Dynamic properties of bound water studied through macroscopic water relaxations in concentrated protein solutions

BIt

ELSEVIER

Biochimica et Biophysica Acta 1208 (1994) 81-88

Biochi~ic~a et BiophysicaA~ta

Dynamic properties of bound water studied through macroscopic water relaxations in concentrated protein solutions M a s a a k i Iino * Department of Physiology I, Nippon Medical School, 1-1-5 Sendagi, Bunkyo-ku, Tokyo 113, Japan Received 4 March 1994

Abstract

Magnetization transfer experiments using an off-resonance irradiation technique were performed on bovine serum albumin solutions by varying the irradiation frequency and concentration. Observed macroscopic magnetizations of water protons were well expressed by the rate equations of populations for spins, which gave the tumbling time of protein protons and the intrinsic relaxation rates of water and protein protons. These parameters conformed to a model of rapid-exchange water system with bound water molecules in the interior of the protein that interact with protein protons. Analysis of the data enabled the separation of relaxation rates into the respective contributions by the interior bound water and water in the hydration layer at the protein surface, and determined the amounts and the average correlation times of these water fractions. The average residence time of the interior bound water with respect to exchange with the bulk water was found to be (5 + 2). 10 - 6 S. The estimation of the hydration layer showed excellent agreement with the amount measured by a thermodynamical method. Keywords: NMR; Magnetization transfer; Cross-relaxation; Bovine serum albumin; Bound water; Residence time

1. Introduction

Proton nuclear magnetic resonance (NMR) is used extensively to probe the molecular dynamics in protein solutions and hydrated biological systems [1-10]. The nuclear spin relaxation rates of water protons may involve mechanisms such as intermolecular dipolar interactions between bound water protons and protein protons [1]. The intermolecular dipolar interaction yields the cross-relaxation between these protons or negative nuclear Overhauser enhancement (negative NOE), which couples to the spinlattice relaxations of water and protein protons, and also contributes directly to the spin-lattice and spin-spin relaxation rates [7]. In an earlier paper in this field [2-7,10], these effects of the intermolecular dipolar interactions have either been ignored or studied using deuterium oxide (D20) in place of water. In the D20 experiment, non-exchanging protons and solvent viscosity change the interactions and correlation times of protons [6]. Two-dimensional NOE studies of hydration of a low molecular weight protein showed in terms of NOE signs [9,10] that there are two qualitatively different types of

* Corresponding author. Fax: +81 3 56853055. 0167-4838/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved

SSDI 0 1 6 7 - 4 8 3 8 ( 9 4 ) 0 0 0 8 4 - T

hydration sites. A small number of bound water molecules in the interior of the protein are in identical locations in crystal structure and in solution [9,10]. A lower limit for the residence time t b of the interior bound water with respect to exchange with the bulk water ( 3 . 1 0 -1° s) was derived from the fact that the observed NOE had a negative sign [9]. An experiment with paramagnetic shift reagents [10] provided an upper limit for t b ( 2 . 1 0 -2 s). These facts do not indicate that the t b values are distributed over a wide range, from 3 . 1 0 - t ° to 2 . 1 0 -2 s. Thus, the dynamic properties of the bound water are still unknown except for these limits. The purpose of this study was to quantitatively determine the dynamic properties of the interior bound water. A two-dimensional NOE spectroscopy cannot determine the value of t b for t b >/xs, because the overall rotational correlation time of protein (tumbling time, r ) must be longer than t b when the spectral line width of the protein proton spreads so far over the chemical shifts of protons that the static magnetic field is less than 100 T. Therefore, conventional macroscopic water relaxation effects [1-8] were observed and analysed instead. Through the analysis, it is possible to obtain dynamical information about unobservable components of heterogeneous biological systems with relevance to imaging studies [11-13].

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M. lino / Biochimica et Biophysica Acta 1208 (1994) 81-88

2. Materials and method

1 . 0 , , , , i , , , ,

2.1. Sample preparation

0.8

Water soluble bovine serum albumin (BSA) was purchased from Nakarai Tesque, Tokyo, Japan (Lot No. M3N8671), and used without further purification. Solutions were prepared without deuteration by adding distilled water to a dried and weighed quantity of BSA powder, and adjusted to pH 7.4 with NaOH. The final amount of salt (NaC1) was adjusted to 10 mM. Within the experimental error limits, measurements with different samples which had been prepared in the same way gave identical results.

Mw

M w ( t ~ , A f , nirr)

Rzew + 4,n-2Af2

Mow

RZw + 4,rrZAf2 + _ 2,2 Rzw ~t O i r r R1---~

(1)

for Rlwt >> 1, where Mow =Mw(0 s, A f, 0 T), y is the proton gyromagnetic ratio, Rlw is the intrinsic spin-lattice

0.6 0.4 0.2 0 -100 -80-60-40-20 0 20 40 60 80 100 FREQUENCY OFFSET (103 Hz)

2.2. N M R measurements The NMR measurements were made on an NMR spectrometer system (JNM-GX270, JEOL, Tokyo, Japan) with a static magnetic flux density of 6.3 T (with Larmor frequency for protons of 2.7 • 108 Hz) using pulsed NMR techniques. All experiments were made at a sample temperature of 297 _+ 0.6 K on a 100 /zl sample volume. The cold sample was allowed to stand at the temperature for 10 h, NMR lock and auto shimming were performed on the D20 solvent (lock solution, 100/zl) which was set close to the sample. The static magnetic field homogeneity was also checked for shimming by detecting the signals from small amounts of protons in the D20 solvent. The apparent spin-lattice relaxation rate of water proton spins in the sample, Ra*, was measured using a conventional inversion recovery sequence as the slowest component of the apparent spin-lattice relaxation. The spin-spin relaxation rate of water proton spins in the sample, R2w, was measured using the standard CarrPurcell-Meiboom-Gill sequence with the computer software procedures supplied with the instrument. The zr/2 7r pulse spacing was 2 • 10 -3 s A gated off-resonance irradiation (closed during free induction decays) was applied through a built-in decoupler channel of the spectrometer during a predelay time t. A sampling pulse of ~-/2 radian was applied for the detection of the magnetization of water proton spins, M w, after the off-resonance irradiation with a spacing of 2 • 10 -3 s to allow for the dead time of the receiver. The first time delay for the sequences was set to 10 s. The off-resonance irradiation field flux density Bir r w a s calculated from the direct saturation effect on water proton spins in a solution of MnC12 (3 • 10 - 4 M) as follows: The effect of a given Bi~r and frequency off-set A f on M w (t, Af, Birr) is given by [14]

~

Fig. 1. Effect of irradiation frequency on the magnetizations of water protons. The irradiation was applied for 10 s with an irradiation field of 1.6.10 -5 T. Open and closed circles are the experimental data at x/(1 - x) = 0.3 and 0.5, respectively.The solid lines represent the best fit results of the present fitting calculations of Eq. (4). Obtained parameters by the fittings are shown in Figs. 3-7. relaxation rate of water proton spins in the absence of magnetization transfer. Since there is no magnetization transfer in an aqueous paramagnetic solution [11], the values of Rlw and R2w in the MnC12 solution were measured using the inversion recovery sequence and the standard Carr-Purcell-Meiboom-Gill sequence respectively. A fitting calculation of Eq. (1) to the data yielded the value of Bir r. The value of Bir r w a s also checked by measuring the pulse width of the irradiation pulse on resonance for 7r radian flip of water proton spins. There was no significant difference between these two methods. We used an off-resonance irradiation field strength of 1.6 • 10 -s T (690 Hz in units of yBirr/2"rr) in the present measurements.

3. Results 3.1. N M R data Macroscopic magnetizations and relaxation rates of water protons were measured for BSA solutions with the concentration up to 7.2.10 - 3 mol kg -1. Fig. 1 shows typical magnetization transfer spectra of Mw(10 s, Af, 1.6- 10 -5 T)/Mw(0 s, A f, 0 T) as a function of x / ( 1 - x), where x is the weight fraction of proteins. As the concentration increased the line shape gradually changed. A typical Gaussian line shape is shown in Fig. 2. Random errors (standard deviations) of the data are 0.02 in the figures (3 samples and 4 times averaging). Figs. 3 and 4 show the experimental results for the values of RI* and R2w, respectively. Each value of the experimental data represents the mean of at least 100 points (3 samples, 2 measurements, 7 delay times, and 4 times averaging). The error is about 3% for RI*w and 4%

M. lino / Biochimica et Biophysica Acta 1208 (1994) 81-88

#-I

83

60

0.8

50

0.6 Mw M-o-0w0.4

40 ~

z

J

30

0.2

0

** (

Fig. 2. Effect of irradiation frequency on the magnetization of water protons at x / ( 1 - x) = 0,9. The irradiation was applied for 10 s with an irradiation field of 1.6.10 -5 T. The solid line represents the best fit result of the present fitting calculations of Eq. (4), which yielded the value of O'.

10 • ~ I ~ 4A 0 0 0.2

•

~ 0.4

! 0.6

0.8

x/(1-x )

for R2w. Since no correction was made for the small concentration of dissolved oxygen to the observed relaxation rates, the relaxation rate of water proton spins included the contribution of the order of 0.1 s-1 for the samples at 297 K under 1 atm air. It is noted that Fig. 4 shows that the linear behavior of R2w holds up below about x / ( 1 - x ) = 0.2. The present values of R~w agree with the D-terms of the dispersion data [3,4].

3.2. Data analysis Assuming the rapid spin diffusion in protein proton spins and the rapid chemical exchange in water proton 2.0

,-'-, 1.6 1.2

0

•

.

0

4

y ~

I

I

I

0.2

0.4 X/(1-x )

0.6

0.8

Fig. 3. Spin-lattice relaxation rates of water protons plotted as a function of x / ( 1 - x). Triangles are the observed values of R~w Circles show the calculated values of Rlw. The line drown through the circles results from a least-squares fitting. Errors for the values shown without error bar are smaller than the size of the symbols.

Fig. 4. Contributions to the spin-spin relaxation rate of water protons. Circles are the observed values of R2w. Triangles show the calculated contribution of the bound water fraction. Squares indicate the residual relaxation rates (the first term of Eq. 26). When triangles are subtracted from circles, they yield squares. The solid line drawn through the circles represents the best fit result of the present model with C r = 0.32 and ~'r = 2.5" 10 -9 s.

spins, an aqueous protein solution can be modeled by a pair of proton spin baths, that is, water proton spin bath and protein proton spin bath coupled magnetically by intermolecular dipolar interactions [11]. Since the effective tumbling time ~- [15,16] of BSA molecules for a dilute solution (-- 10 -7 s) are sufficiently shorter than the reciprocal of the rigid line width of the protons ( = 10 -5 s), the NMR line shape for protein proton spin bath should be a Lorentzian line shape at least for sufficiently dilute solution. For fairly concentrated solution the effective tumbling of BSA molecules slows down. As the concentration increases the spectral line width rises up to the rigid line width tr of the protein proton spins. The line width saturates above a critical concentration, (6.2 ___0.1). 10 -3 mol kg -1, where the tumbling still slows down, and the line shape changes to a Gaussian line shape [17]. When ~exceeds t b, it is possible to estimate the value of t b. Although magnetization transfer between water proton spins and a semisolid component of tissue with a Gaussian line shape have recently generated interest in NMR imaging under the designation of magnetization transfer contrast [12,13], a limitation of the simple off-resonance irradiation technique for semi solid saturations has also been pointed out [11]. There is no longer a solution of Bloch equations at trz > 1, because a single spin-spin relaxation rate cannot give rise to anything other than a Lorentzian line shape. When tr2rt >> 1 and R2wt >> 1, the transverse coherence of the spins will be destroyed, and populations of

84

M. lino / Biochimica et Biophysica Acta 1208 (1994) 81-88

Zeeman energy states can be defined. In the presence of the irradiation and the magnetization transfer, rate equations for the populations of spins are given by [18] dn w dt = - (kw + glw)(nw -- nw°) + kp(nP - no°)

- 2nwP ~

(2)

dnp

dt = - ( k ° + R a p ) ( n P - n°°) + kw(nw - nw°) - 2npP p

(3)

where n w and n o are the population differences between water proton spins and protein proton spins, respectively, with components parallel and antiparallel to the static magnetic field; n ~ and no0 are the values of n w and np, respectively, corresponding to thermal equilibrium; Pw and Pp are the transition probabilities per unit time for water proton spins and protein proton spins, respectively, due to the irradiation; R~p is the intrinsic spin-lattice relaxation rate of protein proton spins in the absence of magnetization transfer; k w is the magnetization transfer rate from water proton spins to protein proton spins, and k_ is the reverse. A steady state is reached in Eqs. (2) and (33, when

Mw(t~,Af, Bi,)

nw

Mow

nw0 Rlw{k p + Rip + 2Pp} + R l p k w

=

(4)

{k w -~"Rlw --[-2Pw} {k 0 +Rap + 2P0} - kwk o

n__&= npo

Rlp{kw + Rlw + 2Pw} +Rlwk p {kw+Rlw+2Pw}{kp+Rlp+2Pp}-kwk p

(5)

Eqs. (4) and (5) are identical with the solutions of Bloch equations at 0.r << 1 [19]. Neglecting the effect of chemical exchanges between protein proton spins and water proton spins on spin-spin relaxations, the transition probabilities (absorption spectra) are given by [17]: PP

2 2 ~ (_1)i "g40.nirre(°'r)2 E i! ( °'q')2i i=0

i o-r + - × 0.r ( i)2 47r~lf 2 0.r+ - + 0.T

(6)

or

2 2

T BirrR2w Pw = 2(R~ w + 41r~lf2)

(7)

and a Lorentzian line shape for o'r << 1, 2Bit2r or 2,./.

Pp = 2(0.%2 + 47r2Af2)

where o"2r is identical with the spin-spin relaxation rate of protein proton spins, R2p, for o'r<< 1. Since a BSA molecule has several tumbling times due to anisotropic rotation [16], Eq. (6) gives an approximation around o-r = 1. It can be considered that there is no effect of the anisotropy at o-r < 1 / 2 and o'r > 2 [16]. In this meaning, r must be an effective tumbling time. It is still hard to directly solve Eqs. (4), (6) and (7) due to the number of parameters. In addition to Eqs. (4)-(7), these parameters are related to the apparent spin-lattice relaxation rate of water proton spins in the absence of the irradiation [20], i.e.,

'{(

R~'w = 5

The inverse magnetization transfer rate, kp, follows from the detailed balance, N . ( 1 - x) k w / W w = Up x k p / W p

irrV 2- e - 2 ~ z a e z / ~ z

20"

(8)

(11)

where W. and W0 are the molecular weight of water and the protein (Wp = 66000 [21]), respectively, and Nw and Np are the number of hydrogen protons consisting of a water molecule and a protein (No = 4390 [21]), respectively. Since the values of Rlw * , R2w and Mw(&, Af, 1.6.10 -5 T) are measured directly, the eleven unknown quantities in the five simultaneous equations, Eqs. (4), (6) and (7), and (10) and (11), are reduced to only three unknown quantities, including o- which is common to BSA solutions and was determined at a high concentration with a Gaussian line shape. From the Gaussian line shape in Fig. 2 the value of 0. was calculated to be (1.26 + 0.32) • 105 s -1. Thus, the fitting calculations yielded all the parameters. The solid lines in Figs. 1 and 2 show the curves fit to Eq. (4), which are consistent with the respective experimental results. Figs. 3, 5 - 7 show the obtained parameters. Fig. 3 shows the values of Rlw plotted against x / ( 1 - x). The linear behavior of Rlw holds up throughout the figure. Figure 5 shows the observed values of k w plotted against x / ( 1 - x ) . The behavior of k w is similar to that of R2w. The values of k 0 and % are shown in Fig. 6. The value of Rip was 2 . 2 + 1.5 s -1, and did not show significant dependence on x / ( 1 - x ) . Fig. 7 shows the values of 0. 2r and r, and the spectral line width of protein proton spins, R'2p, which is represented by the solution of [12] R~-

PP

Rip + kp + RI. + tw)

}

Eq. (6) gives a Gaussian line shape for 0.r >> 1, ~2B2

(9)

41n2

2 - 1 [ rrrR'2p 1 o- tan , j

4--i n2

(12)

Errors for the fitted parameters were taken as square roots of the diagonal elements of the covariance matrix.

M. lint/Biochimica et Biophysica Acta 1208 (1994) 81-88

85

12

7 50

10 8

40

°r~

5

6

%

4 ~,

r-.4

.-, 30

4

~t"4

3 2

20 I

0,- ~- -" • 0 0.2

n

,

0.4

0.6

i

2

0.8 10

x/(1-x )

i

1

Fig. 5. Cross-relaxation rate from water protons to protein protons as a function of x / (1 - x).

00 Assuming that the water molecules exchange between the hydration regions and the bulk of free water more rapidly than any local relaxation rate, one mean relaxation rate in the absence of magnetization transfer is obtained as

O0

•

I

I

0.2

0.4

0.6

0.8 0

x/(1-x ) Fig. 7. Calculated parameters of protein relaxation plotted as a function of BSA concentration. Circles are the values of R'2p calculated from Eq. (12). Triangles show the values of o" 2r and ~'.

x

R'w = 1 - x

Y ' ~ C i ( R l i - R ' f ) +Raf

(13)

i x

n

~x2" = 1 - x

Y'~ Ci( R2i - R2f) + R2f

(14)

i

where C i is the water content in phase i per dry weight of the sample; R l i and R2/ are the intrinsic spin-lattice and spin-spin relaxation rate, respectively, for phase i in the absence of magnetization transfer; and R l f and R2f are the spin-lattice and spin-spin relaxation rates for the free water. When Rli , R 2 i , and C~ do not change with x for

any i, the relaxation rates of water protons are linear functions of x / ( 1 - x ) . The extrapolated values to x / ( 1 - x ) = 0 are consistent with all the observed values for pure water in Figs. 3-4. From Eqs. (13) and (14), we obtain the spin-lattice relaxation rate of free water as Rlf = 0.30 ___0.02 s-1, and the spin-spin relaxation rate of free water as R2r = 0.3 _ 0.1 s -1 Since no correction was made for the small concentration of dissolved oxygen to the observed relaxation rates, the relaxation rate of water proton spins included this contribution of the order of 0.01 S-1

30 25 ,-. 20

2.5

4. D i s c u s s i o n

2

We have presented data for the magnetization transfer rates and the intrinsic relaxation rates of water protons in the absence of magnetization transfer in BSA solution. We first discuss the chemical exchanges between water protons and protein protons. Although the observed magnetization transfer includes the contributions of magnetic dipolar interactions (cross-relaxation) and chemical exchanges, Eqs. (6) and (7) do not include the effects of the chemical exchange. Cross-relaxation increases only the spin-lattice relaxation of water proton spins, while the chemical exchange increases both spin-lattice and spin-spin relaxation rates. Assuming no chemical shift difference between the water proton spin and protein proton spin resonances,

uo0

15

1.5G

10 i

0.5

06

I

I

0.2

0.4

i

0.6

0.8 0

x/(1-x ) Fig. 6. Cross-relaxation rate from protein protons to water protons as a function of x / ( 1 - x). The values of r b were obtained from Eqs. (11), (23), (24), and (26).

86

M. lino / Biochimica et Biophysica Acta 1208 (1994) 81-88

the increase of the spin-spin relaxation rate of water proton spins, AR2w, is given by [12,20]

Fp AR2w

~

,

t

for R2p >> kp, where Fp is the proton fraction of proteins,

and k'p is the rate constant of the chemical exchange from protein proton spins to water proton spins. Since the value t of kp may be independent of the concentration of proteins, the value of k'p is expected to be less than the lowest value of the observed value kp. The parameters for the samples, Fp < 0.3 and k'p < 1.7 s -1 < kp, yield AR2w of less than 0.5 s -~, which can be ignored. Therefore, R2w is regarded as the intrinsic relaxation rate of water proton spins in the absence of the chemical exchange. Similarly, o. 2r and R'2p are regarded as the intrinsic values of protein proton spins. It is clear, mainly from the apparent spin-lattice relaxation dispersion data for tO0/2~" < 5 • 10 7 Hz [2-4], that the magnetic dipolar relaxation of bound water protons reflects the rotational Brownian motion of proteins (/b > 10 .7 s). Yet severe inconsistencies arise when interpreting relaxation data in terms of hydration models. We follow in Edzes and Samulski's steps [7] to separate the relaxation rates into inter- and intramolecular contributions for a high resonance frequency. Let us picture a small amount of C b of the water molecules which are bound in the interior of the protein for a residence time t b [9,10]. The correlation time of the interior bound water is given by r b = 1/

+

(16)

given by the slow limit expressions for rotational motion of a pair of like spins [22], Rlbb = 4 ° ' 2 / 3 W2rb

(19)

RZbb = %2%

(20)

for O"w < l / ' r b << tOo, where o-w is the rigid line width for a pair of protons in a water molecule (o.w = 1.3 • 105 s-a), and tOo is the nuclear angular resonance frequency (1.7. 109 s -1) The interrnolecular dipolar relaxation rates are given by the slow limit expressions for pairs of unlike spins [22], glb p =

4no.2p/3to2%

(21)

5 2 b R2b p = ~no.bp'J-

(22)

for o.bp < 1/% << too, where n is the average number of protein protons which interact with a bound water proton, and o.bp is the average value of the rigid line width for the instantaneous pair of these protons. The spin-lattice relaxation of the bound water protons and the protein protons are now coupled by a cross-relaxation. The cross-relaxation rate from the bound water protons to the protein protons, k b, is given by [22] 2 2 b k b = ~nO'bpT

(23)

for o.bp < 1/% << too" The values of k W and k b are related through the detailed balance, (1 - x)k w = xCbk b.

(24)

Of particular importance is the region where o.w << 1/% << too, where the following analyses are valid. Assuming C b << 1, we can rewrite Eqs. (13) and (14) as X

The interior bound water molecules exchange at a given rate (the reciprocal of t b) with the water molecules in the hydration layer. The water molecules in the hydration layer exchange rapidly with free water molecules. When these water molecules exchange more rapidly than any local relaxation rate, we have unique Raw and R2w characterizing water protons in BSA solution, as given in Eqs. (13) and (14). Then the spin-lattice and spin-spin relaxation rate of the bound water protons, Rlb and Rzb , should include the contributions due to the intermolecular dipolar interaction between the bound water and the protein protons, Rlb p and R2b p respectively, in addition to the contributions due to the intramolecular dipolar interaction in the bound water protons, Rlb b and Rzbb, respectively. Thus: R l b = Rlb b + R l b p

(17)

Rzb =

(18)

R2b b + R2b p

From the investigation of the anisotropic motion of bound water in BSA solution [2], we assume that the motion of the bound water is isotropic. Ignoring the contribution of translational relaxation (translational correlation time >> 1/tO 0) [1], the intramolecular dipolar relaxation rates are

Rlw ~

E

1 -x

i*b

X

R2w ~

Ci{Rli-Rlf}

q-Rlf 9

E CiR2i + ~{1 + -ga}kw 1 - x i=~b

(25)

(26)

where

a ~ o'wZ/no.bZp The first term of Eq. (25) corresponds to the D-term [3] of the NMR dispersion data at a low frequency in the absence of magnetization transfer. Thus, the inter- and intramolecular contributions of the bound water are replaced by the obtained value of k w. If there is anisotropy on the bound water, the first term of Eq. (26) may not be affected, because the slow limit expressions of Eqs. (20), (22) and (23) consist of the same correlation function of spins [22] and the effects compensate one another. With the help of the linear manner of Rlw shown in Fig. 3, we can assume that C i, Rli, and Rzi for i 4: b do not change with x / ( 1 - x). Among the parameters presented above, only r may have a strong dependence on x / ( 1 - x ) , because this value reflects the macroscopic motion of the protein. The others may have no significant

M. lino / Biochimica et Biophysica Acta 1208 (1994) 81-88

dependence because these are determined by the microscopic nature of the molecules. We therefore adjusted the parameters as c~ = 0.33 + 0.05 so that the first term of Eq. (26) has a linear dependence on x/(1- x). Fig. 4 also shows the contributions obtained by the fitting calculation. To estimate C b and ~'b, we assume that the value of ~'b at a small x/(1- x) is equal to r (Eq. 16). The estimated value is C b = (6 + 1) • 10 - 4 , which corresponds to 2.2 _ 0.4 water molecules per BSA molecule. The value of % depends on x/(1 -x) as shown in Fig. 6. With 0.27 nm as a representative distance between water proton spins in a bound water molecule and protein proton spins in near the protein-water interface [2], tYbp is 2.5 • 10 4 S-1 Since the probability that a water proton spin and a nearby protein proton spin have opposite spin polarization is 1/2, and the probability that the slowly exchanging water proton spin will leave the protein surface with a reversed spin polarization is 1/2, then we can say [2] that k b = nOrbp/4 f o r t b O'bp >> 1. Since the observed value of k b is somewhat smaller t h a n n O ' b p / 4 , the value of t b c a n be estimated from Eq. (16) as t b = (5 ___ 2 ) . 10 -6 S. From the fitting calculation of Eq. (16) there was no significant difference in the residence times between the two bound water molecules per protein. We shall next discuss the residual relaxations in the first term of Eqs. (25) and (26). It is assumed that a single component of water fraction contributes the residual relaxation rates. If the spin-lattice and spin-spin relaxation rates of the residual fractions, Rlr and R2r, are given by the intramolecular dipolar relaxation rates for isotropic motion [22], then:

2 2 /

1

4

Rlr = 30"wTr/ 1 + (.O2Tr2 "[- 1 @ 4tO2"/'r2

1 2{ Rzr = 3Orw7r 3

+

_5 _2 ) 1 + to2"rr 2 + 1 + 4o~2"rr2

87

ized by four bonds would have a life of (4-13). 10 - 6 S. Similarly, water localized by two bonds would have a life of (0.3-1.1). 10 -9 s. These two values are so close to their respective present estimated values of t b and ~'r that it is reasonable to infer that the correspondences have some validity. Although a similar argument [4] suggests that t b is about 1 • 10 - 6 S, the residence time of 1 • 10 -6 S cannot explain the present experimental results. The interpretation of the data in terms of hydrogen bonds is decidedly simplistic. A general dynamical characterization of hydration is therefore an interesting project for future research.

5. Conclusion We have obtained macroscopic values for magnetization transfer rates, intrinsic relaxation rates of both water and protein proton spins, and tumbling time of protein molecules. An analysis of magnetization transfer rates yielded quantities and average correlation times for interior bound water and water in the hydration layer. The residence time of interior bound water was estimated to be (5 ~ 2 ) " 10 -6 S. Although the hydration sites were not individually assigned, this work may serve as an experimental framework for the rational design of studies on a heterogeneous biological system and its imaging.

Acknowledgments

(27)

The author is grateful to all staff at NMR laboratory for their untiring help during this research.

(28)

References

for crw < 1 / 7 r. The least-squares fitting lines in Figs. 3 and 4 were obtained when ~'r = ( 2 . 5 _ 0.3). 10 - 9 sand Cr = 0.32 +__0.03 which corresponds to 1170 + 110 water molecules per BSA molecule. Although this analysis of the residual relaxations is highly model-dependent, the value of C r shows excellent agreement with the content of water of hydration (0.32) measured by a thermodynamical method [16]. The value of 7~ is larger than the reported values for a polypeptide chain [23]. Water molecules at the protein-water interface can be localized for a substantial period of time by hydrogen bonding. Assuming non-interacting bonds, the vibrational stretch frequency of a hydrogen bond is in the range of 60 to 2 0 0 c m -1 [24], which gives a period of ( 3 - 9 ) . 10 -14 S. The activation enthalpy is about 11.7 k J / m o l [24], which corresponds to a Boltzmann factor of 109 per hydrogen bond in the probability of dissociation by thermal excitation. Thus, assuming non-interacting bonds, water local-

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