Journal of Biomechanics 46 (2013) 329–337

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Journal of Biomechanics

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Fluid dynamic model of invertebrate sperm chemotactic motility withvarying calcium inputs

Sarah D. Olson n

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA

a r t i c l e i n f o

Article history:

Accepted 9 November 2012In a marine environment, invertebrate sperm are able to adjust their trajectory in response to a gradient

of chemical factors released by the egg in a process called chemotaxis. In response to this chemical

Keywords:

Regularized Stokeslets

Fluid–structure interaction

Calcium dynamics

Chemotaxis

Sperm

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x.doi.org/10.1016/j.jbiomech.2012.11.025

þ1 508 831 4940; fax: þ1 508 831 5824.

ail address: sdolson@wpi.edu

a b s t r a c t

factor, a signaling cascade is initiated that causes an increase in intracellular calcium (Ca2þ). This

increase in Ca2þ causes the sperm flagellar curvature to change, and a change in swimming direction

ensues. In previous experiments, sperm swimming in a gradient of chemoattractant have exhibited

Ca2þ oscillations of varying peaks and frequency. Here, we model a simplified sperm flagellum with

mechanical forces, including a passive stiffness component and an active bending component that is

coupled to the time varying Ca2þ input. The flagellum is immersed in a viscous, incompressible fluid

and we use a fluid dynamic model to investigate emergent trajectories. We investigate the sensitivity of

the model to the frequency of Ca2þ oscillations. In this coupled model, we observe that longer periods

of Ca2þ oscillation corresponds to circular paths with greater drift. In contrast, shorter periods of Ca2þ

oscillations corresponded to tighter search patterns. These outcomes shed light on the relation between

Ca2þ oscillations and different searching trajectories and strategies that invertebrate sperm may utilize

to reach and fertilize the egg in a marine environment.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Fertilization of the egg is crucial for life, yet we still do nothave a complete understanding of how the sperm reaches the egg.The undulatory motion of the sperm flagellum allows the spermto propel itself forward, and is an emergent property of a complexsystem that couples chemical signaling, active and passive elasticproperties of the sperm structure, and the external fluid dynamics(Fauci and Dillon, 2006; Kaupp et al., 2008; Woolley, 2010). In amarine environment, sperm are guided to the egg in a processcalled chemotaxis. In response to a chemoattractant, a signalingcascade is initiated that causes an increase in calcium (Ca2þ)(Kaupp et al., 2006, 2008). This increase in Ca2þ provides asteering mechanism for the sperm to reorient itself along atrajectory that will bring it towards the egg. Chemical commu-nication between the sperm and the egg, eliciting Ca2þ transients,is a fundamental aspect of fertilization in a marine environment.

Freely swimming marine invertebrate sperm, when not in thepresence of a chemoattractant, will have a low or basal level ofintracellular Ca2þ . This coincides to a flagellum beating with awaveform that has constant amplitude (symmetrical waveform).The path of a free swimming sperm is a spiral or helical trajectory

ll rights reserved.

in three dimensions (3-D), corresponding to a linear or circulartrajectory in two dimensions (2-D) (Cosson et al., 1984). Anillustration of an idealized helical trajectory, tracking one pointon the flagellum, is illustrated with the black helix in Fig. 1(A).The average path or centerline of the corresponding helix is a line.A drifting helical path in 3-D will then correspond to a circulartrajectory in 2-D . In response to a chemoattractant, the intracel-lular Ca2þ increases and the amplitude of the waveform will vary,leading to asymmetrical flagellar waveforms and drifting circular

paths in 2-D (Brokaw, 1979). Ca2þ spikes within the flagellumwill control the trajectory, giving turns and runs, with loop-likemovements (Alvarez et al., 2012; Bohmer et al., 2005). When asperm is swimming in a plume of chemoattractant, increasingflagellar asymmetry corresponds to increases in Ca2þ and turns,while a decrease flagellar asymmetry corresponds to a lower orbasal Ca2þ and straighter swimming (Guerrero et al., 2011). lnexperiments performed by Kaupp et al. (2003) on the sperm ofArbacia punctulata (sea urchin), trajectories and flagellar wave-forms were tracked before and after the release of resact, thechemoattractant. The three columns in Fig. 1(B) correspond toincreasing amounts of resact. In the bottom row of Fig. 1(B), thegray traces correspond to the initial trajectory and the blacktraces are after the release of resact, which has initiated asignaling pathway that increases Ca2þ within the cytosol of theflagellum. These trajectories are put into a plane for the figure andeach dot corresponds to tracking one point along the flagellum in

Fig. 1. (A) An idealized 3-D helical trajectory, corresponding to tracking one point

on the sperm flagellum, is depicted as the black curve. The corresponding 2-D

curve in the plane is a linear trajectory. (B) Experimental results of Kaupp et al.

(2003). In the bottom row, the gray traces are the initial trajectory and the black

traces are after the release of resact (chemoattractant at (i) 10 nM, (ii) 100 nM, and

(iii) 1 M from left to right). The resact was caged and after the UV flash is able to

initiate the chemotactic signaling pathway. The corresponding waveforms in

the top row are 80 ms apart near the turn denoted by a % in the bottom row.

(1) Corresponds to before the turn, (2) is at the turn, and (3) is the waveform after

the turn. (Reproduced with permission from Kaupp et al. (2003).)

S.D. Olson / Journal of Biomechanics 46 (2013) 329–337330

time. The corresponding waveforms in the top row of Fig. 1(B) are80 ms apart near the turn denoted by a % in the bottom row. Thewaveform (1) corresponds to before the turn, (2) is at the turn,and (3) is the waveform after the turn. Overall, the increase inintracellular Ca2þ modifies the flagellar waveform and overalltrajectory, allowing the sperm to orient its swimming directionwith the chemoattractant gradient.

Since sea urchin sperm have been studied as a model organismto understand the chemotactic pathway, they will be the focus ofthis paper. The chemoattractants for sea urchin sperm are shortpeptides contained in the jelly layer of the egg coat, which arereleased into the surrounding fluid. For A. punctulata sperm, aspecies of purple-spined sea urchin, resact has been identifiedto be the chemoattractant (Kaupp et al., 2008). It is wellestablished that resact binds to a guanylyl cyclase (GC) receptoron the plasma membrane of the sperm flagellum (Hansboughand Garbers, 1981; Shimomura et al., 1986; Singh et al., 1998),causing the rapid synthesis of cyclic guanosine monophosphate(cGMP) (Shimomura et al., 1986; Singh et al., 1998; Ward et al.,1985). The increased cGMP is then able to bind to the cyclicnucleotide binding domain of the potassium (Kþ) selective cyclicnucleotide gated channel (KCNG) (Strunker et al., 2006). This thencauses Kþ to be removed from the cytosol of the sperm, resultingin a hyperpolarization. As a result, the hyperpolarized activatedcyclic nucleotide gated channel (HCN) opens, causing sodium ions(Naþ) to rush into the cytosol of the sperm flagellum (Gauss et al.,1998). At or shortly after the peak of the hyperpolarization, Ca2þ

channels open and the intracellular Ca2þ increases greatly (Kauppet al., 2003, 2006, 2008). The general pathway is known, but notall of the feedback mechanisms, including adaptation, have beenidentified. The exact concentration of chemoattractant in an odorplume that a sperm comes in contact with in nature can also varygreatly.

In recent years, many experiments have been completed tounderstand the relationship between resact, Ca2þ , flagellar asym-metry, trajectories, and the fluid environment. After Ca2þ

increases within the flagellum, it has been shown that there is a250–350 ms delay between the alteration of waveform (Guerrero

et al., 2010). Additionally, the frequency of Ca2þ oscillations inresponse to the chemoattractant has found to be dose dependent(Guerrero et al., 2010) and sperm are sensitive to a range of resactconcentrations, from a single resact molecule to mM concentra-tions (Kaupp et al., 2003; Strunker et al., 2006). Of note, it hasbeen shown that sperm swimming in a gradient of resact willexhibit phasic or oscillatory calcium responses with different fre-

quencies and amplitudes (Alvarez et al., 2012; Guerrero et al.,2010; Kaupp et al., 2008). In a marine environment, the spermwill be presented with attractant plumes, which may be dynami-cally changing due to a laminar shear background flow (Zimmerand Riffell, 2011). Due to the variation in chemoattractant pre-sented to the sperm and the variations in intracellular Ca2þ

oscillations, we will use a specified, time dependent, Ca2þ inputin this work.

Models have been developed specifically for chemotaxis inmarine invertebrate sperm, where the sperm are described aspoints and the trajectory is dependent on the relevant biochem-istry in a phenomonological manner. In Ishikawa et al. (2004),the radius of curvature of the trajectory was dependent onthe chemoattractant concentration. A theoretical description ofchemotaxis and sperm movement has also been developed byFriedrich and Julicher (2007). The time evolution of the spermtrajectories was given by the Frenet–Serret equations, where thechemoattractant concentration was accounted for as a stimulusthat was coupled to a dynamical system to incorporate adaptationand relaxation of the system. Both of these models are able toreproduce trajectories similar to those observed in experiments,however, they do not account for the relevant fluid dynamics.

Invertebrate sperm are low Reynolds number swimmers,Re� 10�6, in the regime where viscous forces dominate andinertial effects are negligible. It is noted that the undulatorymotion of the flagellum corresponds to a periodic actuation, andbreaks the constraints of Purcell’s scallop theorem, resulting inthe generation of a net propulsive force when swimming in aviscous fluid. Classical work in the area of fluid dynamics has usedthe linearity of the Stokes equations to study periodically actu-ated filaments with prescribed kinematics (Brennen and Winet,1977; Gray and Hancock, 1955; Higdon, 1979; Lighthill, 1975;Phan-Thien et al., 1987). We note that many fluid dynamicmodels have been developed to study and understand a varietyof aspects of sperm motility through computation and analysis(Dillon et al., 2003, 2006; Elgeti et al., 2010; Fauci and McDonald,1995; Friedrich et al., 2010; Fu et al., 2007; Gadelha et al., 2010;Gillies et al., 2009; Lauga, 2007; Smith et al., 2009a; Teran et al.,2010). Recently, Olson et al. (2011) have developed computa-tional models of mammalian sperm motility, where the Ca2þ iscoupled to the kinematics of the waveform. This Ca2þ coupling isnow extended from previous work to study invertebrate spermchemotaxis.

The aim of this current work is to investigate how thefrequency and amplitude of the cytosolic flagellar Ca2þ concen-trations couple to trajectories of sperm using a computationalfluid model of a sperm swimming in a viscous, incompressiblefluid. The time dependent bending kinematics of the flagellarwave will be coupled to a time varying Ca2þ input, representativeof the Ca2þ transients observed in experiments. This model willaccount for the active bending components, which are coupled tothe Ca2þ dynamics, and the passive elastic components due to theaxonemal structure. Since this is a fluid–structure interactionmodel, the kinematics and trajectories that are achieved willdepend on the surrounding viscous fluid and the properties of theelastic filament. We investigate the sensitivity of sperm trajec-tories to the frequency of Ca2þ oscillations, accounting for therelevant hydrodynamics. In this coupled model, we observe thatlonger periods of Ca2þ oscillation corresponds to circular paths

Fig. 2. A cartoon depiction of a sperm in (A) where the tail is the flagellum and the

circular structure is the cell body or head. The internal structure of the flagellum

is called the axoneme and a cross section in the plane is depicted in (B).

The axoneme consists of nine sets of microtubule doublets around the circular

structure and a central pair. Dynein arms reach to attach to the next microtubule

doublet creating crossbridges. The simplified flagellar geometry used in the model,

the flagellar centerline is depicted in (C).

S.D. Olson / Journal of Biomechanics 46 (2013) 329–337 331

with greater drift. In contrast, shorter periods of Ca2þ oscillationscorresponded to tighter search patterns.

2. Methods

The interior of the sperm flagellum is a circular structure consisting of nine

microtubule doublets surrounding a central pair (Fig. 2(B)). The sperm geometry,

depicted in Fig. 2, will be simplified by modeling an elastic flagellum denoted by

Xðr,tÞ ¼ ðxðr,tÞ,zðr,tÞÞ, corresponding to the centerline, where r is a parameter

initialized as arc length and t is time. We simplify the flagellum by only capturing

the centerline, corresponding to a 1-D filament that is restricted to swimming in

the xz plane. This corresponds to planar flagellar bending, which can be assumed

for sea urchin sperm (Guerrero et al., 2011). We note that we are not accounting

for the cell body, which will cause small changes in fluid velocity, swimming

speed, and trajectories as shown previously in experiments and computational

models (Brokaw, 1965; Gillies et al., 2009; Friedrich et al., 2010). The flagellum is

assumed to be elastic and is immersed in a viscous, incompressible 3-D fluid. In

the following subsections, we will first describe the kinematics and mechanical

properties of the flagellum, then the coupled fluid–structure interaction model

will be detailed.

2.1. Curvature model

The undulatory bending of marine invertebrate sperm flagella has been

reported to be sinusoidal (Rikmenspoel and Isles, 1985) and previous experi-

mental results using high speed imaging have shown that human sperm flagella

propagate a curvature wave (Smith et al., 2009b). For a flagellar waveform

corresponding to a simple sine wave with xðr,tÞ ¼ r and zðr,tÞ ¼ b sin ð2p=lÞr�otÞ,

the preferred (signed) curvature function is

kðr,tÞ ¼�2pl

� �2

b sin2pl

r�ot

� �, ð1Þ

where l is the wavelength, o the beat frequency, and the amplitude b is assumed

to be small. This type of curvature wave has been used previously in computa-

tional models of flagellar motility (Fauci and McDonald, 1995; Olson et al., 2011;

Teran et al., 2010). Here, we assume that the wavelength and beat frequency of

the sperm flagellum are constant, as reported in Brokaw (1979), and we couple

the intracellular Ca2þ to the amplitude b in Eq. (1). Since we want to capture

the transition from low Ca2þ motility to higher Ca2þ motility due to a resact

(chemoattractant) mediated pathway, we let the amplitude bðr,tÞ vary as follows:

bðr,tÞ ¼Mb 1þkoCaðtÞ

CaðtÞþkb

� �, ð2aÞ

kb ¼

kb,1 �2pl

� �2

sin2pl

r�ot

� �40

kb,2 �2pl

� �2

sin2pl

r�ot

� �r0

8>>>><>>>>:

, ð2bÞ

ko ¼0 CaðtÞrCn

b CaðtÞ4Cn ,

(ð2cÞ

where Mb is the maximal amplitude of bending at the homeostatic or resting level

of Ca2þ . In Eq. (2a), when CaðtÞrCan , ko¼0 and we have a sperm flagellum beating

at a preferred curvature corresponding to a fixed beat frequency and constant

amplitude. When Ca(t) increases above Cn, the amplitude and magnitude of the

curvature increases. Note that we choose ko and Cn in Eq. (2c) to account for

experimental observations that the curvature of the flagellum takes msecs to

change after the Ca2þ has increased and that Ca2þ can be above the basal level CB

with no increase in the amplitude of the flagellum (Alvarez et al., 2012; Guerrero

et al., 2010).

In Eq. (2a), we account for a Ca2þ dependent change in the preferred

amplitude. When the intracellular Ca2þ increases, an asymmetrical waveform

has been observed, corresponding to an increased angle in the principle bend

direction and a decreased angle in the reverse bend direction (Brokaw, 1979). At

high Ca2þ , this asymmetrical waveform is no longer truly sinusoidal. In this

model, we are able to account for this asymmetric bending as the Ca2þ increases

by choosing kb in Eq. (2b) based on the sign of the preferred curvature. When

kb,1 4kb,2, this defines the principle bending direction as the one with positive

curvature. The actual location of where Ca2þ is acting within the axoneme to

modify bending is not completely known. Dyneins (Fig. 2(B)), chemo-mechanical

ATPases or force generators, cause sliding of the internal microtubule doublets,

which is converted into bending (Lindemann and Lesich, 2010; Woolley, 2010). It

is hypothesized that Ca2þ acts by binding to proteins such as calmodulin, which

interact directly or indirectly with dynein complexes, to cause a conformational

change in the dyneins that results in an asymmetric change in dynein force

generation, causing asymmetrical flagellar bending (Lindemann, 2007; Smith,

2002; Smith and Yang, 2004; Yang et al., 2001). We are assuming planar flagellar

waves for the sperm by restricting forces to the xz plane, a valid assumption for

sperm (Guerrero et al., 2011). For planar bending, this corresponds to dyneins on

one side (corresponding to microtubule doublets 1–4) being active in one bend

direction, while the dyneins on the other side (corresponding to microtubule

doublets 6–9) are only active during the generation of flagellar bending in the

opposite direction (Guerrero et al., 2011). We model the affect of Ca2þ binding to

proteins to modify bending via a modified Hill equation in Eq. (2a) where the

parameter kb corresponds to a different Ca2þ binding affinity on each side of the

flagellum.

2.2. Ca2þ input

We wish to investigate the sensitivity of our model to different Ca2þ inputs.

As described earlier, depending on the particular sperm and concentration

of chemoattractant, Ca2þ transients of varying frequency and amplitude are

observed (Alvarez et al., 2012). Therefore, we will study a time varying Ca2þ

input. The Ca2þ transient will have a period P that involves a fast increase, slower

decrease, and resting at a basal concentration before the next period begins. The

equation for one period is

CaðtÞ ¼

g�a cospZ1P

t

� �, 0rtoZ1P,

gþa� 2aPðZ2�Z1Þ

ðt�Z1PÞþxwðtÞ, Z1PrtoZ2P

CB Z2PrtoP,

8>>>>><>>>>>:

, ð3Þ

where CB is the basal Ca2þ concentration, Z1 is the fraction of the period spent in

the increase, Z2 is the fraction of the period spent above CB in each period, the

maximum Ca2þ concentration is 2aþCB , and g is the sum of a and CB. Here, w(t) is

a noise factor drawn from a uniform random distribution, as Ca2þ transients are

often observed to have small scale oscillations in concentration. The parameter xis a scale factor to control the strength of the small scale Ca2þ oscillations. This

Ca2þ oscillation will be repeated via the use of the modulo operation. Note that at

each time step, Ca(t) will be used as a constant value along the length of the

flagellum, where the only spatial variation in the preferred curvature will be due

to the asymmetric parameter kb in Eq. (2a).

2.3. Energy

We define a non-negative, translation and rotation invariant energy for the

bending filament G¼Xðr,tÞ based on linear elasticity. The passive tensile energy

Etens and active bending energy Ebend are components of the energy functional E,

given by

E¼

ZGðEtensþEbendÞ dr, ð4Þ

Etens ¼ K1@X

@r

���������1

� �2

, Ebend ¼ K2@y@r�kðr,tÞ

� �2

, ð5Þ

where J � J is defined as the Euclidean norm (Fauci and Peskin, 1988; Fauci and

McDonald, 1995; Olson et al., 2011; Teran et al., 2010).

The passive tensile energy Etens corresponds to an inextensibility constraint

for the microtubules of the flagellum; the energy functional is minimized when

S.D. Olson / Journal of Biomechanics 46 (2013) 329–337332

J@X=@rJ¼ 1. The parameter K1 is the passive stiffness coefficient, corresponding to

the material properties of the microtubule doublets and other passive structures

such as the nexin links and radial spokes. Due to a linear taper in the radius of the

flagellum from the tip to the base (Omoto and Brokaw, 1982), we set K1 as a linear

decreasing function of arclength parameter r as in previous models (Olson et al.,

2011).

The active bending energy Ebend is driving the dynamics of the movement,

corresponding to the dyneins creating crossbridges and actively generating force.

The bending energy depends on the shear angle @y=@r and the preferred curvature

k. The shear angle corresponds to the rate of change of the angle between the first

point (head) of the flagellum and the given point (with respect to arclength

parameter r). The preferred curvature k drives the motion of the flagellum and is

coupled to the Ca2þ input concentration as detailed in Sections 2.1– 2.2. The

parameter K2 is the bending stiffness constant and is taken to be constant along

the length of the flagellum.

The derivative of the elastic energy functional E with respect to the position Xof the flagellum gives the force per unit length gs

gs ¼�d

dXðEbendþEtensÞ: ð6Þ

We take the negative of the derivative since these are the forces that the sperm

will be exerting on the surrounding fluid that it is immersed in. Note that since the

organism is self propelled and is not influenced by any external forces, this

formulation requires that total force on the organism is equal to zero.

2.4. Fluid equations

The aim of this paper is to investigate swimming speeds and trajectories of a

sperm flagellum using a computational fluids model. Sperm motility is in the

regime where viscous forces dominate, i.e. zero Reynolds number. Therefore, we

assume that the fluid is governed by the Stokes equations. Since we want to solve

this problem in a computationally efficient manner, we will make use of

fundamental solutions, Stokeslets. For a point force in an unbounded fluid domain,

a Stokeslet is an exact solution for the velocity field, which is singular at the

location of the point force. We wish to solve the velocity of the surrounding fluid

as well as at the centerline, therefore we will regularize the point forces to remove

the singularity at the centerline using the method of regularized Stokeslets

(Cortez, 2001; Cortez et al., 2005). The Ca2þ oscillations will be coupled to the

preferred curvature, which will drive the dynamics of the bending flagellum.

Specifically, the fluid–sperm interaction problem will be governed by the

incompressible Stokes equations and coupled to the mechanics of the flagellar

bending through the term fs , the force density exerted by the sperm on the

surrounding fluid. The mathematical model is given as

0¼�rpþmDvþfs , ð7aÞ

0¼r � v, ð7bÞ

@X

@tðr,tÞ ¼ vðXðr,tÞ,tÞ, ð7cÞ

where v is the fluid velocity, p is the pressure, and m is the fluid viscosity. Eq. (7b)

is the incompressibility condition and Eq. (7c) is the no-slip condition that

enforces the sperm centerline X to move at the local fluid velocity. The force

density fs will be derived in Section 2.5 by regularizing the force per unit length gs

calculated in Eq. (6), based on the functional energy in Eq. (4).

Table 2Non-dimensional calcium parameters. (Table values are non-dimensionalized

2.5. Non-dimensional model and numerical methods

The characteristic scales detailed in Table 1 are based on parameters from sea-

urchin sperm and are used to define the following non-dimensional variables of

Table 1Characteristic scales and parameters.

Parameter Value

l (wavelength of flagellum) 3� 10�5 m (30 mm)

L (length of flagellum) 1:5 � l mm (Eshel et al., 1990)

L (characteristic length) lo (beat frequency) 35 Hz (s�1) (Cosson et al., 1984)

T (characteristic time) 1=om (viscosity) 1� 10�3 kg m�1 s�1

V (characteristic velocity) LoF (characteristic force) mVL

P (characteristic pressure) mV=LC (characteristic concentration) 1 mM

the Stokes equations:

~v ¼v

V, ~p ¼

p

P, ~f

s¼

fsL3

F, ~t ¼

t

T, ~X ¼

X

L: ð8Þ

Using these non-dimensional variables, the Stokes equations become

0¼�r ~pþD ~vþ ~fs , ð9aÞ

0¼r � ~v , ð9bÞ

@ ~X

@~tðr, ~t Þ ¼ ~vð ~Xðr, ~t Þ, ~t Þ, ð9cÞ

where the force density ~fs

is appropriately non-dimensionalized . The preferred

curvature and Ca2þ input in Eqs. (1) and (3), respectively, are also non-

dimensionalized using the characteristic scales in Table 1.

To solve the coupled fluid–structure interactions, we will use the method of

regularized Stokeslets in 3-D, which was derived in Cortez et al. (2005). This

method has been used successfully to model different aspects of sperm motility

(Gillies et al., 2009; Olson et al., 2011). Due to the linearity of the Stokes equations,

we are able to solve for the contributions of each of the forces to the surrounding

fluid as a superposition of fundamental solutions. Specifically, we discretize the

flagellar centerline ~X j , j¼ 1, . . . ,N. The force density of the sperm ~fs

at a point x in

the fluid is calculated as

~fs¼XN

j ¼ 1

~gsjfEð ~x�

~X jÞDr, ð10aÞ

fEð ~x�~X jÞ ¼

15E4

8pðJ ~x� ~X jJ2þE2Þ

7=2, ð10bÞ

where fE is a cutoff or regularization function with regularization parameter E. In

order to solve the velocity at the centerline, regularization of the point forces is

used to remove the singularity at the centerline. The regularized distribution of

point forces on the centerline now has a support region with most of its support

in a sphere of E, corresponding to a ‘virtual’ radius. Therefore, we choose the

numerical parameter E to correspond with the non-dimensional radius of the

flagellum as in Olson et al. (2011).

Using the method of regularized Stokeslets (Cortez, 2001; Cortez et al., 2005),

the unique and everywhere incompressible solution to the Stokes Eqs. (9a)–(9c)for

the regularized forces and cutoff function given in Eqs. (10a)–(10b)is

~vð ~xÞ ¼1

8pXN

j ¼ 1

~gsj ðJ ~x�

~X jJ2þ2E2Þþ½ ~gs

j � ð ~x�~X jÞ�ð ~x� ~X jÞ

ðJ ~x� ~X jJ2þE2Þ

3=2: ð11Þ

We note that this is 3-D infinite fluid domain where zero velocity at infinity is

required. The hydrodynamic pressure ~p can also be solved for using the method of

regularized Stokeslets, where the pressure is determined by satisfying the

incompressibility condition in Eq. (9b). Given a location of the discretized flagellar

centerline ~X at a given time point, the full fluid–structure interaction problem can

be solved using the method of regularized Stokeslets as follows: ð1Þ Evaluate the

forces ~gs in Eq. (6) by discretizing Eq. (5). The energy depends on the curvature

and Ca2þ concentration defined in Eqs. (1) and (3), respectively. ð2Þ Eq. (11) is used

to calculate the velocity at the flagellar centerline. ð3Þ Use Euler method to update

the locations using the no-slip condition in Eq. (9c). Numerical parameters that

were fixed include the time step set at Dt¼ 10�5, regularization parameter set at

E¼ 0:01, and the number of immersed boundary points to discretize the centerline

was set at 100. Additional model parameters include active stiffness coefficient set

to K2¼0.05 and the passive stiffness coefficient K1 with a linear taper ranging from

50 to 150 (Olson et al., 2011).

from literature values).

Parameter Value

CaB (resting Ca2þ) 0.1 (Wennemuth et al., 2003)

Mb (amplitude at CaB) 0.4 (Rikmenspoel and Isles, 1985)

Cn (Ca2þ threshold) 0.2

kb,1 (Ca2þ at half max amp, pr) 0.3

kb,2 (Ca2þ at half max amp, rev) 0.5

a (Ca2þ midpoint) 0.45

b (amplitude factor) 1

x (strength of Ca2þ noise) 0–0.3

P (period of Ca2þ oscillation) 2–7 s (Alvarez et al., 2012)

Z1 (increase phase) 0.1–0.2� P s

Z2 (time above CB) 0.7–5/6� P s

amp, amplitude; pr, principle; rev¼reverse.

Fig. 3. The time varying input Ca2þ concentration, Eq. (3), is shown for three

values of P, controlling the frequency of the oscillations. The portion of the Ca2þ

oscillation period spent increasing is set to Z1 ¼ 1=6 and the portion of period

spent above resting Ca2þ concentration CB is set to Z2 ¼ 5=6. The magnitude or

strength of the small scale Ca2þ oscillation, x is varied. The other parameters are

set based on the values in Table 2.Fig. 4. Sample fluid velocity fields in the plane of flagellar bending. Vortices

around the flagellum progress down the flagellum as the beat cycle continues. The

circle corresponds to the head side of the flagellum. The colormap corresponds to

the given Calcium concentration at that time point. In (A), an earlier time point

corresponding to lower Ca2þ and symmetrical bending. In (B), a later time point

where the Ca2þ is increasing and there exists a slight difference in the amplitude

of the bends, causing the sperm to swimming in a drifting circular trajectory.

S.D. Olson / Journal of Biomechanics 46 (2013) 329–337 333

3. Results

In this study, we wished to investigate the coupling of thefrequency and shape of the Ca2þ oscillation on resulting trajec-tories in a fluid–structure interaction model. The Ca2þ concentra-tion directly relates to the preferred curvature in Eq. (1) andcorresponds to Ca2þ oscillations observed in experiments wheresea urchin sperm are swimming in a concentration gradient ofchemoattractant (Alvarez et al., 2012; Guerrero et al., 2010).Sample Ca2þ inputs that were used are shown in Fig. 3 forspecific choices of parameter values in Eq. (3). Each of the Ca2þ

inputs included a transient that includes a fast increase, a slowerdecrease in concentration back to the basal concentration CB, anda recovery time spent at CB before the next Ca2þ increase. Thesethree pieces all occur in each period. For the simulations ran, wefixed the minimum Ca2þ concentration to CB and set the max-imum Ca2þ concentration to 1 for most simulations, correspond-ing to biologically relevant ranges. In the Ca2þ fluorescenceintensity results of Alvarez et al. (2012), they observed skewedsmall scale oscillations in the Ca2þ concentration in the decreasephase. We choose to also investigate the role of small scaleoscillations in this range, corresponding to the parameter x.

In this model, we are tracking the entire flagellar centerline,which is restricted to beat in the xz plane. Olson et al. (2011) havepreviously investigated emergent flagellar waveforms and swim-ming speeds for a model of hyperactivated mammalian spermmotility. This work is an extension of the mammalian case tochemotaxis in marine invertebrate sperm, therefore we arefocusing specifically on the trajectories of the sperm. (We notethat emergent swimming speeds match well with the data andare in the range of 1002250 mm s�1 for the parameters usedhere.) The centerline is initialized as a straight line and the initialfluid velocity is set to zero. In Fig. 4, sample velocity fields andcorresponding flagellar waveforms are shown at two time points.As the Ca2þ concentration increases, a slight asymmetry inthe amplitude of the bending wave can be seen in Fig. 4(B).The degree of this asymmetry will be controlled by the follow-ing parameters: the amplitude factor b, Ca2þ at half-max ampli-tude in the principle bend direction kb,1, and Ca2þ at half-maxamplitude in the reverse bend direction kb,2. Due to the asymmetry

in the flagellar bending, the sperm starts to swim in a driftingcircular trajectory. In Fig. 5, sample trajectories are given for threedifferent periods of Ca2þ oscillation and are shown for a timeperiod of 9 s.

Using this fluid-dynamic model, we can capture emergenttrajectories by plotting the trajectory of one of the points on thesperm flagellum. In time, as the flagellar wave progresses, eachpoint on the flagellum will oscillate due to the propagating sinewave. For each trajectory, we will track the first point on theflagellum. As the sperm flagellum starts propagating a sinusoidalwave of curvature along the flagellum, the centerline starts tobeat and the centerline starts to slowly move up and down, asdepicted by (i) in Fig. 5(A)–(C). The centerline continues topropagate a wave, interacting with the surrounding fluid, andstarting off in a linear trajectory. In (ii), Fig. 5(A)–(C), thiscorresponds to the portion of the period where Ca2þ quicklyrises to its peak concentration, and then slowly decreases backdown to the basal concentration CB. This phase, (ii), correspondsto a more curved trajectory, what is often referred to in theliterature as drifting circular paths (Cosson et al., 1984). The nextstage, (iii) in Fig. 5(A)–(C), is a constant Ca2þ concentration that ismaintained for a short portion of the period before the next Ca2þ

increase occurs, corresponding to a linear trajectory again. Thenext labeled portion in Fig. 5(A)–(C) is (iv), corresponding to thestart of the second period where the Ca2þ concentrations starts toincrease again.

The difference between the graphs in Fig. 5 is the period of theCa2þ oscillation; period¼3 s in (A), 5 s in (B), and 7 s in (C). Withincreased period, or increased time with higher Ca2þ concentra-tion, the sperm’s trajectory stays in a circular path longer duringeach period (Fig. 5(C)). The curvature of the path correspondswith the Ca2þ concentration, as observed in experiments (Alvarezet al., 2012). It is interesting to note that as the flagellum isinteracting with the fluid, we observe that even with oscillationsvarying from 3 to 7 s, the sperm are exploring similar territory,

Fig. 5. Trajectories are shown for varying the period of the Ca2þ oscillation P: (A)

P¼3, (B) P¼5, (C) P¼7. The trajectory lines correspond to tracking the first

immersed boundary point, the tip of the flagellum (head). Parameters used are

given in Table 2. The Ca2þ noise parameter is set to zero, i.e. x¼ 0. The portion of

the Ca2þ oscillation period spent increasing is set to Z1 ¼ 1=6 and the portion of

period spent above resting Ca2þ concentration CB is set to Z2 ¼ 5=6. The other

parameters are set based on the values in Table 2. In (A)–(C), (i) is the initial low

Ca2þ and start of the fluid–structure interaction, (ii) circular arc due to increased

Ca2þ above CB, (iii) is the linear path associated with low constant Ca2þ at CB, and

(iv) is the start of a new period and increase in Ca2þ .

Fig. 6. Trajectories are shown for varying the parameter x, the small scale noise of

the Ca2þ oscillation: (A) P¼2, (B) P¼7. The trajectory lines correspond to tracking

the first immersed boundary point, the tip of the flagellum (head). Parameters

used are given in Table 2 and Z1 ¼ 1=6, and Z2 ¼ 5=6.

S.D. Olson / Journal of Biomechanics 46 (2013) 329–337334

but they are headed off to search in different directions at the endof the trajectory, corresponding to 9 s in Fig. 5. With the results inFig. 5, it is clear that the period of the Ca2þ oscillation could havea large effect on how the sperm orients itself and the direction ofsearching. If the sperm were swimming in a dynamic plume ofchemoattractant, each turn and increase in Ca2þ concentration

would correspond to a trajectory that aligns the sperm withthe egg.

In addition to studying the effect of the Ca2þ oscillation periodon the trajectory, we also wanted to explore the role of small scalefluctuations or noise in the Ca2þ oscillations in the slower Ca2þ

decrease portion of the period. Since Ca2þ is coupled to theflagellar waveform through the preferred curvature and flagellarwaveforms are emergent properties of the coupled fluid–struc-ture interaction problem, we wished to investigate whether smallscale Ca2þ oscillations were able to alter the trajectory. In Fig. 6,representative results are shown for a 2 s period in (A) and a 7 speriod in (B) with four levels of noise. From Fig. 6, it can be clearlyseen that even small scale oscillations in the force, due to Ca2þ

oscillations, can effect the trajectory. In Fig. 6, for a 2 s periodof Ca2þ oscillation, as the noise increases, the sperm swim ona tighter or narrower circular path. At this shorter Ca2þ periodin (A), the difference between the ending points at 9 s are notdramatically changed. However, for the longer Ca2þ period of 7 sin Fig. 6(B), a tighter circular path is also observed with increasingnoise, but the end points vary greatly. This is due to a longeramount of time with higher Ca2þ , causing a greater difference inthe path.

Fig. 7. Trajectories are shown in (A) varying the resting Ca2þ concentration with

each periodic Ca2þ oscillation by 0.1 and constant resting concentration of

0.1 when no noise is accounted for (x¼ 0) using P¼2. In (B) trajectories comparing

the change in varying Z1 and Z2, corresponding to the portion of the period spent

increasing and the portion spent above resting concentration CB, respectively. The

noise of the Ca2þ oscillations is x¼ 0:3 and the period of the Ca2þ oscillation is

P¼7.

S.D. Olson / Journal of Biomechanics 46 (2013) 329–337 335

In experiments, it has also been observed that the Ca2þ

concentration does not return to a basal concentration CaB

between Ca2þ oscillations (Alvarez et al., 2012). We next studythe case where the Ca2þ concentration increases by 0.1 with eachsuccessive oscillation period. In Fig. 7(A), results of increasing CB

with each period by 0.1 are shown for the case of a 2 s Ca2þ

oscillation. As a result, the sperm trajectory follows a muchtighter circular arc. This is a trajectory with greater path curva-ture than achieved by adding small scale noise to the Ca2þ inFig. 6(A). Additionally, when the period of the Ca2þ oscillation isvaried in terms of the parameter Z1 and Z2, corresponding to thelength of time spent where Ca2þ is increasing and the length oftime above CB, there is also a pronounced change in the trajectory.In Fig. 7(B), the length of time spent in the increase phase wasdecreased and the length of time spent in the decrease phase wasincreased, resulting in a wider circular arc. The sperm flagellum isable to sample a greater distance in space when the period ischanged in this way.

4. Discussion

In this model, we are able to study how Ca2þ concentrationscouple to emergent trajectories in a coupled fluid–structureinteraction problem. The Ca2þ concentrations are a dynamicinput, based on Ca2þ levels and time courses observed in seaurchin sperm swimming in a chemoattractant gradient (Alvarezet al., 2012; Guerrero et al., 2010). Sperm motility is complex andinvolves the coupling of chemical signaling with the mechanics ofthe sperm flagellum and the external fluid dynamics. Here, wewished to couple a Ca2þ input signal to understand how thiscould alter the active bending moments of the sperm and thetrajectories. At each time step of the simulation, given a Ca2þ

input, we are able to solve for the preferred curvature of theflagellum, solve for the energy and bending moment, and deter-mine the new updated location based on the local fluid velocity.

Through variations of the Ca2þ input, we have observed that inthis model, a sperm is able to alter its trajectory based on anumber of factors. Several key parameters emerge from thisstudy. When a sperm is searching for the egg in a dynamicchemoattractant plume, the sperm may want to cover a largeamount of area, tracing curves of circular arcs with periods oflinear trajectories. In Fig. 5(C), it can be seen that a longer periodof Ca2þ oscillation corresponds to circular paths with greaterdrift. In contrast, shorter periods of Ca2þ oscillations corre-sponded to tighter search patterns in Fig. 5(A). We also observedthat small scale Ca2þ fluctuations are not damped out in thisfluid–structure interaction problem, illustrated in Fig. 6. There-fore, the noise or small scale fluctuations in Ca2þ could beimportant in modulating the searching pattern and trajectory ofmarine invertebrate sperm.

In this current work, we have used an accurate fluid solver tostudy emergent trajectories coupled to a dynamic Ca2þ input. Incomparison to mathematical models of sperm chemotaxis derivedby Friedrich and Julicher (2007) and Ishikawa et al. (2004), thismodel is able to capture emergent trajectories and flagellarwaveforms. Additionally, due to the coupling between the fluidand varying Ca2þ input, we are able to see drifting circular arcs aswell as clear turns. These turns and drifting circular arcs arecharacteristic of marine invertebrate sperm chemotaxis whenswimming in a chemoattractant gradient and when presentedwith a large chemoattractant concentration after photorelease(Alvarez et al., 2012; Guerrero et al., 2010; Kaupp et al., 2003). Wealso find in this current work that Ca2þ concentration changescan allow a sperm to search for the egg, as suggested by Guerreroet al. (2010).

In a marine environment, there are many organisms that usechemotaxis, as well as ambiant background flows to reach a givenenvironment or to find a mate. Even though these organisms mayhave to travel very different paths and experience different localfluid flows, each organism has developed a searching pattern andaccessory structures to achieve their goal (Atema, 1996; Crimaldiet al., 2002; Koehl, 2010; Kanso, 2011; Weissburg et al., 2002). Forexample, bottom dwelling (benthic) marine invertebrates releasemicroscopic larvae that are able to respond to chemical cues andare subject to the currents in the ocean (Crimaldi et al., 2002;Koehl, 2010). Similar to sperm chemotaxis, a male copepod(T longicornis, zooplankton) is able to detect a female throughchemical and/or hydrodynamic signals (Doall et al., 1998; Kanso,2011). Recently, Kanso and Yen (Kanso, 2011) have developed amodel to track the male copepod as a point, redirecting orienta-tion in 2-D based on a chemoattractant sensed in two directions.In this model, the role of fluid flow and chemical signaling isinvestigated.

Sperm chemotaxis in a marine environment is complex andthe sperm will be subject to different local fluid flows. Currently,

S.D. Olson / Journal of Biomechanics 46 (2013) 329–337336

in this model we are assuming zero background flow initially. Itwill be interesting in the future to investigate the role of fluid flowon trajectories in combination with a varying Ca2þ input. Addi-tionally, there are several other areas for additional investigationwithin the current modeling framework investigated here. Wewould like to be able to study chemotaxis of marine invertebratesperm where flagellar waveforms and trajectories are based onthe local resact concentration in the fluid. However, in order to dothis, a better understanding is needed to model resact concentra-tion with Ca2þ concentrations and/or the relation between resactand path curvature on swimming trajectories. The current modelis also using a 3-D fluid solver and we are restricting the beatingof the sperm flagellum to a plane. To fully understand trajectoriesof sperm undergoing chemotaxis, we will need to capture fully 3Dtrajectories seen in experiments (Su et al., 2012). In order toaccount for the twist of the flagellum and a component of out ofplane bending, we have been working on the development of anumerical algorithm to capture these helical trajectories (Olsonet al., 2012).

5. Conclusion

We are able to study, in a phenomological manner, howchemoattractant induced Ca2þ oscillations couple to trajectoriesin marine invertebrate sperm. In this work, we focused oninvestigating the potential role of the period of Ca2þ oscillationand the role of small scale oscillations. We have idealized thesperm flagellum as a centerline, restricted to beat in a plane.Through the use of the method of regularized Stokeslets, aLagrangian method, we are able to solve this coupled fluid–structureinteraction problem in a computationally efficient manner. Wecould investigate emergent trajectories for time scales on the orderof 10 s. In this model, the dynamic Ca2þ input is coupled to thepreferred curvature of the flagellar centerline, driving the dynamicsof the bending. We also capture the passive and active bendingcomponents through the use of an energy functional. This modelis able to capture the coupling of Ca2þ oscillations to trajectories,shedding light on strategies for maneuvering in a dynamic plume ofchemoattractant.

Conflict of interest statement

The author S. Olson confirms that there are no known conflictsof interest associated with the research detailed in this manuscript.

Acknowledgments

The work of S. Olson, in part, was supported by NSF DMS1122461. The author would like to thank L. Fauci for helpfuldiscussions.

Appendix A. Supplementary material

Supplementary data associated with this article can be foundin the online version at http://dx.doi.org/10.1016/j.jbiomech.2012.11.025.

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