Journal of the Royal Society of New Zealand

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On the structure of DNA R. H. T. Bates , R. M. Lewitt , C. H. Rowe , J. P. Day & G. A. Rodley To cite this article: R. H. T. Bates , R. M. Lewitt , C. H. Rowe , J. P. Day & G. A. Rodley (1977) On the structure of DNA, Journal of the Royal Society of New Zealand, 7:3, 273-301, DOI: 10.1080/03036758.1977.10419429 To link to this article: http://dx.doi.org/10.1080/03036758.1977.10419429

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Date: 25 January 2017, At: 07:57

Journal of the Royal Society of New Zealand, 1977, Vol. 7, No.3, pp. 273-301, 13 figs

On the Structure of DNA

R. H. T. BATES, R. M. LEWITT, C. H. ROWE Department of Electrical Engineering J. P. DAY* and G. A. RODLEY Department of Chemistry, University of Canterbury [Received by the Editor, 3 November 1976]

Abstract Evidence for the structure of double-stranded DNA and related duplexes is considered in relation to the existing double helix model and a possible alternative to it. It is shown by mathematical analysis that an alternative side-by-side (SBS) model is capable of producing the gross features of the X-ray diffraction pattern of moist DNA fibres. This is complemented by model-independent calculations based on X-ray data for paracrystalline B-DNA that suggest the duplex molecule may not have tne circular (axial projection) symmetry expected for a regular double helix structure. In addition, the application of the SBS model to other areas of DNA structure and function are considered. Long-standing problems assocIated with the highly intertwined feature of the helix model, esrecially the problem of unwinding, do not arise for the SBS model. In particular, electron micrograph evidence for the formation of "bubble" regions during the melting of DNA raises serious questions about earlier attempts to rationalise kinetic data for strand separation.

INTRODUCTION The Watson-Crick double helix model for DNA and related structures appears to be very well established from a wide range of physical studies. As far as we know the basic right-handed helical character of the model has never been seriously challenged, although doubts have been expressed about the reliability of the model as deduced from X-ray studies (Donohue 1969, 1970). Perhaps the most important feature of the double helix model is its precise description of the manner in which corresponding bases of the two strands bind to each other. It is a remarkable fact that the hydrogen-bonded AT and GC pairs produce almost identical distances between the points of attachment to the sugar phosphate backbone. This is one of the underlying reasons why a regular double-helical structure can be constructed. However the Watson-Crick model (Watson and Crick 1953a) is also based on a second, effectively unrelated, premise. It is that the sugar-phosphate backbone prefers a particular conformation that forces the base pairs to twist in a right-handed manner with respect to each other. The invariance of this arrangement means that the DNA molecule as a whole is an extended double helix having the two strands intimately intertwined with each other. The outstanding success of the double helix model has been largely independeij.t of its description of the disposition of the two strands (except for their antiparflllel nature). It has been the detailed picture of base pairing which has made the model of such great importance in biology over the past two decades. But during this time, in spite of many attempts to rationalise the highly intertwined nature of the model, this feature has remained a mystery. When studied in detail, intertwining appears to present major problems in understanding the behaviour of both in vitro and in vivo DNA. The discovery of nicking and splicing enzymes has to a large extent allayed concern in this

* On leave from Department of Chemistry,

University of Manchester, England.

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JOURNAL OF THE ROYAL SOCIETY OF NEW ZEALAND

area. Nonetheless a very high degree of molecular organisation and some wellcoordinated stereospecific reactions in the replication and transcription processes are still demanded by the intertwined double helix feature. In fact, breaking and rejoining of strands would appear to complicate rather than simplify these processes, and it is fair to say that problems associated with the presence of interacting protein in chromosome structures are often ignored. In the course of exploring the possibility of a non-intertwined structure for DNA we have devised a simple alternative model. We refer to this as a side-by-side (SBS) structure to indicate the basic arrangement of the phosphate chains. Although we have not yet been able to explore all the possibilities or shortcomings of this model, we feel it does have sufficiently attractive features for it to be presented for public scrutiny. The details of the SBS structure have already been published (Rodley, Scobie, Bates, and Lewitt 1976). Here we outline reasons for believing that the SBS structure could be a viable alternative to the Watson-Crick model. This involves not only a description of the advantages of the new model but also a discussion of deficiencies of the existing one. The SBS model retains all the essential features, apart from continuous helicity, of the Watson-Crick structure (i.e., base pairing of complementary antiparallel strands with a constant distance between links to the sugar phosphate strands). The model has broad and narrow grooves of a character similar to those of the refined double-helix model. One particular side-on view of the SBS structure is difficult to distinguish (in projection) from a side-on view of the Watson-Crick structure. All these features illustrate the close similarity of the two models. This is significant, because it has to be assumed that if an alternative still exists it must be difficult to distinguish it from the double helix. We show mathematically that the SBS structure is capable of producing the characteristic X-ray pattern from moist fibres of DNA (see also Rodley, Scobie, Bates, and Lewitt 1976). This indicates the two models may not be readily distinguished by X-ray diffraction alone, and points to the possibility that the exact nature of DNA may be too elusive for it to be possible to make an either-or choice between basic models. We first of all outline details of the SBS model, reported more fully elsewhere (Rodley, Scobie, Bates, and Lewitt 1976). Questions about the interpretation of X-ray data in terms of a double-helix model, and a discussion (supported in Appendix B) of the mathematical calculation of the expected X-ray fibre diffraction pattern for SBS molecules, are presented in the succeeding section. In the final two sections we consider the unwinding problem and other applications of the SBS model. The analysis (presented in Appendix A) of the diffraction patterns of paracrystalline B-DNA suggests that a helical structure is only consistent with observation if the errors in the best available X-ray data approach half the measured values. BROAD FEATURES OF THE SBS MODEL Most simply, the model may be described as an alternating arrangement of righthanded and left-handed double helical sections. Equal lengths of five base pairs have been chosen for the initial model to satisfy the ten base pair repeat feature ofB-DNA. Figures 1-4 illustrate details of the model from different points of view. Essential to the construction of the SBS model is the introduction of bends in the phosphate backbone in order to effect the change in rotational sense of each fifth base pair. Our model has two types of bends. One (p) resembles the Crick-Klug "kink" (Crick and Klug 1975) in that it involves a change from a nucleotide gauche-gauche (gg) conformation (used for right-handed sections) to the gauche-trans (gt) arrangement required for left-handed portions and rotation about the C 4 '-C S' bond to the sugar ring. The other bend (q) uses the trans-gauche (tg) conformation to produce the change in helical sense; the sequence of conformations being gg----tg----gt at this bend.

BATES ET AL.-On the Structure of DNA

275

c

? l.-ld~lired I

t

~ \

.......-...-2a--

(a)

-

-

F'G.

drnw;,'", of aJ

~

"j elementary Watson-Crick mode and

~

'~

2a

(b)

-

~

(b) an elementary side-by-side (mS) structure (Rodley, Scobie, Bates, Lewitt 1976). c = axial length of the repeat unit ; a = approximate radius of phosphate strands; C2 = two-fold rotation (dyad) axis (perpendicular to paper) . which relates the p and q bends in the following manner p C q. q 2p

Detailed models indicate that the 5:5 arrangement will produce a net long-range right-handed twist. This is not unexpected, as the geometry of the right-handed region allows greater twisting to occur. The intermeshed SBS structure appears to provide a greater possibility for maximising base plane stacking, and a greater ease of bending and compacting of doublestranded molecules, than for the more rigid double helix. Another attractive feature is the greater accessibility of the base pairs due to the open groove lying along one side of the molecule rather than twisting around it in a helical fashion. The new structural features used in the SBS model, notably variable sugar co¢ormations and the gauche-trans (gt) and trans-gauche (tg) sugar-phosphate configurations, have all been observed in tRNA structures (Jack, Klug, and Ladner 1976) and must now be considered to be generally available to nucleic acids. Although the tRNA structure, having only right-handed double helical regions, may be cited as evidence against the SBS model, the general nature of the backbone structure indicates it can be distorted quite readily; primarily it seems to accommodate maximum base stacking. The same may be considered to be the case for the SBS structure; base stacking is maximised by appropriate adjustment of the backbone configuration. By contrast, the double helix model, as indicated above, is effectively based on the premise that it is the

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JOURNAL OF THE ROYAL SOCIETY OF NEW ZEALAND

backbone structure which dominates; it remains regular and forces the strands to intertwine.

FIG. 2a.-Stereo view of side-by-side (SBS) model viewed from the narrow groove side (Rodley, Scobie, Bates, and Lewitt 1976).

FIG. 2b.-Same model viewed at a position 90° from Figure 2a.

BATES ET AL.-On the Structure of DNA

a

277

FIG. 3.-Sketches of a simplified model of the side-by-side (SBS) structure: (a) side-view corresponding to Figure 2(b), (b) an angular side-on view (Rodley, Scobie, Bates, and Lewitt 1976).

A set of coordinates has been presented for a representative SBS model (Rodley, Scobie, Bates and Lewitt 1976). It is noted that this illustrative example contains a number of close intramolecular contacts. However, we consider the SBS structure to be sufficiently flexible to allow adjustments to be made to relieve these contacts. They mainly concern terminal (phosphate) oxygen atoms whose coordinates were not accurately estimated at the time. This meant that the closeness of some of the contacts was not appreciated and therefore that they were unintentionally omitted in the original report. INTERPRETATION OF X-RAY DATA FOR DNA AND RELATED DUPLEXES A wide range of X-ray data is available for DNA and related duplex structures. Nonetheless only a limited amount of data can be obtained for anyone form because of the paracrystalline and/or fibrous nature ofthe samples. Donohue (1969, 1970) has

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TOURNAL OF THE ROYAL SOCIETY OF NEW ZEALAND

FIG. 4.-Idealised view along axis of side-byside (SBS) structure. Shaded regions indicate the phosphate strands in projection. (Rodley, Scobie, Bates, and Lewitt 1976).

argued that it is therefore difficult to make thoroughly reliable structural deductions. The success of the basic double helix model in explaining with appropriate adjustment such a wide range of different forms has been cited as a major reason for being confident about the correctness of the model (Hamilton 1968). On the other hand, an alternative model with broadly similar features to the double helix could also be varied from one form to another to give a similar, satisfactory agreement with X-ray data. Even when difficulties of analysis, because of poor data and uncertainty about solvent structure, are taken into account, the agreement factors obtained for double helix models remain sufficiently low for it to be doubtful whether these models can be truly placed in the "proven" class. In this section we comment on other features of the results which indicate that the double helix may not be the correct basic model for the interpretation of X-ray data for double stranded polynucleotides. In addition we refer to calculations we have made which show that the SBS model is compatible with the fibre diffraction pattern and is therefore an acceptable alternative model on which to base an analysis of the available paracrystalline data. Evidence on the Shape of the DNA Molecule Study of unit cell dimensions for different forms of DNA and related structures in relation to molecular packing indicates that the DNA molecule does not always have a circularly symmetrical structure when viewed in projection. Moreover in some cases the unit cell parameters require short distances between centres of adjacent doublestranded molecules e.g. 19.0 A (Marvin, Spencer, Wilkins, and Hamilton 1961) and 19.3 A (Langridge, Wilson, Hooper, Wilkins, and Hamilton 1960). Such contacts may be possible through coordination to bridging metal ions as found in the structure of sodium adenoxyl-3',5'-uridine phosphate (Rosenberg, Seeman, Jung, Park, Suddath, Nicholas, and Rich 1973) but it is not immediately apparent why this neighbour group interaction turns out to be directionally non-uniform as, for example, in A-DNA (cf. Fuller, Wilkins, Wilson, Hamilton, and Arnott 1965).

More importantly, we have been able to apply certain function-theoretic aspects of Fourier theory to the B-DNA paracrystalline data in order to study more directly the shape of the DNA molecule. Although the "noise" in the observations (the magnitude of this is difficult to assess) may exceed the uncertainties in our data-processing procedures, our results indicate that the DNA molecule apparently does not have the highly symmetrical shape expected for a double helix. (The calculations are described in detail in Appendix A.) This does not necessarily mean that the Watson-Crick model i& incom:ct, but it dOCf S Jggcst that if the structure really is a double helix, it is signifi-

BATES ET

AL.-On the Structure of DNA

279

cantly more distorted than has been assumed to date. It should be mentioned that certain features of paracrystalline X-ray patterns could also be cited as evidence for distortion of the double helix, notably the observation of meridional reflections in the hollowed-out double-wedge regions in B-DNA (Langridge, Wilson, Hooper, Wilkins, and Hamilton 1960), RNA (Arnott 1970), and poly (dA-dT) (Arnott, Chandrasekaran, Hukins, Smith, and Watts 1974). The possibility of significant distortion of the double helix raises important questions. Refinements of X-ray data for duplex forms based on the double helix model have effectively been possible only because of the assumption of regularity of structure. This has reduced the number of variable parameters, thus giving a sufficiently favourable parameter : data ratio for impressive standard deviations for positional coordinates to be obtained. If the molecule is in fact a distorted double helix, the level of accuracy claimed becomes questionable, as too does the implication that successful refinement confirms the double helix model. Certainly, discussions of such fine details as sugar conformation (Arnott and Hukins 1973) do not appear to be warranted. On the other hand, instead of concluding that the various features cited above are due to the distortion of a double helix structure, one might equally well suggest that they provide indirect support for the SBS proposal. Evidence from Cylindrically-Averaged Patterson Functions Some of the most direct information available from X-ray studies is that supplied by Patterson functions. These have been used to provide evidence for helicity of the phosphate strands (Franklin and Gosling 1953; Sato, Kyogoku, Higuchi, Mitsui, Iitaka, Masamichi, and Miura 1966). Yet a closer study indicates certain difficulties with the Watson-Crick model. For molecular radii in the region of 19-20 A, intramolecular PO -P04 cross-vectors at approximately this spacing should appear in the cylindrically averaged Patterson maps roughly perpendicular to the molecular c-axis. Although there may be questions about the reliability of the maps from limited data, these cylindrically-averaged maps give little indication of a peak in the expected region. A further difficulty with the Watson-Crick model is that two peaks would be expected at c ~ 3/8, c ~ 5/8 (r ~ 0) for the refined structure of the A form (Arnott and Hukins 1972) whereas only one peak at c = 1- is observed (Franklin and Gosling 1953). Although the resolution of the map is low, some indication of this separation might have been expected. Feasibility of the SBS Structure as a Basis Model In Appendix B we discuss the theory of X-ray diffraction by fibre specimens of DNA. All DNA specimens exhibit the characteristic "cross" X-ray diffraction pattern. The observed diffraction is usually small in the double wedge region about the meridian (Figure 5). The intensity shows up as discrete spots for crystalline specimens, and as continuous bands for fibre specimens. The spots are smeared because even the best crystalline specimens of DNA are composed of fibrous crystallites which are far from perfectly aligned with each other.

The early workers on the structure of DNA seem to have been overly impressed by the fact that the observed X-ray patterns could have been caused by diffraction by double-helical structures. It was realised that other structures were possible (Wilkins, Seeds, Stokes, and Wilson 1953) but there is no evidence that alternatives were examined in depth. The apparent simplicity of the helical conformation caused it to be accepted as scientific dogma in a matter of a few years. Using simple representations of the SBS and Watson-Crick models we calculate (in Appendix B) fibre patterns which may be compared with the observed data. These results show that the SBS model accounts for the characteristic double wedge fibre pattern as satisfactorily as does the Watson-Crick structure. We assert that the intensity differences between corresponding regions of the fibre patterns of the crude models are comparable with experimental error. This persuades us that the SBS model is worth a (Hailed investigation.

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JOURNAL OF THE ROYAL SOCIETY OF NEW ZEALAND

5.-X-ray diffraction pattern of moist DNA fibres-reproduced, with permission, from Dickerson (1964).

FIG.

The representative structures analysed in Appendix B are described conveniently using cylindrical polar coordinates. Our expressions for the diffraction from these structures consequently contain Bessel functions. It has been suggested to us that this alone could account for the qualitative agreement of ollr calculated diffraction patterns with the famous "cross" patterns observed with fibres of DNA. The implication here is that the work reported in Appendix B may be of little significance because the conclusions we draw may be inescapable. However, we would like to emphasise that the occurrence of Bessel functions in no way guarantees that the diffraction pattern has the form of a cross. To clarify this point, we must anticipate the detailed analysis developed in Appendix B and refer directly to Equation B.l5. The cross pattern occurs if, for any particular value of I, the terms in the summation are negligible for m < ml, where the integer ml is proportional to I. For an arbitrary structure, even if it can be conveniently expressed in cylindrical polar coordinates, this proportionality of ml to 1does not hold. Consequently, we feel that our analysis is very significant because it shows that the SBS structure belongs to the limited class of structures having diffraction patterns in the appearance of a cross. The only way to use the measured X-ray data to make a critical assessment of the Watson-Crick and SBS models is to calculate the structure factor amplitudes for crystalline specimens, employing accurate atomic coordinates and realistic electron densities. This is a considerable undertaking which we have just embarked upon. We intend to make comparisons with what appear to be the best observational data and the most highly refined version of the Watson-Crick model (Arnott and Hukins 1973). STRAND SEPARATION AND THE UNWINDING PROBLEM

The separation and reassociation of polynucleotide strands are essential processes in the in vivo functions of DNA, i.e., replication, transcription, and recombination (Watson 1976). The detailed mechanisms of these processes are at present unclear. A

BATES ET

AL.-On the Structure of DNA

281

number of suggestions for the mechanisms of strand separation and reassocation have been made from in vitro studies and although these studies may not be directly relevant to in vivo problems, they possibly provide information on the structure of the DNA molecule in solution. We will first examine the situation in vivo. DNA Functions in vivo

The process of DNA replication requires separation of the strands of the parent molecule, synthesis of complementary strands, and assembly of the twin daughter molecules, the whole taking place in a limited time. The problem of discerning a plausible mechanism for unwinding and rewinding the double helix fast enough, and "without everything getting tangled", was first noted by Watson and Crick (l953b). Certain aspects remain puzzling. The main point is that to separate the two strands of a double helix, with no strand breaks, requires the rotation of one strand about the other (equivalent to rotation of one end of the helix with respect to the other) -w times, where + w is the number of turns of the double helix. This topological requirement is independent of the detailed mechanism and can lead to rather implausible conclusions (Gorski 1975). Thus, the replication of E. coli DNA (molecular weight ca. 2 X 109 daltons, w ~ 3 X 105 turns) must occur in less than 30 minutes, the time for cell division under optimum conditions. If the DNA were present as one segment, this would require an overall unwinding rate greater than 150 turns per second. Particularly in higher cells, the problem is further increased by the compacting of the DNA, by the attachment of histones and other proteins, and by the involvement of enzyme systems, e.g. at the replication forks. The unwinding problem would be reduced by single or double strand breaks in the molecule. Breaking of single strands could provide free-rotational "swivels", and segments between swivels could presumably rotate independently. At least one such swivel would in any case be required for the unwinding of circular DNAs, e.g. the E. coli DNA mentioned earlier. The current view (e.g. Watson 1976, with particular reference to circular DNA) suggests that there is a periodic formation and repair of single-strand breaks under the control of swivelase enzymes. However, the cutting and splicing of template strands would require a high degree of molecular organisation to ensure that replication proceeded smoothly and accurately; the process would presumably stop if a single-strand break were encountered at a replication fork. The details of the operation of the swivelases have yet to be established. It seems clear to us that the intertwined nature of the double-helix model still presents difficulties in understanding the replication process, and it would seem that these difficulties are increased when transcription and recombination are considered. The mechanistic problem of strand separation is greatly reduced when the SBS model is examined. As the strands are not highly intertwined, formation of a replication "bubble" can readily be envisaged-strand separation can occur over moderate lengths without generating the resistance from the neighbouring duplex that must occur if strand separation is accompanied by unwinding. Although long-range twisting of the molecule is likely even on the SBS model, it is clear that the unwinding requirement is considerably less than for the double helix. Similar simplification of mechanism would also be likely in transcription and recombination. For example, in transcription, the base sequence of one of the strands of the duplex could be "read" from one side of the molecule; there is no requirement for the mRNA to intertwine with the DNA to the extent required by the double-helix model. DNA in vitro: The Double Helix In vitro it is possible to study the thermodynamics and kinetics of strand separation and recombination in isolation from in vivo complications and the usual techniques of

physical chemistry can be applied to the determination of structure and mechanism (Bloomfield, Crothers, and Tinoco 1974). DNA in solution "denatures" at elevated temperatures, i.e. the twin strands separate in a co-operative process occurring over a raJ1'3~ of a few degrees (mid-point T m, the melting temperature, between 40 and 90°C

282

JOURNAL OF THE ROYAL SoCIETY OF NEW ZEALAND

depending on pH, ionic strength, etc.). Denatured DNA will eventually renature if favourable conditions are restored, and renatured DNA can retain much of the biological activity of the original, indicating a more or less exact return to the initial duplex structure. Experimentally, denaturation/renaturation can be followed by physicochemical methods, e.g. optical density change. The kinetics of denaturation (Spatz and Crothers 1969; Cohen and Crothers 1971) following an "instantaneous" temperature increase are complex but may be described in terms of a large number of simultaneous and sequential first-order reactions, with relaxation times (·t") ranging from ca. 10-6 to 103 s (i.e. for DNA of molecular weight, M ~ 105 to 107 daltons). If denaturation is continued until complete strand separation occurs, renaturation consists of a slow secondorder step (initial recombination, or nucleation, of separated strands) followed by a series of rapid first-order changes (interpreted as the "zipping-up" of the double helix). Suggested mechanisms for denaturation have concentrated on the requirement of unwinding the double helix. Unwinding requires that one end of the molecule rotates with respect to the other. The usual supposition is that if the helix is long enough the rate of denaturation will be limited by frictional resistance to turning the molecule in solution. The earlier models proposed a simple unwinding at both ends (Kuhn 1957; Longuet-Higgins and Zimm 1960; Fig. 6a) or at one end (Fixman 1963; Freese and Freese 1963; Fig. 6b), proceeding by a diffusional process but driven by the lower free energy of the separated strands (entropy increase). Thus, although positive and negative rotation of the strands will be generated by random collisions of solvent molecules (Brownian motion), the process is biased in favour of net unwinding by the free energy change. A general theoretical conclusion was that the unwinding time t(u) should vary approximately with some power of molecular length (tu ex. Mn, n ~ 3/2 to 3). Predicted values for tu varied considerably and depended on the detailed mechanism suggested, values assumed for frictional parameters, etc. (0)

: I

non - revolving revolving

\

1\

\,

revolving

\C)

,"" ;)()OOCOOOOCXXXXXXXXXX~

t,)

q

;

:

\ ..... '

(b)

, -,,,

q

revolving

,/

non - revolving

q ..• _. ~x::(\ \./ FIG.

6.-Unwinding models for DNA as described in the text.

More recently, it has become apparent that these. models ~re ove~-simp~ified (Crothers 1964). As the temperature increases, DNA meltmg starts mAT nch regIOns, which are more or less randomly spaced along the molecule (Bloomfield, C~others, and Tinoco 1974). The GC rich regions are more stable and melt a few degrees higher (much of the evidence for this comes from the examination of progressively denatured samples

BATES ET AL.-On

the Structure of DNA

283

of DNA by electron microscopy). Thus, the model for double helix unwinding must accommodate intermediate stages such as the one shown in Figure 6c, in which three distinct types of region may be identified: (p) double-helical sections, assumed to be torsionally rigid (i.e. analogous to a speedometer cable), (q) unwound sections comprising independent single strands and with no torsional rigidity ("random coil" sections) within the molecule, and (r) a terminal random coil section or sections. In effect this model is the one used in a recent comprehensive theoretical study (Crothers and Spatz 1971). However, we feel that one crucial problem has been ignored in that study, namely consideration of the mechanism which would transfer the twist of "internal" double-helical sections (e.g. xy) to the ends of the molecule. This mechanistic problem did not arise, in the same form, in the earlier models. In Figure 6a the central helix unwinds through diffusional movement of the terminal random-coil sections, biased towards unwinding by the favourable free energy change. This model was later discarded in favour of Figure 6b, in which the double helix itself revolves: frictional resistance is reduced if the random-coil does not rotate. Depending on values chosen for unwinding parameters, unwinding times ca. 10- 1 to 103 s were deduced (Longuet-Higgins and Zimm 1960; Fixman 1963) for DNA of molecular weight 107 daltons, in reasonable accord with experimental values (Table 1). However, a model such as Figure 6c requires explaining how regions such as q grow as the molecule dissociates. For example, the unwinding of a helical turn at point x must result either: (i) in two strand "cross-overs" appearing in the adjacent coil section xz, or (ii) in the winding up of a helical turn at y, or (iii) in the appearance of a "super-coil" turn in the double-helix section xy. TABLE

Mol. wt./daltons 6.0 x 105 2.3 X 106 2.5 X 107 1.3 X 108

Notes: (a) " 0, IAe, I (x, k/b)/Ae, 1(0, k/b)1 is larger than it would be for any other choice of e(x, y, z). lA" I(X, k/b)1 has the form of a triangle with a base of length 2L. Because interpolation becomes theoretically impossible when L = a, we must compare the fractions of the area of the triangle lying outside the intervals -L/2 < x < L/2 and -a/2 < x < a/2, in order to estimate the error-which we find to be 9%-inherent in our assumption that Ae,I(X, k/b) can be replaced by Ae,I(X, k/b). We see that 9% is also the error in IE1(u, k/bW when it is assumed that (A. 12) IF hkt/ 2= IE I(h/2L, kjb)i2 provided that no structure factors of significant magnitude are missing. Because significant structure factors are likely to be missing from even the best available data, we must compare IE1(u, k/b)12 with the smooth interpolations shown in Figure 8. The crosses on Figure 8 indicate the values of IEI(u, k/b)j2 at the marked points on the circles in Figure 9. The differences between the crosses and the smooth curves are estimates of the errors in the curves. Table 2 lists these interpolation errors and the spread of values of IEI(u, V)12 marked on each of the circles shown in Figure 9. Since the average amplitude of each of the curves shown in Figure 8 is of the order of unity, the maximum interpolation error is of similar magnitude to the upper bound on the error involved in assuming that Ae. I(X, k/b) is essentially the same as A e , ,(x, k/b). The combination of these two errors (assuming them to be independent) is also listed in Table 2. The spread of values of IE (u, v)j2 marked on the circles shown in Figure 9 is observationally significant because it is about three times the combined error on the first and third circles, and is about twice the combined error on the second and fourth circles.

294 TABLE

bR I

2

3 4

JOURNAL OF THE ROYAL SOCIETY OF NEW ZEALAND

2. Errors inherent in interpolation procedure, and spreads of values marked on Figure 9. Interpolation error average maximum 0.10 0.10 0.10 0.10 0.06 0.08 0.27 0.42

Combined error 0.12 0.15 0.11 0.43

Spread of values of IE,(u, v)12 marked on Figure 9 0.39 0.27 0.32 0.97

Since the double-helical model of B-DNA requires IE,(u, vw to be independent = lib and R = 2/b, it seems that the observed X-ray diffraction patterns are not compatible with this model. If the spreads listed in the final column of Table 2 exceed the measurement errors, then our results suggest very strongly that the Watson-Crick model is incompatible with the available observed data. An important practical point which does not seem to have been commented on specifically before is that there exists no good way of estimating the error levels in the best available data. Because of its wide potential consequences, our approach should be used with other appropriate data. If a large number of accurate structure factor magnitudes could be obtained for crystals for which the matrix in which the DNA is embedded induces intermolecular forces of such magnitude as to ensure that a ~ 2L, then the sampling theorem would apply exactly to IE,(u, k/b)12 and there would be no error in the assumption that Ae.,(x, k/b) and Ae, ,(x, k/b) are the same. Any appreciable differences between IE,(u, k/b)12 and the smooth interpolations would then indicate unambiguously the existence of significant structure factors missing from the observed data. The A form of DNA satisfies the a ~ 2L requirement, but we cannot apply the method to this form because d is thought to be zero. R. D. B. Fraser has suggested to us that departure from helical symmetry might be expected when a helix with a particular screw axis is involved in interactions (as in a crystal) which do not have the same helical symmetry. For instance, there is much evidence that in the crystal structures of synthetic polypeptides the ex-helix can be appreciably distorted (Fraser and MacRae ofljJ on the circles R

1973).

An interesting aspect of our approach is that we are able to deduce the inconsistency of the double-helical model from the intensities of the structure factors observed in the first layer plane, even though (as D. A. D. Parry has pointed out to us) the observed intensities are heavily contaminated in low-order layer planes by diffraction from the matrix in which the DNA molecules are embedded. The point is that if the molecule does indeed consist of nucleotide pairs, with their associated sugars and phosphate groups, which repeat every clIO along the fibre axis with a rotation (always in the same sense) of n15, then the parts of the matrix displaced and distorted by the DNA must themselves consist of sub-units which repeat every clIO with a rotation of

n15.

ApPENDIX B We set up cylindrical polar coordinates r, cP, z and R, 1jJ, w in real and reciprocal space respectively. We consider a structure which is periodic, with period c, in the z-direction, so that its electron density per, cP, z) satisfies per, cP, z + c) = per, cP, z) (B.1) the chief consequence of which is that the X-ray diffraction pattern exists only in "layer" planes, spaced by l/c in the w-direction in reciprocal space. The complex amplitude E, = E,(R, 1jJ) of the diffraction pattern (rem~mber that it is only the intensity IE,12 that is measurable) is given by (cf. Klug, Cnck, and Wyckoff 1958)

cE,(R, 1jJ)

=

J JJper, cP, z) exp(i2n[Rr cos('f; - ef» + lz/cJ)r def> dr dz

e/2 en 2r.

~e/2

Q Q

(B.2)

BATES ET

AL.-On the Structure of DNA

295

where I is any integer (we say that EI is the diffraction pattern in the lth layer plane). The factor c is included on the left-hand side as a convenient normalisation. Fibre specimens of DNA consist of bundles of crystallites randomly oriented with respect to the azimuthal angle cpo Their fibre axes (i.e. their z-axes) are roughly parallel, but they are in relative thermal motion. It follows that the observed X-ray diffraction from a fibre specimen is effectively proportional to the angular average of the intensity of the diffraction pattern of a single molecule. For any value of I, this angular average depends only on R. Therefore, fibre patterns are sometimes said to exist on "layer lines", which are the projections of the layer planes onto any plane in reciprocal space containing the w-axis. The fibre pattern on the lth layer line is written here as DI(R)

=

2~

Io IEI(R, tfo)12 dtfo

27
, z) = 8(e/> - tp - 2nz/c) (B.6) g(lS)(e/>, z) = 8(e/> - 6 - b sin(2nz/c + {}» (B.7) where tp, 6, {} and b are arbitrary constants (note that the first three can be looked on as phase angles). If DNA is represented as a pair of filamentary helices, so that fer) = 8(r - a), then E~2W)(R, 1{1) is proportional to M2nRa), the ordinary Bessel function of the first kind of order I. This results in there being very little diffracted intensity within a wedge-shaped region centred on the meridional axis (layer line axis), a fact that seems to have impressed itself so firmly on those working on DNA in the nineteen-fifties that it was early decided that DNA must have a helical form. Figure 12a shows the fibre pattern for a double-stranded, filamentary helical model for which g(2W)(.) has the form g(2W)(e/>, z) = 8(4) - 2nz/c) + 8(e/> - 57t/4 - 2nz/c). (B.8)

BATES ET

AL.-On the Structure of DNA

7-·--------~6-

297

711---------611--------

-

5~--=~ .. =--

5,--

.---- ~ +---------=---2

,,

"v

1~,--!

",

....

wl--'-' ,-

J,

R,A-'

",

o --~-- 6-1---+

f-

12a

1-----1

02

R,A-'

12b FIG. 12.-Fibre-diffraction patterns for filamentary models of double-stranded DNA; with fer) = oCr - a), a = 10 A and g( '1', z) having the forms shown in equations (B.8) and (B.9): (a) Watson-Crick model (dashed curve is divided by 2), (b) SBS model (dashed curve is divided by 5).

Note that the intensity is low on the fourth layer line in Figure l2a, in keeping with observation for B-DNA. The same is true in Figure lOa. In Figure lOb, however, it is the intensity on the fifth layer line that is low. The point is that the simplicity of the double-helix model makes it easy to adjust it to take account of observational features. The lower symmetry of the SBS model makes it more difficult to do this. But even our first crude atomic representation of the SBS model gave results (Figure lOb) having the general character of the observed fibre patterns. By adjusting our atomic representation we find that we can vary the relative intensities on the different layer lines. But we do not want to make too much of this because it will only be after detailed refinement of a complete atomic representation of the SBS model that we will be able to assess fully the relative standings of the two models. In this appendix we merely wish to emphasise the potential of the SBS model. Figure l2b shows the fibre diffraction pattern for a double-stranded, filamentary representation of the SBS model, for which g(2S)( .) has the form g(2S)(4>, z) = 8(4) - 0.7 sin(21tz/c» + 8(4) - 2.27 - 0.7 sin(21tz/c» (B.9) Note that the diffracted intensity is again small within a wedge-shaped region centred on the meridional axis. It should be noted that the reason for the regions in Figure 10 clear of diffracted intensity being triangular, where the corresponding regions in Figure 12 are wedgeshaped, is that the discrete nature of the "atomic" representations (to which Figure 10 applies) forces the diffraction pattern to be periodic in the direction of the meridional axis. Consequently the patterns shown in Figure 10 look more like observed patterns than those shown in Figure 12, which correspond to continuous representations of the models. It is, of course, useful to study continuous representations because they display many of the features exhibited by more realistic representations. Substituting equations (B.S) and (B.6) into equation (B.2), invoking a standard integral representation of Bessel functions and making use of Weber's first exponential integral (Watson 1958), we find that EPW)(R, if) = iIL(o:,~, y, I, R) exp(i/[if - rp]) (B.IO) in which L(o:, ~, y, I, R) = 1t~[o: exp( _1t 2 R2 o:2/2) I1/2(1t 2 R2o:2/2) - ~ exp( _1t2R2~2/2) I,d1t2R2~2/2)]/2y (B.ll)

298

JOURNAL OF THE ROYAL SOCIETY OF NEW ZEALAND

I 7" i

i

.. ------

7~

6:'---

6

5~~~~------------------

5

I

" 4L--~------

"'l

l~--~l \.

o-

1 ---t---:OO:l:01',·---r----.....I,--f-----::1:: 02 O

R.A-'

-

4

-

3-----~~~~--------

2~=-=··-·- - - - - - - - - - - - - - - -

o

.... , ---,01-----02-+----+-----='03::--

R.A-'

13a

13b

FIG. 13.-Fibre-diffraction patterns for double-stranded models of DNA: with fer) having the form shown in Figure 11 and g{ - b sin(27tz(c»

+ o(e{> -

8 - b sin(27tz(c

+ '!?»)

(B.l2)

which, when substituted together with equations (BA) and (B.5) into equation (B.2) and after equation (B.3) is invoked, leads, on recognising that the ifi-integration reduces to standard integral representations for Bessel functions, to Q?S (R)

= (27ty)-2

X

co

21


O.

The nature of Bessel functions is such that J1(x) is small for Ixl < III - 2. Consequently, the summation on the right-hand side of equation (B.15) begins effectively at the integer closest to (Ill - 2)/b. It is then clear that Q~2S)(R) is small in a doublewedge region centred on the meridional axis. Figure 13b shows the fibre pattern for a particular set of values of e, 1} and b. As in Figure 13a, the diffracted intensity has significant magnitude within only a restricted interval of each layer line. REFERENCES

AHLBERG, J. H.; NILSON, E. N.; WALSH, J. L. 1967. The Theory of Splines and Their Applications. Academic Press, New York. ARNOTT, S. 1970. The geometry of nucleic acids, in Progress in Biophysics and Molecular Biology 21 : ed. Butler, J. A. V., and Noble, D. Pergamon Press, Oxford, pp. 265-319. ARNOTT, S.; CHANDRASEKARAN, R.; HUKINS, D. W. L.; SMITH, P. J. C.; Watts, L. 1974. Structural Details of a Double-helix observed for DNAs containing alternating Purine and Pyrimidine sequences. Journal of Molecular Biology 88: 523-533. ARNOTT, S., and HUKINS, D. W. L. 1972. Optimised parameters for A-DNA and B-DNA Biochemica Biophysica Research Communications 47: 1504-1509. ARNOTT, S., and HUKINS, D. W. L. 1973. Refinement of the structure of B-DNA and implications for the analysis of X-ray diffraction data from fibres of biopolymers. Journal of Molecular Biology 81: 93-105. BATES, R. H. T. 1969. Contributions to the Theory of Intensity Interferometry. Monthly Notices of the Royal Astronomical Society 142: 413-428. BLOOMFIELD, V. A.; CROTHERS, D. M.; TINOCO, 1. 1974. Physical Chemistry of Nucleic Acids. Harper and Row, New York. BOND, P. J.; LANGRIDGE, R.; JENNETTE, K. W.; LIPPARD, S. J. 1975. X-ray fibre diffraction evidence for neighbour exclusion binding of a platinum metallointercalation reagent to DNA. Proceedings of the National Academy of Sciences of the United States 72: 4825-4829. BRACEWELL, R. N. 1965. The Fourier Transform and Its Applications. McGraw-Hill, New York. CAIRNS, J. 1963. The bacterial chromosome and its manner ofreplication as seen by autoradiography. Journal of Molecular Biology 6: 208-213. CRAIG, M. E.; CROTHERS, D. M.; DOTY, P. 1971. Relaxation kinetics of dimer formation by self complementary oligonucleotides. Journal of Molecular Biology 62: 383-401. CRICK, F. H. c., and KLUG, A. 1975. Kinky Helix. Nature 255: 530-533. COHEN, R. J., and CROTHERS, D. M. 1971. The rate of unwinding small DNA. Journal of Molecular Biology 61: 525-542. CROTHERS, D. M. 1964. The kinetics of DNA denaturation. Journal of Molecular Biology 9: 712-733. - - - 1969. On the mechanism of deoxyribonucleic acid unwinding. Accounts of Chemical Research 2: 225-232. CROTHERS, D. M., and SPATZ, H. C. 1971. Theory of friction-limited DNA unwinding. Biopolymers 10: 1949-1972. DICKERSON, R. E. 1964. X-ray analysis and protein structure, in The Proteins, Vol. II (H. Neurath ed.), 2nd edn. Academic Press, New York and London. DONOHUE, J. 1969. Fourier analysis and the structure of DNA. Science 165: 1091-1096. - - - 1970. Fourier series and difference maps as lack of structure proof: DNA is an example. Science 167: 1700-1702. BrGEN, M., and DE MAEYER, L. C. 1963. Relaxation methods, in Friess, S. L., Lewis, E. S. and Weissberger, A. (eds), Technique of Organic Chemistry, Vol. VIII, Part II, Interscience, New York.

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R. H. T. BATES Electrical Engineering Department University of Canterbury Private Bag Christchurch New Zealand