10949
J. Phys. Chem. 1994, 98, 10949-10956
Peptide Dynamics in Triglycine: Coupling of Internal Bond Rotations and Overall Molecular Tumbling Vladimir A. Daragant and Kevin H . Mayo* Department of Biochemistry, Biomedical Engineering Center, University of Minnesota Health Sciences Center, 420 Delaware Street, SE, Minneapolis, Minnesota 55455 Received: May 16, 1994; In Final Form: July 29, 1994@
As a model system, triglycine (Gl-G2-G3) permits a thorough investigation of peptide backbone motional dynamics by using 13C- and 15N-NMR relaxation. Previously, rotational model analyses of the nonterminal glycine, G2, could not adequately explain 13C-NMRrelaxation data (Daragan, V. A.; Mayo, K. H. Biochemistry 1993, 32, 11488). In this study, 15N-NMR relaxation measurements on 15N-enriched triglycine provide additional motional vectors for more complete rotational model analyses. The inadequancy in describing G2 internal motions with models of anisotropic or restricted rotational diffusion is overcome by using a rotational jump model which has been parameterized with a semiempirical coefficient for backbone recoil rotation. Effectively, this recoil action couples internal bond rotations and overall molecular tumbling. Stochastic dynamics computer simulations using this recoil coefficient allow calculation of triglycine 13CH and 15NH autocorrelation times and 13CH2cross-correlation times. Good agreement between experiment and theory is found only when strong recoil coupling is taken into account.
Introduction
H
Recently, peptidelprotein backbone and side-chain motional dynamics have been the subject of many biophysical studies. 15N- and 13C-NMR relaxation data provide the most reliable information about rotational correlation functions. CH and NH bond rotational autocorrelation functions and CH2 and CH3 cross-correlation functions of mutual motions of different CH bonds provide the necessary information to derive various motional Short peptides are very useful for such studies since many relaxation parameters can be determined easily and theoreticallmodeling calculations can be performed and compared to experiment. One limitation to an experimentally well-determined system is the motional (rotational) model used to analyze the various motions that can contribute to observed relaxation phenomena. Triglycine exemplifies a small peptide for which all 13C autocorrelation and cross-correlation times ( T C H and T H C H , respectively) have been determined.' Although terminal glycine internal and overall tumbling motions usually can be understood by using simple rotational models, nonterminal glycine, G2 (see Figure l), motions could not be described adequately by any commonly used rotational model. Moreover, on comparison of di- and triglycine C-terminal glycine motional analyses, results were contradictory. The model of internal rotational fluctuations within potential wells, for example, gave pl-bond fluctuation amplitudes in triglycine (zt34') greater than those in diglycine (f29'),' whereas the model of restricted rotational diffusion' gave larger internal rotational correlation times, ti, for diglycine (T,= 18.1 ps at 303 K) than for triglycine (zi = 13.6 ps at that same temperature). One possible explanation for this apparent contradiction in di- and triglycine terminal glycine motions and the inability of rotational models to explain nonterminal glycine motions is the coupling of overall molecular tumbling and internal rotations. Autocorrelation times, TO, for reorientation of the C,-N G2
* Corresponding author. On leave from the Institute of Chemical Physics, Russian Academy of Sciences, 117977 Moscow, Russia. Abstract published in Advance ACS Abstracts, October 1, 1994. @
0022-365419412098-10949$04.50/0
H
Figure 1. Triglycine backbone illustrated with atoms labeled as discussed in the text.
bond for di- and triglycine are equal to 49 and 59 ps, respectively, at 303 K. Although these correlation times do not reflect protation, they can be used, nevertheless, to characterize rotational motions of the N-terminal segment. In this case, overall molecular tumbling in diglycine is faster than in triglycine. For the more massive triglycine, it seems obvious that internal rotations at the C-terminal G3 position do not significantly affect those of the N-terminal GI methylene. For diglycine, however, C-terminal G2 rotations probably have a significant effect on N-terminal G1 rotations. In this case, G2 (or G1) rotations may induce an opposing rotational motion (recoil) of G1 (or G2j. This recoil e f f e ~ tin ~ ,fact ~ couples overall molecular tumbling and internal rotation and increases as the masses of the N- and C-terminal segments become equal. Recoil contributions do not change the correlation time of the diglycine N-C, bond rotations but rather increase the rotational anisotropy of the N-terminal segment in the direction of the N-C, bond. For any rotational model analysis, not taking this recoil effect into account may lead to large errors in determining internal rotation parameters. In triglycine, this effect may be significant for v)2- and q2-rotations since these rotations occur about bonds which separate comparably sized peptide segments. This generates additional 'anisotropy' in the nonterminal glycine rotations along the p2- and q2-backbone bonds (see Figure 1). Recoil effects, therefore, should be taken into account when considering NMR relaxation data, especially for short, linear peptides and denaturedunfolded proteins. There are several approaches in the literature aimed at describing the coupling of overall molecular tumbling and internal rotation. The main approach has been to consider the multivariable diffusion equation where internal bond torsion 0 1994 American Chemical Society
10950 J. Phys. Chem., Vol. 98, No. 42, I994
Daragan and Mayo
angles are included as well as the three Euler angles for overall molecular reorientations?-” Budb7 and Coffey and coworkers12-15 have used such an approach to describe dielectric relaxation. Knauss and Evans” studied the rotational dynamics of butane by using the Riemann-Kirkwood-Fixman diffusion equation with a simple bead model and concluded that internal torsional and overall rotational degrees of freedom are coupled by an internal angular momentum. Doug and Richards16 considered coupling of overall tumbling and internal rotations in alkyl chains by using conformation-dependentdiffusion axes. mor^^,^ developed the theory of coupled rotor recoil effects. Under the assumption that angular velocities relax much faster than the angular position, Mor0 demonstrated that recoil rotations are determined by frictional effects in the system and that these are unaffected by the internal potential. This paper analyzes I3C- and I5N-NMR relaxation data on triglycine at different pH values and temperatures. A simple, semiempirical approach to describe the influence of recoil effects on NMR relaxation data is presented along with stochastic dynamics calculations to illustrate the recoil model assumptions used in this paper. Theory The general expression for autocorrelation and cross-correlation spectral densities can be written as
s1
yo Figure 2. Recoil parameters illush.ated far +-band rotation as discussed in the text.
protein, recoil rotation can occur. In this case, the correlation function with respect to the molecular frame in terms of segment A C d f )= 4
~ ( ~ ~ ~ ( e ~ ~ ( f )=) ~ ~ ~ ( e ~ ~ ( o ) ) ) 4nC(D2*om(S2~~(f))D20m,,(S1,,(0)) X m,”
Job(@)
= 4nh- (y~o(eL,(f))y20(eLb(o)))COS(Wf)df (1)
where YZOis the second-rank spherical harmonic and OL&) is the angle between some vector a and the direction of the static magnetic field. The superscript L denotes the laboratory frame. Job(W) stands for autocorrelation (a = h) and cross-correlation (a t b) spectral densities. Autocorrelation and cross-correlation times are defined as = Job(0)
(2)
When a = h, Le., autocorrelation, a simplified, single-subscript notation, e.g., z, or J., will be used. To separate internal bond rotations from overall molecular tumbling, a molecularflaboratory frame transformation is usually performed by using the Wignerl’ rotation matrix DZ,,,(R) y20(eL0(f)) = CD20,(RDL(f))Y*z,(e,(f),@~,(f)) (3) m
RDL(~) defines the set of Euler angles which determines the transformation from the molecular (diffusive) frame to the laboratory frame (called LD transformation). Sa(?) and @&) are polar angles for vector a in the molecular frame. These angles are time dependent when internal bond rotations occur. It should be mentioned that the word ‘diffuse’ frame rather than ‘molecular’ frame is used here to emphasize the fact that this frame is not rigid with respect to the molecule hut rather reflects overall (usually diffusive) molecular tumbling. Consider a peptide molecule with only one internal rotation axis. For example, the N-C, bond torsion angle @ is defined in Figure 2. [Although the following discussion exemplifies @-bond rotational motions, similar arguments can be made for (or even X) bond rotations.] The labels A and B denote the N- and C-terminal sides, respectively, of this peptide. If the mass of A is much greater than that of B, one may define the molecular frame in terms of A and consider internal rotations of B by using the LD transformation given in eq 3. However, when the mass of A is about the same as that of B and segments A and B do not spatially interact as, for example, in a folded
y Z m ( e ~ ( f ) , ~ ~ ( f ) ) ~ 2 m ’ ( e b ( o ) , ~ b ((4) o)))
can not be factored due to coupling of RDL(f) and 6’&),4&) rotations. To avoid this difficulty, a ‘diffusive’ molecular frame which is independent of internal rotation should be defined (see Figure 2). The internal rotation angle A@ can he determined from the coordinates of the four atoms: A, N, C,, and B. The ZDaxis is directed along the axis of internal rotation (along the N-C, bond), and the xD-axis falls in a plane as shown in Figure 2. By definition, A@ = 0” when A and B are eclipsed and lying in the XD-ZD plane. Starting at A@ = O”, molecular rotation may cause either A or B to move out of the XD-ZD plane. To check the orientation of this ‘diffusive’ molecular frame with respect to the laboratory frame, one can refer hack to the zD-axis with A@ = 0’ at any time during this rotation. Deviation from the initial orientation would mean that overall tumbling has occurred during this time interval. If the initial angular velocity of overall rotation were zero, solvent interactions would be responsible for the angular deviation. Use of such a ‘diffusive’ molecular frame allows easy separation of internal and overall rotations. By neglecting any dependence of the overall diffusion coefficient on molecular conformation, the correlation function for isotopic overall rotation (with correlation time 20)can he written as Cob(f) = exp(-f/zO)cab(f)
(5)
where 2
d a b ( ? )=
C ( Y z , ( e , ( f ) , ~ ~ ( f ) ) Y * , , ( B b ( O ) , ~ b ( 0(6) ))) m=0
is the correlation function for internal rotations. Cob(0)= C,b(O) = 1 is a = b. For tetrahedral carbon geometry with a f b, Ciab(0) = Cob(O)= -V3. a and h are the CH bonds of a methylene or methyl group. In general,
Triglycine Peptide Dynamics
J. Phys. Chem., Vol. 98, No. 42, 1994 10951
1 C,(O) = -(3 C0S2(Bab)- 1) 2
where gab is the angle between a and b vectors. If a and b belong, for example, to molecule part A, eq 6 can be rewritten as
dab
= 3 cos 8,
a2,b
COS
e, sin e, sin Ob cos(q,
- qb)
= 0.75 sin2 e, sin2 8, cos 2(qa - qb)
where e,, Ob, qa,and P)b are the polar angles for vectors a and b in the diffusive frame. It is important to realize that since only the difference (pa - Q)b) appears in eqs 7 and 8, polar angles in the ‘diffusive’ frame are the same as those in the molecular frame. The angle of intemal rotation, @, can be measured in the laboratory frame or in some molecular frame defined by part A or B. In the ‘diffusive’ frame, the angle @ may be separated into @A and aB(see Figure 2). These angles are related by
A@ = QA - QB
(9)
Empirical coefficients, K , may be introduced to account for possible rotational recoil effects:
KA = ICPAI/IA@I
(10)
2 = I 0, rotational amplitudes of the C-terminal segment are reduced and compensatory recoil rotations of the N-terminal segment occur in addition to overall tumbling. Equations 16 for JNH(W)and JCH(O)are similar to the Lipari and Szabo21,22equation:
J ( o ) = S2
z;
+ (1 - S2)1 + (w'i)2 1 + (wto)2 50
(18)
with order parameters
Due to a different definition used here for the spectral density function, the coefficient of 2/5 used by Lipari and Szabo21,22 does not appear in these equations. In our opinion, the definition for J a b given in eq 1 is more convenient due to the simplified relationship between the spectral density and the correlation time as shown in eq 2. For cross-correlation spectral densities, JHCH(W), one could write a similar eq based on eq 16:
,
.
,
,
, " .;..."
,
'"...
-0.4
0.0
0.2
0.4
0.6
0.8
1 .o
K,
Figure 3. Dependence of order parameters Sb on the recoil coefficient for +-bond rotation calculated for NH and C,H bond rotation.
It should be mentioned that it is possible to write a generalized equation combining eqs 18 and 20 as
Equation 22 can be used to describe the behavior of autocorrelation and cross-correlation spectral densities without any model assumption^.^^-^^ Here, PZ(COS 6ab) is the second-order Legendre polynomial, and Oab is the angle between vectors a and b. For a = b (autocorrelation spectral density), P2(Oaa) = 1 and eq 22 becomes the Lipari and Szabo21,22equation with 9 = Sob. Maximum rotational restriction is observed when the cross-correlation 'order' parameter is Sab = PZ(COS e&). In this case, eq 22 describes isotropic overall tumbling in the absence of intemal motions. The standard Lipari and Szabo squaredorder parameter was replaced in eq 22 by the cross-correlation order parameter, Sob, since Sob is dependent on geometry and can be negative. For a two-state rotational jump with jumps between two minima separated by an angle of 2y, the SHCH can be written as
where K stands for either KAp,P wPp, , or KAV. pRotations in di- and triglycine, however, need to be described by a threestate rotational jump model with one major central minimum and two equivalent minor side minima.' The angular displacement between the major central and minor side minima, y , equals 90", and WOand W1 define transition rates from central to side minima and vice versa, respectively. When Wo
