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Calculation of protein volumes: An alternative to the Voronoi procedure
J. Mol. Biol. (1982) 161, 305-322
Calculation of Protein Volumes : An Alternative to the Voronoi Procedure B. J.
University
GELLATLYt
AND
J. L.
FINNEY
Department of Crystallography, Birkbeck College of London, Malet Street, London, WClE 7HX, U.K.
(Received 18 March 1982, and in revised form
17 June 1982)
All the methods that have been applied to assessing local volume occupation or packing in proteins have particular defects. For example, the Voronoi method used by Richards (method A) and by Finney misallocates both non-bonded and covalent contacts in a geometrically rigorous, though chemically inconsistent, manner. Richards’ method B, in which covalent and non-bonded contacts are partitioned in chemically sensible ways, is unfortunately not completely rigorous, in that every polyhedron vertex has associated with it a small vertex-error polyhedron, which is not allocated to any atom. We present here a generalization of the Voronoi method that is particularly suited to multicomponent assemblies such as proteins. This radical plane partitioning of volume is completely rigorous; it gives rise to no vertex error, and handles the more numerous non-bonded contacts realistically. Its application to RNAase-S is described and the results compared with both Voronoi’s method and Richards’ method B. A particular advantage of both the radical plane and Richards’ methods is a relative insensitivity to the treatment of the surface, a problem that has plagued other approaches to describing packing in proteins. Although the radical plane is seen to misallocate volume chemically between covalent+bonded neighbours, this problem vanishes when groups of atoms in sidechain residues are considered.
1. Introduction Although globular proteins are often described in terms of their secondary structure, such a characterization is not well-suited to showing regularities that may be present at the lower atomic level of organization. Attempts have therefore been made to produce alternative ways of rationalizing these structures. One such approach uses the concept of geometrical packing, which has been very successful in describing the properties of condensed phases of other systems (Finney, 1970,1975b; Barker et al., 1975). The volume occupation of component atoms, groups and residues provides a systematic way of analysing protein structural data. Amongst other things, this gives information on local volume variations, which may be related to local flexibility (Richards, 1974; Finney, 1975a). t Present
address:
0022-2836/82/300305-18
The Emmbrook $03.00/O
School,
Wokingham, 305
Berkshire, 0
U.K
1982 Academic
Press Inc.
(London)
Ltd.
306
R. J. GELLATLY
AND
J. L. FIN%EY
Previous work using this approach has been hampered by two major difficulties : (1) defining the protein-solvent boundary adequately ; and (2) finding an accurate method of volume partitioning that is physically reasonable for multicomponent systems. We discuss here an alternative approach to this latter problem, using a modification of our previous method to position the protein-solvent boundary. The most widely used partitioning method is due to Voronoi (1908) : however, this is not satisfactory for systems containing different-sized atoms. Richards (1974) developed an alternative to this, using covalent or van der Waals’ radii to define the partitioning, depending on the type of interaction involved. In an attempt to overcome the problem of unallocated volume inherent in this method, we have adopted a related method based on the radical plane construction (Fischer & Koch, 1979). The radical plane division is defined in terms of both the van der Waals’ radii of the atoms and the distance between them. Because of this, \ve expect some problems in dealing with covalent and hydrogen-bonding interactions. where some misallocation of volume will inevitably occur. However, this procedwe is completely rigorous, with no regions of space left unallocated. The technique has been used successfully in the characterization of binary non-crystalline systems such as amorphous metal alloys (Gellatly & Finney, 1982).
2. Materials
and Methods
(a) C’olumr partitioning The Voronoi (1908) method of partitioning the space occupied by an array of atoms gives a non-arbitrary space-filling of polyhedra, formed from planes bisecting the interatomic vectors perpendicularly. A polyhedron vertex is determined as a point equidistant from 4 non-coplanar atoms, which is closer to no other atom in the assembly. Hence the solutions of the equations : (zi-~)2+(yi-y)2+(Zi-z)2
= LE. i = 1.4..
.(l)
give the co-ordinate of the vertex (zyz) and the vertex-centre distance (L,). (zigizi) are the coordinates of the four atoms, i = 1,4 (see Fig. 1 (a)). Each vertex is checked to ensure that no other atom is closer than the current vertex-centre distance L,. For neighbouring atoms of unequal size, the dividing plane may lie well inside the sphere representing the larger atom. Hence too much volume is allocated to the smaller atom and too little to the larger one (Fig. l(b)). The method devised by Richards (1974) attempts to overcome this difficulty and provides a physically reasonable volume partitioning. The dividing plane of covalently bonded atoms (Fig. Z(a)) is drawn perpendicular to the interatomic vector and positioned to divide it in proportion to the assumed covalent radii. For non-bonded interactions van der Waals’ radii are used to determine the position of the plane (Fig. 2(b)). However, this method is nonin that the assembly of polyhedra does not completely fill space; the rigorous to neighbourhood of each vertex contains a volume (the “vert,ex error”) that is unallocated any polyhedron except in the case of exactly touching spheres (see Fig. 3). This vertex error can, however, be avoided, and the rigour of the Voronoi construction can be generalized for unequal spheres. To do this, we use a partitioning method based on radical planes (Fischer & Koch, 1979). The radical plane of 2 spheres is the locus of points from which the tangent lengths to the 2 spheres are equal (see Fig. 4(a)). (z~z) and (zivizi) are defined as for the Voronoi method, with vwi the assigned radius for each atom and L, the vertex-tangent length. The required equations now become : (~~-z)~+(y~-y)~+(.q-z)~--r~~
=L:,
i = 1,4,
CALCULATION
OF
PROTEIN
307
VOLUMES
(0)
FIG. I. Voronoi construction (2 dimensions): distances to the 4 spheres are equal, d, = d/,; atoms.
(a) a vertex is the point such that the vertex-centre (b) positioning of dividing planes for 2 unequal-sized
with the distance check now being for the vertex-tangent length. The definition handles non-overlapping, touching (common tangent), and overlapping (common chord) atoms equally well with an apparently reasonable partitioning of volume between different sized atoms (Fig. 4(b)). (b) Surface treatment The volume of a surface atom will normally be undefined by any of these spacepartitioning procedures, and it is therefore necessary to provide “hypothetical solvent“ positions to define vectors for the normals of the polyhedron faces. In this work these were placed at a uniform density on the surfaces of spheres centred on the surface atom as described previously (Finney et al., 1980). A maximum of 24 such points were added at relative positions corresponding to a snub cube arrangement (Mackay et al., 1977). In addition “tetrahedral” surface points were placed at the surface sphere intersections as described previously (Finney, 1975a). The sphere radius was taken as the sum of the van der Waals’ radius of the surface atom and that of the surface probe. The values of van der Waals’ radii were as used before (Finney, 1975a) (see Fig. 5).
FIG. 2. Richards’
construction
(a) covalent interactions: (b) Non-bonded
interactions:
(2 dimensions) :
drc L d, = -_ r,,+r,, d, = r,, +
d-r,,
2
-rwz
,
where rCi and rwi are the covalent and van der Waals’ radii, respectively,
of atom i.
B. J. GELLATLY
308
AND
J. L. FINNEY
(b)
FIG. 3. Vertex error in Richards’ method associated with: (a) covalent and non-bonded interactions; (b) non-bonded interactions. (c) There is no vertex error for the special case of “touching” non-bonded atoms. performed
(c) Calculations
To allow comparisons to be made between the 3 methods, calculations using Voronoi’s (V), Richards’ method B (B), and radical planes method (R) were performed on the ribonuclease S co-ordinates used previously (Wyckoff et al., 1970). Identical surface treatments (the point
addition described in section (b), above) and van der Waals’ radii were used. For the Richards’ method calculation, the covalent radii reported previously (Richards, 1974) were used (see Fig. 5). Surface probes of radii 1.4 A and 1.7 A were used in separate calculations to examine the sensitivity of the methods to assumptions concerning the probe size. Thus, 6 data sets designated: V4, V7, B4, B7, R4 and R7 were obtained. The letters V, B and R refer to Voronoi, Richards’ B, and radical methods, respectively, while the integer denotes the radius of the surface probe.
Gp
(a)
tb,
FIG. 4. Radical plane construction (2 dimensions): tangent distances to the 4 spheres are equal : d, = rw12-rw2 ‘+d’ 2d (b) positioning
(a) a vertex is the point such that the vertex-
;
of radical planes for unequal-sized
atoms.
CALCULATION
OF PROTEIN
309
VOLUMES
3. Results (a) Atom volumes Figure 5 shows the mean polyhedron volumes for each of the sets of calculations. Several points emerge. First, the Voronoi volume varies strongly with the probe radius by up to 20 to 25%. This is to be expected as the position of the partitioning plane between a surface atom and the probe will be dependent upon the probe radius, being further from the surface atom centre for the larger probe. In contrast, volumes calculated by both Richards’ and radical methods depend much less on the probe. This insensitivity to probe radius reflects the placing by both methods of the dividing plane in the common tangent position for two touching unequal spheres (Figs 3(c) and 4(d)). Considering the problems of surface treatment (Richards, 1974: Finney, 1975a), this relative insensitivity of both the radical and Richard’s B techniques is a major advantage over the Voronoi subdivision. The residual small differences within one method can be explained by the finer surface definition consequent on the use of the smaller probe. Secondly, for the atoms or atom groups in the centre of Figure 5, (say, ring C to O/OH), Voronoi volumes are approximately constant, while both Richards’ and radical procedures produce mean volumes that vary by greater than a factor of two. This is consistent with the Voronoi procedure, misallocating volume between neighbouring unequal atoms in such a way as to smear out the difference between the larger and smaller atoms (Fig. l(b)). I n effect, the Voronoi partition fails to reflect adequately the packing constraints within the molecule, a serious defect if we are using the procedure to examine effects depending on packing considerations. The differences between the mean volumes given by radical and Richards’
35 30 s
25-
E s 20> 15 IO -
I CH, r.
G
NH;
2.0 2.0 0.77 0.70
tic&
0
Pro CH,
OH
CHf
CH,
2.0 1.4 I.6 0.77 0.66 0.66
2.0 0.77
2.0 2.0 0.77 0.77
+
Atom
O/NH
O/OH
I.6 1.5 0.70 0.66 type
His C/N
N
Cl+@
C’
I.6 I.7 2.0 0.70 0.70 0-77
2.0 0.77
f
+
&I
C
1.7 1.7 0-77 0.77 9
FIG. 5. Mean volumes for atom groups (‘I, V7 ; A, V4 ; l , R7 ; n , R4 ; 0, B7 ; 0, B4). The van der Waals’ (r,) and covalent (r,) radii used are given below each atom type. Arrows indicate main-chain atoms. Radical and Voronoi values for C carbonyl groups are superimposed.
d
Lcl -
0-
-u-l
0-
huanbaq
y.
In
0-
IJ-l
312
B. J. GELLATLY
AND
,J. I,. FINNEY
methods are generally (though not always) smaller than the difference between either and Voronoi’s method. Moreover, for atoms with assigned van der Waals’ radii greater than about 1.7 A (an approximate average van der Waals’ radius for the protein atoms or atom groups), radical tends to yield the larger volume, while for smaller atoms the reverse is true. This suggests the results might be reflecting the different sensitivity to local internal environment of the two methods. A detailed examination, however, shows that these differences are quite subtle, and focus on problems of reconciling chemical reasonableness with geometrical consistency. This point is discussed usefully in connection with volumes allocated to the main-chain groups as follows. (b) Main chain atoms (i) Volume distributions The main chain atoms form the largest group of atoms with approximately constant covalent environment. Therefore, the volume distributions of Figure 6 reflect the variations in packing efficiency (including hydrogen bonding) as assessed by the three methods. We would hence expect that the superiority of one method over another would show up as a significantly reduced volume distribution, as is found when the radical method is applied to binary alloy glasses (Gellatly & Finney, 1982). Looked at in terms of hypothetical experiment that might measure volume distributions, each volume partitioning method can be considered as having a “resolution function” with which the “measured” volume distribution is convoluted to yield the distribution of Figure 6. The “resolution function” of Voronoi’s method will be related to the misallocation of volume between unequal neighbours, while we would expect the differences between Richards’ and the radical method to be assigned to either the vertex error (Fig. 3) or the slight difference in partitioning non-touching non-bonded contacts (including hydrogen bonding) or both. Reference to the spreads of the distributions of Figure 6, however, shows no significant difference between the standard deviations of the radical and Richards’ methods. Moreover, and more surprisingly, only for main chain oxygen (Fig. 6(d)) is the Voronoi distribution significantly worse than either of the other “improved” methods. Even here, when the distributions are normalized to the mean volume, it could be argued that the relative spread (estimated as o/V) is not significantly worse for the Voronoi method. We interpret these results as showing that the variations in packing efficiency about these main chain atoms are significantly greater than the smearing effects of the effective resolution functions of all three methods. The poor shape of the Voronoi oxygen distribution (Fig. 6(d)) rather than the normalized width, argues against the use of the Voronoi method. Even though its effective resolution function is expected to be poor, the variations in local volume around main chain atoms are large enough to almost mask this defect. Although the statistics are poorer, this conclusion is generally consistent with the volume distributions of sidechain atoms. Exceptions for which Voronoi distributions are significantly broader than either of the other two methods (proline CH2, CH,, and ring carbons) argue
CALCULATION
OF PROTEIN
further against the use of the Voronoi method B gives broader distributions carbons) may possibly reflect the larger assigned van der Waals’ radii, though point.
313
VOLUMES
method. The two cases in which Richards’ than the radical method (CH, and ring vertex error expected for atoms with large no convincing data can be offered on this
(ii) Mean volumes Figures 5 and 6 both illustrate the differences in mean volumes P assigned by the three .methods. In order of descending values for van der Waals’ radius, the following generalizations can be made, bearing in mind the distribution widths : C” (r,
= 2.0 A) :
PR > Ps z P”
c (r, = 1.7 A): N (r, = 1.7 A):
PR - PB - P” %-tfs-~w though perhaps a small tendency for :
0%.
6(a))
0%. 6(b))
Pv > PB > PR P,> I&- PR.
(Fig. 6(c)) (Fig. 6(d)) Considering that the different volume partitionings depend on the assigned radii of neighbouring atoms, we might look for a rationalization of the above rather unclear pattern in terms of the van der Waals’ radii of the atom and of its average surroundings. For example, most main chain oxygens (r, = 1.4 A) are hydrogenbonded to a neighbouring NH group (r, = 1.7 A). Hence the Voronoi method will place the plane further from the oxygen (cf. Fig. l(b)) than either radical or Richards’ B methods and thus give a larger volume, as is observed (Figs 5 and 6(d)). These two latter methods treat this contact as a slightly overlapping van der Waals’ contact (Figs 2(a) and 4(b)) and hence will place the partitioning plane in similar positions. Applying a similar argument to the main chain nitrogen would suggest 7s - ps > pv. If anyth’ mg, however, the average Voronoi volume is slightly greater than those of radical or Richards’ B methods (Fig. 6(c)). The situation is apparently not quite as simple, and a more detailed examination requires us to look at the differences between the three methods in partitioning bonded contacts. 0 (r, = 1.4A):
(c) The “covalent
error”
(i) Main chain The three methods partition a covalent bond differently. Voronoi places the plane midway between atoms, while Richards’ B makes use of the ratio of assigned “covalent radii” obtained from fitting to an extensive data set (Richards, 1974). An inspection of the covalent radii used (excluding sulphur) shows a variation from 666 A to 0.77 A; this relatively small variation of about 17% will mean that the covalent partitioning for both Richards’ B and Voronoi will generally not be very different. In order to retain geometrical rigour, the radical method must use a van der Waals’ criterion to partition a chemical bond. The variation in van der Waals’ radii is much greater (-43%) than in covalent radii, and hence the positioning of the covalent plane will vary more. Moreover, the form of the equation (Fig. 4(a)) is such that for smaller separations of covalently bonded atoms, this “movement” of the radical partitioning plane is even greater (see also Fig. 10).
314
B. .J. GELLATLY
,4NI)
J. L. FINNEY
C”
INH
FIG. 7. B comparison of the partitioning produced by the 3 methods for main-chain (-), radical (. .), Richards’ B (- - - ).
atoms: Ywonoi’s
The relative effects of these differences on covalent partitioning in the main chain are shown in Figure 7. The similarity of the Richards’ and Voronoi partitioning is clear, as is the significant difference in the radical plane position for the C”--C, C=O and N-C” covalent links. Note that the equality of assigned van der Waals’ radii leads to identity between Voronoi and radical planes for C-N, but the Richards’ B plane is in a different position (unequal covalent radii). We now reconsider the nitrogen volume distributions of Figure 6(c) in the light of Figure 7. Whereas previously we considered only the placing of the plane between the NH and a neighbouring hydrogen-bonded O=C, Figure 7 shows the different covalent partitions must also be taken into account. With respect to covalent links, Richards’ B will tend to give a smaller “partial” volume than Voronoi, while the partitioning of the hydrogen bond will tend to enhance Richards’ over Voronoi’s method. This is consistent with an approximate equality in the Richards’ and Voronoi volumes observed (Fig. 6(c)): the two opposing effects (together with other close approaches not considered here) appear to cancel each other approximately. A comparison of the Richards’ and radical volumes will presumably be dominated by the N-C” radical plane being much closer to the N than the C” (Fig. 7). Thus, we would expect the radical volume to be lower than either Richards’ or Voronoi’s volumes. That this is indeed the case is seen in Figure 6(c), though the apparent smallness of the volume shift is presumably due to compensating effects from other non-covalent neighbours. A reconsideration of the mean volumes of the remaining main chain atoms further illustrates the influence of the covalent partitioning on the different meanvolume results from the three methods. For the oxygen, the O=C radical plane is closer to the oxygen than either Richards’ or Voronoi planes, and would suggest that 0, is less than both ps and pv, as is observed. That the difference between pa and vu is not greater must be due again to compensating effects (perhaps including relative weightings of certain planes in their contribution to total volume) from other non-bonded neighbours.
CALCULATION
OF PROTEIN
VOLUMES
315
For C”, Richards’ B and Voronoi’s covalent-link planes are in similar positions, suggesting Vv - VB, which is in fact observed (Figs 5 and 6(a)). The different positioning of the N-C” plane suggests a slightly larger volume from Richards’ B. an effect that is consistent with observation (Figs 5 and 6(a)). For the radical partitioning, both covalent-link planes are significantly further from the C”. suggesting the much larger VRvalue, as is observed (Figs 5 and 6(a)). We might note in passing that the atoms in the /3 position on all side-chains except glycine have both van der Waals’ and covalent radii equal to those assigned to C” and hence will not effect the C” volumes. Finally, the carbonyl carbon volumes are notable for their equality between methods (Figs 5 and 6(b)). This is consistent with the closeness of the Voronoi and Richards’ planes (Fig. 7). The relative positions of the radical planes (closer to the neighbouring oxygen and further from the C”) presumably lead to approximate compensation in the final total volumes.
(ii) ,L-CH2 As the main chain atoms have (apart from C” in glycine) constant covalent neighbourhoods, use of any partitioning method can, in principle, give information on atom-volume distributions that reflect local packing efficiency. For side-chain atoms, however, covalent neighbourhoods are in general more variable, and unless care is taken to consider only groups of atoms whose covalent neighbourhoods are equivalent with respect to the partitioning method used (identical covalent radii for Richards’ B. identical van der Waals’ radii for radical), some of the volume distribution spread will be due to the different covalent environments. An indication of the relative contributions to the distribution width from covalent and non-bonded interactions can be obtained from considering the fl-CH, volumes of Figure 8. Considering all /3-CH2 groups together, we obtain relatively broad distributions for all three methods. The spread of the distributions reflects : firstly, the differences in local packing as assessed by each method; and secondly, the different kind of atom in the y position (the (Yatom being a constant @). From the point of view of the radical method, there will be a contribution to the spread of the distribution from the difference in van der Waals’ radius between /KHz (2.0 A) and the y atom (between 1.6 A and 2-OA). For example, for a carboxyl or amide carbon, the relative placings of the three different planes will be similar to the C-C planes shown in Figure 7. Hence, the /LCH, in, e.g. Asp and Asn, will tend to have a larger radical volume than in Glu and Gin. Although the statistics are necessarily limited, this is illustrated in the two radical distributions shown in Figure 8, corresponding to y position atoms of van der Waals’ radius 2.0 A (solid line) and between 1.6 A and 1.8 A (broken line). The two sub-groups form separate distributions with different means and reduced widths. A similar effect would be expected for Richards’ method from grouping y atoms with respect to their covalent radii, though the differences would probably be small enough to be masked by the variability of the non-bonded neighbourhoods.
316
B. J. GELLATLY
r = 16.3
o- = 2.6
AND
(15.6%)
J. L. FINNEY
v = 18.6
I5
(I = 3.1 (IS-S%)
I
v4 10.
15
20
25
30 Volume (X3)
Fx. 8. Volume distributions for /3-CH2 atoms. The radical distributions have been subdivided whether the &CH2 atom is adjacent to an atom with radius 1.6 to 1.8 .k (- - - ) or 29 A ( -).
by
(d) Vertex error As discussed in the Introduction, the Voronoi and radical procedures are rigorous in that all space is allocated to the atoms in the assembly. Richards’ method, however, gives rise to small tetrahedral volumes at each vertex that are unallocated to any atom (Fig. 3(a) and (b)). Th us, the assembly of polyhedra does not completely fill the space, the missing volume being attributable wholly to the vertex error. Sample calculations by Richards (1974) suggested that this error is generally very small ( + l%), though in a few cases involving ring systems the error may be 2 to 3%, and very occasionally as high as 10%. An estimate of the total cumulative error for lysozyme based on calculations on Gly-Phe-Gly was of the order of 1%. Consideration of the total volumes obtained using identical surface treatments for the radical and Richards’ methods allows us to make a quantitative estimate of the error. As we sum over all internal atoms, the differences in covalent treatment are completely cancelled. Moreover, as the two methods deal with touching unequal spheres in an identical manner, all surface planes will be identical, and hence enclose an identical volume. Table 1 shows the calculated total volumes. Comparing totals for Richards’ and
CALCULATION
OF PROTEIN TABLE
317
VOLUMES
1
Total volumes (in A3) of ribonuclease S calculated using three methods Voronoi’s
Radical
Richards’
Bt
v4
V7
R4
R7
I34
B7
13456
14919
14925
15383
14316
14809
Additional
calcul&d
volumes for comparison
Richards (1974), using ice surface
17450 A3
Finney (1975a,b), using
15106 A” 16500 A”
(i) “minimum” surface (ii) “typical” surface
t Slight technical problems with the algorithm highly exposed groups. The resulting Voronoi or radical conditions;
consequently
errors given in the text are therefore
used in this program result in small errors for certain
error can be estimated
by comparing
performance
under the
these values should be reduced by about 40 A3. The vertex
underestimated
by -0.3%.
the radical methods, and remembering that the radical values are “correct” for a given surface, the vertex error amounts to 4.1% and 3.7% for I.4 A and 1.7 A surfaces, respectively. The larger value for the l-4 A surf+ce is expected, as the greater difference between the surface probe radius and the average van der Waals’ radius of protein atoms ( N 1.7 A) will result in larger associated vertex-error tetrahedra. Thus, the vertex error amounts to about 4% of the protein volume, an error that is greater than the error expected in a measurement of partial molar volume. Without a much more detailed analysis, we can say little about how this error is distributed, or about how important it may or may not be in looking at volumes of internal protein atoms or groups. Table 1 also lists volumes calculated using other surface treatments, serving to emphasize the sensitivity of calculated volume to surface treatment. The fractional range (20 to 30%) is much larger than the experimental measuring error.
4. Discussion (a) Properties of the three partitioning methods The discussion of main-chain mean volumes (Results section (b) (ii)) involved several competing effects in the three partitioning methods, which were difficult to rationalize clearly. The major differences between the methods are summarized in Figure 9; the arrows show how changes in partitioning method lead to “adverse” effects. We can summarize the major differences as follows: Voronds method misallocates non-bonded volume in a way that seems unphysical, though for most atom groups considered the volume distributions
318
B. J. GELLATLY
c
AND
Non-bonded
J. L. FINNEY
partitioning
FIG. 9. The differences in the 3 volume-partitioning methods. Cancelling errors are shown inside, noncancelling outside. For the sake of this discussion the covalent partitioning of radical and Voronoi have been taken to be “inferior” to Richards’ B method. The quantitative effect for Voronoi is generally small. Radical and Richards’ method B treat non-bonded partitioning equally well.
describing the variability in local packing are broad enough to mask the effect. The treatment of covalent links is only slightly different from that of Richards’ method. The partitioning is rigorous. Richards method B appears to allocate non-bonded volumes in a physically reasonable way, and treats covalent links in terms of assigned covalent radii. If the physical reasonableness of the covalent radius is accepted, this is a satisfactory procedure. Unfortunately, the procedure is not rigorous, and a vertex error amounting to 4% for ribonuclease is unavoidable as the method stands. Radical planes method treats non-bonded contacts in an apparently reasonable way (though different from Richards’ method). The main problem is that a consequence of the method’s rigour (there is no vertex error) is a covalent-bond partitioning scheme based upon van der Waals’ radii. Morover, the mathematical form of the partitioning (Fig. 4) results in a plane positioning which is a non-linear function of separation distance. This point seems worth further clarification. Two atoms separated by a distance d will give rise to partitioning planes at a distance d, from atom 1 as described by the equations in Figures 2(b) (Richards’ B) and 4(a) (radical). These equations can be rewritten : Richards : Radical :
% @I =
dz+r,12-rw22 2d2
The apparent forms of the equations are similar : radical is clearly the more rapidly varying function, and this varies more rapidly as d is reduced. The placing of the partitioning planes by the three methods is illustrated as a function of d in Figure 10, for the main chain G-N covalent link. The relatively rapid variation of the radical plane position, as d is reduced, is shown clearly. Two other interesting possibilities are also illustrated. Firstly, the use of van der Waals’
CALCULATIOS
OF PROTEIN
319
VOLUMES
;I___________-_ _ _ _ - _ _ - _ _ - - ----- - ----__ - - -_ e
0.50.45 0.45 I.0
1 I.5
Voronoi I 2.0
I 2.5
I 3-O
,. 3.5
I 4.0
I 4.5
I 5.0
d
FIN:. 10. A more det,ailed comparison of the partitioning n. radical:
d2+rw,2-r,,2 d,/d = 2&
h. H non-bonded: c. H covalent
d,/d =
of CO--N produced bp the 3 methods:
-;
d+rw,-rw, 2d
’
(using van der Waals’ radii): d,/d = -: rw1 frwa
d. R covalent
(using covalent radii): d,/d = A: rc1
e. Voronoi: .4tom
d,/d
1 = (‘: atom
+Tc2
= dJz. 2 = N: rw, = 20 A;r,,
= 1.7 A; r,, = 077 A; rc2 = 0.7 A
radii rather than covalent radii in fixing Richards’ covalent partitioning plane makes little difference, which is not surprising when the rough monotonic relationship between assigned van der Waals’ and covalent radii is noted (Fig. 5). Secondly, an interesting compromise might be to use Richards’ method B but treating covalent bonds identically to non-bonded interactions. Within the overlap region (d,/d < 3.7 A), d,/d varies much more slowly than for the radical method, and although a vertex error would still arise, we might intuitively expect it to be less than for the method in its current application.
(b) Choice of method In the light of the properties of the partitioning schemes examined, we can make some comments on the application of the various methods to calculations of volume in proteins. Such calculations are made at several levels: protein molecule or subunit total volumes; mean volumes and distributions for atom groups (e.g. main chain, side-chains) ; mean volumes and distributions of single atom groups. For total volume calculations, both Richards’ method B and the radical planes method give results that are almost independent of the surface probe radius used :
B. J. GELLATLY
320
AND
J. L.
FINNEY
87
r = 46.9
o- = 3.7 (8-O %I
L l-l
R4
ci=48.1
ii = 51.9
Q= 5.2 (10.7%)
R7
CTq 4.2
(8.2%) v7
IO 5 d!!!!L 40
50
60
Volume (A3) FK.
11. Total
main-chain
atom
volume
distributions
this seems to us sufficient argument to reject the use of Voronoi’s methodi. In comparing the use of Richards’ B and radical methods for total volumes, all differences in covalent and non-bonded partitioning are cancelled in the summation ; the only remaining difference is the vertex error inherent in Richards’ method. This error amounts to about 4% for RNAase S, an error that is outside the errors of experimental volume measurement. Therefore, unless the vertex errors are individually calculated and corrected for (including subLallocating the surfaceinvolved error tetrahedra between protein and solvent) we argue in favour of the use of the radical method. When considering volume calculations over atom groups, however, the choice is less clear : absolute volumes are of less interest than deviations of occupied volumes from the mean, and therefore as long as the vertex error is reasonably uniformly distributed this problem is less significant. Again we would reject the use of Voronoi’s method because of the non-physical partitioning of non-bonded t This discussion is wholly within the framework of the use of van der Weals’-type surface probes to handle the surface problem. Other moPe chemically consistent methods that are under development may cause the conclusions of this section to be modified (B. J. Gelletly, J. P. Bouquiere & J. L. Finney, unpublished data).
CALCULATION
OF PROTEIN
VOLUMES
321
interactions, even though in some cases the consequent volume spread is masked by the local packing variations. Provided groups are chosen with constant covalent environments (e.g. main chain atoms except glycine, whole side-chains) the differences in covalent treatment between radical and Richards’ (and also Voronoi’s) methods are completely cancelled, and as both Richards’ and radical methods partition non-bonded interactions reasonably, there is little to choose between them. This is illustrated in Figure 11, where the volume distributions for the main chain atoms are shown. There is no significant difference between the standard deviations for the radical and Richards’ distributions. The slightly smaller P value for Richards’ method might be explained almost entirely by a uniformly distributed 4% vertex error. In this case, the difference in non-bonded treatments between the radical and Richards’ B methods would amount to less than 1%. Only if absolute mean volumes are required does the radical method have any practical advantage over Richards’ method B. For calculations of occupied volumes and volume distributions for single atoms (or atom groups such as CH,), we would argue that no procedure is satisfactory unless the atoms are grouped together with a constant covalent environment, in which case the same considerations apply as for the larger groups such as main chain and side-chains. For a variable covalent environment, the spread of the resulting volume distributions will be significantly influenced by the placing of the covalent partitioning planes. The effect will be present for both Richards’ and radical methods, though the form of the equation is such that the effect will be greatest for radical. Clearly, it is impossible to devise a volume partitioning procedure that is both rigorous and consistent with the different chemical constraints in proteins. We can handle a system of interacting van der Waals’ atoms rigorously, using radical planes, but as soon as we have to deal with covalent interactions, we must either use van der Waals’ criteria to partition a covalent bond or abandon geometrical rigour. We argue that a discussion of the preferability of using radical or Richards’ method for examining packing efficiency of an atom or group of atoms with variable covalent environment would be largely academic and of little value. If we ask questions about packing efficiency, then covalent-bond partitioning is physically irrelevant, the identity of the repulsive electron shell between the two atoms having been lost in the covalent interaction. Therefore, any discussion of packing efficiency and variations for atoms or groups with a variable covalent environment must necessarily consider data that is perturbed by volume variations that are not due to the packing constraints being investigated. The perturbations will be smaller for Richards’ than for the radical method, so if such comparisons are required, then Richards’ method B is to be preferred over the radical planes method. We thank Professor F. M. Richards for supplying his volume calculation program, which made these comparisons possible; Cyrus Chothia for discussions; and Alan Mackay for drawing our attention to the original paper of Fischer & Koch on the radical plane construction. The computing services of the Daresbury National Physical Laboratory of the
322
R. J. GELLATLY
AND
J. L. FINNEY
Science and Engineering Research Council, and of Cambridge University Computing Centre are gratefully acknowledged. Barry Gellatly was supported by a Birkbeck College Research Officer post.
REFERENCES Barker. J. A., Hoare, M. R. & Finney, J. L. (1975). 1Vature (London), 257, 1209122. Finney. J. I,. (1970). Proc. Roy. Rot. sec. A, 319. 495-507. Finney, J. L. (1975a) J. Mol. Biol. 96, 721-732. Finney, J. L. (1975b). J. de Phys. 36 (Colloque C2, suppl. 4) C-2-1-11. Finney. .J. L., Gellatly, B. J., Golton, I. C. & Goodfellow, J. M. (1980). Biophys. J. 32, 17731. Fischer, W. & Koch, E. (1979). 2. K&all. 150, 245-260. Gellatly, B. J. & Finney. J. L. (1982). J. Non-Cry&. Solids, in the press. Mackay, A. L., Finney, J. L. & Gotoh, K. (1977). Aeta Crystallogr. sect. A, 33, 98-100. Richards. F. M. (1974). J. Mol. Biol. 82, l-14. Voronoi, G. F. (1908). J. Reine Anger. Math. 134. 198-287. Wgckoff, H. W., Tsernoglou, D., Hanson, A. W.. Knox. J. R., Lee, B. & Richards, F. M. (1970). J. Biol. Chem. 24.5, 305318.
Edited
by R. Huber
Calculation of Protein Volumes : An Alternative to the Voronoi Procedure B. J.
University
GELLATLYt
AND
J. L.
FINNEY
Department of Crystallography, Birkbeck College of London, Malet Street, London, WClE 7HX, U.K.
(Received 18 March 1982, and in revised form
17 June 1982)
All the methods that have been applied to assessing local volume occupation or packing in proteins have particular defects. For example, the Voronoi method used by Richards (method A) and by Finney misallocates both non-bonded and covalent contacts in a geometrically rigorous, though chemically inconsistent, manner. Richards’ method B, in which covalent and non-bonded contacts are partitioned in chemically sensible ways, is unfortunately not completely rigorous, in that every polyhedron vertex has associated with it a small vertex-error polyhedron, which is not allocated to any atom. We present here a generalization of the Voronoi method that is particularly suited to multicomponent assemblies such as proteins. This radical plane partitioning of volume is completely rigorous; it gives rise to no vertex error, and handles the more numerous non-bonded contacts realistically. Its application to RNAase-S is described and the results compared with both Voronoi’s method and Richards’ method B. A particular advantage of both the radical plane and Richards’ methods is a relative insensitivity to the treatment of the surface, a problem that has plagued other approaches to describing packing in proteins. Although the radical plane is seen to misallocate volume chemically between covalent+bonded neighbours, this problem vanishes when groups of atoms in sidechain residues are considered.
1. Introduction Although globular proteins are often described in terms of their secondary structure, such a characterization is not well-suited to showing regularities that may be present at the lower atomic level of organization. Attempts have therefore been made to produce alternative ways of rationalizing these structures. One such approach uses the concept of geometrical packing, which has been very successful in describing the properties of condensed phases of other systems (Finney, 1970,1975b; Barker et al., 1975). The volume occupation of component atoms, groups and residues provides a systematic way of analysing protein structural data. Amongst other things, this gives information on local volume variations, which may be related to local flexibility (Richards, 1974; Finney, 1975a). t Present
address:
0022-2836/82/300305-18
The Emmbrook $03.00/O
School,
Wokingham, 305
Berkshire, 0
U.K
1982 Academic
Press Inc.
(London)
Ltd.
306
R. J. GELLATLY
AND
J. L. FIN%EY
Previous work using this approach has been hampered by two major difficulties : (1) defining the protein-solvent boundary adequately ; and (2) finding an accurate method of volume partitioning that is physically reasonable for multicomponent systems. We discuss here an alternative approach to this latter problem, using a modification of our previous method to position the protein-solvent boundary. The most widely used partitioning method is due to Voronoi (1908) : however, this is not satisfactory for systems containing different-sized atoms. Richards (1974) developed an alternative to this, using covalent or van der Waals’ radii to define the partitioning, depending on the type of interaction involved. In an attempt to overcome the problem of unallocated volume inherent in this method, we have adopted a related method based on the radical plane construction (Fischer & Koch, 1979). The radical plane division is defined in terms of both the van der Waals’ radii of the atoms and the distance between them. Because of this, \ve expect some problems in dealing with covalent and hydrogen-bonding interactions. where some misallocation of volume will inevitably occur. However, this procedwe is completely rigorous, with no regions of space left unallocated. The technique has been used successfully in the characterization of binary non-crystalline systems such as amorphous metal alloys (Gellatly & Finney, 1982).
2. Materials
and Methods
(a) C’olumr partitioning The Voronoi (1908) method of partitioning the space occupied by an array of atoms gives a non-arbitrary space-filling of polyhedra, formed from planes bisecting the interatomic vectors perpendicularly. A polyhedron vertex is determined as a point equidistant from 4 non-coplanar atoms, which is closer to no other atom in the assembly. Hence the solutions of the equations : (zi-~)2+(yi-y)2+(Zi-z)2
= LE. i = 1.4..
.(l)
give the co-ordinate of the vertex (zyz) and the vertex-centre distance (L,). (zigizi) are the coordinates of the four atoms, i = 1,4 (see Fig. 1 (a)). Each vertex is checked to ensure that no other atom is closer than the current vertex-centre distance L,. For neighbouring atoms of unequal size, the dividing plane may lie well inside the sphere representing the larger atom. Hence too much volume is allocated to the smaller atom and too little to the larger one (Fig. l(b)). The method devised by Richards (1974) attempts to overcome this difficulty and provides a physically reasonable volume partitioning. The dividing plane of covalently bonded atoms (Fig. Z(a)) is drawn perpendicular to the interatomic vector and positioned to divide it in proportion to the assumed covalent radii. For non-bonded interactions van der Waals’ radii are used to determine the position of the plane (Fig. 2(b)). However, this method is nonin that the assembly of polyhedra does not completely fill space; the rigorous to neighbourhood of each vertex contains a volume (the “vert,ex error”) that is unallocated any polyhedron except in the case of exactly touching spheres (see Fig. 3). This vertex error can, however, be avoided, and the rigour of the Voronoi construction can be generalized for unequal spheres. To do this, we use a partitioning method based on radical planes (Fischer & Koch, 1979). The radical plane of 2 spheres is the locus of points from which the tangent lengths to the 2 spheres are equal (see Fig. 4(a)). (z~z) and (zivizi) are defined as for the Voronoi method, with vwi the assigned radius for each atom and L, the vertex-tangent length. The required equations now become : (~~-z)~+(y~-y)~+(.q-z)~--r~~
=L:,
i = 1,4,
CALCULATION
OF
PROTEIN
307
VOLUMES
(0)
FIG. I. Voronoi construction (2 dimensions): distances to the 4 spheres are equal, d, = d/,; atoms.
(a) a vertex is the point such that the vertex-centre (b) positioning of dividing planes for 2 unequal-sized
with the distance check now being for the vertex-tangent length. The definition handles non-overlapping, touching (common tangent), and overlapping (common chord) atoms equally well with an apparently reasonable partitioning of volume between different sized atoms (Fig. 4(b)). (b) Surface treatment The volume of a surface atom will normally be undefined by any of these spacepartitioning procedures, and it is therefore necessary to provide “hypothetical solvent“ positions to define vectors for the normals of the polyhedron faces. In this work these were placed at a uniform density on the surfaces of spheres centred on the surface atom as described previously (Finney et al., 1980). A maximum of 24 such points were added at relative positions corresponding to a snub cube arrangement (Mackay et al., 1977). In addition “tetrahedral” surface points were placed at the surface sphere intersections as described previously (Finney, 1975a). The sphere radius was taken as the sum of the van der Waals’ radius of the surface atom and that of the surface probe. The values of van der Waals’ radii were as used before (Finney, 1975a) (see Fig. 5).
FIG. 2. Richards’
construction
(a) covalent interactions: (b) Non-bonded
interactions:
(2 dimensions) :
drc L d, = -_ r,,+r,, d, = r,, +
d-r,,
2
-rwz
,
where rCi and rwi are the covalent and van der Waals’ radii, respectively,
of atom i.
B. J. GELLATLY
308
AND
J. L. FINNEY
(b)
FIG. 3. Vertex error in Richards’ method associated with: (a) covalent and non-bonded interactions; (b) non-bonded interactions. (c) There is no vertex error for the special case of “touching” non-bonded atoms. performed
(c) Calculations
To allow comparisons to be made between the 3 methods, calculations using Voronoi’s (V), Richards’ method B (B), and radical planes method (R) were performed on the ribonuclease S co-ordinates used previously (Wyckoff et al., 1970). Identical surface treatments (the point
addition described in section (b), above) and van der Waals’ radii were used. For the Richards’ method calculation, the covalent radii reported previously (Richards, 1974) were used (see Fig. 5). Surface probes of radii 1.4 A and 1.7 A were used in separate calculations to examine the sensitivity of the methods to assumptions concerning the probe size. Thus, 6 data sets designated: V4, V7, B4, B7, R4 and R7 were obtained. The letters V, B and R refer to Voronoi, Richards’ B, and radical methods, respectively, while the integer denotes the radius of the surface probe.
Gp
(a)
tb,
FIG. 4. Radical plane construction (2 dimensions): tangent distances to the 4 spheres are equal : d, = rw12-rw2 ‘+d’ 2d (b) positioning
(a) a vertex is the point such that the vertex-
;
of radical planes for unequal-sized
atoms.
CALCULATION
OF PROTEIN
309
VOLUMES
3. Results (a) Atom volumes Figure 5 shows the mean polyhedron volumes for each of the sets of calculations. Several points emerge. First, the Voronoi volume varies strongly with the probe radius by up to 20 to 25%. This is to be expected as the position of the partitioning plane between a surface atom and the probe will be dependent upon the probe radius, being further from the surface atom centre for the larger probe. In contrast, volumes calculated by both Richards’ and radical methods depend much less on the probe. This insensitivity to probe radius reflects the placing by both methods of the dividing plane in the common tangent position for two touching unequal spheres (Figs 3(c) and 4(d)). Considering the problems of surface treatment (Richards, 1974: Finney, 1975a), this relative insensitivity of both the radical and Richard’s B techniques is a major advantage over the Voronoi subdivision. The residual small differences within one method can be explained by the finer surface definition consequent on the use of the smaller probe. Secondly, for the atoms or atom groups in the centre of Figure 5, (say, ring C to O/OH), Voronoi volumes are approximately constant, while both Richards’ and radical procedures produce mean volumes that vary by greater than a factor of two. This is consistent with the Voronoi procedure, misallocating volume between neighbouring unequal atoms in such a way as to smear out the difference between the larger and smaller atoms (Fig. l(b)). I n effect, the Voronoi partition fails to reflect adequately the packing constraints within the molecule, a serious defect if we are using the procedure to examine effects depending on packing considerations. The differences between the mean volumes given by radical and Richards’
35 30 s
25-
E s 20> 15 IO -
I CH, r.
G
NH;
2.0 2.0 0.77 0.70
tic&
0
Pro CH,
OH
CHf
CH,
2.0 1.4 I.6 0.77 0.66 0.66
2.0 0.77
2.0 2.0 0.77 0.77
+
Atom
O/NH
O/OH
I.6 1.5 0.70 0.66 type
His C/N
N
Cl+@
C’
I.6 I.7 2.0 0.70 0.70 0-77
2.0 0.77
f
+
&I
C
1.7 1.7 0-77 0.77 9
FIG. 5. Mean volumes for atom groups (‘I, V7 ; A, V4 ; l , R7 ; n , R4 ; 0, B7 ; 0, B4). The van der Waals’ (r,) and covalent (r,) radii used are given below each atom type. Arrows indicate main-chain atoms. Radical and Voronoi values for C carbonyl groups are superimposed.
d
Lcl -
0-
-u-l
0-
huanbaq
y.
In
0-
IJ-l
312
B. J. GELLATLY
AND
,J. I,. FINNEY
methods are generally (though not always) smaller than the difference between either and Voronoi’s method. Moreover, for atoms with assigned van der Waals’ radii greater than about 1.7 A (an approximate average van der Waals’ radius for the protein atoms or atom groups), radical tends to yield the larger volume, while for smaller atoms the reverse is true. This suggests the results might be reflecting the different sensitivity to local internal environment of the two methods. A detailed examination, however, shows that these differences are quite subtle, and focus on problems of reconciling chemical reasonableness with geometrical consistency. This point is discussed usefully in connection with volumes allocated to the main-chain groups as follows. (b) Main chain atoms (i) Volume distributions The main chain atoms form the largest group of atoms with approximately constant covalent environment. Therefore, the volume distributions of Figure 6 reflect the variations in packing efficiency (including hydrogen bonding) as assessed by the three methods. We would hence expect that the superiority of one method over another would show up as a significantly reduced volume distribution, as is found when the radical method is applied to binary alloy glasses (Gellatly & Finney, 1982). Looked at in terms of hypothetical experiment that might measure volume distributions, each volume partitioning method can be considered as having a “resolution function” with which the “measured” volume distribution is convoluted to yield the distribution of Figure 6. The “resolution function” of Voronoi’s method will be related to the misallocation of volume between unequal neighbours, while we would expect the differences between Richards’ and the radical method to be assigned to either the vertex error (Fig. 3) or the slight difference in partitioning non-touching non-bonded contacts (including hydrogen bonding) or both. Reference to the spreads of the distributions of Figure 6, however, shows no significant difference between the standard deviations of the radical and Richards’ methods. Moreover, and more surprisingly, only for main chain oxygen (Fig. 6(d)) is the Voronoi distribution significantly worse than either of the other “improved” methods. Even here, when the distributions are normalized to the mean volume, it could be argued that the relative spread (estimated as o/V) is not significantly worse for the Voronoi method. We interpret these results as showing that the variations in packing efficiency about these main chain atoms are significantly greater than the smearing effects of the effective resolution functions of all three methods. The poor shape of the Voronoi oxygen distribution (Fig. 6(d)) rather than the normalized width, argues against the use of the Voronoi method. Even though its effective resolution function is expected to be poor, the variations in local volume around main chain atoms are large enough to almost mask this defect. Although the statistics are poorer, this conclusion is generally consistent with the volume distributions of sidechain atoms. Exceptions for which Voronoi distributions are significantly broader than either of the other two methods (proline CH2, CH,, and ring carbons) argue
CALCULATION
OF PROTEIN
further against the use of the Voronoi method B gives broader distributions carbons) may possibly reflect the larger assigned van der Waals’ radii, though point.
313
VOLUMES
method. The two cases in which Richards’ than the radical method (CH, and ring vertex error expected for atoms with large no convincing data can be offered on this
(ii) Mean volumes Figures 5 and 6 both illustrate the differences in mean volumes P assigned by the three .methods. In order of descending values for van der Waals’ radius, the following generalizations can be made, bearing in mind the distribution widths : C” (r,
= 2.0 A) :
PR > Ps z P”
c (r, = 1.7 A): N (r, = 1.7 A):
PR - PB - P” %-tfs-~w though perhaps a small tendency for :
0%.
6(a))
0%. 6(b))
Pv > PB > PR P,> I&- PR.
(Fig. 6(c)) (Fig. 6(d)) Considering that the different volume partitionings depend on the assigned radii of neighbouring atoms, we might look for a rationalization of the above rather unclear pattern in terms of the van der Waals’ radii of the atom and of its average surroundings. For example, most main chain oxygens (r, = 1.4 A) are hydrogenbonded to a neighbouring NH group (r, = 1.7 A). Hence the Voronoi method will place the plane further from the oxygen (cf. Fig. l(b)) than either radical or Richards’ B methods and thus give a larger volume, as is observed (Figs 5 and 6(d)). These two latter methods treat this contact as a slightly overlapping van der Waals’ contact (Figs 2(a) and 4(b)) and hence will place the partitioning plane in similar positions. Applying a similar argument to the main chain nitrogen would suggest 7s - ps > pv. If anyth’ mg, however, the average Voronoi volume is slightly greater than those of radical or Richards’ B methods (Fig. 6(c)). The situation is apparently not quite as simple, and a more detailed examination requires us to look at the differences between the three methods in partitioning bonded contacts. 0 (r, = 1.4A):
(c) The “covalent
error”
(i) Main chain The three methods partition a covalent bond differently. Voronoi places the plane midway between atoms, while Richards’ B makes use of the ratio of assigned “covalent radii” obtained from fitting to an extensive data set (Richards, 1974). An inspection of the covalent radii used (excluding sulphur) shows a variation from 666 A to 0.77 A; this relatively small variation of about 17% will mean that the covalent partitioning for both Richards’ B and Voronoi will generally not be very different. In order to retain geometrical rigour, the radical method must use a van der Waals’ criterion to partition a chemical bond. The variation in van der Waals’ radii is much greater (-43%) than in covalent radii, and hence the positioning of the covalent plane will vary more. Moreover, the form of the equation (Fig. 4(a)) is such that for smaller separations of covalently bonded atoms, this “movement” of the radical partitioning plane is even greater (see also Fig. 10).
314
B. .J. GELLATLY
,4NI)
J. L. FINNEY
C”
INH
FIG. 7. B comparison of the partitioning produced by the 3 methods for main-chain (-), radical (. .), Richards’ B (- - - ).
atoms: Ywonoi’s
The relative effects of these differences on covalent partitioning in the main chain are shown in Figure 7. The similarity of the Richards’ and Voronoi partitioning is clear, as is the significant difference in the radical plane position for the C”--C, C=O and N-C” covalent links. Note that the equality of assigned van der Waals’ radii leads to identity between Voronoi and radical planes for C-N, but the Richards’ B plane is in a different position (unequal covalent radii). We now reconsider the nitrogen volume distributions of Figure 6(c) in the light of Figure 7. Whereas previously we considered only the placing of the plane between the NH and a neighbouring hydrogen-bonded O=C, Figure 7 shows the different covalent partitions must also be taken into account. With respect to covalent links, Richards’ B will tend to give a smaller “partial” volume than Voronoi, while the partitioning of the hydrogen bond will tend to enhance Richards’ over Voronoi’s method. This is consistent with an approximate equality in the Richards’ and Voronoi volumes observed (Fig. 6(c)): the two opposing effects (together with other close approaches not considered here) appear to cancel each other approximately. A comparison of the Richards’ and radical volumes will presumably be dominated by the N-C” radical plane being much closer to the N than the C” (Fig. 7). Thus, we would expect the radical volume to be lower than either Richards’ or Voronoi’s volumes. That this is indeed the case is seen in Figure 6(c), though the apparent smallness of the volume shift is presumably due to compensating effects from other non-covalent neighbours. A reconsideration of the mean volumes of the remaining main chain atoms further illustrates the influence of the covalent partitioning on the different meanvolume results from the three methods. For the oxygen, the O=C radical plane is closer to the oxygen than either Richards’ or Voronoi planes, and would suggest that 0, is less than both ps and pv, as is observed. That the difference between pa and vu is not greater must be due again to compensating effects (perhaps including relative weightings of certain planes in their contribution to total volume) from other non-bonded neighbours.
CALCULATION
OF PROTEIN
VOLUMES
315
For C”, Richards’ B and Voronoi’s covalent-link planes are in similar positions, suggesting Vv - VB, which is in fact observed (Figs 5 and 6(a)). The different positioning of the N-C” plane suggests a slightly larger volume from Richards’ B. an effect that is consistent with observation (Figs 5 and 6(a)). For the radical partitioning, both covalent-link planes are significantly further from the C”. suggesting the much larger VRvalue, as is observed (Figs 5 and 6(a)). We might note in passing that the atoms in the /3 position on all side-chains except glycine have both van der Waals’ and covalent radii equal to those assigned to C” and hence will not effect the C” volumes. Finally, the carbonyl carbon volumes are notable for their equality between methods (Figs 5 and 6(b)). This is consistent with the closeness of the Voronoi and Richards’ planes (Fig. 7). The relative positions of the radical planes (closer to the neighbouring oxygen and further from the C”) presumably lead to approximate compensation in the final total volumes.
(ii) ,L-CH2 As the main chain atoms have (apart from C” in glycine) constant covalent neighbourhoods, use of any partitioning method can, in principle, give information on atom-volume distributions that reflect local packing efficiency. For side-chain atoms, however, covalent neighbourhoods are in general more variable, and unless care is taken to consider only groups of atoms whose covalent neighbourhoods are equivalent with respect to the partitioning method used (identical covalent radii for Richards’ B. identical van der Waals’ radii for radical), some of the volume distribution spread will be due to the different covalent environments. An indication of the relative contributions to the distribution width from covalent and non-bonded interactions can be obtained from considering the fl-CH, volumes of Figure 8. Considering all /3-CH2 groups together, we obtain relatively broad distributions for all three methods. The spread of the distributions reflects : firstly, the differences in local packing as assessed by each method; and secondly, the different kind of atom in the y position (the (Yatom being a constant @). From the point of view of the radical method, there will be a contribution to the spread of the distribution from the difference in van der Waals’ radius between /KHz (2.0 A) and the y atom (between 1.6 A and 2-OA). For example, for a carboxyl or amide carbon, the relative placings of the three different planes will be similar to the C-C planes shown in Figure 7. Hence, the /LCH, in, e.g. Asp and Asn, will tend to have a larger radical volume than in Glu and Gin. Although the statistics are necessarily limited, this is illustrated in the two radical distributions shown in Figure 8, corresponding to y position atoms of van der Waals’ radius 2.0 A (solid line) and between 1.6 A and 1.8 A (broken line). The two sub-groups form separate distributions with different means and reduced widths. A similar effect would be expected for Richards’ method from grouping y atoms with respect to their covalent radii, though the differences would probably be small enough to be masked by the variability of the non-bonded neighbourhoods.
316
B. J. GELLATLY
r = 16.3
o- = 2.6
AND
(15.6%)
J. L. FINNEY
v = 18.6
I5
(I = 3.1 (IS-S%)
I
v4 10.
15
20
25
30 Volume (X3)
Fx. 8. Volume distributions for /3-CH2 atoms. The radical distributions have been subdivided whether the &CH2 atom is adjacent to an atom with radius 1.6 to 1.8 .k (- - - ) or 29 A ( -).
by
(d) Vertex error As discussed in the Introduction, the Voronoi and radical procedures are rigorous in that all space is allocated to the atoms in the assembly. Richards’ method, however, gives rise to small tetrahedral volumes at each vertex that are unallocated to any atom (Fig. 3(a) and (b)). Th us, the assembly of polyhedra does not completely fill the space, the missing volume being attributable wholly to the vertex error. Sample calculations by Richards (1974) suggested that this error is generally very small ( + l%), though in a few cases involving ring systems the error may be 2 to 3%, and very occasionally as high as 10%. An estimate of the total cumulative error for lysozyme based on calculations on Gly-Phe-Gly was of the order of 1%. Consideration of the total volumes obtained using identical surface treatments for the radical and Richards’ methods allows us to make a quantitative estimate of the error. As we sum over all internal atoms, the differences in covalent treatment are completely cancelled. Moreover, as the two methods deal with touching unequal spheres in an identical manner, all surface planes will be identical, and hence enclose an identical volume. Table 1 shows the calculated total volumes. Comparing totals for Richards’ and
CALCULATION
OF PROTEIN TABLE
317
VOLUMES
1
Total volumes (in A3) of ribonuclease S calculated using three methods Voronoi’s
Radical
Richards’
Bt
v4
V7
R4
R7
I34
B7
13456
14919
14925
15383
14316
14809
Additional
calcul&d
volumes for comparison
Richards (1974), using ice surface
17450 A3
Finney (1975a,b), using
15106 A” 16500 A”
(i) “minimum” surface (ii) “typical” surface
t Slight technical problems with the algorithm highly exposed groups. The resulting Voronoi or radical conditions;
consequently
errors given in the text are therefore
used in this program result in small errors for certain
error can be estimated
by comparing
performance
under the
these values should be reduced by about 40 A3. The vertex
underestimated
by -0.3%.
the radical methods, and remembering that the radical values are “correct” for a given surface, the vertex error amounts to 4.1% and 3.7% for I.4 A and 1.7 A surfaces, respectively. The larger value for the l-4 A surf+ce is expected, as the greater difference between the surface probe radius and the average van der Waals’ radius of protein atoms ( N 1.7 A) will result in larger associated vertex-error tetrahedra. Thus, the vertex error amounts to about 4% of the protein volume, an error that is greater than the error expected in a measurement of partial molar volume. Without a much more detailed analysis, we can say little about how this error is distributed, or about how important it may or may not be in looking at volumes of internal protein atoms or groups. Table 1 also lists volumes calculated using other surface treatments, serving to emphasize the sensitivity of calculated volume to surface treatment. The fractional range (20 to 30%) is much larger than the experimental measuring error.
4. Discussion (a) Properties of the three partitioning methods The discussion of main-chain mean volumes (Results section (b) (ii)) involved several competing effects in the three partitioning methods, which were difficult to rationalize clearly. The major differences between the methods are summarized in Figure 9; the arrows show how changes in partitioning method lead to “adverse” effects. We can summarize the major differences as follows: Voronds method misallocates non-bonded volume in a way that seems unphysical, though for most atom groups considered the volume distributions
318
B. J. GELLATLY
c
AND
Non-bonded
J. L. FINNEY
partitioning
FIG. 9. The differences in the 3 volume-partitioning methods. Cancelling errors are shown inside, noncancelling outside. For the sake of this discussion the covalent partitioning of radical and Voronoi have been taken to be “inferior” to Richards’ B method. The quantitative effect for Voronoi is generally small. Radical and Richards’ method B treat non-bonded partitioning equally well.
describing the variability in local packing are broad enough to mask the effect. The treatment of covalent links is only slightly different from that of Richards’ method. The partitioning is rigorous. Richards method B appears to allocate non-bonded volumes in a physically reasonable way, and treats covalent links in terms of assigned covalent radii. If the physical reasonableness of the covalent radius is accepted, this is a satisfactory procedure. Unfortunately, the procedure is not rigorous, and a vertex error amounting to 4% for ribonuclease is unavoidable as the method stands. Radical planes method treats non-bonded contacts in an apparently reasonable way (though different from Richards’ method). The main problem is that a consequence of the method’s rigour (there is no vertex error) is a covalent-bond partitioning scheme based upon van der Waals’ radii. Morover, the mathematical form of the partitioning (Fig. 4) results in a plane positioning which is a non-linear function of separation distance. This point seems worth further clarification. Two atoms separated by a distance d will give rise to partitioning planes at a distance d, from atom 1 as described by the equations in Figures 2(b) (Richards’ B) and 4(a) (radical). These equations can be rewritten : Richards : Radical :
% @I =
dz+r,12-rw22 2d2
The apparent forms of the equations are similar : radical is clearly the more rapidly varying function, and this varies more rapidly as d is reduced. The placing of the partitioning planes by the three methods is illustrated as a function of d in Figure 10, for the main chain G-N covalent link. The relatively rapid variation of the radical plane position, as d is reduced, is shown clearly. Two other interesting possibilities are also illustrated. Firstly, the use of van der Waals’
CALCULATIOS
OF PROTEIN
319
VOLUMES
;I___________-_ _ _ _ - _ _ - _ _ - - ----- - ----__ - - -_ e
0.50.45 0.45 I.0
1 I.5
Voronoi I 2.0
I 2.5
I 3-O
,. 3.5
I 4.0
I 4.5
I 5.0
d
FIN:. 10. A more det,ailed comparison of the partitioning n. radical:
d2+rw,2-r,,2 d,/d = 2&
h. H non-bonded: c. H covalent
d,/d =
of CO--N produced bp the 3 methods:
-;
d+rw,-rw, 2d
’
(using van der Waals’ radii): d,/d = -: rw1 frwa
d. R covalent
(using covalent radii): d,/d = A: rc1
e. Voronoi: .4tom
d,/d
1 = (‘: atom
+Tc2
= dJz. 2 = N: rw, = 20 A;r,,
= 1.7 A; r,, = 077 A; rc2 = 0.7 A
radii rather than covalent radii in fixing Richards’ covalent partitioning plane makes little difference, which is not surprising when the rough monotonic relationship between assigned van der Waals’ and covalent radii is noted (Fig. 5). Secondly, an interesting compromise might be to use Richards’ method B but treating covalent bonds identically to non-bonded interactions. Within the overlap region (d,/d < 3.7 A), d,/d varies much more slowly than for the radical method, and although a vertex error would still arise, we might intuitively expect it to be less than for the method in its current application.
(b) Choice of method In the light of the properties of the partitioning schemes examined, we can make some comments on the application of the various methods to calculations of volume in proteins. Such calculations are made at several levels: protein molecule or subunit total volumes; mean volumes and distributions for atom groups (e.g. main chain, side-chains) ; mean volumes and distributions of single atom groups. For total volume calculations, both Richards’ method B and the radical planes method give results that are almost independent of the surface probe radius used :
B. J. GELLATLY
320
AND
J. L.
FINNEY
87
r = 46.9
o- = 3.7 (8-O %I
L l-l
R4
ci=48.1
ii = 51.9
Q= 5.2 (10.7%)
R7
CTq 4.2
(8.2%) v7
IO 5 d!!!!L 40
50
60
Volume (A3) FK.
11. Total
main-chain
atom
volume
distributions
this seems to us sufficient argument to reject the use of Voronoi’s methodi. In comparing the use of Richards’ B and radical methods for total volumes, all differences in covalent and non-bonded partitioning are cancelled in the summation ; the only remaining difference is the vertex error inherent in Richards’ method. This error amounts to about 4% for RNAase S, an error that is outside the errors of experimental volume measurement. Therefore, unless the vertex errors are individually calculated and corrected for (including subLallocating the surfaceinvolved error tetrahedra between protein and solvent) we argue in favour of the use of the radical method. When considering volume calculations over atom groups, however, the choice is less clear : absolute volumes are of less interest than deviations of occupied volumes from the mean, and therefore as long as the vertex error is reasonably uniformly distributed this problem is less significant. Again we would reject the use of Voronoi’s method because of the non-physical partitioning of non-bonded t This discussion is wholly within the framework of the use of van der Weals’-type surface probes to handle the surface problem. Other moPe chemically consistent methods that are under development may cause the conclusions of this section to be modified (B. J. Gelletly, J. P. Bouquiere & J. L. Finney, unpublished data).
CALCULATION
OF PROTEIN
VOLUMES
321
interactions, even though in some cases the consequent volume spread is masked by the local packing variations. Provided groups are chosen with constant covalent environments (e.g. main chain atoms except glycine, whole side-chains) the differences in covalent treatment between radical and Richards’ (and also Voronoi’s) methods are completely cancelled, and as both Richards’ and radical methods partition non-bonded interactions reasonably, there is little to choose between them. This is illustrated in Figure 11, where the volume distributions for the main chain atoms are shown. There is no significant difference between the standard deviations for the radical and Richards’ distributions. The slightly smaller P value for Richards’ method might be explained almost entirely by a uniformly distributed 4% vertex error. In this case, the difference in non-bonded treatments between the radical and Richards’ B methods would amount to less than 1%. Only if absolute mean volumes are required does the radical method have any practical advantage over Richards’ method B. For calculations of occupied volumes and volume distributions for single atoms (or atom groups such as CH,), we would argue that no procedure is satisfactory unless the atoms are grouped together with a constant covalent environment, in which case the same considerations apply as for the larger groups such as main chain and side-chains. For a variable covalent environment, the spread of the resulting volume distributions will be significantly influenced by the placing of the covalent partitioning planes. The effect will be present for both Richards’ and radical methods, though the form of the equation is such that the effect will be greatest for radical. Clearly, it is impossible to devise a volume partitioning procedure that is both rigorous and consistent with the different chemical constraints in proteins. We can handle a system of interacting van der Waals’ atoms rigorously, using radical planes, but as soon as we have to deal with covalent interactions, we must either use van der Waals’ criteria to partition a covalent bond or abandon geometrical rigour. We argue that a discussion of the preferability of using radical or Richards’ method for examining packing efficiency of an atom or group of atoms with variable covalent environment would be largely academic and of little value. If we ask questions about packing efficiency, then covalent-bond partitioning is physically irrelevant, the identity of the repulsive electron shell between the two atoms having been lost in the covalent interaction. Therefore, any discussion of packing efficiency and variations for atoms or groups with a variable covalent environment must necessarily consider data that is perturbed by volume variations that are not due to the packing constraints being investigated. The perturbations will be smaller for Richards’ than for the radical method, so if such comparisons are required, then Richards’ method B is to be preferred over the radical planes method. We thank Professor F. M. Richards for supplying his volume calculation program, which made these comparisons possible; Cyrus Chothia for discussions; and Alan Mackay for drawing our attention to the original paper of Fischer & Koch on the radical plane construction. The computing services of the Daresbury National Physical Laboratory of the
322
R. J. GELLATLY
AND
J. L. FINNEY
Science and Engineering Research Council, and of Cambridge University Computing Centre are gratefully acknowledged. Barry Gellatly was supported by a Birkbeck College Research Officer post.
REFERENCES Barker. J. A., Hoare, M. R. & Finney, J. L. (1975). 1Vature (London), 257, 1209122. Finney. J. I,. (1970). Proc. Roy. Rot. sec. A, 319. 495-507. Finney, J. L. (1975a) J. Mol. Biol. 96, 721-732. Finney, J. L. (1975b). J. de Phys. 36 (Colloque C2, suppl. 4) C-2-1-11. Finney. .J. L., Gellatly, B. J., Golton, I. C. & Goodfellow, J. M. (1980). Biophys. J. 32, 17731. Fischer, W. & Koch, E. (1979). 2. K&all. 150, 245-260. Gellatly, B. J. & Finney. J. L. (1982). J. Non-Cry&. Solids, in the press. Mackay, A. L., Finney, J. L. & Gotoh, K. (1977). Aeta Crystallogr. sect. A, 33, 98-100. Richards. F. M. (1974). J. Mol. Biol. 82, l-14. Voronoi, G. F. (1908). J. Reine Anger. Math. 134. 198-287. Wgckoff, H. W., Tsernoglou, D., Hanson, A. W.. Knox. J. R., Lee, B. & Richards, F. M. (1970). J. Biol. Chem. 24.5, 305318.
Edited
by R. Huber