Quantum theory of the structure and bonding in proteins

Journal of Molecular Structure (Theochem), 109 (1984) 149-169 Blsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

QUANTUM THEORY OF THE STRUCTURE PROTEINS

AND BONDING IN

Part 17. The unionised-aspartic acid dipeptide

DAVID PETERS and JANE PETERS Department of Chemistry, (Gt. Britain)

Royal Holloway

College, Egham Hill, Egham, Surrey TW20 OEX

(Received 23 January 1984)

ABSTRACT The computed results on the unionised aspartic acid dipeptide are compared with experimental results for the aspartic acid residue in some globular proteins. There is generally good agreement between the two sets of results. The agreement is particularly good where the Ramachandran map is concerned, as with the other dipeptides. The computed results for the dihedral angle x, also fit the experimental results satisfactorily. The results for the dihedral angle x1 are more complicated because there is an uncertainty of 180” in the numerical values of this angle as determined by experiment but it is possible to demonstrate a measure of agreement between experiment and theory for this angle also. INTRODUCTION

The work in this paper on the neutral aspartic acid dipeptide of Fig. 1 follows on from that of the preceding paper on the asparagine dipeptide. The two cases are alike in many respects but there is one major difference in that it is often unclear from experiment whether or not the carboxyl group is ionised in the case of the acid. It is, of course, true that at physiological pH the ionisation of a simple carboxylic acid is effectively complete in simple aqueous solution, but the environment of the protein molecule, even on the protein surface, may be sufficiently different from that in water to make the comparison unreliable. For present purposes, unless stated to the contrary, we assume that the carboxyl group is unionised. It is also true that it cannot normally be decided from direct experimental information which oxygen atom carries the proton, so there is an uncertainty of 180” in the numerical value of the dihedral angle x2 (see below). The neutral carboxyl group has a preferred conformation in the syn form [ 21 as shown in Fig. 1 and this conformation is thought to be some 4-8 kcal mol-’ lower in energy than the corresponding tram form. Judging by previous experience with these molecules, it is unlikely but not impossible that such an energy difference will be outweighed by other factors in such a way as to 0166-l 280/34/$03.00

o 1984 Elsevier Science Publishers B.V.

150

Fig. 1. The unionised aspartic acid dipeptide. The drawing shows the backbone in the psheet conformation and the side chain in the standard conformation (xl = 240”, x1 = 270”). The oxygen atom of the carbonyl group of the carboxyl group is the reference atom for the definition of the dihedral angle x2 in the IUPAC convention [ 31.

force the carboxyl group into the trans form, and so we assume throughout this work that the unionised carboxyl group is always in the syn form. The work was begun by setting up the dipeptide as in Fig. 1 with the side chain in the standard conformation (xi = 240”, x2 = 270” in the standard IUPAC notation‘ [3] ). The oxygen atom of the carbonyl group is the reference atom for the definition of the dihedral angle x2. The Ramachandran map was then computed in the usual way and the results are shown in Fig. 2. In the next stage, the backbone was held fixed in the (Yand (Yehelical conformations and the energy was computed as a function of the dihedral angles x 1 and x2. The results are shown in Figs. 5 and 6. The p sheet region was dealt with as in Part 16 by assuming that x1 is restricted to the values +60”, 180” and -60” and that the only important conformations are those in which the backbone and the side chain are close together. The results are shown in Figs. 7(a) and 7(b). To compare the results with experiment, the recorded data on four globular proteins, (Ychymotrypsin [4], y chymotrypsin [ 51, elastase [6] and bovine pancreatic trypsin and its inhibitor [7] were used; the results are shown in Figs. 3 and 4. Finally, for purposes of comparison, the values of the two torsion barriers (labelled x1 and x2 for present purposes) in the propionic acid molecule were computed and the results are shown in Fig. 8. These results show that, as expected [8], the torsion barrier about x1 is 3-4 kcal mol-’ and that about x2 is -1.5 kcal mol-’ in the staggered conformation about x 1. These two numerical values recur throughout the results, particularly in Figs. 5 and 6, as would be expected if the torsion barriers were local phenomena within their own bonds. We have made frequent attempts to estimate the accuracy of the experimental data used in this work but all such attempts have led to inconclusive

151

-16d

-SO’

+

0'

60’

Fig. 2. The Ramachandran map (6, J/ plot) of the unionised aspartic acid dipeptide of Fig. 1. The side chain is fixed in the standard conformation (x, =‘ 240”, x1 = 270’). The zero of energy is the point @I= -BOO, $ = +80” at which a C, hydrogen bond is formed. The absolute value of the zero of energy is -594.45901 hartrees. The numerical values of the fixed geometrical parameters such as bond lengths and bond angles are those used earlier [ 1 ].

results and we are obliged to take the recorded values as they stand. It would be very helpful for work of this kind if the authors of the experimental data were able to offer some guidance on this matter. COMPUTATIONS

The Ramachandran map This map [9, lo] for the unionised aspartic acid dipeptide of Fig. 1 is effectively identical to the corresponding maps ‘for all the other dipeptides except glycine and proline. The experimental data for the aspartic acid residues in the four globular proteins obey the map accurately, as the results of Fig. 3 show. It follows that the important factors which determine the Ramachandran map are confined to the backbone and C? of the side chain. It must be remembered that the computed Ramachandran map refers only to the standard conformation of the side chain, in which the carboxyl group is turned away from the backbone as far as possible, and that when the side chain is close to the backbone then the interactions shown in Figs. 5-7 come into play. It is evident, however, that the latter effects are much smaller than the

152

-160'

160'

Fig. 3. Experimental data for the aspartic acid residue in (X ) (Ychymotrypsin, (0) elastase, (0) BPTI both the trypsin and the inhibitor and (0) -y chymotrypsin. Several of the points have been moved about 5” for clarity and the original literature must be consulted for accurate values of Q and J, .

I

X,=+&W180'

ofhelix

0.

x,

so

/’

0”

,,’

O

,

,

x\ /’ I

0

0

,.’

‘\\ ‘0 160”

(b)

31

Fig. 4. Experimental values of X, and X,. No points occur in the QL region in thii case. (X ) Q! chymotrypsin, (0) elastase, (0) BPT and its inhibitor and (0) 7 chymotrypsin. (a) X, values in the majority; (b) XI values in the minority.

153

0”

Xl

180"

360°

Fig. 5. The x,x; map for the aL helix. The zero of energy is the energy of the CYLpoint of the Ramachandran map (-694.46232 hartrees in absolute terms). This diagram may be thought of as being superimposed on the QL point of the Ramachandran map.

0”

Xl

180°

360°

Fig. 6. The x,x; map for the Q helix. The zero of energy is the energy of the a helical point of the Ramachandran map (-694.46130 hartrees in absolute terms). This diagram may be thought of as being superimposed on the or-helical point of the Ramachandran map.

Fig. 7. (a) The X,@Jand (b) x,$ diagrams for the p sheet region. The number shown in parentheses is the energy of the point on the Ramachandran map. Each of the sixteen diagrams belongs to a single point of this map in this sense. The dotted lines would be the zeros if a single zero was used (the C, point with o = -8O”, J, = +80”) for all sixteen diagrams.

Fig. 8. The torsion barriers in propionic acid. The labelling is that used in the dipeptide of Fig. 1 with x, being the CH,-CH, barrier (0” is the eclipsed conformation) and X, the CH,-CO,H barrier (0” is the conformation with the CO of CO,H syn(cis) to the C-C bond.

155

non-bonded repulsions which determine the map (say, 5 kcal mol-’ as against 20 kcal mol-‘) and so the agreement between experiment and theory at the larger energy level is unaffected by the smaller interactions. This agreement between experiment and the computed Ramachandran maps has been extensively documented in the last two decades [lo, 111 and we may express the situation formally by saying that the obedience of the Ramachandran map is a necessary but not sufficient condition on all the amino acid residues of a polypeptide chain, in order that the resulting threedimensional structure shall be a globular protein in the usual sense. The condition is not sufficient to determine the three-dimensional structure uniquely; if we select a given amino acid residue to a real globular protein and change its backbone conformation from, e.g., (Yhelical to oL helical, then the entire structure of the molecule may or may not change radically. The outcome of this exercise will depend on which residue is chosen and on whether other residues are changed at the same time. The side chain conformations in the (Ye region of the backbone In the preceding paper we dealt with the (YL region of the backbone first and we do the same here, although this region is less useful as an example in this case. The relevant computations were carried out as in the preceding paper and as outlined above. The results are shown in Fig. 5. The Ramachandran map (Fig. 2) shows that this region is character&d as usual by a pronounced minimum in the energy surface. The general nature of this minimum, its depth in particular, are much the same as those of other dipeptides [l] . The computed x1x2 map of Fig. 5 suggests that the minimum energy occurs when both x 1 and x2 have the value of -60”. There is a second minimum in the energy surface when x1 has the value of 180’ but this minimum is some 2 kcal mol-’ higher in energy than the former minimum. We would expect to find from the computational results that both angles have the value of -60” in nearly all cases, providing that no external complication arises, e.g., ionisation of the carboxyl group. The experimental data for the aspartic acid residue in the four test cases show no 01~ residues, despite the fact that the asparagine residue occurs in the 01~ conformation 11 times in the same molecules and with a comparable total number of residues. So we can make no comparison between experiment and theory in the case of the 01~region of the backbone and the aspartic acid dipeptide. It is natural to ask why there is no aspartic acid residue in the 01~region in reality. Our estimates of the total energy of the dipeptide in this region throw no light on the question; the results of Fig. 2 and Fig. 5 both suggest that there is no peculiarity in the energy of the model dipeptide molecule. We can make no formal comment on the question at this stage. It is apparent, however, from a simple model, that it would be difficult to create hydrogen

156

bonds either directly or via water molecules between the side chain and the backbone in this conformation of the backbone, which may well explain the reluctance of the aspartic acid residue to adopt the eL conformation in reality. Were the carboxyl group ionised, the same comment would apply with increased force. By contrast, both the a! helical and p sheet conformations of the backbone will allow the formation of hydrogen bonds either directly or indirectly. We return to this question in the Discussion section below. The side chain conformations in the a! helical region of the backbone The relevant computations were carried out as before and the results are shown in Fig. 6. The experimental results are reported in Figs. 4(a) and 4(b). The general pattern of the computed results for the (Yhelical region is more complicated than for the eL region of the backbone. Looking at the x1x2 energy surface of Fig. 6, we see that there are several minima, all quite close together in energy. The global minimum occurs when x1 is -60” and x2 is f 90” and the other minima which occur with x 1 values of + 60” and 180” are about 1 kcal mol-’ higher in energy. The experimental results of Fig. 4(a) and (b) do not, however, show such a complicated pattern as might have been expected from the computed results. Thus, of the 14 examples, 11 do show a x1 value of -60” (Fig. la) and the accompanying x2 values fall into two groups, one at about -30” and one at about + 120”. The computed x2 values are *go”. The difference between -30” and -90” is perhaps larger than we would expect and there may be a case for re-optimising the values of # and J/ here to explore this discrepancy. No such re-optimising work has been carried out to date. The remaining three cases in which x1 is not -60” are displayed in Fig. 4(b) but there is little to say about them except that the reported value of x 1 of 120’ is probably erroneous, as is the x2 value of 180”. The overall result in this case of the (Yhelical region is very similar to that in asparagine, with a preponderance of the -60” value for x1 and the x2 situation being inconclusive. The side chain conformations in the p sheet region of the backbone As mentioned in Part 16, there are additional difficulties in this region of the backbone as compared with the OLhelical and (Ye helical regions. We try to avoid these difficulties, as before, by fixing the value of x1 at +60”, 180” or -60” and also by studying only those conformations in which the side chain is close to the backbone. The computed results for the 0 sheet region again take the form of a series of diagrams of the energy as a function of x2 and either 4 or 9. These are shown in Fig. 7(a) and (b). We again use as the main zero the corresponding point on the Ramachandran map, so that each of the 16 diagrams has a

157

different zero. The dotted lines on Figs. 7(a) and (b) would be the zeros if we use a single zero, the zero of the Ramachandrsn map at the C, point, for all 16 diagrams. These latter dotted lines are the zeros which should be used for direct comparisons with experiment. For some purposes such direct comparison with experiment, bypassing the Ramachandran map, are more helpful, but for other purposes it is more convenient to use the two-stage comparison first with the Ramachandran map and then with the diagrams of Fig. 7(a) and (b). The experimental results for the p sheet region are straightforward as far as x1 alone is concerned (Figs. 4a and 4b). In about two thirds of the cases, x1 is 180” and the remaining cases are divided equally between the values of +60” and -60”. Looking at the computed results of Figs. 7(a) and (b) and using the dotted lines as the zero, it is clear that the lower diagram of Fig. 9 contains the global zero, with a x1 value of 180” and 4, $ values of -100” and +90”. The energy of this global zero is about -4 kcal mol-‘. Moreover, the results for the other two values of x1, +60” and -6O”, shown in Fig. 7(a), contain minima of about -3 kcal mol-‘. Thus both of these values would be expected to occur experimentally, as they do. The situation concerning the x1 values is again more complicated than that of the xl values. Looking at the results in Fig. 4(a), where x 1 is X30”, there are two main values for x2 of 0” and 180”. The two values are indistinguishable by experiment but the computed results of Fig. 7(b) suggest that this value should be 0” and not 180”. So there is reasonable agreement between experiment and theory for cases where xl is 180”. At the same time, it must be kept in mind that the uncertainty of 180” in the x2 values, plus the difficulty of knowing what the experimental error really is with this dehedral angle, make it difficult to know quite how real is the apparent agreement between experiment and theory. The results for the other two values of x 1, +60” and -6O”, are shown in Fig. 4(b), but with so few cases to study it is difficult to draw any firm conclusions. There is perhaps an indication that both of these xl values are accompanied by a x2 value of 180” while the computed results of Fig. 7(a) suggest a value of about 120” (or 300”) for x2. This is as far as the analysis can be taken at this time. To summarize the /I sheet work, there are inherent difficulties within this region of the Ramachandran map but there does seem to be a measure of agreement between experiment and theory; this is particularly true of the values for the dihedral angle xl which determines the orientation of the whole side chain. DISCUSSION

The present results for the unionised aspartic acid dipeptide are fairly satisfactory. The success of the Ramachandran map in uniting experiment and theory is quite unambiguous and the computed and experimental values

158

of the dihedral angles x1 generally agree. This agreement is found equally in asparagine and in aspartic acid dipeptides. There is, however, a lack of clarity where the results for the dihedral angle x2 is concerned, and this is due in the main to the uncertainty of 180” in the numerical values of this angle from experiment. There is the additional uncertainty about the state of ionisation of the carboxyl group in the present work. We have stressed the point about the dihedral angle x2 because it has a significance beyond the comparison between experimental and computed values. In particular, the behaviour of the side chain and its interactions with the backbone might well be revealed clearly if we could overcome the difficulties which surround the x2 work. Such interactions, presumably via hydrogen bonding, may be important for the structure of the globular proteins and as a thorough analysis of the point would be very helpful. The neutron diffraction experiment [ 121 is of major importance for this purpose. It is also important to remember that the computations are carried out on small model molecules, while the experimental information comes from large globular protein molecules and that the connections between the two sets of molecules may be more subtle than we have supposed. The size of the small model molecules may be the limiting factor in the effectiveness of this work generally. If so, then the external effects coming from outside the dipeptide molecule may be as important as those within the dipeptide and, as suggested elsewhere, such longer range effects may well control the large scale structure of the globular proteins. As we commented above, we suspect that the striking difference between the behaviour of the asparagine and the aspartic acid residues in the oL helical region has its origins in hydrogen bonding which is external to the dipeptide molecule. There is one further question concerning the large p sheet region of the Rarnachandran map; that residues with different side chain conformations might occur in different parts of this region. There is some indication that this does happen. For example, in both the asparagine and the aspartic acid examples, those cases in which the x1 value is +60” or -60” tend to occur near the top of the map (9 -150’ ) and those cases in which the x 1 value is 180” tend to occur near the middle of the 0 sheet region (J/ -90”). This wilI be studied for all the amino acid residues at a later stage. Finally, the state of ionisation of the carboxyl groups in a globular protein is often uncertain and we have treated only the unionised form here. We have carried out some of the computations for the ionic form, in connection with other work, and when these have been extended to cover the whole of the work on the dipeptide they will be reported and compared with the present results in order to find out whether ionisation of the side chain carboxyl group has a substantial effect on the backbone conformation and related properties of the molecules.

169 REFERENCES 1 D. Peters and J. Peters, J. Mol. Struct., 60 (1978) 133(a); 63 (1979) 103(b); 62 (1980) 229(c); 64 (1980) 103(d); 68 (1980) 243(e); 68 (1980) 266(f); 69 (1980) 249(g); 86 (1981) 107(h); 86 (1981) 257(j); 86 (1981) 267(k); 86 (1982) 341(l); 88 (1982) 137(m); 88 (1982) 167(n); 90 (1982) 306(p); 90 (1982) 321(q), 117 (1984) 137(r). 2 I. G. Csizmadia, M. R. Peterson, C. Kozmuta and M. A. Robb, in S. Patai (Ed.), The Chemistry of Acid Derivatives, Part 1, Wiley, New York, NY, 1979, p. 23. 3 IUPAC-IUB Commission on Biochemical Nomenclature, J. Mol. Biol., 62 (1970) 1. 4 J. J. Birktoft and D. M. Blow, J. Mol. Biol., 68 (1972) 187. 5 G. H. Cohen, E. W. Silverton and D. R. Davies, J. Mol. Biol., 148 (1981) 449. 6 L. Sawyer, D. M. Shotton, J. W. Campbell, P. L. Wendell, H. Muirhead, H. C. Watson, R. Diamond and R. C. Ladner, J. Mol. Biol., 118 (1978) 137. 7 R. Huber, D. Kukia, W. Bode, P. Schwager, K. Barteis, J. Diesenhoferand W. Steigemann, J. Mol. Biol., 89 (1974) 73. 8 Cf. ref. 2, pp. 24,31. 9 G. N,. Ramachandran and V. Sasisekharan, Adv. Protein Chem., 23 (1968) 283, 237. 10 B. Pullman, in B. Puiiman (Ed.), Quantum Mechanics of Molecular Conformations, Wiley, London, 1978, p. 324. 11 E.g., E. N. Baker, J. Mol. Biol., 141 (1980) 441; M. N. G. James and A. Sielecki, J. Mol. Biol., 163 (1983) 299. 12 J. C. Hanson and B. P. Schoenbom, J. Mol. Biol., 163 (1981) 117.