AQVEOUS SOLUTION
Vol. 3, No. 5, Septeniber-October 1970
PROPERTIES OF
AMYLOSE655
A Theoretical Interpretation of the Aqueous Solution Properties of Amylose and Its Derivatives David A. Brant* and William L. Dimpfl Depurtment of CIiL-inistry, Uaicersitj*of Culifbrfiiu,Irrinr, C d i f h i a 92664. Receiced April 30, 1970
ABSTRACT: The observed dependence upon degree of polymerization and temperature of the unpertubed coil dimensions of two derivatives of amylose in aqueous solution has been interpreted using the statistical mechanical theory of polymer configuration, and the results are used to provide a description of the aqueous configuration of amylose. The model accounts for interdependence of bond rotations at each glycosidic bridge but assumes the rotations at each bridge to be independent of those at neighboring bridges due to suppression of intramolecular hydrogen bonding by solvation in aqueous medium. Structural models based on the crystal structures of cyclohexaamylose an3 methyl 8-maltoside have been employed. Configurational partition functions have been established using approximate conformational energy calculations for an appropriately chosen chain segment. Excellent agreement is achieved between theory and experiment when the cyclohexaamylose structural geometry is used and for reasonsble values of all structural and potential function parameters. It is concluded that amylose in aqueous solution is a statistical coil without identifiable helical character in the absence of complexing agents. e.g.. iodine, butanol. No correlations of long range in the chain sequence, ix., five to ten residu-s. of the type expected to characterize h4icil structures are incorporated in the model; and it is shown that near-neighbor correlations do not confer helical chatacter on the chain dsspite severe restrictions on the domain of conforma:ion space accessible to the backbone units
T
he unperturbed coil dimensions of numerous polymers have been interpreted successfully in terms of skeletal geometry and hindrances to rotations about the chemical bonds of the skeletal backbone through application of the statistical mechanical theory of polymer chain configuration. Synthetic polymers from a wide selection of classes have been treated. Among polymers of biological interest the proteins and their synthetic polypeptide analogs have received extensive consideration,2--’ and the solution configuration of single-strand polynucleotide chains6 and certain polysaccharides’ have been subjected to similar but less extensive analyses. We have undertaken to investigate the connection between skeletal structure and configuration in polymers of the polysaccharide class. Not only are relationships between polysaccharide configuration and biological function of interest; long-standing questions concerning the solution configuration of polysaccharidesxpgremain which may be resolved by the means now a t hand. Moreover, this class of polymers, possessing numerous members interrelated by known structural and stereochemical differences, provides an ideal milieu for demanding tests of structural hypotheses and postulates concerning the origins of rotational
hindrance potentials. In this paper we seek to interpret experimental results for amylose1” and its derivativesll, l 2 in aqueous solution using the statistical mechanical theory of chain cmfigurationl in conjunction with structural information from crystallographic analyses of low molecular weight analogs of the amylose ~ k e l e t o n and ~ ~ ,estimates ~~ of the conformational energy of the amylose structural unit, maltose.
Configurational Statistics
* To u hotn correspondence should be addressed. ( I ) P. J . Flory, “Statistical Mechanics of Chain Moleculcs,” Interscience Publishers, N e w York, N. Y . , 1969. (2) D. A. Brant and P. J. Flory, J . Amer. Cheni. Soc., 87, 2791
The skeletal geometry of amylose has been discussed a t length in a previous paper,” where the dimerie repeat unit, hereinafter called the niultose unit, is shown as Figure 1. We proceed here under the assumption that the pyranose ring of every glucose residue in the amylose chain is in the C1 conformation. We further take as constant all bond lengths and valence angles and all torsion angles for those bonds which comprise the glucopyranose ring. Responsibility for the tortuosity of the random amylose coil is thereby allocated solely to torsions about the bonds C1-O1 and 01-C4’ of the glycosidic linkage. This constitutes an approximation somewhat more serious than has been made in prior treatments of polymer chains possessing simpler structural geometries. Constancy of bond length and angle, when previously assumed, has been justified on the grounds that distortions from the selected mean values of these parameters will
(1965). (3) W. G. Miller, D. A . Brant, and P. J. Flory, J . .Mol. Biol., 23, 67 (1967). (4) P. R . Schimniel and P. J . Flory, ibid., 34, 105 (1968). (5) W. G. Miller and C. V. Goebel, Biochemistr),,7,3925 (1968). (6) H. Eisenberg and G. Felsenfeld, J . Mol. B i d , 30, 17 (1967); R. A. Scott, Biopolj,mers, 6, 625 (1968). (7) V. S . R. Rao, N. Yathindra, and P. R. Sundararajan, Biopol~niers,8, 325 (1969); D. A . Rees, J . Chern. SOC.A , 217 (1969). (8) M. Kurata and W. H . Stockmayer, Fortsciir. Hochpolj~ni.Forsch., 3, 196 (1963). (9) P. J. Flory, Makromol. Chem., 98, 128 (1966).
( I O ) W. Banks and C. T. Greenwood, Carbohjd. Res., 7, 349 (1968). (11) D. A . Brant and B. I z-m
has been calculated a t four temperatures for the HRW maltose geometry as a function of the valence angle 0 at the glycosidic bridge. The results, shown as solid curves in Figure 1, were obtained using potential functions described previously, l S which include coulombic terms and repulsive parameters ujk evaluated The using van der Waals radii augmented by 0.10 points corresponding to 0 = 117' at each temperature follow directly from the conformational energy V(+,+) plotted as a contour diagram in Figure 3 of the previous paper': through application of eq 5 and 8. Summations in eq 5 were carried out at intervals of 10" in both and $, it having been ascertained that further refinement of the summation interval produced no significant change in the results for C,. Calculations were performed in analogous fashion to yield V(+,$) and C , , for 0 = 113, 114, 115, and 119" to define the solid curves in Figure 1. A calculation similar t o those yielding the solid curves was carried out with neglect of coulombic terms in the potential function, and the results are plotted for purposes of comparison at a single temperature as a dashed curve in Figure 1 . The conformational energy diagram corresponding to the point at 0 = 117" on this latter curve is shown also in Figure 3 of the previous paper,15 with replacement of the solid contour for - 2 kcal mol-I by the
A.
+
(26) P. J. Florr, 0 . I)could necessitate choice of somewhat different values of 0 to provide a satisfactory fit to the data. Moreover, rather large systematic uncertainities in the experimental values of C, have also been acknowledged. l 2 Thus, the angle 0, among c7ther parameters, must be accepted as an adjustable parameter of the calculation t o which there attaches only indirect physical significance. That the theoretical treatment is rooted in a physical model which is fundarnentally correct is nevertheless verified by the agreement of theory with experiment for reasonable values of ull parameters of the calculation, including 0. Additional support for the general features of this model is provided by results which we now discuss of calculations employing alternative theoretical parameters. Calculations parallel to those which yielded the solid curves of Figure 1 have been carried out using the C J glucose geometry in place of the H R W geometry set. The calculated curves of C, cs. 0 at several temperatures shown in Figure 4 disclose the same strong dependence of C , in 0 revealed in Figure 1. In the present case, however, values of 0 greater than 121" are required in order to obtain theoretical values of C, within the experimental range. The temperature coefficient calculated in this case for 0 = 121" is, moreover, only one-half the experimental result which was satisfactorily reproduced by the theory using the H R W glucose geometry. The origin of these differences between results based on the two geometry sets is to be found in the details of the respective conformational energy surfaces and the reflection of these features in the characteristics of the corresponding mean maltose units. A cornparison of conformational energy maps (not shown) for the two geometries computed for a common bridge angle, 0 = 117", discloses that despite the similar appearance of the surfaces, as dictated by the dominant repulsive interactions, the mean maltose unit for the CJ geometry has a conformation lying near the contour for /7 = -3.0 A whereas that for the H R W geometry possesses a conformation for which / I N -1.5 A. This circumstance, which accounts for the appreciable difference in C, a t 0 = 117" evident from comparison of Figures 1 and 4, does not arise entirely because of a difference in the coordinates of the mean maltose unit in +,$ space. Indeed, it results primarily from significant differences in the pattern in +,$ space of the contours for /7. These have been presented for comparison elsewhere,15 where it may be seen that the contours for h # 0 lie much closer to the contour for /I = 0 in the domain of the energy minimum for the CJ geometry than for the H R W geometry with the consequence that any point within this domain lies at a higher '/z, for the former geometry. That a smaller temperature coefficient is calculated using the CJ geometry can be understood from a close comparison of the conformational energy surfaces in the neighborhood of the minimum for the C J glucose geometry with 0 = 121" and for the HRW glucose geometry with 0 = 114", these respective values of 0
AQUEOUS SOLUTION PROPERTIES OF AMYLOSE661
yielding results for C,,, in the range observed for DEAEA. HCI. The latter map, which resembles closely the one shown in Figure 7, has its minimum near = 330", $ = 160". From the minimum, the energy increases least rapidly in the direction of the trough which runs across the line for / I = 0. The population of conformations in this trough is therefore most sensitive to changes in temperature with consequences for the temperature coeflicient which which have already been described. The map for the C J glucose geometry with 0 = 121" is similar to that shown in Figure 8 except that the narrow trough of low energy which runs into the region near 290, 140" in Figure 8 is much broader, and the contour lying 1.5;!kcal mol-' above the minimum extends into this area to 280, 130". Moreover, the domain of energies less than -2.0 kcal mol-' appears as a more nearly circular well lying wholly below the line for /z = 0. As the temperature is raised, conformations outside this well increase in population. The effect of increasing the population of conformations lying above the line for h = 0 is, however, substantially counterbalanced by an increase in the population of the region near 280, 130", where / I is large and negative, Le., -4 A. Hence, although a negative temperature coefficient is predicted for C,, it is only one-half as large as the experimental value calculated from the H R W geometry set. Inasmuch as the calculated results for C, and its temperature dependence are sensitive to the shape of the contour surface in the vinicity of the energy minimum and particularly to the location of the minimum relative to the line for /? = 0, the effect o n C, of changes in the potential functions, in addition to alterations affecting the coulombic terms already discussed, have been investigated. Parameters ujk of the potential function terms for van der Waals repulsion between atoms j and k have been assigned for the previous calculations in accordance with procedures L 9 , 2 1 which recognize the existence described of attractive interactions between atoms in covalent connection to a t o m s j and k . This has been accomplished using a procedure shown to be adequate for conformational energy calculations for polypeptides2 and hydrocarbon^'^ whereby the van der Waals potential function (repulsion plus London attraction) for each atom pair was minimized at a nuclear separation distance 0.20 A greater than the sum of the conventional van der Waals radii of a t o m s j and k . Longer range attractions involving atoms connected to the atoms in question then work to generate a minimum in the total potential function for the interaction of atoms j and k und their connected atoms at a distance equal to the sum of the conventional van der Waals radii of the abutting a t o m s j a n d k . Given the generally larger number of neighboring atoms which are rigidly connected to a particular polysaccharide atom in comparison t o the number connected t o a polypeptide atom, calculations were carried out for the H R W geometry set with potential functions, including coulombic terms, in which the repulsive parameters ujx.
+
(29) A . Abe, R. L. Jernigan, and P. J. Flory, J . Amer. Chenz. Soc., 88, 631 (1966).
662 BRANT,DIMPFL
Mmromalerules
were assigned using van der Waals radii augmented by m9re than 0.1 A. The results are shown in Figure 5 in which C is plotted L'S. temperature for 0 = 117" and for several values of the increment applied to the van der Waals radii. The upper curve in Figure 5, calculated with augmentation of the normal van der Waals radii by 0.20 displays values of C, similar to the experimental results for NaCMA. The temperature coefficient corresponding to this curve is -0.0068 deg-1 in agreement with those characterizing the curves in Figure 2. It is clear that the experimental data may be fit using the H R W geom-try equally as well as in Figures 2 and 3 using valence angles 0 in the range 116-117", which most satisfactorily characterizes the known crystal structures of oligosaccharide analogs, and using larger van der Waals radii to assign the parameters u 3 k . Choice of van der Waals radii 0.10-0.20 A greater than the normal values for this purpose is readily defensible on grounds already stated.
A,
Conclusions The occurrence in amylose of a narrowly confined accessible domain in conformation space in coincidence with modest unperturbed coil dimensions represents the most unusual feature of the preceding analysis. Indeed, the fraction of conformation space in Figure 7 which contributes 90% of the partition function, i.e., the area within the dashed contours, is only 1.1 %. Conformational constraint of this severity is normally associated with a stiff and extended coil configuration owing to the squential repetition of similar monomer conformations which causes the chain to propagate with helix-like regularity. The conformational energy map appropriate to cellulose, which differs from amylose only in the stereochemistry at CI in each glucose, discloses also a s ngle accessible domain but yields predicted vslues for the characteristic ratio exceeding those for amylose by about a factor of 10 when all glucose rings are in the C1 conformation.3 o The much larger value for C, for cellulose is found despite the somewhat greater breadth of its low energy conformational domain. A similar example is found in the poly-L-proline I1 chain which possesses a single accessible conformational minimum and a predicted characteristic ratio exceeding 100. The relatively very small coil dimensions which are associated with the extreme conformational constraint in the amylose backbone have been ascribed above to the low pitch of the helix produced by sequential repetition of the mean maltose unit conformation. Although the mean maltose unit falls within the domain of conformstions which, when repeated in sequence, produce helices of left-handed screw sense, the present theoretical model should in no sense be construed to provide support for the existence of helical sequences in amylose chains in aqueous solution as has been suggested by Rao, et a1.l Conformational fluctuations occur within the domain of right-handed helices so as to destroy completely any rectilinear propagation of the chain resulting from repetition of the more frequently occurring conformations with left chirality.
Thus, a fraction of the maltose unit partition function equaling one-sixth derives from this domain in Figure 7, which we may reasonably take to approximate the conformational energy surface for maltose in unsubstituted amylose. Alternatively, 77 % of the partition function arises from the region in Figure 7 below the line for 12 = 0 which is bounded by the dashed contour. The probability that any maltose unit is contained in a sequence of six units, all with conformations in this domain, is therefore 0.21. Note, however, that this domain spans a range of helical parameters from Iz = 0 to h = -3.5 and from t = 56' to t = 75°.15 Hence, although 21 of the maltose units are contained in sequences which might be construed to comprise one turn of left-handed helix, this nascent helix is entirely devoid of regularity. The probability of observing sequences identifiable as several turns of left-handed helix, with sufficient integrity to bind iodine for example, is vanishingly small. Description of the amylose coil in aqueous medium as a broken or jointed helix is evidently improper and misleading if the theoretical model invoked here is accepted. A tendancy toward self-intersections at relatively short range in the chain sequence is not a characteristic unique to the amylose chain. The least energetic conformation of poly(dimethy1siloxane) is a self-intersecting helix of zero pitch.31 I n that case, however, more extended alternative conformations occur at only slightly higher energies with the consequence that the temperature coefficient of the characteristic ratio is positive. The negative temperature coefficient of C, for amylose finds explanation not in the existence of other discrete conformational alternatives to the least energetic one but rather in the detailed features of the energy surface within the single accessible minimum. This brings to full view the critical nature of the choice of geometry and potential function required to provide a successful theoretical interpretation of the unperturbed polymer coil dimensions. As demonstrated above, calculations using energy maps based o n the CJ geometry set, although superficially very similar to maps based on the H R W set, failed to reproduce the experimental solution properties with reasonable choices of the adjustable parameters. It is evident that experimental results are not reproduced by theory for all reasonable choices of input parameters and that in the present case the choice of geometric parameters is a critical factor. It is perhaps not surprising that the H R W geometry set, which is based upon the average properties of three interior cr-l,4linked glucose rings observed in the crystal structure of cyclohexaamylose, provides a better representation of amylose properties than does the C J set. which is based on a single terminal a-glucose ring in methyl /3-maltoside. Prediction of the characteristics of ordered polymer structures using approximate conformational energy calculations often provides a less stringent test of geometric and potential function parameters than does treatment of solution properties, presumably because the former characteristics are determined largely by (31) P.J. Flory, V. Crescenzi, and J . E. Mark, J . Arne?. Chem.
(30) I
