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Using coherent light, we analyze the temporal diffraction at a single point from real-time living
*C. elegans* locomotion in three-dimensional space. We describe the frequency spectrum of single swimming nematodes in an optical cuvette at a single sampling point in the far-field diffraction pattern. An analytical expression of the double slit is used to model the frequency spectra of nematodes as oscillating segments. The frequency spectrum in the diffraction pattern expands discretely and linearly as a multiple of the fundamental frequency with increasing distance from the central maximum. The frequency spectrum of a worm at a single point in the frequency spectrum contains all the frequencies involved in the locomotion and is used to characterize and compare nematodes. The occurrence of resonant frequencies in the dynamic diffraction pattern increases with the distance from the central maximum. The regular spacing of the resonant frequencies is used to identify characteristic swimming frequencies.

Tracking the three-dimensional (3D) movement of microscopic species over larger distances is developing with the availability of optical components and increasing computing power [

We have used the transparent nematode, Caenorhabditis elegans, to develop methodologies for examining microscopic organismal shape and locomotion in real-time in 3 dimensions without using microscopes [

The analysis of the locomotion of microscopic species typically involves the filming of the species followed by a computationally intense processing of the video [

Χ k → = ∑ n → = 0 N → − 1 e − 2 π i k → ⋅ n → N → x n → (1)

where Χ k → is the element in Fourier space denoted by k → , n → is the element in object space, N represents the total number of elements in a particular dimension and x n → is the element in Fourier space denoted by n → . The phase sensitivity is on the order of the wavelength of the coherent radiation.

In this paper, we present the single point temporal diffraction model explaining the frequency spectra as related to the locomotion of the nematode; i.e., frequency and amplitude of worm oscillations. The relationship between the worm thickness and thrashing amplitude is related to the number of resonance frequencies as a function of the single point location, where the single point simulates a photodiode (PD) placed in the dynamic diffraction pattern.

The C. elegans width-to-length ratio is about 1/10 [

I ( p ) ∝ | 2 d sinc ( π p d ) | 2 (2)

where d is the slit width and p is the position in reciprocal space. Two segments of a C. elegans can then be represented by using the shift theorem [

F ( x + a ) + F ( x − a ) ⇋ 2 Φ ( p ) cos ( 2 π p a ) (3)

where F is the function in object space and Φ its Fourier transform; while x represents position in object space, p the reciprocal variable in Fourier space and a is the distance by which the function F has been displaced. In the case of a dynamic system a is time-dependent a(t).

C. elegans tend to move using a sinusoidal waveform [

I ( p , t ) ∝ ( 2 d sinc ( π p d ) cos ( 2 π p a 0 sin ( 2 π f t ) ) ) 2 (4)

where a_{0} is the amplitude of oscillation and f the frequency at which the oscillations of the segments occur.

Placing a photodiode (PD) in the diffraction pattern fixes the position p in the diffraction pattern. The time-dependent signal at p = 0 will not vary as the frequency inside the cosine term (Equation (4)) equals zero. Holding the oscillation amplitude steady as the distance of the PD from the central maximum increases the frequency range increases matching the increments of the fundamental frequency; i.e., integer multiples of the frequency at which the slits are oscillating. The amplitude can also increase the frequency of in the diffraction pattern since larger amplitudes will span a larger number of wave-fronts.

Placing a photodiode (PD) in the diffraction pattern fixes the position p in the diffraction pattern. The time dependent signal at the central maximum, p = 0, will not vary as the frequency inside the cosine term (Equation (4)) equals zero. Holding the oscillation amplitude steady as the distance of the PD from the central maximum increases the frequency range increases matching the increments of the fundamental frequency; i.e., integer multiples of the frequency at which the slits are oscillating since the sine term inside the cosine term (Equation (4)) in the diffraction pattern dominates. The oscillation amplitude of the worm can also increase the frequency of in the diffraction pattern since a larger amplitude will span a larger number of wave-fronts.

The expected number of higher dominant frequencies is indicated by n in the spectrum can then be determined using the location of the PD at point p is then a factor of the worm thickness, sinusoidal locomotion amplitude and the value of p:

n ≥ a 0 p / ( π d ) (5)

The number of frequencies is indicated by n if n is an integer; otherwise the next larger integer indicates the number of fundamental frequencies included in the signal from the PD. The remaining fraction in n indicates the amplitude of the highest frequency contributing to the cosine while the sine function (Equation (4)) provides the envelope, which can be attributed to the nematode’s width.

Overall, the number of frequency components increase with the phase angle as indicated in

The optical setup was constructed to collect data at one point of the diffraction pattern (

The distance from the nematode to Mirror 2 is 3.65 ± 0.05 cm. Mirror 2 is located 18.75 ± 0.05 cm from the photodiode so that the total distance from the diffracting worm to the photodiode is 22.4 ± 0.1 cm. The experimental data sets in

〈 f 〉 = ∑ f n ⋅ P S D n ∑ P S D n (6)

where f is the frequency, PSD represents the Power Spectral Density and n indicates the matrix element. The experimental spectra are not as distinct in their resonances as the simplified sinusoidal model; nevertheless, the resonances are consistent with known experimental locomotory frequencies of swimming and crawling C. elegans [

to determine the fundamental frequency. Destructive interference can eliminate resonance peaks so that placing the PD strategically allows for a maximum signal. Various worm movements such as head movement and the changing in direction or speed of the nematode can create frequencies. Some of these movements tend to show up in the slower frequencies, diffusing the dominant swimming or crawling frequency.

In this case study, we have demonstrated an efficient method for measuring dominant frequencies by strategically placing the PD in a dynamic diffraction pattern. Using this method, the locomotion of various phenotypes and in various environments can be characterized by frequency. A point in the diffraction pattern is used to record the superposition of all locomotory oscillations of the nematode allowing for the analysis of all frequencies embedded in the worm motion through a single time series. The intensities are sensitive to motion on the order of the wavelength of light used in the experiment so that subtle movements that might be missed in a microscopic analysis are detected and measured. Even the newest work in locomotion of C. elegans involves traditional microscopy rather than diffraction so that the work published by this group is the only work involving live diffraction of C. elegans [^{ }

Collecting the signal at a distance from the central maximum allows for enough resonant peaks in the FT spectrum to use the spacing to measure the fundamental frequency. The regular spacing of the resonant peaks in the FT helps to identify dominant frequencies in the locomotion precisely. Further studies on separating movements of various worm parts are promising future projects. Dynamic diffraction using power spectral analysis complements traditional microscopic methods. Frequency filters can be employed to isolate various movements and switching mechanisms.

Magnes, J., Congo, C., Hulsey-Vincent, M., Hastings, H. and Raley-Susman, K.M. (2018) Live C. elegans Diffraction at a Single Poi. Open Journal of Biophysics, 8, 155-162. https://doi.org/10.4236/ojbiphy.2018.83011