Fig 1.
A schematic of the model and an example of a FRAP experiment.
Panel (a) shows the schematic representation of a cell migrating to the right with several cytoplasmic IFs. Selected velocities are denoted with purple arrows under the filaments. The right endpoints of the IFs are denoted x0 and one is labeled with its length ℓ and velocity v. The right endpoints are distributed between 0 and F. Panels (b)-(g) show data from a FRAP experiment. Panels (b)-(d) show the profile curves for (e)-(g) respectively. The curves in (b)-(d) come from data in a subregion of (e)-(g); for example the yellow box in (f). Panels (e)-(g) show fluorescence images taken from a FRAP experiment performed on vimentin-EGFP expressing astrocytes, located at the edge of a wound, 1h after wounding of the monolayer. The rear-to-front polarity axis similar to the direction of migration is indicated by an arrow in (a), (e) and (h). Data are before (b) and (e), just after (c) and (f), and 2 minutes after bleaching (d) and (g). Panels (h)-(j) are blowups of the region indicated by the yellow box in (f) and show the domain and setup for the simulations. For the clarity, the vertical coordinate of each filament stays the same. The photobleached region (shown in white in (i) and (j)) is between y0 and y1. Several filaments are shown with their initial right endpoint depicted by a dot in (h)-(j). The red filaments are not relevant for our mathematical analysis and simulations since we consider only right moving filaments. Of the remaining filaments, the green ones are unbleached and the blue ones are partially bleached. In (i) the squares denoted show where the new right endpoints will be after bleaching. In the mathematical analysis and simulations it is as if all the filaments to the right of y0 are bleached since we only consider right moving filaments.
Fig 2.
A diagram depicting the different theory and simulation types and cases solved theoretically.
In each panel, kymographs of three typical filaments are shown. Panels (a)-(c) depict the three types of simulations where all the filaments have random velocities and random lengths. Panel (a) shows type 1 where the filaments have random velocities which are constant for the duration of the simulation. Panel (b) shows type 2 simulations where the filaments have random velocities which instantaneously change at random times. Finally, (c) shows type 3 simulations where filaments have random velocities and exhibit a pausing behavior between velocity changes. Panel (d) shows the particular situation/case of Type 1 simulations theoretically solved in Eq (4) where all filaments have the same fixed velocity but random (time-independent) lengths. Panel (e) the particular situation/case of type 1 simulations theoretically resolved in Eq (5) where all the filaments have the same length but different (time-independent) velocities that are randomly selected.
Fig 3.
Impact of filament length on the density of filaments.
This figure shows how filament length affects density measurements of the initial setup (before bleaching, all filaments are fluorescent). The mean length of filaments in (a) is 50 microns and in (b) 0.5 microns. One filament is highlighted in cyan in (a). The red lines show the density of filaments computed from the data shown above (calculated using bin sizes of 0.1 in (a) and 0.09 in (b)), normalized using the maximum bin value (a) and average non-zero bin value (b), and the dashed blue line shows the theoretical density of filaments using Eq (4), where y0 is set to be greater than F (this is the one exception to the assumption that y0 ≤ F), thus there is no bleached region in the panels. The filaments’ right endpoints are uniformly distributed on [0, 200] and the filaments have lengths which come from a Gaussian distribution with standard deviation 5 in (a) and 0.05 in (b). Eq (4a) is for w-values less than 0, where the density is lower since the right endpoints of filaments are not initially placed to the left of 0 (see panel (a)). For panel (a), if F − w > 60 (the mean length plus 2 standard deviations), the w coordinate is far enough to the left of F = 200 that the boundary effects (due to placement of the right endpoints) do not affect density. If y0 is in the plateau region, the front of the traveling wave will be sharp. Different length distributions show the same qualitative features. Panel (c) shows the standard deviation of the filament density for different lengths of filaments and for four different values of total number of filaments. The simulations in (c) have filament lengths which are Gaussian distributions with varying length and the standard deviation is one tenth the length. The standard deviation is taken only for data on the plateau.
Fig 4.
Impact of velocity distributions and filament length on FRAP intensity profile curves from the theory and type 1 simulations.
The solid curves in (a)-(c) are plots of a scaled version of Eq (4) and in (d)-(l) they are a scaled version of Eq (5) such that ‖f‖∞ = 1. The x’s are simulated results. The mean velocity for all simulations is 1 micron per minute and for (g)-(l) the standard deviation of the velocity is 0.25. In (a)-(c) the velocity is fixed and the insets show the simulated profile curves for the entire domain before bleaching and the region to be bleached is shown in grey. The bleached region is 50–60, 120–130, and 150–160 respectively. In (d)-(f) the velocity is uniformly distributed, (g)-(i) have Gaussian distributions, and (j)-(l) have gamma distributions. Panels (d), (g), and (j) have filaments with length 10 microns; (e), (h), and (k) have filaments with length 0.5 microns; and (f), (i) and (l) shows the average of 50 simulations (each a different realization) with filament length 0.5 microns. In these simulations, each filament has a velocity which does not change for the duration of the simulation. Each simulation in (a)-(c) has 200,000 filaments and in (d)-(l) 20,000. The right endpoints are uniformly distributed in (a)-(c) from 0 to 100 and in (d)-(l) from 0 to 470. The bleached region in (d)-(l) is from 200 to 210. The curves and x’s represent times 0.5 (blue and red), 1 (light blue and orange), 3 (lighter blue and light orange), and 5 minutes (cyan and yellow). The y axis is fluorescence intensity (a.u.) in all panels. Length: Row one—Gaussian, μ = 50, σ = 5 microns. Rows 2,3, and 4—fixed.
Fig 5.
Impact of length distributions on FRAP intensity profile curve data (type 1).
This figure compares the simulated results with random velocity and different length distributions. In panel (a) the type of length distribution changes, in panels (c)-(e) the length changes, and in (f)-(h) the mean length changes. Panel (b) is a blowup of the boxed region in panel (a). In panel (a) the average length for all distributions is 10 microns and for the Gaussian and gamma distributions the standard deviation is 0.2. In panels (c)-(e) the solid curves are plots of scaled Eq (5) such that ‖f‖∞ = 1 and the error bars are type 1 simulated results. The length of the error line is twice the standard deviation of the 50 realizations centered at the average of the realizations. The curves and error bars in (c)-(e) are profiles at times 0.5 (blue and red), 1 (light blue and orange), 3 (lighter blue and light orange), and 5 minutes (cyan and yellow). The y axis is fluorescence intensity (a.u.) in all panels. Length: (a)—varied distributions, (c)-(e)—fixed, (f)-(h)—uniform. Velocity: (a),(b), (f)-(h) gamma μ = 1, σ = 0.25; (c)-(e)—uniform on interval [0, 2].
Fig 6.
Impact of velocity changes and pauses on intensity profile curves.
This figure shows how simulations differ between type 1, 2, and 3. The solid curves are plots of scaled versions of Eq (5) such that ‖f‖∞ = 1 and the x’s are simulated results. The curves and x’s represent times 0.5 (blue and red), 1 (light blue and orange), 3 (lighter blue and light orange), and 5 minutes (cyan and yellow). Panels (a), (d), and (h) show type 1 simulations with x’s and theoretical curves from Eq (5) (panels are the same as (d), (g), and (j) of Fig 4). Panels (b), (e), and (i) show type 2 simulations with x’s where the filaments change velocity after a random time chosen from a uniform distribution with mean 1 minute (the theoretical curves are for comparison with column 1). In (c), (f), and (j) the blue x’s show results from type 3 simulations where the stop time comes from a uniform distribution with mean 0.5 (on average of the initial filaments are moving,
) and the yellow x’s show results from type 2 for comparison (with the mean velocity
of the comparable type 3 simulation). In (a)-(c) the velocity is uniformly distributed, in (d)-(f) it has a Gaussian distribution, and in (h)-(j) it has a gamma distribution. All simulations have filaments with length 10 microns. The other parameters are the same as in Fig 4. The y axis is fluorescence intensity (a.u.) in all panels. Length: All—fixed.
Fig 7.
Comparison of how well the different types of simulations fit the experimental data.
The results of the optimization are shown. The box color indicates velocity distribution used: red—uniform, green—Gaussian, and purple—gamma. The line in the box gives the average of the objective function, for the best fit (out of 30) for each of the 13 data sets, the width of the box shows the upper and lower quartile, and the whisker lengths are about ±2.7σ where σ is the standard deviation. The circles are outliers. Using the Wilcoxon signed rank test: * for p < 0.022;** for p < 0.0012; *** for p < 0.0005. All possible pair combinations are statistically significant except the 6 possible pairings of Theory uniform, Theory Gaussian, Type 1 Uniform, and Type 1 Gaussian. In addition the difference between Theory gamma vs Type 3 uniform and Theory gamma vs Type 3 Gaussian are not statistically significant. The theory uses Eq (5). Length: Theory—fixed with μ = 10, Type 1 and 3—uniform with μ = 5.025 microns.
Fig 8.
Fitting simulations to the experimental data.
This figure compares the simulated results (red shades) with experimental data (blue shades) for one data set. The figure has been separated into three regions showing results from the theoretical model, type 1 simulations, and type 3 simulations. Panels (a)-(e), (j), and (k) show all the time data together, whereas the rest show the time data in individual panels. When optimizing only the data in the unshaded area is used. The minimum value of the objective function is 0.0511, 0.0525, and 0.0139 for type 1 and for type 3 0.0415, 0.0404, and 0.0131 for uniform, Gaussian, and gamma distributed velocity respectively. The y axis is fluorescence intensity (a.u.) in all panels. Length: All—uniform with mean 5.025 microns.
Fig 9.
Information on filament dynamics approximated from 13 experimental data sets.
Panel (a) shows the non-truncated velocity distributions as predicted by type 3 simulations (giving the minimum cost function of the 30 realizations) for each data set in a different color. All simulations used a truncated gamma distribution for the velocities. The black dotted line shows the gamma distribution with parameters which are the average of the parameters for the 13 other curves. Panel (b) shows the mean velocities. The squares to the left are velocities for moving filaments von and the squares to the right are velocities for all the filaments i.e., . Panel (c) shows the mean on and off times predicted by type 3 simulations for the 13 data sets considered. The squares to the left are the mean off times, τoff, and the squares to the right are the mean on times, τon. The black circles are the averages of mean off and on times over the 13 data sets. The grey region shows the percent of filaments which are stopped (the horizontal coordinate is randomly perturbed for viewing purposes). Both on and off times are uniformly distributed. The lines connect values from the same data sets. The colors indicate the same data set. Length: All—uniform with mean 5.025 microns.