- Email: [email protected]
Quantum theory of the structure and bonding in proteins
Journal of Molecular THEOCHEM Elsevier
Scientific
QUANTUM PROTEINS
85 (1981)
Structure,
Publishing
THEORY
Company,
107-123 Amsterdam
OF THE STRUCTURE
.- Printed
in The Netherlands
AND BONDING
IN
Part 8. The alanine dipeptide
DAVID
PETERS
and JANE
PETERS
The Bourne Laboratory, Department of Chemistry, Royal Holloway College, University of London, Egham Hill, Egham, Surrey TW20 OEX (Gt. Britain) (Received
19 September
1980)
ABSTRACT The conformational energy map of the dipeptide is constructed for the alanine residue by the ab initio molecular orbital theory and the results are compared with those for the glycine residue. The findings are mostly straightforward because the only change from the glycine map is the addition of the non-bonded repulsions between the methyl group and the rest of the molecule. One consequence of these repulsions in the alanine dipeptide is that one of the two C, hydrogen bonds present in the glycine case is no longer a low energy region; another is the pleating of the p sheet structures. The results offer an explanation of why polyglycine does not form 01 helices while polyalanine does. They also suggest that the left handed Q helix may be locked into its conformation and thus be stable against conformal changes. The low energy conformation of the methyl group is that of the simple staggered form with respect to CF. A curve is constructed for the non-bonded repulsion between oxygen and hydrogen. The results of the ab initio computations agree with those from the PCILO method in many but not all respects. INTRODUCTION
The work so far in this series of papers [l] on the quantum mechanics of the structure and bonding in proteins has been concerned in the main with the glycine residue as the constituent amino acid of the di-, tri- and tetrapeptides. Using ab initio molecular orbital theory in the standardised form of the Gaussian 70 and 76 packages [2], we have established the glycine conformal energy map at the short base level of approximation and demonstrated the existence of hydrogen bonds, torsion barriers, non-bonded repulsions and other effects in these small peptides. One important result from these computations is the direct and purely quantum mechanical demonstration of the existence of 0 bends in proteins via the C,, hydrogen bonds in the tripeptide (Parts 3, 5 [l] ) and the existence of the (Yhelical hydrogen bond in the tetrapeptide (Part 7 [l] ). This work is now extended to the dipeptide built from the alanine residue (Fig. 1) and its conformal energy map is computed (Fig. 2). Alanine occurs widely in proteins and it may be the case that amino acids with other side 0166-1280/81/0000-0000/$02.75
0 1981
Elsevier
Scientific
Publishing
Company
108
Fig. 1. The dipeptide of L-alanine. This is the extended conformation (o = -180”, i! = -- 180’). The standard bond lengths and angles are: NC = 1.380, CO = 1.220, CH (of CO) = 1.110, CC = 1.519, NH = 1.022, CN = 1.459, CH (of CH,) = 1.124, HNC = 120”, HNH = 120”, NC0 = 120”, NCC = 113”, NCH = 113”, OCC = 127”. Further discussion of the geometry is given in Part 1 (1). A positive rotation corresponds to H,, moving downwards and 0 j moving upwards (note that the positive rotations were wrongly defined in Part 2 (1)).
chains do not behave very differently from alanine in some respects because the additional atoms in their side chains are some way away from the backbone of the protein. The first step in this work is to construct the dipeptide map for the alanine residue in the usual way (Part 2 [ 11). There is of course no symmetry in the case of alanine as there is for glycine so it is necessary to vary both dihedral angles over 360”. In addition, there is the question of the orientation of the methyl group in the alanine residue. It is expected from the results of the
Fig.
2. The conformational
putations at 20” intervals. minima or saddles.
energy There
map of L-alanine. is no regular
contour
The map is constructed interval.
Squares
mark
from
com-
maxima,
109
earlier PCILO computations [3] and from general experience on torsion barriers that the staggered conformation of the methyl group with respect to the C” atom and its bonds will form the low energy conformation of the methyl group. Accordingly, we constructed the entire map with this staggered conformation as the standard arrangement of the methyl group and then checked the assumption by optimising the rotation angle of the methyl group at various important points of the map and found it to be correct. The alanine map is then used as the basis for comparisons with experiment with the glycine map and with the PCILO map. Some further analysis of nonbonded repulsions and of the significance of populations is also reported. THE AB INITIO
MAP OF L-ALANINE
The ab initio map of L-alanine is shown in Fig. 2 and the corresponding glycine map in Fig. 3. The latter map has been redrawn from the original (Part 2 [l] ) to match the alanine as well as possible. The zero of energy used in the alanine map is the C7 hydrogen bond on the left of the map ($J = --SO”, J, = 80”) and th’is is the hydrogen bond sometimes called [3] “equatorial”. The choice of the zero of energy is unimportant when working with a single map but it becomes important when comparing two maps and then it is by no means a simple matter (see below). There are numerous different comments and comparisons which may be made with this alanine map and it is convenient to divide these into the sections which follow.
Fig. 3. The conformal energy map of glycine. The map is constructed from computations at 20” intervals. There is no regular contour interval. The squares mark maxima, minima or saddles. This is a redrawn version of the map in Part 2 [ 11.
110 COMPARISON
BETWEEN
EXPERIMENT
AND THE ALANINE
MAP
The first task is to show how well the results correspond to reality. It is possible to cover more than a small fraction of the available evidence if we work with the broad view that alanine will represent all the amino acids except glycine so we are concerned here only with the objective of showing that there is general agreement between the theory and the experiment and not with specific cases. The first such comparison is that of Fig. 4 which reproduces in modified form the well known [ 41 Ramachandran plot of some experimental and general results versus the ab initio results. It is clear that all this information fits very well into the ab initio map. Thus, the p sheet structure at the top left (see below), the 310, CY~and 77helices, the C7 hydrogen bond and even the left handed CYhelix on the right of the map all fall into the low lying regions of the map. This degree of success, which was not so obvious with the glycine map (Part 2 [l] ), demonstrates the validity of the whole procedure and represents a notable achievement for the ab initio method which, being free from assumptions and parameters (in the usual sense of semi-empirical theory), cannot generate false agreement between theory and experiment. Going on to a more detailed comparison between theory and experiment, we take the X-ray data on lysozyme [ 51, a-chymotrypsin [ 61 and insulin [ 71 and compare this data with the alanine map. Taking the smallest molecule, insulin, first the results (Figs. 5 and 6) for both the A and B chains of insulin
-%C”
-126
-eo-
_&
d
id
a6
.zo”
i6G
Fig. 4. Dihedral angles in proteins and the alanine map. Taken from ref. 4 p. 27. The dotted lines define the areas available when a hard sphere model and either of two sets of internuclear distances is used. Details are given in the reference.
111
Fig. 5. The insulin A chain and the alanine map. All of the residues for the A chain are included on this map. Data from ref. 7 where further details are given. There is no glycine in this molecule other than the first residue.
Fig. 6. The insulin B chain and the alanine map. The triangles denote glycine residues. The proline residue is omitted. Further details in ref. 7. The squares and circles refer to chemically identical but crystallographically different molecules. See text concerning the glycine residues.
112
are the work of Blundell, Dodson, Hodgkin and Mercola and each chain occurs in two crystallographically different environments. The A chain contains no glycine residues and all the results for the other residues fit onto the alanine map with reasonable accuracy bearing in mind that uncertainties of at least 25” are inevitable in the theory and perhaps similar errors will arise in the experimental values. The B chain contains three glycine residues and we have put these onto the alanine map to show that they do not fit this map at all although they do fit the glycine map (Fig. 3). The other residues fit onto the alanine map reasonably well. The lysozyme data (Fig. 7) are those reported by Phillips [ 51 and these also fit the alanine map reasonably well when the glycine residues are excludec There is the exception of the phenylalanine residue on the right hand side which occurs in a region far removed from any low energy area so it seems doubtful that this point can be correct. The glycine residue results are collected together in Fig. 8 and again there is good agreement with the numerical results. The data for the largest of the three molecules, a-chymotrypsin, are those reported by Birktoft and Blow [6]. The glycine residue results are again separated out and shown in Fig. 9 while the results for all the other residues are shown together in Fig. 10. There seems to be a generally good fit between experiment and computation in this case as well as in the other two cases. There are of course many other collections of relevant experimental data which could be compared with these maps for glycine and alanine but these
-16$
-120’
-so”
-40°
0'
4d
SB
120°
160°
Fig. 7. Lysozyme and the alanine map. All residues other shown. The point to the right (0 = 120” ) is phenylalanine. details are given.
than glycine and proline are Data from ref. 5 where further
113
Fig. 8. Glycine residues details are given.
Fig. 9. Glycine
residues
in lysozyme
and the glycine
in ol-chymotrypsin
map. Data from
and the alanine
map. Data
ref. 5 where
from
ref. 6
further
-lEOO
-Ima
-xl”
-40’
on
4.0’
ad
Fig. 10. ol-Chymotrypsin and the alanine glycine and proline are included here.
120’
1fxJ~
map. Data from ref. 6. All residues
other
than
three sets of data contain over four hundred pieces of experimental information and, in so far as we are concerned to establish the general validity of the computational results, such a test seems more than adequate. We propose to defer a complete analysis of all the available data until some computations on other residues are completed. This is essential for the proline residue of course but some of the bulky side chains also need examination. The p sheet is widespread in silks [ 81 although a few cases of helical structures in some special cases, including a number of insect silks, are known. In these special cases, however, the glycine content is much lower (ref. 8, p. 337) and this is consistent with the result that glycine favours sheets or bends rather than helices. The pleating of the sheets is also consistent with the alanine map. At the same time, it must be appreciated that matters are not by any means as simple as these preliminary comments might suggest. For example, the p silks do contain large amounts of glycine so we cannot assume that sheets are free from glycine to any extent. It is plain that much detailed study of the other residues as well as glycine and alanine from both experimental and theoretical points of view will be required before clear generalisations are possible. Another necessary reservation is that the X-ray results themselves contain elements of prejudgement and preselection of the probable structures of the molecule which is being examined - usually via some kind of model building to guide the analysis - so that agreement between theory and experiment is a two way affair and the results of the computations are helpful in guiding
115
the experimental work. Again, we must wait for detailed computations on at least some of the additional residues, particularly the bulky and charged ones, before some of the well known [8] preferences for a given conformation by given amino acid residues can be completely understood and analysed in depth to the point of seeing the influence of the molecular structure on the secondary and tertiary structures. COMPARISON
BETWEEN
THE ALANINE
AND GLYCINE
MAPS
The glycine map (Fig. 3) has been redrawn from the earlier publication (Part 2 [l]) but it is still based on the extended form rather than the C7 hydrogen bond as the zero of energy. To convert to t.he hydrogen bond as the zero of energy, subtract 1.0 kcal mall’ from all the contours of the glycine map. It is helpful to look at the glycine and alanine maps side by side because many of the features occur on both maps and this is useful in confirming the internal consistency of the method and the absence of trivial errors. It also greatly simplifies the interpretation of the maps, particularly that of alanine, since, if the established chemical idea that a methyl group will do little or nothing other than add repulsions at certain conformations is correct, then it should be possible to construct a difference map between the two maps. Such a difference map should be everywhere positive and give a quantitative estimate of the repulsive interactions. This map is shown in Fig. 11 and it is clear that this expectation is fulfilled with remarkable accuracy. For example, a model suggests that in the top left hand corner of the map no repulsive terms are expected and over the whole of this region the difference map has a value of less than 1.0 kcal mol-’ (after adjustment for the difference in zeros of the alanine and glycine maps). Moreover, the bottom left hand comer shows the hydrogen--hydrogen repulsions between the methyl group and the amino group while the top right hand corner shows the oxygen-hydrogen repulsions between the methyl group and the carbonyl group. The bottom right hand comer is the sum of the two kinds of repulsion as expected. The clarity and directness of these results is pleasing and we take up the discussion of the quantitative analysis of the non-bonded repulsions in the section below. Looking at the glycine and alanine maps themselves now, the left hand C, hydrogen bond at about @J= -SO”, $ = + 80” occurs in both maps and so does the low energy region close to the extended form in the left hand corners. The hydrogen-hydrogen repulsion at the middle left of the maps and the very high energy central regions of repulsion between non-bonded atoms occur in both maps and these results are well known and well understood [3]. We have said that the new features of the alanine map are those imposed by the repulsive interactions between the methyl group and the rest of the molecule. One consequence of the presence of these repulsions is that the C7
Fig. 11. Alanine minus glycine difference map adjusted for the difference drawn from the numerical values not from the contours. No information the hatched areas.
in zeros. It is is available for
hydrogen bond on the right of the map at about Q = SO”, $ = ---SO” is no longer a low energy point. It is shown below that the populations demonstrate that the hydrogen bond is still present, as expected, but the nonbonded repulsions of some 12 kcal mol-’ mask the hydrogen bond energy of some 3 to 4 kcal molF so that the hydrogen bond is hardly visible on the aldnine energy map. Another new result of the alanine map is that the extended form is no longer the minimum of energy because the repulsions created by the methyl group move the minimum of energy to about 4 = -160”, $ = 160”. In addition, this minimum is now above the C, hydrogen bond in energy rather than just below it as in the glycine case. This result seems to be confirmed [4, 81 by the experimental information on the fl pleated sheet of silk and other fibrous proteins so we may reasonably claim to have an explanation of why such sheets show the well known pleating effect. The arguments of the last paragraph can be extended if we accept the validity of the result that in glycine the extended form is of lower energy than the C, hydrogen bond while the converse is true in alanine. This provides a tentative explanation of the fact that glycine residues tend to form sheets rather than helices while alanine residues do not form sheets as readily as they form helices [S] . In other words, the methyl group’s presence raises the energy of the p sheet structure but leaves the helix unaffected. If this explanation is correct (and it must be tentative because the energy differences are small (cf., Part 4 [ 1 ] )) we have an explanation
117
of why polyglycine forms sheets while polyalanine and other such polymers form helices [4, 81. There are some other comparisons between the alanine and glycine maps but these are best made in the later sections in other contexts. SPECIAL POINTS CONCERNING
THE ALANINE
MAP
We commented earlier that the conformal energy maps are potential sources of information about the energy changes which result from changes of shape of the peptide and protein molecules. Such changes of shape are thought to be important in such processes as complex formation, chemical reactions generally, the allosteric effect and other physical processes [ 91. The glycine map offered no clear examples of this point at least in a qualitative sense because all points of the map may be reached from all other points with small increases (about 5 kcal mol-’ at most) in the total energy. Such energy quantities correspond to several quanta of vibrational energy and are readily available within molecules. The situation is different in the case of alanine because the map shows a “locked” point at about @= 40”, ii, = 60”. By the term “locked” is meant a situation in which a region of the map is completely surrounded by a barrier much higher in energy than the locked region. In the present case, th% region is surrounded by a barrier at least 10 kcal mall’ higher in energy than itself. There is some scope for a conformational change in which ic, increases in value towards the top of the map but this valley is closed. Thus it is possible that a molecule, once formed within this conformational region, would be unable to leave the region by a simple change of conformation although it could of course leave the region by a reaction with another molecule. Equally, the inaccessibility of this region may be why we never find the left handed helix. We know of no clear example of such a process in the experimental sense but it is interesting to notice that this region of the alanine map is close to the left handed helix and relevant experimental evidence would be very useful. There may be differences in chemical reactivity between the right handed and the left handed helix, for example, which are due to the different conformational situations of the two helices. The other special point about the alanine molecule is the question of the conformation of the methyl group itself. As noted above, the staggered conformation of the methyl group with respect to the bonds from CO was chosen as the standard arrangement since this is the lowest energy conformation in simple cases. The latter was verified by optimising the rotation angle at various points of the alanine map (see Table 1). In only one of the eight conformations examined is there appreciable departure from the staggered conformation and this is a point of very high energy (about 25 kcal mall’) close to a steeply rising region of the map where the results are somewhat uncertain. The other seven points are all of lower energy and are consistent in giving an optimised rotation angle close to the staggered
118 TABLE
1
Optimised
rotation
I$’ = -40 o =-180 57.0 XI=
angles (in degrees)
80 -80 65.3
-40 -60 60.5
for the methyl
-120 -140 60.0
group
of alaninea 60 40 61.2
160 -160 59.4
90 0 64.0
-100 0 47.5
aThe starting value of x, is always 60” and the angle used in the optimisation is 30” The energy change associated with the optimisation is always negligibly small.
value of 60”. Thus there is little or no reason to expect the operation of any kind of cogwheel effect between the methyl group and the other groups in this molecule. This is presumably due to the fact that the methyl group is more nearly cylindrically symmetrical than the conventional formulation as CH3 would imply. It should perhaps be pointed out that the above test is not foolproof since we do not simultaneously optimise all three angles but it seems unlikely that this is a serious reservation. NON-BONDED
REPULSIONS
AND REPULSIVE
CURVES
Earlier work showed that it is possible to extract from the dipeptide map (Part 2 [ 11) estimates of the non-bonded repulsions between hydrogen and hydrogen and between oxygen and oxygen atoms. The same information is repeated within the alanine map but there is also new information contained in the difference map (Fig. 11) which deals with hydrogen--hydrogen again and also with hydrogen-oxygen atoms. There is now the complication that interactions with the methyl group involve two of the hydrogen atoms of this group so the procedure is not straightforward.
Fig. 12. Hydrogen-hydrogen repulsion. The open circles are the data for the glycine dipeptide from Part 2 [ 11. The triangles and squares are the data from the repulsion of the methyl group and the amino group of alanine. The new data have been added assuming a zero of 4 kcal mall’ as the plateau of the earlier map. Fig. text
13. Hydrogen-oxygen (cf. Table 2).
repulsion.
The construction
of this figure
is discussed
in the
119
The hydrogen-hydrogen repulsions arise between the methyl group and the amino group and these prove to be additive in a simple way so that we may add them directly to the curve obtained earlier as in Fig, 12. In placing the new points onto this curve, they are referred to the zero of 4 kcal mall’ of the original curve and it is clear that they fit this curve accurately. The hydrogen-oxygen repulsions arise between the methyl group and the carbonyl unit and they involve two hydrogen atoms of the methyl group. The relevant numerical values are reported in Table 2 and the separation of the overlapping effects of the two hydrogens is less straightforward than in the hydrogen-hydrogen case above. We begin by assuming that the extreme values (1, 2 and 8, 9) are straightforward because one of the two distances is very large. This gives two points on the graph of Fig. 13. We may now deduce that points 3 and 7 are simple because one of the two distances is 2.90 A and we know the interaction energy is zero ab this distance from the first two points. Then we take the points 4 and 6 and, reading the energy value corresponding to 2.38 A from the graph, we conclude that a distance of 1.68 A corresponds to about 21 kcal mol-‘. This deals with all the points save 5 and this cannot be accounted for by simply dividing by two as one might expect (the cross on the graph of Fig. 13). We cannot account for the failure of point 5 to fit the graph unless the simple idea of atom--atom repulsion is breaking down and three body effects are important here. In any event, this result is a warning against too simple an interpretation of the nonbonded atom idea. This latter result apart, we have constructed a simple version of a hydrogen-oxygen repulsion curve. ATOM
POPULATIONS
AND HYDROGEN
BONDING
In the earlier papers of this series, we paid a great deal of attention to the atomic populations not only because they may represent in some sense the chemical concept of the number of electrons associated with an atom but TABLE
2
Non-bonded
repulsion
between
hydrogen
and oxygen
atoms=
Serial no.
9 (“) G,,-H,, G,,-H,, Energy
1
2
3
4
5
6
I
8
9
40 3.88 2.64 0.5
60 3.42 2.13 2.1
80 2.90 1.77 9.6
100 2.38 1.68 22.2
120 1.93 1.93 25.2
140 1.68 2.38 22.2
160 1.77 2.90 9.7
180 2.13 3.42 2.0
200 2.64 3.88 0.5
aThese values all correspond to 4 = 60”. The lengths are in A, the energies in kcal mall’. The atom numbering is that of Fig. 1. The table is symmetrical left right but is printed in full to match the figure clearly. The method of converting these values into Fig. 13 is discussed in the text.
120
also because they are a summary of the form of the wave function. Neglecting the populations is, in effect, neglecting the wave function itself as a possible source of information. And since the populations are closely connected with the large diagonal elements of the one-particle density matrix the use of populations is connected with the study of the wave function through the density matrix (which is the invariant of the Fock theory). The earlier studies of the populations were in some ways disappointing, perhaps because in these molecules we are faced with a combination of hydrogen bonding and polarisation effects which cannot be easily disentangled. The major positive result from the earlier work was that the changes in populations of the nitrogen, hydrogen and oxygen atoms on hydrogen bond formation do seem to be reproduced in a systematic manner in different molecules. These changes may be summarised by saying that when the hydrogen bond forms the populations change as follows 7.435
N to 7.448
0.810
H to 0.786
8.267
0 to 8.285
Considering alanine, Table 3 contains the populations of the atoms at four different conformations, the low point near the LYhelix, the extended form, the hydrogen bond to the left of the figure and the other hydrogen bond to the right of the figure whose presence is masked in the energy sense by the non-bonded repulsions. The underlined values are those of the hydrogen bond. It is clear that the low point near the helical region has “normal” populations, the extended form shows intermediate populations as before and the two hydrogen bond cases show values of the populations which fit the earlier values summarised above. This is quite straightforward in the case of the left hand hydrogen bond but in the case of the masked hydrogen bond it is only the populations which reveal its existence clearly. This is an unusual application of the population idea and it suggests that despite some difficulties with the concept of populations we should persist in using it with these molecules. TABLE Atom c,
c2
3 populations c,
H,
6.154 5.688 0.922 5.944 6.154 5.690 0.923 5.944 6.151 5.697 0.925 5.946 6.172 5.698 0.925 5.943
in L-alaninea 0,
N,
H,
H,
8.283 0.799 7.436 0 810 8.290 0.796 7.433 0.807 8.295 0.807 7.446 0.784 -8.293 0.807 7.446 0.785 --
N,
Hz,, C,, O,z
5.755 7.373 8.264 0.813 5.754 7.370 8.280 0.796 5.746 7.365 8.287 0.803 5.741 7.365 8.289 0.801
H,, N,,
H,; H,,
0.944 0.944 0.940 0.929 0.943 0.941 0.930 0.948 0.938 0.935 0.944 0.934 0.934 0.899 0.950 0.950
01helix region extended form hydrogen bond (left) hydrogen bond (right)
aThe 01helix region is 0 = 40”, + = -6O”, the extended form is @ = -160”, IJJ= 160”: the left hydrogen bond is @ = --8O”, ti = 80” : the right hydrogen bond is 0 = 80”, + = -80”. The atom numbering is given in Fig. 1.
121 COMPARISON
BETWEEN
THE PCILO MAP AND THE AB INITIO MAP
It is not possible to make detailed comparisons with all the published maps for the alanine residue at this stage but there are interesting points in the comparison with the PCILO map [ 31. This latter kind of computation was the pioneering work on the quantum mechanical study of these conformation problems and there are many points of similarity between this map and the ab initio one. There are also some serious differences, however, and these are discussed below. The many points of similarity between the two maps include the raised area at the bottom left of the maps due to hydrogen-.-hydrogen repulsion, the low energy of the extended structure and the presence of the C, hydrogen bond at about 4 = --80”, $ = 80”. Also, in a general sense, the increase in energy of the right hand side of the map as one goes from glycine to alanine is reproduced in the PCILO map of alanine. When one looks at the PCILO and the ab initio maps in more detail, however, some major differences emerge even in a qualitative sense. For example, the increase in the energy of the right hand side of the map is very much larger in the ab initio map than in the PCILO map and, in particular, the right hand hydrogen bond which is masked by non-bonded repulsions of about 10 to 15 kcal mall’ in the ab initio results is still a low energy point in the PCILO map. Indeed, the PCILO results suggest that the two C, hydrogen bonds are of nearly the same energy in the alanine map. Another difference between the two kinds of map is the presence or absence of the two channels between the central high energy regions. These are at about o = 0” and ic/ = -80” or + 80” and although we know of no immediate practical significance of these channels they are clearly of importance in the questions of changes in conformation (see above on the locked conformation). The PCILO results suggest that these channels are present in the alanine map and have an energy only 2 to 3 kcal mol-’ above the minimum energy in the region of the saddle point. The ab initio map shows these channels but it suggests that they are about 16 and 24 kcal mall’ above the minimum energy. This is a major difference between the two maps and it could have the practical result of making the left handed helix either accessible or not accessible to conformational changes. A third difference between the two maps is concerned with the methyl group. The ab initio map suggests that the presence of this group moves the minimum of the extended form away from the symmetrical structure at --180”) --180” to the pleated structure at about -160”, 160”. The PCILO map does not show this effect of the methyl group. To summarise the general position then it seems that there are many points of agreement between the PCILO map and the ab initio map and it is certainly true that the former is very much less expensive in terms of computer time. On the other hand, where the two methods disagree it is very difficult to think that the PCILO results can be correct. Both the agreement
122
with experiment and the intrinsic reliability of the theoretical method favour the ab initio results quite clearly. The general picture which emerges from the comparison between PCILO and ab initio methods is that the PCILO method underestimates the importr ante of the non-bonded repulsions within a molecule. All of the points of disagreement set out above would disappear were it possible to increase the magnitude of the non-bonded repulsions within the framework of the PCILO method. Until this has been done, however, it must be borne in mind that the PCILO method may generate false minima in these conformal energy maps and in similar situations. DISCUSSION
AND SUMMARY
This conformal energy map of the alanine residue is important both for the alanine residue itself and particularly because it seems to be possible to use the alanine residue to facsimilate many of the other residues, excluding amino acids with functional groups in the side chain particularly if these are charged (NH,+, CO*-). Within th is reservation, however, the generalisation leads to useful results as the comparisons with experiment in the insulin, lysozyme and a-chymotrypsin cases show clearly. The general effectiveness of the ab initio alanine map in reproducing the experimental facts has been described briefly but there exists a large amount of information with which these results could be compared and the authors would be pleased to hear of such comparisons. The implications of these alanine results for the Cl0 and CL3 hydrogen bonds discussed in the earlier parts of this series of papers (Parts 3 and 5 [ 11) are important and will be taken up elsewhere. It is important to appreciate that computations of the present kind are genuinely quantum mechanical in their basic nature and so are free from built in assumptions about physical factors, such as hydrogen bonding and non-bonded repulsions, in any direct sense. This is not true of the empirical or partitioned potential energy methods which do contain such built in assumptions together with disposable parameters whose values may be adjusted to suit preconceived ideas about the results. At the same time, it must be pointed out that the Gaussian 76 method in the short base level of approximation is very far from basis-set independent as we have discussed at some length in an earlier paper (Part 4 [l] ) in connection with molecules of this kind. All that can be said at the moment in a practical sense is that if any one result or numerical value is particularly important for some purpose, it should be checked carefully with a long base, with polarisation functions or with other refinements of the method to be sure that the value obtained with the short base is not too far from the experimental reality. ACKNOWLEDGEMENT
We are indebted to one of our students, with the computations.
Mr. M. Wickens,
for assistance
123
REFERENCES 1 D. Peters and J. Peters, J. Mol. Struct., 50 (1978) 133 (Part 1): 53 (1979) 103 (Part 2); 62 (1980) 229 (Part 3); 64 (1980) 103 (Part 4); 68 (1980) 243 (Part 5); 68 (1980) 255 (Part 6); 69 (1980) 249 (Part 7). 2 W. J. Hehre, W. A. Latham, R. Ditchfield, M. D. Newton and J. A. Pople, GAUSSIAN 70 and 76, QCPE 236, Indiana University, Bloomington, IN, 1971 and 1976. 3 B. Pullman in B. Pullman (Ed.), Quantum Mechanics of Molecular Conformations, Wiley, London, 1977, Ch. 4: A. Pullman and B. Pullman, Adv. Prot. Chem., 28 (1974) 347. 4 R. E. Dickerson and I. Geis, The Structure and Action of Proteins, Benjamin/Cummings, Menlo Park, California, U.S.A., 1969. 5 D. C. Phillips, Proc. Nat. Acad. Sci. U.S.A., 57 (1967) 484. 6 J. J. Birktoft and D. M. Blow, J. Mol. Biol., 68 (1972) 187. 7 T. L. Blundell, G. G. Dodson, D. C. Hodgkin and D. A. Mercola, Adv. Prot. Chem., 26 (1972) 279. 8 R. D. B. Fraser and T. P. MacRar, Conformations in Fibrous Proteins, Academic Press, London, 1973, Chs. 9 and 10. 9 K. J. Laidler and P. S. Bunting, The Chemical Kinetics of Enzyme Action, 2nd edn., Clarendon Press, Oxford, 1973, Ch. 7 and p. 256.
Scientific
QUANTUM PROTEINS
85 (1981)
Structure,
Publishing
THEORY
Company,
107-123 Amsterdam
OF THE STRUCTURE
.- Printed
in The Netherlands
AND BONDING
IN
Part 8. The alanine dipeptide
DAVID
PETERS
and JANE
PETERS
The Bourne Laboratory, Department of Chemistry, Royal Holloway College, University of London, Egham Hill, Egham, Surrey TW20 OEX (Gt. Britain) (Received
19 September
1980)
ABSTRACT The conformational energy map of the dipeptide is constructed for the alanine residue by the ab initio molecular orbital theory and the results are compared with those for the glycine residue. The findings are mostly straightforward because the only change from the glycine map is the addition of the non-bonded repulsions between the methyl group and the rest of the molecule. One consequence of these repulsions in the alanine dipeptide is that one of the two C, hydrogen bonds present in the glycine case is no longer a low energy region; another is the pleating of the p sheet structures. The results offer an explanation of why polyglycine does not form 01 helices while polyalanine does. They also suggest that the left handed Q helix may be locked into its conformation and thus be stable against conformal changes. The low energy conformation of the methyl group is that of the simple staggered form with respect to CF. A curve is constructed for the non-bonded repulsion between oxygen and hydrogen. The results of the ab initio computations agree with those from the PCILO method in many but not all respects. INTRODUCTION
The work so far in this series of papers [l] on the quantum mechanics of the structure and bonding in proteins has been concerned in the main with the glycine residue as the constituent amino acid of the di-, tri- and tetrapeptides. Using ab initio molecular orbital theory in the standardised form of the Gaussian 70 and 76 packages [2], we have established the glycine conformal energy map at the short base level of approximation and demonstrated the existence of hydrogen bonds, torsion barriers, non-bonded repulsions and other effects in these small peptides. One important result from these computations is the direct and purely quantum mechanical demonstration of the existence of 0 bends in proteins via the C,, hydrogen bonds in the tripeptide (Parts 3, 5 [l] ) and the existence of the (Yhelical hydrogen bond in the tetrapeptide (Part 7 [l] ). This work is now extended to the dipeptide built from the alanine residue (Fig. 1) and its conformal energy map is computed (Fig. 2). Alanine occurs widely in proteins and it may be the case that amino acids with other side 0166-1280/81/0000-0000/$02.75
0 1981
Elsevier
Scientific
Publishing
Company
108
Fig. 1. The dipeptide of L-alanine. This is the extended conformation (o = -180”, i! = -- 180’). The standard bond lengths and angles are: NC = 1.380, CO = 1.220, CH (of CO) = 1.110, CC = 1.519, NH = 1.022, CN = 1.459, CH (of CH,) = 1.124, HNC = 120”, HNH = 120”, NC0 = 120”, NCC = 113”, NCH = 113”, OCC = 127”. Further discussion of the geometry is given in Part 1 (1). A positive rotation corresponds to H,, moving downwards and 0 j moving upwards (note that the positive rotations were wrongly defined in Part 2 (1)).
chains do not behave very differently from alanine in some respects because the additional atoms in their side chains are some way away from the backbone of the protein. The first step in this work is to construct the dipeptide map for the alanine residue in the usual way (Part 2 [ 11). There is of course no symmetry in the case of alanine as there is for glycine so it is necessary to vary both dihedral angles over 360”. In addition, there is the question of the orientation of the methyl group in the alanine residue. It is expected from the results of the
Fig.
2. The conformational
putations at 20” intervals. minima or saddles.
energy There
map of L-alanine. is no regular
contour
The map is constructed interval.
Squares
mark
from
com-
maxima,
109
earlier PCILO computations [3] and from general experience on torsion barriers that the staggered conformation of the methyl group with respect to the C” atom and its bonds will form the low energy conformation of the methyl group. Accordingly, we constructed the entire map with this staggered conformation as the standard arrangement of the methyl group and then checked the assumption by optimising the rotation angle of the methyl group at various important points of the map and found it to be correct. The alanine map is then used as the basis for comparisons with experiment with the glycine map and with the PCILO map. Some further analysis of nonbonded repulsions and of the significance of populations is also reported. THE AB INITIO
MAP OF L-ALANINE
The ab initio map of L-alanine is shown in Fig. 2 and the corresponding glycine map in Fig. 3. The latter map has been redrawn from the original (Part 2 [l] ) to match the alanine as well as possible. The zero of energy used in the alanine map is the C7 hydrogen bond on the left of the map ($J = --SO”, J, = 80”) and th’is is the hydrogen bond sometimes called [3] “equatorial”. The choice of the zero of energy is unimportant when working with a single map but it becomes important when comparing two maps and then it is by no means a simple matter (see below). There are numerous different comments and comparisons which may be made with this alanine map and it is convenient to divide these into the sections which follow.
Fig. 3. The conformal energy map of glycine. The map is constructed from computations at 20” intervals. There is no regular contour interval. The squares mark maxima, minima or saddles. This is a redrawn version of the map in Part 2 [ 11.
110 COMPARISON
BETWEEN
EXPERIMENT
AND THE ALANINE
MAP
The first task is to show how well the results correspond to reality. It is possible to cover more than a small fraction of the available evidence if we work with the broad view that alanine will represent all the amino acids except glycine so we are concerned here only with the objective of showing that there is general agreement between the theory and the experiment and not with specific cases. The first such comparison is that of Fig. 4 which reproduces in modified form the well known [ 41 Ramachandran plot of some experimental and general results versus the ab initio results. It is clear that all this information fits very well into the ab initio map. Thus, the p sheet structure at the top left (see below), the 310, CY~and 77helices, the C7 hydrogen bond and even the left handed CYhelix on the right of the map all fall into the low lying regions of the map. This degree of success, which was not so obvious with the glycine map (Part 2 [l] ), demonstrates the validity of the whole procedure and represents a notable achievement for the ab initio method which, being free from assumptions and parameters (in the usual sense of semi-empirical theory), cannot generate false agreement between theory and experiment. Going on to a more detailed comparison between theory and experiment, we take the X-ray data on lysozyme [ 51, a-chymotrypsin [ 61 and insulin [ 71 and compare this data with the alanine map. Taking the smallest molecule, insulin, first the results (Figs. 5 and 6) for both the A and B chains of insulin
-%C”
-126
-eo-
_&
d
id
a6
.zo”
i6G
Fig. 4. Dihedral angles in proteins and the alanine map. Taken from ref. 4 p. 27. The dotted lines define the areas available when a hard sphere model and either of two sets of internuclear distances is used. Details are given in the reference.
111
Fig. 5. The insulin A chain and the alanine map. All of the residues for the A chain are included on this map. Data from ref. 7 where further details are given. There is no glycine in this molecule other than the first residue.
Fig. 6. The insulin B chain and the alanine map. The triangles denote glycine residues. The proline residue is omitted. Further details in ref. 7. The squares and circles refer to chemically identical but crystallographically different molecules. See text concerning the glycine residues.
112
are the work of Blundell, Dodson, Hodgkin and Mercola and each chain occurs in two crystallographically different environments. The A chain contains no glycine residues and all the results for the other residues fit onto the alanine map with reasonable accuracy bearing in mind that uncertainties of at least 25” are inevitable in the theory and perhaps similar errors will arise in the experimental values. The B chain contains three glycine residues and we have put these onto the alanine map to show that they do not fit this map at all although they do fit the glycine map (Fig. 3). The other residues fit onto the alanine map reasonably well. The lysozyme data (Fig. 7) are those reported by Phillips [ 51 and these also fit the alanine map reasonably well when the glycine residues are excludec There is the exception of the phenylalanine residue on the right hand side which occurs in a region far removed from any low energy area so it seems doubtful that this point can be correct. The glycine residue results are collected together in Fig. 8 and again there is good agreement with the numerical results. The data for the largest of the three molecules, a-chymotrypsin, are those reported by Birktoft and Blow [6]. The glycine residue results are again separated out and shown in Fig. 9 while the results for all the other residues are shown together in Fig. 10. There seems to be a generally good fit between experiment and computation in this case as well as in the other two cases. There are of course many other collections of relevant experimental data which could be compared with these maps for glycine and alanine but these
-16$
-120’
-so”
-40°
0'
4d
SB
120°
160°
Fig. 7. Lysozyme and the alanine map. All residues other shown. The point to the right (0 = 120” ) is phenylalanine. details are given.
than glycine and proline are Data from ref. 5 where further
113
Fig. 8. Glycine residues details are given.
Fig. 9. Glycine
residues
in lysozyme
and the glycine
in ol-chymotrypsin
map. Data from
and the alanine
map. Data
ref. 5 where
from
ref. 6
further
-lEOO
-Ima
-xl”
-40’
on
4.0’
ad
Fig. 10. ol-Chymotrypsin and the alanine glycine and proline are included here.
120’
1fxJ~
map. Data from ref. 6. All residues
other
than
three sets of data contain over four hundred pieces of experimental information and, in so far as we are concerned to establish the general validity of the computational results, such a test seems more than adequate. We propose to defer a complete analysis of all the available data until some computations on other residues are completed. This is essential for the proline residue of course but some of the bulky side chains also need examination. The p sheet is widespread in silks [ 81 although a few cases of helical structures in some special cases, including a number of insect silks, are known. In these special cases, however, the glycine content is much lower (ref. 8, p. 337) and this is consistent with the result that glycine favours sheets or bends rather than helices. The pleating of the sheets is also consistent with the alanine map. At the same time, it must be appreciated that matters are not by any means as simple as these preliminary comments might suggest. For example, the p silks do contain large amounts of glycine so we cannot assume that sheets are free from glycine to any extent. It is plain that much detailed study of the other residues as well as glycine and alanine from both experimental and theoretical points of view will be required before clear generalisations are possible. Another necessary reservation is that the X-ray results themselves contain elements of prejudgement and preselection of the probable structures of the molecule which is being examined - usually via some kind of model building to guide the analysis - so that agreement between theory and experiment is a two way affair and the results of the computations are helpful in guiding
115
the experimental work. Again, we must wait for detailed computations on at least some of the additional residues, particularly the bulky and charged ones, before some of the well known [8] preferences for a given conformation by given amino acid residues can be completely understood and analysed in depth to the point of seeing the influence of the molecular structure on the secondary and tertiary structures. COMPARISON
BETWEEN
THE ALANINE
AND GLYCINE
MAPS
The glycine map (Fig. 3) has been redrawn from the earlier publication (Part 2 [l]) but it is still based on the extended form rather than the C7 hydrogen bond as the zero of energy. To convert to t.he hydrogen bond as the zero of energy, subtract 1.0 kcal mall’ from all the contours of the glycine map. It is helpful to look at the glycine and alanine maps side by side because many of the features occur on both maps and this is useful in confirming the internal consistency of the method and the absence of trivial errors. It also greatly simplifies the interpretation of the maps, particularly that of alanine, since, if the established chemical idea that a methyl group will do little or nothing other than add repulsions at certain conformations is correct, then it should be possible to construct a difference map between the two maps. Such a difference map should be everywhere positive and give a quantitative estimate of the repulsive interactions. This map is shown in Fig. 11 and it is clear that this expectation is fulfilled with remarkable accuracy. For example, a model suggests that in the top left hand corner of the map no repulsive terms are expected and over the whole of this region the difference map has a value of less than 1.0 kcal mol-’ (after adjustment for the difference in zeros of the alanine and glycine maps). Moreover, the bottom left hand comer shows the hydrogen--hydrogen repulsions between the methyl group and the amino group while the top right hand corner shows the oxygen-hydrogen repulsions between the methyl group and the carbonyl group. The bottom right hand comer is the sum of the two kinds of repulsion as expected. The clarity and directness of these results is pleasing and we take up the discussion of the quantitative analysis of the non-bonded repulsions in the section below. Looking at the glycine and alanine maps themselves now, the left hand C, hydrogen bond at about @J= -SO”, $ = + 80” occurs in both maps and so does the low energy region close to the extended form in the left hand corners. The hydrogen-hydrogen repulsion at the middle left of the maps and the very high energy central regions of repulsion between non-bonded atoms occur in both maps and these results are well known and well understood [3]. We have said that the new features of the alanine map are those imposed by the repulsive interactions between the methyl group and the rest of the molecule. One consequence of the presence of these repulsions is that the C7
Fig. 11. Alanine minus glycine difference map adjusted for the difference drawn from the numerical values not from the contours. No information the hatched areas.
in zeros. It is is available for
hydrogen bond on the right of the map at about Q = SO”, $ = ---SO” is no longer a low energy point. It is shown below that the populations demonstrate that the hydrogen bond is still present, as expected, but the nonbonded repulsions of some 12 kcal mol-’ mask the hydrogen bond energy of some 3 to 4 kcal molF so that the hydrogen bond is hardly visible on the aldnine energy map. Another new result of the alanine map is that the extended form is no longer the minimum of energy because the repulsions created by the methyl group move the minimum of energy to about 4 = -160”, $ = 160”. In addition, this minimum is now above the C, hydrogen bond in energy rather than just below it as in the glycine case. This result seems to be confirmed [4, 81 by the experimental information on the fl pleated sheet of silk and other fibrous proteins so we may reasonably claim to have an explanation of why such sheets show the well known pleating effect. The arguments of the last paragraph can be extended if we accept the validity of the result that in glycine the extended form is of lower energy than the C, hydrogen bond while the converse is true in alanine. This provides a tentative explanation of the fact that glycine residues tend to form sheets rather than helices while alanine residues do not form sheets as readily as they form helices [S] . In other words, the methyl group’s presence raises the energy of the p sheet structure but leaves the helix unaffected. If this explanation is correct (and it must be tentative because the energy differences are small (cf., Part 4 [ 1 ] )) we have an explanation
117
of why polyglycine forms sheets while polyalanine and other such polymers form helices [4, 81. There are some other comparisons between the alanine and glycine maps but these are best made in the later sections in other contexts. SPECIAL POINTS CONCERNING
THE ALANINE
MAP
We commented earlier that the conformal energy maps are potential sources of information about the energy changes which result from changes of shape of the peptide and protein molecules. Such changes of shape are thought to be important in such processes as complex formation, chemical reactions generally, the allosteric effect and other physical processes [ 91. The glycine map offered no clear examples of this point at least in a qualitative sense because all points of the map may be reached from all other points with small increases (about 5 kcal mol-’ at most) in the total energy. Such energy quantities correspond to several quanta of vibrational energy and are readily available within molecules. The situation is different in the case of alanine because the map shows a “locked” point at about @= 40”, ii, = 60”. By the term “locked” is meant a situation in which a region of the map is completely surrounded by a barrier much higher in energy than the locked region. In the present case, th% region is surrounded by a barrier at least 10 kcal mall’ higher in energy than itself. There is some scope for a conformational change in which ic, increases in value towards the top of the map but this valley is closed. Thus it is possible that a molecule, once formed within this conformational region, would be unable to leave the region by a simple change of conformation although it could of course leave the region by a reaction with another molecule. Equally, the inaccessibility of this region may be why we never find the left handed helix. We know of no clear example of such a process in the experimental sense but it is interesting to notice that this region of the alanine map is close to the left handed helix and relevant experimental evidence would be very useful. There may be differences in chemical reactivity between the right handed and the left handed helix, for example, which are due to the different conformational situations of the two helices. The other special point about the alanine molecule is the question of the conformation of the methyl group itself. As noted above, the staggered conformation of the methyl group with respect to the bonds from CO was chosen as the standard arrangement since this is the lowest energy conformation in simple cases. The latter was verified by optimising the rotation angle at various points of the alanine map (see Table 1). In only one of the eight conformations examined is there appreciable departure from the staggered conformation and this is a point of very high energy (about 25 kcal mall’) close to a steeply rising region of the map where the results are somewhat uncertain. The other seven points are all of lower energy and are consistent in giving an optimised rotation angle close to the staggered
118 TABLE
1
Optimised
rotation
I$’ = -40 o =-180 57.0 XI=
angles (in degrees)
80 -80 65.3
-40 -60 60.5
for the methyl
-120 -140 60.0
group
of alaninea 60 40 61.2
160 -160 59.4
90 0 64.0
-100 0 47.5
aThe starting value of x, is always 60” and the angle used in the optimisation is 30” The energy change associated with the optimisation is always negligibly small.
value of 60”. Thus there is little or no reason to expect the operation of any kind of cogwheel effect between the methyl group and the other groups in this molecule. This is presumably due to the fact that the methyl group is more nearly cylindrically symmetrical than the conventional formulation as CH3 would imply. It should perhaps be pointed out that the above test is not foolproof since we do not simultaneously optimise all three angles but it seems unlikely that this is a serious reservation. NON-BONDED
REPULSIONS
AND REPULSIVE
CURVES
Earlier work showed that it is possible to extract from the dipeptide map (Part 2 [ 11) estimates of the non-bonded repulsions between hydrogen and hydrogen and between oxygen and oxygen atoms. The same information is repeated within the alanine map but there is also new information contained in the difference map (Fig. 11) which deals with hydrogen--hydrogen again and also with hydrogen-oxygen atoms. There is now the complication that interactions with the methyl group involve two of the hydrogen atoms of this group so the procedure is not straightforward.
Fig. 12. Hydrogen-hydrogen repulsion. The open circles are the data for the glycine dipeptide from Part 2 [ 11. The triangles and squares are the data from the repulsion of the methyl group and the amino group of alanine. The new data have been added assuming a zero of 4 kcal mall’ as the plateau of the earlier map. Fig. text
13. Hydrogen-oxygen (cf. Table 2).
repulsion.
The construction
of this figure
is discussed
in the
119
The hydrogen-hydrogen repulsions arise between the methyl group and the amino group and these prove to be additive in a simple way so that we may add them directly to the curve obtained earlier as in Fig, 12. In placing the new points onto this curve, they are referred to the zero of 4 kcal mall’ of the original curve and it is clear that they fit this curve accurately. The hydrogen-oxygen repulsions arise between the methyl group and the carbonyl unit and they involve two hydrogen atoms of the methyl group. The relevant numerical values are reported in Table 2 and the separation of the overlapping effects of the two hydrogens is less straightforward than in the hydrogen-hydrogen case above. We begin by assuming that the extreme values (1, 2 and 8, 9) are straightforward because one of the two distances is very large. This gives two points on the graph of Fig. 13. We may now deduce that points 3 and 7 are simple because one of the two distances is 2.90 A and we know the interaction energy is zero ab this distance from the first two points. Then we take the points 4 and 6 and, reading the energy value corresponding to 2.38 A from the graph, we conclude that a distance of 1.68 A corresponds to about 21 kcal mol-‘. This deals with all the points save 5 and this cannot be accounted for by simply dividing by two as one might expect (the cross on the graph of Fig. 13). We cannot account for the failure of point 5 to fit the graph unless the simple idea of atom--atom repulsion is breaking down and three body effects are important here. In any event, this result is a warning against too simple an interpretation of the nonbonded atom idea. This latter result apart, we have constructed a simple version of a hydrogen-oxygen repulsion curve. ATOM
POPULATIONS
AND HYDROGEN
BONDING
In the earlier papers of this series, we paid a great deal of attention to the atomic populations not only because they may represent in some sense the chemical concept of the number of electrons associated with an atom but TABLE
2
Non-bonded
repulsion
between
hydrogen
and oxygen
atoms=
Serial no.
9 (“) G,,-H,, G,,-H,, Energy
1
2
3
4
5
6
I
8
9
40 3.88 2.64 0.5
60 3.42 2.13 2.1
80 2.90 1.77 9.6
100 2.38 1.68 22.2
120 1.93 1.93 25.2
140 1.68 2.38 22.2
160 1.77 2.90 9.7
180 2.13 3.42 2.0
200 2.64 3.88 0.5
aThese values all correspond to 4 = 60”. The lengths are in A, the energies in kcal mall’. The atom numbering is that of Fig. 1. The table is symmetrical left right but is printed in full to match the figure clearly. The method of converting these values into Fig. 13 is discussed in the text.
120
also because they are a summary of the form of the wave function. Neglecting the populations is, in effect, neglecting the wave function itself as a possible source of information. And since the populations are closely connected with the large diagonal elements of the one-particle density matrix the use of populations is connected with the study of the wave function through the density matrix (which is the invariant of the Fock theory). The earlier studies of the populations were in some ways disappointing, perhaps because in these molecules we are faced with a combination of hydrogen bonding and polarisation effects which cannot be easily disentangled. The major positive result from the earlier work was that the changes in populations of the nitrogen, hydrogen and oxygen atoms on hydrogen bond formation do seem to be reproduced in a systematic manner in different molecules. These changes may be summarised by saying that when the hydrogen bond forms the populations change as follows 7.435
N to 7.448
0.810
H to 0.786
8.267
0 to 8.285
Considering alanine, Table 3 contains the populations of the atoms at four different conformations, the low point near the LYhelix, the extended form, the hydrogen bond to the left of the figure and the other hydrogen bond to the right of the figure whose presence is masked in the energy sense by the non-bonded repulsions. The underlined values are those of the hydrogen bond. It is clear that the low point near the helical region has “normal” populations, the extended form shows intermediate populations as before and the two hydrogen bond cases show values of the populations which fit the earlier values summarised above. This is quite straightforward in the case of the left hand hydrogen bond but in the case of the masked hydrogen bond it is only the populations which reveal its existence clearly. This is an unusual application of the population idea and it suggests that despite some difficulties with the concept of populations we should persist in using it with these molecules. TABLE Atom c,
c2
3 populations c,
H,
6.154 5.688 0.922 5.944 6.154 5.690 0.923 5.944 6.151 5.697 0.925 5.946 6.172 5.698 0.925 5.943
in L-alaninea 0,
N,
H,
H,
8.283 0.799 7.436 0 810 8.290 0.796 7.433 0.807 8.295 0.807 7.446 0.784 -8.293 0.807 7.446 0.785 --
N,
Hz,, C,, O,z
5.755 7.373 8.264 0.813 5.754 7.370 8.280 0.796 5.746 7.365 8.287 0.803 5.741 7.365 8.289 0.801
H,, N,,
H,; H,,
0.944 0.944 0.940 0.929 0.943 0.941 0.930 0.948 0.938 0.935 0.944 0.934 0.934 0.899 0.950 0.950
01helix region extended form hydrogen bond (left) hydrogen bond (right)
aThe 01helix region is 0 = 40”, + = -6O”, the extended form is @ = -160”, IJJ= 160”: the left hydrogen bond is @ = --8O”, ti = 80” : the right hydrogen bond is 0 = 80”, + = -80”. The atom numbering is given in Fig. 1.
121 COMPARISON
BETWEEN
THE PCILO MAP AND THE AB INITIO MAP
It is not possible to make detailed comparisons with all the published maps for the alanine residue at this stage but there are interesting points in the comparison with the PCILO map [ 31. This latter kind of computation was the pioneering work on the quantum mechanical study of these conformation problems and there are many points of similarity between this map and the ab initio one. There are also some serious differences, however, and these are discussed below. The many points of similarity between the two maps include the raised area at the bottom left of the maps due to hydrogen-.-hydrogen repulsion, the low energy of the extended structure and the presence of the C, hydrogen bond at about 4 = --80”, $ = 80”. Also, in a general sense, the increase in energy of the right hand side of the map as one goes from glycine to alanine is reproduced in the PCILO map of alanine. When one looks at the PCILO and the ab initio maps in more detail, however, some major differences emerge even in a qualitative sense. For example, the increase in the energy of the right hand side of the map is very much larger in the ab initio map than in the PCILO map and, in particular, the right hand hydrogen bond which is masked by non-bonded repulsions of about 10 to 15 kcal mall’ in the ab initio results is still a low energy point in the PCILO map. Indeed, the PCILO results suggest that the two C, hydrogen bonds are of nearly the same energy in the alanine map. Another difference between the two kinds of map is the presence or absence of the two channels between the central high energy regions. These are at about o = 0” and ic/ = -80” or + 80” and although we know of no immediate practical significance of these channels they are clearly of importance in the questions of changes in conformation (see above on the locked conformation). The PCILO results suggest that these channels are present in the alanine map and have an energy only 2 to 3 kcal mol-’ above the minimum energy in the region of the saddle point. The ab initio map shows these channels but it suggests that they are about 16 and 24 kcal mall’ above the minimum energy. This is a major difference between the two maps and it could have the practical result of making the left handed helix either accessible or not accessible to conformational changes. A third difference between the two maps is concerned with the methyl group. The ab initio map suggests that the presence of this group moves the minimum of the extended form away from the symmetrical structure at --180”) --180” to the pleated structure at about -160”, 160”. The PCILO map does not show this effect of the methyl group. To summarise the general position then it seems that there are many points of agreement between the PCILO map and the ab initio map and it is certainly true that the former is very much less expensive in terms of computer time. On the other hand, where the two methods disagree it is very difficult to think that the PCILO results can be correct. Both the agreement
122
with experiment and the intrinsic reliability of the theoretical method favour the ab initio results quite clearly. The general picture which emerges from the comparison between PCILO and ab initio methods is that the PCILO method underestimates the importr ante of the non-bonded repulsions within a molecule. All of the points of disagreement set out above would disappear were it possible to increase the magnitude of the non-bonded repulsions within the framework of the PCILO method. Until this has been done, however, it must be borne in mind that the PCILO method may generate false minima in these conformal energy maps and in similar situations. DISCUSSION
AND SUMMARY
This conformal energy map of the alanine residue is important both for the alanine residue itself and particularly because it seems to be possible to use the alanine residue to facsimilate many of the other residues, excluding amino acids with functional groups in the side chain particularly if these are charged (NH,+, CO*-). Within th is reservation, however, the generalisation leads to useful results as the comparisons with experiment in the insulin, lysozyme and a-chymotrypsin cases show clearly. The general effectiveness of the ab initio alanine map in reproducing the experimental facts has been described briefly but there exists a large amount of information with which these results could be compared and the authors would be pleased to hear of such comparisons. The implications of these alanine results for the Cl0 and CL3 hydrogen bonds discussed in the earlier parts of this series of papers (Parts 3 and 5 [ 11) are important and will be taken up elsewhere. It is important to appreciate that computations of the present kind are genuinely quantum mechanical in their basic nature and so are free from built in assumptions about physical factors, such as hydrogen bonding and non-bonded repulsions, in any direct sense. This is not true of the empirical or partitioned potential energy methods which do contain such built in assumptions together with disposable parameters whose values may be adjusted to suit preconceived ideas about the results. At the same time, it must be pointed out that the Gaussian 76 method in the short base level of approximation is very far from basis-set independent as we have discussed at some length in an earlier paper (Part 4 [l] ) in connection with molecules of this kind. All that can be said at the moment in a practical sense is that if any one result or numerical value is particularly important for some purpose, it should be checked carefully with a long base, with polarisation functions or with other refinements of the method to be sure that the value obtained with the short base is not too far from the experimental reality. ACKNOWLEDGEMENT
We are indebted to one of our students, with the computations.
Mr. M. Wickens,
for assistance
123
REFERENCES 1 D. Peters and J. Peters, J. Mol. Struct., 50 (1978) 133 (Part 1): 53 (1979) 103 (Part 2); 62 (1980) 229 (Part 3); 64 (1980) 103 (Part 4); 68 (1980) 243 (Part 5); 68 (1980) 255 (Part 6); 69 (1980) 249 (Part 7). 2 W. J. Hehre, W. A. Latham, R. Ditchfield, M. D. Newton and J. A. Pople, GAUSSIAN 70 and 76, QCPE 236, Indiana University, Bloomington, IN, 1971 and 1976. 3 B. Pullman in B. Pullman (Ed.), Quantum Mechanics of Molecular Conformations, Wiley, London, 1977, Ch. 4: A. Pullman and B. Pullman, Adv. Prot. Chem., 28 (1974) 347. 4 R. E. Dickerson and I. Geis, The Structure and Action of Proteins, Benjamin/Cummings, Menlo Park, California, U.S.A., 1969. 5 D. C. Phillips, Proc. Nat. Acad. Sci. U.S.A., 57 (1967) 484. 6 J. J. Birktoft and D. M. Blow, J. Mol. Biol., 68 (1972) 187. 7 T. L. Blundell, G. G. Dodson, D. C. Hodgkin and D. A. Mercola, Adv. Prot. Chem., 26 (1972) 279. 8 R. D. B. Fraser and T. P. MacRar, Conformations in Fibrous Proteins, Academic Press, London, 1973, Chs. 9 and 10. 9 K. J. Laidler and P. S. Bunting, The Chemical Kinetics of Enzyme Action, 2nd edn., Clarendon Press, Oxford, 1973, Ch. 7 and p. 256.