r39LaRelaxation in Protein Solutions
1357
Longitudinal Relaxation in Spin 7/2 Systems. Frequency Dependence of Lanthanum-I 39 Relaxation Times in Protein Solutions as a Method of Studying Macromolecular Dynamics JacquesReuben'andZeevLuz lsotope Department, Weizmann lnstifute of Science, Rehovot, lsrael
(Received October 9, 1975)
A matrix for the longitudinal relaxation in spin 7/2 systems is derived and solved numerically. The numerical solution is approximated by an analytical expression which is used in the interpretation of lanthanum-139 relaxation times in dilute solutions of bovine serum albumin (BSA). The protein induces frequency dependent enhancements in the 139Larelaxation rate through the rapid exchange of La3+ between its aquo and BSA complexes. The La3+ ions associate with the free carboxylates of the protein and as a result the quadrupole coupling constant of 139Lais increased. In the BSA complex the interaction between the electric field gradient on the '"La nucleus and the quadrupole moment is modulated by the isotropic rotational motion s. The dissociation conof the protein molecule which is characterized by a correlation time T~ = 3.7 X s < T M < 2.1 stant of the La3+-BSA complex is K D = 0.46 M and its mean lifetime is bracketted: 3.7 X
x
10-6 s.
Introduction It has recently been suggested that nuclear relaxation rates of lanthanum-139 can be used to study the molecular dynamics of proteins in solution.* In this communication we present results of the application of the method to aqueous solutions of bovine serum albumin (BSA) and provide a detailed analysis of the 139Lalongitudinal relaxation times in terms of the La3+-protein binding equilibrium and the rotational diffusion of the macromolecular complex. Lanthanum-139 (natural abundance 99.9%) is a nucleus of spin I = 7/2. In diamagnetic systems its dominant nuclear relaxation mechanism is the modulation of the nuclear quadrupole interaction by molecular motion. In solutions of La3+ salts where the cation is symmetrically solvated by the solvent molecules the relaxation rate is relatively slow because of the small quadrupole interaction and also the short correlation times modulating the quadrupole interaction in these systems. Upon complex formation, e.g., ion pairing or binding to proteins, the asymmetric environment of the La3+ ions gives rise to an increase in the quadrupole coupling constant and usually also to longer correlation times. As a result there is a considerable increase in the 139Lanuclear relaxation rate.lS2 Thus by studying the effect of complexing agents on the '"La relaxation rate and comparison with theoretical prediction, information on both the binding equilibria of the La3+ ions as well as on the molecular dynamics of their complexes may be obtained. In the theoretical section below we derive expressions for the longitudinal relaxation rate of a spin I = 7/2 system by modulation of the quadrupole interaction. It is shown that the longitudinal relaxation rate can be expressed (to a good approximation) in terms of a single exponent over the whole range of the relaxation theory and an approximate analytical expression is derived for 7'1 in a convenient form. This expression is then used in the analysis of the frequency dependence of the '"La relaxation rates in BSA solutions. The binding sites for the La3+ ions are believed to be free carboxylates of the BSA molecules. In order to obtain an estimate for their number as well as for the value of the quadrupole
coupling constant, the effect of acetate ions on the 1:39Larelaxation rates was studied.
Relaxation Theory for Spin 7/2 Systems In this section we derive expressions for the longitudinal relaxation of a nucleus of spin I = 7/2 by modulation of the quadrupole interaction using Redfield's theory. The corresponding derivation of the transverse relaxation (for the equivalent case of the zero field splitting interaction) was discussed previously and used in the interpretation of electron spin resonance spectra of Gd3+ complexes in solution."-5 The quadrupole Hamiltonian can be written as a direct product of the irreducible second rank tensor operators, T 2 p , of the spin part and F 2 p , of the spatial part: 2
. H g = p=-2
(-)PF2PT2-P
(1 1
The components of the tensors in their corresponding principal coordinate systems [laboratory fixed for T 2 P and molecule fixed ( P 2 P ) for F 2 p ] are summarized in Table I where
A=
e29Q
4Z(2I - 1)
1 = -e 2 9 Q 84
(for I = 7/2)
(2)
and 9 is the asymmetry parameter of the quadrupole interaction. The longitudinal relaxation behavior of the spin system is obtained from the so-called relaxation matrix, the elements of which in the basis of the M I spin state a,b are Raahb
=
2Jabab/h'
paaaa
=
- b#a
a Z b Raabh
(34 (3b)
The spectral densities for rotational diffusion are
The Journal of Physical Chemistry, Vol. 80, No. 12, 1976
1358
Jacques Reuben and Zeev Luz
TABLE I
-
IV
where
1
- [31z2- 1(1 + l)]
&A
0
is the eigenvector corresponding to the eigenvalue and Z is a vector with components ( a l M a a ) . In the present case the longitudinal relaxation is given by four decaying functions; the amplitudes of the four other eigenvalues vanish. This can be seen by noting that the 8 X 8 relaxation matrix can be transformed into two uncoupled matrices (11) corresponding to the basis [ M I ( - M I ) ] and [ M I (h
1/Tlh,
1
i
\
v 0 0 6 t h
I
I
+
4
24A'.
-!
E2
A + @ -A -a -A A + B + P -B -0 -B B+C+a+y 0
001
01
IO
IO
wort
Fl ure 1. Plots of the longitudinal relaxation rates, l / T l i , in units of , of the corresponding amplitudes, as ( e q Q / f ~ ) ~ [ l ( q 2 / 3 ) ] ~ ,and function of W O T ~for I = 7/2. The plots were calculated from the relaxation matrix as explained in the text. Note that the amplitudes for lines I, 11, and 111 are plotted on an expanded vertical scale.
9
where
and 00 and T~ are respectively the Larmor frequency and the correlation time for the rotational diffusion. With these definitions relaxation matrix I is derived where A = 2lj(wo)
a = 7j(200)
R = 16J(w0)
B = 15j(2~0)
C = E?j(00)
y = 2Oj(200)
The decay of the longitudinal magnetization consists of a superposition of a number of exponentially decaying func, the eigenvalues tions, the relaxation rates of which, 1 / 7 ' ~are of the relaxation matrix, and their amplitudes are proportional to ( Z * (d*/(2Z + 1)
A+a
E'
-A A+B+P
(7)
-a
-CTr
0
-P
-
(11)~
C+P+Y
This function is plotted in the lower part of Figure 2. As can be seen it is practically identical with line IV of Figure 1.
-ff
-P
-Y
0
c + p + r -c
-P
-Y
-C
-B A+B+P -A
-Y
H
The Journal of Physical Chemistry, Vol. 80, No. 12, 7976
~
- ( - M I ) ] ,where the upper and lower sign correspond to the symmetric and antisymmetric basis functions, respectively. The required relaxation times and amplitudes can be obtained directly by diagonalizing the 4 X 4 matrix of the antisymmetric basis set. The results for the various l/Tli's and the corresponding amplitudes are plotted in Figure 1 as functions of the parameter COOT,. I t is seen that three of these functions have very ~ small amplitudes and are negligible over the whole W O T range. Thus in practice we expect the magnetization to decay with a single exponent, corresponding to line IV in Figure 1. There is no analytical expression for this curve (except when WOTC
