Eur. Phys. J. E (2021) 44 :49https://doi.org/10.1140/epje/s10189-021-00016-x

THE EUROPEANPHYSICAL JOURNAL E

Regular Article - Living Systems

Lagrangian mechanics of active systemsAnton Solovev and Benjamin M. Friedricha

TU Dresden, Dresden, Germany

Received 30 September 2020 / Accepted 12 January 2021 / Published online 8 April 2021© The Author(s) 2021

Abstract We present a multi-scale modeling and simulation framework for low-Reynolds number hydro-dynamics of shape-changing immersed objects, e.g., biological microswimmers and active surfaces. Thekey idea is to consider principal shape changes as generalized coordinates and define conjugate gener-alized hydrodynamic friction forces. Conveniently, the corresponding generalized friction coefficients canbe pre-computed and subsequently reused to solve dynamic equations of motion fast. This frameworkextends Lagrangian mechanics of dissipative systems to active surfaces and active microswimmers, whoseshape dynamics is driven by internal forces. As an application case, we predict in-phase and anti-phasesynchronization in pairs of cilia for an experimentally measured cilia beat pattern.

Biological hydrodynamics Biology provides ample exam-ples of active shape changes in fluid environments: Bac-teria like E. coli rotate helical prokaryotic flagella toswim [1], other bacteria like Spiroplasma propagatetwist waves along their flexible body [2], sperm cells andmotile algae posses slender cell appendages termed cilia(or eukaryotic flagella), whose regular bending wavespropel these cells in a fluid [3,4]. On epithelial surfaces,collections of beating cilia transport biological fluidssuch as mucus in airways, cerebrospinal fluid in brainventricles, and oviduct fluid in the Fallopian tubes [5,6].In addition to their important role in self-propulsionand fluid transport, these model systems enable us tolearn about internal force generation mechanisms inthese cells, such as the collective dynamics of molecularmotors inside cilia [7–10]. On larger scales, the interac-tion of many shape-changing units leads to the spon-taneous formation of spatiotemporal patterns, e.g., indense suspensions of microswimmers [11], or collectionsof cilia exhibiting metachronal coordination [12].

These examples represent a class of fluid–structureinteraction problems, where shape-changing active struc-tures exert forces on the surrounding fluid, while thesurrounding passive fluid exerts hydrodynamic frictionforces back on these active structures. These hydro-dynamic forces may affect the active shape dynam-ics; examples include the torque–velocity relationshipof rotating prokaryotic flagella [13], the load responseof beating cilia and eukaryotic flagella [10,14], as wellas minimal model swimmers [15–17]. Closed feedbackloops between passive fluids and active structures canlead to emergent dynamics; examples include sponta-neous pattern formation in dense microswimmer sus-

a e-mail: benjamin.m.friedrich@tu-dresden.de (corre-sponding author)

pensions [11,18] or (hydrodynamic) synchronization ofbeating cilia and flagella [12,19–23].

Common hydrodynamics methods at low Reynolds num-bers At the relevant length and time scales, viscousdrag dominates inertia, corresponding to low Reynoldsnumbers [24–26]. In the limit of zero Reynolds numbers,the Navier–Stokes equation of hydrodynamics simpli-fies to the Stokes equation. Although the Stokes equa-tion is linear, hydrodynamic computations can stillbe costly, because hydrodynamic interactions are long-ranged [27].

In the past, different computational methods of dif-ferent degrees of approximation have been used in thecommunity, including resistive force theory for slenderfilaments, which includes short-range, but not long-range hydrodynamic interactions [28–30], the morerefined method of slender-body theory, which considersa line distribution of hydrodynamic singularities (pointforces) along a filament [31–33], or multi-particle colli-sion dynamics, which replaces the continuum descrip-tion of the Stokes equation by the stochastic dynamicsof a large number of “fluid particles” [34–36]. Despite itsapplicability for large-scale problems [37], the stochasticnature of the MPCD algorithm introduces algorithm-specific fluctuations, which can be impractical if onewants to study the role of biological noise. LatticeBoltzmann methods similarly rely on fictitious “fluidparticles”, for which in each time step both a streamingand a collision step is performed [38]. Finally, bound-ary element methods convert the problem of solvingthe Stokes equation in three-dimensional space to atwo-dimensional boundary integral problem of findinga surface distribution of forces on a moving bound-

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ary surface. Boundary element methods are similar inspirit to slender-body methods, but less susceptible toissues of regularization, since a two-dimensional distri-bution of forces is used. Modern algorithms use fastmulti-pole methods that solve a tree of hierarchicallycoarse-grained subproblems instead of solving a singlelarge linear system when computing the force distribu-tion on a surface [39–41].

Irrespective of the hydrodynamic computation methodused, it can be computationally costly to calculate asolution of the Stokes equation in every time step,while simulating the dynamics of a shape-changingmicroswimmer or an active surface.

Lagrangian mechanics In this article, we present amulti-scale simulation framework, where the Stokesequation has to be solved only in an initial step for asmall set of principal shape modes of a shape-changingsurface. The resultant surface distributions of hydro-dynamic friction forces define generalized hydrody-namic friction coefficients by a projection method ofLagrangian mechanics [10,42–48]. These scalar frictioncoefficients are independent of the velocity of the mov-ing surface. Once tabulated, these friction coefficientsprovide a look-up table for subsequent fast simulationsof shape dynamics and active motion. Specifically, weview principal shape changes of an active surface as gen-eralized coordinates, for which we compute conjugategeneralized friction forces. We obtain effective equa-tions of motion for the generalized coordinates from aforce balance between these generalized friction forcesand active driving forces. These active driving forcescoarse-grain the internal active processes that drive theactive shape changes of the surface (such as the collec-tive dynamics of molecular motors). Importantly, thesea priori unknown active driving forces can be calibratedfor a reference case (e.g., using experimental data) andthen used to extrapolate to other application cases ofinterest. Therefore, our framework extends Lagrangianmechanics of dissipative systems to active surfaces andactive microswimmers, whose shape dynamics is drivenby active forces.

1 Notation: Stokes equation andhydrodynamic dissipation

Fluid dynamics at the scale of individual biological cellsis characterized by low Reynolds numbers, i.e., viscouseffects commonly dominate over inertia [24–26]. Corre-spondingly, fluid flow is described by the Stokes equa-tion, which reads for an incompressible Newtonian fluidin the absence of body forces in the bulk [27]

0 = −∇p + μ∇2u, (1)

with incompressibility condition ∇ · u = 0. Here, u(x)denotes the flow velocity, p(x) the pressure field, and μthe dynamic viscosity of the fluid.

The total stress tensor σ for an incompressible fluiddepends on both the hydrostatic pressure p and thesymmetrized strain rate tensor Δ [27]

σ = −p1+ 2μΔ, Δ =12

[∇ ⊗ u + (∇ ⊗ u)T]. (2)

Thus, the Stokes equation, Eq. (1) could be equivalentlywritten as 0 = ∇ · σ in the bulk of the fluid. Specialconditions apply at boundaries.

No-slip boundary condition for an active surface Weconsider a surface S immersed in the fluid that changesits shape as a function of time. For example, S may rep-resent the outer surface of a shape-changing microswim-mer, or even the combined surface for a collectionof microswimmers. We introduce the surface velocityv(x, t) for each point x ∈ S at time t.

We impose a no-slip boundary condition at this sur-face, i.e., require that the local velocity u(x) of fluidflow matches the local velocity v(x) of the surface foreach surface point

u(x) = v(x) for all x ∈ S. (3)

Hydrodynamic friction forces A shape change of thesurface S induces a flow field u(x) with correspond-ing stress tensor field σ(x). The stress σ(x) determinesthe surface density of forces f(x) exerted by the surfaceon the fluid (with units of a stress N/m2, also calledcontact force, or traction force density)

f(x) = −σ · n for all x ∈ S, (4)

where n is the surface normal pointing into the fluid.Correspondingly, −f(x) is the surface density of hydro-dynamic friction forces exerted by the fluid on the sur-face. The total force exerted by the surface on the fluidis simply the surface integral of f(x)

F =∫

Sd2x f(x). (5)

Analogously, the total torque (with respect to a refer-ence point x0) exerted by the surface on the fluid isgiven by

M =∫

Sd2x (x − x0) × f(x). (6)

Superposition principle The linearity of the Stokesequation of low-Reynolds number flow, Eq. (1), impliesa superposition principle for hydrodynamic frictionforces, which will be pivotal for the modeling ansatzpresented here. Specifically, we consider a boundary

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condition with rate of displacement v that is given as alinear combination of velocity distributions v1 and v2

asv = α1 v1 + α2 v2, (7)

with real coefficients α1, α2 ∈ R. Then, the resultantflow field u is given by u = α1u1 + α2u2, while thesurface density of hydrodynamic friction forces f isf = α1f1 + α2f2, where ui and fi denote the flow fieldand the surface density of hydrodynamic friction forcescorresponding to boundary condition vi, respectively,for i = 1, 2.

Hydrodynamic dissipation We introduce the rate ofwork R(h) exerted by the surface on the fluid

R(h) =∫

Sd2xv(x) · f(x). (8)

For incompressible Newtonian fluids at zero Reynoldsnumber, R(h) equals the instantaneous rate of hydrody-namic energy dissipation within the fluid [27]. Indeed,let us consider the local dissipation rate, which is givenby Φ = 2μΔ : Δ, where Δ : Δ =

∑i,j ΔijΔij denotes

tensor contraction. The dissipation rate can be rewrit-ten as Φ = ∇ ·(u ·σ) using Eqs. (1), (2) and the incom-pressibility condition ∇ · u = 0. Gauss divergence the-orem now gives [27] (using u(x) = v(x) for x ∈ S)

∫

Sd2xv(x) · f(x)

︸ ︷︷ ︸power exerted by surface

=∫

V

d3xΦ(x)︸ ︷︷ ︸

hydrodynamic dissipation in bulk

.

(9)Here, V denotes the three-dimensional fluid domainwith boundary surface S. At finite Reynolds numbers,R(h) still equals the rate of work exerted by the surfaceon the fluid, yet this injected energy would be dissi-pated as heat with a delay, such that Eq. (9) wouldonly hold for time averages.

2 Lagrangian mechanics: generalizedcoordinates

We consider a shape-changing surface S(t). While adescription of all possible shape changes of S wouldrequire an infinite number of degrees of freedom, inimportant application cases, we can restrict ourselvesto a constrained set of shape changes characterized by asmall number of shape coefficients, or generalized coor-dinates, q1, . . . , qn.

Examples for minimal model swimmers include undu-lating sheets with a finite set of admissible wavelengths[49], bead distances as in Najafi’s three-sphere swim-mer [50], or lever arm angles in Purcell’s the three-linkswimmer [51] and Dreyfus’ rotator [52], see Fig. 1. Anexample for a biological microswimmer would be the

rotation angle ϕ of an idealized rigid helical prokary-otic flagellum. Similarly, the regular traveling bendingwaves of cilia and eukaryotic flagella can be describedby an oscillator phase ϕ that characterizes the currentposition in a periodic shape cycle [45,53–55]. Elasticdegrees of freedom arising from waveform compliancecan be incorporated in such a framework as additionalamplitude degrees of freedom [10,48].

We introduce the state vector, q = (q1, . . . , qn). Theshape dynamics of the active surface S(t) = S[q(t)] isthus entirely described by the dynamics of q(t). In par-ticular, the local rate of surface displacement dependslinearly on the generalized velocities qi as

v(x) = w1(x;q) q1 + w2(x;q) q2 + · · · + wn(x;q) qn,(10)

where the normalized velocity fields wi(x) = ∂x/∂qidepend on q(t) but not on q(t). In fact, Eq. (10) simplygeneralizes Eq. (7) to the case of generalized coefficientsαi = qi with units of a generalized velocity. Correspond-ingly, the surface distribution of hydrodynamic frictionforces f(x) is given as a linear combination

f(x) = g1(x;q) q1 + · · · + gn(x;q) qn, (11)

where the normalized force densities gi(x;q) = fi(x)/αi

correspond to the surface density of hydrodynamic fric-tion forces fi(x) induced by the velocity field vi(x) =αi wi(x;q). An example of a surface velocity field withcorresponding surface density of hydrodynamic frictionforces is shown in Fig. 2.

The formalism allows to include also rigid body trans-formation such as translations and rotations of the sur-face S in the set of generalized coordinates. Therefore,the self-propulsion of shape-changing microswimmerscan be described using the same formalism, see the sec-tion of rigid body transformations below.

3 Generalized hydrodynamic friction forces

We introduce generalized hydrodynamic friction forcesPi conjugate to the generalized coordinates qi, followingthe convention of Lagrangian dynamics of dissipativesystems [42], see also [43,46]

Pi =∫

Sd2xwi(x) · f(x), i = 1, . . . , n. (12)

The superposition principle for the shape changeswi(x), Eq. (10), allows us to rewrite the total hydro-dynamic dissipation rate R(h) as a sum of products ofgeneralized velocities times their conjugate generalizedfriction force

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A Undulating sheet

F Swimming bacterium (E. coli) G Swimming sperm cell

10 um

q6+1

D Purcell’s three-link swimmer

q6+1

q6+2

1 umq6+1

r0

E Dreyfus’ rotator

q6+1

q6+2

q6=α

Sq1

q2

C Najafi’s three-sphere swimmer

q6+1

q1=x

q6+2

B Rigid body transformationsq1 q2 q3

q4 q5 q6

Fig. 1 Generalized coordinates: Examples. a Undulatingsheet with two wave modes. The amplitudes q1, q2 of thewave modes represent generalized coordinates of the shape-changing surface S. b Rigid body motion of a microswimmerin three-dimensional space is characterized by three transla-tional and three rotational degrees of freedom, correspond-ing to six generalized coordinates: qi for translations parallelto the ei-axis, and qi+3 for rotations around the ei-axis,i = 1, 2, 3, respectively. c Najafi’s three-sphere swimmerconsists of three collinear spherical beads with time-varyingbead distances [50], corresponding to two internal degrees offreedom, q6+1 and q6+2, in addition to the generalized coor-dinates of rigid body motion. d Purcell’s three-link swim-mer consists of three connected segments [51], whose rela-tive angles q6+1 and q6+2 can be treated as two generalized

coordinates. e Similarly, Dreyfus’ rotator consists of threesegments connected at a single joint; the relative angles q6+1

and q6+2 again define generalized coordinates. This shape-changing microswimmer exhibits pronounced rotation in theplane in addition to translational motion, hence its name. fSimplified geometry of the bacterium E. coli with a singleprokaryotic flagellum. A rotary motor inside the cell wallcan spin the helical flagellum around its central axis; thisinternal rotational degree of freedom defines a single gen-eralized coordinate q6+1 with periodicity of 2π. g Proto-typical flagellar beat pattern of a sperm cell, parameterizedby a 2π-periodic phase variable, which defines a generalizedcoordinate q6+1. For the amplitude of regular flagellar bend-ing waves and mean flagellar curvature, we used parametersfrom [30]

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y, μm

z, μ

m

A Surface velocity

Generalized friction coefficient

B Force density C Flow field

Fig. 2 Hydrodynamic friction forces: Example of ciliumduring power stroke. a Surface velocity v1(x) on a shape-changing surface S, here given by a slender cilium (blue)attached to a no-slip boundary surface (gray); the ciliumprogresses with phase velocity q1 = ϕ1 along its periodicbeat cycle. For visualization, the three-dimensional shapeof the cilium as well as v1(x) was projected on the yz-plane (see Fig. 3a for a three-dimensional representation).b Corresponding surface distribution of hydrodynamic fric-tion forces f1(x) exerted by the active, shape-changing sur-face on the surrounding viscous fluid. The force distribu-tion is obtained by solving the Stokes equation, Eq. (1), seealso Multi-scale modeling: numerical implementation sec-tion for details. From the velocity and force distributions,v1(x) and f1(x), we can compute a generalized hydrody-

namic friction coefficient, Γ11, which is proportional to thephase-dependent rate of energy dissipation in the surround-ing fluid. In the general case of n generalized coordinatesq1, . . . , qn, we obtain a n × n matrix Γij , see also Eq. (15).c Flow field induced by the active shape change of the cil-ium, here shown as two-dimensional section at x = 0. Thecolor represents the magnitude |u(x)| of three-dimensionalvelocity vectors, whereas white arrows represent the pro-jections of u on the yz-plane. The flow field was com-puted as convolution of the fundamental solution of theStokes equation with the force distribution f1(x). Ciliumphase corresponding to Fig. 3a: ϕ1 = 1.4 π, cilium beat fre-quency: ω0/(2π) = 32 Hz [12], dynamic viscosity of fluid:μ = 10−3 Pa s (corresponding to viscosity of water at 20 ◦C)

R(h) =∑

i

Pi qi. (13)

Note that the different generalized coordinates qi mayhave different physical units, in which case also allderived quantities will have different units; nonetheless,all vector and matrix operations of the formalism areconsistent unit-wise.

In the special case, where some of the qi denote arigid body transformation of an immersed microswim-mer, i.e., a rigid body translation or rotation, the conju-gate generalized force simply corresponds to the respec-tive components of the total force F or total torque Mexerted by the swimmer on the fluid, respectively, seethe section on rigid body motion below.

Generalized hydrodynamic friction coefficients Usingthe superposition principle of Stokes flow, we can con-veniently express the generalized hydrodynamic frictionforces as linear function of the generalized velocities qiby introducing generalized hydrodynamic friction coef-ficients

Pi =n∑

j=1

Γij qj , i = 1, . . . , n. (14)

The generalized friction coefficients can be computed asscalar products between the (normalized) velocity pro-files wi(x), and the (normalized) force profiles gj(x),see also Fig. 2

Γij =∫

Sd2xwi(x) · gj(x), i, j = 1, . . . , n. (15)

Alternatively, we could express Γij in terms of par-tial derivatives with respect to the generalized veloc-ities qi as Γij =

∫Sd2x (∂ v(x)/∂ qi) · (∂ f(x)/∂ qj).

We refer to diagonal elements Γii of the generalizedhydrodynamic friction matrix Γ as self-friction coef-ficients. Off-diagonal elements Γij , i �= j, or cross-friction coefficients, characterize a coupling betweendifferent degrees of freedom (e.g., a coupling betweentranslational and rotational degrees of freedom for chi-ral objects or direct hydrodynamic interactions betweendifferent sub-objects that can, in principle, move inde-pendently).

The rate of hydrodynamic dissipation can thus beexpressed as a quadratic form in the generalized veloc-ity q

R(h) = q · Γ · q =∑

i,j

Γij qiqj . (16)

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The hydrodynamic dissipation rate R(h) plays therole of a Rayleigh dissipation function for Lagrangianmechanics of dissipative systems [42]. Specifically, wecould have equivalently defined the generalized forcesas 2Pi = ∂R(h)/∂qi. (Following standard notation, theRayleigh dissipation function is actually R(h)/2 [42].)

The matrix Γ is symmetric, which represents a spe-cial case of Onsager reciprocity [56]. The proof followsdirectly from the Lorentz reciprocal theorem [27]: Letvi, σi and vj , σj denote the flow field and stress tensorassociated with a change of only qi with rate qi, or achange of only qj with rate qj , respectively, while allother generalized coordinates are kept constant; then,

qi Γij(q) qj = −∫

Sd2xvi · σj · n

(∗)= −

∫Sd2xvj · σi · n = qj Γji(q) qi, (17)

where we used the Lorentz reciprocal theorem at (∗).The matrix Γ is also positive semi-definite, consis-

tent with the fact that the rate of energy dissipationshould be nonnegative. (In fact, Γ should be positivedefinite, except maybe at singular points q in config-uration space, where wi = ∂x/∂qi, i = 1, . . . , n arelinearly dependent.)

In addition to hydrodynamic dissipation as charac-terized by R(h), internal dissipative processes can beincluded in our framework, provided the correspondingdissipation function is likewise a quadratic form of thegeneralized velocity [10,48].

4 Equation of motion

Balance of generalized forces We introduce active driv-ing forces Qi, i = 1, . . . , n that coarse-grain internalprocesses that drive the active shape changes of theactive surface. Previous minimal models of flagella syn-chronization considered spheres moving along circu-lar orbits driven by a tangential force [44,57–59]. Ouractive driving forces Qi generalize the active drivingforces considered in these models.

We postulate a balance of generalized forces betweendriving forces and hydrodynamic friction forces

Qi = Pi, i = 1, . . . , n. (18)

We emphasize that Eq. (18) is simply an instance ofNewton’s second law and thus does not involve any newassumptions. Simplifying modeling assumptions haveonly been made in constraining the shape dynamics toa finite number of degrees of freedom q1, . . . , qn and inthe choice of the active driving forces Q1, . . . , Qn.

From the force balance equation, Eq. (18), andEq. (12) expressing the generalized forces Pi, we obtainequations of motion for the generalized velocities q

q = Γ−1 · Q, (19)

where Q = (Q1, . . . , Qn)T is the vector of active forces.

Calibration of active driving forces Because each driv-ing force Qi characterizes internal processes, it is plausi-ble to assume that Qi only depends on the correspond-ing degree of freedom qi, but not on the other qj , j �= i,i.e., we may assume Qi = Qi(ϕi). This assumption willhold in particular in applications, where the index ienumerates different microswimmers or different cilia.In principle, Qi may additionally depend on the frictionforce Pi itself, i.e., if the internal active processes maychange under load [17]. In this case, Eq. (18) becomesa self-consistency equation that has to be solved usingmethods for implicit equations. For a number of biolog-ical application cases, it was sufficient to assume thatQi is independent of load [10,45,48]. In this case, theactive driving forces can be uniquely calibrated froma reference dynamics, ideally known from experiments.Once this is done, one can extrapolate to alternativedynamic scenarios.

As an example for this calibration procedure, previ-ous work used experimental data of in-phase synchro-nized beating in the biflagellate green alga Chlamy-domonas, which allowed to predict the response to per-turbations of this synchronized state [45]. Similarly,measured cilia beat patterns in the absence of externalflow have been used to calibrate active driving forcesand predict the response to external flow [10]. In sec-tion “Application: pair of interacting cilia”, we considerthe dynamics of an isolated cilium with constant phasespeed to calibrate its active driving force. We then usethis model to predict synchronization dynamics for apair of cilia. In all these cases, the driving forces Qi

coarse-grain internal active processes.Additionally, the formalism allows to incorporate

internal elastic degrees of freedom qi and the corre-sponding elastic restoring forces Qi in a formally equiv-alent manner. An example includes the waveform com-pliance of flagellar bending waves [10,48]. Similarly,one can include external forces acting on self-propelledshape-changing microswimmers, as discussed in thenext section.

5 Rigid body motion of a self-propelledmicroswimmer

The above formalism includes the important applica-tion case of shape-changing microswimmers and theirself-propulsion in a viscous fluid. For that aim, we intro-duce rigid body transformation and include these in theset of generalized coordinates.

Specifically, we consider a microswimmer with outersurface S and introduce a material frame of thismicroswimmer consisting of a reference point x0 anda set of orthonormal vectors e1, e2, e3.

A rigid body motion of the swimmer is character-ized by a translation of its reference point, x0 = v0 =

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v1 e1 + v2 e2 + v3 e3, and a rotation of its materialframe with ek = εijkΩjei, where εijk denotes the Levi-Cevita symbol and we use Einstein summation conven-tion. The components v1, v2, v3 and Ω1, Ω2 and Ω3

of the translational and the rotational velocity vectorwith respect to the basis e1, e2, e3, respectively, rep-resent the six degrees of freedom of rigid body motionand satisfy v1 = v0 · e1, v2 = v0 · e2, v3 = v0 · e3,and Ω1 = e2 · e3 = −e3 · e2, Ω2 = e3 · e1 = −e1 · e3,Ω3 = e1 · e2 = −e2 · e1.

We choose these velocity components as the six gen-eralized velocities

q1 = v1, q2 = v2, q3 = v3, q4 = Ω1, q5 = Ω2, q6 = Ω3.(20)

Formally, the coordinates q1, . . . , q6 are elements of theLie group se(3) = R3 × so(3) of rigid body transforma-tion [60].

For the special case, where the generalized velocitiesrepresent rigid body motion as in Eq. (20), the conju-gate generalized hydrodynamic friction forces defined inEq. (12) are simply given by the components of the totalhydrodynamic friction force F and the total hydrody-namic friction torque M, respectively

P1 = F · e1, P2 = F · e2, P3 = F · e3,

P4 = M · e1, P5 = M · e2, P6 = M · e3. (21)

In this case, the 6 × 6 matrix of generalized hydrody-namic friction coefficients Γ reduces to the well-knownhydrodynamic friction matrix (inverse mobility matrix)of an arbitrary-shaped rigid object. For a collection ofrigid objects (e.g., a collection of rigid spheres as con-sidered in [61]), we recover the inverse of the grandmobility matrix.

We can describe active shape changes of the microswim-mer using coordinates x′

1, x′2, x′

3 relative to the swim-mer’s material frame for each point x ∈ S on the surface

x = x0 + x′1 e1 + x′

2 e2 + x′3 e3. (22)

We introduce the time-dependent rigid body transfor-mation that maps the material frame of the swimmerto the laboratory frame, such that the reference pointr0 of the swimmer is mapped to the origin 0 ∈ R3, andthe material frame vectors are mapped to the standardunit vectors, respectively. The coordinates x′

1, x′2, x′

3are then just the coordinates of the image x′ of a pointx ∈ S under this transformation, i.e., the coordinatesof the surface after it has been brought into a referencecondition [62]. Equation (22) allows us to decomposethe displacement velocity v(x) of the surface into a con-tribution stemming from the rigid body motion and acontribution stemming only from any shape change

v(x) = x = x0 +3∑

j=1

x′j ej

︸ ︷︷ ︸rigid body motion

+3∑

j=1

x′j ej

︸ ︷︷ ︸shape change

. (23)

The superposition principle of low-Reynolds numberflow, Eq. (11), implies that the surface density f(x) ofhydrodynamic friction forces can be written as a super-position of contributions due to rigid body motion anda contribution fact(x) due the active shape change

f(x) = v1 g1(x) + v2 g2(x) + v3 g3(x) + Ω1 g4(x)+Ω2 g5(x) + Ω3 g6(x) + fact(x), (24)

where fact(x) depends only on the shape change x′, butnot on the translational velocity v0 nor the rotationalvelocity Ω.

Since inertia is assumed negligible, the total force andtotal torque acting on a microswimmer must equal anyexternal force or torque acting on the swimmer, F =Fext, M = Mext [27]. It follows that a microswimmerthat is free from external forces or torques does notexert any net force or torque on the surrounding fluiditself

F = 0, M = 0. (25)

Equation (25) holds in particular for a neutrally buoy-ant biological microswimmer (a good approximation formany biological microswimmers).

The surface density of hydrodynamic friction forcesdue to active shape changes, fact(x), gives rise to a con-tribution Fact =

∫Sd2x fact(x) to the total force, as well

as an analogous contribution Mact to the total torque.The force and torque balance equations, Eq. (25), thusprovide an inhomogeneous system of six linear equa-tions for the six components of the translational androtational velocity, v0 and Ω.

We emphasize that Eq. (18) is very general, andincludes the following application cases of microswim-mer motion:

• External forces or torques: For example, externalforces Fext or external torques Mext are capturedby corresponding external forces Qext

i . Examplesinclude gravitational force for a non-buoyant swim-mer or torques exerted by an external rotatingmagnetic field on an artificial microswimmer withnonzero magnetic dipole moment.

• Prescribed shape dynamics: For a prescribed shapedynamics, say of shape coordinate qi with prescribeddriving protocol qi(t), one would omit the corre-sponding force balance equation Qi = Pi from theset of equations Eq. (18), and solve for the equa-tion of motion of the other coordinates with pre-scribed qi(t). The conjugate hydrodynamic frictionforce Pi nonetheless appears in the formula for thetotal hydrodynamic dissipation rate R(h), wherePiqi equals the rate of work required for the shapechange with rate qi. A number of classical theorypublications on self-propelled biological microswim-mers considered prescribed shape dynamics [28,49–52,62].

• Constrained motion: Several applications consid-ered constrained swimmers, for example, biologi-cal microswimmers clamped in micropipettes con-strained from translational motion [19,20,22]. For-

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mally, this is a special case of a coordinate qi withprescribed dynamics for the coordinates q1, . . . , q3

representing rigid body translation, enforcing qi =0. The conjugate hydrodynamic friction force Pi

equals the external constraining force required toimpose the constraint. Similarly, to constrain amicroswimmer from rotational motion requires aconstraining torque M = P4 e1 + P5 e2 + P6 e3. Asa historical note, in their classical 1955 paper, Gray& Hancock considered self-propulsion of sperm cellswith constrained rotational motion to simplify thecalculation [28].Finally, clamped microswimmers exposed to uni-form external flow with flow velocity u0 far from theswimmer as considered in [10] can be incorporatedinto our formalism by switching to a co-moving ref-erence frame in which the fluid is at rest. In the co-moving frame, the clamped swimmer is “dragged”through the fluid, corresponding to a constraint forrigid body translation, qi = −u0 ·ei, i = 1, 2, 3. Cor-respondingly, the total hydrodynamic friction forceF = P1 e1+P2 e2+P3 e3 represents the constrainingforce required to clamp the microswimmer in suchan external flow.

6 Multi-scale modeling: numericalimplementation

To solve for the dynamics of an active surface accordingto Eq. (19), it suffices to compute the generalized hydro-dynamic friction matrix Γ for a set of reference config-urations q and save this as a look-up table; the frictionmatrix Γ(q) for arbitrary q can then be found by inter-polation. This allows to solve the equation of motionEq. (19) fast, using pre-computed hydrodynamic fric-tion coefficients. We outline the numerical implementa-tion of this general program.

While Eq. (15) may look abstract, all quantities canbe directly obtained from numerical computations forarbitrary surfaces S. Assume the surface S is repre-sented by a triangulated mesh. The triangular faces (or“elements”) shall be enumerated by k ∈ I with mid-points xk and respective areas Ak.

In a first step, we compute a (normalized) surfacedistribution of velocities wi(xk), k ∈ I for each gen-eralized coordinate i = 1, . . . , n, either by computingthe derivative wi(xk) = ∂xk(q)/∂qi analytically or byevaluating the finite difference quotient

wi(xk) =xk(q + ε Δi) − xk(q)

ε, (26)

for each midpoint xk, k ∈ I, where Δi is the unit vectorwhose components are all zero, except the ith compo-nent, and ε is a small number.

We can use boundary element methods to numeri-cally compute a surface density of hydrodynamic fric-tion forces f(xk) with physical units of a stress, given an

arbitrary surface distribution of velocities v(xk) spec-ified at each midpoint xk, k ∈ I. Specifically, in theapplication example below, we use the fast multi-poleboundary element method fastBEM [39,40].

In the next step, we compute the surface densityfj(xk) = αj gj(xk) of hydrodynamic friction forces,corresponding to the velocity distribution vj(xk) =αj wj(xk). Here, αj is an arbitrary constant to ensureproper physical units of a velocity for vj . We thusobtain n surface distributions of (normalized) hydro-dynamic friction forces gj(xk), j = 1, . . . , n, one foreach generalized coordinate qj . These force distribu-tions gj(xk) depend on q, but not on q. Finally, wecompute the components Γij of the generalized hydro-dynamic friction matrix Γ by taking the scalar prod-uct of the ith (normalized) velocity distribution wi(xk),and the jth (normalized) force distribution gj(xk)

Γij =∑

k∈Iwi(xk) · gj(xk)Ak, i, j = 1, . . . , n, (27)

where Ak was the area of the kth triangle. We can inter-pret αjgj(xk)Ak at the total force acting on the kthelement (with proper physical units of a force) if thegeneralized coordinate qi would change at a rate αi.

Importantly, it suffices to compute the generalizedhydrodynamic friction matrix Γ only for a set of refer-ence configurations and save this as a look-up table. Ifm discrete values are used for each of the n generalizedcoordinates, the Stokes equation needs to be solved atotal of nmn times, as we need to change each of the qj ,j = 1, . . . , n for mn different choices of q. By exploit-ing symmetries, as well as translational and rotationalinvariance for individual microswimmers, this numbercan be reduced further. The friction matrix Γ(q) forarbitrary q can then be found by interpolation. Forexample, cubic spline interpolation, low-order polyno-mials, and (double) Fourier series were used in previousapplications [10,45,48].

In principle, different hydrodynamic simulation meth-ods could be used to solve the Stokes equation and com-pute the force distribution f(xk). Deterministic latticeBoltzmann solvers may be suitable, provided the effec-tive Reynolds numbers are sufficiently small. An earlyapplication represented the surface of a microswim-mer not by a triangulated mesh, but as a collection ofequally sized spheres, and computed the grand mobil-ity matrix for these spheres using the hydrolib pack-age [63]. In the application example below, we employthe fast multi-pole boundary element method fastBEM[39,40], available for download at [64]. The open-sourceimplementation of the fast boundary element methodSTKFMM directly incorporates the fundamental solu-tion of the Stokes equation close to a no-slip bound-ary wall [65] and thus relieves the need for an explicitrepresentation of the boundary as a triangulated mesh,yet currently only supports the computation of velocityfields from force distributions [66,67].

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7 Application: pair of interacting cilia

We demonstrate our LAMAS modeling frameworkusing the example of hydrodynamic synchronizationin pairs of cilia. We therefore reconsider the questionof in-phase and anti-phase synchronization previouslyaddressed by Vilfan et al. [57], yet, instead of a mini-mal model of spheres orbiting on circular trajectories,we employ in our simulations a realistic cilia beat pat-tern obtained from previous experiments.

We digitalized and reconstructed three-dimensionalshapes of a beating cilium on the surface of the uni-cellular ciliated protist Paramecium [12] as presentedin [68]. The cilia beat is periodic, and we can thusdescribe the shape of the cilia centerline as a periodicshape sequence parameterized by a 2π-periodic phasevariable ϕ, see Fig. 3a. For unperturbed beating, thephase speed equals the angular frequency of the ciliabeat, ϕ(t) = ω0.

7.1 Equation of motion for a pair of cilia

We consider two identical cilia beating in the samedirection attached to a no-slip boundary wall, seeFig. 3a and b. We describe each cilium by a single phasevariable that parameterizes its periodic sequence of cen-terline shapes. The two phase variables ϕ1 and ϕ2 fullycharacterize the dynamics of the two beating cilia, andrepresent a set of generalized coordinates with state vec-tor q = (ϕ1, ϕ2).

For our example, the force balance equation, Eq. (18),takes the form

Q1(ϕ1) = Γ11(ϕ1, ϕ2)ϕ1 + Γ12(ϕ1, ϕ2)ϕ2

Q2(ϕ2) = Γ21(ϕ1, ϕ2)ϕ1 + Γ22(ϕ2, ϕ2)ϕ2. (28)

This equation can be further simplified. The symmetryrelation Eq. (17) implies Γ12(ϕ1, ϕ2) = Γ21(ϕ1, ϕ2) forthe hydrodynamic interactions. Numerical computationof Γ11(ϕ1, ϕ2) shows that this self-friction coefficient ofthe first cilium is virtually independent of the phase ofthe second cilium, and almost does not change whenthe other cilium is not present at all. An analogousstatement holds for the second cilium. Therefore, wecan replace the two self-friction coefficients in Eq. (28),Γ11(ϕ1, ϕ2) and Γ22(ϕ1, ϕ2), by the self-friction coeffi-cient for a single cilium to very good approximation.This approximation allows us to define the active driv-ing forces using the case of a single cilium.

Calibration of active driving force We require that asingle cilium should beat at a constant phase speedϕ1 = ω0, where ω0 denotes the intrinsic beat frequencyof the cilium if there are no interactions with othercilia. This requirement uniquely determines the activedriving force Q1(ϕ1). Specifically, for a single cilium,the force balance equation reads, Q1(ϕ1) = Γ11(ϕ1) ϕ1.We conclude Q1(ϕ1) = ω0 Γ11(ϕ1); Fig. 3c displaysthe phase dependence of Γ11(ϕ1). Since both cilia are

assumed identical with same intrinsic beat frequencyω0, this also specifies the active driving force Q2(ϕ2) ofthe second cilium.

Equation of motion Using the force balance equation,Eq. 28, and the calibrated driving force, we obtain theequation of motion

ϕ1 = ω0 − C1(ϕ1, ϕ2) ϕ2, C1(ϕ1, ϕ2) =Γ12(ϕ1, ϕ2)

Γ11(ϕ1)

ϕ2 = ω0 − C2(ϕ1, ϕ2) ϕ1, C2(ϕ1, ϕ2) =Γ12(ϕ1, ϕ2)

Γ11(ϕ2).

(29)

Equation (29) describes a pair of coupled phase oscilla-tors.

In the following, we use Eq. (29) and pre-computedfriction coefficients to analyze in-phase and anti-phasesynchronization of the two cilia depending on their rel-ative position. Details on the numerical computation ofΓij(ϕ1, ϕ2) can be found in the appendix. An exampleof the generalized friction coefficient Γ12(ϕ1, ϕ2), whichcharacterizes hydrodynamic interactions between thetwo cilia, is shown in Fig. 3d; Fig. 5 shows Γ12(ϕ1, ϕ2)for additional cilia orientations.

7.2 Results: in-phase and anti-phasesynchronization as function of direction

Hydrodynamic interactions decay as 1/d3. For largeseparation distances d between the two cilia, hydro-dynamic interactions between the two cilia as char-acterized by Γ12(ϕ1, ϕ2) decay as 1/d3, see Fig. 3e.This asymptotic scaling is consistent with the expectedleading-order singularity of the flow field induced by asingle cilium. Specifically, the flow field induced by apoint force parallel to a no-slip boundary wall is givenby the Blake tensor [65] and decays as 1/d3 for pointsat a constant height from the boundary, which is therelevant case for the interaction between cilia [69].

Linear stability analysis Since both cilia were assumedidentical, the in-phase synchronized state with ϕ1(t) =ϕ2(t) is always a periodic solution of Eq. (29). To assessthe linear stability of this in-phase synchronized state,we monitored the evolution of a small perturbation ofthe phase difference δ(t) = ϕ2(t)−ϕ1(t) during one beatcycle. Specifically, we integrated Eq. (29) with the ini-tial condition ϕ1(t = 0) = −δ0/2 and ϕ2(t = 0) = δ0/2for a small perturbation |δ0| � 1 up to time T definedby ϕ(T ) = [ϕ1(T ) + ϕ2(T )]/2 = 2π (corresponding tothe completion of a full beat cycle), and recorded thenew phase difference δ1 = δ(T ).

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A Cilia beat pattern

F Linear stability G Steady-state��

C Self-friction D Hydro. interaction E Scaling

H Scaling

B Relative orientation

d

y

x�

y

x

z

stroke

Recoverystroke

We define a dimensionless Lyapunov exponent as

λ = log |δ1/δ0|, (30)

which characterizes whether the initial perturbationdecays or grows. The in-phase synchronized state is lin-early stable if |δ1| < |δ0| (hence λ < 0) and linearlyunstable if |δ1| > |δ0| (hence λ > 0).

Figure 3f shows λ as function of relative cilia position.Here, the first cilium is located at the origin, while thesecond cilium is located at the position of the respectivecolored dots.

The symmetry of Eq. (29) implies that the synchro-nization behavior is invariant under a point reflection,which swaps cilia 1 and 2. Whether in-phase synchro-nization is stable or not only depends on the directionof the separation vector between the two cilia (whereλ > 0 for direction angles ψ = 2π/3 and 5π/6, in whichcase the cilia synchronize anti-phase, as discussed next).

Additionally, we analyzed the steady-state dynamicsof Eq. (29) and identified phase differences δ∗ that cor-respond to stable periodic solutions, see Fig. 3g. As atechnical point, δ(t) may weakly oscillate during eachcycle; we therefore define δ∗ as the phase difference attimes for which ϕ is an integer multiple of 2π.

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� Fig. 3 In-phase and anti-phase synchronization in a pairof interacting cilia. a Cilia beat pattern from unicellularParamecium [12] as reported in [68], shown as sequenceof three-dimensional shapes parameterized by a 2π-periodicphase variable ϕ (color code). Spacing of square grid: 2µm.b We consider a pair of cilia with respective phases ϕ1 andϕ2, whose base points are separated by a distance d alonga direction that encloses an angle ψ with the x axis (wherethe y axis is set by the direction of the effective stroke ofboth cilia). c Self-friction coefficient Γ11(ϕ1) of a single cil-ium as function of its phase variable ϕ1, obtained by solvingthe Stokes equation of three-dimensional flow (blue dots), aswell as continuous representation as Fourier series (orangeline). In the case of a single cilium, Γ11 is proportional to thephase-dependent active cilia driving force Q1(ϕ1). d Gener-alized hydrodynamic friction coefficient Γ12(ϕ1, ϕ2) charac-terizing hydrodynamic interactions from the second ciliumto the first cilium, see also Eq. (28). Positive values (redcolors) imply that the motion of the second cilium causesthe first cilium to beat slower, while negative values (bluecolors) imply that the first cilium beats faster. Cilia distanced = 18µm, orientation angle ψ = 2π/3. e The magnitude ofhydrodynamic interactions, here quantified by the L2-normof Γ12, decay as ∼ 1/d3, consistent with the theoretical scal-ing expected from the Blake tensor [65]. Different curves cor-respond to different separation directions between the twocilia (ψ = 0: dark blue, ψ = π/3: light green, ψ = 2π/3: teal;also indicated by the direction arrows.) f We characterizethe stability of the in-phase synchronized state, defined byϕ1(t) = ϕ2(t), for different relative orientations of the twocilia by a Lyapunov exponent λ, see Eq. (30). Colored dotsat respective positions in the xy-plane represent the value ofλ if the second cilium is positioned at the position of the dotand the first cilium is located at the origin. Negative valuesimply that in-phase synchronization is linearly stable (greencolors, λ < 0), while positive values imply that in-phasesynchronization is linearly unstable (red colors, λ > 0). gWe determined the steady-state phase difference δ∗ betweenthe two cilia for different relative cilia positions, analogousto panel F. While δ∗ = 0 for cilia orientations with stablein-phase synchronization (cyan), we observe anti-phase syn-chronization with δ∗ ≈ π for cilia orientations with λ > 0(red colors). For relative cilia orientation aligned with thedirection of the effective stroke of the cilia beat (ψ = π/2),we observed cases of multi-stability (bicolored dots). h Con-sistent with the far-field scaling of hydrodynamic interac-tions as shown in panel E, we find that also the Lyapunovexponent λ, which represents an effective synchronizationstrength, likewise decays as 1/d3 with distance d betweenthe two cilia. Different curves represent different separationdirections, analogous to panel E. Frequency of cilia beat:ω0/(2π) = 32Hz [12], fluid viscosity: μ = 10−3 Pa s

When the in-phase synchronized state is linearly sta-ble for a given cilia configuration, we obviously haveδ∗ = 0. If, however, the in-phase synchronized state islinearly unstable, we approximately find δ∗ ≈ π, corre-sponding to anti-phase synchronization. For a few ciliaconfigurations, we observe multi-stability, characterizedby two different values of the phase differences δ∗ thatcorrespond to stable periodic solutions; these configu-rations are indicated as bicolored half circles in Fig. 3g.

The magnitude |λ| of the Lyapunov exponentsdecreases as 1/d3 with distance d between the two

cilia, see Fig. 3h, consistent with the far-field scalingof hydrodynamic interactions shown in panel E. Thissuggests that short-range interactions between close-by cilia may dominate emergent behavior in carpets ofmany cilia.

8 Discussion

Summary We presented a multi-scale modeling andsimulation framework for active surfaces immersed inviscous fluids. This includes self-propulsion of shape-changing microswimmers as a special case. The key ideais to constrain the shape dynamics to a small numberof principal deformation modes. These modes representgeneralized coordinates, for which generalized hydrody-namic friction coefficients are defined according to theformalism of Lagrangian mechanics. To actually com-pute these friction coefficients, the Stokes equation issolved for an infinitesimal change of each generalizedcoordinate in an initial step. For subsequent dynamicsimulations, a generalized force balance between hydro-dynamic friction forces and active driving forces issolved in each time step. This is sufficiently fast sincethis second step does not involve any hydrodynamiccomputations, but uses the pre-computed hydrody-namic friction coefficients.

Our formalism generalizes classical Lagrangian dynam-ics of dissipative systems [42] to active systems. Therate of work exerted by the active surface on the sur-rounding fluid provides a Rayleigh dissipation functionR(h), which defines generalized friction forces Pi con-jugate to each generalized coordinate qi as a partialderivative 2Pi = ∂R(h)/∂qi. Numerically, the general-ized friction forces are computed from a surface densityof hydrodynamic friction forces using a Lagrangian pro-jection method. Active driving forces coarse-grain inter-nal active processes, such as the dynamics of molecularmotors inside cilia and flagella. These active drivingforces can be calibrated from a reference data set, forwhich the dynamics is already known or prescribed.

Our approach shares the idea of multi-scale model-ing with recent developments of reduced-order mod-els, which likewise propose a decomposition of bio-logical hydrodynamics problems with multiple queriesinto an initial setup phase during which the Stokesequation needs to be solved for example configurations(“offline phase”), and an subsequent phase of param-eter space exploration (“online phase”) [41]. However,our approach does not require an affine dependence ofhydrodynamic quantities on model parameters.

Potential applications We applied our general frame-work to a model example of mutual synchronizationbetween two cilia, using an experimentally measuredcilia beat pattern. Future work will generalize thisapproach to cilia carpets with many cilia [70], whichpreviously had been either studied using detailed simu-lations with many degrees of freedom [71,72], or using

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minimal models [69,73–75]. A key simplifying assump-tion will be to approximate many-body hydrodynamicinteractions between many cilia as a superposition ofpairwise interactions. A similar approach can be appliedto study self-organized pattern formation in suspensionof shape-changing microswimmers, using the approx-imation of pairwise interactions between microswim-mers, which is valid for dilute suspensions.

An important feature of our modeling framework isthat biological noise can be incorporated in a naturalway. Beating cilia are known to exhibit both phasefluctuations (frequency jitter), as well as amplitudefluctuations [20,53,76]. This active noise jeopardizessynchronization of cilia by hydrodynamic interactions.Additionally, noise randomizes the motion of biologicalmicroswimmers. While thermal noise causes noticeablerotational diffusion of micrometer-sized bacteria suchas E. coli [77], amplitude fluctuations of cilia beatingaffect the swimming of tenfold larger eukaryotic swim-mers [47]. In our framework, active noise is incorporatedby using stochastic active driving forces. In previouswork, adding additive Gaussian white noise with noisestrengths calibrated from experiments was sufficient toaccount for effective diffusion of swimming sperm cells,or noisy synchronization of coupled cilia [53]. For sim-ulations accounting for biological noise, it is beneficialto use a deterministic solver for the Stokes equation asdone here, in order to not confound physically relevantnoise and fluctuations from a stochastic hydrodynamicsimulation method.

Next, our modeling framework helps to conceptualizethe load response of cilia and flagella, which beat slowerif the hydrodynamic load opposing their beat increases.Classical work showed how cilia and flagella beat slowerat increased viscosity of the surrounding fluid [12,14];likewise, external flows change the speed of cilia beat-ing [10,17,23]. The load response of cilia and flagellais a prerequisite for putative mechanisms of synchro-nization by hydrodynamic or mechanical interactions.We propose that the generalized hydrodynamic frictionforce defined here can serve as a proxy for the effec-tive hydrodynamic load acting on an actively shape-changing structure such as a beating cilium.

Limitations Our approach is restricted to the limit ofzero Reynolds numbers, because it crucially relies onthe superposition principle for Stokes flow. In a lami-nar flow regime at finite Reynolds numbers, we expectthat computations of self-friction are still accurate, butlong-range hydrodynamic interactions become increas-ingly less accurate with increasing distance if the Stokesequation is used. Nonetheless, our approach should stillserve as a reasonable approximation, since any long-range hydrodynamic interactions that are incorrectlypredicted by the Stokes equation will be very weakalready.

In principle, a similar framework could be developedusing the linearized Navier–Stokes equations insteadof the Stokes equation used here, but only in Fourier

space. The linearized Navier–Stokes equation providesa refined approximation for long-range hydrodynamicinteractions if the Reynolds number for oscillatorymotion becomes appreciable. In this case, a superposi-tion principle applies for time-periodic flows. However,working in frequency space instead of the time domainwill make the practical solution of dynamic problemsmore difficult.

Another limitation of our approach is that it isinherently restricted to Newtonian fluids. While certainimportant biological fluid dynamics problems involveviscoelastic fluids, the lack of a superposition principlein this case implies that other methods need to be used.

Conclusion Our modeling and simulation frameworkLAMAS can be complimentary to existing methods.Our approach is particularly suited to screen exten-sive parameter ranges, provided the modified parame-ters concern the dynamical model (such as active driv-ing forces or effective elastic constants [48]), and donot require re-computation of the generalized hydro-dynamic friction coefficients. Likewise, our approachallows to compute multiple stochastic realizations of thesame problem fast.

Supplementary information The online version con-tains supplementary material available at https://doi.org/10.1140/epje/s10189-021-00016-x.

Acknowledgements AS and BMF are supported by theGerman National Science Foundation (DFG) through theMicroswimmers priority program (DFG Grant FR3429/1-1and FR3429/1-2 to BMF), as well as through the Excel-lence Initiative by the German Federal and State Govern-ments (Clusters of Excellence cfaed EXC-1056 and PoLEXC-2068). BMF acknowledges a Heisenberg grant (DFGgrant FR3429/4-1).We thank Andrej Vilfan for fruitful discussions. BMF devel-oped the original idea of LAMAS, AS performed all numer-ical computations. AS and BMF wrote the manuscripttogether.

Funding Open Access funding enabled and organized byProjekt DEAL.

Data Statement This manuscript has no associated dataor the data will not be deposited. [Authors’ comment: Nonew data were created or analyzed in this study.]

Open Access This article is licensed under a Creative Com-mons Attribution 4.0 International License, which permitsuse, sharing, adaptation, distribution and reproduction inany medium or format, as long as you give appropriate creditto the original author(s) and the source, provide a link tothe Creative Commons licence, and indicate if changes weremade. The images or other third party material in this arti-cle are included in the article’s Creative Commons licence,unless indicated otherwise in a credit line to the material. Ifmaterial is not included in the article’s Creative Commonslicence and your intended use is not permitted by statu-tory regulation or exceeds the permitted use, you will need

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to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Appendix: numerical methods

We present additional details on the numerical computa-tions for the application case of a pair of interacting cilia.

Mesh generation We generated a triangulated mesh for thecombined surface S of cilia and boundary surface using acustom-built Python package (available upon request), seealso Fig. 4. We represent the digitalized shapes of the ciliacenterline as a family of space curves r(s, ϕ) parameterizedby arclength s with 0 ≤ s ≤ L, where L = 10µm is thelength of the cilium, and a 2π-periodic phase variable ϕ.The centerline shapes of the two cilia are thus given by

r0,1 + r(s, ϕ1) and r0,2 + r(s, ϕ2), where the base points r0,1

and r0,2 have a distance d. The separation vector r0,2 − r0,1

encloses an angle ψ with x-axis (where y axis is set by theeffective stroke of both cilia), see Fig. 3b.

We generate a triangulated mesh for the each cilium bytreating the cilium as a bent cylinder of radius 0.125µm,using 8 node points in azimuthal direction, and 61 nodesin longitudinal direction, as well as one apex node at theproximal and distal ends, respectively. For numerical stabil-ity, the proximal apices of each cilium mesh have a distanceof 0.25µm from the boundary surface. A smaller distancevirtually does not change the computed friction coefficients,but can cause convergence issues.

The hydrodynamic solver fastBEM requires closed sur-faces, which prompted us to use a circular disk of finitethickness (radius 60µm, thickness 1.5µm) instead of a planesurface. Initial simulations confirmed that using a largerdisk radius virtually did not change results. Disk faces weremeshed using the Python triangle package (minimum trian-gle angle 20◦, maximum triangle area 2µm2 on the upper

A Triangulated mesh for cilia pair B Close-up

20 μm 2 μm

Fig. 4 Triangulated mesh for pair of cilia attached toboundary surface. a Entire mesh consisting, of two cilia rep-resented as bent cylinders (red, cyan), as well as a thin diskof radius 60µm representing the boundary surface (gray),corresponding to approximately 1000 triangular elements

per cilium and 7500 elements for the boundary surface. bClose-up view on a single cilium. Nodes on the bottom ofthe surface are hidden from view. Cilia distance d = 18µm,orientation angle ψ = 2π/3

Fig. 5 Hydrodynamic interaction as function of ciliaphases for different cilia orientations. Generalized hydrody-namic friction coefficient Γ12(ϕ1, ϕ2) as in Fig. 3d for dif-ferent cilia orientation angles: left: ψ = π/2 (direction of

effective stroke), middle: ψ = 2π/3 (oblique to direction ofeffective stroke), right: ψ = π (perpendicular to direction ofeffective stroke). Cilia distance: d = 18µm. Note the differ-ent color scale compared to Fig. 3d

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surface of the disk up to a distance of 50µm from the diskcenter, 20µm2 otherwise). Additionally, to improve the con-vergence of the solver, we refined the mesh in a small areabelow the proximal apices of the cilia (maximum trianglearea 0.04µm2 up to a distance of 0.625µm from cilia basepoints). In total, each meshed cilium consists of 975 trian-gles, while the meshed disk consists of approximately 7600triangles.

Hydrodynamic computations We employed a fast multi-pole boundary element method termed fastBEM [39,40,64]to solve the inverse problem of finding the surface distribu-tions of hydrodynamic friction forces f(x) for given velocityfields v(x) on the combined surface S of both cilia and theboundary surface. To solve this inverse problem, the algo-rithm employs an iterative linear GMRES solver (toleranceparameter used here, tol = 5 · 10−4). In principle, the solverwould allow also for mixed boundary conditions that specifya combination of forces and velocities on different parts ofthe surface, which is, however, not needed here.

Initial tests showed that the self-friction is virtually inde-pendent of the phase the other cilium, allowing us to approx-imate Γ11(ϕ1, ϕ2) ≈ Γ11(ϕ1) and Γ22(ϕ1, ϕ2) ≈ Γ11(ϕ2),where Γ11(ϕ1) corresponds to the simulation result for asingle cilium. For the smallest distance tested here, 14µm,the difference was at most 2%. Thus, computation of Γ11

required m hydrodynamic computations for m = 20 equidis-tant phase values. The symmetry relation Eq. (17) givesΓ21(ϕ1, ϕ2) = Γ12(ϕ1, ϕ2); thus, it is sufficient to computeonly Γ12 (i.e., perform only computations where cilium num-ber 2 moves, while cilium number 1 is static). To computeΓ12(ϕ1, ϕ2), we performed m2 = 400 hydrodynamic com-putations for m2 pairs of phase values on a equidistant(ϕ1, ϕ2)-grid, with mean CPU time of about 102 secondsper computation. We repeated these computations for 42different relative positions of cilia as shown in Fig. 3f.

Interpolation From the generalized friction coefficientscomputed for a discrete set of (ϕ1, ϕ2)-values, we obtainedin a final step a continuous representation in the form of a(double) Fourier series truncated after order 4 (correspond-ing, e.g., to (2 · 4 + 1)2 = 81 Fourier terms for Γ12(ϕ1, ϕ2)).

Dynamical equation The system of coupled ordinary dif-ferential equations, Eq. (28), was solved with Python(method scipy.integrate.solve_ivp, tolerance 10−8, scipy ver-sion 1.5.0). In each time step, we compute the inverse ofmatrix Γ.

Lyapunov exponents For the computation of Lyapunovexponents λ shown in Fig. 3f, we used a small perturbationδ0 = 10−2 of the in-phase synchronized state. Preliminarysimulations using a smaller perturbation δ0 = 10−3 gavevirtually identical results.

Steady-state phase difference We computed the steady-state phase difference δ∗ between the two cilia as a fixedpoint of the Poincare map L : δ0 → δ1. Specifically, we

computed L(δ0) for 30 equidistant values of δ0 in the inter-val [0, 2π) by integrating Eq. (28) using initial conditionsϕ1(t = 0) = −δ0/2 and ϕ2(t = 0) = +δ0/2. We then numer-ically solved for fixed points L(δ∗) = δ∗, using monotoniccubic spline interpolation of L. The periodic solution cor-responding to a steady-state phase difference δ∗ is stable ifthe numerical derivative dL/dδ0|δ0=δ∗ is smaller than 1.

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