**Some Macromolecules form Fibers rather than Crystals**

Many biological macromolecules will not or cannot crystallize.
However, an important group of fibrous macromolecules such as DNA or
many of the components of the cytoskeleton form orientated fibers in
which the axes of the long polymeric structures are parallel to each
other. Often, as in the case of muscle fibers, the orientation is
intrinsic; sometimes the long molecules can be induced to form
orientated fibers by pulling them from a gel with tweezers, sometimes
by flowing a gel through a capillary tube, or even by subjecting them
to intense magnetic fields.

The experimental set-up is rather simple: the orientated fiber is
placed in a collimated x-ray beam at right angles to the beam and the
"fiber diffraction pattern" is recorded on a film placed a
few cm away from the fibre.

Fibers show helical symmetry rather than the three-dimensional
symmetry taken on by crystals. By analysing the diffraction from
orientated fibers one can deduce the helical symmetry of the molecule
and in favourable cases one can deduce the structure. In general this
is done by constructing a model of the fiber (as in DNA) and then
calculating the expected diffraction pattern. By comparing the
calculated and observed diffraction patterns one eventually arrives at
a better model.

**Fiber diffraction patterns fall into two main classes: crystalline
and non-crystalline. **

In the crystalline case (e.g. A-form of DNA) The long fibrous
molecules pack to form long thin micro-crystals which share a common
axis (usually referred to as the c-axis). The micro-crystals are
randomly arranged around this axis. The resulting diffraction pattern
(on left of figure h1) is

equvalent to taking one long crystal and spinning it about its axis during the x-ray exposure. All Bragg reflexions are registered at one time. The reflexions are grouped along "layer-lines" which arise from the repeating structure along the c-axis. However, particularly at high resolution the Bragg reflexions tend to fall on top of each other. If the Bragg reflexions could be separated and measured out to high resolution then the standard methods we have described for crystals could be used. Unfortunately this is never the case and model building must be employed - as is generally the case for non-crystalline fibers.

In non-crystalline fibers (e.g. B-form of DNA) the long fibrous
molecules are arranged parallel to each other but each molecule takes
on a random orientation around the c-axis. The resulting diffraction
pattern (right of figure h1) is also based on layer-lines, which
reflect the periodic repeat of the fibrous molecule. The intensity
along the layer-lines is continuous and can be calculated via a
"Fourier-Bessel Transform" of the repeating structure of the
fibrous molecule (the Fourier-Bessel Transform replaces the Fourier
transform used in standard crystallography - The Fourier-Bessel
transform arises because of the cylindrical symmetry).

*Fig h1 Diffraction from the A and B forms of DNA*

**Diffraction from a helix**

Calculating the x-ray diffraction pattern from a helix was of central
significance in the development of molecular biology. It was first
described by Francis Crick in his doctoral thesis. He wished to
understand the diffraction to be expected from an [alpha]-helix.
However, the theory was very quickly applied to determining the
structure of DNA.

Crick showed that the diffraction from a helix occurs along a series
of equidistant *lines* rather than the Bragg *spots* one
obtains from a three dimensional crystal. These lines (known as
layer-lines) are at right angles to the axis of the fiber and the
scattering along each layer-line is made up from *Bessel functions*.
In helical diffraction Bessel functions take the place of sines and
cosines one uses for crystals: Bessel functions (written *J*_{n}*(x),*
where *n* is called the order and* x* the argument) are the
form that waves take in situations of cylindrical symmetry (e.g. the
waves you get if you throw a pebble into the middle of a pond). Bessel
was a German astronomer who calculated accurately the orbits of the
planets. Fourier used Bessel functions to calculate the flow of heat in
cylindrical objects. Bessel functions characteristically begin with a
strong peak and then oscillate like a damped sine wave as *x *increases.
The position of the first strong peak depends on the order* n* of
the Bessel function. A Bessel function of order zero begins in the
middle of the pattern, a Bessel function of order 5 has its first peak
at about *x* = 7, a Bessel function of order 10 does everything
roughly twice as far out.

*(Fig h2 plots of J*_{0}*(x) to J*_{15}*(x))*

Crick showed
that for a continuous helix the order of Bessel function *n*
occuring on a certain layer line is the same as the layer line number *l *(counted
from the middle of the diffraction pattern). In Fig *h3* we show a
continous helix and its diffraction pattern. Because the order of
Bessel function increases with layer line number so does the position
of the first strong peak. which then form the characteristic
"helix cross". The position of the first strong peak is also
inversely proportional to the radius of the helix. The spacing of the
layer-lines is reciprocal to the pitch* (P)* of the helix. There
is a reciprocal relationship between the layer line separation and the
pitch- small separation large P, large separation small P.

*Fig h3: A continuous helix and its diffraction pattern*

However, real helicies are not continuous, rather they consist of
repeating groups of atoms or molecules.The symmetry of a discontinuous
helix can be defined in a number of ways: the most general is to quote
how far one goes along the axis from one repeating subunit to the next
in the macromolecule (the rise per residue *p*) and by what angle
you turn (*[phi]*) between one subunit and the next. This is
enough to define a helix. Directly derivable from these parameters is
the pitch *P*. The main effect of shifting from a continous to a
discontinuous helix is to introduce new helix crosses
with their origins diplaced up and down the axis (*meridian*) of
the diffraction pattern by a distance 1/*p.* The diffraction
pattern of a discontinous helix (with 10 subunits in one turn) is shown
in Fig *h4. *Note that the layer-lines can be grouped into two
kinds: those which are strong on the meridian of the fiber diffraction
pattern (*meridional*) and those which have no intensity on the
meridian (*non-meridional*). For a simple helix which repeats in
one turn the fundamental layer line repeat is* 1/P*. The distance
out along the meridian of the *first meridional *layer-line (not
counting the origin) gives *1/p*.

*h4 A discontinuous helix and its diffraction pattern*

**Helix Selection Rule **

For more complicated helices which repeat after two or more turns *n*
and *l *are related by the *helix selection rule*

*l* = t*n* + u*m*

The selection rule is an integer equation which makes use of an
alternative definition of helix symmetry: there are t subunits in u
turns of a repeat. m can take all positive and negative integer values.
As an example, for an [alpha] helix t=5 and u=18 i,e, there are 18
subunits arranged on 5 turns per repeat. Solutions to the selection
rule tell you which Bessel functions will turn up on which layer lines.
Bessel functions with very large orders can be forgotten since they
will occur so far out in the diffraction pattern (at such high
resolution) that they will not be visible. Converseley, if one can
figure out which Bessel functions turn up on which layer lines one
knows the symmetry. The effects helical symmetry are in fact very
useful for an analysis of fiber diffraction patterns. Without helical
symmetry all Bessel functions would turn up on all layer lines, which
would be a mess. The helical symmetry limits the allowed Bessel to one
or two per layer-line which renders such problems tractable.

**How this applies to DNA**

DNA-B form is a simple helix which repeats in one turn. It has 10 base
pairs per turn so that the angle turned ber base [phi] is 36°. The
spacing between the bases is 3.4Å (i.e. *p =* 3.4 Å)
and the pitch is ten times greater (i.e. *P* = 34Å. For low
order layer lines the order of the Bessel function* n* which
occurs on the* l*'th layer line is *l*. Because of their mass
the phosphate groups are the dominant scatterers in a nucleic acids.The
phosphate oxygens show up as prominent white spheres in the atomic
model (Fig h5). The spacing of the layer lines in Fig h1 corresponds to
34Å - the helix repeats in 34Å. In this case this is also
the pitch *P*. Knowing this and Francis Crick's formula for the
scattering of a helix, Jim Watson was able to glean the radius of the
phosphate groups from the look of the helix cross in Rosalind
Franklin's fiber diffraction patterns of DNA . Furthermore, the strong
meridional reflexion (see Fig *h1*) which has a Bragg spacing of
3.4Å, must correspond to 1/*p, (*i.e. the spacing between
bases was 3.4Å). These pieces of information went a long way
towards defining the essential parameters of the Watson-Crick model.

*Fig h5 A space filling model of DNA-B form (W. Fuller*)

Impressum. K. C. Holmes Jan 98 (copywrite 98)