ABSTRACT
An early x-ray photograph of DNA taken by W. T. Astbury in the 1930s. Two characteristic features stemming from the stacked bases are indicated by arrows. (Adapted from [5, 6]).
Shortly after the pioneering introduction of x-ray techniques by van Laue and Bragg, structural analyses were soon devoted to the study of complex organic polymers able to condense in a partially ordered phase. Thus, cellulose (the main component of the cell wall in most plants) or keratin (a protein present in natural hair or wool …bers) were considered in the 1920s and 1930s; and diverse DNA samples in the late 1940s and early 1950s. In this way, it was progressively realized that helical arrangements may play a signi…cant role in a growing number of …brous polymers. In Fig.6.1 we show one of the …rst di¤raction pictures of a nucleic acid taken by William T. Astbury (1898-1961) in 1936. Although the image is quite blurry, two characteristic features at the north and south regions of the meridian axis suggest a spacing of about 3.34 Å along the …ber axis, which was properly interpreted as
revealing a close succession of ‡at nucleotides standing out perpendicularly to the long axis of the molecule to form a relatively rigid structure. [5]
This was not still a helix, however, but in 1949 Sven Furberg, a research student working under John Bernal, proposed a DNA structure based on a
(with the sugar units perpendicular to the bases) in his Ph. orientation of the units in a nucleotide subsequently proved a considerable help to Francis H. C. Crick (1916-2004) and (1928) in their …nal model.[8] A speci…c helical di¤raction required in order to properly interpret the obtained x-ray The demand for a theoretical framework was pressing models, based on a judicious combination of x-ray dif-stereochemical information about acceptable bond lengths, hydrogen bonding interactions, were proposed for several by Pauling and his collaborators. of di¤raction by helices, including a proper formula for the
…rst reported and successfully applied to explain the dif-of certain synthetic polypeptides by W. Cochran, F. Crick, A version of the formalism was independently developed by (1919-2003), but it was published in 1955,[10] after the discovery years.[11] In what follows, we shall brie‡y introduce some basic results of the helix di¤raction theory, which are convenient to properly understand the most signi…cant features of DNA structure and biological functionality. Let us consider a wire of in…nitesimal thickness curled around to form a uniform helix of in…nite length, constant radius R, and pitch P , given by the equations
x = R cos'(z) (6.1)
y = R sin'(z)
z = z
where '(z) = 2z=P , and z measures the distance along the helix axis. The value of the Fourier transform at a point q in the reciprocal space is given by
F(q) = A Z f(r)eiqrdr; (6.2)
where A is an appropriate constant, f(r) is the electron density through the scattering helix, r =xi+yj+zk is the position variable given by Eq.(6.1), and the integral is taken over all space. Assuming f(r) f0 for a uniform helix, and expressing the volume element as dr = R2dz; Eq.(6.2) can be written in the form,
F(q) = Af0R2 Z exp [i(qxR cos'+ qyR sin'+ zqz)] dz: (6.3)
In order to exploit the cylindrical symmetry of the helix it is convenient to express the components of the reciprocal space vector q in cylindrical coor-
dinates, so that qx = qr cos and qy = qr sin , where qr = q q2x + q
tan = qy=qx. In this way, Eq.(6.3) can be rewritten as[9]
F(q) = Af0R2 Z exp [i fRqr cos(' ) + zqzg] dz: (6.4)
in
can be seen as a composition of a translation along the Z rotation about this axis. Since the helix repeats itself in the Z direction, the scattering is analogous to that of a di¤raction grating with spacing P . Therefore, the di¤rac-a uniform helix in reciprocal space occurs along a series of (whose spacing is determined by the helix pitch) rather than one obtains from a three-dimensional crystal. These lines, lines, are at right angles to the Z axis in reciprocal space, by the series (Fig.6.2),
qz(n) = 2 n
P ; (6.5)
Eq.(6.4) can be rearranged in the form [12]
F(q) = F0 X n
In qz 2 n
P
; (6.6)
where F0 Af0PR2=2, and we have introduced the auxiliary integral
In = ein Z 2 0
eiu cos(' )ein(' )d'; (6.7)
where u Rqr. This integral can be evaluated by using the identityZ 2 0
eix cos eind = 2inJn(x); (6.8)
where Jn(x) denotes the nth-order Bessel function of the …rst kind.[13] Thus, adopting ' in Eq.(6.7), and taking into account the identity in = ein=2, one …nally obtains[9]
F(q) = 2F0 X n
Jn(u)e in( +=2)
qz 2 n
P
: (6.9)
For a given value of n; Eq.(6.9) gives the amplitude and phase of the xray scattering on the nth layer line. The intensity of the di¤raction peaks is given by In(q) = jFn(q)j2 / jJn(u)j, which is independent of the angle and the resulting pattern has cylindrical symmetry. In addition, from the mathematical properties Jn(u) = (1)nJn(u), and J2n(u) = J2n(u), n = 0; 1; 2 : : :, one realizes that the di¤raction pattern In(q) / jJn(u)j will be symmetric with respect to the qr and qz axes, respectively (Fig.6.2). Thus, the presence of Bessel functions is a natural consequence of the cylin-
drical symmetry in helical di¤raction, where they replace the trigonometric functions one …nds in usual di¤raction by three-dimensional lattices. Bessel functions characteristically begin with a strong peak and then oscillate like a damped sine wave as the argument increases. The position of the …rst strong
Di¤raction pattern of a continuous helix of radius R and pitch P . Main di¤raction spots are arranged along a series of lines (layer lines) labeled by integer values n. The lines are perpendicular to the meridian axis qz and are separated by a distance 1=P . The characteristic cross-shaped pattern stems from the symmetry properties of Bessel functions.
