Dynamics of filamentous actin structures in cells:

(A) Diverse filamentous actin structures (red) in cells with different functions: (top left and clockwise) stereocilia in hair cells for mechanotransduction, actin cables in budding yeast for intracellular transport, filopodia in motile cells for local environment sensing, and microvilli in epithelial cells for absorption of extracellular chemicals. (B) These filamentous structures all consist of parallel actin filaments (red, zoomed in view of black box) bundled by crosslinking proteins (green). To describe changes of their length over time we model these linear structures as a single polar filament consisting of building blocks (pink chevron tile) of length a, in monomer units. (D) In “balance point models” of length control, assembly and disassembly of the polymer, which abstracts the filamentous structure, is described as the stochastic addition and removal of individual building blocks with rate constants k+(L)and k−(L), which can depend on the length of the polymer (L). (D) A steady state length, L* is achieved when the two competing rates match each other, at the intersection of the red and blue lines, representing length-dependent assembly and disassembly, respectively. (E) The length versus time schematic shows a typical output from stochastic simulations of the balance point model: the length quickly grows from L = 1 until it reaches a steady state L*and characteristic fluctuations around the steady state length are observed. (F) These steady state length fluctuations define the probability distribution function P*(L), which can be characterized by its mean and variance (σ2), where σ is the standard deviation of P*(L). (The length vesus time graph in (E) was obtained from model where and k−(L) = k−; see Figure 2A.)

Universal length fluctuations in balance point models:

(A-D) Results for a balance point model with a length dependent rate of assembly, k+(L) = κ +/L, and length-independent disassembly, k−(L) = k−. Different steady state lengths, L*, are achieved by tuning the assembly parameter, ; . In all simulations the length-independent rate of disassembly, . (B) Steady state length distributions from stochastic simulations for different values of k+. (C) The length distributions from (B) collapse to a Gaussian distribution (black line) centered around zero with a standard deviation of one, when the lengths are rescaled by the mean and standard deviation of each individual length distribution. (D) The variance of the length distributions scales linearly with the mean length (error bars are standard deviations). (E-H) Results for a balance point model with a length independent rate of assembly, k+(L) = k+, and length-dependent disassembly k−(L) = k−L. Different steady state lengths, L*, are achieved by tuning the assembly rate; . In all simulations the disassembly rate parameter . (F) Steady state length distributions for different values of κ+, obtained from stochastic simulations. (G) The length distributions from (F) collapse to a Gaussian distribution (black line) centered around zero with a standard deviation of one, when the lengths are rescaled by the mean and standard deviation of each individual length distribution. (D) The variance of the length distributions scales with the mean length to the power 1.02 ± 0.03. (Error bars are standard deviations). In all the simulations the subunit size is a = 10 monomers.

Universal scaling of variance with mean length for filamentous actin structures in vivo.

(A) Row 3 (shortest) stereocilia in mouse inner (blue) and outer (orange) hair cell bundles show length variation when treated with drugs such as amiloride and benzamil that block the flow of calcium ions in the tip links connecting the shorter rows to the taller ones. The log-log plot of the variance versus the mean of stereocilia lengths has a slope of 2.47 ± 0.46 (R2 = 0.86). (B) Actin cables (red) in budding yeast have lengths that scale with the length of their mother cells. The log-log plot of the variance versus the mean cable length has a slope of 2.47 ± 0.56 (R2 = 0.82). (C) Exogenous actin bundling proteins caused length variation in brush border microvilli of epithelial cells (green). The log-log plot of the variance vs mean length of microvilli has a slope of 1.92 ± 0.51 (R2 = 0.74). (D) Different filopodia lengths are achieved by genetic, chemical, and mechanical perturbations. The log-log plot of the variance vs mean of filopodia lengths have a slope of 2.19 ± 0.42 (R2 = 0.89). (E) Combined plot of the data for all the different actin structures in (A)-(D). The data for stereocilia, cables, and microvilli fall on a single power-law line with a slope of 2.09 ± 0.10 (R2 = 0.98). The data for filopodia follows a similar power law scaling as the rest (the gray line is parallel to the black line) but with a larger prefactor. All the errors in the slope are standard errors. Data points show mean ± standard error (orange, blue, red, and purple) except green which is mean ± standard deviation (see Methods for details about error estimates).

Bundled-filament model of length regulation:

(A) We consider parallel actin bundles to be made of N actin filaments (red), held together in a bundle by crosslinking proteins (green). Individual filaments have lengths: li=1,…,N, which are assumed to be exponentially distributed. The bundle length is defined as the maximum of all the filament lengths, L = max {l1, l2, …, lN}. (B) Length distribution of a bundle of N = 25 filaments obtained from a simulation that generated 10,000 bundles. Individual filaments within a bundle are sampled from an exponential distribution with mean lengths: < l > = 0.1 μm (red), < l > = 0.5 μm (green), < l > = 1.0 μm (orange), and < l > = 2.0 μm (blue). (C) The distributions of bundle lengths collapse onto a single curve when the lengths are normalized by the mean bundle lengths, which coincides with the theory prediction, Equation 14 (black line). (D) The log-log plot of the variance as a function of the mean length from the distributions showed in (B) gives a power law exponent of 2.00 ± 0.02.

Theoretical predictions of the bundled-filament model:

(A) Average length of the bundle, < L > as a function of the number of filaments in the bundle, N for different average lengths of the exponentially distributed individual filament within the bundle, < l > = 0.1 μm (blue), < l > = 0.5 μm (orange), < l > = 1 μm (green). (B) The variance in bundle lengths is independent of the number of filaments in the bundle when < l > is fixed. (C) The square of the coefficient of variation is plotted as a function of the mean length for all the data in Figure 3E, following the same color code. The parallel gray lines are predictions from Equation 13 for different number of filaments in a bundle, (N): the top line corresponds to N = 10 and the line below corresponds to N = 500.