Measurement of the persistence length of cytoskeletal filaments using curvature distributions

Abstract

Cytoskeletal filaments, such as microtubules and actin filaments, play important roles in the mechanical integrity of cells and the ability of cells to respond to their environment. Measuring the mechanical properties of cytoskeletal structures is crucial for gaining insight into intracellular mechanical stresses and their role in regulating cellular processes. One of the ways to characterize these mechanical properties is by measuring their persistence length, the average length over which filaments stay straight. There are several approaches in the literature for measuring filament deformations, such as Fourier analysis of images obtained using fluorescence microscopy. Here, we show how curvature distributions can be used as an alternative tool to quantify biofilament deformations, and investigate how the apparent stiffness of filaments depends on the resolution and noise of the imaging system. We present analytical calculations of the scaling curvature distributions as a function of filament discretization, and test our predictions by comparing Monte Carlo simulations with results from existing techniques. We also apply our approach to microtubules and actin filaments obtained from in vitro gliding assay experiments with high densities of nonfunctional motors, and calculate the persistence length of these filaments. The presented curvature analysis is significantly more accurate compared with existing approaches for small data sets, and can be readily applied to both in vitro and in vivo filament data through the use of the open-source codes we provide.

Significance

Cytoskeletal filaments define the shape of the cell and are essential for various cellular processes, such as cell division and cell motility. To gain a deeper understanding of such processes, it is crucial to develop an accurate way to measure the persistence length, a mechanical property quantifying the bending resistance of filaments. Here, we developed and validated a novel method to measure the persistence length of biofilaments, such as actin filaments or microtubules, based on their curvature distributions. We further demonstrate the advantages of our approach for in vivo experiments, where the data set sizes are often limited.

Introduction

The cytoskeleton is a filamentous network found in cells, which maintains cell shape, aids in cell motion, and plays a key role in intracellular transport and cell division (1); it is made up of microtubules, actin filaments, and intermediate filaments. Microtubules are the rigid tubular structures in cells that, in addition to their structural role, act like molecular highways to transport cellular signals and cargo in a wide variety of cellular processes (2). While microtubules are hair like in terms of their aspect ratio, they have a large resistance to bending—similar to plexiglass (3). Actin filaments, on the other hand, are significantly softer and easier to bend compared with microtubules. They also serve a variety of roles in cells, including formation of protrusions at the leading edge of a crawling fibroblasts and contraction of the cytokinetic machinery (1). Utilizing cytoskeletal associated proteins and molecular motors, both microtubules and actin filaments can also form active or passive bundles, such as flagella of swimming organisms or microvilli on the brush-border cells lining the intestine (1). Thus, it is important to study the mechanical behavior of these filaments to understand their function as dynamic mechanical components in living cells, and gain insight into the intracellular mechanical stresses (4).

Inside a cell, microtubules and actin filaments continuously bend and buckle, resisting forces as they perform their function; therefore, a filament’s resistance to bending is an important mechanical property to measure. One way to characterize the bending deformations of these filaments is to measure the persistence length, L_p, the length scale at which thermal fluctuations can cause the filaments to spontaneously bend (5). One of the first estimations of microtubule persistence length in vitro made use of statistical analysis of the contour lengths and end-to-end distances in dark-field microscopic images, which resulted in very small values, of about 75 μm (6). The measured microtubule persistence length was an order of magnitude greater than that of actin filaments; nevertheless, it was much lower than the accepted values for microtubules of 1–10 mm in recent literature (7). Since then, microtubule and actin flexural rigidity (therefore persistence length) has been further estimated from dynamic video images using various techniques, including spectral analysis of thermally fluctuating filaments (3,8, 9, 10, 11), end fluctuations (12,13), and tangent correlations (11,14,15). There have also been nonequilibrium measurements, including hydrodynamic flow (8,12,16) optical trapping (17, 18, 19, 20, 21), and atomic force microscopy (22).

Despite numerous studies on persistence length of biofilaments in vitro in the past three decades (3,6,14,15,21,23) far fewer measurements exist in vivo. Some of the existing approaches are not readily applicable in vivo due to the complexity of forces and the boundary conditions involved (24). The crowded environment of the cell also limits the number of filaments that can be traced in a given microscopy image. It is also a challenge to obtain long temporal resolution from an in vivo experiment to achieve accurate persistence length measurement (25). In addition, in living cells, these filaments can potentially have defects and associated proteins that can change local resistance to bending (26), complicating the interpretation of global methods such as Fourier analysis (3). It is, therefore, crucial to develop new approaches that focus on local deformations to measure the persistence length L_p accurately, particularly applicable to small data sets.

To overcome some of these limitations, Odde et al. (27) used curvature distributions to describe the behavior of microtubules in the lamella of fibroblasts, and characterized the breaking mechanisms of microtubules. Since then, curvature distributions have been used to characterize biopolymers, for example, to analyze DNA flexibility (28), to quantify the bias in branching direction of actin filaments (29), and to study microtubule bending under perpendicular electric forces (30). Bicek et al. (31) used curvature distributions of microtubules to study the anterograde transport of microtubules in LLC-PK1 epithelial cells, where exponential distributions of curvatures were observed in vivo when microtubules were driven by motor forces. Similar exponential distributions were observed in the analysis of actively driven actin filaments (32). However, obtaining quantitative results of the persistence length remains a challenge, given the effects of spatial sampling and experimental noise on the observed curvature distributions (4).

In this work we show how these challenges can be overcome by using a subsampling approach, combined with analytical calculations of the scaling of form for curvature distributions. We validate our approach using simulated, model-convolved (4,33) filaments and compare it with the popular approach, Fourier analysis (3). We apply our curvature analysis to measure the persistence lengths of microtubule and actin filaments immobilized on glass surfaces with nonfunctional motors. Finally, we illustrate how our approach is more accurate than Fourier analysis when applied to typical in vitro and in vivo scenarios with varying length filaments and small data sets (<20 filaments).

Materials and methods

Worm-like chain model and curvature distributions

In the continuum description, a semiflexible polymer, such as an actin filament or microtubule, can be modeled using the worm-like chain model, with the Hamiltonian given by

$$\mathcal{H}=\frac{1}{2}EI\int {\kappa }^2\left(s\right)ds,$$ (1)

where EI is the flexural rigidity (E, Young’s modulus; I, second moment of cross-sectional area), and the curvature is defined as $\kappa \equiv d\theta /ds$. Assuming that bending energy is the dominant energy, one can show from Boltzmann distribution that, for a discretized version of the filament with equal bond lengths $\text{Δ}s$ (as illustrated in Fig. 1 a), the curvatures κ are going to be distributed according to

$$P\left(\kappa \right)=\sqrt{\frac{L_p\text{Δ}s}{2\pi }}{e}^{-L_p\text{Δ}s{\kappa }^2/2}.$$ (2)

Here, the persistence length L_p is related to the flexural rigidity via L_p = EI/(k_BT), where k_B is Boltzmann’s constant and T is the temperature (2). For small enough deformations, i.e., relatively stiff chains, the curvature is related to the local angle change as $\kappa =\theta /\text{Δ}s$.

Figure 1.

Illustration of the subsampling process. (a) Schematic showing curvature calculations for m = 1, where curvatures are obtained directly from the original coordinates of the filament. Here, black lines shows the original backbone, and red points denote the nodes used in the curvature calculation for m = 1. (b) Example curvature distribution for m = 1, where standard deviation is $1/\sqrt{L_p\text{Δ}s}$. (c) Schematic showing curvature calculations for m = 2, where new filaments are formed out of every other point. Here, orange line and points, overlaid over the original filament, denote the new subsampled filament for m = 2. (d) Example curvature distribution for m = 2, where standard deviation is $1/\sqrt{\frac{4}{3}{L}_p\text{Δ}s}$. The angle changes used in the curvature calculation are denoted by ${\theta }_i^{\left(m\right)}$, where i shows the location of the angle change on the backbone, and m corresponds to the decimation level. The curvature distributions were measured from 1000 Monte-Carlo-generated filaments with L_p = 10 μm, L = 10 μm, and Δs = 100 nm. To see this figure in color, go online.

Simulations of filaments

To compare the performance of the curvature analysis with the other existing techniques, 10 μm long filaments with bond spacing of 100 nm were generated using Monte Carlo sampling of the Gaussian curvature distribution (see Worm-like chain model and curvature distributions). Filaments, however, are often traced from experimental images, and the point spread function of light, together with contamination from detector noise and optical aberrations, make accurate collection of x-y coordinates difficult. To better compare with experimental data, we convolved the coordinates of simulated filaments with the point spread function, and added experimental noise—an approach called model convolution microscopy (4,33,34) (see also Section S1).

Subsampling of curvature distributions

Persistence length is a global parameter depicting polymer flexural over a distance, hence the information about curvature change over certain distances is needed to accurately determine persistence length. Hence, a subsampling procedure, as described in (4) is necessary for not only eliminating noise, but also to avoid curvature distribution measurements being dominated by information only over short distances. This subsampling can be done via selecting coordinates at equal distances (i.e., keeping only every k^th sample), either once or in a repetitive manner, i.e., by taking every other point to obtain a new filament, and once again decimating the backbone by taking every other point, etc. Let us denote m = 1 as the original filament coordinate set (with no subsampling), as shown in Fig. 1 a. The curvature values are then measured at every point along the chain, except at the first and last points (red symbols in Fig. 1 a). The resulting curvature distribution will be Gaussian, as shown in Fig. 1 b. The next step of subsampling corresponds to m = 2, where the first and second segments are formed by connecting the first coordinate pair to the third coordinate pair, and the third coordinate pair to the fifth coordinate pair, and so on (orange symbols and lines in Fig. 1 c). The newly formed pair of line segments can then be used to calculate the curvature at the third, fifth, etc., coordinate locations on these subsampled filament coordinates. The resulting curvature distribution of all curvatures in this subsampled filament is then given by (see Fig. 1 d)

$$P\left({\kappa }^{\left(1\right)}\right)=\sqrt{\frac{4L_p\text{Δ}s}{3\pi }}{e}^{-\frac{4}{3}{L}_p\text{Δ}s{\left({\kappa }^{\left(1\right)}\right)}^2}.$$ (3)

The details of this calculation are given in Section S2. The generalized curvature distribution for an arbitrary subsampling level m can be derived as follows (see Section S3)

$$P\left({\kappa }^{\left(m\right)}\right)=\sqrt{\frac{{\text{Λ}}^{\left(m\right)}}{2\pi }}{e}^{-{\text{Λ}}^{\left(m\right)}{\left({\kappa }^{\left(m\right)}\right)}^2/2},$$ (4)

where the exponent ${\text{Λ}}^{\left(m\right)}$ is

$${\text{Λ}}^{\left(m\right)}=L_p{\mu }^{\left(m\right)}\text{Δ}{s}^{\left(m\right)},$$ (5)

and ${\mu }^{\left(m\right)}$ for m = 1,2,… is found to be

$${\mu }^{\left(m\right)}=\frac{m}{\left(2m-1\right)\left(1-1/m\right)/3+1}.$$ (6)

Plots of μ as a function of m for L_p = 10 μm and L_p = 3000 μm are shown in Fig. 2, and the agreement of simulations with the theoretical predictions for physiologically relevant persistence lengths is excellent. Note that in the limit ${\text{lim}}_{m\to \infty }\mu =3/2$, which can be interpreted as a rescaled persistence length L_p → (3/2)L_p. This demonstrates that the effects on decimation can be significant, especially if one is interested in quantitatively measuring the persistence length of the underlying chain. Note that the μ plot will start deviating from this convergent behavior and curves will start falling off as a function of m if the filaments are much softer than a typical actin filament, i.e., $L_p/\text{Δ}s\lesssim 20$ (data not shown). This is because, at such short persistence lengths, the filaments will start forming loops, resulting in the approximations in the theory presented here to break down and therefore smoothening of the actual curvature of such short filaments. The slope of a plot of ${\text{Λ}}^{\left(m\right)}$ (here ${\text{Λ}}^{\left(m\right)}$ represents the inverse variance of the curvature distribution) as a function of ${\mu }^{\left(n\right)}\text{Δ}{s}^{\left(m\right)}$ (Eq. 5) can be used to calculate the value of the persistence length, L_p, for a given semiflexible polymer.

Figure 2.

Validations of the analytical calculation for μ, i.e., the scaling factor of the persistence length due to subsampling, via Monte Carlo simulations. Plots of μ as a function of subsampling level, m, is shown for simulated filaments with given persistence length of (left) 10 μm and (right) 3000 μm, comparable with the persistence lengths of actin filaments and microtubules, respectively. The blue circles represent the simulated values. The black solid line shows the theoretical predictions for m (Eq. 6), while the red squares are the theoretical predictions for n (Eq. S26). The filaments are 20 μm long and generated with Δs = 100 nm. To see this figure in color, go online.

Typically, the pixel size for experimental or simulation imaging data depends on the instrument used (or modeled); hence, it is already at some m level of subsampling with μ > 1. Since μ has an upper limit of 3/2 with m approaching infinity, the larger the subsampling is, the less the deviation is from this limiting value. In all the analysis of experimental data in this paper, we assumed an unknown but enough level of subsampling is done, and used the limit value μ = 3/2 in the calculation of persistence lengths. For simulated data, actual functional form has been used, namely Eq. 6, although we did not see a noticeable difference in results even if μ = 3/2 is used while analyzing simulated data. It is also important to note, however, that while analyzing experimental data sets with scarce data, subsampling can further reduce the number of data points. To remedy this, we calculate the curvature values recursively, as described in detail in Section S4.

Cumulative curvature distributions

Even though the persistence length can be measured via probability density curvature distribution (as described in Subsampling of curvature distributions), we use cumulative distribution function instead to avoid the need of an appropriate bin size. The cumulative distribution function, $C\left({\kappa }^{\left(n\right)}\right)$, can be derived from Eq. 4 using the following relation

$$C\left({\kappa }^{\left(n\right)}\right)={\int }_{-\infty }^{{\kappa }^{\left(n\right)}}P\left({{\kappa }^{\prime }}^{\left(n\right)}-{{\kappa }^{\prime }}_0^{\left(n\right)}\right)d{\kappa }^{\prime },$$ (7)

where ${{\kappa }^{\prime }}_0^{\left(n\right)}$ is the mean of the distribution. Using Eqs. 4 and 7 it can be shown that

$$C\left({\kappa }^{\left(n\right)}\right)=\frac{1}{2}\left[1+\text{erf}\left(\sqrt{\frac{\text{Λ}}{2}}\left({\kappa }^{\left(n\right)}-{{\kappa }^{\prime }}_0^{\left(n\right)}\right)\right)\right].$$ (8)

Preparation and imaging of actin filaments

Chicken skeletal muscle actin was purified from acetonic powder using standard methods developed to purify actin from skeletal muscle (35) and gel filtered according to (36). Fractions containing actin were rapidly frozen in liquid nitrogen as 50 μL pellets and stored at −80°C. Myosin S1 fragment was prepared from chicken skeletal muscle (37) and the S1 preparation was also stored as 50-μL frozen pellets at −80°C. To visualize F-actin, actin filaments polymerized with 100 mM KCl and 5 mM MgCl₂ from a 2 μM solution of monomers were decorated with a molar excess of Alexa Flour 488 Phalloidin (Molecular Probes, Eugene, OR, USA) overnight, and diluted before use to 20 nM with AB buffer (25 mM imidazole-HCl, 25 mM KCl, 4 mM MgCl₂, 1 mM EGTA, 1 mM DTT [pH 7.4]) plus 100 mM DTT (38). Coverslips were affixed to a slide by water droplets and coated with 0.2% (w/v) nitrocellulose in isoamyl acetate, and allowed to dry. Fifty to 100 μL of S1 solution diluted in AB buffer at a various concentrations was pipetted onto parafilm on top of ice. The coverslip was placed with the nitrocellulose facing down onto the sample and incubated for an hour. The coverslip was then washed with 100 μL of AB buffer supplemented with 0.5 mg/mL BSA to block any unoccupied nitrocellulose. A drop of 20 nM phalloidin-decorated actin was added to a slide and the coverslip containing the bound S1 was placed on the drop on the slide. Imaging was performed using a Zeiss Axiovert 200M microscope equipped with a CoolSNAP fx CCD camera. A filter cube for FITC/GFP was used with a 63× objective DIC Plan-APO CHROMAT NA 1.4. The microscope and camera were driven by Zeiss Axiovision software.

Preparation and imaging of microtubules

Cy-5-labeled microtubules were polymerized from bovine brain tubulin as previously reported (39) and stabilized with 10 μM taxol. A gliding assay was prepared using 500 pM rigor (R210A) full-length Drosophila melanogaster kinesin-1 heavy chain. Casein (0.5 mg/mL) was used both for blocking and for binding the rigor kinesin to the glass coverslip, as reported previously (40). The final imaging solution was 0.5 mg/mL casein, 10 μM taxol, 20 mM glucose, 20 μg/mL glucose oxidase, 8 μg/mL catalase, 1:200 β-mercaptoethanol, and 2 mM ATP in BRB80 (80 mM PIPES, 1 mM EGTA, 1 mM MgCl₂ [pH 6.8]). Experiments were performed under total internal reflection fluorescence using a Nikon TE2000 inverted microscope and a Melles Griot 632 nm Helium-Neon laser. Static images were captured using a Cascade 512 EMCCD camera (Roper Scientific, Thousand Oaks, CA, USA) and MetaVue software (Molecular Devices, San Jose, CA, USA) using a 400-ms exposure time. The pixel size was 108.1 nm/pix.

Filament tracing

The model-convolved and experimental filaments were traced with two types of tracing algorithms: an in-house developed Gaussian scan (GS) approach, and the ImageJ plug-in JFilament (11). GS was implemented in MATLAB (The MathWorks, Natick, MA) closely following the approach by Brangwynne et al. (10). In brief, the images are processed through noise reduction (median filter), followed by skeletonization. The filament backbones are then fit to a fourth-order polynomial, which is then used to calculate the perpendicular angle along the backbone of the filament, and a scan along this axis is performed on the original image to obtain an intensity profile. Finally, these profiles were fit to a Gaussian function and the mean values of the functions were taken as the coordinates of the filaments (see Fig. 3 a).

Figure 3.

Validation of curvature analysis using model-convolved Monte-Carlo-simulated filaments with persistence lengths of 10 μm (left, soft) and 3 mm (right, stiff), traced by Gaussian scan (GS) tracing. (a) Example of filament tracing where the red line represents recorded coordinates. Scale bar, 2 μm. (b) Cumulative curvature distribution of 100 filaments measured using every 20th point along the backbone (m = 20) for the soft filaments, and every 30th point along the backbone (m = 30) for the stiff filaments. The insets show the curvature distributions without subsampling (m = 1). The dashed black line denotes simulation data, and the solid green line is a fit to Eq. 8. (c) Plots of ${\text{Λ}}^{\left(m\right)}$ as a function of ${\mu }^{\left(m\right)}\text{Δ}{s}^{\left(m\right)}$, calculated using 100 filaments. Points represent data calculated using m = 1–20 (left panel), and m = 1–30 (right panel). Blue solid lines represent the fits of all data points to Eq. 5, resulting in persistence length of 10.6 μm (left panel) and 3040 μm (right panel). Black dashed lines represent input persistence length of 10 μm (left panel) and 3000 μm (right panel) from the simulations. To see this figure in color, go online.

In contrast, JFilament is based on stretching open active contours or “snakes”—parametric curves that deform to minimize the sum of an external energy derived from the image and an internal bending and stretching energy (11) (see Fig. S6 a). Two types of forces are introduced from the external energy: 1) forces that pull the contour toward the central bright line of a filament and 2) forces that stretch the contour toward the ends of bright ridges. To trace a filament, initial snake is given by hand-clicking along the filament, and it is then deformed to match the filament shape on the image. For further details, the reader is referred to (11).

Often times, the coordinate spacing of the filaments obtained from the tracing software can be slightly nonuniform. We, therefore, performed the linear interpolation procedure adapted from Ott et al. (14) on the traced filaments to convert them into filaments with uniform coordinate spacing (see Section S5 in supporting material for details).

Fourier analysis

The persistence length, L_p, of a biopolymer can be calculated using a Fourier analysis approach introduced by Gittes et al. (3). In this approach, one can express the filament shape, characterized by φ(s), as a superposition of Fourier modes. Gittes et al. showed that, for a thermally driven polymer, the variance of the mode amplitudes scales as ∼n⁻². They also calculated the error of the variance due to pixellation in experimental images. This approach is often used in practice to calculate the persistence length.

However, the approach, invented by Gittes et al., was specifically designed to analyze multiple frames of a thermally fluctuating taxol-stabilized microtubule, which always has the same filament length throughout the frame. This approach would not be suitable to analyze multiple filaments with varying lengths, or a filament with dynamic length. To make a fair comparison with the curvature approach that we apply to filaments with different lengths, one can calculate a modified length-dependent variance that takes into account length variation, namely,

$$⟨{\left(\frac{a_n-a_n^0}{L}\right)}^2⟩\simeq \frac{1}{{\pi }^2{L}_p}\frac{1}{n^2}+⟨\frac{{\epsilon }_k^2}{L^3{N}^2}\left(N-1\right)⟩{\pi }^2{n}^2+⟨\frac{4{\epsilon }_k^2}{L^3}⟩,$$ (9)

where a_n is the mode amplitude, $a_n^0$ the amplitude in the absence of applied or thermal forces, L the total length of the filament, n the mode number, N the number of coordinates on the filament, and ${\epsilon }_k$ the error in coordinate position due to pixellation. Note that this expression is valid for small n values. Nonetheless, since experimental noise is dominant at high n values, this approximation is valid for most practical situations.

Bootstrapping

To measure the persistence length accurately for small data sets, we adapted the bootstrapping approach proposed by Hawkins et al. (23). Forty and 50 filaments were randomly picked out of a given set of actin filaments (160 total) and microtubules (200 total), respectively. Picking the same filament again is allowed. Those selected filaments then underwent curvature and Fourier analyses to estimate the persistence length. This process was repeated 5000 times to construct a histogram of persistence lengths, and the mean persistence length and the associated standard deviation were estimated by fitting the histogram to a log-normal distribution.

Results

Curvature at short length scales is affected by experimental noise

While for ideal thermally driven polymers the curvature distribution is given by Eq. 2, due to limits in resolution and inherent noise, and errors introduced by tracing algorithms, the curvature distribution for experimentally observed filaments often deviates from a Gaussian distribution. To use curvature distributions to measure L_p quantitatively, some form of subsampling is required to avoid measurement noise, either due to pixellation or tracing algorithm-induced errors (4). As described in the Materials and methods and the supporting material, starting with the Hamiltonian of the worm-like chain model, we analytically calculated how the curvature distribution rescales as a decimation procedure (as a way to subsample) when applied on traced biopolymer coordinates. We then obtained a theoretical expression for the variance of the curvature distribution that can be used to calculate the persistence length L_p quantitatively.

To recapitulate a typical experimental measurement, we performed model convolution (4,33) (see Section S1) on the Monte-Carlo-generated filaments with input persistence lengths of 10 μm and 3 mm. These values were chosen to mimic actin- and microtubule-like filaments, respectively (3). The convolved filament images were then analyzed with two types of tracing algorithms: GS and JFilament (see Materials and methods). The cumulative curvature distributions were measured recursively (see Section S4) throughout the filaments for each subsampling level, m. As shown in Fig. 3 b insets, the cumulative curvature distributions deviate from a Gaussian distribution for both soft (left panel) and stiff (right panel) GS-traced filaments. Similar results were obtained when JFilament tracing was used for soft filaments (see Fig. S6 b, left), showing that subsampling is required to avoid measuring curvature in the noise-dominated regime. It is important to note that, while the distribution has a Gaussian form for JFilament traces of soft filaments (Fig. S6 b, left), the cumulative curvature distribution does not produce the theoretically expected variance. The cumulative curvature distributions of each subsampling level were then fitted to Eq. 8 to obtain ${\text{Λ}}^{\left(m\right)}$. As observed in Fig. 3 c, the cumulative curvature distributions for small subsampling levels were often affected by noise at low ${\mu }^{\left(m\right)}\text{Δ}{s}^{\left(m\right)}$ (product of scaling factor μ and bond length $\text{Δ}s$ at subsampling level m) values, and deviate from a line. Here, $\text{Λ}$ corresponds to the inverse variance of the curvature distribution at a given subsampling level, and therefore, higher values indicate narrowing of the curvature distribution.

The accuracy of curvature analysis was investigated over four different sampling sizes, namely, 10, 20, 50, and 100 filaments per set. For each sampling size, the cumulative curvature distribution for each subsampling level was measured for ${\text{Λ}}^{\left(m\right)}$, and the measured values for ${\text{Λ}}^{\left(m\right)}$ were fit to Eq. 5 to obtain L_p. We also measured the persistence length using a modified Fourier analysis approach to allow for the analysis of filaments with varying lengths (see Materials and methods). Fig. S7 illustrates the variance plot of the Fourier mode amplitudes for simulated soft (left) and stiff (right) filaments. We observed that the pixellation noise (concave-up tail) is more apparent on the variance plot for stiff filaments compared with the soft ones, as expected (Fig. S7, right panel).

Fig. 4 shows the comparison between curvature and Fourier analysis for different sample sizes. The bar charts are each averaged over 10 independent sets of filaments, with error bars representing the standard deviations. The curvature analysis performed as accurately as the Fourier approach on large data sets as expected, but significantly better when analyzing smaller sets of filaments. Cumulative curvature distribution also provides more consistent results as indicated by smaller error bars. We observed no significant difference in results when performing these analyses on the same filaments traced using JFilament (Fig. S8).

Figure 4.

Accuracy of curvature analysis (blue) compared with Fourier approach (red) for different size data sets. All filaments are generated via Monte Carlo simulations, convolved with the point spread function of the microscope, and traced by GS tracing. Each bar is an average over 10 sets of measurements with error bars representing standard deviation. The dashed lines represent the input persistence length. To see this figure in color, go online.

Curvature distributions of experimental actin filaments and microtubules

The curvature analysis was then performed to measure the persistence length of actin filaments and microtubules in vitro, using filaments immobilized on glass surfaces with relatively high density of rigor (“nonfunctional”) motors (see Materials and methods). Effectively adhering the filaments to the surface allowed for image acquisition at slower frame rates, and helped avoid blurry images, in addition to preventing filament bundling over time. The filaments were traced using GS tracing and JFilament, as demonstrated in Figs. 5 a and S9 a.

Figure 5.

Curvature analysis applied to actin filaments (left panels) and microtubules (right panels) adhered to glass substrates with rigor/nonfunctional motors. All filament tracing is done via GS tracing. (a) Illustration of tracing of actin filaments and microtubules. Scale bar is 1 μm (for the actin) and 2 μm (for the microtubule). (b) Plots of ${\text{Λ}}^{\left(m\right)}$ as a function of ${\mu }^{\left(m\right)}\text{Δ}{s}^{\left(m\right)}$, calculated using 160 filaments for actin filaments and 200 filaments for microtubules. ${\mu }^{\left(m\right)}$ is assumed to be 1.5 for all m. Blue dots represent data points calculated from actin filaments (left panel) using m = 1–20, and microtubules (right panel) using m = 1–50; blue lines represent fits to Eq. 5. Note that here a fixed value of μ = 3/2 is used instead of Eq. 6. (c) Histograms show distributions of the measured persistence length values obtained from bootstrapping 5000 data sets. The distributions were fit to the log-normal distribution to calculate the mean and standard deviation values. To see this figure in color, go online.

Since the results from our simulations suggested that both the curvature and Fourier analyses are at most accurate when analyzing large data sets, to assess the accuracy of our measurements, we implemented the bootstrapping method described by Hawkins et al. (23) (see Materials and methods). For each bootstrapped set, we measured the cumulative curvature distribution of those filaments at different subsampling levels and constructed the ${\text{Λ}}^{\left(m\right)}$ plot to find the persistence length (Figs. 5 b and S9 b) using μ = 3/2 (see Materials and methods). The distributions of measured L_p values over 5000 bootstrapped sets are shown in Figs. 5 c and S9 c, where the histograms were fit to the log-normal distribution to calculate the mean and standard deviations. We found the persistence length of the actin filaments and microtubules traced via GS tracing to be 9.9 ± 1.1 and 310 ± 90 μm, respectively. We obtained similar results from JFilament traced filaments as shown in Fig. S9 c.

The filaments were also analyzed via Fourier analysis (Fig. S10), and the measured L_p values obtained using both approaches were found to be comparable for both types of filaments traced using GS (Fig. S11). While the L_p value for actin filaments, traced with JFilament, was in agreement with the curvature analysis results (Fig. S12, also in agreement with GS tracing), the comparison for microtubules demonstrated a difference between two approaches. Even though the curvature analysis gave similar results between two tracing approaches, the Fourier analysis performed poorly, as demonstrated by the variance plot in Fig. S13 a (right panel), resulting in an underestimation of the persistence length (Figs. S11 and S12, right panels).

Effect of length distribution and small data sets on persistence length measurement

It is often challenging to obtain a large number of filaments from fluorescence images in vivo, and in particular it is usually difficult to trace a given filament for a long enough time to acquire uncorrelated data for persistence length analysis. In addition, experimentally observed filaments often have length distributions, as shown in Fig. S14. To study the effect of length variation on persistence length analysis, we generated actin filaments and microtubules (L_p = 10 and 300 μm, respectively) with experimentally observed length distributions in vitro, and used model convolution to generate realistic images (33). After tracing these filaments with GS tracing and JFilament, we applied curvature and our length-corrected Fourier analyses (see Materials and methods). The measured persistence lengths from both analysis approaches and tracing methods are in excellent agreement with the input L_p values for large number of filaments (>50 filaments per analysis), suggesting that the curvature approach and our modified Fourier analysis can provide satisfactory results when applied to filaments with length variation (see Figs. S15 and S16).

To investigate the effect of small number of filaments on analysis, we created 10 data sets, each containing 10 filaments generated and traced using the approaches described above. Insets of Fig. 6, a and b show samples of such sets for actin- and microtubule-like filaments, respectively. As shown in Fig. 6, a and b (and also in Figs. S15 c and S16 c), the curvature analysis clearly outperformed Fourier analysis under these conditions, making it more suitable for in vivo applications, especially when it is difficult to obtain large number of filaments for analysis.

Figure 6.

Curvature approach (C) applied to small data sets of simulated actin filaments and microtubules of varying length, compared with length-corrected Fourier analysis (F). Ten sets each containing 10 independent Monte-Carlo-generated filaments were used, and the contours were convolved with the point spread function of the microscope to create realistic images (samples shown in the insets). The filaments followed the same length distribution as the experimental filaments, but ones longer than 15 μm were discarded. Bars show the corresponding L_p values measured using both approaches. The filaments were traced using the GS tracing. Scale bars, 5 μm. To see this figure in color, go online.

Discussion

In this work, we developed a novel quantitative approach that relies on local curvature to measure the persistence length of cytoskeletal filaments. While, in the past decade, we and others have used curvature analysis to characterize filament deformations (4,31,41), to the best of our knowledge this is the first time that curvature distributions have been used to quantify the persistence length of cytoskeletal filaments. More specifically, we analytically derived the scaling form of curvature distributions as a function of subsampling, a necessary step to avoid measurement noise. The accuracy of this theory was verified through model-convolved (33,34) Monte-Carlo-simulated actin- and microtubule-like filaments. We also investigated various factors, such as tracing algorithms (GS or JFilament (11)) and data set sizes. We compared our findings with the popular Fourier analysis approach (3,9,23,25,42) and found the results in excellent agreement.

When applied to an ensemble of filaments with varying filament lengths, we found that our curvature analysis was more accurate than existing approaches. It is important to note that, to use the Fourier analysis (3,23) with filaments of varying lengths, we had to modify the variance formulas in (3) to account for such length changes. Overall, we found that, when there are sufficient filament coordinates, i.e., large data sets, both curvature and Fourier analysis approaches showed excellent agreement regardless of the tracking techniques used and the stiffness of the filaments. When our curvature analysis was applied to actin filaments and microtubules in in vitro gliding assays, the calculated persistence lengths were in good agreement with literature values (14,18,43, 44, 45). It is also important to note that, for very short and stiff filaments, i.e., L ≪ L_p, the accuracy of our approach will be limited as there is no long-distance curvature change information (as is the case for Fourier and tangent correlation approaches). In addition, depending on the subsampling used, our approach may result in smoothing out of actual curvature values due to the presence of loops for very soft filaments; therefore, we recommend a minimum spacing 20-fold smaller than the expected persistence length to avoid inaccuracies.

Persistence length of a biological filament is an important quantitative descriptor of its resistance to bending, key to understanding the forces exerted on or by cytoskeletal filaments. While numerous approaches have been developed and used to measure persistence length of actin and microtubules in the past several decades (3,6,14,21,30,46), there is considerable variability in reported values (4,7) even from in vitro experiments. Many factors contribute to this discrepancy, including poor signal-to-noise ratios in fluorescence images giving rise to insufficient temporal resolution, variability in experimental conditions, limitations of measurement or filament tracking techniques, broad filament length distributions, and limited data set sizes. Primarily due to these limitations, measurements of persistence length in vivo are scarce, particularly for stiff filaments such as microtubules (4).

The strength of the curvature analysis is its ability to provide significantly better accuracy for low number of filaments. This is particularly important for in vivo data, where obtaining large data sets under the same physiological conditions is often challenging (31,42) or when investigating questions, such as the length-dependent rigidity of microtubules (7,46)—where binning filaments with respect to length reduces the number of data points significantly. It can also be used with electron microscopy images; for instance, when determining the bending rigidity of individual microtubule protofilaments. While the example applications of the curvature analysis in this paper are limited to the two major cytoskeletal filaments, the approach can also be used to study bending deformations of any type polymers, such as DNA/RNA, flagella, and cilia of motile organisms. It is also important to note that our approach, while providing considerable improvement in measurement of persistence length, will still suffer in accuracy with short and stiff filaments when L ≪ L_p due to lack of long-distance information on curvature change. To make this analysis more accessible, open-source curvature and Fourier analysis software is made available on github at https://github.com/tuzellab.

Author contributions

P.W. developed the simulations and GS tracing software, performed the calculations and image tracing, and wrote the ImageJ plug-in, MATLAB codes, and the manuscript. E.T. conceived the study, wrote the manuscript, and supervised the overall study. L.V. and K.J.M. performed the actin and microtubule experiments. W.O.H. supervised the microtubule experiments. W.O.H., L.V., and E.T. contributed to the writing of the manuscript.

Declaration of interests

The authors declare no competing interests.

Editor: Jonathon Howard.