Multispin Cross-Correlated Transverse Dipolar NMR Relaxation in Solution

In this paper, we want to consider what would be involved in calculating the R 2 relaxation of amide protons in a protein caused by dipolar interactions with nearby protons, for which there are many. NMR textbooks give analytical equations and sometimes derivations for solution NMR relaxation due to dipolar interactions between two spins. There are also closed equations for dipolar interactions between three spins, which include relaxation interference, also known as cross-correlated cross-relaxation. We here derive an expression for interference between four spins. For larger systems, such as amide protons in a protein, we develop a local-ﬁeld methodology, from which solution relaxation interference can be computed for a basically limitless number of interacting spins.


Introduction
Several researchers have been interested in investigating whether proton-proton dipolar interaction in proteins can be utilized to obtain structural [1,2] or dynamical information [3]. Here, we want to consider what would be involved in calculating the R 2 relaxation of amide protons in a protein caused by dipolar interactions with other protons and, ultimately, compare those with experiment [4]. NMR textbooks give closed equations and sometimes derivations for solution NMR relaxation due to dipoledipole relaxation between two spins and for chemical shift anisotropy (CSA) relaxation.
ere are also closed equations for relaxation due to interactions between three dipoles [5] and two dipoles and the CSA [6]. In these cases, relaxation interference, also known as cross-correlated cross-relaxation, has been taken into account. Dipolar-dipolar relaxation interference has been described very early in the history of NMR [7,8] and has been exploited to achieve line narrowing [9] and structural information [1,2]. CSA-dipolar cross-correlated relaxation is at the core of the line narrowing in the TROSY experiment [10].
Here, we derive an analytical expression for dipolar interference between four spins. For larger systems, such as amide protons in a protein, we develop a local-field methodology, from which solution relaxation interference can be computed for a basically limitless number of interacting spins.

Theory
Let us start by considering the R 2 relaxation of spin I in a rigid molecule with three protons, S-I-Q, subject to fluctuating dipolar interactions IS and IQ. We assume here that I, S, and Q have different chemical shifts and are thus similar, but "unlike." e first step is to add the relaxation rates due to the IS and IQ dipolar interactions, as given by equation (A.24) of the Appendix. (2) However, in a rigid molecule, the stochastic fluctuations of the IS and IQ dipolar interactions are 100% correlated and relaxation interference occurs. Relaxation interference is also called cross-correlated cross-relaxation or cross-correlated relaxation.
Interference between two dipolar mechanisms can be qualitatively understood from Figure 1. Here, we consider a linear three-spin system with spin I in the center and spin S and Q equidistant from I.
In this case, there are two relaxation rates for spin I, one for molecule A, where the dipolar fields of S and Q at the location I cancel independent of orientation with respect to the magnetic field, and another for molecule B, where the dipoles reinforce each other and fluctuate strongly depending on orientation. is is expressed as Of course, the dipoles flip by R 1 processes. For macromolecules, R 1 rates are much slower than R 2 rates. us, for R 2 cross-correlations, one may assume that orientations of the dipoles remain set during the typical R 2 relaxation time.
If the different dipolar permutations also give rise to different resonances, by either scalar or residual static dipolar coupling, one observes four NMR lines for I. e two outer lines have equal linewidths, and the two inner lines also have equal, but different, linewidths. Which set will be the broader one depends on the molecular geometry and the sign of the (scalar) coupling. If no such coupling exists, one will observe a superposition of a broader and narrower line for I.
In order to obtain quantitative values for the dipolardipolar cross-correlation, one starts with expanding the double commutator master equation for R 2 relaxation, as shown in equation (A.16) of the Appendix. We mostly follow a formalism as introduced by [5,11].
For two R 2 relaxation mechanisms IS and IQ, modulated by the same motions, the relaxation master equation is where T mIS are the tensor operators of the perturbing Hamiltonian, which for dipolar interaction between spins I and S are given by with similar equations for T mIQ . j m ω m are the spectral densities of the molecular motion at the frequencies of the bilinear spin operators including magnitude terms.
In the expansion, the "diagonal" terms such as m�+2 m�− 2 and their complex conjugates will give rise to the autocorrelation rates R IS 2 and R IQ 2 , as shown in equation (A.22) of the Appendix.
In the cross terms, and their complex conjugates, only spin operators with exactly the same precession frequency will be relevant, i.e., only the pair (I z S z , I z Q z ) and the four terms I + S z , I − S z , I + Q z , and I − Q z . We obtain the following equation from equations (6a) and (6b): − d〈I + 〉 dt � 〈4 I + , I z S z , I z Q z 〉j cc 0 (0) where the ½ term is dropped because there are two equivalent cross terms, as given in equations (7a) and (7b). e cross-correlation spectral densities are with, for isotropic motion, where the P 2 cos(θ IS− IQ ) � 1/2(3 cos 2 (θ IS− IQ ) − 1) term originates from transforming the IQ dipolar vector into the IS frame or vice versa and using the addition theorem of spherical harmonics. Here, θ IS− IQ is the angle between vectors IS and IQ and τ c is the (rotational) correlation time.
In contrast to the R 2 autocorrelation, the double commutators drive cross-relaxation One ends up with a set of coupled differential equations: Here, R IP 2I and R AP 2I are the in-phase (IP) R 2 rate for the <I + > coherence and antiphase (AP) R 2 rate for the 4 <I + S z I z > coherence.
e antiphase terms relax faster because of the extra S z and Q z R 1 relaxation. For macromolecules, the difference can be neglected because R 1 is small as compared to R 2 , and one can easily diagonalize the rate matrix and obtain relaxation "eigenvectors" d dt with e total R 2 relaxation for proton i, as stated before in equation (3), is then given by One notices that 1 H IS-autorelaxation contains a 5J(0) term (equation (2)), while the dipolar cross-correlation has a 4J(0) term (equation (14)). e extra J(0) in autorelaxation is due to the fact that J(0) and J(ω I − ω S ) are for two protons similar enough for the spectral densities to add (see Appendix), but (ω I − ω S ) and (ω I − ω Q ) are different enough for the spin operators to diphase and not crosscorrelate. us, there will not be a complete cancelation of relaxation, as suggested in the situation of Figure 1. It can still happen, if the differences in distances IS and IQ would make up for the 4/5 term.
If one cannot make the approximation that R IP 2I and R AP 2I are equal, the rates become Let us now extend this formalism to a 4-spin system, with a central spin I and 3 other spins S, Q, and P in its vicinity.
From the master equation, we arrive at three cross terms:

Concepts in Magnetic Resonance Part A, Bridging Education and Research
leading to terms such as 4I + S z Q z , 4I + S z P z , and 4I + P z Q z , driven by the rates is appears to lead to eight different relaxation rates ). However, that cannot be right; adding one more dipole P to the S-I-Q situation should just give rise to two more rates, four in total. So, one should not stop here and construct the full relaxation matrix.
Also, one must take cross-relaxation between the different four-spin terms into account (here only showing some and three reverse processes. e differential equations in matrix form will now become d dt Here, we are neglecting the differences between R 2 inphase and antiphase relaxation. is matrix has four eigenvalues, which correspond to what must be physically correct. While the exact eigenvalues of this matrix could potentially be analytically obtained, it becomes harder when differences between R 2 in-phase and antiphase relaxation have to be considered as well. However, because matters will quickly become more complicated when even more spins interact, we will not attempt to derive an analytical solution for the four-spin case here. For instance, in a protein, there are typically at least 20 protons around every amide proton in a 6Å sphere which would all have to be taken into account. In such a case, one would need to diagonalize a 262144 * 262144 matrix (2 20 − 2 ).
We must take another approach. To arrive at an estimation for the effects in a multiproton spin system, we will start from a "solid-state NMR" point of view. We calculate B Ω loc(i) , the net local magnetic field at center proton i due to the M surrounding protons j [12] for a certain orientation of the molecule in the external magnetic field: Here, θ ij is the angle between the internuclear vector ij and the magnetic field direction Ω in the molecular frame. D represents a particular distribution of the signs of the z-components of the dipoles j. If one varies the magnetic field direction according to a sphere distribution and adds the results, one obtains the powder pattern for that particular distribution D. Subsequently, one coadds all powder patterns for different values of D and normalizes to arrive at the "cross-correlated" dipolar powder pattern for the 1

HN under consideration.
It is the time dependence of B loc , as caused by molecular motion, that drives the solution NMR dipolar relaxation. e R 2 relaxation is then obtained as the second moment of the (cross-correlated) powder pattern [12]: where the brackets indicate average overall orientations. In practice, we permute only the signs of the eight closest protons (256 different distributions D) and treat the spins further out with a single random distribution where spins up and down are on average equal. So, each distribution then gives rise to an individual resonance for i with its own R 2 rate. e sum of all of those creates the inhomogeneous sumline for resonance i.

Verification
We verified the algorithm with a three-atom arrangement (see Table 1) with results in Figure 2. In this figure, the green points were calculated from equations (14) and (15) for dipolar-dipolar cross-correlation, while the drawn black line was calculated using the "solid-state" approach. e results are identical. In red is a relaxation curve calculated from the straight addition of R 2 (IS) and R 2 (IQ) (see equation (1)). It is clear that the cross-correlation cannot be neglected, except when one considers the first part of the curve. Fitting a single exponential against a complete cross-correlated relaxation curve (dashed line) yields an erroneous rate.
In Figure 3(a), we show calculated R 2 relaxation rates for the HN of Asp45 of the protein GB1. is amide has 21 proton neighbors within a sphere of 6Å. Here, the difference between relaxation curves with and without cross-correlations is smaller than the example case in Figure 2. From this, one would tend to conclude that many protons cancel each other's cross-correlations. However, that is not necessarily true; in Figure 3(b), we show calculated R 2 relaxation rates for the HN of Asn35 of the protein GB1. is amide has more (36) proton neighbors within a sphere of 6Å, yet the difference between relaxation curve with and without crosscorrelations is larger than for Asp45. is happens because the R 2 relaxation of Asn35 is dominated by two close-by protons.
is manuscript was written in anticipation of the interpretation of experimental T 1rho data for protein amide protons, which we hope may contribute to more complete understanding of protein molecular dynamics (E.R.P. Zuiderweg and D.A. Case, in preparation). We conclude from the current analysis that one can still fit a single exponential to the beginning of the T 1rho relaxation curves, even when many protons interact. is result is the same as for a 3-spin network, but that was not previously demonstrated.
However, how far down a relaxation curve can go varies from proton to proton environment. Judging from Figure 3(b), one should limit the recording of T 1rho relaxation curves to values larger than 0.8xI 0 , where I 0 is the initial value of the decay curve.

Description of the Codes
e computer program requires as input a "protonated" PDB file, the radius of the sphere of protons around the amide protons one is interested in, the rotational correlation time, and the spectrometer frequency. Basically, the program consists of four nested loops: amides, protons around amides, permutation of dipole signs of these surrounding protons, and rotation of the magnetic field vector in the molecular frame.
A set of 10 nested loops permutes the dipolar signs of the closest 10 hydrogens (1024 distributions). e more remote hydrogens in the sphere (if any) have their dipolar signs assigned according to a 50% random chance. e local dipolar field at a certain 1 HN due to the surrounding protons in a certain permutation D of surrounding dipoles is calculated according to equation (20). Here, the program takes the differences between "like" and "unlike" spins (see Appendix) into account. en, the program calculates, according to equation (22), the R 2 relaxation rate due to that permutation, by rotating the external field Table 1: Coordinates of an arbitrary three-spin arrangement (Å). d IS � 2Å, d IQ � 2.236Å, and P 2 cos(θ IS− IQ ) � 0.70.  (14) and (15) for dipolar-dipolar cross-correlation, while the solid black line was calculated using the "solid-state" approach (equations (21) and (22)). Fitting a single exponential against a complete cross-correlated relaxation curve (dashed line) yields an erroneous rate. In red is a relaxation curve calculated from the straight addition of R 2 (IS) and R 2 (IQ) (equation (1)). A (isotropic) rotational correlation time τ c of 10 ns was used in all calculations. direction (the z-axis) through the molecular frame using an isotropic spherical distribution (5000 orientations) (http://corysimon.github.io/articles/uniformdistn-onsphere/). e relaxation rate for that permutation D is then used to compute a R 2 relaxation curve. e computed R 2 relaxation curves for all different permutations D are then added to obtain the complete R 2 relaxation curve, as shown in Figures 2 and 3. e program is written in Fortran 90 and contains no references to outside libraries. e source code is available from the author.

Appendix
is appendix presents a refresher on dipolar R 2 relaxation, mostly following the formalism, as developed by Goldman [11] and further extended by [13], to help follow the algebra in the main body of the paper.
When the density operator σ is subject to both a timeindependent (eigen) Hamiltonian H 0 and a time-dependent perturbing dipolar Hamiltonian H 1 (t), its evolution, in units of Z, is given by Transforming to the rotating frame of the time-independent Hamiltonian, one obtains with the time-dependent Hamiltonian also transformed to the rotating frame. One keeps in mind that this equation describes an ensemble average. First, one imposes on equation (A.2) that the density operator cannot evolve ("relax") when it was not perturbed in the first place (e.g., by a r.f. pulse). us, one makes the substitution where σ eq is the density operator at equilibrium, which is not evolving under H 0 (i.e., the usual "high temperature approximation with σ eq ∼ I z ). One uses the Hausdorff expansion to integrate equation (A.3):

(A.4)
Since the dipolar Hamiltonian has zero average in solution, the second term in equation (A.4) vanishes. e fourth and higher-order terms also vanish because the perturbing Hamiltonian causes only small changes of the density operator, allowing the expansion to converge rapidly. One thus obtains the formal relaxation equation: e perturbing Hamiltonian fluctuates not only by the fluctuating dipole-dipole interactions between spins I caused by random molecular motion captured in the terms F DD m (t) but also by frequencies of the spin operators I, S, and I + S − (ωI, ωS, ωI− ωS, etc.).
It can be abbreviated as T mIS are the tensor operators of the perturbing Hamiltonian, which for dipolar interaction between spins I and S are given by   (21) and (22)), while the red curves are computed as the sum of the R 2 rates (equation (1)). A (isotropic) rotational correlation time τ c of 10 ns was used in all calculations.
On expanding, this becomes Here, μ 0 is the permittivity of vacuum, Z is the Planck's constant divided by 2π, c′s are the gyromagnetic ratios, and r IS is the distance between the dipoles. e angles θ(t) and − ω m t″)t contribute. is is called the Redfield kite, named after the author of one of the earliest papers on NMR relaxation [14]. It suffices to just take the five diagonal terms into account and drop the second summation. One arrives at the "master equation" of relaxation [11] Concepts in Magnetic Resonance Part A, Bridging Education and Research