A review of statistical structures in synthetic polypeptides and biological macromolecules

A review of statistical structures in synthetic polypeptides and biological macromolecules A. Miller Laboratory of Molecular Biophysics, Department of Zoology, University of Oxford, South Parks Road, Oxford, UK

and D. A. D. Parry Departmentof Chemistry, Biochemistry and Biophysics, Massey University, Palmerston North,

New Zealand (Received 26 February 1974) INTRODUCTION

The relationship between the structure and function of biological macromolecules is fundamental to our understanding of molecular biology. It is generally true that those proteins which are rich in charged residues have a high axial ratio (fibrous proteins) and those which are low in charged residues tend to have a small axial ratio (globular proteins). Rod-shaped proteins have a larger surface area and smaller internal volume per unit volume than do the globular proteins. The surface charge per unit area is reduced by the protein adopting a cylindrical shape. Also, a larger percentage of apolar residues can be accommodated in the centre of a globular unit than in a cylindrical unit. This review is primarily concerned with attempting to understand how the long, rod-shaped molecules found in fibrous proteins pack together in three dimensions in a regular or quasi-regular manner to produce functionally useful filaments or fibrils such as are present in muscle, tendon, cartilage, skin etc. Since it is often possible to produce synthetic filaments by self-assembly of the long molecules from solution, the structure of the molecular aggregate must be specifically related to the packing of the individual molecules. Electron microscopy has allowed the visualization of orderly arrays of filaments in muscle and hair but technical limitations still prevent easy recognition of regular packing of the molecules themselves. X-ray diffraction, however, conclusively shows the existence of such regular assemblies of molecules in the native state of many biological fibres. Long rodshaped macromolecules are specifically designed by nature to aggregate. The kind of molecular interactions in the aggregate are normally dictated by thermodynamic considerations of free energy. In the event of the possible formation of several structures of similar free energy, that which is biologically useful must be preferred. The purpose of a structural investigation of a fibrous protein is to see how the molecular structure can lead to specific interactions which produce regular aggregates and also how these interactions bestow properties appropriate to the biological role on the resulting aggregate. In this review we draw attention to a particular

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aspect of regular arrays of fibrous proteins, their synthetic analogues and also the nucleic acids, viz. statisticality. We first describe the various possible categories o f statisticality and show how the X-ray diffraction patterns from such structures may be calculated. Then follows a review o f the known occurrences of statisticality in assemblies of biopolymers and their analogues and a discussion of the biological implications. We also point out the characteristic features of X-ray diffraction patterns from structures which exhibit statistical ordering. TYPES OF STATISTICALITY Statistical structures are arrangements of molecules (or molecular aggregates)which exhibit a limited and highly specific form of disorder. Whilst in a crystal lattice the contents of the unit cells are identical, in many statistical structures there are two or more unit cells of differing contents which occur throughout the lattice. The differences between these unit cells is brought about in the cases we shall consider, by the molecules or the molecular aggregates packing together in two or more different ways. The crystal lattice parameters remain closely similar in different unit cells, but the unit cell contents may be one of a finite number of kinds. Two general types of statisticality are geometrically possible. In the first type we imagine each unit cell can be one of t different kinds, each of which may be unrelated to the kind of their neighbours. For example, suppose a unit cell contains a molecule which can either point 'up' or 'down' (i.e. t = 2). In the first type of statisticality the occurrence of molecules in 'up' and 'down' directions will be randomly distributed throughout the crystal lattice, i.e. a molecule found in the 'up' direction will not be determined by the direction of the neighbouring molecules. The unit cell of the statistical structure will be the same as that of a perfect lattice with all the molecules pointing in the same direction. However, in the former case each unit cell can be thought of as containing a statistical molecule which comprises a half weighted 'up' and 'down' molecule at the same lattice site. In the second type of statisticality, the particular kind of one unit cell

A review of statistical structures in polypeptides: A. Miller and D. A. D. Parry may affect that of neighbouring cells. The kinds of unit cells are 'coupled' in some manner. This usually results in the unit cell of the statistical structure being larger than that of the perfect lattice with all unit cells of one type. EFFECT OF STATISTICAL ORDERING ON THE FOURIER TRANSFORM If a 'unit' is a molecule or molecular aggregate, then in order to define that unit fully, its position and nature must be specified. If both of these parameters are statistically distributed in an independent manner (as will be the usual case in the types of system which we will discuss), then the scattering power of a set of N units which have scattering factors f j ( j = 1. . . . N) will be1: N

I~'(s) =ZJ}2+ Z Y J)f/cos(2~rs.rjj') 1 j,aj,

(1)

where s and r are the vectors in reciprocal and real space respectively and r # = r j ' - r j . It follows directly from equation (1) that the observed intensity will be the average value of equation (1) determined over all possible states, i.e. IN(s)

= Nfj ~ + Z Yfj.J),cos(2~s. r#,) = Nj~ -2 + ZZJ~J},cos(2,s. r # ) =

N [jj2 _ (j)) 2] + (fj) 2[N + Z Zcos(ecrs. rjj')]

(2)

Now [jje_ (./~)2] may be expressed as: 1

i=N

1

j=N

i

and (j~)z as:

7v~

fJ

The second term in equation (2) is instantly recognizable as the diffracted intensity for a group of N identical units with scattering factor f j and will therefore give rise to discrete Bragg reflections. However, the first term arises from the differences in the scattering units. As such, its magnitude will in general be small and it will not be confined to discrete points in reciprocal space but will vary relatively smoothly along the layer lines. This feature in a diffraction pattern is often called 'a layer line streak'. Particularly clear examples may be seen in patterns obtained from nylon-6 (e-form) 2 and a-poly(e-alanine)a, 4. If a structure exhibits 'pure' randomness, then the intensity of the Bragg reflections will be zero (as E f t = 0 ) . The corresponding intensity of the layer line streaks will be proportional to: 1

i--N

j=37

1

i

In the case where the molecules are coaxial but not otherwise ordered with respect to one another, only continuous layer line streaks (i.e. no discrete Bragg reflections) will be present in the X-ray diagram. EXAMPLES OF STATISTICAL ORDERING

Polarity disorder Using X-ray diffraction methods, Brown and Trotter 5 investigated the structure of highly crystalline fibres

of a-poly(L-alanine). The equatorial reflections were readily accounted for by a hexagonal cell of side 8.55 A, which can contain only one molecule on density considerations. Brown and Trotter also measured a meridional reflection of spacing 1.495/k which is characteristic of an a-helical structure. However, they were unable to obtain quantitative agreement between an a-helical model and the observed data. Three years later, Elliott and Malcolm a recognized the source of the problem and suggested that an incorrect choice of crystal structure had been made. Polypeptide chains exhibit chain sense - N H . C H R . C O - and if a-helical molecules point 'up" or "down' on a random basis, then the molecular site in the unit cell may be considered a half weighted "up' and 'down' molecule, i.e. the unit cell still contains only 'one' molecule. This arrangement was investigated by Elliott and Malcolm using optical diffraction and consequently better agreement between the observed data and the model was achieved. This synthetic polypeptide therefore exhibits polarity staffsticality at the molecular level. The structure was finally refined by Arnott and Wonacott a using the method devised 6-s by them. Furthermore, they were able to show that if methyl groups were assumed to be spherical, the packing energy of two 'up' a-helices of poly(Lalanine) would be very similar to that of an 'up' and 'down' pair. Even though some energy differences were noted when the approximation to the methyl group was replaced by the individual hydrogen atoms, it would seem likely that in this case the statistical structure arises as a consequence of the energy balance between the alternative modes of packing. The transition from an 'up' to a 'down' molecule or aggregate of molecules is not unique. The change is made by a rotation about some axis perpendicular to the long axis of the molecule or molecular aggregate. The molecule or molecular aggregate may then be translated in the direction of the fibre axis and rotated about that axis by amounts consistent with optimum packing and interactions. These two parameters were refined by Arnott and Wonacott who eventually reduced the R factor for their structure determination of a-poly(Lalanine) to the satisfactory value of 0.206. As we have previously mentioned, statistical structures often give rise to continuous layer line streaks (in addition to discrete Bragg reflections) in the X-ray diffraction patterns. ~-Poly(L-alanine) shows two such streaks on the 13th and 21st layer lines (axis repeat= 70.4 A, see Figure la). Arnott and Wonacott calculated both the positions and intensities of streaks expected from their statistical model and found an extremely satisfactory correlation with the data, thus removing any doubt as to the validity of their conclusions. One point should be emphasized. The introduction of statisticality into a structure determination implies that as more parameters are used in the refinement procedure, the 'fit' between model and data is likely to be improved. This is certainly not necessarily the case and is well illustrated in the case of poly(L-proline)II. This system was investigated by Cowan and McGavin 9 who showed that the polypeptide formed left-handed three fold helices. Furthermore, they postulated that all the helices were similarly directed with respect to one another. Arnott s studied a statistical structure where the axes of the helices were parallel but randomly directed. Despite the fact that the statistical structure

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A review of statistical structures in polypeptides: A. Miller and D. A. D. Parry had more degrees of freedom, the resulting refinement showed that the non-statistical structure not only fitted the X-ray data more satisfactorily but also led to the undefined parameters refining to physically more reasonable values. Consequently, we may conclude that the introduction of statisticality to a structure determination does not imply a reduction in the difficulty of fitting a structure to the observed data. Hamilton 1° has discussed the problem of whether the incorporation of extra parameters in a structure refinement leads to a meaningful improvement of the fit of the model to the data. He has shown that the arbitrary definition of 'fit' as a function of the number of refillable parameters can be put on a firm mathematical basis thus illustrating the point that the most satisfactory structure may not be the one with the lowest R factor. It is worth noting that nylon-6(~, forln)(polycaproamide) exhibits a polarity statisticality of a different sort. In this case, it is not the individual chains which show randomness of direction but rather the sheets of parallel similarly directed molecules. With this arrangement, good agreement (R factor=0.15) between the model and the observed X-ray diffraction data was obtained by Bradbury et al. 2. The strong seventh layer line streak observed in the X-ray pattern (Figure lb)was the only streak predicted from this statistical structure and therefore served to confirm the findings of Bradbury et aL

Screw disorder Considerable effort has been put into structural studies on tobacco mosaic virus (TMV). Early work by Bernal and Frankuchen 11 indicated that the TMV particles were rod-shaped with a diameter of about 150/k. Caspar lz showed that the TMV is built up of units which are packed in a helical manner and that the surface of the TMV may be thought of as a series of grooves and protuberances. The maximum diameter of the particle has now been determined as about 185/~13. As adjacent TMV units are able to pack closer to one another than might be expected, some sort of interlocking of the grooves and protuberances must occur. X-ray patterns from oriented gels of TMV show a continuous distribution of layer line intensity which means that the individual particles are scattering X-rays in an independent manner 14, 15. The results of Bernal and Frankuchen 11 imply that TMV particles are rotating or oscillating about their long axes over a complete range of concentration. When the gels are gradually dried, the particles oscillate less and less, become more closely packed and eventually form a hexagonal array. This type of statistical structure, commonly known as screw disorder, is therefore one in which the relative translation and rotation of one molecule (or molecular aggregate) with respect to its neighbour is precisely determined by the grooves on the outer surface of the molecule (or molecular aggregate) and is not dependent on the actual structure of the grooves. Franklin and Klug were the first to observe the presence of a screw disorder in T M V I L Although it is not the intention of this review to discuss statistical structures found in polymers, we have included that found in polytetrafluoroethylene (PTFE) because of its unusual character. Bunn and Howells 1~ showed that the X-ray pattern of P T F E changed when

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the temperature was raised above 25°C. The reflections on the equator and on the 6th and 7th layer lines (c = 16-8 A) remain discrete whilst those on the remaining layer lines become diffuse. Klug and Franklin 14 disagreed with Bunn and Howells' interpretation that the change in order was due to either a random displacement of the molecules about their long axis or a random rotation about the axis and they suggested that the combination, i.e. a screw disorder was more likely. This case was particularly interesting as two screw disorders must be present which would have similar pitch length but opposite hands. PTFE, like TMV, packs in a hexagonal array. In order to be able to understand the effect of screw disorders on an X-ray diffraction pattern, part of the theory of diffraction from helical molecules put forward by Cochran et aL 17 must be stated. They showed that the structure factor per asymmetric unit of helix can be expressed as:

F(R, ~, 1/c ) = ZGn, l(R)expin(~b + rr/2) n

where

J J9 = scattering factor of atom ], rjq~j, z3"= cylindrical coordinates of atom j, R, ~, 1/c =reciprocal space cylindrical coordinates, J~ = Bessel function of the first kind of order n. Furthermore, if P is the pitch length of the helix and h is the axial rise per subunit on that helix, then the diffraction is confined to those axial regions in reciprocal space which are satisfied by: n

m

c P

I

h

-

where l is the layer line index, c is the axial repeat and n and m are integers. A continuous helix is one in which m =0. The layer line maxima fall approximately on the axes of a cross originating from the centre of the diffraction pattern. The angle that the arms of the cross make with the R axis depends on the pitch angle of the helical molecule, where the pitch angle is given by arctan (2,triP), r being the radius of the continuous helix measured from its long axis. A discontinuous helix may produce a diffraction pattern in which there are a series of these 'crosses' originating not only from the centre of the pattern (m=0) but also from points at (l/h) intervals along the 1/c axis. Klug et al. is have shown that screw disorders lead to a characteristic X-ray diffraction pattern in which the inner part remains crystalline and consists of discrete reflections (on layer lines where m--0) whilst the outer part (where m # 0 ) comprises only diffuse layer line streaks. Klug and Franklin 14 postulated that this type of statistical ordering may also be found in the structures of the nucleic acids. The X-ray patterns from C-DNA 19, A"-RNA z0 and the triple stranded polynucleotide poly(U), poly(A), poly(U) 21 (see Figure l c ) have been shown to have the features expected of a system with screw disorder. In each case, the macromolecules pack in a hexagonal array.

A review of statistical structures in po/ypeptides: A. Miller and D. A. D. Parry

Figure 1 (a) X-ray diffraction pattern from an oriented ~-poly(L-alanine) fibre showing a strong layer line streak on the 13th layer line (c axis repeat 70.4,&.) and a weaker streak on the 21st layer line. This plate has been photographically modified to reduce the intensity of the inner part of the pattern in order to show the 13th and 21st layer line with their Bragg maxima and continuous layer line streaks on the same print. The inset is a weaker exposure of the 13th layer line and clearly shows the arced Bragg reflection superimposed on the continuous layer line streak. (We are indebted to Dr A. Elliott for providing this plate.) (b) X-ray diffraction pattern from nylon-6(y-form) (polycaproamide) taken at room temperature after heating to 195°C in vacuo.The specimen was tilted by 18° from the normal in order to show the strong 7th layer line streak (c=16-7~k). (c) X-ray diffraction pattern from oriented fibres of the sodium salt of poly(U).poly(A).poly(U) at 75%r.h. The pattern shows discrete maxima near the centre but continuous layer line diffraction elsewhere. (We are indebted to Professor S. Arnott for providing this plate.) (d) X-ray diffraction pattern from glycerinated water-bug flight muscle in rigor. This shows sampling on the m = 0 set of layer lines but not clearly on layer lines for which m:Y=0.Rodger23 has observed sampling on layer lines where m # 0 (see text)

It is clear that screw disorders may be found in molecules or molecular aggregates of vastly different structure and function. In all cases, the molecules or molecular aggregates have an outermost surface showing pronounced helical ridges and troughs. In general, the stability of any system will be enhanced when the assemblies are packed closest together and it is therefore

likely that there will always be the maximum interlocking of the ridges and grooves of neighbouring molecules or molecular aggregates. This type of statistical ordering implies that the molecules or molecular aggregates do not take up a specific azimuthal orientation (or axial translation) with respect to one another, possibly as a consequence of the number of units (or residues) per

P O L Y M E R , 1974, V o l 15, N o v e m b e r

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A review of statistical structures in polypeptides: A. Miller and D. A. D. Parry pitch length being non-integral. In other words, when the pattern of contacts does not repeat between particles at pitch length intervals then screw disorders are more likely to occur. This is undoubtedly an over simplification and it is certainly not true in the case of poly(U). poly(A).poly(U) which has an eleven fold axis. In all the cases discussed, the molecules or molecular aggregates pack in a hexagonal array which corresponds to the closest packing of cylindrical rods. The angle the grooves make with the long fibre axis direction of the molecules or molecular aggregates varies considerably in different species. It is small in the case of TMV and large for the nucleic acids. Also the nature of the grooves and protuberances greatly varies between that for the nucleic acids, where the protuberances are continuous phosphate ester chains and that for PTFE, where the protuberances are discontinuous and comprise discrete fluorine atoms. In the former structure, the pitch length and hand of the screw disorder are unique but in the latter arrangement, several possible screw dislocations are possible though it would appear that the screw disorders of shortest pitch length are more likely. Lateral, axial or azimuthal disorder An example of azimuthal statisticality occurs in insect flight muscle. The thick filaments of muscle are arranged on the 61 axes of a hexagonal lattice of plane group P6. This plane group has diad axes midway between the 61 axes and three-fold axes at the trigonal points. In muscles of different types the thin filaments are arranged in different ways in the P6 lattice. In vertebrate striated muscles the thin filaments lie on the trigonal points and in insect flight muscle they lie on the diads. The thin filament consists of actin, tropomyosin and troponin. The actin helix may be regarded as a two stranded rope of pitch 770/k. The axial separation between globular actin molecules in one strand is 54 A and the two strands are related to each other by an axial shift of 27 A and an azimuthal rotation of almost 180 °. There is therefore, no way in which the symmetry of thin filaments centred on the trigonal points can conform to the symmetry of the P6 lattice, and X-ray diffraction patterns from vertebrate striated muscle in rigor show no sampling of any of the layer lines originating from thin filaments marked by myosin heads. However, thin filaments placed on the diads can conform to the symmetry of the P6 lattice but only partly. If the thin filament is considered as a two strand rope where the strands are continuous, then it has a 2-fold axis and can sit on the diad axis of the P6 lattice. However, when the globular actin molecules are considered in the two strand rope, the thin filament no longer has a two-fold axis and so can no longer conform to the symmetry of the P6 lattice. These geometrical considerations may be followed in the X-ray diffraction pattern from water-bug flight muscle 22 (Figure ld). It may be observed that the layer lines m = 0 are sampled by the hexagonal reciprocal lattice indicating that, considered as continuous helices, the thin filaments conform to the P6 lattice. Layer lines with m ¢ 0 are not clearly sampled and show diffuse layer line streaks. Miller and Tregear 22 pointed out that a lattice could be constructed in which the thin filaments were statistically distributed between two azimuthal orientations separated by ~-. Various types of distribution are possible and

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POLYMER, 1974, Vol 15, November

were mentioned by Miller and Tregear e2, some of which require a super lattice to index all the observed reflections. In order to define the type of statisticality which exists in water-bug flight muscle it is necessary to have detailed knowledge of the intensity distribution along the layer lines from the thin filaments with m # 0 . Improved X-ray diffraction patterns have recently been obtained by Rodger 23 which show discrete reflections on the 59 and 70/~ layer lines in addition to diffuse scatter. These reflections may be indexed on the simple hexagonal reciprocal lattice. This suggests that there is no super-lattice in water-bug flight muscle but that in the small cell the thin filaments are equally distributed between the two azimuthal orientations. We note that this is a special type of screw disorder; it is screw disorder where only one of two defined azimuthal orientations are possible as distinct from all azimuthal orientations which occur in complete screw disorder. /3-Poly(L-alanine) is another example of this type of statistical organization. There had been considerable doubt for many years as to whether the polypeptide chains in the/3-pleated sheet were arranged in a parallel or anti-parallel manner. Infra-red (i.r.) evidence favoured an anti-parallel arrangement 24-26 whilst the X-ray data showed evidence of a one molecule unit cell which was taken at the time as indicating parallel chains. Brown and Trotter 5 studied the X-ray pattern but were unable to suggest a model in agreement with all the X-ray results. The problem was neatly solved by Arnott et al. 27 who suggested that the fl-sheets consisted of antiparallel chains and that adjacent sheets were randomly displaced from the orthogonal packing position by +_a/2 in a direction perpendicular to the chain axes and in the plane of the pleated sheet (where a is the distance between chains in this direction). In this model, each molecular site in the unit cell can be considered as a half 'up' and a half 'down' chain. By finding the way in which sheets pack together, Arnott et al. were able to refine the molecular structure of fl-poly(Lalanine) to a satisfactory level (R factor=0.136). Although fl-poly(L-alanine) is a statistical structure, there do not appear to be any strong layer line streaks in addition to the usual Bragg maxima. We may generalize this point by stating that whilst the presence of layer line streaks and discrete reflections infer some type of statistical structure, the absence of layer line streaks cannot be taken as evidence for the absence of 'statisticality'. The intensity distribution in the X-ray pattern from Tussah silk is the same as that from fl-poly(L-alanine) 5, 87, 2s. This type of statistical ordering must therefore be found in at least one protein and is not confined to synthetic polypeptides. fl-Keratin has a similar history to that of /3-poly(Lalanine) in that there had been no general agreement on whether the fl-chains were parallel, anti-parallel or randomly directed with respect to each other. Fraser et al.29, 3o re-investigated the entire problem using quantitative X-ray and i.r. data. They considered two particular models. In the first the fl-sheets were assumed to contain a perfect arrangement of anti-parallel chains. These sheets were then displaced randomly from the orthogonal packing position by +3(4-7)/k in a direction perpendicular to the chain axes and parallel to the plane of the sheet directly analogous to that proposed by Arnott et al. for fl-poly(L-alanine). In the second model, the

A review of statistical structures in polypeptides : A. Miller and D. A. D. Parry fl-sheets were considered to comprise extended chains which were either 'up' or 'down' on a random basis. Fraser et al. showed that both models could be refined to a high level of significance as regards the X-ray diffraction data alone (R factors of 0.133 and 0.129 respectively) but the inter sheet disorder model was more likely in terms of the minimum energy calculations and the i.r. data. In two appendixes to their paper, Fraser et al. determined explicitly the effect on the X-ray patterns of both of these types of ordering. For the inter sheet disorder model, reflections whose Miller indices h were even would be unaffected whilst those with odd values of h would be reduced in magnitude by a factor A~t, where M was defined as the average number of sheets per crystallite. It is now apparent why the fl-poly(L-alanine) pattern showed no evidence of a two molecule unit cell, i.e. the intensities of all reflections with odd values of h (those that would indicate a unit cell of twice the measured size) were multiplied by 1/3~t where 3~t was extremely large. Layer line streaks characteristic of a statistical structure of this type will have no intensity at lattice points where h is even. For the intra sheet disorder model, reflections with both even and odd h indices would be affected by factors which are dependent on the probability (P) that a particular chain is 'correctly' directed with respect to the preceding chain (P = 0 and P =1 correspond to parallel and antiparallel chain pleated sheets respectively). With both fl-poly(L-alanine) and fl-keratin, the pleated sheet is a perfect arrangement of anti-parallel chains. Shifting the fi-sheets laterally by +_a/2 with respect to each other implies that each position is energetically favourable and is equally likely to occur on a statistical basis. In the case of fl-poly(L-alanine), k~r is large and the side chain methyl group is small (and approximately spherical). For fl-keratin, the side chains are not all the same and 3~t is only 2 or 3. This may imply that there is some relationship between the size and type of side chains involved in the energy stabilization of a structure exhibiting this type of statisticality and the mean number of sheets in a crystallite. Even so, the two positions of minimum energy appear to be sufficiently pronounced to dominate the overall stability of the aggregate. Because of the existence of this type of statistical ordering, Fraser et al. were able to conclude that there seemed no compelling reason for believing in the existence of parallel (or random) chain fi-pleated sheets in any fibrous protein or synthetic polypeptide that has yet been investigated. It would appear that the anti-parallel fi-sheet is a more stable entity than its 'parallel' or random analogue. The X-ray pattern from the lithium salt of D N A (hexagonal C-form) shows streaks on the 1st and 3rd layer lines (c =30-8A) and discrete reflections on adjacent layer lines. This is consistent with the D N A molecules being randomly displaced by _+e/2 in the fibre axis direction with respect to one another 14, 31, 32 In general, therefore, if adjacent molecules are axially staggered with respect to one another on a random basis by _+c/q, (where c is the fibre axis repeat length and q is an integer) then the X-ray diffraction pattern will show discrete reflections on layer lines which are multiples of q and diffuse layer line streaks elsewhere. The extreme case of axial disorder is one in which molecules or molecular aggregates remain on lattice

sites but are completely free to move axially. This has the effect of maintaining the lateral sharpness of the equatorial reflections and producing layer line streaks elsewhere. Such a situation often occurs for synthetic polypeptides in solution as for example in the case of poly(7-benzyl-L-glutamate) in m-cresol 33. However, this situation should not be confused with statistical ordering in which only a finite number of discrete axial displacements are allowed.

DISCUSSION We can analyse the structures of these various biological macromolecules and synthetic polypeptides and attempt to determine the type of interactions that may be important. What are the properties of a biological molecule or aggregate that enable it to function optimally and why does statisticality play such an important part? There appears to be two main reasons. The ideal arrangement of molecules or molecular aggregates will be one in which the greatest number of satisfactory interactions between molecules or molecular aggregates can arise. This will occur in general when the individual units are related by symmetry elements. However, such an arrangement may be one which cannot be physically satisfied by all members of the group in the same way. If only a few of these molecules or molecular aggregates cannot satisfy (say) a space group symmetry requirement, it appears that these exceptions may adopt an alternative position in keeping with as many symmetry elements as possible. It should be noted that it is not usually steric hindrance which prevents the attainment of full symmetry but rather the difficulty (or even impossibility) of being able to position all the molecules or molecular aggregates in equivalent environments. In other words, statisticality may arise as an attempt by the molecules or molecular aggregates to arrange themselves with a higher degree of order than is physically possible. Possibly the best example is given by the actin filaments in water-bug flight muscle. At low resolution, these have a diad axis but at high resolution the globular units along each strand are resolved and the true symmetry is no longer 2-fold, i.e. at high resolution, the space group symmetry no longer holds strictly. When the energy differences between specific but alternative modes of packing or organization are small, then statistically any mode is equally likely. Anisotropy will consequently be reduced at this level at least. The importance in biological systems of a self-assembly mechanism 34 has now been well established. Such a mechanism implied that the molecules have a single set of primary aggregating interactions which can organize the molecules into some three-dimensional aggregate. Having reached this level the various protein assemblies will then (in general) interact with one another in a specific manner. In most cases, the symmetry of an aggregate is such that the surrounding assemblies will not all interact in the same way. What must the molecular aggregate do in this case? Presumably, an equilibrium must be set up where the optimum number of ideal relationships are made and it is at this level that 'statisticality' is most often encountered. For instance, a molecular aggregate might have to be in two different axial positions simultaneously

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A review of statistical structures in polypeptides: A. Miller and D. A. D. Parry in order to satisfy the necessary symmetry conditions inherent in the system. The problem may be partly overcome if it is assumed that a statistical relationship exists where either position is equally favoured. It should be emphasized that the biological entity may be equally satisfied irrespective of which of the statistical possibilities arises in practice. The energies of the alternative structures will be the same or very similar unless the function of the assembly so demands a difference to exist. The invariant set of in vivo primary aggregating interactions enable the myosin molecules to build a thick filament, the actins to form a thin filament, the tropocollagens to form a microfibril, the extended chains to produce an ordered pleated sheet and so on. Once this degree of organization has been achieved and the molecules are in regular three-dimensional form, a second set of aggregating interactions can be made. It is equally important to realize that whilst biological systems may have one type of statisticality, other types are functionally unallowable. For instance, the thick and thin filaments in muscle exhibit polarity and no directional variability can be introduced without the mechanism of muscle contraction being lost. The collagen fibre also exhibits strict polarity as may be seen in the electron microscope using, for example, positively stained sections from rat tail tendon collagen. In summary we have shown that statisticality is becoming an increasingly more evident and important facet of structure determination. We believe that the concept of statisticality helps to explain how flexible and dynamic protein assemblies may be able to interact in a more highly symmetrical manner than the symmetry of the molecule or molecular aggregate theoretically allows.

ACKNOWLEDGEMENTS This project is part of the programme of the M R C Research Group in Molecular Biophysics. One of us (D. A. D. P.) acknowledges the support of the Science Research Council during the initial part of this work.

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