modeling paramecium swimming in a capillary...

Scientia Iranica B (2016) 23(2), 658{667

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Modeling Paramecium swimming in a capillary tube

A.N. Sarvestani, A. Shamloo� and M.T. Ahmadian

School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran.

Received 7 December 2014; received in revised form 8 February 2015; accepted 31 May 2015

KEYWORDSParamecium;Swimming;Modeling;Modi�ed boundaryelement.

Abstract. In certain types of biomimetic surgery systems, micro robots inspired byParamecium are designed to swim in a capillary tube for gaining access to internal organswith minimal invasion. Gaining insight into the mechanics of Paramecium swimmingin a capillary tube is vital for optimizing the design of such systems. There are twoapproaches to modeling the physics of micro swimming. In the envelope approach, whichis widely accepted by researchers, Paramecium is approximated as a sphere, self-propelledby tangential and normal surface distortions. However, not only is this approach incapableof considering the speci�c geometry of Paramecium, but it also neglects short rangehydrodynamic interactions due to beating cilia, thus, leading to dissimilarity betweenexperimental data and simulation results. The aim of this study is to present a sublayer approach to modeling Paramecium locomotion that is capable of directly applyinghydrodynamic interactions due to beating cilia onto the Paramecium boundary. In thisapproach, the Paramecium boundary is discretized to hydrodynamically independentelements and, at each time step of swimming, a speci�c function is �t to the Parameciumboundary. Then, element coordinates are extracted and uid dynamic equations are solvedto model the physics of micro swimming.© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

In ow regimes pertinent to Paramecium locomotion,viscous forces dominate inertial forces; consequently,speci�c symmetry breaking events are needed to causesuccessful propulsion [1]. Paramecium propulsion isachieved via coordinated motion of as many as 4000non-symmetrically moving organelles called cilia [2].So, accurate simulation of hydrodynamic interactionsbetween cilia is invaluable in determining the mech-anism of Paramecium locomotion. Micro swimmingrequires complex mechanisms of propulsion which can-not be understood by modeling the symmetry breakingevents due to beating cilia via a sub layer approach. Inthe case of micro robots used for minimally invasivemedicine, optimization of the locomotion mechanism

*. Corresponding author. Fax: +98 21 66165599E-mail addresses: [email protected] (A. Shamloo);[email protected] (M.T. Ahmadian)

could help reduce energy consumption, which is of greatinterest among researchers [3].

Attempts to �nd a mathematical or numericalmodel for swimming micro organisms date back to theearly 1950s. In a pioneering work, Lighthill derived aclosed mathematical form for modelling low-Reynoldsnumber locomotion, called the squirmier model [4].Later, in 1971, Blake did some corrections to the closedform derived by Lighthill and developed the sphericalenvelope approach for modeling the squirming mo-tion [5]. In this approach, a micro swimmer is modelledas a sphere able to propel itself using wave-like surfacedeformations. These wave-like surface deformationsoriginate from the coordinated motion of cilia tipscalled the metachronal wave. The metachronal waveis nonsymmetric as a result of nonsymmetric deforma-tions of beating cilia. The ciliary beat is divided intoe�ective and recovery strokes. The e�ective stroke isfor propelling the uid in a forward direction, whilethe recovery stroke, which lasts longer, returns the

A.N. Sarvestani et al./Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 658{667 659

cilium to its initial condition with minimum backwardpropulsion [6]. Many analytical and numerical studieshave been undertaken for improving the accuracy ofthe squirmer model. For example, one of the mostrecent contributions to this model is the work of Wangand Ardekani who solved the problem of unsteadysquirming motion [7]. Their great contribution to thesquirming model made this model capable of consider-ing added mass and rotation along the swimming axis,a feature that is observed in experimental studies [8].

The most e�cient stroke of surface deformationsfor propulsion was characterized by de�ning an e�-ciency function for surface deformations and maximiz-ing this function [9]. The speci�c surface deformationstroke for optimal feeding of micro organisms wascharacterized by �nding the stroke pertinent to theoptimal ow of nutrients towards the cell surface [10].Also, it has been identi�ed that only hydrodynamicinteractions cause Paramecium to swim in a helicaltrajectory in a capillary tube [11].

Another trend in numerical simulation of low-Reynolds swimming was inspired by the pioneeringwork of Gueron and Liron [12]. Due to the developmentof computers with high computational capacity in the1990s, this class of model, termed a sublayer model,was implemented by direct measurement of all hydro-dynamic interactions due to beating cilia [13]. Theproblem with this approach is its high computationalcost [14].

Although it has been proved that for modelinga densely packed arrangement of cilia, the envelopemodel is the best approach, this method has someproblems with the speci�cation of boundary conditionspertinent to the motion of cilia tips; therefore, ithas been unable to correctly predict the maneuveringmechanism of ciliates with densely packed arrangementof cilia. In order to overcome this problem, a modi�edboundary element method for simulation of Parame-cium swimming seems to be appropriate.

In the current work, it is intended to present amodi�ed boundary element method for simulation ofParamecium locomotion in two dimensions. In thisapproach, Paramecium is discretized to boundary ele-ments which are hydrodynamically independent fromeach other. In other words, because hydrodynamicinteractions in low-Reynolds number regimes are shortranged [15], it is possible to increase element length sothat the boundary elements become hydrodynamicallyindependent of each other. As a result, creeping owequations can be solved for each element independently.This leads to great reductions in computational cost.In the modi�ed boundary element method, insteadof considering a wavy surface for Paramecium andspecifying boundary conditions pertinent to a travellingwave [16], its membrane, at many time steps, isdiscretized to boundary elements that contain 5 cilia

beating together. The data for the ciliary beat areextracted from the bio-inspired uid lab at VirginiaTech, which consist of the coordinates and velocities ofeach point on Paramecium at each time step (see Fig-ure S2 and associated table in Supplementary Material(Section 6)) [16]. For each boundary element, at eachtime step, normal and tangential components of viscousforces are extracted via a creeping ow simulation.Integrating the normal and tangential viscous forceson the Paramecium boundary leads to extraction ofpropulsive force at each time step. This leads toextraction of the swimming velocity at each time step,and, thus, the swimming trajectory can be de�ned.The swimming trajectory obtained in this study is theresult of time dependent simulation of Parameciumswimming in a capillary tube and is in agreementwith experimental data, proving that hydrodynamicinteractions play a key role during the swimming ofParamecium in a capillary tube.

2. Modeling cilium

With regard to previous studies [6], the cilium, at eighttime steps of beating (see Figure 1(a)), is modeled asa set of rigid elements. For minimizing computationalcost, it is initially needed to �nd the minimum numberof rigid elements for approximation of the cilium'sstructure with the least possible error. In order

Figure 1. Modeling each cilium: (a) Cilium geometry atspeci�c time steps. Four of these time steps pertain to thee�ective, and four of them to the recovery, strokes; (b) d1j ,d2j and d3j show the normal distance between ciliumgeometry and rigid elements by which a cilium isapproximated; and (c) approximation error in alogarithmic scale, which shows that �ve rigid elements canlead to an accurate approximation of cilium geometry.

660 A.N. Sarvestani et al./Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 658{667

to quantify the approximation error, each cilium isdiscretized to sections with similar lengths in 8 timesteps (Figure 1(b)). In Figure 1(b), cilium geometryat time step j is discretized to three rigid elements.The approximation error can be characterized by sum-ming the maximum normal distances between ciliumsections and the rigid elements, which are shown asd1j , d2j and d3j . Initially, it is aimed to �nd theminimum number of elements which can give the bestapproximation (i.e. the least approximation error) ofthe cilium pro�le. Thus, a function was de�ned toshow the sum of the maximum distances betweencilium sections and the approximating elements. Thisfunction (F ) is numerically quanti�ed using Eq. (1)and plotted in Figure 1(c). This plot shows that formore than �ve elements, the quantity of function (F )remains unchanged. Considering that computationalcost is linearly proportional to the number of theseelements, the minimum number of elements that givesan acceptable approximation for modeling cilia (�veelements) was chosen for further simulations. The Ffunction is de�ned as:

F =Xi

Xj

dij ; (1)

where i shows the element number, j shows the stepnumber and dij shows the normal distance between theith section on the cilium and its corresponding element,in the jth time step of simulation.

3. Numerical simulation

In order to characterize di�erent physical parametersof micro swimming, the Boundary Element Method(BEM) is widely used [17,18]. According to themethodology introduced by Pozrikidis [19], in theBEM, boundaries are discretized and elements areconstructed based on discretization of the boundaries.Lauga and Riley developed a theoretical frameworkfor modeling micro swimming in low Reynolds number ow regimes in complex uids [20,21]. Their resultshave been veri�ed by new methods for measuring ow characteristics presented by Valarde et al. [22].Based on this framework, it has been shown thatimplementing the boundary element method for mod-eling the physics of micro swimming is one of thesuitable available options [23-25]. In the current work,the Paramecium boundary is discretized to boundaryelements containing a speci�c number of cilia, and uidequations are solved for each element independently.Modi�cation of the BEM in this study originates fromthis independence, which is due to the fact that elementdimensions are chosen to be so long that hydrodynamicinteractions between adjacent elements are negligible.

One of the main challenges in modeling theciliary beat in Paramecium is determining the element

length for discretizing the Paramecium boundary. Infact, it is vital to determine the maximum possibleelement length, because longer elements lead to lowercomputational cost.

An important feature in low-Reynolds uid me-chanics is that hydrodynamic interactions are e�ectiveat low ranges. Gueron et al. showed that hydro-dynamic interactions between two cilia, at a distanceof about 1.2 times cilium length, are negligible [12].Accordingly, it is possible to discretize the Parameciumboundary to elements that are hydrodynamically inde-pendent. In other words, we have to �nd the minimumpossible element size for which low-Reynolds equationscan be solved independently from other elements. Theprocedure for determining maximum possible elementlength is presented in Section 3.2.

Discretizing the Paramecium membrane to hy-drodynamically independent elements leads to a con-siderable decrease in computational cost. In thismethod, low-Reynolds equations for each element canbe solved separately, and the required parameters, suchas drag force, can be integrated onto the Parame-cium boundary. By integrating the viscous drag onthe Paramecium boundary, it is possible to calculatethe propulsive force and swimming velocity. Aftercharacterizing the element length, a time dependentsimulation was conducted in order to determine theswimming trajectory of Paramecium, its speed, andthe e�ciency of swimming. Simulation results werethen compared to experimental data [16] for validatingthe numerical method. Figure 2(a) shows the owchart for the numerical simulation used in determiningthese parameters. The �rst step in the simulation ofeach time step of swimming is to read the coordinatesand velocities of the cilium at each time step (seeSupplementary Material, Figure S2. and associatedtable (Section 6)). The second step is the constructionof boundary elements and determination of their coor-dinates in the co-moving reference frame. Then, the uid dynamic solver gives the normal and tangentialcomponents of the viscous force, which are integratedover the Paramecium boundary for extraction of thepropulsive force and swimming velocity. Finally, inorder to combine the di�erent time steps and achievea time dependent simulation, the swimming velocitycalculated during the previous time steps is speci�edas an initial condition by de�ning an inlet and anoutlet for simulating each boundary element. In otherwords, if we consider each boundary element on aswimming Paramecium, the uid passing close to eachelement could be replaced by an inlet and an outlet(see Section 3.3).

3.1. Simulation of ciliary beatsIn our simulations, the uid is regarded as viscousand incompressible, and the �laments that comprise


Figure 2. Numerical simulation: a) The simulation ow chart; and b) uid velocity distribution at a speci�c time step ofswimming.

nine microtubule pairs of the axoneme are consideredrigid boundaries immersed within the uid. In orderto �nd the velocity and pressure distribution in the uid domain, Navier-stokes equations for creeping oware solved. For the case of low-Reynolds swimming,Navier-stokes equations can be simpli�ed to Stokesequations. Eq. (2) shows the momentum and conti-nuity equations in creeping ow:

�rp+ �r2u = 0; r:u = 0; (2)

where � is uid viscosity, u represents the velocityvectorized �eld, and p accounts for the pressure scalar�eld. After solving the Stokes equations on the uiddomain, we can �nd the pressure distribution fordi�erent uid nodes. In order to characterize theaverage hydrodynamic force per length applied ontothe cilia as a function of time (S(t)), we can integratethe viscous stress (�) over the cilia deformed surface:

S(t) =R�:n̂dsLcilium

; (3)

� = �pI + ��ru+ (ru)T

�; (4)

where p is the uid pressure, n is the unit normal vectorto the cilia, s is the length parameter on the movingwalls, and I is the identity matrix [11]. Simulationis time dependent by considering the variation ofvariables with time. Parameters for the mathematicalmodel for Paramecium swimming and their referencesare presented in Table 1 in the Supplementary Material(Section 6).

In this study, it is required to �nd the largestamount of time scale that can ensure numerical con-vergence. If the time scales are chosen too small,

Figure 3. Simulation data (horizontal stress component).These plots show that 84 time steps are enough forconvergence in simulation. In other words, in the criticalregions around which a circle has been drawn, we see thatthe plot is not smooth for 21 time steps (the yellow plot);however, the plot is smooth for smaller time steps (84 timesteps - the blue plot). This shows the increased chance ofconvergence for smaller time steps.

the computational cost is increased drastically. Hence,simulations at 16, 21, 28, 42, and 84 time steps wereundertaken, and the plots for the hydrodynamic forceper length versus time were obtained.

In Figure 3, the vertical axis shows the averagehorizontal component of the stress applied to thecilium, which is de�ned as:

S(t) =R�xdslc

; (5)

where �x is the horizontal component of the viscousstress applied to the cilium, ds is the length of theelement, and lc is equal to the cilium length. Ac-cording to this �gure, the diagrams of total stress


at di�erent time steps can be considered a measurefor the convergence of the simulation. From this�gure, it is obvious that at some critical regions of thee�ective and recovery strokes, the results of simulationsat di�erent time steps, in which cilia are located atdi�erent positions, are signi�cantly di�erent from eachother. The diagram in Figure 3 shows that if wediscretize duration of the ciliary beat to 84 time steps,the measured hydrodynamic properties at each timestep are so close to the data for the next time stepthat each time step can be an initial condition for thenext time step. This means that 84 time steps can beconsidered an optimum number for our simulations. Inthe simulation of objects moving in uids, the objecthas di�erent positions at di�erent time steps. If, in thesimulation procedure, time is not discretized to smallenough steps, the moving object may be at positionsfar from each other in di�erent time steps. This leadsto inaccurate results which may not be in agreementwith experimental data.

3.2. Coordinate system and characterizationof element length

In order to model Paramecium locomotion, it is re-quired to discretize its boundary so that hydrodynamicinteractions between discrete elements are negligible.Hydrodynamic interactions between each beating cil-ium and its neighbors are signi�cant in a �nite domaincalled the cilium in uence range. Gueron and Liron.proved that this in uence range is 1.2 times ciliumlength [12]. In this study, this in uence range isinvestigated by measuring the viscous stress appliedby each cilium onto its neighbors. In other words,simulations with one to eight beating cilia are con-ducted and time averaged drag is applied to one of them(the �rst from the left is measured using Eqs. (3) and(4), and plotted in Figure 4(a)). This �gure indicatesthat hydrodynamic e�ects are negligible between �ve

adjacent cilia. Because ciliary spacing is 3 �m andcilium length is 15 �m, the cilium in uence rangeis calculated as 1.11 times cilium length. Therefore,according to Figure 4(a) and (b), we chose �ve cilia oneach boundary element.

3.3. Simulation of ciliary propulsion inParamecium

The numerical simulation procedure (Figure 2(a))begins with the construction of boundary elementsand determination of their coordinates in a Cartesianreference frame that is attached to the capillary tubewall (this is the simulation input data which depends onthe diameter of the capillary tube and the dimensionsof the chosen Paramecium) (see Figure 5). Each

Figure 5. Time dependent simulation of Parameciumswimming. In various time steps of swimming, functionsrepresenting the geometry of the Paramecium boundaryare extracted. These functions lead to determination ofboundary element coordinates at each time step.Coordinates for boundary elements are then transformedto the co-moving reference frame by de�ning the twoparameters, Y and �. Y is the normal distance betweenthe element and the corresponding tube wall, and � is theangle of rotation of the speci�c boundary element withrespect to its corresponding wall. Parameters, Y and �,are used to simulate the ciliary beat in each speci�celement by setting them as input parameters for theCOMSOL model pertinent to each element.

Figure 4. Methodology for determining the element length: a) In this �gure, the viscous force applied on the �rst ciliumfrom the left is plotted for con�gurations of one to eight cilia beating together. For con�gurations of more than �ve cilia,slight variations in the measured force are observed. In other words, hydrodynamic interactions between the �rst and �fthcilium are negligible; and b) each element on the Paramecium membrane contains �ve cilia; consequently, elements arehydrodynamically independent of each other.

boundary element contains 10340 tetrahedral elementsin COMSOL. The chosen solver is the time dependentSPOOLES, which is based on direct LU factorizationof the sparse matrix. Direct LU factorization is usuallychosen for modeling uid ow within micro uidicsdue to its stability and good convergence rate. In-deed, because we intended to simulate Parameciumthat changes its geometry while swimming, we shoulddetermine the coordinates of di�erent points on itsmembrane at each time step. Coordinates of di�erentpoints on the Paramecium membrane are extractedfrom the video of a swimming Paramecium takenfrom the bio-inspired lab at Virginia Tech. Also, forinvestigating propulsive force and swimming velocity,coordinates of di�erent points on its boundary areessential. The swimming velocity is calculated byintegrating the function pertinent to the propulsiveforce and using zero initial conditions.

The next step starts by transferring boundaryelement coordinates from the Cartesian reference frame(x; y) to the moving reference frame (Y; �). For tubediameter Dtube = 200 �m, at the initial time step(t = 0), if we choose a point on the Paramecium'supper boundary with coordinates (x0; Y0up), it can beobserved that 34 < x0 < 304 �m. Then, it is possibleto �t function f(x0) on the upper boundary, so thatwe have, for each point, Y0up = f(x0) and @Y0up

@x =@f(x0)@x = _f(x0). In the moving reference frame, Y is

the normal distance between each boundary elementand its corresponding tube wall. Therefore, in orderto calculate the normal distance, the equation for theline normal to an element with coordinates (x0; Y0up)is given in:

y � yup0 =

�dxdf(x0)

(x� x0): (6)

for calculating the normal distance, the coordinate ofthe point at which this line intersects with the upperboundary (x1; 200) can be calculated from Eq. (7):

�(200� f(x0)) _f(x0)� x0 = x1: (7)

Hence, the normal distance is equal to the distancebetween the two points, (x0; Y0up) and (x1; 200), whichis given in Eq. (8):

Y =q

(x1 � x0)2 + (200� yup0 )2: (8)

The second component in the element coordinates inthe moving reference frame is �, which is characterizedfrom Eq. (9):

�0 = cos�1�

200� yup0

y0

�: (9)

After adjusting the element coordinates to the moving

Figure 6. The COMSOL model for extraction of pressuredistribution around each boundary element. As previouslydiscussed, boundary elements are hydrodynamicallyindependent; consequently, for each individual element, itis possible to conduct an independent CFD(computational uid dynamics) analysis by which we canextract the functions pertinent to the normal andtangential components of viscous force on each element,which is speci�ed by special parameters, Y and �. Byintegrating these functions over the Parameciumboundary, we can calculate the propulsion force, whichcould be shown by a set of two time dependent functions;one function for quantity and another for the direction ofpropulsion force. Also, it may be possible to derivefunctions for the propulsion torque, which is of interest forfurther research.

reference frame, cilia coordinates, and velocities atdi�erent time steps, along with transferred coordinates(see Figure S1 and its associated table in the \Sup-plementary material" (Section 6)), the details of whichwere extracted from the experimental data provided bythe bio-inspired uid lab at Virginia Tech, [16]), wereinserted into the COMSOL model for calculating theviscous force applied onto each boundary element (seeFigure 6). Then, functions representing the normaland tangential components of the viscous force as afunction of the element coordinates were extractedfor calculating the propulsive force and swimmingvelocity (see Figure S1 in the Supplementary Material(Section 6)). At the beginning of unsteady swimming,the velocity of the organism is equal to zero. Formodeling the initial time step of the swimming, thementioned procedure is accurate enough. However,for modeling swimming at time steps in which theParamecium has a bulk velocity, the uid near eachboundary element is moving at a speci�c velocity. Thephysics of this process could be investigated by de�ningan inlet and an outlet boundary condition for theCOMSOL model (see Figure 6).

For validation of the simulation procedure, theexperimental data of Ishikawa and Hota. were

Figure 7. Comparison of the simulation data withexperimental data of Ishikawa and Hota [18]. This plotshows the velocity distribution around a swimmingParamecium. Each element of the Paramecium boundaryhas its unique angle; therefore, the horizontal label depictsthe angle of each boundary element with respect tohorizontal axes attached to the capillary tube wall.

used [18]. Ishikawa measured the uid velocity dis-tribution around a swimming Paramecium. We foundthat our simulation results were in agreement withthese data. Figure 7 shows the uid velocity arounda swimming Paramecium versus surface angle. Forboth simulation and experimental data, uid velocitylimits to zero for the anterior and posterior sectionsof the swimming Paramecium. For the middle sectionof the Paramecium, uid velocity reaches its maxi-mum size. In the middle sections, a slight di�erencebetween simulation results and experimental data isobserved. This slight di�erence is due to the fact thatthey have considered an axisymmetric geometry for aswimming Paramecium, whereas its geometry is notaxisymmetric. Their method for obtaining the uidvelocity is particle velocimetry. In this method, theexact geometry of Paramecium could not be takeninto consideration and an approximate geometry isinvestigated.

Also, simulation results of the Paramecium swim-ming in a capillary tube are compared to Jana'sexperimental data [16]. These data include swimmingvelocity, swimming trajectory and swimming e�ciency.Discussions about these data are presented in theresults and discussions section.

4. Results and discussions

Recently, Jana et al. (Bio-inspired uid lab at VirginiaTech) presented a paper in which they intended toinvestigate the hydrodynamic e�ects of con�nementon the Paramecium swimming trajectory [16]. Forthis purpose, they prepared capillary tubes of variousdiameters and observed the helical trajectory followedby Paramecium swimming in them. Since we areinterested in developing a numerical model for explain-

ing hydrodynamic e�ects on Paramecium propulsion,their accurate experimental data were used for ourbenchmark test.

In order to run the benchmark test, simulationswere run for di�erent capillary tube diameters. In thesimulation procedure, initially, boundary elements areconstructed and their coordinates are extracted. Then,for each time step of swimming, the propulsive forceand swimming velocity and trajectory are extracted.Results were compared to Jana's experimental results.

4.1. Discussions on Paramecium swimming ina capillary tube

The Paramecium swimming trajectory is a helix whichmay be characterized by its wavelength and amplitude(see Figure 8). In this work, only a speci�c sectionof this helix is extracted. Jana et al. observed thatthe amplitude of the introduced helix increases nearlylinearly with an increase in capillary tube diameter(Figure 8(a) and (b)). Thus, the numerical simulationalgorithm can be validated if its results are in goodagreement with the experimental data presented byJana et al. The wavelength and amplitude could bedetermined by measuring the distance between twopoints of the wave corresponding to the swimmingtrajectory, with the slope parallel to the capillary tubewall. According to Figure 8(b), there is good agreementbetween the experimental data and simulation results.

Figure 8(c) depicts the results of another bench-mark test. In this test, swimming velocity, whichis non-dimensionalized by the velocity calculated byBlake [5], is measured. In this �gure, the black lineshows the swimming velocity from the experimentaldata. From this �gure, it is evident that for R=c < 2:1,simulation and experimental results show increasingtrends and converge with each other for very large R=cvalues [16].

4.2. Swimming e�ciencyThere are various de�nitions for swimming e�ciency.The traditional de�nition for low-Reynolds swimminge�ciency is presented in the following equation. Thisde�nition is widely used to determine the most e�cientswimming kinematics:

� = T �

: (10)

In the above equation, T � denotes the force needed topull a rigid body with similar geometry to Parameciumat constant time-averaged velocity, , while is the mean energy dissipation while pullingthe body. In di�erent studies, di�erent methods forcalculating T � have been presented. Michelin andLauga [9] derived T � = 6� from Stokes' formulafor the speci�c case of a squirmer. However, Jana etal. [16] considered the following equation for calculatingT � for the speci�c case of Paramecium:


Figure 8. Time dependent simulation of Paramecium swimming: a) Schematic representation of helical swimmingparameters: c is the Paramecium width, A is the amplitude of the swimming trajectory, R is the radius of the capillarytube, and � is the wave length of the swimming trajectory; b) plot of the non-dimensional amplitude of the swimmingtrajectory versus the non-dimensional radius of the capillary tube. Simulation data are in agreement with experimentaldata and show an almost linear increase of amplitude with the increase in capillary tube radius; and c) comparison ofresults from simulation using a modi�ed boundary element method with experimental data described by Jana et al. [16].The horizontal axes show the non-dimensional radius of the capillary tube, and vertical axes show the normalizedswimming velocity.

T � = �Uswim

�; (11)

where � denotes the uid viscosity, and � denotes thewavelength of the metachronal wave propagated bycilia. In order to compare our simulation results withthe experimental results of Jana et al. [16], we usedEq. (10) for calculating the swimming e�ciency.

Figure 9 shows a comparison of Jana et al.'s exper-imental data [16] with our simulation results. In fact,Jana et al. presented di�erent data for the e�ciencyof anterior and posterior parts of Paramecium. Inour simulation results, we only extracted e�ciency forthe swimming trajectory of the Paramecium's centerof gravity, which is the average of anterior and pos-terior e�ciencies. Simulation results reveal that theswimming e�ciency is limited to 4%, while increasingthe diameter of the capillary tube in silico. Michelinand Lauga proved that the theoretical e�ciency for thesquirmer limits to 22.22% [9]. This is in contrast to oursimulation results, indicating that the e�ciency of realParamecium is far di�erent from a squirmer. Thus,the trajectory of a swimming Paramecium may not becalculated by optimizing the e�ciency of the squirmer.

5. Discussion

In this study, it is primarily intended to develop a sublayer approach for modeling Paramecium locomotion

Figure 9. Swimming e�ciencies of various trajectoriesfollowed by anterior and posterior portions of Parameciumcompared to swimming e�ciency obtained from numericalsimulation. In numerical simulations, ratio of R=c wasvaried from 1.3 to 3.6. Simulation results are in goodagreement with experimental results of Jana et al. [16].

with less computational cost. In fact, the presentedmethodology is capable of coupling the internal mecha-nisms of cilia with external uid dynamics, which is notonly vital for investigating Paramecium locomotion,but may also be useful in determining the causes ofcilia related diseases, such as primary cilia dyskinesia.For modeling Paramecium locomotion, its boundary


was discretized into elements with lengths of 15 �m.This length was chosen because it enabled us to simu-late Paramecium with hydrodynamically independentelements and a reduced computational cost. In fact, itis an inherent property of low-Reynolds uid mechanicsthat hydrodynamic interactions are e�ective at lowranges. This is the key distinction of low-Reynoldsswimming, which enables application of computationalmodels with comparatively less computational cost.Hydrodynamical independence of adjacent elementsis, of course, an approximation. However, resultsof the developed method depict that this approxima-tion is more close to reality than analytical meth-ods.

At each time step of swimming, the coordinatesof each boundary element were extracted as a functionof Y (normal distance between each element and itscorresponding wall) and � (angle of rotation of elementwith respect to its corresponding wall). These twoparameters, as well as the other parameters describ-ing cilia coordinates (see Figure S2 and associatedtable in Supplementary Material (Section 6)), andthe element velocities, were given to our numericalmodel for extracting the normal and tangential com-ponents of viscous forces applied onto each boundaryelement. By integrating these functions over theParamecium boundary, propulsion force, swimmingvelocity and swimming trajectory were determined.The results of this study can be useful for determin-ing Paramecium locomotion subjected to uid mo-tion within speci�c con�nements. For example, datafor a swimming trajectory can be useful in utilizingParamecium as a micro robot for minimally invasivetherapies.

6. Supplementary materials

Supplementary material of this article can be found at:https://jumpshare.com/v/pxrIRH8vmU2onBRN32rf

7. Competing interests

The authors declare that they have no competinginterests.

Acknowledgment

The authors would like to thank members of the bio-inspired uid lab at Virginia Tech for experimentaldata and discussions.

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Biographies

Ali Nematollahi Sarvestani received his MSc degreefrom Sharif University of Technology, Tehran, Iran.His research interest includes uid ow simulation withbiological applications. He has participated in IMECEa number of times and has performed research into�nite element simulations, bioengineering and micro airvehicles.

Amir Shamloo received his PhD degree from Stan-ford University, CA, USA, in 2010. He is now As-sistant Professor in the Mechanical Engineering De-partment of Sharif University of Technology, Tehran,Iran. His main research interest involves the designand fabrication of micro uidic devices and their ap-plication in biology, as well as simulating biologicalprocesses.

Mohammad Taghi Ahmadian is Professor in theDepartment of Mechanical Engineering at Sharif Uni-versity of Technology, Tehran, Iran. His articles havebeen cited more than 2000 times and he is a reviewer formany scienti�c journals. His research mainly focuses onmicro and nano sciences, specializing in dynamics andvibrations.

modeling paramecium swimming in a capillary...

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