Averaged implicit hydrodynamic model of semiflexible filaments

Preethi L. Chandran and Mohammad R. K. Mofrad*Molecular Cell Biomechanics Laboratory, Department of Bioengineering, University of California, Berkeley, California 94720, USA

�Received 7 June 2009; revised manuscript received 15 November 2009; published 26 March 2010�

We introduce a method to incorporate hydrodynamic interaction in a model of semiflexible filament dynam-ics. Hydrodynamic screening and other hydrodynamic interaction effects lead to nonuniform drag along evena rigid filament, and cause bending fluctuations in semiflexible filaments, in addition to the nonuniformBrownian forces. We develop our hydrodynamics model from a string-of-beads idealization of filaments, andcapture hydrodynamic interaction by Stokes superposition of the solvent flow around beads. However, insteadof the commonly used first-order Stokes superposition, we do an equivalent of infinite-order superposition bysolving for the true relative velocity or hydrodynamic velocity of the beads implicitly. We also avoid thecomputational cost of the string-of-beads idealization by assuming a single normal, parallel and angularhydrodynamic velocity over sections of beads, excluding the beads at the filament ends. We do not include theend beads in the averaging and solve for them separately instead, in order to better resolve the drag profilesalong the filament. A large part of the hydrodynamic drag is typically concentrated at the filament ends. Theaveraged implicit hydrodynamics methods can be easily incorporated into a string-of-rods idealization ofsemiflexible filaments that was developed earlier by the authors. The earlier model was used to solve theBrownian dynamics of semiflexible filaments, but without hydrodynamic interactions incorporated. We vali-date 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 afluctuating actin filament between two stationary lateral filaments. Finally, preliminary examinations suggestthat a large part of the observed velocity in the interior segments of a fluctuating filament can be attributed toinduced solvent flow or hydrodynamic screening.

DOI: 10.1103/PhysRevE.81.031920 PACS number�s�: 87.16.Ka, 87.16.Ln, 87.15.A�

I. INTRODUCTION

Hydrodynamic interaction refers to the force communica-tion between two particles mediated via the solvent. The mo-tion of a particle causes a solvent flow around it, which inturn exerts a drag on a neighboring particle. This drag forceon a particle from the solvent flow induced by a neighboringparticle is referred to as hydrodynamic force.

Semiflexible filaments are polymers whose contourlengths are on scale of their persistence length, so that theirthermally driven bending fluctuations are resisted by theirelastic bending stiffness �1,2�. Semiflexible filaments makeup the structural scaffold of cell and tissue matrices �actinand microtubules in cells, collagen in tissue�, and under-standing their dynamic behavior is critical to understandingforce transmission and remodeling in cell and tissue matri-ces.

A semiflexible filament is subject to hydrodynamic forcesdue to solvent flows induced by the bending and diffusivemotions both within itself and from its neighbors. However,dynamic models of semiflexible filaments typically neglectthe drag forces arising from the induced solvent flow. Forinstance, consider a single semiflexible filament in dilute so-lution. Typical dynamic models assume the friction coeffi-cient to be uniform along the filament length, and equal to

that of a unit rigid cylinder of same diameter �2–6�. How-ever, Lagamarsino et al. �7� showed that when a rigid fila-ment is modeled as a string-of-beads and the hydrodynamicinteraction between beads accounted for, the drag force andfriction coefficient along the filament is not uniform evenwhen under a uniform force. The drag force is much greaterat the beads toward the filament ends, and decreases at thebeads toward the filament center �7�. While such nonuniformdrag might not affect the overall motion of a rigid filamentunder uniform force, it will cause a semiflexible filament tobend under uniform force-a phenomenon that will be missedby semiflexible filament models assuming constant frictioncoefficient.

The nonuniform drag along a rigid filament is due to hy-drodynamic screening. The solvent flow induced by the mo-tion of the filament ends carry the central region of the fila-ment with it. Since the central region is now partly drifting inthe induced solvent flow, the relative velocity �observed ve-locity minus solvent velocity�, and therefore drag experi-enced by it, is reduced. The phenomenon is akin to draftingin bicycle or car racing.

The observations of nonuniform drag due to hydrody-namic screening and other hydrodynamic interaction effectssuggest that a part of the bending fluctuations of a semiflex-ible filament may be caused by the nonuniform drag, in ad-dition to the nonuniform Browian forces. Therefore, a modelof semiflexible filament dynamics that is used to interpretexperimental observation needs to include the effects of hy-drodynamic interaction, originating from within itself andfrom neighboring filaments.

The common approaches to capturing hydrodynamic in-teraction in rigid cylinders tend to employ two conceptual

*Corresponding author. Present address: Department of Bioengi-neering, University of California Berkeley 208A Stanley Hall#1762 Berkeley, CA 94720-1762; FAX: �510� 642-5835;mofrad@berkeley.edu

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ideas. The first is to idealize the cylinder as a string of beads�8,9�, and the second is to determine the hydrodynamic in-teraction between the beads by first-order superposition ofStokesian solvent fields �10�. Idealizing a filament as a stringof beads has the advantages that the solvent flow equationsaround a moving bead are well established and that the ge-ometry between interacting beads is relatively easy tohandle. The chief disadvantage, however, is that solving forall beads in a string-of-beads approach makes it computa-tionally expensive to model even a single filament �9�. Nowconsider the second idea. Stokesian dynamics is commonlyused to describe solvent flow in these settings because of thesmall particle diameters and small velocities involved �11�.Stokesian flow equations are linear. Therefore the solventvelocity at any point in an assembly of beads can be esti-mated, to first order, by summing the solvent flows due to themotion of each bead alone. The hydrodynamic drag on abead can then be determined from its observed velocity rela-tive to the solvent velocity at its center. The first-order su-perposition approach to approximating induced solvent flowhas the advantage that it is relatively simple to implement.However the disadvantage is that it neglects the reflection ofthe induced solvent flow between the beads, a higher-ordersuperposition effect. As a result, the first-order superpositionoverlooks the hydrodynamic influence from neighboring par-ticles which are stationary, and overestimates the hydrody-namic influence from neighboring particles that are drifting.

The aim of this paper is to introduce a method for includ-ing hydrodynamic interaction effects on a fluctuating semi-flexible filament. The proposed method is still based on thetwo conceptual ideas discussed above and therefore retainstheir advantages. But it avoids the disadvantages associatedwith them: We retain the underlying ideas of idealizing thefilament as a string of beads and of determining the hydro-dynamic drag on the beads by Stokes superposition of thesolvent fields around each bead. However instead of a first-order superposition, we do an infinite-order superposition bycasting the relative velocity of a bead �observed velocity mi-nus solvent velocity at bead center� as a separate variable andsolving for it implicitly. We call the relative velocity of thebead as its hydrodynamic velocity, because the hydrody-namic drag on the bead is proportional to this relative veloc-ity. The one-step implicit solution is equivalent to superpos-ing many back and forth reflections of the solvent velocityfields, and therefore eliminates the errors of first-order Stoke-sian superposition. We then reduce the computation cost as-sociated with the string-of-beads idealization by grouping allbut the end beads into sections, and assuming uniform trans-lational and rotational velocities for each section. This re-duces the number of variables and equations to be solved. Ingrouping the velocities of the interior beads while still sepa-rately solving for the velocities of the end beads, we makeuse of the observations that the hydrodynamic drag typicallychanges sharply at the filament ends and changes slowly overthe filament interior �see Sec. II B�.

The rest of the paper is organized as follows. In Sec. II weshow the theoretical development of the proposed approachorganized in three parts. In part A we show the formulationof the implicit string-of-bead hydrodynamics. The hydrody-namic drag on each bead in a string-of-beads idealization is

determined by implicitly solving for its relative or hydrody-namic velocity. In part B we show the formulation of theaveraged implicit string-of-beads hydrodynamics. By assum-ing sections of interior beads to have a representative trans-lational and rotational velocity, the implicit hydrodynamics issolved over fewer beads. In part C we show how the aver-aged implicit hydrodynamics method can be incorporatedinto an earlier model of a semiflexible filament in Brownianmotion, which had assumed uniform friction coefficientalong the filament �6�. An overall schematic of the proposedapproach is shown in Fig. 1. In Sec. III we use the newmodel to simulate the Brownian fluctuation of a free andconfined actin filament. We show that hydrodynamic interac-tion may have systematic effects on the Brownian conforma-tions assumed by the filament, warranting detailed follow-upstudies. We also show that hydrodynamic interaction withneighbors appears to confine a filament by increasing thefilament’s persistence length.

II. METHODS AND VALIDATION

A. Implicit string-of-beads hydrodynamics

The hydrodynamic drag on a bead is determined from itstrue relative velocity or hydrodynamic velocity, which is itsobserved velocity relative to the solvent flow at the beadcenter. The first-order Stokesian superposition approach ob-tains the solvent velocity at a bead as the summation of thesolvent velocities due to each neighboring bead. The Methodof Reflections improves on the first-order estimate by itera-tively adding corrections so that the superposed solvent

(a)

(b)

(c)

FIG. 1. Implicit and averaged hydrodynamics of semiflexiblefilaments. �a� In a string-of-beads realization of a semiflexible fila-ment, the hydrodynamic drag on each bead can be determined bysolving for the true relative velocity or hydrodynamic velocity ofeach bead. �b� Since the hydrodynamic velocity/drag is much largerat the end beads of a filament, and since the hydrodynamic veloci-ties of the interior beads change slowly, sections of the interiorbeads �shown within rectangles� can be approximated by hydrody-namic velocities of representative beads within them �gray beads�.Therefore instead of solving for the hydrodynamic velocities of the23 beads in �a�, we need to only solve for the hydrodynamic veloci-ties of the two end beads and the three representative beads in �b�.�c� The implicit hydrodynamic velocity technique is combined withan earlier string-of-rods idealization developed by the authors �6�.In it, the elastic response of a filament is calculated by consideringeach section of the filament as a cantilever beam, and by solving forthe forces and bending degrees of freedom at the cantilever inter-sections. The drag force on each cantilever is determined from thehydrodynamic velocities of its representative bead and any associ-ated end bead.

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flows are induced by the hydrodynamic velocities and not theobserved velocities of the neighbors �10�. We use the sameidea to improve on the first-order superposition estimate ofthe solvent field. However, instead of iteratively correctingfor the hydrodynamic velocities as in the Method of Reflec-tions, we solve for them implicitly. We illustrate this in moredetails below in the context of the hydrodynamic interactionbetween two beads moving alongside each other.

1. First-order linear superposition

Fig. 2 shows the two-dimensional solvent velocity fieldaround a bead moving in a fluid at rest. The solvent fieldaround the bead can be resolved into radial and tangentialvelocities, which can be determined at any point using thedistance to the bead center and the angle relative to the beadvelocity. Now consider two beads, I and II, moving withvelocity v1 and v2 perpendicular to the line joining theircenters �Fig. 3�a��. A simple first-order superposition of theirsolvent velocity fields would estimate their relativevelocities1 �v1

H and v2H� as,

v1H = v1 −

3a

4�r12�v2, �1a�

v2H = v2 −

3a

4�r12�v1. �1b�

However, if bead II were stationary �v2=0�, then by Eq. �1a�,bead I would experience no hydrodynamic influence from it.The equation does not capture the hydrodynamic drag onbead I due to its solvent flow being reflected back by bead II.Alternatively if bead II were drifting in the solvent flow in-duced by bead I �v2=v1� 3a

4�r12� ��, then by Eq. �1a�, bead Iwould still experience a hydrodynamic influence from beadII, even though it is the solvent flow from its own motionthat is causing bead II to move. Higher order superpositionsof the solvent field would correct for these effects, and theMethod of Reflections applies them iteratively �12,13�.

2. Method of reflections

The Method of Reflections is an iterative technique thatestimates the relative velocities and therefore hydrodynamicdrag of particles in a Stokesian solvent, by explicitly addinghigher-order corrections to the solvent field in each iteration�14�. Mathematically, each iteration serves to correct the sol-vent velocity field so that the no-slip boundary condition isenforced at the particle center �see Fig. 3�b� for more de-tails�. Physically, this translates to iteratively correcting the

1Throughout the paper we will assume that the overall solventvelocity is zero. We will use the term “relative velocity” to refer tothe particle velocity relative to the local solvent velocity, and notrelative to the overall solvent velocity.

v

v�

vrr

�

FIG. 2. Two-dimensional solvent field around a bead movingwith velocity v in a fluid at rest. A bead moving in a dilute Stoke-sian solvent at rest induces a solvent flow around it. The inducedsolvent flow at any point can be determined as radial �vr� and tan-gential �v�� velocities, which are functions of the radial separation rand angle from the flow �. In this study, the cubic-order term in thesolvent velocity equations is neglected.

r

2a

v1 v2

2a

I II

‘First reflection’

‘Second reflection’

‘Third reflection’

(b)

(a)

FIG. 3. Method of Reflections. �a� Consider the case of twobeads, radius a and distance r apart, moving at velocities v1 and v2

perpendicular to the line joining their centers. Let A and B be thelocation of the centers of beads I and II. �b� Method of Reflections:For ease of description, consider v2=0, and neglect the cubic-orderterm in the solvent velocity field of a bead �Fig. 2�. The no-slipboundary condition requires that the solvent velocity and the beadvelocity match at bead center. Due to the first-order superposition ofthe solvent field induced by bead I, the solvent velocity at bead II isno more zero. The Method of Reflections corrects for it by super-posing a negative solvent field at B, so that the net velocity at BeadII is zero. Physically it means that the bead II is moving opposite tothe local solvent flow around it, so that the observed velocity iszero. The corrective solvent field at bead II is now felt at bead I as�3 /2a /rv1�3 /2a /r, leading to a mismatch again between the sol-vent and bead velocity at bead I. The Method of Reflections againcorrects for the mismatch by adding on another solvent field at beadI. Physically the correction means that bead I is now moving rela-tive to a local solvent velocity of �3 /2a /rv1�3 /2a /r, instead of alocal solvent at rest. Each solvent correction in the Method of Re-flections can be seen as updating the relative/hydrodynamic velocityof a bead to ensure that solvent flow at it is induced by the recentlyupdated hydrodynamic velocity of its neighbor.

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hydrodynamic influence on a particle to originate from thetrue relative velocity and not the observed velocity of itsneighbors. For instance in the case of two beads movingalongside �Eqs. �1��, the next order correction in the Methodof Reflections would be to the hydrodynamic velocity ofbead I,

v1H = v1 −

3a

4�r12��v2 −

3a

4�r12�v1� , �2a�

where the term within the brackets is but the first-orderhydrodynamic/relative velocity of bead II �Eq. �1b��. Thecorrection is equivalent to saying that it is the relative veloc-ity of bead II, and not its observed velocity, that causes thesolvent flow at bead I. Similarly, the next order correction ofthe hydrodynamic velocity of bead II would appear as

v2H = v2 −

3a

4�r12��v1 −

3a

4�r12��v2 −

3a

4�r12�v1� , �2b�

where the term in the outer brackets is the previously up-dated hydrodynamic or relative velocity of bead I �Eq. �2a��.To summarize, the iterative corrections of the Method ofReflections ensure that it is the updated hydrodynamic ve-locities that induce solvent flow and exert hydrodynamic ef-fects. However, such an explicit iterative scheme becomesdifficult to implement once the number of particles increases,even if only two reflections or correction orders are consid-ered.

3. Implicit hydrodynamics

Instead of iteratively updating the hydrodynamic velocitywith each reflection, the hydrodynamic velocity can be con-sidered as a separate variable and solved for implicitly. Forthe case of two beads moving perpendicular to their centerline �Fig. 3�a��, the implicit version of Eqs. �2� would be

v1H = v1 −

3a

4�r12�v2

H �3a�

v2H = v2 −

3a

4�r12�v1

H �3b�

By solving these two equations simultaneously, the hydrody-namic velocities, and therefore the hydrodynamic drags, ofthe two beads can be determined.

Figure 4 shows the predictions of the implicit techniquefor the change in the drag of a bead in the presence of an-other bead. Both beads are of the same radius, moving withthe same velocity perpendicular and parallel to the linethrough their centers �see insets in Fig. 4�. Note that theprefactor 3/4 in Eq. �3� changes to 3/2 for two beads movingparallel to their center line �see solvent field equations in Fig.2�. The drag correction is determined as the ratio of thebead’s hydrodynamic velocity to its apparent relative veloc-ity, vH /v. The predictions of the implicit method show abetter match to rigorous calculations �15,16� and experimen-tal observations �17,18�, than the predictions of a first-ordersuperposition. The implicit technique predicts a smaller dragreduction than the first-order superposition technique. This is

because a component of a bead’s velocity comes from drift-ing in the solvent flow induced by its neighbor. In the im-plicit technique, that component does not exert a drag-reducing hydrodynamic influence on the bead’s neighbor.Finally, note that the solution of the first-order superpositionis the same as that given by the first reflection in the Methodof Reflections, and the solution of the implicit technique isthe same as that given by infinite reflections in the Method ofReflections. The error between the implicit and theexperimental/rigorous solutions in Fig. 4 is not because ofStokes superpositions, but comes from the approximations inthe equations describing solvent flow around a bead �Fig. 2��10�.

4. Implicit hydrodynamics for a rigid rod in dilute solution

The hydrodynamic drag along a rigid rod can be esti-mated by idealizing it as a string of beads, and solving forthe hydrodynamic velocities of the beads. We resolve thevelocities of a rigid rod and a bead in the following way. The

(a)

(b)

FIG. 4. The correction in the drag on a single bead from thepresence of a neighboring bead moving at the same velocity, per-pendicular �a� and parallel �b� to the center line. The plots show thedrag correction predicted by the first-order superposition techniqueand by the implicit hydrodynamics technique. The predictions arecompared against detailed calculations of the drag correction byWakiya �15� and against experimental observations by Faxen �16�.

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two-dimensional motion of a rigid rod can be described com-pletely by three velocities: the translational velocity of therod center in the direction normal to the rod �Vn�, the trans-lational velocity of the rod center in the direction parallel tothe rod �Vp�, and the rotational velocity about the rod center�W�. The motion of any bead within the rod can be describedcompletely by two velocities: the translational velocity nor-mal to the rod �vn�, and the translational velocity parallel tothe rod �vp�. We use upper-case letters to denote rod veloci-ties, and lower-case letters to denote bead velocities. Therelation between the velocities of the ith bead and that of therigid rod are,

vin = Vn + Wric, �4a�

vip = Vp, �4b�

where ric is the distance of the ith bead from the rod center c.The true relative velocity or the hydrodynamic velocity of

the ith bead can be determined by extending Eq. �3� to in-clude the hydrodynamic influence from the rest of the beadswithin the rod. Using appropriate prefactors for the hydrody-namic influence,

�vipH

vinH � = � Vp

Vn + ricW� −

j=1,j�i

M �3aj

2�rij�3aj

4�rij�� • �v j

pH

v jnH � . �5�

Here M is the number of beads composing the rod, and vipH

and vinH are the hydrodynamic velocities of the ith bead,

parallel and normal to the rod.The total drag force experienced by a rigid string of M

beads is known to be much smaller than the total drag forceexperienced by M separate beads. This is due to hydrody-namic screening—the solvent flow set up by the motion ofone bead reduces the relative velocity and therefore dragexperienced by a neighboring bead. Figures 5�a� and 5�b�show the hydrodynamic velocity profile of the beads along arigid rod in pure normal and rotational motion, scaled re-spectively by the normal and rotational velocity of the rod.The profile was determined by solving Eq. �5� simulta-neously for all beads within the rod �for i=1 to M�. Theprofiles are shown for rod sizes of 20, 60, and 100 beads. Fora rigid rod in pure normal translation, the normal velocity ofeach bead is uniform along the rod; but the normal hydrody-namic velocity �Fig. 5�a�� is higher at the rod’s ends anddecreases toward the interior, giving a “U” shape. For a rigidrod in pure rotation, the normal velocity of each bead in-creases linearly in opposite directions away from the rodcenter. The normal hydrodynamic velocity �Fig. 5�b��, how-ever, increases faster at the rod ends than in the rod interior.The highest hydrodynamic velocities, and therefore drag, oc-cur on the bead at each rod end. We refer to them as “endbeads,” and refer to the rest of the beads as “interior beads.”The hydrodynamic velocities in the interior beads decreasetoward the center of the rod. It appears that the beads in thecentral region of the rod are nearly drifting in the solventflow setup by the motion of their neighbors on either side,and therefore experience a decreased relative velocity and

drag. The end beads, on the other hand, do not experience thefull benefit of hydrodynamic screening and have larger hy-drodynamic velocities.

For a rigid rod in pure parallel and tangential motion, theparallel velocity of each bead is uniform along the rod. InFigs. 5�c� and 5�d�, we show the parallel hydrodynamic ve-locities of the beads, for rods of 20 and 100 bead sizes. Thehydrodynamic velocities are scaled by the parallel velocityof the rod. The profile shows large bead-to-bead fluctuationsin hydrodynamic velocities, with no definite overall trend.We found these fluctuations to be mathematical artifacts.They disappear upon inclusion of the cubic term in theStokes solution for the solvent field around a bead �Fig. 2�,and the profile takes on a U shape similar to that of thenormal hydrodynamic velocities in Fig. 5�a� �data notshown�. However, we omit the cubic-order term and the at-tendant increase in mathematical complexity because �1� theomission of the cubic-order term leads the less than 5% errorin the total drag calculations for a rigid filament, and �2� thefluctuations disappear when the hydrodynamic velocities aredetermined as section averages �see Sec. II B�.

In Fig. 6 we show validations of the filament hydrody-namic profile obtained with the implicit method. First, the

(a) (b)

(c) (d)

FIG. 5. Profile of bead hydrodynamic velocities along a rigidrod in normal, rotational, and parallel motion. �a� The normal hy-drodynamic velocity profile of beads constituting a rigid rod in purenormal translation. The hydrodynamic velocities are scaled by thenormal velocity of the rod. Three rod sizes of 20, 60, and 100 beadsare shown, with the profiles displaced by 50, 30, and 10 beads,respectively. �b� The normal hydrodynamic velocity profile of beadsconstituting a rigid rod in pure rotation. The hydrodynamic veloci-ties are scaled by the rotational velocity of the rod. Three rod sizesof 20, 60, and 100 beads are shown, with the profiles displaced by5, 10, and 15 beads, respectively. �c,d� The parallel hydrodynamicvelocity profile of beads constituting a rigid rod in parallel transla-tion. The hydrodynamic velocities are scaled by the parallel veloc-ity of the rod. Rod sizes of 20 �Fig. 5�c�� and 100 �Fig. 5�d�� beadsare shown, with the profiles displaced by 50 and 10 beads,respectively.

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increased drag at the ends for a rigid rod in normal transla-tion is in agreement with the observations of Lagamarsino etal. �7�. Figure 6�a� shows that the normal hydrodynamic ve-locity profile for a string of 101 beads compares well againstthat determined by Gluckman et al. �14� using detailed cal-culations. In Figs. 6�b� and 6�c�, we show the total frictiondrag predicted for a rigid rod in translation and rotation, bythe implicit method and by standard equations �Eqs.

�6a�–�6d��. The friction drag is scaled by the velocity of therigid rod. In the implicit method, the total friction drag is

calculated as i=1M vi

jH

Vj with j=n , p ,r for rods in normal, par-allel and rotational motion. The predictions of the implicitmethod compare well against standard expressions for� j /�sph �Eqs. �6a�–�6d��, where � j is the parallel, normal androtational friction coefficient of a rigid cylinder �for j= p ,n ,r, respectively�, and �sph is the friction coefficient of asphere.

�p =8��L

ln�L

d+ �p , �6a�

�n =4��L

ln�L

d+ �n , �6b�

�r =��L3

3 ln�L

d+ �r , �6c�

and �sph = 6��a , �6d�

where K, T, and � are the Boltzmann constant, temperatureand solvent viscosity, respectively. �p, �n, and �r are correc-tion terms added to the frictional coefficient equations toinclude the effects of the cylinder ends on the cylinder dif-fusion. The correction terms are equal to −0.114, 0.886, and−0.447, respectively �19,20�. It is interesting to note that theimplicit technique intrinsically accounts for these end correc-tions.

5. Generalized equations for implicit hydrodynamics

In Sec. II A 2 we described the implicit method for twobeads moving parallel and perpendicular to their center line.The equations in Sec. II A 2 can be easily extended to cap-ture the hydrodynamic influence between beads moving atarbitrary velocities and directions. We demonstrate for thecase of two beads which are parts of two different rods sepa-rated by an arbitrary angle and distance �Fig. 7�.

Let beads I and II be part of two separate rods. LetV1

p ,V2p and V1

n ,V2n be the parallel and normal translational

velocities of the rods, and let �1 ,�2 be their orientationangles �Fig. 7�. Let v1

p ,v2p and v1

n ,v2n be the corresponding

parallel and normal velocities of the beads of interest, and let�12 and r12 be the angle and distance between them �Fig. 7�.The hydrodynamic influence of bead II on bead I can bedetermined as

�v1pH

v1nH � = �V1

p

V1n � − H12�v2

pH

v2nH � , �7�

where

(a)

(b)

(c)

FIG. 6. Validation of the implicit hydrodynamic technique. �a�The scaled frictional drag or hydrodynamic velocity profile along arigid rod of 101 beads in normal translation, compared against thepredictions of Gluckman et al. �14�. �b� The total frictional drag onrigid rods of different lengths in normal and parallel motion, com-pared against standardized equations. The vertical axis shows thetotal frictional drag of the rod scaled by that of a single bead. Thehorizontal axis shows the number of beads constituting the rod. Inthe absence of hydrodynamic screening between the beads of therod, the translational frictional drag would fall along the dashed lineof slope 1. �c� The scaled frictional drag on rigid rods of differentlengths in rotational motion, compared against standardizedequations.

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H12 = �cos��12 − �1� − sin��12 − �1�sin��12 − �1� cos��12 − �1�

�

��3a

2�r12�cos �12

3a

2�r12�sin �12

−3a

4�r12�sin �12

3a

4�r12�cos �12

� �8�

H12 is a matrix describing the hydrodynamic influence ofbead II on bead I. It has two component matrices �Eq. �8��.The right-side matrix determines the radial and tangentialsolvent field due to bead II at the location of bead I. Multi-plication by the left-side matrix projects the radial and tan-gential solvent velocities at bead I in the directions of itsparallel and normal velocities. Following trigonometric sim-plification, H12 can be reduced to

H12 =3

4

a

�r12�� cos �1 sin �1

− sin �1 cos �1�

�� 1 + cos2 �12 cos �12 sin �12

cos �12 sin �12 1 + sin2 �12� . �9�

The right-side matrix gives the solvent velocity that is in-duced by bead II at bead I, projected along the x and ydirections. The left-side matrix projects that solvent velocityinto directions parallel and normal to bead I.

Figure 8 illustrates the use of Eqs. �7� and �9� to deter-mine the change in the hydrodynamic velocity profile of arigid rod in normal translation �rod I�, from the presence ofneighboring rigid rod which is stationary �rod II�. The plotsare determined for three different angles and distances be-tween the rods �Fig. 8�d��. The normal translation of rod I isopposed by the presence of the stationary rod II, and corre-spondingly the normal hydrodynamic velocities �and there-fore drag� increases in regions close to rod II �Fig.8�a�–8�c��. For instance, the bead hydrodynamic velocities ofrod I increase in the region away from its center, due to aparallel neighboring rod located there; and the increase isgreater toward the rod end �Fig. 8�a�, and case �a� in Fig.8�d��. When the neighboring rod is located perpendicular to-ward the center of rod I, the normal hydrodynamic velocities

increase locally there �Fig. 8�b�, and Fig. 8�d� case �b��. Asexpected, the hydrodynamic velocity profile due to a neigh-boring rod that is positioned at an angle, appears to be somecombination of the profiles due to a parallel and perpendicu-lar neighbor �Fig. 8�c�, and Fig. 8�d� case �c��. It is interest-ing that the increase in hydrodynamic velocities occurs onlyin regions adjacent to the neighbor, and the increase becomesnegligible by 50 bead diameters of separation �not shown�.

We also compared the sedimentation velocity of a tet-ramer �4 beads at the corner of a square, and touching eachother� that is predicted by the implicit technique, against thatmeasured experimentally in Swanson et al. �21�. The implicithydrodynamics technique experimental method gave a sedi-mentation velocity of 0.4447�0.0004 relative to that of afree bead �21�.

B. Averaged implicit hydrodynamics

While the implicit method is relatively straightforwardand captures the end effects and hydrodynamic screeningthat govern the diffusion of rigid rods, it is computationallyexpensive. The technique introduces at least two new hydro-dynamic variables per bead, and the computational cost of animplicit matrix solution scales as N2, where N is the totalnumber of variables.

1. Averaged implicit hydrodynamics for a rigid rod

The normal and parallel hydrodynamic velocity profilesalong a rigid rod in pure translation or rotation �Figs. 5�a�

�12

r12

Bead I

Bead II

�1

�2Rod II

Rod I

FIG. 7. Generalized hydrodynamic influence of bead II on beadI.

100 bead diameters

(c)

(b)

(a)

Neighbor rods

50 beaddiameters Separation distance

Rod I

Rod II

(a) (b)

(c) (d)

FIG. 8. The change in the normal hydrodynamic velocity profileof a rigid rod in normal translation from presence of a neighboringstationary rod. Panels �a�, �b�, and �c� show the change in the nor-mal hydrodynamic velocities of rod I, for three angles of the sta-tionary neighboring rod II: 0°, 45°, and 90°, respectively. Panel �d�shows the relative positions of the two rods for the three cases. RodI 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.

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and 5�b�� suggests that sections of the interior beads can beapproximated by a single normal, parallel and rotational hy-drodynamic velocity. That is, the normal and parallel hydro-dynamic velocity profile over a section of the interior beadscan be approximated from the normal, parallel and rota-tional hydrodynamic velocity of the central bead, c, in thesection. For the ith bead in a section, the approximated nor-mal and parallel hydrodynamic velocities are

vinH = vc

nH + wcHric, �10a�

vipH = vc

pH. �10b�

We refer to the central bead in a section as the representativebead of that section. The parallel hydrodynamic velocity pro-file of the interior beads fluctuate over a small length scalewith no definite trend �Fig. 5�c� and 5�d��. Therefore no use-ful information is lost when fluctuating velocities are re-placed by a constant value. In all, by assuming each sectionof interior beads to have constant hydrodynamic velocities,the implicit hydrodynamic equations need to be written andsolved for only one representative bead per section, and foreach end bead. This considerably reduces the cost of theimplicit solution. Also, by not considering the end beads aspart of a section, the large localized drag on them is stillcaptured, and the hydrodynamic profile over the rod is moreeffectively approximated.

The angular hydrodynamic velocity of a representativebead cannot be solved in a form similar to the normal hydro-dynamic equation. It will result in a trivially zero equation ifa representative bead were located in the center of the rod�rcc=0�. Instead, we determine the hydrodynamic angularvelocity about the representative bead as the differential of itsnormal hydrodynamic velocity profile there. That is, we de-termine it as the change in normal hydrodynamic velocityexperienced by the bead, if it translated an infinitesimal dis-tance along the rod. If c be the representative central bead ofa section,

wcH =

1

2a

d

di�vi

nH��c. �11�

For simplicity, we first demonstrate the averaging techniqueby grouping the interior beads of a rigid rod into a single

section. The hydrodynamic equation for the normal, paralleland rotational hydrodynamic velocities about bead c can bewritten using Eqs. �5� and �11�, and grouping beads with thesame hydrodynamic velocities,

�vcpH

vcnH

wcH � · �

1 + �j=2,j�c

M−13a

2�rcj�

1 + �j=2,j�c

M−13a

4�rcj�

1 + �j=2,j�c

M−1 � 3a

4�rcj�� �

Hydrodynamic influence of sectionon its representative bead

= �Vp

Vn

W� − �

j=1,M �3a

2�rcj�0 0

03a

4�rcj�0

03a

4�rcj�20�

Hydrodynamic influence of end beadon representative bead of section

· �v jpH

v jnH

0�

�12�

The summation in the left hand side of Eq. �12� is over allinterior beads comprising the section. The summation in theright hand side of the equation �Eq. �12�� is over the endbeads, j=1,M. The first matrix gives the hydrodynamic in-fluence on bead c from within its section, and the secondmatrix gives the hydrodynamic influence on bead c from theend beads. We use the symbols HS and HSE to refer to thetwo matrices, respectively.

Similarly the hydrodynamic equations for each end beadcan be written in the single-section averaging scheme. Notethat the angular hydrodynamic velocity equation is not re-quired for the end beads. Writing Eq. �5� for i=1,M andgrouping the interior beads with the same hydrodynamicvelocities,

�vipH

vinH � = � Vp

Vn + Wric� −��

j=2

M−13a

2�rij�0 0

0 �j=2,

M−13a

4�rij��j=2

M−1

rjc3a

4�rij��

Hydrodynamic influence of section on end bead �HES�

�vcpH

vcnH

wcH � −

��vkpH

vknH �k=1,M

k�i�

3a

2�rik�0

03a

4�rik��

Hydrodynamic influence of endbead on end bead �HEE� �13�

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The summation in the first matrix is over all interior beadscomprising the section of beads. The first matrix gives thehydrodynamic influence of the section on the end bead�HES�, and the second matrix gives the hydrodynamic influ-ence of one end bead on another �HEE�.

To summarize, the averaging procedure therefore consistsof two ideas. The first idea is that the normal and parallelhydrodynamic velocity profiles over sections of interiorbeads can be approximated by the normal, parallel and rota-tional hydrodynamic velocity of the central bead in that sec-tion. The second is that the angular hydrodynamic velocityabout the representative bead can be determined by the dif-ferential of the normal hydrodynamic velocity profile there.It is important to note that in this averaging technique, thehydrodynamic equation written for a bead still includes thehydrodynamic influences of all its neighboring beads. How-ever, the number of beads for which the hydrodynamic equa-tions need to be solved is reduced, thereby decreasing thecomputational cost.

In Figs. 9�a�–9�d� we show the single-section averagingof the hydrodynamic velocity profiles of Figs. 5�a�–5�d�. Theinterior beads have uniform normal, rotational and parallelhydrodynamic velocities for the respective cases of a rod inpure normal, rotational, and parallel motion; and the endbeads show much higher hydrodynamic velocities. The totaldrag calculated by summing the averaged profile matchesthat predicted by standard equations and by the implicitstring-of-beads method �see Figs. 6�b� and 6�c��.

2. Averaged implicit hydrodynamics for multiple sectionsand rigid rods

The above averaging technique can be extended to theinteraction between multiple sections within a rod, to the

interaction between multiple rods. The hydrodynamic equa-tions now require generalized expressions for the hydrody-namic influence of one end bead or section of beads on therepresentative bead of a section or on an end bead. The hy-drodynamic influence of an end bead on another end bead orrepresentative bead can be determined using Eq. �9� of Sec.II A 4. The equation describes the hydrodynamic influence ofone arbitrarily placed bead on another. Here we describe thecalculation of the hydrodynamic influence of one section ofbeads on the representative bead of another section. The hy-drodynamic influence of a section on an end bead is similarlycalculated, except that the self interaction matrix HS is unityfor an end bead �the summation terms in HS of Eq. �12� donot exist for an end bead�.

Consider two sections, S1 and S2, which are parts of twoseparate rigid rods �Fig. 10�. Let V1

p ,V2p and V1

n ,V2n be the

parallel and normal translational velocities of the sections; �1

and �2 be their orientation angles; and c1 and c2 be theirrepresentative central beads. Let �c1j and rc1j be the angleand distance between c1 and the jth bead of section S2 �Fig.10�. Let sections S1 and S2 be composed of M interiorbeads. Using the previously described notation for the hydro-dynamic velocities, the equation for the hydrodynamic

(a) (b)

(c) (d)

FIG. 9. Approximation of the bead hydrodynamic velocity pro-file of Fig. 5 by assuming all interior beads to have uniform normal,parallel, and rotational hydrodynamic velocities �that of the repre-sentative beads�. The bead hydrodynamic velocity profiles areshown in gray and their approximations are shown in black.

(b)

c1

S2

S1

c2 Bead j

(a)

c2

�1

�2

S1

c1

S2Bead j

FIG. 10. Capturing the hydrodynamic influence of the beads ofsection S2 on the representative bead of section S1. �a� The distancerjc1

and the angle � jc1between c1 and bead j of section S2, vary for

each bead when integrating through the beads of S2. �b� A simplerotation of frame so that section S2 lies along the horizontal axis,makes rjc1

and � jc1a function of only one variable xjc1

.

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influence of sections S1 and S2 on the representative bead ofsection S1 is given as

vc1

pH

vc1

nH

wc1

H � • HS1= V1

p

V1n

W1� − HS1S2 vc2

pH

vc2

nH

wc2

H � , �14�

where

HS1=

3

4a�

1 + j=1,j�c

M3a

2�rc1j�

1 + j=1,j�c

M3a

4�rc1j�

1 + j=1,j�c

M3a

4�rc1j�

�j�S1

, �15�

HS1S2= � cos �1 sin �1 sin �1

− sin �1 cos �1 cos �1

− sin �1 cos �1 cos �1�

��− �

j=1

M �1 + cos2 �c1j

rc1j�dj − �

j=1

M � sin �c1j cos �c1j

rc1j�dj − �

j=1

M

rc2j� sin �c1j cos �c1j

r1j�dj

− �j=1

M � sin �c1j cos �c1j

rc1j�dj − �

j=1

M �1 + sin2 �c1j

rc1j�dj − �

j=1

M

rc2j�1 + sin2 �c1j

rc1j�dj

d

ds�

j=1

M � sin �c1j cos �c1j

rc1j�dj

d

ds�

j=1

M �1 + sin2 �c1j

rc1j�dj

d

ds�

j=1

M

rc2j�1 + sin2 �c1j

rc1j�dj�

j�S2

, �16�

whered

ds=

1

2a� d

di�

c. �17�

HS1 is the matrix showing the hydrodynamic influence of S1on its central representative bead c1, and is similar to HS inEq. �12�. HS1S2 is the matrix showing the hydrodynamic in-fluence of the beads of S2 on the representative bead c1. It isderived from the matrix describing the generalized hydrody-namic influence of one bead on another �Eq. �9��, but isintegrated over all beads of section S2 to capture the hydro-dynamic influence of section S2 on c1. Also, HS1S2 includesthe hydrodynamic influence of section S2 on the averagedangular hydrodynamic velocity of S1—obtained as the dif-ferential change in the normal hydrodynamic velocity profilealong S1 at c1, induced by the presence of section S2.

The integrals in HS1S2 are difficult to evaluate as both rc1jand �c1j vary from bead to bead along section S2. Instead, weevaluate them by rotating the frame so that S2 now lies alongthe horizontal axis �Fig. 9�b��. This makes rc1j, sin �c1j andcos �c1j within the integral a function of only one variable,xc2j and the constants X12 and Y12,

rc1j2 = �X12 − xj�2 + Y12

2 , �18a�

cos �c1j =�X12 − xj�

rc1j, �18b�

sin �c1j =Y12

rc1j, �18c�

where xc2j is the distance between the jth bead of section S2and its central bead c2. X12 and Y12 are the horizontal andvertical distance between the central beads in the new refer-ence frame. It is noted that the subscript j indicates a beadbelonging to section S2 in the above three equations.

Fig. 11 shows the averaged hydrodynamic profile of the100 bead rod in Fig. 8 for the case of a 10 bead separationfrom its neighbor. The rods were averaged using 10 and 20sections per rod. As shown in Fig. 11, the averaging tech-nique reproduced the profile of the bead hydrodynamic ve-locities to a good degree, with the fit improving with increas-ing sections per rod.

C. Incorporating implicit hydrodynamics in Browniandynamics of semiflexible filaments

We now demonstrate how the implicit bead hydrodynam-ics can be incorporated into a string-of-beads idealization ofa semiflexible filament, and then show how the averaged im-plicit hydrodynamics can be included in a string-of-rods ide-alization of a semiflexible filament that was earlier devel-oped by the authors �6�.

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1. String-of-beads idealization of semiflexible filament

In a string-of-beads representation, the Brownian dynam-ics of a semiflexible filament is usually determined by solv-ing the particle form of the Langevin dynamics equation overeach bead. The Langevin dynamics equations govern the bal-ance between its frictional drag force, conservative force�FP�, and random Brownian forces �FB�. At each bead i,

�vi�t� = FiP�t� − Fi

B�t� , �19�

where � is the frictional coefficient of a bead, and v is thebead velocity. Typical conservative forces �FP� are bendingand stretching forces which depend on the relative positionsof the beads. The Brownian force �FB� at each bead is ob-tained from a Gaussian distribution whose variance is givenby the fluctuation dissipation theorem as

�FB�t�FB�t��� = 4KT��t − t�� . �20�

In this Langevin dynamics setting, we incorporate hydrody-namic interaction by having the frictional drag on a bead bedetermined by its hydrodynamic or true relative velocity �vH�rather than its observed velocity v.

�viH�t� = Fi

P�t� − FiB�t� . �21�

Solving Eqs. �21� and �7� �equation for the bead hydrody-namic velocity� for all beads simultaneously, the Browniandynamics of the semiflexible filament can be determinedwith the hydrodynamics included.

2. String-of-rods idealization of semiflexible filament

As discussed previously, solving for the hydrodynamicsimplicitly in a string-of-beads idealization introduces at leasttwo new variables per bead, increasing the size of the com-putational problem. Instead, we can use the idea of groupinginterior beads in sections and solving for the hydrodynamicvelocities of the representative bead alone �see Sec. II B 2� ina string-of-rods idealization of a semiflexible filament �6��Fig. 1�d��. We briefly describe the string-of-rods idealizationbelow, and then show how the averaged implicit hydrody-namics can be easily included in it.

In a string-of-rods idealization, a semiflexible filament istreated like a string of contiguously bending rods or seg-ments �Fig. 12�a��, and the Brownian dynamics determinedby solving equations at the segment intersections. The mainideas of the formulation are �6�:

�i� The semiflexible filament is divided into segments�Fig. 12�a��. At each segment intersection or node the follow-ing six variables are solved for: the x and y displacements ofthe intersection, the filament angle � and the curvature d� /dsat the intersection, and the x and y forces arising at the cutface of the intersection. From the x and y forces, the differ-ential of the filament curvature, d2� /ds2, at the intersectioncan be determined.

�ii� The dynamics of a segment is determined by the bal-ance of three forces acting on it: the Brownian and dragforces acting along its length, and the forces arising at the cutface of its ends �6�.

�iii� The Brownian forces distributed along the curvedsegment are resolved in the following way: the normal/

parallel Brownian forces along the segment can be projectedin two mutually perpendicular directions �u1- and u2-�, andthe projections can then be resolved into normal/parallel re-sultant forces and couples acting at the center of each pro-jection �Fig. 12�b��. The variance of the resultant forces andcouples can be shown determined by the friction coefficientsof rigid cylindrical rods �Eqs. �6��.

�iv� The drag forces distributed along the segment lengthare similarly resolved. The normal/parallel drag forces alonga curved segment is projected in two mutually perpendiculardirections �u1- and u2-�, and the projections can then be re-

(a) (b)

(c)

FIG. 11. Averaged hydrodynamic profile of the 100-bead rod inFigs. 8�a�–8�c� shown only for the case of a 10 bead separationfrom its neighbor. The rods were averaged using 10 and 20 sections.

u1u2

(a)

(b)

FIG. 12. Rods-on-string idealization of semiflexible filaments.�a� The rods-on-string idealization treats a semiflexible filament as astring of continuously bending rods or segments. The filament vari-ables at the rod intersections or nodes are solved for. �b� The forceson a curved segment are resolved into their projections in two mu-tually perpendicular directions. The mutually perpendicular direc-tions are taken to be along the line joining the segment ends �u1�,and along the line perpendicular to it �u2�. The projected forces ineach direction are further resolved into a resultant force and couplein that direction. Solving for the force and moment balance equa-tions over an instantaneously rigid rod requires only knowing theresultant forces and couples. In panel �b� only the distributed forcesnormal to the curved segment are considered.

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solved into a normal/parallel resultant forces and couplesacting at the center of each projection. The normal/paralleldrag forces along the segment are calculated from the dis-placements along the segment, normal and parallel to it.

�v� For such a force resolution, the force and momentbalance equations in the large-strain Euler bending theorycan be used to solve for the filament curvature, d� /ds, andthe x and y forces at the segment intersections. Also, theintegral of the Euler moment-balance equation can be used tosolve for the filament angle, �, at the segment intersections.

�vi� For each segment, �, d� /ds, and d2� /ds2 at its twoends are known. Therefore, using a fifth-order polynomialinterpolation, ��s� can be determined as a function the seg-ment’s contour, s. Then by integrating the sine and cosineprojections of ��s�, the x- and y-length projections of thesegment can be determined. From the change in the x- andy-length projections at each time-step, the x and y displace-ments of the segment intersections are solved for.

�vii� At each time-step, the six equations are assembledfor each segment and the entire equation set for the filamentis solved simultaneously.

�viii� The time evolution of the semiflexible filament dy-namics is solved by backward Euler time stepping. This alsomeans that the segment configuration at the beginning of thetime-step determines the resultant Brownian forces andcouples for that time step, and it also determines the direc-tions considered normal and parallel to the segment.

The computational advantage of the above formulation isthat many beads are replaced by a single rod as the unit ofcoarse graining. This reduces the overall size of the equationset and therefore the cost of computing. The formulation alsocaptures the physics of the Brownian bending better: the Eu-ler beam equations are solved without discretizing, the con-tinuous curvature of the filament is reproduced, and filamentinextensibility is intrinsically preserved since the � profile isdetermined as a function of the segment contour s.

Within the setting of the string-of-rods formulation, theaveraged implicit hydrodynamics are incorporated in the fol-lowing way. First, in the string-of-rods idealization the dragforces along a segment need to be calculated as resultantforces and couple in two mutually perpendicular directions�Fig. 12�b��. Usually, one of these directions is taken to bealong the straight line joining the two segment ends �u1-�,and the other direction is taken along the line perpendicularto it �u2-� �Fig. 12�b��. Since our averaged implicit hydrody-namic model is currently only applicable for straight sectionsof a filament, we use it to determine the resultant drag forceand couple over the segment projection u1-only �Fig. 13�.Therefore, the u1-projection of a curved segment is equiva-lent to a “section” in the averaged implicit hydrodynamicsmethod. We retain the previously described method �6� todetermine the resultant drag force and couple in theu2-projection. The u2-segment projections are typically muchsmaller that the u1-segment projections �much less than 5%for the segment resolutions in the simulations in this paper�.Hence, we do not expect the u2-drag to have a significanteffect on the overall segment hydrodynamics. Secondly, thehydrodynamic velocities of the filament ends are also solvedfor separately. That is, the filament ends are treated like theend beads of the averaged implicit hydrodynamics method

�Fig. 13�. The hydrodynamic drag forces arising at the fila-ment ends are treated in the string-of-rods idealization as endforces �6�.

The resultant drag force and couple on the u1-segmentprojection due to the hydrodynamic velocities of its repre-sentative bead are

FikH = �sphvi

kHLiu1

2rk = n,p , �22�

FiwH = �sphwi

HLiu1

2r, �23�

FikH and Fi

wH are the resultant drag force and couple on theu1-projection of segment i. The superscripts n , p indicate thedirections normal and parallel to the u1-segment projection.vi

nH, vipH and wi

H are the normal, parallel and angular hydro-dynamic velocities of the representative bead of theu1-projection of segment i, and Li

u1 is the projection of thecontour length of segment i in the u1-direction.

The end force due to the hydrodynamic velocities of thefilament ends appear as

FbkH = �sphvb

kH k = n,p , �24�

where FbkH is the drag force due to the filament end bead b,

and vbnH and vb

pH are the normal and parallel hydrodynamicvelocities of the filament end.

At each time step of the semiflexible filament simulation,the hydrodynamic equations for the representative and endbeads are solved simultaneously with the equations for thenode degrees of freedom. In keeping with the backward Eu-ler time-stepping solution, all the hydrodynamic interactionterms are calculated based on the filament geometry at thebeginning of the time step.

FIG. 13. Incorporating averaged implicit hydrodynamics in thestring-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 averageimplicit hydrodynamics is used to solve the hydrodynamic veloci-ties of the end beads �circles with solid outlines�, and that of thecentral representative beads �circles with broken outlines� of theu1-projections of the segments. The u1-projection of each segmentis shown in thin black lines and is along the direction joining thetwo ends of the segment. Therefore the u1-projection of each seg-ment constitutes a section in the implicit hydrodynamics technique.The hydrodynamic drag force due to the hydrodynamic velocities ofan end bead enters the string-of-rods formulation as an end force onthat segment. The hydrodynamic drag due to the hydrodynamicvelocity of a representative bead enters the string-of-rods formula-tion as the resultant drag force/couple on the u1-projection of thatsegment.

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III. RESULTS AND SIMULATIONS

A. Brownian fluctuations of a semiflexible filament.

We simulated the Brownian fluctuation of a 12 �mphalloidin-stained actin filament with bending stiffness EI=7.3e−26 N m2, in water at 25 °C �5�. The filament wasmodeled as a string of 15 segments of length 0.8 �m each,for 25 s in time steps of 0.0025 s. The results of these simu-lations are shown in Fig. 14. For each time step, the filamentcontour was decomposed into Fourier sine series and theamplitudes of the first nine Fourier bending modes are cal-culated �5�. The variance of these bending amplitudes is plot-ted in Fig. 14�a�. The variances compare well against theo-retical predictions obtained by equating the bending energy

of each mode to KT /2 �Equipartition theorem� �5,22�, vali-dating the Brownian model. We also compare the time evo-lution of the bending amplitudes of the first four bendingmodes against that predicted for a semiflexible filament un-der small bending strains and having a uniform friction co-efficient along its length �23�. The plot �Fig. 14�b�� indicatesthe time evolution behavior to be in the proximity of thetheoretical approximation. It also shows that the simulationreaches equilibrium for all the bending modes.

Experimental validation of fluctuation dynamics.Bernheim-Groswasser et al. �24� used fluorescence correla-tion spectroscopy to measure the time evolution of the mean-squared-displacement �MSD� of fluorescent-labels on actinfilaments in Brownian fluctuation, for over five decades intime �20 �s–2 s�. They found that only filament fluctuationtheories which included hydrodynamic interaction could pre-dict the experimental observations. However beyond 10 mstime intervals, the predictions degraded for the hydrody-namic theories without fitting parameters. We simulated theexperiment using actin filament dimensions and experimen-tal conditions reported in the paper �24� �6 �m filament,17 �m persistence length, 7 nm diameter, and 1 mPa solventviscosity�. We performed the simulation for 10 s at 0.0005 stime intervals. Figure 15 shows the time evolution of theMSD of the filament nodes during the simulation. The simu-lation data are plotted against the experimental determina-tions of the MSD of homogenously and nonhomogenouslylabeled actin filaments. The simulation results compare wellagainst the experimental result, even in time intervals above10 ms. Note that we did not simulate MSD evolution at timeintervals below 0.0005 s.

B. Hydrodynamic screening and Brownian fluctuations of asemiflexible filament

For a preliminary understanding of how hydrodynamicscreening and nonuniform friction coefficient affected theBrownian fluctuations of a semiflexible filament, we exam-

(a)

(b)t (sec)

FIG. 14. Validation of the rods-on-string idealization of semi-flexible filaments with averaged implicit hydrodynamics. The simu-lation results for the Brownian fluctuations of a phalloidin-stainedactin filament are shown �12 �m length, 7.3e−26 N m2 persis-tence length, 0.0025 s simulation time interval�. �a� The variancesof the amplitude �an� of the first nine bending modes �n� are shown.The amplitudes of the bending modes are calculated from the simu-lation results by decomposing the filament conformation at eachtime step into Fourier series. Theoretical estimates of the amplitudevariances can be made by assuming each bending mode to haveKT /2 energy �equipartition theorem�. The variances determinedfrom the simulation closely match the theoretical estimates. �b� Thetime evolution of the variance of the bending amplitude for the firstfour modes is shown in black. Shown in gray are the theoreticalestimates of the time evolution obtained by assuming small bendingfluctuations and a constant hydrodynamic friction. The simulationresults lie in the vicinity of the theoretical estimates.

������

�����

����

���

�

��

���� ��� � �� ��� ���� �����

���� ����

��

��

�� �

���

���

��

����

m2)

����� ����

���������� � � �

���������� � � �

(µm

2)

FIG. 15. Experimental validation of model dynamics. The timeevolution of the mean-squared displacement of an actin filament isplotted as a function of time interval, t. The MSD was determinedas �r2�t-t��. The simulation result for a 6 �m actin filament isshown with gray diamonds. The experimental observation for ho-mogenously and nonhomogenously labeled actin filaments is shownwith unfilled and filled black diamonds, respectively. The experi-mental data were obtained from Bernheim-Groswasser et al. �24�.

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ined the contribution of solvent velocity �induced by thebending fluctuations� to the observed velocity along the fila-ment �see Eq. �1��. Only velocities normal to a segment wereconsidered, although the discussion is applicable to the par-allel and rotational velocities as well. We simulate theBrownian fluctuations of the actin filaments at 0.01 sec timesteps, and with all other simulation parameters the same asSec. III A. Shown in Fig. 16�a� are the x /y positions of thefilament nodes at the first four simulation times �0, 0.01,0.02, 0.03 s�. The initial conformation of the filament at 0secis that of a straight rod. We examined the observed and hy-drodynamic velocity along the filament at each representa-tive bead and at the end beads for each simulation time in-terval �Figs. 16�b�–16�d��. The difference between theobserved and hydrodynamic velocities would be the velocitycontribution from the induced solvent flow �see Eq. �1��.

In the three time intervals shown, the hydrodynamic ve-locities are much lower than the observed velocities, exceptat the filament ends �Figs. 16�b�–16�d��. Conversely, the sol-vent velocity profile appears to closely match the observedvelocity profile, except at the filament ends. Therefore, thefollowing inductions can be made by comparing the plots inFigs. 16�b�–16�d�. The larger hydrodynamic velocities ordrag occur at the filament ends. The solvent flow or hydro-dynamic screening contributes to a large part of the observedvelocity in the interior of the filaments. In fact, the solventflow contributes to most of the observed velocity at the sidesof the convexities in the normal velocity profile �black dia-monds in Figs. 16�b�–16�d��, and at points where the hydro-dynamic or relative velocity is zero �gray circles in Figs.16�b�–16�d��. To the most part, the solvent contribution ap-pears to magnify convexities in the hydrodynamic velocityprofiles. These observations suggest that the solvent flow in-duced by hydrodynamic interactions may have a systematicinfluence in the Brownian conformations assumed by a semi-flexible filament.

C. Confined fluctuations of a semiflexible filament betweentwo stationary filaments

We show that our model captures the lateral hydrody-namic confinement of a fluctuating semiflexible filament be-tween two stationary filaments. We repeat our previous actinfilament simulations, but with two stationary actin filamentsplaced alongside at three confinement widths of 3, 1.5, and1 �m �Fig. 17�. Also, to better capture the confinement weuse the central difference instead of the backward time step-ping to calculate the hydrodynamic velocities. Calculation ofthe hydrodynamic velocities requires knowing the relativepositions of the filaments. In the backward Euler steppingscheme, the relative positions at the beginning of the timestep is used. Using the filament positions at the beginning ofthe time step may overestimate or underestimate the frictiondrag experienced by a fluctuating filament, especially if con-fined. For instance, it would underestimate the drag on afilament approaching a neighbor, leading to crossover. It willalso overestimate the drag on a filament moving away from aneighbor, leading to the filament being stuck in the vicinityof the neighbor. Decreasing the simulation time intervalwould alleviate these issues, but we chose to use the central-difference time stepping instead.

Figure 17 shows the Brownian fluctuations of the 12 �mactin filament between two other stationary actin filamentsseparated by 3, 1.5, and 1 �m. The filaments undergo fluc-tuations of decreasing amplitude with increasing confine-ment. At 1.2 s, the unconfined filament shows a larger trans-lational and rotational diffusion than the confined filaments.These simulation results show that our model is able to cap-ture the two-dimensional confinement of the fluctuating fila-ment that occurs due to hydrodynamic interaction. Moreover,we calculated the variances of the bending amplitude for thedifferent bending modes, for each case of confinement. Fig-ure 18�a� suggests that confinement occurs as a decrease inthe amplitude of the bending modes. Figure 18�b� shows thecorrelation of the contour angles as function of filament

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FIG. 16. Hydrodynamic contribution to the Brownian fluctua-tions of a semiflexible filament. �a� The Brownian conformation ofa 12 �m actin filament at 0, 0.01, 0.02, and 0.03 s. The x and ypositions of the filament nodes are plotted. �b,c,d� The normal ve-locities of the end and representative beads in the time intervals of0–0.01 s �a�, 0.01–0.02 s �b�, and 0.02–0.03 s �c�, plotted against thex positions of the representative/end beads at time t=0 s.

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length. The decay in correlation along the filament contour ismuch slower with increasing confinement, and the overalldegree of correlation is higher with increasing confinement.The angle correlation profiles suggest that the filament per-sistence length increases with increasing confinement �Fig.18�b��. A similar increase in persistence length was observedby Koster et al. �25� for actin filaments fluctuating in confin-ing microchannels of decreasing width. Figures 18�c�–18�f�show the time correlation profile of the first four bendingmodes for the different degrees of confinement. While theplots clearly show the equilibrium bending variance to de-crease with increasing confinement, we do not have conclu-sive data yet on how the time relaxation for bending changeswith confinement.

IV. SUMMARY AND DISCUSSION

Semiflexible filaments, constituting the chief structural in-gredients of the cytoskeleton �26� while contributing to themechanical signaling pathways in the cell �27,28�, are gen-erally subject to hydrodynamic forces due to solvent flowsinduced by the bending and diffusive motions within the fila-ments themselves and from their surrounds filaments. Theaim of this paper was to include the effect of hydrodynamicinteraction in a dynamic model of semiflexible filament. Thedrag along a filament is altered by the solvent flow inducedby the bending and diffusive motions along the filament, andalong neighboring filaments. For example, the drag along arigid filament in uniform motion is not uniform due to thehydrodynamic screening within the filament. The nonuni-form drag would have caused a semiflexible filament to

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FIG. 17. �Color online� The Brownian fluctuations of a 12 �mactin filament in the unconfined and laterally confined state. Thefilament properties are similar to that in previous simulations. Thesimulations were performed in time intervals of 0.0025 s. The con-fined filament fluctuates between two other stationary filamentsseparated by 3, 1.5, and 1 �m �from the left to right in figure�.From top to down, the fluctuations of the filament at every 0.1 sintervals are shown. The filament conformation is obtained by fifth-order interpolation between the filament nodes �shown as blackdots�. The filament contour length between two nodes �black dots�is 0.8 �m.

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FIG. 18. Dynamics of laterally confined fluctuating actin fila-ments. �a� Variance of the first four bending amplitudes for differentdegrees of confinement. �b� Angle correlation along the filamentlength for different degrees of confinement. �c� Time correlation ofthe amplitude of the first bending mode �a1� for different degrees ofconfinement. �d� Time correlation of the amplitude of the secondbending mode �a2� for different degrees of confinement. �e� Timecorrelation of the amplitude of the third bending mode �a3� fordifferent degrees of confinement. �f� Time correlation of the ampli-tude of the fourth bending mode �a4� for different degrees ofconfinement.

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bend. Therefore, hydrodynamic interactions do not only af-fect the bending fluctuations of a filament in nondilute sys-tems, but also affect filaments in dilute systems due to thealtered drag from hydrodynamic screening effects. It is there-fore important for realistic models of semiflexible filamentsdynamics to include hydrodynamic interaction.

Due to the induced solvent flow, the true relative velocityalong the filament is different from the observed velocity. Werefer to the true relative velocity along the filament as thehydrodynamic velocity. The common way to capture therelative velocities along the filament is to idealize it as astring-of-beads, and use the linearity of Stokes flow to deter-mine the induced solvent field by superposing the separatesolvent fields due to each bead. However, the first-orderStokes superposition neglects the back reflection of solventflow. We corrected for this by casting the true relative veloc-ity of the bead as a new variable and solving for it implicitly.We call the true relative velocity of the bead as its hydrody-namic velocity. The one-step implicit solving of hydrody-namic variables is equivalent to infinite superpositions of thebead solvent fields or infinite reflections in the Method ofReflections because �1� each superposition serving to enforcethe no-slip boundary condition at a bead can be mathemati-cally interpreted as requiring that solvent flow be induced bythe hydrodynamic velocity of a particle alone, �2� infiniteorders of explicit iteration are equivalent to one implicit so-lution. The predictions of the implicit technique deteriorateat very small distances between beads. This is not because ofthe Stokesian superpositions, but because the presence of abead in the solvent is represented by a point perturbation inthe velocity field. In other words, the equations for the sol-vent flow around a bead �Fig. 2� do not respect the impen-etrability of the bead. However, in spite of this limitation, thepredictions of an implicit or infinite-order superposition isvastly superior to that of a first-order superposition, espe-cially in that it captures the hydrodynamic influence of sta-tionary particles while being relatively easy to implement.

A string-of-beads approach is traditionally limited by thecomputational size of even modeling a single filament. Theproposed method addresses this limitation by assuming sec-tions of beads, excepting the beads at the ends of filaments,to have the same average hydrodynamic velocities. We donot include the end beads in the averaging because the largerhydrodynamic drags are typically concentrated at the fila-ment ends. By solving for their relative velocities separately,we are able to better resolve the hydrodynamic profile alongthe filament. The averaging is performed by approximatingthe normal and parallel hydrodynamic velocities of the beadsin a section by the normal, parallel, and angular hydrody-namic velocity of the central representative bead in the sec-tion. This is equivalent to approximating the hydrodynamicprofile in the section by a linear normal hydrodynamic pro-file and a constant parallel hydrodynamic profile. As a resultof such averaging, the hydrodynamic equations need to beonly solved for the representative equations, but the hydro-dynamic influence from all beads is included. We do notwrite the equations for the angular hydrodynamic velocity ofthe representative bead in a form similar to its normal hydro-dynamic velocity. Doing so would result in the equationsbeing trivially zero. Instead we write the angular hydrody-

namic equation as a differential of the representative bead’snormal hydrodynamic velocity equation. In other words, wesolve for the angular hydrodynamic velocity by differentiat-ing normal hydrodynamic velocity profile of the section, atthe representative bead.

We incorporated the averaged implicit hydrodynamicsinto an earlier proposed string-of-rods idealization of a semi-flexible filament �6�. In that formulation, the semiflexiblefilament is divided into continuously curved segments. Thedrag force distributed along on a curved segment is resolvedinto resultant forces and couples acting on two mutually per-pendicular projections of the segment. One projection �u1-�is taken to be the line joining the segment ends. We treat thisprojection as equivalent to a “section” in the averaged im-plicit technique. The hydrodynamic velocity of the centralrepresentative bead of the projection is used to determine theresultant drag forces and couple on it. The hydrodynamicequations of the filament end beads and the hydrodynamicequations of the representative beads of the u1-projectionsare solved simultaneously with the equations of the string-of-rods idealization. We found that the time evolution of thebending amplitudes lies in the vicinity of the theoretical ap-proximation. The theoretical approximation assumes that thefilament bending is small and that the friction coefficientalong the filament is uniform, i.e., it neglects hydrodynamicinteraction. We also simulated experimental data on the timeevolution of the mean-squared displacement �MSD� of fluc-tuating actin filaments. A good prediction was observed evenfor time intervals above 10 ms, where models of transversefilament fluctuation, hydrodynamic interaction included,fared less well �24�. The time evolution of the filament MSD,observed in the experiment and in our simulation, was muchlarger than the predictions of filament models with no hydro-dynamic interaction. This suggests that hydrodynamic inter-action could possibly decrease the time-relaxation constantsfor the bending of semiflexible filaments.

Examination of the contribution of hydrodynamic forcesto the bending fluctuations of a semiflexible filament showedthat at point along the filament, the difference between theobserved and hydrodynamic velocity gives the part of thepoint’s motion that occurs from drifting in the solvent flowinduced by its neighbors. We found that a large part of theobserved velocities along a filament can be attributed todrifting in induced solvent flow. And as expected, the ob-served velocities at the filament ends received much lesscontribution from the induced solvent flow. Thus the hydro-dynamic drag profiles along the fluctuating semiflexible fila-ment are reminiscent of those observed along rigid filaments�see Fig. 5�a� and Ref. �7��, except that the nonuniformitiesare less pronounced. This is because a semiflexible filamentcan bend to equalize the nonuniform drag forces along it.More study is required to understand how hydrodynamic in-teractions systematically affect the bending fluctuations of asemiflexible filament.

We tested the ability of our model to show the confine-ment of a fluctuating filament between neighboring filamentsdue to hydrodynamic interaction. For ease of demonstration,we chose the neighboring filaments to be stationary. We notethat a first-order superposition would miss the hydrodynamicconfinement from stationary filaments. In order to capture

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the fast-changing drag on a confined filament as it movestoward and away from its neighbors, we chose to solve thehydrodynamic equations alone by central-difference timestepping instead of the backward time stepping used in therest of the model. A central time stepping takes the averageof the filament positions at the beginning and end of the timestep, and therefore gives a better estimate of the average dragexperienced by the filament during the simulation time inter-val. The simulation results show the persistent length of thefilament increases with increasing confinement, suggestingthat the filament confinement occurs as change in persistencelength. We also show the time evolution of the bending am-plitudes of the confined filaments that was observed in oursimulations. More rigorous studies are required, however, toquantify the time-relaxation behavior. Finally we note thatthe crossing over of filaments is not prevented in our simu-lations. Since the hydrodynamic equations are only solved atrepresentative and end beads and since the hydrodynamicforces only increase by r−1 as the r �distance of separation�decreases, the filament is free to crossover each other. Thesimulations only break down when the representative or end

beads intersect. Accounting for the van der Waals steric hin-drance forces would alleviate this problem to a large extent.However, we chose not to include the steric forces in ourtwo-dimensional model, because a two-dimensional �2D� ap-proximation of a three-dimensional �3D� system by itselftends to overestimate the confinement effect.

To conclude, we presented a model of semiflexible fila-ments that accounts for the hydrodynamic effects originatingboth within itself and from interaction with its neighbors.The model offers great potential for understanding the rheol-ogy of nondilute semiflexible filament systems, which in turnis useful for understanding the mechanics of cell and tissuematrices �26�.

ACKNOWLEDGMENTS

The authors are thankful to members of Molecular CellBiomechanics Laboratory for their invaluable input. Finan-cial support from National Science Foundation �CBET-0829205 and CAREER-0955291� is gratefully acknowl-edged.

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M. R. K. Mofrad and R. D. Kamm �Cambridge UniversityPress, Cambridge, U.K., 2006�.

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�28� G. Bao, R. D. Kamm, W. Thomas, W. Hwang, D. A. Fletcher,A. J. Grodzinsky, C. Zhu, and M. R. K. Mofrad, Cellular andMolecular Bioengineering �to be published�.

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