9556
J . Phys. Chem. 1991, 95, 9556-9561
overlapping and separate Previously, a modified Flory-Krigbaum theory has been applied successfully to represent the free energy change due to mixing solvent and surfactant tail molecules anchored to the micellar interface., This theory may also be used to calculate the pair internal energy of the tail regions. The pair internal energy due to mixing solvent and surfactant tails for two nonoverlapping micelles is3' u(r) = (2xkQku,/ur)(l
- Pku,) d V
(A21
where x is the solvent-tail Flory interaction parameter, pk is the segment probability distribution function, us is the volume of a tail segment, and uI is the volume fraction of solvent molecule. The integral is performed over the micelle tail region. When micelles overlap, a tail segment k from one micelle, a tail segment 1 from the other micelle, and solvent mix in the overlap region. The pair internal energy for overlapping micelles is u(r) = l(xkT(pkus
+ Pps)/u/)(l - pkus - plus) d V
(A3)
The total potential between two micelles is obtained by taking the difference between the above two equations. Thus u(r) = (-2XkTu,Z/UI)JPnP/
dV
('44)
This expression can be simplified if the following observations are made. The segment probability distribution function, p, has been obtained for flat35444and curved35surfaces. It is a function of tail segment length, r, and A,, but A, can be factored out so that p = p'/A,, where p' is independent of A,. The thin region of overlap (about 2.5 A thick in decane42) can be approximated as flat because the micelle radius varies from about 30 to 100 A. Thus d V = A, dr, where A, is the area of contact. From geometry, A, = 17[(u/2 A)z - ( ~ / 2 ) ~ where ], A is the distance of overlap
+
(44) Hesselink, F. Th.J . Phys. Chem. 1969,73, 3488.
between the two spherical micelles and u is the hard-core contact distance between two micelles. After including these approximations, eq A4 can be written as u(r) = -2T[(0/2
+ A)Z - ( U / 2 ) 2 ] X k ~ ~ , 2 / ( A s Z u / ) ~ P dr' k ~ ' / (A51
Equation A5 is similar to a square well potential, except the attractive well is not exactly constant in the region of overlap. To use the physical basis of eq A5 along with the known square well depths from a previous SANS study,22we fit eq 5 to the data to obtain the following expression: cm5)8 + 1.696
t / k T = (3.781 x
(A6)
where 6=
+ AI2 - (./2)21x/(A,z~/)
('47)
The value of A = 2.5 A at r = u is from SANS studies of AOT reverse micelles.22~28 As is calculated by satisfying the internal equilibrium criterion of the molecular thermodynamic model. The Flory interaction parameter x can be related to the solvent and surfactant tail solubility parameters if the geometric mean assumption is applied to the unlike pair parameters kTx =
U/(6,
- 6,)2
(A8)
where 6, and 6, are the solubility parameters of the oil and surfactant tail, respectively. The correlation, shown in Figure 10, will be used to calculate w with eq A l . The linearity of the data suggests that the integral in eq A5 is relatively constant over this region. It is also worth noting that t is more linear with respect to 0 than solvent density, a correlating variable that has been suggested p r e v i o ~ s l y .It~ is~ ~convenient ~ that this correlation is accurate for large values o f t where w is on the order of w,. Here, the calculation of phase behavior is sensitive to t. Registry No. AOT, 577-1 1-7; NaC1, 7647-14-5; propane, 74-98-6.
Multiple-Site Tltration Curves of 'Proteins: An Analysis of Exact and Approximate Methods for Their Calculation Donald Bashfordf and Martin Karplus* Department of Chemistry, Haruard University, Cambridge, Massachusetts 021 38 (Receiued: April 17, 1991)
The exact calculation of a multisite titration curve, given the intrinsic pK,s of the sites and the site-site interaction constants, grows exponentially in the number of sites and becomes prohibitive for molecules having more than 15 or 20 sites. A commonly used approximation method, introduced by Tanford and Roxby (Biochemistry 1972, 11, 2192) gives large computational savings and is routinely employed for proteins. This method is shown to be a mean field approximation to the multiple-site problem. It gives significant errors when strongly interacting sites titrate at similar pH values. An alternative method, the reduced-site approximation, is introduced. Both approximations are tested against the exact method on a set of randomly generated hypothetical molecules. The reduced-site approximation is found to have much greater accuracy throughout most of the titration curve. I t is recommended that the reduced site approximation be used when computationally feasible.
Introduction A molecule with N titratable sites has 2N possible protonation states which should be treated as such in calculations of hydrogen ion dependent properties of the system (e.g., evaluation of the net charge as a function of pH). Since proteins typically contain several tens to hundreds titratable sites, calculations that explicitly include all protonation states may be impractical, even with current computers. Before the detailed three-dimensional structures of 'Present address: Department of Molecular Biology, The Scripps Research Institute, La Jolla, CA 92037.
0022-3654/91/2095-9556$02.50/0
proteins became known, this problem was dealt with through approximations that were justified, in part, by the uncertainty of the basic model. Groups of the same chemical type were assumed to have identical intrinsic pK, values,'-3 and the electrostatic interactions between groups were treated by simplified procedures. An early example of this type of approach is the model proposed (1) Linderstrom-Lang, K. Compt.-rend Lab. Carlsberg 1924, 15, 1-29.
(2) Ljnderstrom-Lang, K.;Nielsen, S. In Electrophoresis; Bier, M., Ed.; Academic: New York, 1959; pp 35-89. (3) Tanford, C.; Kirkwood, J. G. J . Am. Chem. Soc. 1957,79,5333-5339.
0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 9557
Titration Curves of Proteins by Linderstrom-Lang, in which the titrating charge is distributed uniformly over the surface of a sphere.'S2 Tanford and Kirkwood used point charges arranged symmetrically inside a ~ p h e r e .The ~ existence of high-resolution structures for proteins, the availability of spectroscopic methods to follow the titration behavior of individual group^,^.^ and the importance of the protonation state of individual residues in protein functions, such as enzyme catalysis,6-' have created a need for theoretical treatments that explicitly take account of site dependencies of the intrinsic titration properties and the interactions of titrating groups. In an attempt to include structural information in the calculation of the effect of sitesite electrostatic interactions on the titration curve of proteins, Tanford and Roxby* introduced the intuitively reasonable approximation that the average influence of site k on site i is proportional to the average charge of site k ; that is, the pK of site i , pKi, was assumed to have the simple form
where A is the fully deprotonated form of the molecule, H represents the protons, v(x) is the number of protons added in bringing the molecule to state x, and AH,(x) is the molecule in protonation state x. It is assumed that the free energy for adding a proton to each particular site consists of two contributions; the first is due to the intrinsic properties of the site and independent of the protonation state of other sites, and the second is due to interactions with other sites. In the following, we describe the interactions as electrostatic in nature, but the same formalism applies to any system with pairwise additive interactions. Following Tanford and K i r k ~ o o d we , ~ introduce the intrinsic proton association standard free energy of site i, AGinlr,irfor the first contribution. It is defined as the standard association free energy that site i would have if all other titrating sites were electrically neutral; fixed charges or partial charges are included in AGinlr,i.The intrinsic pK, of site i, Kintr,i.is then
with
The second contribution can be written in terms of a matrix whose elements, Wij,are the electrostatic interactions between unit charges placed at sites i and j . The diagonal elements of W are zero since electrostatic self-energies are taken to be part of AGinw We then have, for the standard free energy of the reaction corresponding to scheme I,
Here 0, is the probability that site i is protonated, q: is the charge of site i in the unprotonated state, and Wik is the interaction free energy between unit charges at sites i and k. The intrinsic pK of site i, pKintr,i,is the pKa the site would have if the charges of all other titrating sites were set to This approximation is similar in spirit to Cannan's9 numerical finding that for a set of sites with identical intrinsic pKas and identical sitesite coupling constants, W,the titration curve is offset from the curve of a system with W = 0 by an amount proportional to WZ,where 2 is the average total charge of the molecule due to the titrating groups. Cannan's approximation was based on Linderstrom-Lang's' proof that in the limit of a large number of titrating sites, such a relation holds in the linear region near the midpoint of the titration curve. Equations la and 1 b can be solved by iteration to self-consistency. Since the size of the calculation grows only like M , the Tanford-Roxby approximation is practical for application to most proteins, and its use has become widespread (see ref 10 for a review). Because eqs la and Ib provide an approximate solution to the multistate titration problem, it is important to determine the limitations of the method and to consider alternative procedures. This paper shows that Tanford and Roxby's approximation can be derived from a mean field approximation. A series of numerical comparisons between this approximation and the exact 2N calculation are presented for model systems of up to 10 titratable sites. It is shown that the mean field approximation often works well but that there are cases in which significant errors arise. An alternative approximation scheme, the reduced-site approximation, is presented and evaluated. The Exact Theory and the Mean Field Approximation The protonation state of a molecule with N titratable sites can be represented by an N-dimensional vector, x, whose elements, x,, take on values of 0 or 1 according to whether the corresponding site is unprotonated or protonated, respectively. The association reaction to form a molecule in the specific protonation state x is A + v(x)H AH,(x) (1)
-
In terms of pH (assuming that the proton activities are equal to the concentrations) the equilibrium between A and AH,(x) can be written [AH,(x)]/ [A] = ~ - B A G ( X ) - ~ X ) ~ . ~ O ~ P H
where p = l/kT. The fractional protonation state of site i is xie-BAG(x)-u(x)2.303pH
1x1
where (x) indicates the summations over all 2N possible states of the protonation vector, x. Thus one may follow the titration behavior of individual sites or, by summing the 8 , that of the molecule as a whole. We now show how the summations over all protonation states can be avoided by a mean field approximation. The exponential in eq 4 can be interpreted as -BAG(AH,(x)-A;pH), where AG(AH,(x)-A;pH) = AG(x)
+ 2.303j7'u(x)pH
(6) is the standard free energy difference between the species AH,(x) and A in an environment at a given pH. Regarding x as the set of variables specifying the microstates of a system with the free energy AG(AH,(x)-A;pH), we can construct a free energy functional C = min C[AG(AH,(x)-A;pH)
P(X)
1x1
+ j3-I
In p(x)]p(x) (7)
where p(x) is the distribution function on x and the form of p(x) that minimizes G is the equilibrium distribution function of the system. For convenience of notation, we introduce the quantity, bj, defined by bj
(4) Imoto, T.; Johnson, L. N.; North, A. C. T.; Phillips, D. C.; Rupley, J. A. In The Enzymes; Boyers, P. D., Ed.; Academic: New York, 1972; pp 665-868. (5) Groenveld, C. M.; Ouwerling, M . C.; Erkelens, C.; Canters, G. W. J . Mol. Biol. 1988, 200, 189-199. (6) Fersht, A. Enzyme Structure and Mechanism; W. H. Freeman & Co.: New York, 1985. (7) Knowles, J . R. CRC Crit. Reo. Biochem. 1976, 4 , 165-173. (8) Tanford, C.; Roxby, R. Biochemistry 1972, 11, 2192-2198. (9) Cannan. R. K. Chem. Reu. 1942, 30, 395-412. (IO) Matthew, J. B.; Gurd, F. R. N . Merhods Enzymol. 1986, 130, 41 3-436.
(4)
= -2.303j71(pKint,j - PH)
+ cW,kqko k
(8)
With eqs 2, 3, 6 , and 8, eq 7 becomes
Up to this point, no approximations have been made and eq 9 together with a normalization condition on p(x) leads to a Boltzmann distribution and the exact protonation formula, eq 5 . The mean field (MF) approximation is introduced by assuming that p(x) has the uncorrelated functional form
9558 The Journal of Physical Chemistry, Vol. 95, No. 23, I991
Since the xi take on only the values 0 or 1, one can write (without loss of generality) P(xi) +
1
[ei
-
Bi
Bashford and Karplus tonation state of each site. The standard free energy for adding a proton to site i will be a maximum when all other sites are protonated since this results in the strongest possible electrostatic interaction opposing protonation; Le., N
AGmax,i = AGin1r.i +
forxi = 0 forxi = 1
C Wj(4j” + 1
e o )
I’ I
(17)
Similarly, the minimum possible free energy occurs when all other sites are deprotonated; i.e.,
where
N A‘min,i
Therefore, p(x) can be expressed entirely in terms of a vector of 8, values, 8. The minimization expression, eq 7, can now be written
where we have used the fact that for a distribution with no correlations, X Y k = 8@k. The functional minimization on p ( x ) becomes a minimization on the set of variables, for which one can use the extremum condition
@,,
From eq 1 I , we write (In p i ( x i ) ) = ( 1
- 8) In (1
- 8)
+ 8 In @
(15)
With this result, eq 14 becomes
which, after substituting eq 8 for 6 , is equivalent to Tanford and Roxby’s formulas, eqs 1 a and 1b. As we show below, the M F approximation works well in many cases, but can lead to significant errors when strongly coupled sites titrate at similar pH values. This is to be expected from the above derivation since it is in such cases that one expects to find significant correlations between the protonation states of titrating sites, and the M F approximation explicitly neglects such correlations. For example, a protein may contain two sites close together so that, in a particular pH range, it is energetically favorable to have one site or the other protonated, but because of electrostatic repulsion it is unfavorable to have both sites protonated. This is the analogue of the simple case of a symmetric dibasic acid (see below). This would give rise to an anticorrelation between the two sites. Since cases like this may be quite common in proteins it is desirable to develop alternative approximation methods for multiple site titration curves that are accurate for strongly coupled cases while remaining computationally feasible for large numbers of titrating sites.
The Reduced-Site Approximation If the pKintrsof the sites are spread over the range of values typical of the chemical groups in proteins (4-12) then for any particular pH, most of the 2Npossible protonation states will make no significant contribution to the summations in eq 5;e.g., if the pH is far above that a t which a particular site titrates, that site will almost always be unprotonated, and states with x having that site protonated will be negligible in the summations. Therefore, the site can be assumed to be unprotonated for this pH value, and the number of possible protonation states is reduced to 2N-1. Similarly, if the pH is far below that at which some other site titrates, that site can be assumed to be protonated. For each site that is fixed in this way the computational effort is reduced by a factor of 2. In applying this conceptionally simple but computationally powerful approximation, the first step is to determine which sites can be regarded as fixed. Comparing the pH to the various pKintri is inadequate since we wish to allow for the possibility that strong couplings may produce large shifts in the pK,s. Instead, we first calculate the maximum and minimum possible fractional pro-
= A‘intr,
+ CWip,O j- I
(18)
At a given pH, the maximum (minimum) possible protonation state of site i is calculated from the minimum (maximum) protonation free energy by
- e-BAG,i.(,.,)r2.303~H omax(min).i - 1 + e-BAGmi.(mx)~2.303PH
(19)
Given @max,i and 8min,ifor each site i, we assume that a site is protonated for Omin > 0.95 and unprotonated for Omax < 0.05.The limits can be adjusted according to the requirements of accuracy and computational economy. It will be seen that the actual errors introduced will usually be less than the chosen limits would imply, since the maximum and minimum calculations are based on the unusual conditions of fully protonated or fully unprotonated molecules. The largest errors occur at the extremes of the titration curve where the limiting protonation states are actually realized. Often these are of little interest because they are outside the stability range of the protein. The fixed and variable sites having been identified for a given pH, it is necessary to adjust the intrinsic free energies of the variable sites to reflect the influence of the fixed charges; that is, we write
where the summation runs only over the fixed sites, j, and f is zero or one according to whether site j is fixed as protonatedor unprotonated. Equations 3 and 5 are now applied to the reduced set of variable sites to determine their titration behavior. At each pH, there can be a different set of fixed and variable sites.
Numerical Tests In this section, the results of several numerical comparisons between the mean field (MF) approximation, the reduced-site (RS) approximation, and the exact expressions are presented. First, M F and exact calculations for a two-site molecule are analyzed to demonstrate the limitations of the M F model. Next MF, RS, and exact calculations for a series of hypothetical 10-site molecules are presented. For a molecule with two cationic sites coupled by an interaction of strength, W, the exact formula, eq 5, becomes 1OPKintr,l-PH + 1O P K ~ . ~ , . I + P K ~ , , ~ - - Z P H - ~ . ~ ~ ~ B W o1 = 1 + 1OPKintr,l-PH + 1OPKintr.2-PH + 1O P K ~ . ~ , . I + P K ~ . , , ~ - Z P H ~ . ~ ~ ~ B W (214 “2
-
1
+ 1OPKimt-PH + 1OPKint.2-PH + 1O P K ~ , I , . I + P K ~ , ~ , , ~ ~ P H ~ . ~ ~ ~ ~ W
(2 1b) Figure 1 compares the results of the M F approximation with the exact expressions for the case of two identical sites. When the site-site coupling is less than one pK unit the approximation is quite good; but, at stronger couplings, the errors become significant. Couplings with strengths on the order of 1 pK unit often occur in proteins. For example, two sites separated by 5 A with an effective dielectric constant of 50 have an electrostatic coupling of approximately 1 pK unit. In addition, the self-consistent iteration scheme8 used to solve eqs l a and l b does not converge well for strong couplings. This can be alleviated somewhat by introducing an improved convergence scheme in which the next
The Journal of Physical Chemistry. Vol. 95, No. 23, 1991 9559
Titration Curves of Proteins
\
I
0 2
I
I
I
1
I
I
3
4
5
8
7
8
8
PH
PH
-T
2
2
3
4
5
6
I-I-
7
1
8
9
2,
2
I
I
I
I
I
3
4
5
8
7
PH
8
9
-
+ O2 vs pH for two identical sites having pK,,,
= 7.0. Solid line, exact calculation. Dashed line, MF approximation. (a, top) Sitesite coupling, W = 2.5 pK units. (b, middle) Sitmitecoupling, W = 1.5 pK units. (c. bottom) Site-site coupling, W = 0.8 pK units. Figure 1. 8,
3
4
5
6
7
8
9
Figure 2. 8, + O2 vs pH for two sites having different pKinwand site-site coupling, W = 2.5 pK units. Solid line, exact calculation. Dashed lines, M F approximation. (a, top) pKinuJ= 6.7, pKintr,2= 7.0. (b, bottom) pKintrj = 6.0, pKinw,2= 7.0.
r.
I
I
0
2
PH
PH
guess for 0 is the average of the B’s calculated in the two previous iterations. In Figure 2, the M F approximation is examined for molecules with two nonidentical (different pKintr)sites. If the sites titrate at widely separated pH values, the errors are small even though the coupling is strong. To test the M F and reduced site (RS)approximations, random sets of pKint,sand W matrices have been generated for a molecule with IO titratable sites. The pKi,,+ are chosen from a uniform probability distribution between 4.0 and 10.0. The sites are taken as cationic or anionic (qo = 0 or -1) with equal probability. Values
of W,are chosen from a uniform probability distribution between 0.5 and 1.5 pK units for “strong coupling“ conditions, and between 0.1 and 0.8 pK units for a “moderate coupling” test. Site-by-site titration curves have been calculated by the exact formulas, the M F approximation, and the RS approximation for 25 strong coupling and 25 moderate coupling cases. The average and maximum errors are shown in the histograms in Figure 3. For the strong coupling cases,the M F approximation can give significant errors throughout the titration curve of the individual site. The errors of the RS approximation are more than an order of magnitude smaller, except a t the end points where the M F approximation works best. Even a t the end points, the RS errors do not exceed 0.05, the selected protonation threshold. For the moderate coupling cases, the MF approximation gives errors smaller by about a factor of 4 than for strong coupling. The errors of the RS approximation are only slightly altered by the coupling strength. It is expected that the M F approximation will give the largest errors for sites that are strongly coupled to other sites that titrate in the same pH range. To test this idea, we define a quantity, A, as a measure of coupling and similarity of titration range, N
W:;
where the pKaPpof a site is defined as the pH at which the site is half protonated, and W,,is given in pK units. Figure 4 shows the distribution of errors at the titration midpoint of each site as a function of A. The titration behavior near the midpoint is important in the definition of quantities such as apparent pK,. The errors of the M F approximation increase with A while the
Bashford and Karplus
9560 The Journal of Physical Chemistry, Vol. 95, No. 23, 195'I 0.I
c
L
e .05 v)
5 0.4
.6
.8
1.0
.8
4
1.2
1.4
1.6
1.8 2.0
1.4
1.6
2.2
2.4
2.6
2.8
L
.05 v)
E
h
.05 ,048 .04
.01
.I
.04
1.0
1.2
1.8
A
Figure 4. Root-mean-squareerrors of MF and RS approximations vs A (see text) in a series of calculations on 25 randomly generated, hypothetical, IO-site molecules. Histogram bars represent sets of sites with X between 0.4 and 0.6, etc. Dark bars, RS approximation. Light bars, MF approximation. (a, top) Strong coupling case. (b, bottom) Moderate 2
.3
.5
.4
.6
.7
.e
S '.e9
coupling case.
9
.ll
e W
5 .os L
W
.01
.I
.2
.3
.4
.5
.s
.7
d
.9
.39
Number of variable sites
Figure 5. Root-mean-squareerror vs number of variable sites in the RS
h
approximation.
05 ,049
04 L
.03 W
.02 .Ol .Ol
.I
.2
.3
.4
e
.5
.6
.7
.e
.9
.99
Figure 3. Average and maximum errors of M F and RS approximations in a series of calculations on 25 randomly generated, hypothetical, 10-site
Number of variable sites
molecules. Titration curve points were calculated at pH increments of
0.2. Histogram bars represent sets of individual site titration points taken over all IO sites in all 25 molecules with exact B values between 0.01 and 0.1, between 0.1 and 0.2, etc. Errors are defined as the difference between the exact B values and the MF or RS B values. Dark bars, rootmean-square errors. Light bars, maximum absolute value of errors. (a,
top) TR approximation, strong coupling case. (b, upper middle) RS approximation, strong coupling case. (c, lower middle) MF approximation, moderate coupling case. (d, bottom) RS approximation, moderate coupling case. Note the difference in scale between the MF and RS plots.
RS errors remain low throughout the whole range of X values. For the RS approximation, one expects the errors to increase as fewer sites are treated as variable in the calculation. Figure 5 shows the distribution of errors at the midpoint as a function of the number of sites included; this graph is based on the previous calculations with appropriate sorting as to number of variable sites. It is seen that, although error increases as the number of variable sites decreases, the rms error at the midpoint remains below 0.01 for all variable site numbers. Figure 6 shows the number of calculations in which the RS approximation at a pH corresponding to the midpoint of one of the sites required inclusion of a given number variable sites. This provides a measure of the computational savings realized by the RS approximation. For the strong coupling cases, 103 out of 250 calculations required the inclusion of all 10 sites, making the RS calculation identical with the exact calculation. Computation savings are greater for the moderate coupling case.
Number of variable sites
Figure 6. Number of occurrences in which the RS approximation re-
quires inclusion of a given number of sites at the titration midpoint of a site. Statisticsare taken over IO sites in 25 molecules. (a, top) Strong coupling case. (b, bottom) Moderate coupling case. The strong coupling case is an extreme test that brings out the weaknesses of both approximation schemes. The MF approximation gives inaccurate results, and the RS approximation fails to realize substantial computational savings. In real protein molecules, small sets of titrating groups may have strong couplings, as in enzyme active sites, but it is unlikely that more than a few titrating sites are strongly coupled. The titration curves of proteins are typically only slightly broadened relative to the curve that one would obtain from an analogous mixture of independently titrating model compounds or from a denatured p r ~ t e i n . ' ~ -In' ~the strong ( 1 1) Hansen, J. P.; McDonald, 1. R. Theory of Simple Liquids, 2nd d.; Academic: London, 1986. (12) Edsall, J. T.; Wyman, J. Biophysical Chemistry; Academic: New York, 1958.
9561
J . Phys. Chem. 1991, 95, 9561-9568
shown that it is a mean field type approximation in which correlations between titrating sites are neglected. Test calculations indicate that, for moderate to weak sitesite couplings, the mean field (MF) approximation is reasonably accurate. However, if strong sitesite couplings occur, the M F approximations can give significant errors. The reduced-site (RS) approximation, which we have introduced here, is much more accurate and remains so even in the presence of strong couplings. As long as the number of strongly coupled sites is not too large, the RS approximation also gives large computational savings over the exact method. It is recommended that the RS approximation be used in place of the M F approximation whenever it is computationally feasible, and that the M F approximation results be regarded with caution in cases where sites have couplings of a pK unit or more to other sites that titrate in the same pH range. A hybrid of the M F and RS methods could be a possibility for future development. Sites that are almost completely protonated or deprotonated could be assigned partial charges by the MF approximation, while the remaining sites are treated exactly. This might reduce the errors given by the RS approximation near the endpoints.
coupling calculations above, the values of pKap range from -4 to 18 even though pKint,ranges only from 4 to 16. The moderate coupling case has a more realistic spread of pKaPpvalues, from 1 to 13, but the occurrence of any strong couplings has been suppressed by the method of selecting the W matrix. The situation in proteins probably has features of both cases. Most sites will have only moderate to weak couplings, and either approximation will give satisfactory results; but, for a few strongly coupled sites, the M F approximation may lead to significant errors. In a recent calculation of the titration behavior of lysozyme with a macroscopic dielectric model,I4 we have found coupling constants between 21 titrating sites ranging from 0.001 to 7.3 pK units. Out of 210 couplings, 35 were greater than 0.1 and only 9 were greater than 0.5. Using the RS method, a maximum of 10 of the 21 sites had to be included as variable at any given pH. Thus, computational savings of at least a factor of 2" over the exact method were obtained. Conclusions
Tanford and Roxby's approximation, which is widely used in studies of protein titration curves, achieves great computational savings over exact calculations for molecules with a large number of titrating sites (unless convergence problems arise). We have
Acknowledgment. It is a pleasure to thank Dr. A. V. Finkelstein for useful and stimulating discussions concerning mean field theory. This work was supported in part by a grant from the National Institutes of Health and by a grant from DARPA.
(13) Nozaki, Y . ;Tanford, C. Merhods Enzymol. 1967, 1 1 , 715-734. (14) Bashford, D.; Karplus, M. Biochemistry 1990, 29, 10219-10225.
X-ray Structure Analyses and Vibrational Spectral Measurements of the Model Compounds of the Liquid-Crystalline Arylate Polymers. 2. Structural Isomorphism In a Series of Model Compounds Jian-an Hou, Kohji Tashiro,* Masamichi Kobayashi, Department of Macromolecular Science, Faculty of Science, Osaka University. Toyonaka, Osaka 560, Japan
and Toshihide Inoue The Research Association of Polymer Basic Technology, Toranomon, Kanda, Tokyo 101, Japan (Received: April 29, 1991)
X-ray structural analysis has been done for the model compounds of liquid-crystalline arylate polymers, Et-OCO-PhOCOPh,COO-Ph-COO-Et. The crystal structure data of the compound with n = 1 are as follows at room temperature: monoclinic, P2,/c-C$, a = 25.430 (9) A, b = 5.4435 (6) A, c = 8.282 (2) A, j3 = 97.63 (3)O, Z = 2, R = 0.064. The structure and packing fashion of the molecules of the nl crystal were found essentially isomorphous to those of the j3 form of the n2 crystal. The X-ray diffraction and infrared and Raman spectral patterns of the n3 compound are very similar to those of the nl and n2-8 crystal forms, making it possible to estimate the crystal and molecular structures of this compound. From the temperature dependence of the X-ray reflection, the anisotropic thermal expansion coefficients were evaluated for these three compounds. They were found to be similar to each other among these compounds, being supported by a direct optical microscopic observation of the dimensional change of the single crystals.
Introduction In the structural study of liquid-crystalline polymers, the poor X-ray diffraction data, Le., the small number and broadness of the reflections, make it difficult to perform detailed structural analysis of the crystalline and liquid-crystalline phases. A utilization of the model compounds may give us useful information for this purpose. We have chosen the following compounds of formula 1 as typical model materials for the arylate liquidcrystalline polymers H5C+O
G o c o~
"
C
~
~
C
O
O
-
c
,
where n = 1,2, and 3 (abbreviated as n l , n2, and n3). In previous papers,'-j we clarified the existence of the two crystalline phases, CY and 0, for the compound n2 (abbreviated as n2-a and n2-8) and analyzed their crystal structures. At the same time, the phase transitional behavior between the CY and j3 forms was investigated and the structure in the liquid-crystalline state was discussed on the basis of the X-ray diffraction and infrared-Raman spectral data. In that study, a local structural change or an internal (1) Tashiro, K.; 1990, 112, 8273. H
1
0022-3654/91/2095-9561$02.50/0
,
Hou,J.; Kobayashi, M.; Inoue, T. J . Am. Chem. Soc.
( 2 ) Tashiro, K.; Hou, J.; Kobayashi, M.; Inoue, T. Polym. Prepr. Jpn. 1990,39,749. (3) Hou,J.; Tashiro, K.; Kobayashi, M.; Inoue, T. Polym. Prepr. Jpn. 1990, 39, 2355.
0 1991 American Chemical Society
