I suggest that you consider ways in which radial distribution functions for helical
structures could be calculated with less effort than has been necessary in the past.
Possibly an analytical method could be used. For example, it might be interesting
to consider the radial distribution function for an 18-residue 5-turn helix, as calculated
by smearing each set of atoms uniformly along a helix, as was done in the calculation
of structure factors by Professor Schomaker. The radial distribution function could
then be expressed in terms of an integral. I am not sure that the integral could be
evaluated.
A somewhat better approximation could be made by assuming that the atoms of each kind
can be represented by a sinusoidal distribution of scattering matter along a helical
xxx curve - that is, a constant plus a single cosine function, such as to make the
density increase from 0 to a maximum and then back to 0 again, this change occurring
18 times in 5 turns along the helix. It might be possible to evaluate the integral
for this case.
The limiting case would, of course, involve an infinite number of sinusoidal terms,
all with the same coefficients, to correspond to point atoms.
Although it is possible that a relatively simple expression for the radial distribution
function can be developed in this way, I am not optimistic about it. I think that
it is likely, however, that a simple machine could be built, for use in calculating
the radial distribution functions for helical structures. You might consider the design
of such a machine. I suggest that it be kept very simple in design, and probably that
it not be planned for construction by the instrument makers.
I suggest that the machine might consist of a framework supporting a rod in vertical
orientation, with a side arm, which would define a point at fixed altitude, radius,
and azimuth. A device should be designed that would permit the rotation of this assembly
through exactly 100°, and at the same time its elevation by the distance corresponding
to 1.50 A. By the repeated operation of this device, the indicated point would be
carried through the successive positions corresponding to the set of atoms defined
by the parameters.
A second point, at the base of the apparatus, would be defined by another set of parameters;
this point would represent another atom in the structure.
For each setting of the first point a measurement of the interatomic distance corresponding
to the relative positions of the two points would be made by means of a scale. I suggest
that the scale should be attached to the assembly representing the first point, and
that it should move passed a fiducial mark at the second point, permitting the operator
to read the distance. Probably the scale could best be an inch scale marked off in
tenths of an inch, with perhaps smaller subdivisions.
I think that a scale of 2 inches to 1 Angstrom would be best. This would permit the
radial distribution function out to 20 Angstroms to be determined, with an instrument
standing 40 inches high (plus the additional expansion of the moving assembly, which
might double the height).
I think that a device of this sort could be constructed very easily that would be
reliable to within about 0.1 inch. This would correspond to the determination of interatomic
distances to within 0.05 A, which is good enough for our purposes. In fact, it might
well be worth while to build the apparatus as described, but to plan to use it on
the scale 1 inch equal 1 Angstrom. This would permit the determination of the radial
distribution function out to 40 Angstrom (over 7 turns of the α helix), with an accuracy
of 0.1 A.
The interesting structures will in general correspond to cylindrical radii not greater
than about 6 A. I suggest that the machine be built with a radius of action of 12
inches, corresponding to 6 A on the scale 2 inches = 1 A, but capable of handling
cases up to 12 A, on the scale 1 inch = 1 A, Perhaps, for practical reasons, it would
be better to keep the size down to about 10 inches radius, rather than 12 inches.
As an example of the operation of the machine, let us consider the 5.2-residue helix.
There are 5 different atoms in a residue (including the β carbon atom), and about
40 residues within the distance of 40 A, Accordingly the radial distribution curve
will involve contributions of approximately 1000 interatomic distances. These would
be measured with the machine in 25 sequences, of , 40 measurements each. If two investigators
worked together, I estimate that three minutes sight be required to set the initial
coordinates for each sequence, and that, with one investigator; operating the rotation-translation
machine and the other reading and recording the distances, the 40 measurements in
the sequence would be made in five minutes, giving a total of eight minutes per sequence.
The 25 sequences would then require about 3 hours. With practice, the investigators
could, I think, decrease this time significantly. For comparison, I may mention that
Dr. Weinbaum worked for some weeks on each of the radial distribution functions that
he obtained by straightcalculation.
A very complicated structure, such as the collagen 3-chain helix, might require two
or three days for calculation of its radial distribution function. I think that this
is a job that should be done.
After the interatomic distances are calculated, a suitable distribution of the scattering
power over adjacent interatomic distances would be made, as was done in the earlier
calculations, and the final distribution curve obtained by adding the numbers. This
is not a difficult job - it should take only an hour or two, without requiring use
of a calculating machine.
Linus Pauling:W
P.S. I have asked Peter Pauling to confer with you on the problem of constructing
this radial distribution, machine - he has had a lot of experience in constructing
objects with the use of the facilities in our shop. I shall have him come in to see
you shortly.
