J. Am. Chem. Soc. 1992,114, 6195-6199
6195
The “CUPID” Method for Calculating the Continuous Probability Distribution of Rotamers from NMR Data hljko DZakula,t William M. Westler, Arthur S. Edison,’ and John L. Markley*J Contribution from the Biochemistry Department, College of Agricultural and Life Sciences, University of Wisconsin-Madison, 420 Henry Mall, Madison. Wisconsin 53706. Received September 30, I991
Abstract We present a new method, Continuous ProbabIlity Distribution of rotamers (CUPID), for determining the distribution of rotamer probability p(x) about a dihedral angle x. This method utilizes measured vicinal homonuclear and heteronuclear spinspin coupling constants (J)and nuclear Overhauser enhancements (NOES) from NMR spectra and demands no prior assumption about the conformations or the degree of flexibility across the bond. To analyze the NMR measurements, it is necessary to know J ( x ) , the angular dependence of each coupling constant, and r ( x ) , the angular dependence of each interatomic distance giving rise to an NOE. With 2N or more measured coupling constants and/or NOES and with known functions J ( x ) and r(X),one determines the probability distribution by solving for the coefficients of the Fourier expansion of p(x) to order N . As a further refinement to p ( x ) , we make the assumption that the distribution functions are sums of Gaussians. The theory underlying CUPID is presented here; the accompanying article (DZakula, Z.; Edison, A. S.;Westler, W. M.; Markley, J. L. J. Am. Chem. Soc., following paper in this issue) demonstrates an application of CUPID to the analysis of simulated data for an a-helix in a protein. 1. Introduction
The dihedral angular dependence of vicinal nuclear spinspin coupling constants, J(O), is well established.I4 This dependence was first described by Karplus’ as J(0) = a cosz (e) b cos (e) c, where a, b, and c are empirical parameters, and t9 is the dihedral angle between two coupled nuclei. If the angle is rigid, 0 can be determined from measurements of two different couplings, with some exception^.^ However, conformationally flexible molecules give rise to average coupling constants weighted by their probability distribution: It is generally impossible to know, a priori, the conformational state of a molecule. To account for conformational averaging, it is usually assumed7-I3 that a set of three discrete values of 0 exhausts all conformational possibilities. The angles are estimated by a “best gum”, “chemical intuition” (leading most often to staggered conformers), or a calculation of rotational potentials. The problem is then expressed as
+
+
where p , (i = 1,2,3) are the unknown populations of the discrete conformers, and (J)is the measured averaged coupling constant. We refer to this as the udiscrete model”. Many difficulties are associated with this model. First, deviations from the commonly used staggered conformers are often found.Iel6 Second, even when nonstaggered angles are recognized, it is still necessary to estimate the actual values of 0 before the problem can be solved. Third, the three conformers are assumed to cover the entire conformational space. Fourth, the discrete probability distribution, defined by delta (6) functions, neglects contributions to ( J ) from dihedral angles with low probabilities but large couplings. Finally, due to the normalization, solutions for piare possible regardless of the assumed angles; thus the approach provides no check on their appropriateness. Clearly, this model is inadequate to describe many molecules (especially biological macromolecules) where torsional motion will broaden the assumed 6 functions and where forces resulting from tertiary structural interactions, such as hydrogen bonds, can cause significant deviations from the assumed discrete distribution.” Vicinal spinspin couplings between a and fl protons in amino acids have been used in conjunction with intra-residue NOES to Permanent address: Institute of Physiology and Biochemistry, Faculty of Biology, School of Sciences, and Biophysical Laboratory, Center for Multidisciplinary Studies, University of Belgrade, Studentski trg 3, 11000 Belgrade, Serbia, Yugoslavia. To whom correspondence should be addressed. 2 Graduate Biophysics Program, University of Wisconsin-Madison.
determine x1rotamer populations in the side chains of peptides and proteins according to a set of rules formulated by Wagner et al. (Figure 4 of ref 18). However, the approach is valid only when the conformation is fixed, Le., in the absence of rotational averaging. In addition, the approach assumes that the dihedral angle x1takes only staggered values. When the NMR data are inconsistent with such predictions, as sometimes happens,I* the model does not distinguish whether the discrepancy is due to several interconverting rotamers (staggered and/or nonstaggered) or one nonstaggered conformer. Wyssbrod5proposed two methods for analyzing peptide coupling data: a intersection-of-sets method (based on direct geometrical analysis of Karplus curves) and conformation-locus diagrams for determining restricted xI conformations of the side chains of amino acids with a /3-methylene group. The conformation-locus method (1) Karplus, M. J. Chem. Phys. 1959, 30, 11-18. (2) Karplus, M. J . Am. Chem. Soc. 1963,85, 2870-2871. (3) Barfield, M.; Grant, D. M. In Advances in Magnetic Resonance; Waugh, J. S., Ed.; Academic Press: New York, 1965; Vol. 1, pp 149-193. (4) Kowalewski, J. In Annual Report in NMR Spectroscopy; Webb, G. A., Ed.; Academic Press: New York, 1982; Vol. 12, pp 81-176. (5) Wyssbrcd, H. R. In Proceedings of the Seventh American Peptide Symposium; Rich, D. H., Gross,E., Eds.; Pierce Chemical Co.: Rockford, IL, 1981; pp 261-265. (6) Three distinct cases of averaging need to be considered:’ ( a ) motion rapid compared with the reciprocal of the coupling constant which yields couplings that are rotationally averaged; (6) motion slower than the reciprocal of the coupling constant with only one conformer detected by NMR; ( c ) motion much slower than the reciprocal of the coupling constant such that each conformer exists long enough to be detected by NMR. In this article, we consider only case ( a ) which most often is the situation in liquids since vicinal coupling constants are generally quite small. (7) Pople, J. A. Mol. Phys. 1958, 1, 3-8. (8) Jardetzky, 0. Biochim. Biophys. Acta 1980, 621, 227-232. (9) Bystrov, V. L.; Portnova, S.L.; Balashova, T. A.; Tsetlin, V. I.; Ivanov, V. T.; Kostetzky, P. V.; Ovchinnikov, Yu. A. Tetrahedron Lett. 1969, 59, 5225-5228. (10) Colucci, W. J.; Gandour, R. D.; Mooberry, E. A. J . Am. Chem. Soc. 1986, 108, 7141-7147. (1 1) Cowburn, D.; Live, D. H.; Fischman, A. J.; Agosta, W. C. In Intramolecular Dynamics; Jortner, J., Pullman, B., Eds.; D. Reidel: New York, 1982; pp 473-480. (12) Cowburn, D.; Live, D. H.; Fischman, A. J.; Agosta, W. C. J . Am. Chem. SOC.1983, 105, 7435-7442. (13) Fischman, A. J.; Wyssbrod, H. R.; Agosta, W. C.; Cowburn, D. J . Am. Chem. Soc. 1978, 100, 54-58. (14) Nagayama, K.; Wiithrich, K. Eur. J. Biochem. 1981, 115,653-657. (15) Fujinaga, M.; Sielecki, A. R.; Read, R. J.; Ardelt, W.; Laskowski, M., Jr.; James, M. N. G.J . Mol. Biol. 1987, 195, 397-418. (16) Langs, D. A.; Smith, G. D.; Stezowski, J. J.; Hughes, R. E. Science 1986, 124C-1242. (17) Fischman, A. J.; Live, D. H.; Wyssbrod, H. R.; Agosta, W. C.; Cowburn, D. J . Am. Chem. SOC.1980, 102, 2533-2539. (18) Wagner, G.; Braun, W.; Havel, T. F.; Schaumann, T.; G6, N.; Wiithrich, K. J . Mol. Biol. 1987, 196, 611-639.
0002-786319211514-6195%03.00/0 0 1992 American Chemical Society
Diakula et al.
6196 J . Am. Chem. SOC.,Vol. 114, No. 15, 1992
uses a correlation diagram of 3Jua versus 3Jua vicinal protonproton couplings (see also ref 19). however, it does not take into account rotational averaging between distinct conformers. At about the same time, Nagayama and Wiithrich14used the same approach to study xI conformers of BPTI. They accounted for rapid small-amplitudevibrations about a single rotamer by assuming simple uniform averaging within a limited interval of angles. Hyberts et al.zosuggested an extension of that method, based on use of H"-H@COSY cross-peak multiplet structure in conjunction with HN-H@NOEs in order to assign stereospecifically the @-resonancesand to determine concomitantly values of x1and 4 torsion angles. The weahess of that approach is that it attempts to determine simultaneously two torsion angles (xl and 4) on the basis of a data set limited to only four measured numbers: two scalar couplings and two NOEs. All these methods are suitable only when there is a single conformer and when no largeamplitude rotations take place. The literature contains references to at least two continuous models used in computing average values of coupling constants. The first, proposed by Bystrov et ala?'assumes no potential bamer and simply averages J over a complete rotation. The second assumes knowledge of the rotational potential energy and uses a Boltzmann distribution to describe the averaged coupling constant (see ref 22-24 and references therein). This approach is useful in principle, but suffers when knowledge of the rotational potential function is lacking (as is the case for most molecules, especially biological macromolecules). As far as rotational averaging of NO& is concerned, an attempt was made by Schirmer et al.25 to reconstruct the continuous probability distribution of glycosyl rotamers in nucleosides from NOEs. They fitted the experimental data to a probability distribution made up of a sum of Gaussian curves. This approach has been difficult to implement and has not gained much popularity. A uniform averaging model was used by Braun et alez6to estimate the upper limits of interproton distances from rotationally averaged NOE data. In that model the distance between the hydrogen atoms was allowed to vary uniformly between a minimum (equal to the van der Waals contact distance of 2.0 A) and an unknown maximum value, which has to be determined from the experiment. Their simple approach formed a conceptual basis for the estimation of interproton distance constraints from NOE data, which then serve as input to distance-geometry analysis. Recently, Torda et alaz7recognized the importance of rotational averaging of NO& and proposed the use of molecular dynamics in order to compute the average proton-proton distances and NOE intensities. This approach, however, is both cumbersome and computationally expensive. As pointed out above, all these approaches have certain weaknesses and fail to identify multiple rotational isomers (staggered or nonstaggered), fail to quantify their probabilities, or do not estimate their mobilities. We present here a method for the analysis of NMR coupling constants and nuclear Overhauser enhancements called Continuous ProbabIlity Distribution of rotamers (CUPID) that yields a continuous angular distribution of probability, p(x). Our approach simultaneously solves the problems of determining the flexibility and the most probable angles. (19) Wagner, G.; Wiithrich, K. Methods Enzymol. 1986,131,307-326. (20) Hyberts. S.G.; MBrki, W.; Wagner, G. Eur. J . Biochem. 1987,164, 625-635. (21) Bystrov, V. F.; Portnova, S.L.; Tsetlin, V. I.; Ivanov, V. T.; Ovchinnikov, Yu.A. Tetrahedron 1969, 25, 493-515. (22) Bystrov, V. F. Prog. Nucl. Magn. Reson. Spectrosc. 1976,10,41-81. (23) Barfield, M.; Hruby, V. J.; Meraldi, J.-P. J . Am. Chem. SOC.1976, 98, 1308-1314. (24) de Leeuw,F. A. A. M.; Altona, C.; Kessler, H.; Bermel, W.; Friedrich, A.; Krack, G.; Hull, W. E. J . Am. Chem. SOC.1983, 105, 2237-2246. (25) Schirmer, R. E.; Davis, J. P.;Noggle, J. H.; Hart, P.A. J . Am. Chem. SOC.1972. 94. 2561-2572. (26) Braun, W.;B k h , C.; Brown, L. R.; Ga, N.; Wiithrich, K. Biochim. Biophys. Acta 1981, 667, 377-396. (27) Torda, A. E.; Scheek, R. M.; van Gunsteren, W. F. J . Mol. Biol. 1990, 21, 223-235.
2. Theory 2.1. Foundatiorm 2.1.1. Angular Probability Distribution. The probability distribution of rotamers, p(x), can be expanded as a Fourier series by making use of the fact that it must be a periodic function of the dihedral angle x m
AX) = PO
m
+ n=E p1 n cos (nx) + n=l E u n sin (nx)
(1)
where po, pm and unare coefficientsof the expansion. After taking into account the normalization condition that the distribution function must obey
and the fact that integrals of functions cos (nx) and sin (nx) over the interval (-T, 7 ) vanish, it is easily seen that
= 1/(2*) (3) In practice, p(x) can be approximated by evaluating eq 1 as a limited sum: Po
N
N
The main goal of this work is to reconstruct p(x) by calculating the 2N coefficients pn and un (n = 1, ..., N) from a set of experimentally measured spinspin coupling constants and NOES. 2.1.2. Rotatio~llyAveraged Spin-Spin Coupling Constants. The general relationship between vicinal nuclear spinspin coupling constants and dihedral angles (Karplus equation) can be expressed as
where J is the coupling constant, 6 is the dihedral angle between the coupled spins, and a, b, and c are empirically derived parameters. Note that the angle 0 may differ in phase from the defined dihedral angle x. For example, as shown in Figure 1 of the accompanying paper?* the standard definition of the dihedral angle about the C"-C@bond in amino acids, x l , is such that x1 =' 0 when the atoms, N, C", C@,and X ' (the heteroatom in y position) are coplanar.z9 However, when x, = Oo, the dihedral angle between the atoms H" and Hfl (about the same c"Lc@bond) is 0 = -120O. When the defined dihedral angle x about a given bond differs from 8, it is convenient to express 8 in terms of x, Le., as x + Q, where Q is a constant that defines the stereochemistry for a particular pair of coupled atoms. Equation 5 then becomes
-a2 COS [2(x + Q)] +b COS (x + n) + c + -a2 = 2-a cos (2Q) cos (2x) - -2a sin ( 2 3 ) sin (2x) +
J(e) = J ( +~Q) =
b cos (Q)cos (x) - b sin (0)sin (x)
+c+2
(6)
Different pairs of atoms coupled across the same bond have different dihedral angles 0. However, the notation x + Q allows one to express all of the couplings a c r m that bond, as well as the probability distribution p, in terms of the same variable, x. If rotation occurs about x at a rate much greater than the coupling constant, an NMR measurement yields an average value for the scalar coupling constant
(7) (28) Diakula, 2.; Edison, A. S.;Wcstler, W. M.; Markley, J. L. J . Am. Chem. Soc., following paper in this issue. (29) IUPAC-IUB Commission of Biochemical Nomenclature. J . Mol. Biol. 1970, 52, 1-17.
Continuous Probability Distribution of Rotamers
J. Am. Chem. SOC.,Vol. 114, No. 15, 1992 6197
Combination of eqs 4, 6, and 7 and use of the orthogonality relations between the functions cos (nx) and sin (mx), where n, m = 0, 1, 2, ..., gives
+ -a2 cos (2R)p2J-* cos2 (2x)dx -a2 sin (2Q)u2$_: sin2 (2x)dx = c + -a2 + a r b [ p l cos ( Q ) - ul sin (Q)]+ - [ p 2 cos (252) - u2 sin (2Q)] 2
the probability distribution (l), one obtains
b sin (Q)u,J-* sin2 (x)dx
--*
-.I
m
(NOE)
N
Note that the Karplus equation truncates eq 4 to the second order. With at least 2N measured coupling constants from different pairs of nuclei across a common chemical bond, a set of 2N equations similar to eq 8 can be used to determine values of the coefficients pn and u, of the Fourier expansion of p(x) to the order N ( N I 2), provided that the coefficients of the corresponding Karplus equations are known. 2.1.3. R o t a t i d y Averaged NOE. We shall assume here that the internal rotation is faster than longitudinal relaxation (Le,, the period of one complete internal rotation, T,, is much smaller than TI)and that the internal rotation is fast on the chemical shift time scale (Le., T, is smaller than the reciprocal of the difference of the chemical shifts of the nuclei that exhibit dipolar coupling). When the internal rotation is slower than overall molecular tumbling (Le., T , is longer than the correlation time of the overall tumbling of the molecule, T ) , the rotational averaging of an NOE between an atom fixed in the coordinate system bound to the molecule and an atom involved in the internal rotation can be expressed as3&33
WOE)
=f(T,Tr)(lr6)
= f ( 7 , T 1 ) J”p(x)r(x)+dx
(9)
where r is the distance between the two interacting nuclei (a function of the dihedral angle x) andfis a function of the correlation times whose explicit form is not important for our study. In the case of fast internal rotation ( T , T,) Fi(zj-1)
Fi(2j) = *Sij
*Gj;
(18a)
where i = 1, ..., M,,; j = 1, ..,,N
which can be rewritten as
a*U = @ The elements of the 2 N X 2 N matrix a-- are M
aij =
or the Fourier coefficients of the function r(x)+ in the case of slow internal rotation
(T
> T,), its elements are
ni = [1/2- Cio
where i = 1, ..., M,,(22a)
where C,o is the zeroth-order Fourier coefficient of the function r ( ~ ) corresponding -~ to the ith pair of cross-relaxing nuclei. When the internal rotation is slow (T 2N) corresponds to the minimum of the sum of the squares of differences between experimental and theoretical couplings and NOEs, given by
pi =
9
C(qt- thea)' +
i- I
M"
[x(t& - :d,&,)2, T I- 1
MJ
9=
>> T ,
(23a)
