Double Helix & Optical Diffraction Pattern

Introduction

In this Post we describe our attempt to replicate the experiment on X-ray diffraction by the DNA molecule.
This experimental research, conducted at the time by Rosalind Franklin, allowed us to understand the structure of the DNA molecule.
Of course working with X-rays and with “crystallized” DNA molecules is not trivial, for this reason our experience will be a sort of simulation made with the visible radiation of a He-Ne laser instead of X-rays and with a spring (a macroscopic spring) instead of the DNA molecule.
Our “simulation” with laser and spring, will allow us to understand how from the diffraction images (an example is shown in the cover image) it is possible to obtain detailed information on the spatial structure of the DNA molecule.

The diffraction and interference of electromagnetic radiation are easily observable phenomena that give direct and tangible evidence of the wave nature of light. Diffraction is the basis of many technologies, scientific techniques and common optical and electromagnetic phenomena. We will not deal with diffraction in detail, for which we refer to the numerous texts and the following previous Posts : Laser & Diffraction Grating, Light as a Wave : Slit Diffraction.

However, let us recall the principle of Babinet, which states that the diffraction pattern produced by an opaque disk of diameter D is identical to that produced by a circular aperture of diameter D. In practice, the diffraction patterns produced by an obstacle and by an aperture having the same linear dimensions are the same.

This principle helps us because the diffraction pattern produced by a set of slits of width w and spaced d is the same produced by a set of wires having a diameter w and spaced d. The theory of slit diffraction is well known.
The figure below shows the diffraction pattern obtained from n slits : the minima with greater spacing corresponds to the width w, while the minima with smaller spacing correspond to the distance d between the slits. The greater the number of slits, the better the definition of minima and maxima.

A front view of a spring is made up of two series of segments having a diameter equal to the diameter of the spring wire, spaced apart on a regular basis, with pitch P, and angled together with angle 2α. The diagram below shows these dimensional parameters of a spring.

Of course we know that the DNA molecule consists of a double helix. A double helix seen from the front consists of four series of segments, as shown in the following image.

In the image below (from Tatiana Latychevskaia e Hans-Werner Fink, Physics Department of the University of Zurich, ,Three-dimensional double helical DNA structure directly revealed from its X-ray fiber diffraction pattern by iterative phase retrieval) we show the links between the diffraction pattern produced by the DNA molecule and the dimensional parameters of the double helix.

• a : “X”-form distribution of the diffraction peaks is an indication of a helical structure. The inclined segments which make the front view of a helix give rise to a inclined line of diffraction peaks. Two series of segments produce two diffraction lines with the same angulation from each other.
• b : The separation between the diffraction maxima (minima) corresponds to the pitch of the helix, this descends directly from the theory of diffraction of n slits and from the Babinet principle.
• c : The broad extended peaks on the top and the bottom are formed by diffraction on small periodical features (the base pairs). This also comes from theory of diffracion and Babinet principle.
• d : The missing diffraction spots is an indication of a double helix. This is interesting, if there were only one helix there would be no missing spots but with two helix, in that position there is destructive interference : thus the missing spot.
• e : The lateral position of the maxima of the diffraction spots is related to the radius of the DNA helix.

The Experiment

The experiment simply consists in carrying out a diffraction test of a laser beam by a spring that serves as a model of the DNA helix. For this purpose we used a He-Ne laser that emits on λ = 632.8 nm, the laser beam is enlarged with a beam-expander so as to obtain a beam with a section of about 1 cm. The spring is positioned in front of the laser beam and the diffraction pattern that is obtained is observed at a distance of 5 – 10 m .

The image below shows the laser and the spring used.

The following image shows an enlarged photograph of the spring used, on which the dimensional parameters are highlighted: the pitch P, the diameter d of the wire and the angle θ.
As already stated in the previous paragraph, seen from the front with respect to the laser beam, the helix appears as two series of “arms”, all parallel, with diameter d and a spacing of a pitch P, these two series of parallel arms have an angle with respect to the normal to the helix axis of a value equal to θ.

The diffraction figure that is obtained is shown, somewhat blurred, in the image below.
What you notice are the two angled lines, which branch off from the center of the “X”. These two lines originate from the diffraction of the two sets of arms that make up the spring and are perpendicular to the latter. The angle between the diffraction lines corresponds to the angle between the arms.

2θ = 18° -> θ = 9°

The diffraction lines then have different minimum and maximum due to the mutual interference of the laser beam waves diffracted by the spring arms. From the theory of diffraction, knowing the spacing, the minor and the major, we know how to calculate the linear dimensions of the obstacles that caused the diffraction.

Distance between the spring and the screen : D = 8.25 m
Wavelength of laser radiation : λ = 632.8 nm
Minor spacing : Δ₁ = 11 mm
Major spacing : Δ₂ = 2.3 mm

The greater spacing is caused by the smaller sized obstacle : d₁ = λD / Δ₁ = 0.47 mm
The smaller spacing is caused by the larger sized obstacle : d₂ = λD / Δ₂ = 2.27 mm

From trivial geometric considerations we can derive the diameter of the spring wire and the pitch, which correspond quite well to the measurements made with the caliper :
d = d₁ = 0.47 mm
P = d₂ / cosθ = 2.3 mm

Conclusions

The experiment of diffraction of a laser beam by a spring proved to be a valid didactic aid for the description of Rosalind Franklin’s historical experiments on X-ray diffraction by DNA molecules that made it possible to understand the structure in detail of DNA molecule.
While replicating the X-ray diffraction on DNA molecules is out of the reach of many laboratories, the use of this simplified model (spring + laser) allows us to explore the diffraction technique and to obtain the dimensional parameters of the helix with a rather good approximation.

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