the 2 × 109 to 2 × 1011 s−1 range for L = 2 (red). They represent fast local fluctuations. The error in D2 is large (∼50%), mainly because rotator 2 is at least 100 times faster than rotator 1. The polypeptide chain segment comprising the β1 strand and the β1/β2 loop exhibits relatively slow local motion and relatively large differences between the L = 1 and L = 2 scenarios. The first part of the αhelix exhibits relatively fast local motion. 3.4. Biologically and Structurally Relevant Implications. The β2 strand has been implicated in the binding of GB3 to its cognate Fab fragment.35 We find that the L = 1 potential is stronger (i.e., c10 is consistently larger) in the β1/β2 loop as compared to the flanking chain segments (Figure 2a, black). S10

Figure 6. Best-fit values of the local diffusion rate, D2 [s−1] obtained using the L = 1, K = 0, 1 potential (black) and the L = 2, K = 0, 2 potential (red), as a function of residue number.

and S11 are virtually constant in the second half of the β1 strand and in the β1/β2 loop (Figure 3, parts a and b; black). In the β1/β2 loop |βOF−DF| is consistently larger for L = 1 as compared to L = 2; this trend is exceptional. In general, the differences between all of the corresponding “L = 1 parameters” and “L = 2 parameters” are particularly large in the second half of the β1 strand and the β1/β2 loop (Figures 3, 4, 5c, and 6). Even trends differ for L = 1 and L = 2  see the values of ⟨S20⟩ and ⟨S22⟩ at the N-, and C-termini of the β1/β2 loop (Figure 4). The emerging picture concurs with the chain segment comprising the β1 strand and the β1/β2 loop being involved in configuring the active site for binding the cognate Fab fragment of GB3 to the β2 strand. Effective binding requires polar/ directional ordering. This factor dominates the form of the local potential within this chain segment. The calculated L = 1 potential is a good approximation, whereas the calculated L = 2 potential is a significantly worse approximation, to this form. 391


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Table 1. Best-Fit Parameters Obtained with SRLS-Based Data-Fitting Using the L = 1 Potential of eq 3 for Residues A26, Y3, and L12a residue

c10

c11

c11/c10

S10

S11

Sxx

Syy

Szz

βOF−DF [deg]

D2 [s−1]

A26 Y3 L12

45.08 61.04 71.67

0.00 6.89 4.92

0.00 0.11 0.07

0.93 0.92 0.95

0.00 0.03 0.01

0.00 −0.021 −0.007

0.07 0.101 0.057

0.93 0.92 0.95

+4.3 −6.4 −21.4

5.6 × 1011 2.1 × 1010 3.3 × 1010

a 1 c0

and c11, potential coefficients; S10 and S11 (Sxx, Syy and Szz), principal values of the irreducible (Saupe) L = 1 ordering tensor defined in terms of c10 and c11; βOF−DF, angle between the principal axis of the local ordering tensor, ZOF, and the principal axis of the magnetic 15N−1H dipolar tensor, ZDF; D2, local diffusion rate. The ratio c11/c10 estimates potential rhombicity.

Table 2. Best-Fit Parameters Obtained with SRLS-Based Data-Fitting Using the L = 2 Potential of eq 4 for Residues A26, Y3, and L12a residue

c20

c22

c22/c20

S20

S22

Sxx

Syy

Szz

βOF−DF [deg]

D2 [s−1]

A26 Y3 L12

15.70 20.03 19.59

0.04 15.00 19.36

0.00 0.75 0.99

0.93 0.92 0.83

0.00 0.04 0.11

−0.465 −0.436 −0.348

−0.465 −0.484 −0.484

0.93 0.92 0.83

+4.3 −5.8 −7.8

2.2 × 1011 5.5 × 1010 3. two ×109

a 2 c0

and c22, potential coefficients; S20 and S22 (Sxx, Syy, and Szz), principal values of the irreducible (Saupe) L = 2 ordering tensor defined in terms of c20 and c22; βOF−DF, angle between ZOF and ZDF; D2, local diffusion rate. The ratio c22/c20 estimates potential rhombicity.

Figure 7. Potentials, u, and corresponding PDFs, exp(−u), for the N−H bond of residue A26 of the α-helix obtained for c10 = 45.08 and c11 = 0.00 (part A) and c20 = 15.70 and c22 = 0.04 (part B). The potentials are shown in Cartesian coordinates in panels a and in spherical coordinates in panels c. The PDFs are shown in Cartesian coordinates in panels b and in spherical coordinates in panels d.

Figure 8. Potentials, u, and corresponding PDFs, exp(−u), for the N−H bond of residue Y3 of the β1-strand obtained for c10 = 61.04 and c11 = 6.89 (part A) and c20 = 20.03 and c22 = 15.00 (part B). The potentials are shown in Cartesian coordinates in panels a and in spherical coordinates in panels c. The PDFs are shown in Cartesian coordinates in panels b and in spherical coordinates in panels d. 392


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Figure 9. Potentials, u, and corresponding PDFs, exp(−u), for the N−H bond of residue L12 of the β1/β2-loop obtained for c10 = 71.67 and c11 = 4.92 (part A) and c20 = 19.59 and c22 = 19.36 (part B). The potentials are shown in Cartesian coordinates in panels a and in spherical coordinates in panels c. The PDFs are shown in Cartesian coordinates in panels b and in spherical coordinates in panels d.

respectively (Table 1). Further investigation is required to determine whether these trends are general. Protocols for protein structure determination based on residual dipolar couplings35,36 are expected to profit from geometric information of the kind exemplified above. How to extend these protocols to do so is yet to be determined. The spherical-coordinate representation of the axial potential of A26 exhibits minimum at θ = 0 and maximum at θ = π (Figure 7Ac). The spherical-coordinate representations of the rhombic potentials of Y3 and L12 exhibit minima and maxima at different locations (Figures 8Ac and 9Ac). The rhombic potentials are deeper than the axial potentials. The forms of the spherical-coordinate representations of the PDFs (Figures 7Ad, 8Ad, and 9Ad) also differ. The dominance of values close to θ = 0 reflects potential strength; nonuniformity along the φcoordinate is associated with potential rhombicity (largest for Y3, shown in Figure 8Ad). These results indicate that polar/orientational ordering quite different in nature prevails at N−H sites of residues A26 (αhelix), Y3 (β1-strand) and L12 (β1/β2 loop) of GB3. Figures 7B, 8B, and 9B are based on the non-polar L = 2 potential of eq 4. All of the Cartesian potentials (panels a) have ZOF parallel to VF. For A26 (axial potential) the populated orientations (colored blue) are distributed symmetrically around z; the xy plane and its vicinity (colored red) are not preferred (Figure 7Ba). For Y3 (moderately rhombic potential) the populated orientations are distributed asymmetrically around z, with x less preferred than z. For L12 (highly rhombic potential) x is nearly as preferred as z. Note that while the rhombicity of the L = 2 potential increases along the series A26, Y3, and L12, according to c22/c20 equal to 0.00, 0.75 and 0.99, respectively, the rhombicity of the L = 1 potential increases along the series A26, L12, and Y3, according to c11/c10 equal to 0.00, 0.07, and 0.11, respectively. The Cartesian L = 2 PDFs exhibit lobes with their main symmetry axes along z. These lobes are symmetric for A26 (Figure 7Bb), somewhat asymmetric with x preferred over y for Y3 (Figure 8Bb), and substantially asymmetric with x preferred to a large extent over y for L12 (Figure 9Bb). The distinctive feature of the spherical-coordinate representations of the L = 2 potentials, shown in Figures 7Bc, 8Bc and

Hence, corresponding parameters emerging from the datafitting processes for the L = 1 and L = 2 scenarios differ substantially. In the remaining part of the polypeptide chain additional factors, common to the L = 1 and L = 2 scenarios, contribute, reducing these differences. An RDC-based investigation of an immunoglobulin-binding domain of protein G detected asymmetric motion perpendicular to the β-strand directions, interpreted to reflect a standing wave across the β-sheet.36 The rhombicity detected herein in the β-sheet, but not in the α-helix (Figures 2b and 3b), is consistent with this finding. 3.5. Potentials and PDF Forms of Typical Residues. The best-fit L = 1 and L = 2 potential coefficients obtained for residues A26 of the α-helix, Y3 of the β1 strand, and L12 of the β1/β2 loop, are shown in Tables 1 and 2, respectively. These parameters have been used to calculate the potentials, u, and the associated PDFs, exp(−u), shown in Figure 7−9. The Figures labeled “A” (“B”) show data associated with the L = 1 (L = 2) potential. Panels a and b (c and d) depict potentials and PDFs in Cartesian (spherical) coordinates. Let us focus on Figures 7A, 8A, and 9A, based on the polar L = 1 potential of eq 3. The Cartesian potential and PDF of A26 (Figure 7Aa,b) have axial symmetry (i.e., c11 = 0). The main symmetry axes of these functions, i.e., the preferred ZOF axis, is parallel to +z, the uniaxial local director frame, called VF (see Supporting Information). Large and positive c10 means that the +z orietation is populated predominantly, as shown clearly by the form of the PDF (Figure 7Ab). The form of the potential (Figure 7Aa) conveys similar information, recalling that “red” designates positive potential values, which are scarcely populated for large and positive c10. The Cartesian potentials and PDFs of Y3 (Figure 8Aa,b) and L12 (Figure 9Aa,b) have rhombic symmetry. The preferred ZOF axes are tilted from VF in the (+x)(+z) plane, to a larger extent for Y3, associated with rhombicity c11/c10 = 0.11 (Table 1), than for L12, associated with rhombicity c11/c10 = 0.07 (Table 1). Thus, ZOF is tilted from VF at angles increasing along the series A26, L12 and Y3. The forms of the potential and associated PDF are centered at the (+x)(+z) plane. On the other hand, ZOF is tilted from ZDF (i.e., the N−H bond) by |βOF−DF| = 4°, 6.4° and 21.4° for A26, Y3, and L12, 393


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The Journal of Physical Chemistry B 9Bc, is their φ-dependence. The axial potential exhibits no φdependence (A26, Figure 7Bc). The moderately rhombic potential exhibits two barriers separated by a valley (Y3, Figure 8Bc). The substantially rhombic potential exhibits higher barriers and a deeper valley (L12, Figure 9Bc). The sphericalcoordinate representations of the PDFs are shown in Figures 7Bd, 8Bd and 9Bd. The φ-dependencies reflect the transition from an axial PDF, through a moderately rhombic PDF, to a substantially rhombic PDF. Thus, the non-polar ordering description also varies among secondary-structure elements. However, the emerging picture differs qualitatively from the corresponding polar ordering description. The following comment is in order. The c10 values are considerably larger than the corresponding c20 values. In general, the strength of a potential is in inverse proportion to its curvature. For axial symmetry (A26, Figures 7Ac and 7Bc) the curvature of the L = 1 potential is larger than the curvature of the L = 2 potential, in agreement with c10 = 45.08 and c20 = 15.70. To examine the case of rhombic symmetry we focus on Y3 (Figure 8, parts Ac and Bc). For φ = 0 and φ = π, the curvature of L = 1 potential is larger than that of the L = 2 potential, in agreement with c10 = 61.04 and c20 = 20.03. Interestingly, for φ = π/2 the curvatures of the L = 1 and L = 2 potentials are the same. Thus, one can determine c10 that “corresponds” to a given value of c20 (20.03 in this case) as follows: one sets φ = π/2 in eq 3 and increases c10 until the curvature of this L = 1 potential is the same as the curvature of the L = 2 potential (eq 4) with c20 = 20.03, where φ was set equal to π/2. 3.6. Comparison with Our Previous SRLS-Based Analysis of GB3. In our previous SRLS analysis of 15N relaxation from GB3 the starting orientation of the main ordering axis, ZOF, was set parallel to Cαi−1 − Cαi ,28 in agreement with the geometry-related perspectives of the MD studies of refs 29 and 30, and the 3D GAF model.31 In the present study the starting orientation of ZOF is set parallel to the N−H bond (ZDF). In simple analyses of spin relaxation, e.g., the model-free (MF) method,32−34 the orientation ZOF∥ZDF is encoded. Hence, with ZOF∥ZDF as starting orientation, the angle βOF−DF represents straightforwardly the departure from the simple MF geometry. Setting at the start the main ordering axis parallel to N−H or Cαi−1 − Cαi , which within a reasonably good approximation are perpendicular, lead to practically the same best-fit potential (cf. refs 5 and 27). The ordering tensor frames determined by these calculations are (nearly) interconvertible by permuting the axes labels, and the pairs of potential coefficients determined transform as irreducible tensor components.27 We carried out these transformations in ref 28; the transformed coefficients are shown in Figure 7 of that article. It can be seen that the red traces in parts a and b of Figure 2 are similar to their counterparts in Figure 7 of ref 28. In both cases the β1, β2, β3, and β4 strands and intervening loops differ from the α-helix of GB3 primarily in the rhombicity of the local potential.

spherical harmonics Y20 and Y22(θ, φ) + Y2−2(θ, φ), or the newly devised polar potential given by the spherical harmonics Y10 and Y1−1(θ, φ) − Y11(θ, φ). Strong local potentials, with coefficients ⟨c10⟩ ∼ 60 for L = 1, and ⟨c20⟩ ∼ 20 for L = 2, characterize the GB3 backbone. In the α-helix both potentials are virtually axially symmetric (i.e., c11 ∼ 0 and c22 ∼ 0). In the chain segment preceding (following) the α-helix both potentials are rhombic with ⟨c11⟩ ∼ 6 for L = 1 and ⟨c22⟩ ∼ 14 for L = 2 (⟨c11⟩ ∼ 3 for L = 1 and ⟨c22⟩ ∼ 7 for L = 2). The SRLS analysis based on the L = 1 potential revealed unique parameter values for the β1 strand and the β1/β2 loop, and in some cases the β2 strand. In addition, the difference between corresponding parameters associated with the L = 1 and L = 2 scenarios is particularly large in the β1 strand and the β1/β2 loop. We interpret these findings to indicate that in the β1 strand and the β1/β2 loop the local ordering is collectively strong and directional, to facilitate the binding of the cognate Fab fragment of GB3 to the β2 strand. The calculated L = 1 potential is a good approximation, whereas the calculated L = 2 scenario is a substantially worse approximation, to the actual scenario; hence these potentials differ. Further research aimed at detecting “functional” ordering is of interest. Fast N−H fluctuations are reflected by the local diffusion rate, D2, which lies in the 2 × 109−5 × 1011 s−1 range. These fluctuations are on average faster for the L = 1 scenario. A recently devised comprehensive scheme for characterizing the local ordering in terms of potential and PDF shapes turned out to be very useful in singling out effects due to ordering polarity, and secondary structure context. An important future prospect is extending the present study to additional proteins. Analyses based on L = 1 and L = 3 potentials, as well as L = 1 and L = 2 potentials, are of interest. We contemplate devising linear combinations of spherical harmonics that approximate well MD-derived potentials of mean force. The development of SRLS fitting schemes based on such potentials (used fixed) is another interesting future prospect.

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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/acs.jpcb.5b10244. Summary of the slowly relaxation local structure approach for 15N relaxation in proteins, for L = 2, K = 0, 2 potentials (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(E.M.) E-mail: [email protected]. Telephone: 972-3531-8049. Notes

4. CONCLUSIONS Mobile protein moieties, e.g., N−H bonds, are attached physically to the protein. Hence their local ordering is polar/ directional. This property is adequately described by a potential given by the real spherical harmonics Y10 and Y1−1(θ, φ) − Y11(θ, φ). 15N relaxation parameters from GB3 acquired previously at five magnetic fields have been analyzed with SRLS using either the traditional non-polar potential given by the

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Israel−U.S.A. Binational Science Foundation (Grant No. 2010185 to E.M. and Jack H. Freed), and the Israel Science Foundation (Grant No. 437/11 to E.M.). 394


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Polar Versus Non-polar Local Ordering at Mobile Sites in Proteins

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