Rosalind Franklin's X-ray diffraction patterns of DNA molecules rendered the important clue that DNA has the structure of a double helix. The most famous X-ray photograph, Photo 51, is still printed in most Biology textbooks. We suggest two optical experiments for undergraduates that make this historic achievement comprehensible for students by using macromodels of DNA and visible light to recreate a diffraction pattern similar to Photo 51. In these macromodels, we replace the double helix both mathematically and experimentally with its two-dimensional (flat) projection and explain why this is permissible. Basic optical concepts are used to infer certain well-known characteristics of DNA from the diffraction pattern.

Rosalind Franklin used X-ray diffraction to determine the structure of DNA molecules. One of her best X-ray pictures is numbered Photo 51 and is shown in Fig. 1(a). This photo was instrumental to J. D. Watson and F. Crick in deducing the double-helix model of DNA. Because of its historical significance, Photo 51 is still printed in most textbooks about X-ray diffraction or genetics. These diffraction patterns were a telltale indicator that DNA is a double helix. In addition, the radius, pitch, pitch angle, and the number of phosphate molecules per pitch of the DNA helix could be determined. Although this photo is used in most biology, genetics, biophysics, or modern physics textbooks, it remains unclear to students how or why these conclusions can be drawn. We suggest two experiments that can help students make these connections. The experiments can be incorporated in both introductory and upper-level courses; however, the modelling is geared toward upper-level courses.

Fig. 1.

(a) The well-known Photo 51, the diffraction pattern from DNA in its so-called B configuration. The dimensions of DNA are: pitch P = 3.4 nm, radius R = 1 nm, and a phase difference between the two helices (sine waves) of ΔP = 3P/8. Several important features include the characteristic X-shape or distorted rhombus, the ten diffracted orders per X, and the missing fourth order. (b) A two-dimensional projection of the phosphate molecules in the DNA backbone. The projection outlines two sine waves. We justify this flat model theoretically in Sec. II.

Fig. 1.

(a) The well-known Photo 51, the diffraction pattern from DNA in its so-called B configuration. The dimensions of DNA are: pitch P = 3.4 nm, radius R = 1 nm, and a phase difference between the two helices (sine waves) of ΔP = 3P/8. Several important features include the characteristic X-shape or distorted rhombus, the ten diffracted orders per X, and the missing fourth order. (b) A two-dimensional projection of the phosphate molecules in the DNA backbone. The projection outlines two sine waves. We justify this flat model theoretically in Sec. II.

Close modal

Franklin (and separately Stokes) worked on creating a mathematical model for diffraction from DNA.1,2 Historically, this pattern was calculated using a lattice approach, where one would look for Bragg diffracting conditions.3 The diffraction pattern was then described mathematically by Watson and Crick, as well as Stokes.2,4 Kittel5 deduced the diffraction pattern of a helical structure using Bragg reflection, the approach used by most crystallographers. The successful structure analysis of the DNA molecule resulting in the double helix model is an example of how an experiment and its analysis changed our view of the world. The achievement was rightfully honored with the Nobel prize in 1962. Unfortunately, Franklin died before the prize was awarded, but her contribution to the effort has been acknowledged by Watson6 and Klug7,8 (Klug later won the Nobel prize in 1982 for work he started with Rosalind Franklin).

It is desirable to discuss this crucial experiment and its outcome with undergraduate students in biophysics, modern physics, and optics courses. Most approaches to make the connection between the structure of DNA and its X-ray diffraction pattern more plausible use mathematics that one can find in solid state physics textbooks.5,9,10 However, a good part of the problems that students have with these approaches comes from their unfamiliarity with complex experiments described by a scientific and mathematical language. Students often fail to associate theory and experimental outcomes when performing a difficult experiment. In this work, we repeat the scattering, but instead of X-rays, as in the original experiment, we use monochromatic light with a much longer wavelength, similar to Lucas et al.11,12 The longer wavelength allows us to recreate the diffraction pattern of Photo 51 by using larger, homemade models of the DNA helix, such as the spring from a ballpoint pen or sinusoidally arranged holes in a piece of cardboard (which represent the heavy phosphate molecules in the DNA backbone). The advantage of this approach is that both the helix and its diffraction pattern can be seen with the naked eye. Students have the benefit of creating the diffracting structure first, and so, they know the structure that produces the diffraction pattern, whereas in real X-ray diffraction patterns, the goal is to determine a structure too small to be seen.

Regarding the mathematical approach, we show that in order to analyze the resulting scattering pattern, we only need Huygens' elementary wave model in the Fraunhofer approximation. Within this approximation, the helix does not need to be three-dimensional; instead, a flat projection of the helix perpendicular to its axis [see Fig. 1(b)] can be used. The diffraction of light from a helical structure thus requires only the Fourier transform as a mathematical tool. Mathematically, we decompose the aperture into three substructures: the heavy phosphates that lay on the DNA backbone (represented by the holes in a piece of cardboard), the arrangement of these holes sinusoidally (representing the helical shape in the flat model), and a function used to offset the helix to two locations so that the single helix becomes a double helix. According to the convolution theorem, the resulting pattern is then readily described as a product of the Fourier transforms of each substructure. To conclude our investigations, we compare the diffraction patterns of our experiments with our theoretical results, as well as the historic pattern in Photo 51. These experiments can be performed in introductory or upper-level courses and as a modelling exercise in upper-level physics courses.

The scattering of light by small objects is calculated from the Huygens-Fresnel integral in a model where elementary spherical waves are stimulated by an incident light wave $u(r→)$ at different points of the object and summed to give at point B:13

$uB=k02πi∫u(r→) ρ(r→)eik→0 ⋅R→BRBdτ.$
(1)

Here, $ρ(r→)$ is the object's scattering density, $k0$ is the wave number of the incident monochromatic plane wave $u(r→)=u0 eik→0 ⋅r→$, and $R→B$ is the vector from the scattering point S, at the location $r→=(x,y,z)$ of the object, to the point of interest B, at location $r→B=(xB,yB,zB)$, is a volume element in real space, so that $R→B=r→B−r→=(xB−x,yB−y,zB−z)$ (see Fig. 2). Integration is over all real space with $dτ=dxdydz$ as volume element.

Fig. 2.

Light scattering by a small object. The vector $R→B=r→B−r→$ is directed from the scattering point S at location $r→$ of the object to the test point B at location $r→B$. Note that $k→0 ⋅r→$ is zero for normal incidence on flat objects.

Fig. 2.

Light scattering by a small object. The vector $R→B=r→B−r→$ is directed from the scattering point S at location $r→$ of the object to the test point B at location $r→B$. Note that $k→0 ⋅r→$ is zero for normal incidence on flat objects.

Close modal

The scattering density $ρ(r→)$ describes the number of scattering centers and their strengths per unit volume; in Secs. II and III, we will only consider opaque-transparent apertures, where $ρ(r→)=0$ for the opaque regions and $ρ(r→)=1$ for transparent regions. The result one gets from Eq. (1) depends on the approximation used to express the distance vector $R→B$. If quadratic terms in $r→$ are included, complex interference effects like the Gouy phase can be calculated. To calculate the far field interference of a monochromatic plane wave scattered by a small object, we only need the linear approximation; for $x,y,z≪xB,yB,zB$ and $xB,yB≪zB$, we find that

$RB≈xB2+yB2+zB2−2 r→⋅r→B≈rB−r→⋅r→BrB=rB−r→⋅r̂B.$
(2)

Substituting Eq. (2) into Eq. (1) and writing the scattering density $ρ(r→)$ as a Fourier-integral, we obtain

$uB=k02πiu0eik0rBrB1(2π)3/2∫ ρ̃(k→) eik→⋅r→dτk∫eik→0⋅r→e−ik0r→⋅ r̂Bdτ=k02πiu0eik0⋅rBrB∫ρ̃(k→) δ(k→−k0r̂B+k→0)dτk=k02πiu0eik0⋅rBrBρ̃(k→B−k→0),$
(3)

where $k→B=k0r̂B$ is the wave vector of the outgoing wave and $k→0$ the wave vector of the incoming wave, with $|k→B|=|k→0|=k0$ as required by conservation of energy. Integration is over all k-space and $dτk=dkxdkydkz$ is a volume element in reciprocal space. Thus, in the far field, the amplitude $uB$ of the scattered wave is proportional to the Fourier transform $ρ̃(k→B−k→0)$ of the scattering density, with $k→=k→B−k→0$ as required by momentum conservation.

The important question to ask is how the depth of the object (in the z-direction) contributes to the amplitude $uB$ in the far field, where $rB$ is very large compared to both the size of the scatterer and the size of the diffraction pattern. If, as is the case of Fraunhofer diffraction, the incident wave propagates in the z-direction along the optical axis, its wave vector has only a z-component: $k→0=(0,0,k0)$. Since $|r→B|=rB≈zB$, the z-component $k0r̂B$ of the outgoing wave vector is compensated in $ρ̃(k→B−k→0)$ by the wave vector of the incoming wave (see Fig. 2) and has only components with $kz=0$

$ρ̃(kx,ky,0)=1(2π)3/2∫ρ(x,y,z)ei(kxx+kyy)dxdydz=1(2π)3/2∫ρ′(x,y)ei(kxx+kyy)dxdy,$
(4)

with $ρ′(x,y)=∫−∞+∞ρ(x,y,z) dz$. Thus, we see that the incoming and outgoing (in the far field) wave vectors have nearly the same z-components, which means that the depth of the scattering object is not important in this approximation, and it can therefore be projected parallel to the z-axis onto the xy-plane. In our experiment, this means that the spring from a ballpoint pen, when used as a diffracting object, can be approximated as a sinusoidal curve (of finite thickness) in the xy-plane. We will use this result in our experiments and in our mathematical modelling when we diffract from flat apertures instead of from three-dimensional structures. Of course, disregarding the depth of the scattering object in Fraunhofer diffraction implies that no noticeable information about the chirality (the sense of the winding) of the spring can be obtained from the far-field spectrum.

The X-shape (or distorted rhombic shape) in Photo 51 was an important hint that DNA is helical with a radius R of 1 nm and a pitch P of 3.4 nm. Also noticeable was a missing fourth layer line [see Fig. 1(a)], which helped in identifying that there is a second helix that is offset from the first by 3/8 of the pitch.

By diffracting light from the spring of a ballpoint pen, students can rework the thought processes that led scientists to draw these conclusions. In Fraunhofer diffraction, the spring creates the characteristic X-shape pattern similar to that of DNA14 [see Figs. 1(a) and 3(b)]. In our approach, the wavelength λ of the light is much smaller than the grating constant d and the depth D (≈d) of the object (the Klein–Cook parameter $Q≪1$), and so, the diffraction from a helix is in the far field like the diffraction from a flat sinusoidal curve. Thus, we can make deductions about the structure and dimensions of DNA from Photo 51 at a very basic level. In particular, the X-shape results from the parallel sections of the projected wire [the two flanks of the sine curve indicated by the dashed lines in Fig. 3(a)], which diffract like two sets of multiple slits or gratings oriented at an angle $2α$ to each other. Using the diffraction pattern, the spacing d between parallel sections of the projected wire can be obtained using the equation mλ = d sin θ, where θ is the angle of the m-th maximum in the intensity pattern on one of the legs of X, with the zeroth order being the center. The pitch angle α can be obtained by simply measuring the half acute angle of the X-shaped pattern in Fig. 3(b).

Fig. 3.

(a) A ballpoint pen spring (length ∼30 mm, used as our helical diffracting object), where about four pitches are being illuminated by coherent laser light (λ = 633 nm, 10 mW) expanded to a diameter of ∼5 mm by a microscope lens and a secondary lens of focal length f = 10 cm. (b) The diffraction pattern as observed on a screen 6 m away, showing the characteristic X-shape (Ref. 14). The intensity maxima on the legs of the X are caused by the grating of the pitch, and the missing fifth, tenth, etc., orders are a result of the thickness of the wire. The horizontal maxima stem from the two-slit interference of the vertical parts of the spring. A polarizer was placed at the zeroth order and adjusted to near total extinction to prevent oversaturation of the image.

Fig. 3.

(a) A ballpoint pen spring (length ∼30 mm, used as our helical diffracting object), where about four pitches are being illuminated by coherent laser light (λ = 633 nm, 10 mW) expanded to a diameter of ∼5 mm by a microscope lens and a secondary lens of focal length f = 10 cm. (b) The diffraction pattern as observed on a screen 6 m away, showing the characteristic X-shape (Ref. 14). The intensity maxima on the legs of the X are caused by the grating of the pitch, and the missing fifth, tenth, etc., orders are a result of the thickness of the wire. The horizontal maxima stem from the two-slit interference of the vertical parts of the spring. A polarizer was placed at the zeroth order and adjusted to near total extinction to prevent oversaturation of the image.

Close modal

Franklin and Stokes created a mathematical model for diffraction from DNA,1,2 but it is not derived in their 1953 Nature publication. A simple approach for students to model the diffraction pattern is to replace the opaque helix mathematically with its two-dimensional projection and to make it transparent. We end up with a flat sinusoidal “slit” aperture and use an area integral to calculate the diffraction pattern. Replacing an opaque object with its inverse (transparent) counterpart is possible because open areas of an aperture contribute mainly with their edges.15 The result is that an (opaque) obstacle has an identical diffraction pattern in the Fraunhofer region as a transparent aperture of the same shape (Babinet's principle),16 except for an additional bright spot in the center (Poisson's spot). Thus, the helical wire is represented by the area between two sinusoidal curves that extend in the x-direction and that are offset by $±a$ with respect to the y-axis. This creates a small “aperture” that simulates the projected figure of a helical wire of thickness ∼2a [see Fig. 4(a)]. It should be noted that the spacing between these two sine curves varies and has only the thickness 2a at the minima and maxima of the sinusoidal wave. Students can now calculate the diffraction pattern with a simple area integral, which they typically learn in calculus classes, and then compare it with the observed pattern from the three-dimensional spring.

Fig. 4.

Apertures and diffraction patterns of single and double helix apertures. (a) A single helix projection and the absolute value of its Fourier transform calculated from the electric field amplitude given by Eq. (6). The single solid sinusoidal aperture creates a characteristic X that is associated with helical diffraction patterns. The parameters used reflect the geometry of the spring from Fig. 3(a): P = 1.8 mm, R = 2 mm, and a = 0.2 mm. (b) A double helix projection and the absolute value of the Fourier transform calculated from the electric field amplitude given by Eq. (6) and multiplied by Eq. (14). Notice that adding a second helix that is 3P/8 out of phase with the first creates destructive interference at the fourth diffracted order. The aperture functions were calculated using Maple as horizontal structures, and their diffraction patterns were rotated to reflect the geometry of our setup.

Fig. 4.

Apertures and diffraction patterns of single and double helix apertures. (a) A single helix projection and the absolute value of its Fourier transform calculated from the electric field amplitude given by Eq. (6). The single solid sinusoidal aperture creates a characteristic X that is associated with helical diffraction patterns. The parameters used reflect the geometry of the spring from Fig. 3(a): P = 1.8 mm, R = 2 mm, and a = 0.2 mm. (b) A double helix projection and the absolute value of the Fourier transform calculated from the electric field amplitude given by Eq. (6) and multiplied by Eq. (14). Notice that adding a second helix that is 3P/8 out of phase with the first creates destructive interference at the fourth diffracted order. The aperture functions were calculated using Maple as horizontal structures, and their diffraction patterns were rotated to reflect the geometry of our setup.

Close modal

If n windings of the spring are illuminated by a uniform electric field $u0$ then Huygens waves emerge from a sinusoidal aperture given by

$ρ(x,y)={1 if R cos(2πPx)−a≤y≤R cos(2πPx)+a0 otherwise.$
(5)

Here, R and P are the radius and pitch of the helix as defined earlier but now have the meaning of amplitude and wavelength for the flat projection. Huygens waves create a diffraction pattern in the far field that is given by the Fourier transform in Eq. (4). Thus $ρ̃(kx,ky,0)$, becomes

$ρ̃(kx,ky)=F{ρ(x,y)}=∫x=−x0x0∫y=R cos(2πPx)−ay=R cos(2πPx)+a exp[i(kxx+kyy)] dydx= 2a sin(kya)kya∫x=−x0x exp {i[kxx+kyR cos(2πPx)]}dx,$
(6)

where $F{ρ(x,y)}$ denotes the Fourier transform and k = 2π/λ. Because the scattering density $ρ$ is uniform within the aperture and zero outside, the limits of the integral are defined by the aperture itself. The limits $±x0$ of the x integral should be chosen to reflect the extent of the illuminated spring. The diffraction amplitude can be calculated using software such as Maple or Mathematica. In order to compare the diffraction pattern of the model with the experiment, one needs to plot the intensity $I∝ρ̃(kx,ky) ρ̃(kx,ky)*$. However, squaring $ρ̃$ would lead to very small intensities in the higher diffracted orders, and so, we instead plotted the absolute value of Eq. (6) [see Fig. 4(a)].

It is valuable to reflect on the two features of this diffraction amplitude. First, the integral with respect to x represents the periodic maxima in the diagonal $±kx,±ky$ directions caused by the interference from multiple illuminated windings. Second, the factor $sin(kya)/kya$ stems from the interference of the wire itself (i.e., single-slit interference) and causes a wavelike modulation of the X-pattern in the $ky$-direction, which leads to missing orders for specific values of a. In our experiment, the thickness-to-pitch ratio is such that the fifth order vanished [see Fig. 3(b)]. However, because in our simple model, the thickness varies across the pitch, the missing orders due to the wire thickness cannot be reproduced correctly.

How can the missing fourth order indicate a second helix? It is instructive and relatively simple to see that the fourth order could be missing as a consequence of the presence of a second helix if this second helix is shifted along the x-axis with respect to the first helix by a specific fraction f of the spacing d. (This fraction can also be defined in terms of the pitch P.) An easy way to understand this is as follows. The multiple slits of the single sinusoidal aperture create maxima at $mλ=d sin θ$, with m being an integer. The m-th maximum is suppressed if it coincides with a minimum of the double slits with slit distance $f d$ between the two sinusoidal aperture [see Fig. 4(b)], i.e., with an odd integer of $λ/2$: $(2l+1) λ/2=f d sin θ=f mλ$, with integer $l. The fourth maximum (m = 4) vanishes for $f=1/8, 3/8, 5/8, 7/8$, which are the fractions of d by which the second helix can be offset with respect to the first. The values $f=3/8, 5/8$ result in the same relative spacing of the two helices. Similarly, $f=1/8, 7/8$ results in the same relative offset, but this offset is probably too small to be feasible for the molecular structure of DNA. If students use these considerations, they can figure out which separation would cause other orders to disappear. On the other hand, it is a nice exercise to see that if $f=3/8$, the 12th, 20th, etc., orders would vanish as well as the fourth.

In order to model the entire diffraction pattern of a double helix as seen in Fig. 4(b), we used the convolution theorem as described in Sec. IV B Eq. (10).

From the diffraction pattern of DNA, Rosalind Franklin also deduced that there were ten phosphates per pitch along the DNA backbone. Precisely how this conclusion can be drawn is now discussed.

That the diffracting object—the helix—can be replaced by its two dimensional projection allows us to create a more realistic and detailed aperture than our previous DNA model. Specifically, we can mimic the ten phosphate groups per pitch. In contrast to our simplified model, DNA is not a wire wound into a helical shape. From the viewpoint of X-rays, there are large areas filled with nothing between the phosphates on the DNA backbone. We could of course use beads to represent these phosphates, but they would have to be arranged on an almost transparent wire or background and affixed at specific locations. Instead, we use the two dimensional projection of the helix and again use Babinet's principle. This approach has the benefit of avoiding Poisson's spot, which generally makes viewing more difficult. Thus, we create an aperture where the phosphate molecules, which contribute mostly to the diffraction of X-rays, are represented by circular holes in black cardboard, rather than beads, arranged in a sine wave pattern. A similar aperture has been suggested by Lucas et al.11

We describe here how students can make their own model and build the mathematical bridge to its diffraction pattern. In the case of DNA, which has ten phosphate groups per turn (pitch P), we generate ten holes per wavelength (see Fig. 5). This pattern was created by first plotting a sine wave using Maple. The ratio of amplitude to wavelength of the sine wave was the same as the ratio of radius to pitch of the DNA. Our helix had a radius of 3.56 mm and a pitch of 11.97 mm. Then, circles of radius a were plotted at previously calculated positions along the sine wave, ten circles per wavelength. (Note that in this two-dimensional projection, the distance between the circles is not equally spaced like the distance between the phosphate molecules on the three dimensional helical DNA strand.) We placed this printed plot over the black cardboard and punched holes using a needle of radius a = 0.3 mm. A smaller needle would have resulted in not enough light getting through, and a larger needle would have led to overlapping holes in the extrema of the sinusoidal arrangement. Although the hole size does affect the diffraction pattern, as long as the holes remained less than 40% of the size of the sinusoidal amplitude (i.e., the DNA's radius), the finer diffraction detail like the X-shape could still be observed.

Fig. 5.

The 2D projection of a helix is a sinusoidal wave. Our apertures were made by punching holes of radius a into cardboard, with the holes representing the phosphate molecules. The laser light (λ = 633 nm, 10 mW) is expanded using a microscope objective and a lens (f = 0.25 m) to cover the entire aperture. The position of the second lens was then adjusted slightly to create a slightly converging beam until the diffraction pattern could be clearly seen on a distant screen (L ∼ 4 m away). A focusing lens placed after the aperture was used to record the diffraction pattern with a CCD camera.

Fig. 5.

The 2D projection of a helix is a sinusoidal wave. Our apertures were made by punching holes of radius a into cardboard, with the holes representing the phosphate molecules. The laser light (λ = 633 nm, 10 mW) is expanded using a microscope objective and a lens (f = 0.25 m) to cover the entire aperture. The position of the second lens was then adjusted slightly to create a slightly converging beam until the diffraction pattern could be clearly seen on a distant screen (L ∼ 4 m away). A focusing lens placed after the aperture was used to record the diffraction pattern with a CCD camera.

Close modal

The double helix in DNA is represented by adding a second sinusoidal arrangement of holes which is shifted by 3/8 of the sinusoid's wavelength [see Fig. 1(b)]. We note that in actuality, we adjusted the offset between the helices for ease of aperture creation purposes. With the offset of 3P/8, there exists an overlap of certain holes, which would have been difficult to recreate when punching them. Thus, for simplification, we adjusted the offset until the overlapping holes completely coincided with each other such that the two holes from the two different helices could now be represented by a single hole, as depicted in Fig. 1(b). After performing some calculations, we found that this adjustment had a slight effect on the diffraction pattern. However, these differences were not important since the main purpose of the offset was to observe a missing fourth layer line in our diffraction pattern, and this result remained unaltered with our simplification.

The apertures were then placed into an expanded laser beam (see Fig. 5). The second lens of the telescopic beam expander was shifted slightly along the optic axis toward the aperture until the diffraction pattern appeared focused at a reasonable distance (L ∼ 4 m). In contrast to the experiment with a spring,14 at first glance, the diffraction pattern does not look like the typical X-shape pattern. Instead, there is a huge Airy disk that is created by the circular apertures. However, a closer look reveals that the zeroth order and the surrounding rings have repeating rhombic structures within them [see Fig. 6(a)], not just one X-shape as in the experiment with the solid helix. It is the height of these rhombuses or X-shapes which contains the information about the number of holes (phosphates) per pitch.

Fig. 6.

Diffraction patterns from sinusoidally arranged holes (representing phosphate molecules). The ring structure is caused by diffraction from the holes; the finer, diagonal structures inside the bright center and inside the rings are due to the arrangement of the holes in a sinusoidal pattern. (a) Photo of the experimentally generated pattern for a single helix; a round polarizer was placed at the zeroth order and adjusted to near total extinction to prevent oversaturation of the camera with the 633 nm laser light. (b) Calculated Fourier transform for a single helix (absolute value of Eq. (15) with f = 0) with parameter values P = 3.4, R = 1, f = 0, a = 0.1 R, and 5 illuminated pitches (these values are the same for all figures except as noted otherwise). (c) A zoomed-in photograph of the experimental diffraction pattern using a black-and-white CCD camera. As a result of the ten phosphates per period, the rhombuses are ten diffracted orders high. (d) Calculated transform of the inner center. (e) The double sinusoidal experimental diffraction pattern recorded with the CCD camera. The weakening of the fourth diffracted order (m = 4) is due to the offset of the double helix by 3P/8. (f) Calculated transform of a double helix made of holes (with f = 3/8).

Fig. 6.

Diffraction patterns from sinusoidally arranged holes (representing phosphate molecules). The ring structure is caused by diffraction from the holes; the finer, diagonal structures inside the bright center and inside the rings are due to the arrangement of the holes in a sinusoidal pattern. (a) Photo of the experimentally generated pattern for a single helix; a round polarizer was placed at the zeroth order and adjusted to near total extinction to prevent oversaturation of the camera with the 633 nm laser light. (b) Calculated Fourier transform for a single helix (absolute value of Eq. (15) with f = 0) with parameter values P = 3.4, R = 1, f = 0, a = 0.1 R, and 5 illuminated pitches (these values are the same for all figures except as noted otherwise). (c) A zoomed-in photograph of the experimental diffraction pattern using a black-and-white CCD camera. As a result of the ten phosphates per period, the rhombuses are ten diffracted orders high. (d) Calculated transform of the inner center. (e) The double sinusoidal experimental diffraction pattern recorded with the CCD camera. The weakening of the fourth diffracted order (m = 4) is due to the offset of the double helix by 3P/8. (f) Calculated transform of a double helix made of holes (with f = 3/8).

Close modal

In order to zoom in on the bright center, we replaced the observation screen with a CCD camera that had its front lens removed, making sure that the exposed sensor was placed exactly on the center of our diffraction pattern. Using a neutral density filter (ND = 2.5), we decreased the intensity of the laser beam so that it did not oversaturate the camera. A second converging lens placed after the aperture brought the diffraction pattern closer and made it smaller so that it was easier to capture its image with the camera [see Figs. 6(c) and 6(e)].

Our apertures (representing a helix and a double helix) are now made of two or three substructures (see Fig. 7): a circular hole $ρO$ simulating the phosphate molecule, the sinusoidal arrangement of the holes $HS$ representing the helix, and the two positions of this sinusoidal arrangement $HP$ creating the double helix. One can describe the aperture as a convolution of these aperture functions16,17

$(ρO*HS*HP)(x″,y″)≡∫y′=−∞∞∫x′=−∞∞∫y=−∞∞∫x=−∞∞ρO(x,y)HS(x′−x,y′−y)HP(x″−x′,y″−y′)dx dy dx′ dy′.$
(7)
Fig. 7.

The convolution theorem states that the Fourier transform of an aperture function constructed by convoluting substructures can be calculated by taking the product after the Fourier transforms of the substructures. The top images show the field distributions across the aperture substructures. From left to right are: (a) the hole ρ0; (b) the sinusoidal position of the holes HS; and (c) the position of the two sine functions for the double helix HP. Panel (d) shows the final aperture function—a double sine function outlined by holes—which is the convolution of the three substructures. The bottom images show the Fourier transforms of substructures in the top row. From left to right: (e) a Bessel function; (f) a rhombus composed of a reciprocal grating with grating constant q/P in the kx direction and a reciprocal grating in the ky direction; and (g) a cosine wave leading to missing orders. Panel (h) shows the final transform, which corresponds to the product of the three substructure transforms.

Fig. 7.

The convolution theorem states that the Fourier transform of an aperture function constructed by convoluting substructures can be calculated by taking the product after the Fourier transforms of the substructures. The top images show the field distributions across the aperture substructures. From left to right are: (a) the hole ρ0; (b) the sinusoidal position of the holes HS; and (c) the position of the two sine functions for the double helix HP. Panel (d) shows the final aperture function—a double sine function outlined by holes—which is the convolution of the three substructures. The bottom images show the Fourier transforms of substructures in the top row. From left to right: (e) a Bessel function; (f) a rhombus composed of a reciprocal grating with grating constant q/P in the kx direction and a reciprocal grating in the ky direction; and (g) a cosine wave leading to missing orders. Panel (h) shows the final transform, which corresponds to the product of the three substructure transforms.

Close modal

The circular aperture $ρ0$ with radius a exposed to a planar electromagnetic wave is a piece-wise step function (i.e., a cylinder with radius a and height 1; see Fig. 7, upper left corner)

$ρO(x,y)={10if x2+y2≤a2if x2+y2>a2.$
(8)

The other apertures are arrays of Dirac delta functions $δ(x)$ arranged in a desired pattern (also called Dirac combs). A convolution of $ρO$ with a Dirac comb creates mathematically an aperture consisting of holes as repeated identical substructures. Convolving the aperture $ρ0$ with the array function

$HS(x,y)=∑n=0Nδ (x−Pqn) δ (y−R cos[2πP(Pn/q)])$
(9)

arranges the holes in a sinusoidal pattern. Here, P is the pitch of the helix, R is the radius of the helix, n is an index for the n-th hole, N is the total number of holes illuminated by the incident light wave, and q is the number of holes (phosphate molecules) per pitch. The axial spacing in the x-direction between the phosphate molecules is equidistant such that there are q of them per period P, resulting in the locations xn = nP/q with respect to the x-axis and a grating constant P/q. However, the projections of the phosphate molecules are sinusoidally—not equally—spaced along the y-axis. For DNA, the dimensions are P = 3.4 nm, q = 10 phosphates/period, and R = 1 nm.

An additional convolution of the aperture $ρO*HS$ with the array

$HP(x,y)=δ(x−123P8) δ(y)+δ(x+123P8) δ(y),$
(10)

shifts the sinusoidal array of holes to the positions $(1/2)(3P/8,0)$ and $−(1/2)(3P/8,0)$, where P is the pitch of the helix, i.e., the wavelength of the sinusoidal aperture. Hence, two sinusoidal arrangements are created that are separated by an axial distance of 3P/8. This is the axial separation between the two helices in DNA.

The convolution theorem states that the Fourier transform of convoluted functions is equal to the product of the Fourier transform of each function, that is,

$ρ̃(kx,ky)=F{ρO*HS*HP}=F{ρO}F{HS} F{HP}.$
(11)

The Fourier transform of a circular hole is16

$F{ρO}=2πa2 [J1(±akx2+ky2)±akx2+ky2],$
(12)

where a is the circle's radius and $ka2=kx2+ky2$. Fortunately, the Fourier transform of $HS$ is rather simple; we find that

$F{HS}=∑n=1N exp(ikxPqn) exp(ikyR cos(2πqn)).$
(13)

Finally, the Fourier transform of $HP$ is

$F{HP}=∫–∞∞∫–∞∞[δ(x−123P8)δ(y)+ δ(x+123P8)δ(y)]exp(ikxx)exp(ikyy)dxdy=2 cos(3P16kx),$
(14)

which is simply a cosine function in the $±kx$ direction of reciprocal space. Putting everything together, the resulting Fourier transform for a double helix is just the product of Eqs. (12)–(14), given by

$ρ̃(kx,ky)=2πa2 [J1(±akx2+ky2)(±akx2+ky2)] ∑n=0N exp(ikxPqn) exp(ikyR cos(2πqn))2 cos(f P2kx),$
(15)

where $f=0$ for the single helix and $f=3/8$ for the double helix mimicking the DNA. The Hermitian product $ρ̃(kx,ky) ρ̃(kx,ky)∗$ then describes the diffraction pattern. The absolute square of Eq. (15) can be plotted [see Figs. 6(b), 6(d), and 6(f)] and compared to the experimental images [Figs. 6(a), 6(c), and 6(e)].

Notice how each term in Eqs. (11) and (15) leads to a specific aspect of the diffraction pattern, each playing a crucial role as graphically presented in Fig. 7:

• $F{ρO}$ leads to the Airy disk and rings.

• $F{HS}$ is a summation over “waves” in reciprocal space. It represents the periodic maxima and minima in the diagonal $±kx,±ky$ directions on the observing screen. Note that the vertical periodicity of the diffraction pattern has the reciprocal grating constant $q/P$, and so, the rhombus (or the X) has a height of $q$ diffracted orders. Thus, counting the diffracted orders reveals the number of phosphates per pitch. In Fig. 6, this can be seen by counting the diffracted orders in the height of a rhombus from 0 to 10. In Fig. 1(b), it can be better seen by counting the diffracted orders in one leg of the X from −5 to +5. In the limit of a solid sinusoidal line, $q→∞$, and so, the rhombus does not repeat and is instead infinitely high (an X-shape) as in the experiment with the helical wire. In the case $q=1$, there is no reciprocal grating in the $±ky$ direction and therefore no X-shape [see Fig. 8(a)].

• Finally, the cosine term in $F{HP}$ modulates the pattern in the $±kx$ direction, which leads to missing orders in the pattern when the argument of the cosine term is $fPkx/2=(2l+1)π/2$ with l = integer or $f=(2l+1)/2m$ because $kx=2π m/P$ for the m-th diffracted order. Figure 4(b) shows a result of multiplying $F{ρ}$ of Eq. (6) with $F{HP}$.

Fig. 8.

First quadrant of the diffraction pattern for (a) one hole (q = 1), (b) two holes (q = 2), and (c) four holes (q = 4) per pitch. The dotted lines and arrows show how the pattern is related to the diffracting structure. (Note the first minimum of the Airy disk that suppresses the pattern in an area outlining a quarter of a circle.) Panels (d)–(g) were all made using P = 3.4, R = 1, and 5 illuminated pitches. The dashed quarter circle outlines the first minimum of the Airy disk. Panel (g) shows that when the holes overlap, the rhombus grows into an X (the scale in this panel was changed to show the full pattern, which makes comparison with Fig. 3(b) easier).

Fig. 8.

First quadrant of the diffraction pattern for (a) one hole (q = 1), (b) two holes (q = 2), and (c) four holes (q = 4) per pitch. The dotted lines and arrows show how the pattern is related to the diffracting structure. (Note the first minimum of the Airy disk that suppresses the pattern in an area outlining a quarter of a circle.) Panels (d)–(g) were all made using P = 3.4, R = 1, and 5 illuminated pitches. The dashed quarter circle outlines the first minimum of the Airy disk. Panel (g) shows that when the holes overlap, the rhombus grows into an X (the scale in this panel was changed to show the full pattern, which makes comparison with Fig. 3(b) easier).

Close modal
Fig. 9.

As the hole size increases, the Airy disk it creates in reciprocal space decreases: (a) a = 0.1 R, corresponding to our experimental setup; (b) a = 0.2 R, the diffraction pattern of the holes begins to “suffocate” the rhombic pattern and makes it less visible in the rings; (c) a = 0.3 R, the tip of the rhombus is now located in the first Airy ring, which is the condition for DNA (where the radii of the phosphate molecules are about one third of the helix's radius); (d) a = 0.4 R, the rhombic pattern still exists outside the Airy disk but is too weak to be seen. All figures were made for P = 3.4, R = 1, and 5 illuminated pitches.

Fig. 9.

As the hole size increases, the Airy disk it creates in reciprocal space decreases: (a) a = 0.1 R, corresponding to our experimental setup; (b) a = 0.2 R, the diffraction pattern of the holes begins to “suffocate” the rhombic pattern and makes it less visible in the rings; (c) a = 0.3 R, the tip of the rhombus is now located in the first Airy ring, which is the condition for DNA (where the radii of the phosphate molecules are about one third of the helix's radius); (d) a = 0.4 R, the rhombic pattern still exists outside the Airy disk but is too weak to be seen. All figures were made for P = 3.4, R = 1, and 5 illuminated pitches.

Close modal
Fig. 10.

As the number of illuminated pitches (N) increases, the diffraction pattern becomes sharper (similar to two-beam versus multiple-beam interference).

Fig. 10.

As the number of illuminated pitches (N) increases, the diffraction pattern becomes sharper (similar to two-beam versus multiple-beam interference).

Close modal
Fig. 11.

As the shift ΔP between the two sinusoidal functions changes, the resulting interference pattern also changes, leading to different missing diffracted orders: (a) ΔP = P/2, all odd orders vanish; (b) ΔP = P/4, the second, sixth, and tenth orders vanish; the 4th and 12th orders are seen to vanish for both (c) ΔP = P/8 and (d) ΔP = 3P/8.

Fig. 11.

As the shift ΔP between the two sinusoidal functions changes, the resulting interference pattern also changes, leading to different missing diffracted orders: (a) ΔP = P/2, all odd orders vanish; (b) ΔP = P/4, the second, sixth, and tenth orders vanish; the 4th and 12th orders are seen to vanish for both (c) ΔP = P/8 and (d) ΔP = 3P/8.

Close modal

It is very instructive for students to adjust (experimentally or by modeling) certain parameters like the hole size a (Fig. 9), the number of illuminated pitches N (by expanding the beam) (Fig. 10), the shift $ΔP$ between the helices (Fig. 11), and the number of holes per pitch q and to observe the effect on the diffraction pattern. Varying the number of holes per pitch will clarify how Rosalind Franklin could conclude that there are ten phosphates per pitch and not two or seven (Fig. 8). Fig. 1(b) can be used as a starter template for the experiments.

One didactical aspect of this work is that for a volume scatterer the far field (Fraunhofer region) is equivalent to the far field of a plane scatterer. And the plane scatterer is constructed from the volume scatterer by projecting all its scattering elements onto a plane perpendicular to the incident beam. This means it makes no difference if we scatter from a helix or a plane, sinusoidal slit. Based on this insight, our two dimensional representations of DNA generated mathematically as well as experimentally enable another didactical aspect of this work: they replicate the famous X-structure in Rosalind Franklin's Photo 51 and its missing fourth order and provide an explanation for the overall structure of the pattern as a product of the Fourier transforms of three substructures: the scattering molecules, their arrangement along one sinusoidal line, and the arrangement of the two sinusoidal lines themselves. Using mathematical modeling or optical experiments, students can experience the interplay between these substructures, which will provide them with new insights into X-ray diffraction and Fourier transformation.

Our structure analysis is of course complex, and it should encourage the readers to do their own investigations and improvements by changing the theoretical and experimental parameters as suggested. It is perhaps the main outcome of this work that only playful experience with the experimental and theoretical background of physical phenomena can lead to a profound understanding of major discoveries and why they are worthy to be honored with a Nobel prize.

This work was funded by the Research Corporation for Science Advancement (Grant No. CC6339) and the Fredrick A. Hauck Research Grant and the Women of Excellence Giving Circle at Xavier University. L. Wessels was supported by the Jonathan F. Reichert Foundation. The authors thank Jessica Murphy for the expert preparation of the figures.

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