3978
J. Phys. Chem. 1981, 85, 3978-3984
frequencies x, which characterize any FC transition. Conclusion In this work a model was considered which, when ignoring reflection, yields reactive T-matrix elements which are proportional to the Franck-Condon overlap coefficients. These terms were applied to derive factorization expressions for reactive vibrational amplitudes and probabilities. The fiial expressions canbe used to analyze both experimental and numerical results. As an example, we studied the two isotopic systems H(D) + Clz. To do that, we applied the factorization for the amplitudes, rather than that for the probabilities. Here, two variables (one pair for each final vibrational state), which were shown to be
linearly dependent, were constructed. The parameters of the resulting straight lines yielded the ratio between the fundamental frequencies of the two oscillators x and their relative shift T . Whereas x for HClz was smaller than the ratio between the two original frequencies of HC1 and Clz (-3 vs. 7), for DC1, the value of x was similar to the corresponding ratio (which was 5). The relative coordinate shift between the reagent and the product oscillators was found to be equal to 3 in the two cases. Acknowledgment. This research was supported by the Deutsche Forschungsgemeinschaft under Sonderforschungsbereich 91 "Energietransfer bei Atomaren und Molekularen Stossprozessen".
Concentration Dependence of the Diffusion Coefficient of a Dimerizing Protein: Bovine Pancreatic Trypsin Inhibitor Peter R. Wills" and Yannls Georgalls Max-Plenck-Institut fur Molekulare Genetik, Abteilung Wittmann, 1000 West Berlin 33 (Dahlem), West Germany (Received: May 1, 198 1; In Final Farm: July 28, 198 1)
We examine the theory of the concentration dependence of the diffusion coefficient of a solution of macromolecules and conclude that an attractive potential, in addition to the usual repulsive hard-core and electrostatic potentials, must be invoked to explain how the diffusion coefficient of spherical protein molecules can decrease with increasing concentration. The case of a narrow square-well potential, modeling dimerization, is considered. The equivalence between the statistical mechanical and thermodynamic descriptions of protein dimerization is demonstrated. It is found that, in the case of the diffusion-controlled dimerization of a protein, the intensity autocorrelation function determined in quasi-elasticlight-scattering experiments may be of single-exponentialform rather than the double-exponentialform predicted by the standard theory. These conclusions allow self-consistent interpretation of results obtained by using this technique to investigate diffusion in solutions of the small protein bovine pancreatic trypsin inhibitor.
Introduction Since Batchelor,l Felderhof? and Jones3presented their resulb pertaining to the hydrodynamic interaction between two spheres suspended in a fluid, it has become possible to build a fairly complete theory of the concentration dependence of the diffusion coefficient of spherical macromolecules."7 The theory is complete in the sense that it uses exact expressions for the particle-pair hydrodynamic interaction tensors and thus allows a result, correct to first order in the concentration, to be obtained, once the nature of any direct pair-interaction potential is specified. For globular proteins, the hard-core potential is minimal and immediatelyallows use of the results for the hydrodynamic interaction between impermeable particles. Electrostatic interactions are usually modeled in terms of a screened Coulombic potential. Any attractive forces between proteins (hydrogen bonding, hydrophobic interactions, or van der Waals forces) are inherently short-range. When the equilibrium thermodynamic properties of protein solutions are investigated, the effects of these forces are normally reckoned as association constants.8 When first-order concentration effects are being investigated, a dimerization constant is all that need be specified from the thermody-
namic point of view. In the Appendix to this paper, we demonstrate the equivalence between this approach and that used by Batchelorl and Lekkerkerker and co-worke r ~ , who ~J~ specify instead the depth and the width of a square-well potential. The experiments using the protein bovine pancreatic trypsin inhibitor (BPTI) which we report here are of some intrinsic interest, since they demonstrate that equipment which is now available in many laboratories may be used to determine accurately and quickly the Stokes radius of proteins as small as BPTI, provided they can be obtained at high enough concentration. The molecular weight of BPTI is only 6500, and the crystallographic datal1 indicate that it is a compact globular protein, roughly pear-shaped, with a length of 2.9 nm and a greatest lateral cross-sec-
* Address correspondence to this author a t the following address: Department of Physics, University of Auckland, Private Bag, Auckland. New Zealand.
(10)C. Van den Broeck, F. Lostak, and H. N. W. Lekkerkerker, J. Chem. Phys., 74, 2006 (1981). (11) R. Huber, D. Kukla, A. Riihlmann, 0. Epp, and H. Formanek, Naturwissenschaften,57, 389 (1970).
(1)G. K. Batchelor, J. Fluid Mech., 52, 245 (1972). (2) B. U. Felderhof, Physica A (Amsterdam),89, 373 (1977). (3) R. B. Jones, Physica A (Amsterdam), 92, 545, 557, 571 (1978). (4) G. K. Batchelor, J. Fluid Mech., 74,1 (1976). ( 5 ) B. U. Felderhof, J. Phys. A: Math. Gen., 11, 929 (1978). (6) R. B. Jones, Physica A (Amsterdam), 97, 113 (1979). (7) P. R. Wills, J. Phys. A: Math. Gen., in press. (8) P. R. Wills, L. W. Nichol, and R. J. Siezen, Biophys. Chem., 11, 71 (1980). (9) R. Finsy, A. Devriese, and H. Lekkerkerker, J. Chem. Soc., Faraday Trans. 2, 76, 767 (1980).
0022-3654/81/2085-3978$01.25/00 1981 American Chemical Society
Protein Diffuslon and Dimerization
tional diameter of 1.9 nm. Our result for the free-particle diffusion coefficient confirms that it is also in a compact conformation in solution. The existence of a dimer has been reported previously12J3and is also confirmed.
1
0.2- \
\
Theory The first-order concentration dependence of the Fick’s law diffusion coefficient, D, for a solution of macromolecules may be expressed by the equation D = Do(l
+ A@)
(1)
where Do is the free-particle diffusion coefficient, X is a constant, and 4 is the volume fraction of the solution which the solute occupies. For rigid spherical macromolecules of radius a, the volume fraction is related to the solute number concentration, c, by the equation
4 = 4aa3c/3
i
-0.4- I b 1
I
1
1
I
(2)
Following the convention adopted by Felderhof,6 the constant X may be separated into contributions due to distinguishable interaction effects =
XV
+ XA + + XO + AD
(3)
Each contribution Ax may be expressed as the integral of a particular function with respect to r, the center-to-center separation of two spheres. These integrals involve the radial distribution function g(r); results correct to first order in the concentration are obtained if g(r) is specified, in terms of the energy of interaction u(r) between the two spheres, by the zeroth-order term in its density expansion gob) = exp[-u(r)/kTl
(4)
Here k is Boltzmann’s constant and T is the absolute temperature. In the ensuing discussion we shall make use of theory which is valid only if u(r) is short-range,6 so it should be born in mind that the results which we obtain will be valid only in the macroscopic (zero wave vector) limit.7 Let us consider the different contributions to the constant A. The sum of the “virial” part Xv and the “Oseen” part Xo is given by eq 5 when “stick” boundary conditions Xv
+ Xo = - ( 3 / a 3 ) S0m [go(r)- l](r2- ar) dr
(5)
are assumed for the hydrodynamic interaction. The integrand in eq 5 is the sum of two terms, one quadratic in r and the other linear in r. Integration of the first (quadratic) term gives Xv and integration of the other term gives ho. For hard-core particles, go(r)is identically zero between r = 0 and r = 2a. Therefore, integration of both terms over this range gives a contribution of +2 to A. If any pair-interaction potential outside of this range is repulsive (u(r) > 0 for r > 2a), then a further positive contribution to X is obtained. The sum of the hydrodynamic “self-interaction”part, XA, and the “short-range”part, As, may be expressed as XA
+ Xs = ( l / a 3 ) S2am[ A ( r )+ S(r)]go(r)r2dr
(6)
where integration of the term involving A(r) gives XA and integration of the term involving S ( r ) gives As. The functions A(r) and S(r) are given in terms of elements Aij and Bij of the hydrodynamic pair-interaction tensors for spheres which have been defined in eq 4.2 of Batchelor4 (12)H.Kraut, W. Korbel, W. Scholtan, and F. Schultz, Hoppe-Seyler’s Z. Physiol. Chern., 321, 90 (1960). (13)F. A. Anderer and S.HOrnle, 2.Naturforsch. E , 20,457(1965).
XA
+ As
=
L1m [A(r)+ S(r)]r2dr + a3
2a
as a function of r in Figure 1 by using the values of the tensor elements Aij and BV supplied in Tables 1 and 2 of B a t ~ h e l o rthese , ~ data being more accurate than values obtained by using the available series Using numerical integration, Batchelor has found the first integral in eq 10 to make a contribution of -1.55 to A. This result has recently been confirmed experimentally by Kops-Werkhoven and Fijnaut.14 Inspection of eq 4 and 10 and Figure 1 reveals that the second term in eq 10 will give rise to an additional positive contribution to X when the potential of interaction is repulsive (u(r) > 0) in the range r > 2a. We are thus able to conclude that the diffusion coefficient of impenetrable spherical macromolecules will increase with increasing concentration, unless, in addition to the usual Coulombic repulsion, there is an attractive potential of interaction between the molecules at some separations r > 2a. This conclusion was recently reached independently by Van den Broeck, Lostak, and LekkerkerkerlO through considerations similar to those which we have presented here. The dimerization of two spherical macromolecules by means of short-range attractive forces (hydrogen bonding, hydrophobic interactions, or van der Waals forces) may be modeled in terms of a narrow square-well potential of energy -E for separations in the range 2a < r I2a + 6. The contributions to X due to the various potentials, ~~~
~~~~
~
~
~~
(14)M.M.Kops-Werkhoven and H. M. Fijnaut, J. Chern. Phys., 74, 1618 (1981).
3980
The Journal of Physical Chemistry, Vol. 85, No. 26, 1981
hard-core (hc),electrostatic (el), and square-well (sw), may be formally separated to give eq 11. In the case of the D = Do[l
+ (Ahc + A")#] + DoXsw#
(11)
contributions due to the virial and Oseen parts, the separation is achieved by integrating the right-hand side of eq 5 over the separate ranges 0 Ir I 2a (hc), 2a < r I 2a + 5 (sw), and 2a + 4 < r < 00 (el). In the case of the contributions due to the self-interaction and short-range parts, the contributions to Xhc are given by the first integral on the right-hand side of eq 10, and the contribution to Xsw and Xel may be found by integrating the second term over the ranges 2a < r 5 2a + 5 and 2a + 5 < r < m, respectively. The contribution due to the dipole part is included in Aha. Now, it is evident that, as 5 becomes smaller, the first term in eq 11 approaches the value D,, the diffusion coefficient of a solution (of volume fraction 6) of monomer molecules which have no tendency to dimerize but are in other ways identical with those under discussion. It is thus desirable to relate the second term in eq 11 to the free-particle diffusion coefficient Dd of a dimer of spheres and calculate the diffusion coefficient as a weighted sum of contributions due to monomers and dimers. The dimer diffusion coefficient is given exactly in terms of the elemenb of the mobility tensors governing the hydrodynamic interaction between two spheres at separation r = 2a by the equation4 Dd = D0[A11(2a)+ 2B11(2~)+ A12(2~)+2B12(2~)]/6 (12a) = 0.719D0
(1%) If we assume that the width 5 of the square-well potential is very small compared with 2a, then we may calculate XBw by evaluating the integrals in eq 5 and 6 at r = 2a. We thus obtain Xaw = 24[exp(E/kT) - l](Dd/Do - l ) t / a
(13)
and the change in the diffusion coefficient which dimerization induces is found, by using eq 2 and 11-13, to be DoXsw#= 32aa2,$[exp(E/kT)- 1](Dd- Do)C (14) Provided E is much greater than thermal energy (kT),we may combine eq 11 and 14 and the expression for the dimerization constant K obtained in the Appendix (eq A6), and so obtain D = D,
+ 4KC(Dd - Do)
(15)
where we have written D, for the corrected diffusion coefficient of a solution of spherical, charged monomers of concentration c (the first term on the right-hand side of eq 11). The reduction to eq 15 from eq 14 solves the problemg of specifying a single measurable parameter to describe the effect of short-range attractive forces on the diffusion coefficient. Equation 15 is the standard expression, correct to first order in the solute concentration, for the z-average diffusion coefficient D, of an equilibrium mixture of monomers and dimers, which is given exactly by Dz= [Dm(l- xd) + 2XdDd]/(1 + xd) (16) where xd is the weight fraction of the solute present as dimers. The concentration-dependent diffusion coefficient D governs the Fick's law expression for the flow of solute dc/dt = DV2c
(17)
In a rigorous approach to the description of diffusion in
Wills and Georgalis
a system containing two solute components (in our case monomers and dimers), one begins with a pair of coupled diffusion equations in the (number) concentrations n, of both species dn,/dt
= DmmVanm + DmdV2nd
dnd/dt = DdmV2nm + Dddv2nd
(184
(18b)
If we were to use such an approach, we could avail ourselves of the results of Jones: who calculated the four diffusion coefficients by using exact expressions for the elements of the mobility tensors describing the hydrodynamic interaction between (spherical) molecules of the different species. For a monomer-dimer equilibrium system, results correct to first order in the total solute concentration c are obtained by neglecting terms which imply interactions within a group of more than two of the basic spherical monomer units. Under these circumstances, the two diffusion equations uncouple to give dn,/dt
= D,V2n,
dnd/dt = DdV2nd (19b) where the meanings of D, and Dd are the same as those already ascribed to them. For a monomer-dimer equilibrium system we can use the formulas, correct to first order in c (see eq A 1 and A7) an,/ac = 1- KC (20a) andlac = 2Kc to derive eq 17 from eq 19a and 19b, and we find that D is indeed the z-average diffusion coefficient. This demonstrates that the flow of matter down a concentration gradient is governed by the z-average diffusion coefficient for a monomer-dimer equilibrium system in contrast to the case when the two components are chemically inert with respect to one another and the weight-average diffusion coefficient is the relevant parameter. Quasi-elastic light scattering may be used to determine the z-average diffusion coefficient of a mixture of monomers and dimers irrespective of whether the two components are in equilibrium with one another. It is the zaverage rather than the weight-average diffusion coefficient which is determined simply because of the increased scattering power of the dimer. (It is demonstrated in the Appendix that this weighting of the scattering function has been preserved in our analysis.) This is known from theory15 and has been demonstrated experimentally.16 However, the standard theory of quasi-elastic light scattering makes a further prediction which is of interest to us here. The theory treats dimers as intact point scatterers which undergo their own independent Brownian motion, and thus the intensity autocorrelation function is expected to be of double-exponentialrather than single-exponential form.15 In contrast, our own analysis does not assume the integrity of the dimer. Rather, by positing no activation barrier for dimer formation, we have implicitly assumed that the rate of association depends only on the frequency of collision between two individual spheres. This raises the practical question as to whether the rate of dissociation may be so fast that dimers will on average separate into two monomers in a fraction of the coherence time of the scattered field and thus fail to manifest themselves as a separate component. (15) B.J. Berm and R. Pecora, "Dynamic Light Scattering", Wiley, New York, 1976. (16) J. D. Harvey, R. Geddes, and P. R. Wills, Biopolymers, 18,2249 (1979).
The Journal of Physical Chemistty, Vol. 85, No. 26, 1981 3901
Protein Diffusion and Dimerization
Smoluchowski's expression17for the rate constant kl of a diffusion-controlled bimolecular reaction may be taken as an upper limit for the rate of the forward reaction. In the case of the dimerization of two spheres of radius a, we have kl = 16mD0
(21)
The equilibrium constant K is the ratio of the rate constant kl for the forward reaction (association) to the rate constant kl for the reverse reaction (dissociation) K = kl/k-l (22) In a quasi-elastic light-scattering experiment, one determines the rate constant r for the decay of the scatteredfield correlation function, and this quantity (in the case of homodyne detection) is related to the diffusion coefficient by the equation
r = q2D
(23)
where q is the magnitude of the scattering vector. Combining eq 21-23 and making the approximation D = Do, we obtain k-l/r = 1 6 r a / ( q 2 K )
(24)
We wish to know under what circumstances the ratio k-l/r may be much greater than unity, and we shall take as an example the experiments conducted by using the protein BPTI which we report in this paper. We see from eq 24 that the desired result may be obtained if K is chosen to be arbitrarily small, so we shall choose a relatively large value by stipulating that, in a 0.1% (volume fraction) solution of the protein, the weight fraction of solute in the dimer form, xd, be 0.2. The radius of the BPTI monomer is 1.4 nm, so, in order to obtain a solution with the specified composition, the dimerization constant would have to be cm3. When q is set equal to 2.3 X as large as 1.1 X IO5cm-l, the value of the ratio k-l/F is found to be 1.2 X IO2, demonstrating that, in realistic cases, the half-life of a protein dimer may be orders of magnitude shorter than the coherence time of the scattered field observed in a typical quasi-elasticlighbscattering experiment. When this is the case, the standard theory which indicates that the z-average diffusion coefficient of the separate species (eq 19) may be determined by using quasi-elastic light scattering will only apply at inaccessibly short times. However, our own analysis, in which the definition of dimers as a separate chemical component is quite arbitrary, is applicable at much longer times. When the rate of dimer dissociation is sufficiently rapid, the quasi-elastic scattering of light is determined by fluctuations in the total solute concentration (eq 17), and, on the experimentally accessible time scale, local equilibrium of the monomer-dimer reaction may be presumed to prevail (eq 20). The intensity autocorrelation function is expected to be of single-exponential form, and its decay rate is expected still to correspond to the z-average diffusion coefficient. Our conclusion is at variance with the work of Berne and Giniger,ls who have predicted that in the fast exchange limit it is the weight-average rather than the z-average diffusion coefficient which determines the half-width of the quasi-elasticlight-scatteringspectrum. We believe that Berne and Giniger18are incorrect. In eq 2.2 of their paper, the time-dependent dielectric fluctuation is expressed as the sum of independent contributions from fluctuations in the monomer and dimer concentrations. However, when (17) M. v. Smoluchowski, Ann. Physik (Leipzg), 48, 1103 (1915). (18) B. J. Berne and R. Giniger, Biopolymers, 12, 1161 (1973).
the rate of exchange between monomer and dimer is sufficiently fast, fluctuations in the concentrations of the two species are not independent on the experimental time scale. Under these circumstances, eq 2.2 of Berne and GinigeP fails to take into account the rapidly established equilibrium and therefore requires modification down lines suggested by our eq 20a and 20b. The result derived by Berne and GinigeP coincides with the better-known result that the local flow of independent components down a concentration gradient is governed by the weight-average diffusion coefficient.
Experimental Section Lyophilized bovine pancreatic trypsin inhibitor (Aprotinin, research grade) was obtained from Serva Feinbiochemica (Heidelberg) and used without further purification. The protein was dissolved in 0.02 M phosphate buffer (pH 7.0) with 0.35 M NaCl and filtered through Nucleopore membranes of 0.2-pm pore size. The optical arrangement and the instrumental specificiation for the quasi-elastic light-scattering experiments have been described e1~ewhere.l~The 514.5-nm laser line wm used, and the light scattered at 90' was detected. All experiments were conducted at a temperature of 20.0 f 0.1 OC. The refractive index of protein solutions was determined with an Abbe refractometer. A minimum of 100 intensity autocorrelation functions was collected at each protein concentration and stored on disk. A channel delay time of 1ps was employed, and each correlation function was collected over a period of 10 s. When the total number of photon counts exceeded the mean for a given protein sample by more than 1% , it was rejected. This rarely occurred but allowed the effect of residual contaminating dust to be minimized. The correlation functions were individually normalized by using the value of the background calculated from the product of the total clipped and unclipped counts. (The mean deviation of the background, calculated in this way, from the approximate background measured by using the long-time delay channels of the Malvern correlator, was of the order of 0.5% for each protein sample.) The mean and the variance of each point in the normalized correlation function were calculated by using the 100 or so sample functions collected at each protein concentration. The variance was reasonably constant across each correlation function but was, in all cases, found to be significantly higher (by a factor of 2) for the first few data points. These measured variances were used to calculate weighting factors for statistical fitting during data analysis, rather than relying on the assumption that the variance in the correlation function was constant at all points. Diffusion coefficients were determined by using the method of cumulants,2° which involves polynomial fitting to the logarithm of the normalized correlation function. This method of data analysis has intrinsic difficulties because the fitted parameters are not coefficientsof terms constituting an orthogonal base set for the data. We therefore adopted what we considered to be an optimal strategy for obtaining reliable estimates of the first two cumulants. In succession, we fitted polynomials of increasing order to the first 8, 16, 24, and 32 data points and then to the whole data set. The F-test criterion was used to decide in each case how many polynomial terms gave the best fit to the data without unduly constraining it. In ~~
(19) W. Doster, B. Hess, D. Watters, and A. Maelicke, FEBS Lett., 113, 312 (1980). (20) P. N. Pusey, D. E. Koppel, D. W. Schaefer, R. D. Camerini-Otero, and S. H. Koenig, Biochemistry, 13, 952 (1974).
The Journal of Physical Chemistry, Vol. 85, No. 26, 1981
3982
lot i
Wills and Georgalis
1 I
I
0.01
0.02
I
0.03
@ Figure 2. Diffusion coefficient D of solutions of hydrodynamic volume fraction 4 of BPTI at pH 7 and 20 “C in the presence of 0.35 M salt. Error bars indicate the range of values obtained by using different estimates of the first cumubnt of the scattered-intensity autocorrelation function. The diffusion coefficient D, of a dimer of BPTI, Calculated from the intercept of the least-squares fit to the data using eq 12b, is indicated.
all cases it was found that a first-order polynomial reasonably represented the first 8 or 16 data points, a second-order polynomial the first 24 or 32 data points, and a third-order polynomial all 68 data points. This strategy gave five separate estimates of the first cumulant and three estimates of the second cumulant, at each protein concentration.
Results and Discussion The concentration dependence of the diffusion coefficient of bovine pancreatic trypsin inhibitor is shown in Figure 2. The error bars represent the range of values obtained for the diffusion coefficient at each protein concentration using different estimates of the first cumulant of the temporal autocorrelation function of the scattered intensity. Linear least-squares fitting to the data in Figure 2 allowed the free-particle diffusion coefficient Do and the constant X defined in eq 1 to be estimated. A value of (15.4 f 0.2) X lo-’ cm2s-l was obtained for D020,w, the diffusion coefficient of the protein in water at 20 “C, when correction was made for the viscosity 77 of the solvent (1.046 X P). Using the Stokes-Einstein relationship
Do = kT/(67r7a) (25) we obtain a radius of 1.39 nm for the BPTI monomer. This value was used in eq 2 to calculate the hydrodynamic volume fraction21of solutions of BPTI. The Stokes radius should be compared with the minimum compact-sphere radius aminof 1.23 nm calculated from the molecular weight M and an estimate (0.73 cm3 g-l) of the partial specific volume d of the protein using the equation2’ urnin = [3Md/(4aL)]1/3 (26) where L is Avogadro’s number. The fact that the Stokes radius of the BPTI monomer is very close to its minimum value and smaller than the maximum dimension seen in the crystal structure1’ indicates that in solution the molecule is still in a tightly folded, globular conformation. It also justifies the use of an impenetrable sphere model in further calculations. A value of -80 f 5 was obtained for the constant A. (A better estimate of the gradient dD/dc at zero concentration (21)A. P.Minton, Biophys. Chem., 12, 271 (1980). (22)C.Tanford, “Physical Chemistry of Macromolecules”,Wiley, New York, 1961.
could not, according to the F-test criterion, be obtained by fitting a quadratic or higher-order polynomial to the data in Figure 2.) The fact that this value is below the hard-sphere value of 1.45 calculated by Batchelor4c o n f i i s that under the chosen experimental conditions there is an attractive potential between the BPTI molecules. We shall assume that this potential can be modeled in terms of the previously detected dimerization reaction.l2J3 In order to calculate an equilibrium constant for the reaction by using eq 11 and 15, it is necessary to estimate Xel, the contribution to A due to electrostatic interactions between the molecules. Following previous ~ o r kwe, shall ~ ~ use ~ ~the mean-sphere approximation and assume that each monomer molecule carries a net charge which is distributed in a spherically symmetric fashion on its surface. We may then use the expression for a screened Coulombic potential suggested by Verwey and OverbeekZ4 u(r) = #&z2e exp[-K(r - 2a)]/r
2a
