Averaged implicit hydrodynamic model of semiflexible filaments

We introduce a method to incorporate hydrodynamic interaction in a model of semiflexible filament dynamics. Hydrodynamic screening and other hydrodynamic interaction effects lead to nonuniform drag along even a rigid filament, and cause bending fluctuations in semiflexible filaments, in addition to the nonuniform Brownian forces. We develop our hydrodynamics model from a string-of-beads idealization of filaments, and capture hydrodynamic interaction by Stokes superposition of the solvent flow around beads. However, instead of the commonly used first-order Stokes superposition, we do an equivalent of infinite-order superposition by solving for the true relative velocity or hydrodynamic velocity of the beads implicitly. We also avoid the computational cost of the string-of-beads idealization by assuming a single normal, parallel and angular hydrodynamic velocity over sections of beads, excluding the beads at the filament ends. We do not include the end beads in the averaging and solve for them separately instead, in order to better resolve the drag profiles along the filament. A large part of the hydrodynamic drag is typically concentrated at the filament ends. The averaged implicit hydrodynamics methods can be easily incorporated into a string-of-rods idealization of semiflexible filaments that was developed earlier by the authors. The earlier model was used to solve the Brownian dynamics of semiflexible filaments, but without hydrodynamic interactions incorporated. We validate our current model at each stage of development, and reproduce experimental observations on the mean-squared displacement of fluctuating actin filaments . We also show how hydrodynamic interaction confines a fluctuating actin filament between two stationary lateral filaments. Finally, preliminary examinations suggest that a large part of the observed velocity in the interior segments of a fluctuating filament can be attributed to induced solvent flow or hydrodynamic screening.

Figure 1

Implicit and averaged hydrodynamics of semiflexible filaments. (a) In a string-of-beads realization of a semiflexible filament, the hydrodynamic drag on each bead can be determined by solving for the true relative velocity or hydrodynamic velocity of each bead. (b) Since the hydrodynamic velocity/drag is much larger at the end beads of a filament, and since the hydrodynamic velocities of the interior beads change slowly, sections of the interior beads (shown within rectangles) can be approximated by hydrodynamic velocities of representative beads within them (gray beads). Therefore instead of solving for the hydrodynamic velocities of the 23 beads in (a), we need to only solve for the hydrodynamic velocities of the two end beads and the three representative beads in (b). (c) The implicit hydrodynamic velocity technique is combined with an earlier string-of-rods idealization developed by the authors [6]. In it, the elastic response of a filament is calculated by considering each section of the filament as a cantilever beam, and by solving for the forces and bending degrees of freedom at the cantilever intersections. The drag force on each cantilever is determined from the hydrodynamic velocities of its representative bead and any associated end bead.

Figure 2

Two-dimensional solvent field around a bead moving with velocity $v$ in a fluid at rest. A bead moving in a dilute Stokesian solvent at rest induces a solvent flow around it. The induced solvent flow at any point can be determined as radial $\left(v_r\right)$ and tangential $\left(v_{\theta }\right)$ velocities, which are functions of the radial separation $r$ and angle from the flow $\theta$. In this study, the cubic-order term in the solvent velocity equations is neglected.

Figure 3

Method of Reflections. (a) Consider the case of two beads, radius $a$ and distance $r$ apart, moving at velocities $v_1$ and $v_2$ perpendicular to the line joining their centers. Let A and B be the location of the centers of beads $I$ and $II$. (b) Method of Reflections: For ease of description, consider $v_2=0$, and neglect the cubic-order term in the solvent velocity field of a bead (Fig. 2). The no-slip boundary condition requires that the solvent velocity and the bead velocity match at bead center. Due to the first-order superposition of the solvent field induced by bead I, the solvent velocity at bead II is no more zero. The Method of Reflections corrects for it by superposing a negative solvent field at B, so that the net velocity at Bead II is zero. Physically it means that the bead II is moving opposite to the local solvent flow around it, so that the observed velocity is zero. The corrective solvent field at bead II is now felt at bead I as $\left(3/2a/r{v}_1\right)3/2a/r$, leading to a mismatch again between the solvent and bead velocity at bead I. The Method of Reflections again corrects for the mismatch by adding on another solvent field at bead I. Physically the correction means that bead I is now moving relative to a local solvent velocity of $\left(3/2a/r{v}_1\right)3/2a/r$, instead of a local solvent at rest. Each solvent correction in the Method of Reflections can be seen as updating the relative/hydrodynamic velocity of a bead to ensure that solvent flow at it is induced by the recently updated hydrodynamic velocity of its neighbor.

Figure 4

The correction in the drag on a single bead from the presence of a neighboring bead moving at the same velocity, perpendicular (a) and parallel (b) to the center line. The plots show the drag correction predicted by the first-order superposition technique and by the implicit hydrodynamics technique. The predictions are compared against detailed calculations of the drag correction by Wakiya [15] and against experimental observations by Faxen [16].

Figure 5

Profile of bead hydrodynamic velocities along a rigid rod in normal, rotational, and parallel motion. (a) The normal hydrodynamic velocity profile of beads constituting a rigid rod in pure normal translation. The hydrodynamic velocities are scaled by the normal velocity of the rod. Three rod sizes of 20, 60, and 100 beads are shown, with the profiles displaced by 50, 30, and 10 beads, respectively. (b) The normal hydrodynamic velocity profile of beads constituting a rigid rod in pure rotation. The hydrodynamic velocities are scaled by the rotational velocity of the rod. Three rod sizes of 20, 60, and 100 beads are shown, with the profiles displaced by 5, 10, and 15 beads, respectively. (c,d) The parallel hydrodynamic velocity profile of beads constituting a rigid rod in parallel translation. The hydrodynamic velocities are scaled by the parallel velocity of the rod. Rod sizes of 20 [Fig. 5c] and 100 [Fig. 5d] beads are shown, with the profiles displaced by 50 and 10 beads, respectively.

Figure 6

Validation of the implicit hydrodynamic technique. (a) The scaled frictional drag or hydrodynamic velocity profile along a rigid rod of 101 beads in normal translation, compared against the predictions of Gluckman et al. [14]. (b) The total frictional drag on rigid rods of different lengths in normal and parallel motion, compared against standardized equations. The vertical axis shows the total frictional drag of the rod scaled by that of a single bead. The horizontal axis shows the number of beads constituting the rod. In the absence of hydrodynamic screening between the beads of the rod, the translational frictional drag would fall along the dashed line of slope 1. (c) The scaled frictional drag on rigid rods of different lengths in rotational motion, compared against standardized equations.

Figure 7

Generalized hydrodynamic influence of bead II on bead I.

Figure 8

The change in the normal hydrodynamic velocity profile of a rigid rod in normal translation from presence of a neighboring stationary rod. Panels (a), (b), and (c) show the change in the normal hydrodynamic velocities of rod I, for three angles of the stationary neighboring rod II: $0°$, $45°$, and $90°$, respectively. Panel (d) shows the relative positions of the two rods for the three cases. Rod I is shown as a black filled rod, and rod II is shown as a gray, unfilled rods. The distances of separation between the rods are 5, 10, 20 and infinite bead diameters.

Figure 9

Approximation of the bead hydrodynamic velocity profile of Fig. 5 by assuming all interior beads to have uniform normal, parallel, and rotational hydrodynamic velocities (that of the representative beads). The bead hydrodynamic velocity profiles are shown in gray and their approximations are shown in black.

Figure 10

Capturing the hydrodynamic influence of the beads of section $S_2$ on the representative bead of section $S_1$. (a) The distance $r_{j{c}_1}$ and the angle ${\theta }_{j{c}_1}$ between $c_1$ and bead $j$ of section $S_2$, vary for each bead when integrating through the beads of $S_2$. (b) A simple rotation of frame so that section $S_2$ lies along the horizontal axis, makes $r_{j{c}_1}$ and ${\theta }_{j{c}_1}$ a function of only one variable $x_{j{c}_1}$.

Figure 11

Averaged hydrodynamic profile of the 100-bead rod in Figs. 8a, 8b, 8c shown only for the case of a 10 bead separation from its neighbor. The rods were averaged using 10 and 20 sections.

Figure 12

Rods-on-string idealization of semiflexible filaments. (a) The rods-on-string idealization treats a semiflexible filament as a string of continuously bending rods or segments. The filament variables at the rod intersections or nodes are solved for. (b) The forces on a curved segment are resolved into their projections in two mutually perpendicular directions. The mutually perpendicular directions are taken to be along the line joining the segment ends $\left(u_1\right)$, and along the line perpendicular to it $\left(u_2\right)$. The projected forces in each direction are further resolved into a resultant force and couple in that direction. Solving for the force and moment balance equations over an instantaneously rigid rod requires only knowing the resultant forces and couples. In panel (b) only the distributed forces normal to the curved segment are considered.

Figure 13

Incorporating averaged implicit hydrodynamics in the string-of-rods idealization of semiflexible filaments. The string-of-rods idealization considers a semiflexible filament (shown in gray) as a continuously curved string of rods or “segments.” The average implicit hydrodynamics is used to solve the hydrodynamic velocities of the end beads (circles with solid outlines), and that of the central representative beads (circles with broken outlines) of the $u_1$-projections of the segments. The $u_1$-projection of each segment is shown in thin black lines and is along the direction joining the two ends of the segment. Therefore the $u_1$-projection of each segment constitutes a section in the implicit hydrodynamics technique. The hydrodynamic drag force due to the hydrodynamic velocities of an end bead enters the string-of-rods formulation as an end force on that segment. The hydrodynamic drag due to the hydrodynamic velocity of a representative bead enters the string-of-rods formulation as the resultant drag force/couple on the $u_1$-projection of that segment.

Figure 14

Validation of the rods-on-string idealization of semiflexible filaments with averaged implicit hydrodynamics. The simulation results for the Brownian fluctuations of a phalloidin-stained actin filament are shown ($12\mu \text{m}$ length, $7.3\text{e}-26\text{N}{\text{m}}^2$ persistence length, 0.0025 s simulation time interval). (a) The variances of the amplitude $\left(a_n\right)$ of the first nine bending modes $\left(n\right)$ are shown. The amplitudes of the bending modes are calculated from the simulation results by decomposing the filament conformation at each time step into Fourier series. Theoretical estimates of the amplitude variances can be made by assuming each bending mode to have $KT/2$ energy (equipartition theorem). The variances determined from the simulation closely match the theoretical estimates. (b) The time evolution of the variance of the bending amplitude for the first four modes is shown in black. Shown in gray are the theoretical estimates of the time evolution obtained by assuming small bending fluctuations and a constant hydrodynamic friction. The simulation results lie in the vicinity of the theoretical estimates.

Figure 15

Experimental validation of model dynamics. The time evolution of the mean-squared displacement of an actin filament is plotted as a function of time interval, $\Delta t$. The MSD was determined as $⟨r^2\left(t\text{−}\Delta t\right)⟩$. The simulation result for a $6\mu \text{m}$ actin filament is shown with gray diamonds. The experimental observation for homogenously and nonhomogenously labeled actin filaments is shown with unfilled and filled black diamonds, respectively. The experimental data were obtained from Bernheim-Groswasser et al. [24].

Figure 16

Hydrodynamic contribution to the Brownian fluctuations of a semiflexible filament. (a) The Brownian conformation of a $12\mu \text{m}$ actin filament at 0, 0.01, 0.02, and 0.03 s. The $x$ and $y$ positions of the filament nodes are plotted. (b,c,d) The normal velocities of the end and representative beads in the time intervals of 0–0.01 s (a), 0.01–0.02 s (b), and 0.02–0.03 s (c), plotted against the $x$ positions of the representative/end beads at time $t=0\text{s}$.

Figure 17

(Color online) The Brownian fluctuations of a $12\mu \text{m}$ actin filament in the unconfined and laterally confined state. The filament properties are similar to that in previous simulations. The simulations were performed in time intervals of 0.0025 s. The confined filament fluctuates between two other stationary filaments separated by 3, 1.5, and $1\mu \text{m}$ (from the left to right in figure). From top to down, the fluctuations of the filament at every 0.1 s intervals are shown. The filament conformation is obtained by fifth-order interpolation between the filament nodes (shown as black dots). The filament contour length between two nodes (black dots) is $0.8\mu \text{m}$.

Figure 18

Dynamics of laterally confined fluctuating actin filaments. (a) Variance of the first four bending amplitudes for different degrees of confinement. (b) Angle correlation along the filament length for different degrees of confinement. (c) Time correlation of the amplitude of the first bending mode $\left(a_1\right)$ for different degrees of confinement. (d) Time correlation of the amplitude of the second bending mode $\left(a_2\right)$ for different degrees of confinement. (e) Time correlation of the amplitude of the third bending mode $\left(a_3\right)$ for different degrees of confinement. (f) Time correlation of the amplitude of the fourth bending mode $\left(a_4\right)$ for different degrees of confinement.