α-Helical coiled coils are usually stabilized by hydrophobic interfaces between the two constituent α-helices, in the form of ‘knobs-into-holes’ packing of non-polar residues arranged in repeating heptad patterns. Here we examine the corresponding ‘hydrophobic cores’ that stabilize bundles of four α-helices. In particular, we study three different kinds of bundle, involving four α-helices of identical sequence: two pack in a parallel and one in an anti-parallel orientation. We point out that the simplest way of understanding the packing of these 4-helix bundles is to use Crick's original idea that the helices are held together by ‘hydrophobic stripes’, which are readily visualized on the cylindrical surface lattice of the α-helices; and that the ‘helix-crossing angle’—which determines, in particular, whether supercoiling is left- or right-handed—is fixed by the slope of the lattice lines that contain the hydrophobic residues. In our three examples the constituent α-helices have hydrophobic repeat patterns of 7, 11 and 4 residues, respectively; and we associate the different overall conformations with ‘knobs-into-holes’ packing along the 7-, 11- and 4-start lines, respectively, of the cylindrical surface lattices of the constituent α-helices. For the first two examples, all four interfaces between adjacent helices are geometrically equivalent; but in the third, one of the four interfaces differs significantly from the others. We provide a geometrical explanation for this non-equivalence in terms of two different but equivalent ways of assembling this bundle, which may possibly constitute a bistable molecular ‘switch’ with a coaxial throw of about 12 Å. The geometrical ideas that we deploy in this paper provide the simplest and clearest description of the structure of helical bundles. In an appendix, we describe briefly a computer program that we have devised in order to search for ‘knobs-into-holes’ packing between α-helices in proteins.
In 1953 Francis Crick  elucidated the formation of coiled coils, where two α-helices wrap around one another in a super-helical structure and are held together by an interface of interlocking hydrophobic amino acids. He showed that a heptad repeat—i.e. a seven-letter repeat—in the amino acid sequence, of the kind where H denotes a hydrophobic (i.e. water-hating) amino acid residue such as leucine, provides a near-longitudinal hydrophobic ‘stripe’ along the length of an α-helix, which enables it to engage a neighbouring α-helix having a similar sequence (figure 1a). In particular, Crick coined the idea of ‘knobs-into-holes’ packing of the amino acids at the interface, with the hydrophobic amino acid ‘knobs’ of one helix burying themselves in the ‘holes’ or gaps between the hydrophobic knobs of the partner helix. Burial of hydrophobic residues in the interior is an important structural principle in the assembly of a protein's polypeptide chain into its final, folded form. This leaves hydrophilic (i.e. water-loving) residues exposed to interaction with the surrounding water. Crick also showed that the angle between the local axes of the paired helices of the coiled coil is a consequence of the spiral form of this near-longitudinal hydrophobic stripe; which in turn depends on the value of the helical repeat—i.e. the number of amino acid residues in one turn of the α-helix—of the amino acids on the polypeptide chain of the α-helices. The stripe would be axial if the helical repeat were equal to 3.5; and it makes a left-handed spiral for a value of 3.6, which is typical for an isolated and relaxed α-helix (figure 1a). The clearest example of Crick's scheme is the long tropomyosin coiled coil, which has an uninterrupted heptad repeat . Coiled coils with other, somewhat irregular, heptad repeats have been discussed in terms of ‘skips’, ‘stutters’ and ‘stammers’ in the pattern of hydrophobic repeats [2,3,4,5]. These features cause bends or kinks in the coiled coil assembly.
It is well known  that α-helices are capable of assembling into a wide variety of structural types. Our aim in the present paper is to study three specific examples of 4-helix bundles by investigating their schemes of knobs-into-holes packing as found in high-resolution atomic structures reported in the literature. To aid this task, we have written a computer program that reads in a Protein Data Bank (pdb) coordinate file, searches for knobs-into-holes interfaces between pairs of α-helices, and outputs the corresponding patterns of interlocking amino acids. Our program, which is outlined in the appendix, is somewhat similar to socket software ; but, unlike socket, it does not presuppose a heptad sequence repeat.
Our three main examples of 4-helix bundles have quite different geometries: the modified GCN4 tetramer of Harbury et al.  has overall left-handed twist, resembling broadly that of the well-known GCN4 simple coiled coil; tetrabrachion, the tetrameric Staphylothermus marinus surface layer protein , hereafter referred to as the ‘archaeon surface protein’, is like a bundle of four parallel pencils, with zero twist; while the tetramerization domain of the Mnt repressor from the Salmonella bacteriophage P22 has a strong overall right-handed twist and, unlike the other two, has anti-parallel packing . Our interpretation of these structures is that each of them displays knobs-into-holes interfaces, but of three distinct varieties, which are related in a simple geometric way to the repeat pattern of hydrophobic residues in the constituent α-helices. (We do not refer to ‘ridges into grooves’ packing of residues  in this paper: that concept is relevant to α-helices that cross over, rather than coil around, one another.)
We begin by reviewing Crick's classical description of a coiled coil between two α-helices; and then we examine each of the three tetramers in turn. Our presentation will be mainly through simplified diagrams: full details of the three structures are available in the structural database.
2. The classical coiled coil
Figure 1b shows an axial view, from the N-terminal end, of an α-helix having the heptad repeat pattern described above: hydrophobic residues at positions a and d—which are shaded in figure 1a—are here represented by black circles, while the remaining residues appear as open circles. In this kind of diagram, a single circle at any of the seven positions represents all n of the corresponding amino acids. (The termini of the polypeptide chain of a protein are conventionally labelled as N (start) and C (end).)
Figure 1c shows the unrolled, plane cylindrical surface lattice of the α-helix, at a radius of about 5 Å. It has been drawn here for a helical repeat equal to 11/3≈3.67 residues in a turn, which is close to the value found in many α-helices. The seven near-longitudinal lattice lines a … g are tilted to the left, just as in figure 1a. The residues are numbered in sequence from 0, and the bold single-start lattice line corresponds to the polypeptide chain that links the successive residues. The arrows labelled 1, 3, 4, 7, 11 and 15 connect residues whose numbers differ by 1, 3, etc. These satisfy the law of vector addition, e.g. 1+3=4. Vector 7 lies along the lines a … g; and in this diagram the hydrophobic stripe is a zig-zag pattern of black circles separated by vectors 4, 3, 4, 3, etc. As drawn here, vector 11 is axial, i.e. perpendicular to the circumferential line 0–0.
If the diagram of figure 1c were to be cut out, creased along the seven lines a … g, and folded into a tube, we would have a twisted tube with a seven-sided cross section, much as in figure 1a. The twist—left-handed—would be a direct consequence of the difference between the helical repeat of 11/3≈3.67 and 3.5, a helical repeat for which the tube would obviously be untwisted. The diagram of figure 1b would be an exact geometrical representation of a helix with a repeat of 3.5; but it can be adapted to the present case if we imagine that it rotates slowly in a counter-clockwise direction as we move along the helix and away from the reader, from the N- towards the C-terminal direction.
Figure 1d shows Crick's pattern of knobs-into-holes interlocking, as in figure 1a, but now viewed from inside the near helix and towards the far one. The far surface lattice is thus viewed from the outside, as in figure 1a, and the hydrophobic residues (black circles) have exactly the same zig-zag pattern as in figure 1c: the lines a, d and their neighbours g, e have been identified by bold letters at the bottom of the picture. The hydrophobic residues of the near helix are viewed from inside the cylindrical lattice, and are shown here by shaded circles; so their pattern is the mirror-image reflection of figure 1c about a line such as a. The lattice lines a, d, etc. of this α-helix are identified by light letters at the top of the diagram. The knobs-into-holes interlocking of the zig-zag pattern of residues in the two hydrophobic stripes (cf. figure 1a) is clear. Note, in particular, that each hydrophobic residue is surrounded by three others on the adjacent α-helix: Crick's ‘holes’ do not presuppose four surrounding residues, as holes do in socket .
In figure 1d the interlocking sets of hydrophobic residues are shown running vertically up the page. In order to prepare this picture, the pattern of figure 1c has been rotated clockwise by about 12°; and the axis of that pattern has been marked in figure 1d by a thick broken line. The near lattice has likewise been rotated, but in the opposite direction; and its axis is also shown, by a thin broken line. The angle between these two broken-line axes is the ‘helix-crossing’ angle at which the α-helices are obliged to cross by the interlocked knobs-into-holes packing; and this in turn—as Crick explained—drives the α-helices to pack into a coiled coil with a left-handed super-helical form. The inclined lines on the unrolled cylindrical surface lattice are not straight, of course, when the lattice is drawn onto a cylinder. Therefore, in order for the packing scheme of figure 1d to extend along the interface, it is necessary for the centre-lines of the α-helices to curve slightly, so that the interface becomes a straight, but slightly twisted, strip. As Crick argued , the energy of such bending of the helices ‘is probably very small’.
An axial view of a small portion of this same interface is shown schematically in figure 1e. Here, the shading of the interlocking hydrophobic residues is different from that shown in figure 1d: residues in the d position, which are side by side at x in figure 1d, are shown black, while residues a, side by side at y, are shown shaded. (In figure 1e, the three-dimensional interlocking of residues would have been shown more clearly if the two α-helices had been drawn closer together; but then the ‘d’s would have obscured the ‘a’s.)
Figure 1d, similar to its original in Crick's paper , is a schematic of the actual knobs-into-holes packing of the hydrophobic residues. Our computer program (see appendix) provides similar diagrams, with the individual amino acids identified, and distances between their Cβ atoms marked. An example of such a pattern, corresponding to the simple GCN4 coiled-coil interface, is shown in figure 1f. Here, the amino acids of the far helix are represented by bold circles, whereas those of the near helix are shown lighter. Vectors 3 and 4, as in figure 1c, are marked. Overall, the pattern consists of alternating ‘squares’ of 3 and 4. Distances are shown in ångström units. Two aspects of such diagrams may be questioned: the precision of such distances, even for high-resolution crystal structures; and also the validity of using Cβ atoms as indicators of the positions of the (sometimes large) amino acids. Nevertheless, these patterns strongly confirm the arrangement of knobs-into-holes packing along the ‘hydrophobic stripe’.
In the interface shown in figure 1f, most of the participating amino acids are leucine (L) or valine (V). But two polar asparagines (N) are also present; and examination of the three-dimensional structure shows that these form hydrogen bonds with one another. The significance of this apparent anomaly has been discussed by Laughton et al. .
Thus, by means of a sequence of simple diagrams, we have represented Crick's three-dimensional concept of the ‘knobs-into-holes’ packing of α-helices by means of the diagram of figure 1f, which includes quantitative data from a specific case.
3. The TolC barrel
Here we may mention briefly, as a minor variant of the classical coiled coil, the formation of a cylindrical barrel of 12 nearly parallel α-helices by an arrangement in which the helices have a more complex heptad repeat, of the kind so that the H and h residues, at a, d and b, f, respectively, form two distinct hydrophobic stripes, as indicated in figure 2a,b . We may note here that each of these α-helices is connected, by knobs-into-holes interface packing, to two neighbours: the result is a 12-helix tubular structure. Later, we shall investigate other examples of α-helices having interfaces with two partner helices.
In comparison with α-helices in conventional two-helix coiled coils, those in the TolC barrel are untwisted; that is, they have a smaller number—approximately 3.55—of residues per helical turn. This illustrates the principle that the helical repeat of an α-helix is not, in general, a function only of the hydrophobic sequence repeat. We shall return to this important point in §7.
4. Modified GCN4 4-helix bundles
The GCN4 α-helix forms a short homodimeric coiled coil—i.e. a coiled coil of two identical α-helices—by means of a classical knobs-into-holes interface, as shown in figure 1f. Alber and co-workers [13,7] modified the amino acid sequence of the hydrophobic stripe of this α-helix and discovered that different modified sequences formed 2-, 3- or 4-helix bundles, respectively. For example, by placing leucine at all a positions and isoleucine at all d positions, they created an α-helix that self-assembled into a 4-helix bundle, with the four helices parallel in an N–C sense.
That arrangement is shown schematically in figure 1g. There, the a and g residues are in black, while the d and e residues are shaded: the black ones are roughly at one ‘level’ and the shaded ones are at the next. Although it appears that residues a, e from one helix and g, d from the next form classical knobs-into-holes interfaces, only one of each of the g and e residues is hydrophobic; stabilization of the bundle takes place by means of a central ‘core’ of hydrophobic residues, formed from alternating ‘rings’ or layers of four leucines at a and four isoleucines at d. Such an arrangement is sometimes described as ‘knobs-to-knobs’ packing ; but the basic phenomenon of neighbouring hydrophobic knobs clustering, but not interpenetrating, is essentially the same as that operating in the knobs-into-holes arrangement as described by Crick.
Hulko et al.  have described a different kind of parallel 4-helix bundle, formed by the HAMP domain of bacterial chemoreceptors. It is shown schematically in figure 1h, where again alternating layers of residues are drawn black and shaded. In this arrangement, one can pick out a distorted version of the classical knobs-into-holes interface between residues a, e of one helix and a, d of the next. However, it is more convincing to focus on the residues a from one pair of opposing helices and d, e from the other pair (e.g. the black residues in figure 1h), which together form a compact knobs-to-knobs rectangular cluster of six hydrophobic residues in a layer; while a similar group—but rotated through 90°—form in the next layer; and so on.
Hulko et al.  have indeed argued that, in the HAMP domain structure, the packing of figure 1h may be transformed to that of figure 1g by means of contra-rotation of the helices through ±26° about their axes, with the knobs playing the part of cogs in a train of four gear wheels. (That would require the residues in the upper and lower helices in figure 1g to be re-labelled, so that a becomes e, etc.) Such a switch or transformation could provide a means whereby the HAMP domain, which appears in many transmembrane proteins—i.e. ones lying through the thickness of cellular membranes—could play a role in signal transmission. Later, we shall argue for another kind of potential bistable switch in the Mnt-repressor 4-helix bundle. Such switches may contribute to allosteric switching in some molecular devices within the cell, which exhibit cooperative behaviour.
Essentially the same two packing patterns of figure 1g,h have also been reported by Deng et al.  in anti-parallel 4-helix bundles of GCN4, but now with other engineered modifications from the native amino acid sequence. Thus figure 1g, but with the order of residues in the upper and lower helices changed from clockwise to counter-clockwise, is the same as fig. 6A of ; while figure 1h, similarly altered, corresponds to figs 6B,C of —the key point here being that the particular heptad repeat results in a hydrophobic stripe of three neighbouring 7-start lattice lines rather than the two as shown in figure 1c.
The question of what heptad-repeating sequences will predispose helices to form either parallel or anti-parallel bundles of four, and packed according to the schemes of either figure 1g or 1h, is a complex matter that lies outside the scope of the present study. In general, the stabilizing effect of a hydrophobic core is reinforced by hydrogen-bonded interactions around the periphery of the bundle (see e.g. ).
However, the main point that we wish to make here is that all of these various 4-helix bundles involve heptad repeats of various sorts in the amino acid sequence; and in consequence all of these bundles have, overall, a left-handed twist about a central axis.
5. The archaeon surface protein 4-helix bundle
All of our examples so far have involved α-helices with variants of the classical heptad amino acid repeat. We now consider an α-helix from an archaeon surface protein  whose amino acid sequence has an 11-repeat, which may be represented as Thus in figure 3a, which shows a schematic axial view along the α-helix from the N end, there is a cluster of hydrophobic residues, shown as black circles labelled p, i, m, q on the circumference.
The corresponding cylindrical surface lattice, shown in figure 3b, has its residues at precisely the same locations as in figure 1c: as before, the helical repeat has been set at 11/3. Since the sequence repeat is now 11, we have labelled the 11-start vertical (axial) lattice lines i, j, k…s, starting with i at residue 1, j at 2, etc. Hence this pattern of hydrophobic residues makes overall a vertical ‘stripe’, parallel to the axis of the α-helix—although it is clearly a more complicated stripe than its inclined counterpart of figure 1c.
Now in figure 1c we may describe the hydrophobic stripe by its constituent vectors, either as (4,3)n or (3,4)n—where the sum of the integers included in the brackets is 7, corresponding to the 7-repeat of the amino acid sequence. Likewise, in the case of figure 3b, we may describe the hydrophobic stripe as (4,3,1,3)n or (3,1,3,4)n, etc. In each case, the sum of the four integers included in the brackets is 11: this indicates a stripe parallel to the axis of the α-helix, as distinct from the left-leaning pattern (4,3)n of figure 1c.
The pattern of knobs-into-holes interlocking along this kind of hydrophobic stripe is shown schematically in figure 3c. Again, as in figure 1d, the ‘far’ residues are shown as black circles, on lattice lines identified in bold letters at the bottom, while the ‘near’ residues, viewed from inside their α-helix, are shown shaded, with their lattice lines identified in light letters at the top.
This pattern is obviously more complex than those we have discussed so far. For example, some circles in figure 3c may seem to be too close to one another; but they do indeed correspond to hydrophobic amino acids in close proximity in the three-dimensional crystal structure, and also as revealed in the output from our computer program.
An axial view of a portion of such an interface is shown in figure 3d. We can see that the residues here marked in black form an interface of a similar kind to that shown in figure 1e—except, of course, that the knobs-into-holes packing occurs in ‘clusters’ along the interface, punctuated by gaps where the shaded residues interlock: the shaded residues are the ‘outliers’ on lattice lines p, q. We may note here that repeating patterns with hydrophobic residues occupying adjacent positions in the sequence (here p, q) have sometimes been described as ‘da layers’. We have chosen our reference-letter system in figure 3 specifically to avoid in this case the heptad lettering system (abcdefg)n, since the 7-letter scheme does not match the actual 11-letter repeat.
So far we have been considering a single interface between two of the four α-helices of the archaeon surface protein bundle. The packing arrangement of the four together is shown schematically in the axial view of figure 3e. Here it is apparent that each α-helix has two parallel stripes of hydrophobic residues, something like the arrangement shown in figure 2, but now with the two interfaces arranged practically at right angles in the view of figure 3e, so as to form a compact square: 11 is sufficiently close to 12 for this to be geometrically feasible. In the unrolled surface lattice (figure 3b) the ‘second stripe’ has been indicated by shaded circles. The two stripes are separated by one residue around the circumference, and indeed they share the residues on lattice line q in figure 3b.
This ‘two-stripe’ sequence-repeat pattern may be written as where H and h denote hydrophobic residues in the respective stripes and * is shared. The shared hydrophobic residues form a ‘ring of four’, as shown by the inner group of shaded residues in figure 3e. These are analogous to the rings shown in figure 1g, except that here they are aligned on the four geometrically parallel lattice lines q.
So far we have described a pattern of interlocking, but unspecified, hydrophobic amino acids. Figure 3f shows the layout of each of the four interfaces seen in figure 3e, but now set out even more schematically than in figure 3c, with the ‘far’ residues marked in bold open circles and the ‘near’ ones lighter. This diagram has been constructed from the output of our computer program. The numbers 1, 3 and 4 along the centre-line of the ladder correspond to the vectors marked in figure 3b. The pattern consists of a sequence of ‘squares’ 3, 4, 3, 1, 3, 4, 3, 1, etc. In this diagram, individual residues are identified. Note particularly that residues 22, 33 and 44, lying on lattice line m, are polar amino acids arginine (R) or asparagine (N); but unlike the asparagine residues in the classical GCN4 coiled-coil interface (figure 1f), they do not form hydrogen bonds with one another.
Note also that the residues that form ‘rings of four’ (lattice line q) are either leucine (L) (26, 37) or isoleucine (I) (48)—just as in the modified GCN4 4-helix bundle (figure 1g). At these locations on the schematic ‘ladder’ of figure 3f, the ‘vertical’ distances marked on the right are actually horizontal in the three-dimensional structure.
Finally, note that between successive ‘rings of four’ are central cavities, which, as Stetefeld et al.  point out, may contain water molecules.
Here, we should also mention a 4-helix bundle with right-handed overall twist formed by the human vasodilator-stimulated phosphoprotein (VASP) tetramerization domain of Kühnel et al. . The parallel α-helical components have a 15-residue repeat, which was successfully predicted to support a stable bundle . As in our previous examples, the interlocking of hydrophobic residues may best be described in terms of ‘layers’ (figures 1g,h and 3e).
Now for the modified GCN4 bundle, there is a repeating set of two distinct layers, each of which adopts the same kind of layout, as shown by the black and shaded circles, respectively, in figure 1g. For the archaeon surface protein, there are three kinds of layer: one involves the residues q, which form a central square; while the other two correspond to more-or-less classical knobs-into-holes packing at the four inter-helix interfaces (figure 3c,e).
For the VASP bundle, with its 15-residue repeat, there are altogether four kinds of layer. This may be deduced from figure 1c, when it is noted that the number of distinct layers is equal to the number of turns of the single-start helix within the repeat of 7, 11 or 15. Examination of the VASP structure shows that one kind of layer, designated ‘de’ by Kühnel et al. , involves a ring of contacts between valine and the hydrophobic moiety of lysine on adjacent helices. In contrast, the other three kinds of layer, designated a, h and l, respectively, involve compact groups of four hydrophobic residues at the centre—leucine, isoleucine, valine or phenylalanine—which are generally similar to the clusters of four shown in figure 1g. In some cases (all of a and one l), these are the only four hydrophobic residues in the layer; while around the periphery are salt bridges between glutamic acid and lysine, glutamine or arginine, respectively. In other cases (both h and one l), the central four hydrophobic residues connect with others in patterns similar to those involving residues marked a, g (or d, e) in figure 1g, without any obvious external salt bridge interaction. In addition, this structure has other stabilizing salt bridges on the exterior of the bundle, between residues in adjacent ‘layers’.
6. The Mnt-repressor 4-helix bundle
The 4-helix bundle formed by the Mnt-repressor α-helix  is more complex, geometrically, than any of the 4-helix bundles discussed so far. In both the modified GCN4 and archaeon surface protein bundles, the helices have been parallel, in the sense that all four N termini are at the same end of the bundle. In contrast, the Mnt-repressor bundle has neighbouring helices around the circumference running anti-parallel, so that their N terminals are at opposite ends of the bundle. This arrangement is consistent with twofold rotational symmetry about an axis perpendicular to the bundle's axis, as was found in the NMR structure.
Figure 4a shows, approximately to scale, a view of the bundle along this twofold axis. The dots represent the Cβ atoms of the side chains, and the portions of the helices that are shown here run between residues 55 and 77. The helices have been labelled, arbitrarily, as A…D, and the N- and C-terminals of the portions shown are indicated by subscripts.
The larger black circles and thicker black lines in the near helices A and C, and the smaller black circles and thinner black lines in the far helices B and D, connect residues whose sequence numbers differ by four; and the thinnest lines in all four helices correspond to the single-start polypeptide chains.
Now in the modified GCN4 bundle the interfaces between neighbouring α-helices run along the 7-start lines of the cylindrical surface lattice (figure 1c); and in the archaeon surface protein bundle they run along the 11-start lattice direction (figure 3b). Here, in the Mnt-repressor bundle, the interfaces all run along the 4-start lattice lines, as shown in the surface lattice of figure 4b—which has the same overall geometry as the lattices of figures 1c, 2b and 3b. In some ways this is the most obvious arrangement of all: for if we were to crease the rolled-out lattice along these four lines and connect the edges to make a tube, we would have a box of square cross section, with a strong right-handed twist—as may indeed be seen in the sketch of figure 4a. Square boxes are obviously convenient for making a bundle of four; and the right-handed twist makes for right-handed supercoils among neighbours, and hence right-handed twist in the entire assembly.
The interfaces between helices A and B on the right and between helices C and D on the left of figure 4a are identical, in consequence of the twofold rotational symmetry. But it is clear that the interfaces between the two near helices A and C and the two far helices B and D are significantly different. Thus, residues B55, 66 and 77 are opposite residues D77, 66 and 55 on the far interface, whereas A55, 66 and 77 are separated longitudinally from C77, 66 and 55, respectively, by a substantial distance on the near interface. Further, while residues 66 are fairly close to one another on the three interfaces AB, BD and DC, they are a long way apart in the fourth interface AC. The same overall effect can be seen in the ‘altitude’ of the alternating end residues 55 and 77 at the top of the helices, as shown in figure 4a: as one goes around the four helices from A to B to D to C, it is a downhill progression.
One would get a similar effect—as shown on the left in figure 4c—by wrapping a piece of paper round a cylindrical rod to make a sleeve; then marking on it four equi-spaced vertical lines; and finally slitting the sleeve between two of these lines and shearing the cut edges so that the lines become right-handed helices. These four lines have essentially the same relationship to one another as the four helices in figure 4a: their N-terminal ends have been labelled A, B, C and D, accordingly.
In the surface lattice of figure 4b, the lattice lines have been labelled t, u, v, w: a four-letter code for a four-repeat of hydrophobic residues. The hydrophobic ‘stripe’ may be described as (3,1)n, by comparison with (4,3)n in figure 1c and (4,3,1,3)n in figure 3b. In figure 4d the layout of a generic knobs-into-holes interface has been shown by the same convention as in figures 1d and 3c. Note, however, that here the u-lines of the two helices coincide, since the helices are anti-parallel, whereas in figure 1d the a-lines of the two (parallel) helices are on opposite sides of the diagram.
The way in which four of these twisted boxes pack into a bundle is shown in the schematic axial view of figure 4e: the view is from the top of figure 4a, and the N- and C-terminal ends are indicated. The corresponding interface diagrams, as produced by our computer program, are displayed in figure 4f, using the same conventions as in figures 1f and 3f. The diagram on the left represents the two identical interfaces AB (viewed from B to A) and CD (viewed from D to C). The AB interface is in the same overall orientation as in figure 4a, but the diagram should be turned upside-down—so that the end-lettering of C and D becomes the right way up—for the CD interface. The twofold axis of the three-dimensional structure passes between the valine (V) residues A66 and B66. In this diagram, the residues are mirror symmetric about this line; but note that the inter-Cβ distances are thus not symmetric.
The middle diagram of figure 4f shows the interface BD, viewed from D to B. It is shorter than the AB/CD interfaces, and the inter-Cβ distances are not, on average, so close. Here the twofold axis passing between the valine (V) residues B66 and D66 is marked; and this diagram is symmetric both in the layout of residues and in the inter-Cβ distances, as expected (from figure 4a). Residues 65, 66 and 67 lie on lattice lines t, u and v, respectively; so residues 66 and 67 feature in interfaces AB/CD, while 65 and 66 feature in interface BD (and AC) (figure 4a).
Finally, the diagram on the right in figure 4f shows details of the interface AC. The interlocking knobs are the same as those of interface BD; but now the twofold axis passes between valine (V) residues A70 and C70. The qualitative interlocking pattern of knobs in interface AC can be obtained from that of interface BD by disengaging the two helices and moving one relative to the other by eight residues, or approximately 12 Å. Thus, for example, residue 66 is opposite to 65 in interface BD, but opposite to 73 in interface AC: a shift of eight residues.
The asymmetric assembly of the Mnt-repressor's bundle of four α-helices of identical sequence suggests that the structure may act as a bistable ‘switch’, with the well-bonded left and right pairs of helices in figure 4a moving as rigid bodies relative to each other. Thus, if the right pair moves down relative to the left pair so that the front interface adopts the packing pattern of the rear interface shown in figure 4a, and vice versa, the switch will have moved between two equivalent conformations. These two equivalent states are shown schematically in figure 4c.
At a geometric level, our main finding in this paper is that Crick's idea of knobs-into-holes packing of hydrophobic residues, to form a stabilizing interface between two α-helices in a coiled coil, may be generalized to three species of 4-helix bundle, with the ‘hydrophobic stripes’ of the constituent α-helices running along the 7-, 11- or 4-start lines, respectively, of the cylindrical surface lattice of amino acids of these helices, with repeat patterns (4,3)n, (4,3,1,3)n and (3,1)n, respectively.
Our geometrical description—Crick's description—of the way in which the orientation of a ‘hydrophobic stripe’ on the (plane) cylindrical surface lattice immediately determines the ‘helix-crossing angle’, and thus the sense—right- or left-handed—of the supercoil, is much clearer and more general than descriptions that are based purely on the reference frame of the sequence repeats [2,17], without any mention of the cylindrical surface lattice. For instance, in discussing the ‘classical’ coiled coil with its heptad hydrophobic repeat, it is stated in  that ‘by giving the right-handed helices a left-handed twist [our italics], coiled coils effectively reduce the number of residues per turn to 3.5 with respect to the supercoil axis, and thus allow the positions of side chains to repeat after two turns (or seven residues).’ In fact the α-helices are not given a left-handed twist: as Crick explained, they are simply given small rigid body rotations, so that their inclined hydrophobic stripes on the cylindrical surface lattice are brought into alignment. Further, description of the ‘classical’ coiled coil as ‘7/2 repeats’ is not clear; and indeed the elaborate diagrams that have been made on the basis of such repeats [2,17] are superfluous, once it has been grasped that the geometry of supercoiling is determined primarily by the ‘stripes’ of hydrophobic amino acids on the cylindrical surface lattice; which is fixed mainly by the sequence repeat of hydrophobic amino acids.
The ‘hydrophobic core’, as studied here, is not a sufficient stabilizing effect in the formation of a bundle: in our examples—including the VASP bundle and the TolC barrel—there are additional specific hydrogen-bonding links between adjacent α-helices on the outer surface of the bundles. Such effects would have to be taken into account in any scheme for ab initio design of 4-helix or other bundles.
One of the motivations for this study was the possible role of α-helical bundles in providing conformational ‘switches’ for various kinds of molecular machinery. (These are broadly analogous, but on a much smaller scale, of course, to ‘snap-through’ problems in engineering structures, as elucidated by Thompson .) For example, a switching action in α-helical tropomyosin coiled coils  allows the formation of reversible kinks in that molecule, which may have a role in the operation of muscle. Another example is the coiled-coil moiety of GCN4, which may act as a switch when DNA is bound ; and a similar effect may provide the longitudinal ‘piston movement’ in the long coiled coils of transmembrane signalling proteins, which has been postulated as part of the signalling mechanism by Yu & Koshland . Another kind of movement is ‘rotary switching’, as described in §4, which has been postulated for the role of the HAMP domain  in transmembrane receptors. Here four α-helices, meshing together with their knobs acting like the cogs of gear wheels, could contra-rotate between two non-equivalent states.
All of these examples of bistable switching mechanisms in α-helical bundles involve small relative movements, of the order of a few ångström, between the hydrophobic residue ‘knobs’ and the holes into which they fit. In contrast, our suggested switch in the Mnt-repressor bundle has a much longer throw, of the order of 10 Å. Unlike these other postulated switches, it would involve a disengagement of the knobs-into-holes interlock, and rearrangement, with the knobs occupying different holes. But there seems to be no fundamental objection to a large-throw switch of this kind.
It may be noted that conformational asymmetry, of the kind found in the right-handed Mnt-repressor bundle, is not found in the modified GCN4 4-helix bundle, which has left-handed twist of about the same magnitude. In other words, the simple diagram of figure 4c, with its potential ‘switching’ implications, does not necessarily apply in general to twisted bundles. There seems to be no fundamental reason for there to be different packing arrangements between our left-handed and right-handed 4-helix bundles, although it does seem obvious that an untwisted bundle, such as the archaeon surface protein, will not be a candidate for switching action.
B.F.L. and J.V.P. were supported by the Wellcome Trust. We thank Dr Charlie Laughton for stimulating and insightful discussions, Hazel Dunn for typing and Dr Mark Schenk for help with figure 1a.
Appendix. Computer algorithm for detecting and mapping knobs-into-holes interfaces
In outline, our computer program proceeds in two parts. The first phase (steps 1–8, as detailed below) consists of a straightforward sequence of operations. The second phase (steps 9–18) is more complex and involves pattern recognition procedures. Numerical constants are required at various stages; these have been determined empirically during studies of various examples, in which the interface diagrams produced by the program have been compared directly with the three-dimensional structures.
Read in atomic coordinates from a pdb-format file, and extract tables of Cα and Cβ atoms.
Detect the presence of α-helices by the use of a well-known algorithm based on the distances between four successive Cα atoms . For each helix, construct an approximate helical axis based on a progression of three successive Cα coordinates. Coordinates of this axis are calculated for every fifth residue of the helix, and these are used to identify pairs of helices that are close enough to form a coiled-coil interaction.
For each pair of α-helices so detected:
— Determine whether the helices are oriented in a ‘parallel’ or ‘anti-parallel’ arrangement.
— Compute a distance matrix between the two helices using the coordinates of their axes. From this matrix, identify regions of the helices where the inter-axial distances are less than 14 Å. When such regions are found, there exist potential coiled-coil interfaces.
— Determine the crossing angle between the axes of the helices in each interface region. The sign of this angle indicates whether the local coiled coil is left- or right-handed. Typically the crossing angles for a coiled coil involving three or more turns of the helices are less than 35° in magnitude.
For each potential interface, compute an inter-Cβ distance matrix, i.e. a matrix whose entries are distances from each Cβ atom of one helix to every Cβ atom of the other.
Figure 5a shows the form of a portion of a typical distance matrix for a canonical left-handed parallel coiled coil , with x and y representing distances less than 12 Åand dots representing distances greater than or equal to 12 Å, which will not be of interest here. Over the length of the coiled-coil interface, the shortest inter-Cβ distances correspond to elements lying along a diagonal of the matrix; and they are separated by alternating ‘squares’ of 4 and 3. The x values on the diagonal correspond to the ‘horizontal’ distance between residues from opposite sides of the ladder in figure 1f—distances that are not marked there or in the displays of figures 3f and 4f, for reasons of clarity. The ‘off-diagonal’ x entries—i.e. numbers x at the other two corners of the squares—correspond to the alternating long and short ‘vertical’ distances on either sides of the ladder of figure 1f.
Distances marked y in figure 5a correspond to close contacts on either side of ladder diagrams, such as figure 1f, but not shown there: for example, residue L260 is close to both L260 on its left and L261 (not shown) on its right.
In this example, the α-helices are ‘parallel’, and the patterns of close contacts lie on the leading diagonal of the distance matrix. For an anti-parallel interface, the corresponding diagonal runs from top right to bottom left of the matrix. In such cases, for ease of coding, we ‘parallelize’ the interface by inverting the order of residues of the second helix (reading it from its C-terminus to N-terminus).
In general, our goal is to convert data in the form of such a distance matrix into knobs-into-holes interface ‘ladder’ patterns of the kind shown in figures 1f, 3f and 4f. Our earliest patterns of this sort, for the TolC barrel structure , were obtained by direct inspection of three-dimensional graphics, and measurement of candidate Cβ–Cβ distances for amino acids seen to be in obvious close contact.
The general problem in creating interface diagrams from distance-matrix data is one of pattern recognition. Faced with a print-out of a large distance matrix, it is usually not difficult to spot by eye a ‘leading diagonal’ of short distances, which is obviously a major factor in locating an interface. But we need to be able to differentiate between ‘3-squares’ and ‘4-squares’ in patterns such as those of figures 1f and 3f; and indeed between 3-, 4- and 1-squares in general. The problem becomes more acute for short interfaces, when it might not be obvious whether a sequence 4, 3 is to be followed by a 4 (as in figure 1f) or a 1 (as in figure 3f).
The problem is further complicated by the fact that inter-Cβ distances may be as short as 3.6 Åfor two contacting alanines, whereas larger amino acids cannot come so close. Also, the distances involved in 4-, 3- and 1-steps tend to be different: for instance, the 1-step involves bigger distances. Thus we cannot use a simple, universal ‘cut-off’ value in order to identify potential close contacts of amino acids. We have therefore devised an empirical scheme for assessing close contacts, using a ‘smooth’, symmetric step-function as shown in figure 5b. The ‘proximity score’ V (i,j) is evaluated for a given distance d(i,j) between Cβ atoms i and j. Function V (i,j) has a high score of about a when d(i,j) is small, and a low score of about 0 when d(i,j) is large; and when d(i,j) is equal to c/b, the score is equal to a/2. The formula we have used is Different values of c are used for the ‘4-square’, ‘3-square’ and ‘1-square’ patterns. The curve in figure 5b has been drawn for a=10, b=1.1 and c=8.5 Å: these values correspond to the search for ‘4-squares’ and ‘3-squares’ in a canonical left-handed coiled coil.
The first step in the pattern recognition process is to replace each ‘close-contact’ distance d(i,j) in the matrix by its corresponding ‘proximity score’ V (i,j). Significant numbers V will, of course, occupy the same locations in the matrix as those marked in figure 5a.
The next stage is to construct, for each element i, j, a ‘score’ to indicate whether or not the element is at the upper-left corner of a ‘4’ pattern as in figure 5a. This is accomplished by summing the V values for all elements marked e, f in figure 5c, where e of the pattern is located at the current element i, j. For example, when this is done systematically for all elements in figure 5a, there will be a score of around 80 when e is located at the upper-left elements of the two ‘4-squares’; but much smaller scores everywhere else.
Similar summations are performed for every element of V (i,j) in order to search for the upper-left elements of ‘3-square’ and ‘1-square’ patterns, by means of the two protocols shown in figure 5d.
It is then a straightforward matter to assign a single number to the leading element of each 4-, 3- or 1-square of the i, j matrix; and to assign a value of 4, 3 or 1 accordingly in another i, j matrix. All other elements are assigned a value of 0.
Once all the matrix elements have been assigned a unique value of 4, 3, 1 or 0, we search for non-zero elements along a diagonal of the matrix. A typical diagonal for a classical left-handed coiled-coil interface will have a pattern … 4 0 0 0 3 0 0 4 0 0 0 3 …. The corresponding patterns for a ‘straight’ interface and a ‘right-handed’ interface are … 4 0 0 0 3 0 0 1 3 0 0 4 0 0 0 3 0 0 1 3 0 … and … 3 0 0 1 3 0 0 1 3 0 0 1 …, respectively.
Other patterns are also possible. For example, the classical heptad-repeat pattern … 4 3 4 3 4 3 … may be varied to … 4 3 4 4 3 4 3 … or … 4 3 4 3 3 4 3 … when there are ‘stutters’ or ‘stammers’, respectively . Neither of these occurs in the cases studied in the present paper.
At this stage a complication can arise, where it is not clear from the respective 4-, 3- and 1-square scores whether a particular group of four residues forms a 4-square or a 3-square followed immediately by a 1-square. For example, in figure 3f, it might appear from the numerical scores that the 3- and 1-squares going from R22 through S25 to L26 are a 4-square. But if they were, R22 and L26 would have to be on opposite sides of the diagram. This can readily be checked by evaluating the scalar product of the vectors and . In this case, the scalar product is positive; so it is clear that we do not have a 4-square. We have written a sub-program to evaluate such scalar products when the occasion demands.
If the length of the interface is not less than 7 (implying that there are at least three turns of the pairs of α-helices coiling around one another), we conclude that it is a coiled-coil interface and output the residues and distances involved, in the format of figure 1f.
One contribution of 17 to a Theme Issue ‘A celebration of mechanics: from nano to macro’.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.