Microtubule dynamics depart from wormlike chain model

Thermal shape fluctuations of grafted microtubules were studied using high resolution particle tracking of attached fluorescent beads. First mode relaxation times were extracted from the mean square displacement in the transverse coordinate. For microtubules shorter than 10 um, the relaxation times were found to follow an L^2 dependence instead of L^4 as expected from the standard wormlike chain model. This length dependence is shown to result from a complex length dependence of the bending stiffness which can be understood as a result of the molecular architecture of microtubules. For microtubules shorter than 5 um, high drag coefficients indicate contributions from internal friction to the fluctuation dynamics.Comment: 4 pages, 4 figures. Updated content, added reference, corrected typo

Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid

At timescales once deemed immeasurably small by Einstein, the random movement of Brownian particles in a liquid is expected to be replaced by ballistic motion. So far, an experimental verification of this prediction has been out of reach due to a lack of instrumentation fast and precise enough to capture this motion. Here we report the observation of the Brownian motion of a single particle in an optical trap with 75 MHz bandwidth and sub-angstrom spatial precision and the determination of the particle's velocity autocorrelation function. Our observation is the first measurement of ballistic Brownian motion of a particle in a liquid. The data are in excellent agreement with theoretical predictions taking into account the inertia of the particle and hydrodynamic memory effects

Motility pattern with two turning events.

<p>During each run, the speed of the cell <i>v</i>₀ is constant. Motion is nearly straight and is affected by the rotational diffusion <i>D</i>_<i>r</i>. The cell changes the direction of its motion during turning events (black dots), where turning angles Δ<i>φ</i>_1,2 are allowed to have two different probability distributions. Importantly, the cell strictly alternates the two types of turning events. When swimming in the gradient of signaling chemicals ∇<i>c</i>, the cell can bias its motion and respond with a drift speed <i>v</i>_<i>d</i> in the direction of the gradient, which we want to calculate.</p

Drift speed as a function of the chemoattractant gradient strength.

<p>Analytically obtained predictions (curve) agree with numerical results (symbols) up to the gradient values of ⋍ 0.5 <i>μm</i>⁻⁴ (<i>D</i>_<i>r</i> = 0.2 rad²s⁻¹, λ = 3.3 s⁻¹, <i>α</i> ≈ −0.34, <i>β</i> = −1).</p

Drift speed without rotational diffusion.

<p>(a) Analytically obtained function <i>v</i>_<i>d</i>(<i>α</i>, <i>β</i>), see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190434#pone.0190434.e020" target="_blank">Eq (11)</a> is shown as a green surface and numerically obtained results as symbols for <i>D</i>_<i>r</i> = 0.0 rad²s⁻¹, |∇<i>c</i>| = 0.05 <i>μm</i>⁻⁴ and <i>W</i> = 0.0458 <i>μ</i>m³. (b) Comparison of analytically (lines) and numerically (symbols) obtained drift speed dependences on the parameter <i>α</i> for four values of <i>β</i>: <i>β</i> = 1.0 (yellow), <i>β</i> = <i>α</i> (red), <i>β</i> = −1.0 (blue), <i>β</i> = 0.0 (purple).</p

Drift speed with rotational diffusion.

<p>(a) Analytically obtained function <i>v</i>_<i>d</i>(<i>α</i>, <i>β</i>), shown as surface, and numerically obtained result as points for <i>D</i>_<i>r</i> = 0.2 rad²s⁻¹, |∇<i>c</i>| = 0.05 <i>μm</i>⁻⁴ and <i>W</i> = 0.0458 <i>μ</i>m³. (b) Comparison of analytically (lines) and numerically (symbols) obtained drift speed dependences on the parameter <i>α</i> for four values of <i>β</i>: <i>β</i> = 1.0 (yellow), <i>β</i> = <i>α</i> (red), <i>β</i> = −1.0 (blue), <i>β</i> = 0.0 (purple).</p

Variability of the drift velocity with the cell body size.

<p>(a) Histograms showing the distribution of flick angles with various body lengths (data from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190434#pone.0190434.ref017" target="_blank">17</a>]). Red histogram represents all measured flick angles—170 events for 20 cells with various body lengths. Flick angles for four particular values of cellular body lengths are shown by narrow color histograms (<i>l</i> = 3.45 <i>μ</i>m, orange; <i>l</i> = 2.31 <i>μ</i>m, yellow; <i>l</i> = 1.72 <i>μ</i>m, green; <i>l</i> = 1.35 <i>μ</i>m, cyan histogram). The data obtained for 20 individuals (each of which is displayed at least 6 flicks in their trajectory) are shown as red points. Small blue ticks show the mean flick angles obtained from individual distributions. Black tick labels of the vertical axis correspond to the histograms of individual cells, whereas the red tick labels correspond to the red histogram for all cells. (b) Analytically obtained drift speed calculated for the mean flick angle of the considered cells, as a function of the angle and the corresponding body length. For results the following parameters were used: |∇<i>c</i>| = 0.5 <i>μ</i>m⁻⁴, λ = 3.3 s⁻¹, <i>v</i>₀ = 45 <i>μ</i>ms⁻¹, <i>D</i>_<i>r</i> = 0.2 rad²s⁻¹, <i>W</i> = 0.0458 <i>μ</i>m³ and <i>β</i> = cos 172°.</p

Analytically obtained drift speed and diffusion constant as functions of the mean flick angle and the corresponding body length.

<p>The curves represent the angle-dependent characteristics obtained from Eqs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190434#pone.0190434.e020" target="_blank">(11)</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190434#pone.0190434.e022" target="_blank">(12)</a>. To plot the drift speed and the diffusion constant as a function of cell size, we use the data of [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190434#pone.0190434.ref017" target="_blank">17</a>] to relate the size to the turning angle, and use that angle in analytical results Eqs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190434#pone.0190434.e020" target="_blank">(11)</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190434#pone.0190434.e022" target="_blank">(12)</a> (these values are shown by symbols). The fact that data based on cell sizes line up with the theoretical curves as functions of angles indicates an approximately linear relation between the cell size and the cosine of the turning angle, which is in agreement with data presented in [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190434#pone.0190434.ref017" target="_blank">17</a>]. The parameters are λ = 3.3 s⁻¹, <i>v</i>₀ = 45 <i>μ</i>ms⁻¹, <i>D</i>_<i>r</i> = 0.2 rad²s⁻¹ and <i>β</i> = cos 172°.</p