This manuscript describes how to distinguish different nematodes using far-field diffraction signatures. We compare the locomotion of 139 wild type and 108 "Roller" C. elegans by averaging frequencies associated with the temporal Fraunhofer diffraction signature at a single location using a continuous wave laser.
This manuscript describes how to classify nematodes using temporal far-field diffraction signatures. A single C. elegans is suspended in a water column inside an optical cuvette. A 632 nm continuous wave HeNe laser is directed through the cuvette using front surface mirrors. A significant distance of at least 20-30 cm traveled after the light passes through the cuvette ensures a useful far-field (Fraunhofer) diffraction pattern. The diffraction pattern changes in real time as the nematode swims within the laser beam. The photodiode is placed off-center in the diffraction pattern. The voltage signal from the photodiode is observed in real time and recorded using a digital oscilloscope. This process is repeated for 139 wild type and 108 "roller" C. elegans. Wild type worms exhibit a rapid oscillation pattern in solution. The "roller" worms have a mutation in a key component of the cuticle that interferes with smooth locomotion. Time intervals that are not free of saturation and inactivity are discarded. It is practical to divide each average by its maximum to compare relative intensities. The signal for each worm is Fourier transformed so that the frequency pattern for each worm emerges. The signal for each type of worm is averaged. The averaged Fourier spectra for the wild type and the "roller" C. elegans are distinctly different and reveal that the dynamic worm shapes of the two different worm strains can be distinguished using Fourier analysis. The Fourier spectra of each worm strain match an approximate model using two different binary worm shapes that correspond to locomotory moments. The envelope of the averaged frequency distribution for actual and modeled worms confirms the model matches the data. This method can serve as a baseline for Fourier analysis for many microscopic species, as every microorganism will have its unique Fourier spectrum.
This method compares experimental and modeled frequency spectra of the locomotion of C. elegans using two strains with very different locomotory patterns. The results show that the frequency spectrum depends on temporal changes as the nematode swims in a water column so that clear microscopic images are not needed for analysis. This method allows for quantitative real-time analysis and provides complementary information to images/videos obtained with traditional microscopes. Fraunhofer diffraction, also called far-field diffraction, provides the basis for obtaining live diffraction data1,2. The light intensity at any single point in the diffraction pattern is the result of superimposing light from every point in the outline of the nematode3. As a result, the light intensity collected over time carries information about the locomotion of the nematode. Analyzing the time-dependent diffraction signal can identify the characteristic motion of the corresponding mutant since analyzing all the frequencies involved in the locomotion complements the traditional video analysis. In this case, the characteristic differences between the locomotion of the "roller" and wild type C. elegans are confirmed by comparing the frequency spectra of the two different strains of nematode.
Some previous characteristics have been confirmed using frequency analysis of diffraction signals such as swimming frequencies2,4. More importantly, this method can be used as a complementary method to traditional microscopy to observe locomotion in real time on a computer screen as the data are being collected. The frequency spectrum of worms with distinct locomotory patterns can be quantified by considering the Fourier transformed signal of the diffraction signal.
The multidisciplinary nature of Fourier-based diffraction in this work involves the fields of biology and physics. Diffraction by under sampling has long been used to investigate crystal structures in biology5 and other fields. In this experiment, however, oversampling6,7 creates the far-field diffraction pattern so that the organism is centered in the laser beam. Oversampling is typically used for lens-less imaging8 in conjunction with a phase retrieval algorithm that reconstructs an image of the original object. Phase retrieval is difficult to achieve when scatterers are present as is the case with a nematode. The temporal diffraction signature is enough to evaluate key frequencies of the worm motion. This method is less computationally taxing and provides an optical way to quantify locomotion. This technique could readily be adapted for analysis of mutations or environmental conditions that alter behavior.
1. C. elegans Growth and Maintenance
2. Optical Setup (Figure 1)
3. Oscilloscope Setup
4. Preparing the Worm and Cuvette for Data Collection
5. Real Time Data-Acquisition of Diffraction Pattern Intensity Changes
6. Fourier Spectrum of Data
7. Modeling of the Fourier Spectrum
The optical experimental setup shown in Figure 1 allows for the study of microorganisms without being tied to a focal plane. The thrashing signal from the photodiode can be observed in real time on the computer screen as the data is collected. Unusual patterns will be visible immediately without having to analyze a video in detail.
Examples of modeled sequential worm movement and corresponding diffraction patterns are shown in Figure 2. The modeled diffraction patterns qualitatively resemble experimental patterns1 and are an initial indication that the simulations successfully model the nematode.
A sample temporal diffraction signature of the two types of C. elegans studied here is shown in Figure 3. It can be seen qualitatively that each nematode thrashes at different rates and amplitudes. Some of the differences can be quantified through curve fitting as was done in a previous publication1. The discrete Fourier transform, however, reveals more details regarding embedded frequencies:
, (1)
where Fk is the digital Fourier transform (FT) and fn is the time-dependent raw diffraction signal with the discrete time variable n and the discrete frequency variable k. N is the total number of data points. The average digital Fourier transform allows for the nematode to be identified by amplitude of its diffraction frequency spectrum (Figure 4). The wild type spectrum is dominated by lower frequencies than the roller motion spectrum.
A model that approximates the wild type versus the roller C. elegans notes the wild type tends to thrash in a wavelike (W or S shape) motion (Figure 2a) while the roller tends to favor one side that roughly resembles an oscillating C shape (Figure 5). This offers some explanation for the different spectra. The roller will mostly form a C to one side while the W oscillation can be thought of as two opposing C motions. For this reason, the W motion is more complex revealing more secondary low frequencies than the C motion. This result is confirmed in the computational model. The W shape has a much higher frequency density than the C shape (Figure 6). This is confirmed in the FFT in Figure 4 where the roller frequencies are more clustered while not completely discrete. The statistics of the roller are skewed since the roller can revert to wild type locomotion temporarily.
The smoothed power spectra of roller type C. elegans shows a broad peak at ~1.5 Hz, while the swimming wild type C. elegans exhibits a multimodal spectrum (including peaks at ~1.0 Hz and 1.75 Hz). The photodiode (PD) has a finite size spreading over several matrix elements. Individual matrix elements or points on the diffraction pattern vary in intensity since constructive and destructive interference varies; nevertheless, the frequencies at which the intensities vary are the same for all matrix elements, as can be seen in Figure 7. Considering the time derivative Eq. 1, it can be seen that the frequency fluctuations do not depend on the phase matrix but only on the original object's fluctuations:
, (2)
As the PD spreads over several matrix elements, the peak locations average to a consistent frequency profile. Some variation can be expected and can give clues about the orientation of the worm. The frequency distribution will change as the gait of the worm changes. The current model is a simple model that only allows for the evaluation of peak locations rather than relative peak heights. Different locomotory patterns will average to different peak locations.
Figure 1. Experimental Setup. The low power laser beam travels through the neutral density filter, is reflected by mirror M1 down through the cuvette containing the worm onto mirror M2, and travels towards the photodiode. Please click here to view a larger version of this figure.
Figure 2. Sequential Worm Shapes and Corresponding Diffraction Patterns. (a) Some select sequential binary images of modeled W shape nematodes and the (b) corresponding sequential diffraction patterns. Please click here to view a larger version of this figure.
Figure 3. Experimental Sample Diffraction Signatures. Diffraction signatures collected for (a) OH7547 "roller" and (b) N2 wild type C. elegans using a single photodiode in the diffraction pattern. Please click here to view a larger version of this figure.
Figure 4. Experimental Averaged Power Spectra of Roller and Wild Type Fraunhofer Diffraction Series. The spectra show the frequencies present in averaged Fourier transform of the time series recorded with the photodiode. A Gaussian filter of standard deviation 0.075 Hz, truncated at 3 standard deviations, is used for smoothing. Note the broad spectral peak at ~1.5 Hz in the smoothed roller spectrum, compared with the multimodal smoothed wild type spectrum (including peaks at ~1.0 and 1.75 Hz). Please click here to view a larger version of this figure.
Figure 5. Diffraction Formation Illustration. Diffraction patterns can be modeled by thinking of each line segment as an infinitesimally small straight line (left). Superimposing these lines (right) demonstrates the construction of far-field diffraction patterns generated by a C shaped nematode. Please click here to view a larger version of this figure.
Figure 6. Simulated Power Spectra of Roller and Wild Type Fraunhofer Diffraction Series. (a) C shape and (b) W shape worms with the photodiode centered at the matrix elements 200 (vertical) and 175 (horizontal). The W shape shows a higher density of frequencies due to the more complex locomotion. Please click here to view a larger version of this figure.
Figure 7. Simulated Power Spectra of Roller and Wild Type Fraunhofer Diffraction Series at Different Photodiode Locations. (a) W shape worm and the (b) C shape worm for single matrix elements at various locations simulating various locations of the photodiode. Peak heights vary for various locations; however, peak locations remain the same for specific shapes. Please click here to view a larger version of this figure.
Including stretches of data with inactivity will skew the results since artificial lower frequencies will be averaged into the results. Saturating the photodiode can be recognized by flat peaks or "cut off" peaks in the raw data. Dividing each raw data set by peak intensities will help with accounting for fluctuations in the laser intensity.
The peak frequencies are an indicator of the overall thrashing frequency; however, complicated motion causes interference at beat frequencies in the diffraction pattern and need to be examined carefully.
This method can be used to investigate the locomotion of other nematodes. The environment can be changed to another medium. Wavelengths may be changed as well. Working in the visible range of the electromagnetic spectrum is easiest and safest.
A more refined model will simulate the diffraction spectra more realistically in the future. A future model may include a worm that can change orientations, which would not affect frequency locations but relative peak heights. A more realistic model would allow for a probabilistic distribution of thrashing frequencies, which would broaden the peaks as in the experimental data. A spread in frequencies would account for variations in thrashing frequencies.
The current worm shape is crude, especially in the head and tail region, which should be more tapered than in the current model. It may be interesting to conduct a detailed analysis of the time series of the signal since it could give clues about the complexity of the locomotion in different mutants.
It is worth considering the practicality of expanding this technique into characterizing multiple nematodes simultaneously. This method should be understood as a complementary method to existing methods using traditional microscopes. This method has an advantage in not requiring a microscope during the data-acquisition so that the worm may move out of the focal plane. The averaged frequency spectra show clear differences in worm motion and can be quantified by the prevalent frequency peaks, which is a novel method in quantifying worm locomotion. Data analysis of the diffraction signatures is in further development and will hopefully lead to an automated identification process of multiple mutants and individuals.
The authors have nothing to disclose.
We thank Juan Vasquez for his computational contributions with this project. We are grateful for the support of the Vassar College Undergraduate Research Summer Institute (URSI), the Lucy Maynard Salmon Research Fund and the NSF award No. 1058385.
Tunable Helium-Neon laser | Research Electro-Optics | 30602 | Four wavelengths can be selected between 543 nm and 633 nm. |
2 Front Surface Aluminum Mirrors | Thorlabs | PF10-03-F01 | |
Photodiode: SI Amplified Detector | Thorlabs | PDA 100A | |
Quartz Cuvette | Starna Cells | 21/G/5 | Plastic cells may be used as well. |
MatLab (Software) | MathWorks | R2016b (9.1.0.441655) | Use the fft command to simulate diffraction |
Excel | Microsoft | 14.7.1 | Used for data analysis of Fig. 4 |
Caenorhabditis elegans Roller | University of Minnesota Caenorhabditis elegans Center (CGC) | Strain: OH7547 Genotype: otIs199. |
https://cbs.umn.edu/cgc/home |
Caenorhabditis elegans Wild Type | University of Minnesota Caenorhabditis elegans Center (CGC) | Strain:N2 Genotype: C. elegans wild isolate | https://cbs.umn.edu/cgc/home |