Parts of this research were funded by a National Institutes of Health R15 Award (National Institute of General Medical Sciences award no. R15GM123420, awarded to R.M.R.-A. and R.J.M.), a Cottrell Scholar Award from the Research Corporation for Science Advancement (award no. 27459, awarded to R.J.M.), and a William M. Keck Foundation Research Grant (awarded to R.M.R-A.). GHK gratefully acknowledges financial support from the Dutch Research Council (NWO; project number VI.C.182.004 of the NWO Talent Programme).

NOTE: A Jupyter Notebook file containing the code to go along with each step in the following protocol can be found on the following GitHub repository, https://github.com/rmcgorty/PyDDM/tree/main/Examples. A PDF of that file is included in&#160;Supplementary File 1. Additionally, a walkthrough of the code and documentation of each function and class can be found on the website,&#160;https://rmcgorty.github.io/PyDDM/.
1. Software installation
<ol>
	<li>To follow along with the example DDM analysis files, install Jupyter Notebook for running the code. Install other required common Python packages, including NumPy and Matplotlib as well. These packages all come bundled with the Anaconda distribution (see&#160;https://www.anaconda.com/products/individual).&#160;</li>
	<li>Install the Python package xarray63. This package is necessary for organizing and storing metadata and analysis parameters. If using the Anaconda distribution, install xarray (along with its recommended dependencies) using the command: 
	conda install -c conda-forge xarray dask netCDF4 bottleneck</li>
	<li>Install the PyYAML package using the command: 
	conda install -c anaconda yaml 
	This package is necessary for reading metadata about the images to analyze and the parameters set by the user for analysis and fitting.</li>
	<li>Install the PyDDM package, by downloading from the GitHub repository or using the git command: 
	git clone&#160;https://github.com/rmcgorty/PyDDM.git</li>
</ol>
2. Planning the imaging sessions
<ol>
	<li>Choose the optimal available imaging modality and optical settings. As mentioned, DDM can be used with a number of microscopy methods.</li>
	<li>To assist in planning the appropriate objective lens and image size to use, determine the range of wavenumbers, q, that will be probed based on the pixel size and total image size. Confirm that the choice of magnification and field of view are optimal for the experiment, based on these calculations. For the images analyzed here, a 60x 1.4 NA objective and an image size of 256 x 256 pixels with a pixel size of 0.83 &#181;m were used for the active actin-microtubule composite network. For the images of beads embedded in a vimentin network, a 100x 1.4 NA objective and an image size of 512 x 512 pixels with a pixel size of 0.13 &#181;m were used. 
	NOTE: The minimum q is set by 2&#960;/N&#916;x, where the image size (assumed to be square) is N&#160;&#215; N pixels with a pixel size of &#916;x. The maximum q is the minimum of&#160;&#960;/&#916;x and 2&#960; NA/&#955;, where NA is the imaging objective&#39;s numerical aperture, and &#955; is the wavelength of light (for brightfield imaging, one can replace NA with (NAobjective + NAcondenser)/2).</li>
	<li>Next, consider the range of timescales to investigate. Typically, DDM analysis is done on sequences of at least 1000 frames.
	<ol>
		<li>To determine the appropriate frame rate, consider the expected time it will take for features in the sample to move a distance on the order of the minimum resolvable length scale (corresponding to the maximum&#160;q).</li>
		<li>In considering the upper limit of the range of timescales probed, recognize that, typically, the power spectrum of hundreds of&#160;image differences of a given lag time&#160;&#916;t are averaged together to provide sufficient statistics to reduce noise. Hence, acquire image sequences longer than the maximum timescale probed. 
		&#8203;NOTE: If an expected diffusion coefficient, D, or velocity, v, is known, then one can estimate expected characteristic decay times using&#160;&#964; = 1/Dq2&#160;or&#160;&#964; = 1/vq along with the range of q, which was determined based on the field of view and pixel size. The range of expected &#964; values over the accessible q-range can help guide the choice of frame rate and the number of frames to acquire.</li>
	</ol>
	</li>
</ol>
3. Sample preparation and image acquisition
NOTE: For details of the sample preparation and imaging settings used for the data presented in the representative results section, see previous publications from the authors11,51,64 and Supplementary File 2.
<ol>
	<li>Based on the consideration of the time and length scales to probe, acquire image sequences of, ideally, over 1000 frames. 
	NOTE: The code will analyze square images or square regions of interest within the image, so adjust the frame size accordingly.</li>
	<li>Save image sequences as a three-dimensional greyscale TIFF stack. Alternatively, the format used by Nikon Instruments systems, ND2 format, can be read by the installed package. If images are saved in some different format, use ImageJ or another imaging processing program to convert the images to a TIFF stack. 
	&#8203;NOTE: If using ND2 files, the package nd2reader from&#160;https://github.com/Open-Science-Tools/nd2reader must be installed.</li>
</ol>
4. Parameter setup
<ol>
	<li>Make a copy of the parameter file example_parameter_file.yml provided in the PyDDM code repository under the examples folder. Open this YAML file with a text editor like NotePad++ or the text editor in JupyterLab. See Supplementary File 2 for an example YAML parameters file used in the analysis of data presented in the representative results section.</li>
	<li>In the copied YAML file, provide the data directory and file name corresponding to the image sequence to be analyzed. Under the metadata section, provide the pixel size and frame rate.</li>
	<li>Under the Analysis_parameters section, provide details for how the DDM matrix should be calculated. Some parameters here are optional.
	<ol>
		<li>At a minimum, provide values for the parameters number_lag_times and last_lag_time. These correspond to the number of different lag times for which to calculate the DDM matrix and the longest lag time (in frames) to use, respectively. For the data of tracer beads in vimentin networks used here, the parameters number_lag_times and last_lag_time were 60 and 1000, respectively. The code will compute the DDM matrix for lag times from 1 frame (or some other minimum lag time if the optional parameter first_lag_time is specified) to the last_lag_time with logarithmic spacing. 
		NOTE: If M frames were acquired, one could calculate the DDM matrix for a lag time as large as M-1. However, with poor statistics at such a large lag time, the data is likely to be noisy. The longest lag time for which to calculate the DDM matrix will depend on the details of the data, but we suggest trying around one-third of the total image series duration.</li>
	</ol>
	</li>
	<li>Provide details for how the DDM matrix or the intermediate scattering function (ISF) should be fit in the Fitting_parameters section. Give the name of the model under the model parameter. Provide the initial guess, lower bound, and upper bound for each of the fitting parameters in the chosen model. 
	&#8203;NOTE: To display a list of the possible fitting models, run the function print_fitting_models. The models can also be found in the online documentation on the PyDDM website.</li>
</ol>
5. Calculating the DDM matrix
<ol>
	<li>Initialize an instance of the DDM_Analysis class. To do so, provide the metadata and analysis parameters discussed above by passing the filename, with the full file path included, of the YAML file to DDM_Analysis. Alternatively, pass the metadata and parameters as a Python dictionary data structure.</li>
	<li>Run the function calculate_DDM_matrix to calculate the DDM matrix. This calculation may take several minutes or longer depending on the frame size and the number of lag times. See Figure 2 for typical run times.</li>
	<li>Inspect the returned data, which will be in a data structure from the xarray package known as a Dataset. This data structure is stored under the attribute ddm_dataset. 
	NOTE: Not only the DDM matrix but also associated variables and metadata will be stored in this data structure. It will also be saved to disk in a Network Common Data Form (netCDF) format.</li>
	<li>Inspect the plots and figures, which will be generated and displayed. These figures are also saved as a PDF file in the data directory.
	<ol>
		<li>See that one of the generated plots shows the ensemble-averaged squared modulus of the Fourier-transformed images, <img alt="Equation031" src="/files/ftp_upload/63931/63931eq031.png" style="margin-bottom: 0px;" />&#160;as a function of q. By default, the code uses this to estimate the background parameter B. Estimate the background from <img alt="Equation031" src="/files/ftp_upload/63931/63931eq031.png" style="margin-bottom: 0px;" /> by assuming that, in the limit of large q, it will approach B/2, where B is the background.</li>
		<li>If <img alt="Equation031" src="/files/ftp_upload/63931/63931eq031.png" style="margin-bottom: 0px;" /> is not reaching a plateau at large q, then use another method for estimating B. To accomplish this, set the parameter background_method in either the YAML file or as an optional keyword argument to the function calculate_DDM_matrix. More details about the methods for estimating B are presented in the representative results section.</li>
	</ol>
	</li>
</ol>
<img alt="Figure 2" class="xfigimg" src="/files/ftp_upload/63931/63931fig2v2.jpg" /> 
Figure 2: Computation time for calculating the DDM matrix. In (A) and (B) the time for calculating the DDM matrix, <img alt="Equation009" src="/files/ftp_upload/63931/63931eq009.png" style="margin-bottom: 0px;" />, is shown. The data used in all cases is a movie of 5000 frames with an image size of 512 x 512 pixels. The DDM matrix was calculated for 30 lag times, logarithmically spaced between 1 frame (0.01 s) and 1000 frames (10 s). The code was run on an Intel i7-10700 2.90 GHz desktop computer with 32 GB RAM. In (A), the effect of varying how many image differences are used in computing the DDM matrix for each lag time is shown. For this, the images are binned to result in an image size of 256 x 256. For each lag time &#916;t, images separated by that&#160;&#916;t are subtracted and the resulting matrix is Fourier transformed. For a given &#916;t, all pairs of images separated by&#160;that&#160;&#916;t can be used (shown in blue), only non-overlapping image pairs can be used (e.g., frames 1 and 10, 10 and 19, etc.; shown in brown), or 300 image pairs or fewer can be used for each &#916;t. In (B), the effect of changing the image size on the computation time is shown. The images were binned either by grouping 2 x 2, 4 x 4, or 8 x 8 pixels, resulting in image sizes of 256 x 256, 128 x 128, or 64 x 64, respectively. For each, about 300 image pairs are used in computing the DDM matrix for each &#916;t. (C) From the DDM matrix, the intermediate scattering function (ISF) can be extracted. This is shown for the three cases in (A). The blue data points (with no offset) correspond to the ISF when the maximum number of image pairs are used for each &#916;t; the brown data points (with an offset of 0.1) correspond to the ISF when non-overlapping image pairs are used for each &#916;t; and the pink data points (with an offset of 0.2) correspond to the ISF when at most 300 image pairs are used for each &#916;t. The ISF found using non-overlapping image pairs shows noisiness at long &#916;t. For that case, few image pairs are used at long&#160;&#916;t (e.g., for&#160;&#916;t of 1000 frames, only 4 image pairs are used). (D) By fitting the ISF to an exponential function, the characteristic decay time, &#964;, for each wavenumber, q, is determined. In pink, results are shown after binning the original images by 2 x 2, resulting in an image size of 256 x 256. In gray, results are shown after binning by 8 x 8, resulting in an image size of 64 x 64. By binning the data, information about the dynamics at higher wavenumbers is lost, but calculating the DDM matrix for the 64 x 64 images is about 16x faster than for the 256 x 256 images. <a href="https://www.jove.com/files/ftp_upload/63931/63931fig2v2large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
6. Fitting the DDM matrix or the ISF
<ol>
	<li>Initialize an instance of the DDM_Fit class. To do so, pass to DDM_Fit the filename of the YAML file containing the image metadata and parameters for fitting.</li>
	<li>Decide which model for the DDM matrix or the ISF to use for fitting the data. List the available models by executing the function print_fitting_models. Specify the model to be used in the YAML parameter file or by using the function reload_fit_model_by_name.</li>
	<li>Set the initial guesses and bounds for each parameter in the chosen model in the provided YAML parameter file. To change the initial guess for any parameter, use the function set_parameter_initial_guess. Set bounds for the parameters with the function set_parameter_bounds. For example, as seen in Supplementary File 2, for the data of tracer beads in the vimentin network, the initial guess for the decay time was 1 s and the bounds on that parameter were 0.01 s and 2000 s.</li>
	<li>Execute the fit with the function fit. Assign a variable to the output of this function to easily access the results. 
	&#8203;NOTE: This function can take many optional arguments. See the code documentation and provided examples for a list of such arguments and when to consider setting them to non-default values.</li>
</ol>
7. Interpreting the fit results
<ol>
	<li>Generate plots for inspecting the fits and the q-dependence of the fit parameters with the function fit_report. 
	NOTE: This function will generate a series of plots, which will also be saved as a PDF. Optional arguments to this function can be used to modify the plots produced.</li>
	<li>Among the generated plots will be a figure with 2 x 2 subplots showing the DDM matrix or ISF (depending on the chosen fitting model) at four q-values (specified as an optional argument to fit_report), along with the calculated DDM matrix or ISF using the model and best fit parameters. To plot the DDM matrix or ISF along with the best fit in an interactive way, use the class Browse_DDM_Fits as shown in the provided examples when the Jupyter Notebook environment is used.</li>
	<li>From the plot of the characteristic decay time&#160;&#964; vs. the wavenumber q, determine whether the dynamics follow diffusive, subdiffusive, ballistic, or some other type of motion. This can be done by looking for the power law relationship between&#160;&#964; and q. 
	&#8203;NOTE: On the log-log plot of&#160;&#964; vs. q generated by the function fit_report, three lines will be shown, corresponding to power law fits over a specified range of&#160;q values. The solid black line corresponds to fitting &#964; vs. q to a power law, &#964; = 1 / Kq&#946;, where K and&#160;&#946; are free parameters. The dashed line in orange corresponds to fitting to simple diffusion, &#964; = 1 / Dq2, where D is a diffusion coefficient. The dot-dashed line in blue corresponds to fitting to &#964; = 1 / vq, where v is a velocity.</li>
</ol>
8. Saving the results
<ol>
	<li>The results of the fit will be saved in a xarray dataset. Use the xarray function to_netcdf or Python&#39;s built-in pickle module to save this data structure to disk. Use the xarray function open_dataset to load these netCDF files.</li>
	<li>Use the function save_fit_results_to_excel to save the fit results, along with the data, to a worksheet file.</li>
</ol>

The authors have nothing to disclose.

The software package described here uses DDM to analyze density fluctuations observed in images acquired using an optical microscope. Representative results from the data of tracer particles embedded in vimentin networks were first shown. The analysis of such data can be used to characterize the mesh size and stiffness of the network similarly to how single-particle tracking has been used in many past studies of cytoskeleton networks6,12,13. An advantage of using DDM over single-particle tracking is that DDM does not require the particles to be localized. Therefore, even in images where the particle density is too high or the particles too small to localize and track, DDM can still determine the dynamics. Where single-particle tracking would be advantageous is when inspecting particle-to-particle variability. With DDM, one finds the ensemble averaged dynamics whereas, with single-particle tracking, one may compute both a single particle&#39;s MSD and the ensemble-averaged MSD. However, DDM can be used to investigate heterogeneous dynamics by analyzing multiple regions of interest within a large field of view.
Next, representative results from data of fluorescently labeled filaments in an active network composed of two differently labeled cytoskeletal filament types were shown11. With this data, the ballistic motion was characterized without needing any localizable features within the image. Since DDM extracts the ensemble averaged dynamics with few user inputs, it makes comparing image series acquired with different conditions straightforward (e.g., comparing samples with different ratios of actin to microtubules or samples with different concentrations of myosin, as done in50). Additionally, using fluorescent imaging, we can investigate the dynamics of different components of a network using multicolor labeling. This was done in11,50, where the dynamics of actin and of microtubules were separately analyzed in an active actin-microtubule composite network using multicolor imaging. In the representative results section here, just the results from the microtubule channel were shown, but in previous work, we compared the dynamics of the microtubule and actin filaments11.
We note that these representative results showcase either passive subdiffusion or active ballistic motion. Importantly, DDM can be used to analyze systems where there is a crossover in the type of dynamics at intermediate time or length scales. As examples, Kurzthaler et al. used DDM with a system of active Janus colloids to explore active directed motion at short time scales and randomization of the orientation at longer time scales59; Giavazzi et al. used DDM with a coarsening foam and found a crossover in the dynamics corresponding to the length scale of a bubble33; and Cho et al. used DDM with colloidal gels and found three distinguishable regimes at different length scales spanning from the fractal clusters to the whole network32.
The data included in the representative results section were acquired with brightfield microscopy and laser-scanning confocal microscopy. However, as previously noted, DDM can be used with many imaging modalities. With any imaging modality, users should consider optical settings such as the degree of optical sectioning or the depth of field. A high degree of optical sectioning may reduce signal from out-of-focus objects, but one will not be able to accurately measure dynamics over timescales greater than the timescale for objects to move out of the depth of field25,28. A more thorough discussion of how the q-dependent depth of field affects DDM analysis can be found in22. For brightfield imaging, users may also need to consider the sample thickness. While for weakly scattering samples, thicker samples may provide more signal42, turbid samples may require modifying the analysis to account for multiple scattering81. Finally, for imaging methods that are not linear space invariant (that is, where the intensity recorded by the camera of an object depends on where that object is in the x-y sample plane), one may need to account for the linear space variance, as demonstrated with dark-field DDM27.
For those getting started with DDM, we wish to emphasize the importance of considering the spatial and temporal resolution. When inspecting the determined decay times as a function of wavenumber, it is important to mark the limits of one&#39;s resolution (i.e., the maximum and minimum lag times and the maximum wavenumber, as done in Figure 5). One should think carefully about these limits prior to collecting data so that the optimal objective lens, image size, frame rate, and movie duration can be selected. The other important consideration is how to estimate the background parameter B. Multiple methods for estimating the background have been used in the literature, and the effects of over- or underestimating B have been described in prior publications62,77. As shown in Figure 10, PyDDM allows users to implement different methods for estimating B, and we suggest new users try these methods and evaluate which are appropriate to use.
A strength of this package is its thorough documentation and walkthroughs featuring example data, the storage and organization of metadata to keep track of how analyses were performed, and the flexibility in how to analyze the DDM matrix (various fitting models, multiple methods for estimating the background parameter B, the ability to find the MSD). However, there are multiple aspects to this code that could be improved. Currently, the code has not been optimized for fast computation speed. Methods for speeding up the computation have been reported61,62, and these will be implemented in future releases. Additionally, we plan to implement recently reported methods to better estimate uncertainties and to employ simulations for guiding users to the appropriate ISF model62. For other improvements, we hope users will contact us with suggestions.

The cytoskeleton is a network of protein filaments that spans across the cytoplasm of eukaryotic cells, connecting the cell surface to the nucleus. It has unique material properties, providing mechanical protection against large and repeated mechanical loads, yet also driving dynamic cell shape changes1. Reconstituted cytoskeleton networks can give rise to a range of interesting dynamic behaviors, from the caging of embedded particles to ballistic motion driven by molecular motors2,3,4,5,6,7,8,9,10,11. Methods to analyze the dynamics of such networks include tracking the motion of embedded tracer microspheres6,7,12,13,14, image analysis to track the size of protein-dense clusters over time8, dynamic light scattering15, particle image velocimetry4,16,17,18,19, computing the power spectral density of images over time19, and kymograph analysis20. As more studies on reconstituted cytoskeleton networks are conducted, whether to understand cellular mechanics or active matter, robust, unbiased, and reproducible methods for characterizing the dynamics are increasingly necessary. Differential dynamic microscopy (DDM)21,22, a relatively new technique that has been used to study cytoskeleton dynamics, is one such technique that efficiently quantifies dynamics with few user-defined parameters. With the software package described here, researchers with little experience in programming or image analysis will be able to leverage DDM for their own work.
DDM is an image analysis technique to extract a sample&#39;s dynamics. Like particle tracking or particle image velocimetry, DDM requires a time series of images (often thousands of images), typically recorded with a microscope. Unlike particle tracking, individual features or tracer beads do not need to be localized (or even be localizable) in the image. Unlike both particle tracking and particle image velocimetry, one recovers the ensemble dynamics with DDM with relatively few user-specified parameters. With DDM, images are analyzed in Fourier space to determine the decay time of density fluctuations&#160;over a range of wavenumbers, q, where q = 2&#960;u, and u is the magnitude of the spatial frequencies, <img alt="Equation004" src="/files/ftp_upload/63931/63931eq004.png" />. One obtains scattering-like information but with real-space images acquired on a microscope21,22,23. Therefore, one can take advantage of the various contrast-generating methods of microscopy, such as widefield fluorescence22,24, confocal fluorescence25, polarized26, dark-field27, or light-sheet fluorescence28 microscopies. Furthermore, images used for DDM analysis may be used for particle tracking or particle image velocimetry to provide complementary information.
This combination of features from dynamic light scattering and optical microscopy makes DDM a powerful and versatile technique. Since its first description by Cerbino and Trappe in 200821, where DDM was demonstrated to measure the diffusion of 73 nm colloidal particles, DDM has been used to measure flowing colloids29, colloidal aggregation30,31, the viscoelasticity of nematic liquid crystals26, the dynamics of colloidal gels32, coarsening foams33, nanoparticles in confined environments34,35,36,37, bacterial motility38,39,40,41, the diffusion of weakly scattering protein clusters42, capillary waves at fluid interfaces43, and other systems. Those looking for a more complete listing of publications employing DDM can refer to thorough review papers on the subject22,23,44,45.
DDM has also been used to investigate the dynamics of biological networks. Drechsler et al. used DDM to measure the dynamics of actin in living Drosophila oocytes46. Burla et al. quantified the dynamics of tracer particles in networks of hyaluronan and hyaluronan-collagen composites47. Several uses of DDM to study the dynamics of tracer particles in reconstituted cytoskeleton networks9,10, the transport of DNA molecules in such networks48,49, and the dynamics of active reconstituted networks have also been documented11,50,51. An advantage of DDM in measuring the dynamics in such systems is that individual particles or molecules do not need to be localized and tracked. So, for example, the dynamics of DNA molecules in crowded environments can be measured with DDM despite the difficulty in tracking such small and non-spherical molecules. Furthermore, with fluorescence microscopy, one can use multi-color labeling to selectively measure the dynamics of individual constituents in a complex composite.
To perform DDM, a sequence of images is taken over time, I(x,y,t). For a given lag time, &#916;t, all (or a subset of) pairs of images separated by that lag time are found. The squared Fourier transform of the difference of each pair,
<img alt="Equation007" src="/files/ftp_upload/63931/63931eq007.png" style="margin-bottom: 0px;" />
is computed and averaged together. This quantity, <img alt="Equation008" src="/files/ftp_upload/63931/63931eq008.png" style="margin-bottom: 0px;" />, is radially averaged, provided that the dynamics are isotropic. This yields the DDM matrix (also referred to as the image structure function), <img alt="Equation009" src="/files/ftp_upload/63931/63931eq009.png" style="margin-bottom: 0px;" />. This process is shown graphically in Figure 1. To determine the sample&#39;s dynamics from this DDM matrix, the DDM matrix is assumed to take the form
<img alt="Equation010" src="/files/ftp_upload/63931/63931eq010.png" style="margin-bottom: 0px;" />
where A is the amplitude, which depends on the details of the microscope and the structure of the sample, B is the background, which depends on the noise in the images, and f&#160;(q, &#916;t) is the intermediate scattering function (ISF), which contains information about the dynamics21,22. In simple cases,
<img alt="Equation014" src="/files/ftp_upload/63931/63931eq014.png" style="margin-bottom: 0px;" />
where&#160;&#964; is a characteristic decay or decorrelation time. Such an ISF has been used in several studies employing DDM on ergodic systems like dilute colloidal suspensions21,24,27,37,40,52. However, other forms of the ISF can be used to model various types of dynamics. For example, one may use a cumulant expansion to model the ISF for polydisperse samples as
<img alt="Equation016" src="/files/ftp_upload/63931/63931eq016.png" style="margin-bottom: 0px;" />
where&#160;&#956; is a measure of the polydispersity42,53; if density fluctuations decay by two separate modes one may use an ISF like
<img alt="Equation018" src="/files/ftp_upload/63931/63931eq018.png" style="margin-bottom: 0px;" /> 26, 54, 55, 56, 57;
other ISFs can be used for swimming micro-organisms or other active particles38,39,40,41,58,59.
<img alt="Figure 1" class="xfigimg" src="/files/ftp_upload/63931/63931fig1v2.jpg" /> 
Figure 1: Overview of DDM analysis. From the time series of images, the Fourier transform of image differences is computed to calculate the DDM matrix. The DDM matrix can be fit to a model to determine the time scale of density fluctuations across a range of q values. <a href="https://www.jove.com/files/ftp_upload/63931/63931fig1v2large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
Here, the use of a DDM analysis software package developed in Python, PyDDM, is described. This software package builds on the work done by our research labs and other published studies over the last several years. Primary motivators for creating this software package include the need for (1) keeping track of and storing metadata and parameters used in the analysis; (2) thorough documentation with detailed examples of analysis from start to finish; and (3) an easy way to employ different (or create new) mathematical models for fitting the data (e.g., adding ISF models, such as ones recently developed for active filaments60, would be straightforward). Other software packages for DDM analysis also exist, though not all are well documented and written in an open-source programming language. For example, there is C++ code with computing on GPUs (https://github.com/peterlu/ConDDM)25, C++ code that uses Fourier transforms in time to speed up calculations (https://github.com/giovanni-cerchiari/diffmicro)61, MATLAB and Python versions (https://github.com/MathieuLeocmach/DDM)40, MATLAB code (https://sites.engineering.ucsb.edu/~helgeson/ddm.html)27, and MATLAB code with uncertainty quantification (https://github.com/UncertaintyQuantification/DDM-UQ)62. As this PyDDM package is well documented and provides a lot of flexibility in how the DDM matrix is calculated and analyzed, it can hopefully be useful to researchers looking to implement DDM regardless of their background in programming or image analysis.&#160;
The protocol shows how this software package can be used to quantify the dynamics of in vitro reconstituted cytoskeleton networks. This is done by using two distinct sets of imaging data: (1) images of submicron tracer particles embedded in a vimentin network taken with brightfield microscopy and (2) images of fluorescently labeled actin and microtubule filaments in an entangled composite network with myosin-driven activity taken with laser-scanning confocal microscopy. The analyses of these two datasets highlight notable strengths of DDM, including its ability to analyze images taken with a variety of imaging modalities (e.g., brightfield or confocal fluorescence), to extract dynamics from either embedded tracers or from labeled filaments, and to quantify a variety of dynamics (e.g., subdiffusive and constrained or ballistic).

<table><tbody><tr><td>CMOS camera, Orca-Flash 4.0</td><td>Hamatsu</td><td></td><td></td></tr><tr><td>F-127 Pluronic</td><td>Sigma Aldrich</td><td></td><td></td></tr><tr><td>Jupyter Notebook</td><td></td><td></td><td></td></tr><tr><td>Nanodrop</td><td>Thermo Fisher</td><td></td><td></td></tr><tr><td>Nikon Ti-Eclipse microscope</td><td>Nikon</td><td></td><td></td></tr><tr><td>PLL-PEG-bio</td><td>SuSos AG, D&amp;uuml;bendorf, Switzerland</td><td></td><td></td></tr><tr><td>Polystyrene beads</td><td>Sigma Aldrich</td><td></td><td></td></tr><tr><td>Protein dialysis mini-cassette</td><td>Thermo Fisher</td><td></td><td></td></tr><tr><td>PyDDM</td><td>University of San Diego</td><td>N/A</td><td>Open source software available from https://github.com/rmcgorty/PyDDM</td></tr></tbody></table>

quantifying cytoskeleton dynamics using differential dynamic microscopy

Cells can crawl, self-heal, and tune their stiffness due to their remarkably dynamic cytoskeleton. As such, reconstituting networks of cytoskeletal biopolymers may lead to a host of active and adaptable materials. However, engineering such materials with precisely tuned properties requires measuring how the dynamics depend on the network composition and synthesis methods. Quantifying such dynamics is challenged by variations across the time, space, and formulation space of composite networks. The protocol here describes how the Fourier analysis technique, differential dynamic microscopy (DDM), can quantify the dynamics of biopolymer networks and is particularly well suited for studies of cytoskeleton networks. DDM works on time sequences of images acquired using a range of microscopy modalities, including laser-scanning confocal, widefield fluorescence, and brightfield imaging. From such image sequences, one can extract characteristic decorrelation times of density fluctuations across a span of wave vectors. A user-friendly, open-source Python package to perform DDM analysis is also developed. With this package, one can measure the dynamics of labeled cytoskeleton components or of embedded tracer particles, as demonstrated here with data of intermediate filament (vimentin) networks and active actin-microtubule networks. Users with no prior programming or image processing experience will be able to perform DDM using this software package and associated documentation.

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Quantifying Cytoskeleton Dynamics Using Differential Dynamic Microscopy

Differential dynamic microscopy (DDM) combines features of dynamic light scattering and microscopy. Here, the process of using DDM to characterize reconstituted cytoskeleton networks by quantifying the subdiffusive and caged dynamics of particles in vimentin networks and the ballistic motion of active myosin-driven actin-microtubule composites is presented.

Cells can crawl, self-heal, and tune their stiffness due to their remarkably dynamic cytoskeleton. As such, reconstituting networks of cytoskeletal ...

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Research

JoVE Journal

Bioengineering

3.4K Views.  Utrecht University. Ons protocol kan de dynamiek in veel systemen kwantificeren met behulp van een reeks optische microscopietechnieken. We benadrukken hoe deze methode in het bijzonder kan helpen bij het karakteriseren van de dynamiek van gereconstitueerde cytoskeletnetwerken. Het belangrijkste voordeel van het gebruik van ons softwarepakket Differential Dynamic Microscopy is dat het goed gedocumenteerd is, meerdere voorbeeldanalysebestanden heeft en gemakkelijk kan worden aangepast om verschillende soorten dyn...

Video: Quantifying Cytoskeleton Dynamics Using Differential Dynamic Microscopy

Simultaneous Visualization of the Dynamics of Crosslinked and Single Microtubules In Vitro by TIRF Microscopy

Microtubules are polymers of &#945;&#946;-tubulin heterodimers that organize into distinct structures in cells. Microtubule-based architectures and networks often contain subsets of microtubule arrays that differ in their dynamic properties. For example, in dividing cells, stable bundles of crosslinked microtubules coexist in close proximity to dynamic non-crosslinked microtubules. TIRF-microscopy-based in vitro reconstitution studies enable the simultaneous visualization of the dynamics of these different microtubule arrays. In this assay, an imaging chamber is assembled with surface-immobilized microtubules, which are either present as single filaments or organized into crosslinked&#160;bundles. Introduction of tubulin, nucleotides, and protein regulators allows direct visualization of associated proteins and of dynamic properties of single and crosslinked microtubules. Furthermore, changes that occur as dynamic single microtubules organize into bundles can be monitored in real-time. The method described here allows for a systematic evaluation of the activity and localization of individual proteins, as well as synergistic effects of protein regulators on two different microtubule subsets under identical experimental conditions, thereby providing mechanistic insights that are inaccessible by other methods.

Simultaneous Visualization of the Dynamics of Crosslinked and Single Microtubules In Vitro by TIRF Microscopy

Here, a TIRF microscopy-based in vitro reconstitution assay is presented to simultaneously quantify and compare the dynamics of two microtubule populations. A method is described to simultaneously view the collective activity of multiple microtubule-associated proteins on crosslinked microtubule bundles and single microtubules.

Microtubules are polymers of &#945;&#946;-tubulin heterodimers that organize into distinct structures in cells. Microtubule-based architectures and ...

Biochemistry

Visualizing Actin and Microtubule Coupling Dynamics In Vitro by Total Internal Reflection Fluorescence (TIRF) Microscopy

Traditionally, the actin and microtubule cytoskeletons have been studied as separate entities, restricted to specific cellular regions or processes, and regulated by different suites of binding proteins unique for each polymer. Many studies now demonstrate that the dynamics of both cytoskeletal polymers are intertwined and that this crosstalk is required for most cellular behaviors. A number of proteins involved in actin-microtubule interactions have already been identified (i.e., Tau, MACF, GAS, formins, and more) and are well characterized with regard to either actin or microtubules alone. However, relatively few studies showed assays of actin-microtubule coordination with dynamic versions of both polymers. This may occlude emergent linking mechanisms between actin and microtubules. Here, a total internal reflection fluorescence (TIRF) microscopy-based in vitro reconstitution technique permits the visualization of both actin and microtubule dynamics from the one biochemical reaction. This technique preserves the polymerization dynamics of either actin filament or microtubules individually or in the presence of the other polymer. Commercially available Tau protein is used to demonstrate how actin-microtubule behaviors change in the presence of a classic cytoskeletal crosslinking protein. This method can provide reliable functional and mechanistic insights into how individual regulatory proteins coordinate actin-microtubule dynamics at a resolution of single filaments or higher-order complexes.

Visualizing Actin and Microtubule Coupling Dynamics In Vitro by Total Internal Reflection Fluorescence TIRF Microscopy

This protocol is a guide for visualizing dynamic actin and microtubules using an in vitro total internal fluorescence (TIRF) microscopy assay.

Traditionally, the actin and microtubule cytoskeletons have been studied as separate entities, restricted to specific cellular regions or processes, and ...

Visualizing Cytoskeleton-Dependent Trafficking of Lipid-Containing Organelles in Drosophila Embryos

Early Drosophila embryos are large cells containing a vast array of conventional and embryo-specific organelles. During the first three hours of embryogenesis, these organelles undergo dramatic movements powered by actin-based cytoplasmic streaming and motor-driven trafficking along microtubules. The development of a multitude of small, organelle-specific fluorescent probes (FPs) makes it possible to visualize a wide range of different lipid-containing structures in any genotype, allowing live imaging without requiring a genetically encoded fluorophore. This protocol shows how to inject vital dyes and molecular probes into Drosophila embryos to monitor the trafficking of specific organelles by live imaging. This approach is demonstrated by labeling lipid droplets (LDs) and following their bulk movement by particle image velocimetry (PIV). This protocol provides a strategy amenable to the study of other organelles, including lysosomes, mitochondria, yolk vesicles, and the ER, and for tracking the motion of individual LDs along microtubules. Using commercially available dyes brings the benefits of separation into the violet/blue and far-red regions of the spectrum. By multiplex co-labeling of organelles and/or cytoskeletal elements via microinjection, all the genetic resources in Drosophila are available for trafficking studies without the need to introduce fluorescently tagged proteins. Unlike genetically encoded fluorophores, which have low quantum yields and bleach easily, many of the available dyes allow for rapid and simultaneous capture of several channels with high photon yields.

Visualizing Cytoskeleton-Dependent Trafficking of Lipid-Containing Organelles in Drosophila Embryos

In the early Drosophila embryo, many organelles are motile. In principle, they can be imaged live via specific fluorescent probes, but the eggshell prevents direct application to the embryo. This protocol describes how to introduce such probes via microinjection, and then analyze bulk organelle motion via particle image velocimetry.

Early Drosophila embryos are large cells containing a vast array of conventional and embryo-specific organelles. During the first three hours of ...

Developmental Biology

Microtubule Plus-End Dynamics Visualization in Huntington's Disease Model based on Human Primary Skin Fibroblasts

Transfection with a fluorescently labeled marker protein of interest in combination with time-lapse video microscopy is a classic method of studying the dynamic properties of the cytoskeleton. This protocol offers a technique for human primary fibroblast transfection, which can be difficult because of the specifics of primary cell cultivation conditions. Additionally, cytoskeleton dynamic property maintenance requires a low level of transfection to obtain a good signal-to-noise ratio without causing microtubule stabilization. It is important to take measures to protect the cells from light-induced stress and fluorescent dye fading. In the course of our work, we tested different transfection methods and protocols as well as different vectors to select the best combination of conditions suitable for human primary fibroblast studies. We analyzed the resulting time-lapse videos and calculated microtubule dynamics using ImageJ. The dynamics of microtubules' plus-ends in the different cell parts are not similar, so we divided the analysis into subgroups - the centrosome region, the lamella, and the tail of fibroblasts. Notably, this protocol can be used for in vitro analysis of cytoskeleton dynamics in patient samples, enabling the next step towards understanding the dynamics of the various disease development.

This protocol is dedicated to the microtubule plus-end visualization by EB3 protein transfection to study their dynamic properties in primary cell culture. The protocol was implemented on human primary skin fibroblasts obtained from Huntington's disease patients.

Transfection with a fluorescently labeled marker protein of interest in combination with time-lapse video microscopy is a classic method of studying the ...

High-resolution Imaging and Analysis of Individual Astral Microtubule Dynamics in Budding Yeast

Dynamic microtubules are fundamental to many cellular processes, and accurate measurements of microtubule dynamics can provide insight into how cells regulate these processes and how genetic mutations impact regulation. The quantification of microtubule dynamics in metazoan models has a number of associated challenges, including a high microtubule density and limitations on genetic manipulations. In contrast, the budding yeast model offers advantages that overcome these challenges. This protocol describes a method to measure the dynamics of single microtubules in living yeast cells. Cells expressing fluorescently tagged tubulin are adhered to assembled slide chambers, allowing for stable time-lapse image acquisition. A detailed guide for high-speed, four-dimensional image acquisition is also provided, as well as a protocol for quantifying the properties of dynamic microtubules in confocal image stacks. This method, combined with conventional yeast genetics, provides an approach that is uniquely suited for quantitatively assessing the effects of microtubule regulators or mutations that alter the activity of tubulin subunits.

Budding yeast is an advantageous model for studying microtubule dynamics in vivo due to its powerful genetics and the simplicity of its microtubule cytoskeleton. The following protocol describes how to transform and culture yeast cells, acquire confocal microscopy images, and quantitatively analyze microtubule dynamics in living yeast cells.

Dynamic microtubules are fundamental to many cellular processes, and accurate measurements of microtubule dynamics can provide insight into how cells ...

Biology

Micromanipulation Techniques Allowing Analysis of Morphogenetic Dynamics and Turnover of Cytoskeletal Regulators

Examining the spatiotemporal dynamics of proteins can reveal their functional importance in various contexts. In this article, it is discussed how fluorescent recovery after photobleaching (FRAP) and photoactivation techniques can be used to study the spatiotemporal dynamics of proteins in subcellular locations. We also show how these techniques enable straightforward determination of various parameters linked to actin cytoskeletal regulation and cell motility. Moreover, the microinjection of cells is additionally described as an alternative treatment (potentially preceding or complementing the aforementioned photomanipulation techniques) to trigger instantaneous effects of translocated proteins on cell morphology and function. Micromanipulation such as protein injection or local application of plasma membrane-permeable drugs or cytoskeletal inhibitors can serve as powerful tool to record immediate consequences of a given treatment on cell behavior at the single cell and subcellular level. This is exemplified here by immediate induction of lamellipodial cell edge protrusion by the injection of recombinant Rac1 protein, as established a quarter-century ago. In addition, we provide a protocol for determining the turnover of enhanced green fluorescent protein (EGFP)-VASP, an actin filament polymerase prominently accumulating at lamellipodial tips of B16-F1 cells, employing FRAP and including associated data analysis and curve fitting. We also present guidelines for estimating the rates of lamellipodial actin network polymerization, as exemplified by cells expressing EGFP-tagged &#946;-actin. Finally, instructions are given for how to investigate the rates of actin monomer mobility within the cell cytoplasm, followed by actin incorporation at sites of rapid filament assembly, such as the tips of protruding lamellipodia, using photoactivation approaches. None of these protocols is&#160;restricted to components or regulators of the actin cytoskeleton, but can easily be extended to explore in analogous fashion the spatiotemporal dynamics and function of proteins in various different subcellular structures or functional contexts.

We describe how micro- and photomanipulation techniques such as FRAP and photoactivation enable the determination of motility parameters and the spatiotemporal dynamics of proteins within migrating cells. Experimental readouts include subcellular dynamics and turnover of motility regulators or of the underlying actin cytoskeleton.

Examining the spatiotemporal dynamics of proteins can reveal their functional importance in various contexts. In this article, it is discussed how ...

Measurement of Microtubule Dynamics by Spinning Disk Microscopy in Monopolar Mitotic Spindles

We describe a modification of an established method to determine microtubule dynamics in living cells. The protocol is based on the expression of a genetically encoded marker for the positive ends of microtubules (EB3 labelled with tdTomato fluorescent protein) and high-speed, high-resolution, live-cell imaging using spinning disk confocal microscopy. Cell cycle synchronization and increased density of microtubules are achieved by inhibiting centrosomal separation in mitotic cells, and analysis of growth is performed using open-source U-Track software. The use of a bright and red-shifted fluorescent protein, in combination with the lower laser power and reduced exposure time required for spinning disk microscopy reduce phototoxicity and the probability of light-induced artifacts. This allows for imaging a larger number of cells in the same preparation while maintaining the cells in a growth medium under standard culture conditions. Because the analysis is performed in a supervised automatic fashion, the results are statistically robust and reproducible.

Here we present a robust and detailed method of microtubule dynamics analysis in cells synchronized in prometaphase using live-cell spinning disk confocal microscopy and MATLAB-based image processing.

We describe a modification of an established method to determine microtubule dynamics in living cells. The protocol is based on the expression of a ...

Probing Structural and Dynamic Properties of Trafficking Subcellular Nanostructures by Spatiotemporal Fluctuation Spectroscopy

Imaging-derived mean square displacement (iMSD) is used to address the structural and dynamic properties of subcellular nanostructures, such as vesicles involved in the endo/exocytotic trafficking of solutes and biomolecules. iMSD relies on standard time-lapse imaging, is compatible with any optical setup, and does not need to dwell on single objects to extract trajectories. From each iMSD trace, a unique triplet of average structural and dynamic parameters (i.e., size, local diffusivity, anomalous coefficient) is calculated and combined to build the "iMSD signature" of the nanostructure under study. 
The potency of this approach is proved here with the exemplary case of macropinosomes. These vesicles evolve in time, changing their average size, number, and dynamic properties passing from early to late stages of intracellular trafficking. As a control, insulin secretory granules (ISGs) are used as a reference for subcellular structures that live in a stationary state in which the average structural and dynamic properties of the whole population of objects are invariant in time. The iMSD analysis highlights these peculiar features quantitatively and paves the way to similar applications at the sub-cellular level, both in the physiological and pathological states.

Imaging-derived mean square displacement (iMSD) analysis is applied to macropinosomes to highlight their intrinsic time-evolving nature in terms of structural and dynamic properties. Macropinosomes are then compared to insulin secretory granules (ISGs) as a reference for subcellular structures with time-invariant average structural/dynamic properties.

Imaging-derived mean square displacement (iMSD) is used to address the structural and dynamic properties of subcellular nanostructures, such as vesicles ...

Measuring Cell-Edge Protrusion Dynamics during Spreading using Live-Cell Microscopy

The development and homeostasis of multicellular organisms rely on coordinated regulation of cell migration. Cell migration is an essential event in the construction and regeneration of tissues, and is critical in embryonic development, immunological responses, and wound healing. Dysregulation of cell motility contributes to pathological disorders, such as chronic inflammation and cancer metastasis. Cell migration, tissue invasion, axon, and dendrite outgrowth all initiate with actin polymerization-mediated cell-edge protrusions. Here, we describe a simple, efficient, time-saving method for the imaging and quantitative analysis of cell-edge protrusion dynamics during spreading. This method measures discrete features of cell-edge membrane dynamics, such as protrusions, retractions, and ruffles, and can be used to assess how manipulations of key actin regulators impact cell-edge protrusions in diverse contexts.

This protocol aims to measure the dynamic parameters (protrusions, retractions, ruffles) of protrusions at the edge of spreading cells.

The development and homeostasis of multicellular organisms rely on coordinated regulation of cell migration. Cell migration is an essential event in the ...

Reconstituting and Characterizing Actin-Microtubule Composites with Tunable Motor-Driven Dynamics and Mechanics

The composite cytoskeleton, comprising interacting networks of semiflexible actin filaments and rigid microtubules, restructures and generates forces using motor proteins such as myosin II and kinesin to drive key processes such as migration, cytokinesis, adhesion, and mechanosensing. While actin-microtubule interactions are key to the cytoskeleton's versatility and adaptability, an understanding of their interplay with myosin and kinesin activity is still nascent. This work describes how to engineer tunable three-dimensional composite networks of co-entangled actin filaments and microtubules that undergo active restructuring and ballistic motion, driven by myosin II and kinesin motors, and are tuned by the relative concentrations of actin, microtubules, motor proteins, and passive crosslinkers. Protocols for fluorescence labeling of the microtubules and actin filaments to most effectively visualize composite restructuring and motion using multi-spectral confocal imaging are also detailed. Finally, the results of data analysis methods that can be used to quantitatively characterize non-equilibrium structure, dynamics, and mechanics are presented. Recreating and investigating this tunable biomimetic platform provides valuable insight into how coupled motor activity, composite mechanics, and filament dynamics can lead to myriad cellular processes from mitosis to polarization to mechano-sensation.

This paper presents protocols for engineering and characterizing tunable three-dimensional composite networks of co-entangled actin filaments and microtubules. Composites undergo active restructuring and ballistic motion, driven by myosin II and kinesin motors, and are tuned by the relative concentrations of actin, microtubules, motor proteins, and passive crosslinkers.

The composite cytoskeleton, comprising interacting networks of semiflexible actin filaments and rigid microtubules, restructures and generates forces ...

Quantifying Cytoskeleton Dynamics Using Differential Dynamic Microscopy

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