A chemomechanical model of sperm locomotion reveals two modes of swimming Chenji Li1Brato Chakrabarti2Pedro Castilla1Achal Mahajan1and David Saintillan1 1Department of Mechanical and Aerospace Engineering

2025-04-27 0 0 7.86MB 15 页 10玖币

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A chemomechanical model of sperm locomotion reveals two modes of swimming

Chenji Li,1Brato Chakrabarti,2Pedro Castilla,1Achal Mahajan,1and David Saintillan1, ∗

1Department of Mechanical and Aerospace Engineering,

University of California San Diego, La Jolla, CA 92093, USA

2Center for Computational Biology, Flatiron Institute, New York, NY 10010

(Dated: October 13, 2022)

The propulsion of mammalian spermatozoa relies on the spontaneous periodic oscillation of their

ﬂagella. These oscillations are driven internally by the coordinated action of ATP-powered dynein

motors that exert sliding forces between microtubule doublets, resulting in bending waves that

propagate along the ﬂagellum and enable locomotion. We present an integrated chemomechanical

model of a freely swimming spermatozoon that uses a sliding-control model of the axoneme capturing

the two-way feedback between motor kinetics and elastic deformations while accounting for detailed

ﬂuid mechanics around the moving cell. We develop a robust computational framework that solves

a boundary integral equation for the passive sperm head alongside the slender-body equation for

the deforming ﬂagellum described as a geometrically nonlinear internally actuated Euler-Bernoulli

beam, and captures full hydrodynamic interactions. Nonlinear simulations are shown to produce

spontaneous oscillations with realistic beating patterns and trajectories, which we analyze as a

function of sperm number and motor activity. Our results indicate that the swimming velocity does

not vary monotonically with dynein activity, but instead displays two maxima corresponding to

distinct modes of swimming, each characterized by qualitatively diﬀerent waveforms and trajectories.

Our model also provides an estimate for the eﬃciency of swimming, which peaks at low sperm

number.

I. INTRODUCTION

The world at low Reynolds number comprises of a large variety of swimming microorganisms [1]. Examples range

from spermatozoa that navigate through the female reproductive tract to fuse with the ovum, to ciliated unicellular

organisms like Paramecium commonly found in ponds, to bacteria found in guts to algae in the oceans [2]. These

microorganisms rely on various mechanisms to break the time reversibility of Stokes ﬂow in order to propel themselves

in the suspending ﬂuid [3]. While bacteria like Escherichia coli use the rotation of their helical ﬂagellar bundle for

propulsion, eukaryotes like sperm cells rely on the propagation of bending waves along their ﬂagella. Even though

the nomenclature of ﬂagellum is used for both prokaryotes and eukaryotes, their structure and origin are distinctly

diﬀerent. Eukaryotic ﬂagella (or cilia) are thin hair-like cellular projections with an internal core known as the

axoneme that has been preserved during the course of evolution [4]. The axoneme has a circular cross-section and is

roughly 200 nm in diameter with 9 pairs of microtubule doublets arranged uniformly along its periphery. The doublets

are connected with each other through a spring-like protein structure called nexin that extends along the entire length

of the axoneme. Thousands of dynein molecular motors act between the microtubule doublets and generate internal

sliding or shear forces in the presence of ATP. Due to structural constraints, the sliding forces are converted to internal

bending moments that deform the ﬂagellar backbone [5,6]. Through a highly coordinated binding and unbinding, the

molecular motors conspire to produce bending waves along the ﬂagellum that help in the propulsion of spermatozoa

[7]. There have been several modeling eﬀorts with varying levels of detail and complexity aimed at elucidating the

biophysical processes that give rise to these spontaneous oscillations in isolated, and ﬁxed ﬁlaments [5,8–20]. While

the basic mechanisms giving rise to spontaneous deformations are now well known, the detailed relationship between

internal dynein actuation, elastohydrodynamics of the ﬂagellum, non-local hydrodynamic interactions, and emergent

waveforms and motility characteristics remains poorly understood. In this work, we present a biophysical model of

sperm locomotion that integrates details of internal elasticity and hydrodynamic interactions with a chemomechanical

feedback loop for dynein activity within an idealized geometry. The model is applied to elucidate the relationship

between internal actuation and the resulting beating patterns, and demonstrates the key role of dynein activity in

controlling the gait and overall motility of the spermatozoon.

The hydrodynamics of swimming sperm has been widely studied, going back to the classical work of G.I. Taylor on

swimming sheets [21]. This has been followed by a series of mathematical analyses of ﬂagellar propulsion [22–25], and

hydrodynamic simulations [26–29]. Recent hydrodynamic studies relevant to sperm motility have addressed the role

of surfaces in sperm accumulation [30,31], viscoelasticity of the medium [32,33], and geometry of the head [34,35].

∗dstn@ucsd.edu

arXiv:2210.06343v1 [physics.flu-dyn] 12 Oct 2022

Almost all [36] of such mathematical models coarse-grain the internal mechanics of the axoneme by prescribing the

kinematics of the ﬂagellum.

However, it is known that a variety of chemical cues related to calcium (Ca2+) signaling along the axoneme

regulate the ﬂagellar beating. Such signaling pathways are responsible for motility [37], hyperactivation [38], and the

reversal of wave-propagation direction along the ﬂagellum [39]. As a ﬁrst step towards understanding such biophysical

phenomena, one needs to construct a model that incorporates the necessary chemomechanical feedback loops giving

rise to sustained ﬂagellar beating, coupled to all the relevant hydrodynamic interactions.

We address this in this paper by building on our previous work on active ﬁlaments used to model spontaneous

oscillations of isolated and ﬁxed cilia and ﬂagella [7]. The proposed biophysical model of a swimming spermatozoon

includes the following: (a) a simpliﬁed model for ﬂagellar beating that accounts for an idealized axonemal structure,

internal elasticity, and dynein activity and kinetics, and (b) detailed non-local hydrodynamic interactions between

the head and the ﬂagellum. The paper is organized as follows. First, in Sec. II we provide a brief description of

the active ﬁlament model, the necessary boundary conditions, and outline the numerical method. We then discuss a

linear stability analysis in Sec. III followed by the analysis of various beating patterns and their properties far from

equilibrium. By characterizing the swimming trajectories and emergent waveforms, we reveal how internal activity

aﬀects the motility and gives rise to two distinct modes of swimming. Using an energy budget, we then highlight the

eﬃciency of the model spermatozoon. We ﬁnally discuss the features of both instantaneous and time-averaged ﬂow

ﬁelds. We summarize and conclude in Sec. IV.

II. SPERMATOZOON MODEL

A. Equations of motion

A mature human sperm head is 5 −6µm long and 3 µm wide [40]. We choose to model it as a rigid spheroid. The

ﬂagellum of a human sperm cell has length L∼30 −50 µm and cross-sectional diameter a∼200 nm. We model this

using an active ﬁlament model [7] that incorporates the necessary structural details of the axoneme, and accounts

for various biophysical active processes that drive spontaneous oscillations. The active ﬁlament model approximates

the 3D axoneme by its 2D projection. As a result, the beating of the ﬂagellum in our model is entirely planar. As

depicted in Fig. 1, the spheroidal head is clamped to the ﬂagellum and is immersed in a 3D inﬁnite ﬂuid bath.

We parametrize the centerline of the active ﬁlament by its arc-length s∈[0, L] and identify any point on it by the

Lagrangian marker xf(s, t) in a ﬁxed reference frame. For an inextensible ﬁlament, we then have

xf(s, t) = xf(0, t) + Zs

t(s0, t) ds0,(1)

where ˆ

t= cos φˆ

ex+ sin φˆ

eyis the tangent to the centerline and φ(s, t) is the tangent angle as depicted in Fig. 1.

We also deﬁne the associated unit normal along the centerline ˆ

n=−sin φˆ

ex+ cos φˆ

ey. The velocity at any point

along the ﬁlament is then given by

v(s, t)≡˙

xf(s, t) = v(0, t) + Zs

φ(s0, t)ˆ

n(s0, t) ds0.(2)

Force and torque balance for a planar elastic rod in the overdamped limit yields [7]

fvis +∂sF= 0,(3)

∂sM+N= 0,(4)

where fvis is the viscous force per unit length exerted by the ﬂuid on the ﬁlament, F=Tˆ

t+Nˆ

nis the contact

force, and M=Mˆ

ezis the contact moment [41] in the active ﬁlament. Since the ﬂagellum is a slender ﬁlament

(=a/L 1), we model its hydrodynamics using non-local slender body theory (SBT) [42,43], which relates viscous

forces to the centerline velocity as

8πν (v− H[fh]) = −M[fvis]≡ M[ff].(5)

Here, νis the ﬂuid viscosity. The term H[fh], where fhis the hydrodynamic traction on the sperm head, denotes the

disturbance velocity due to the motion of the head and is obtained as a single-layer boundary integral equation [44]:

H[fh](s) = ZZDh

G(xf(s),xh)·fh(xh) dS(xh),(6)

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(a) Schematic of the swimming sperm (b) Free body diagram

M(0)

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F(0)

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xf(s, t)

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FIG. 1. (a) Schematic of a swimming spermatozoon. In our model, a spheroidal head (shown in green) is clamped to a ﬂexible

ﬂagellum described as a planar spacecurve with centerline xf(s,t). (b) Free body diagram illustrating the balance of forces and

moments at the head-ﬂagellum junction.

where G(x,x0) is the 3D free-space Green’s function for Stokes ﬂow, and Dhdenotes the 2D surface of the head. The

right-hand side in Eq. (5) involves the force per unit length exerted by the ﬂagellum on the ﬂuid ff≡∂sF=−fvis

through a mobility operator Mwith two contributions: M=L+K[42]. The local part L[ff] accounts for drag

anisotropy along the ﬂagellum and is given by

L[ff](s) = 1

ξ⊥

n(s)ˆ

n(s) + 1

ξk

t(s)ˆ

t(s)·ff(s),(7)

where ξ⊥= (2 −c)−1and ξk=−(2c)−1are resistance coeﬃcients in the normal and tangential directions, and

c= ln(2e) <0. Non-local hydrodynamic interactions between distant ﬂagellar sections are captured by K[ff] deﬁned

K[ff](s) = ZL

0"I+ˆ

R(s, s0)ˆ

R(s, s0)

|R(s, s0)|·ff(s0)−I+ˆ

t(s)ˆ

t(s)

|s−s0|·ff(s)#ds0,(8)

where R(s, s0) = xf(s)−xf(s0) and ˆ

R=R/|R|.

B. Active ﬁlament model

Here, we provide a concise overview of the active ﬁlament model for the ﬂagellum, which directly follows our past

work on clamped ﬁlaments [7] as well as a prior model by Oriola et al. [19]. These build on an earlier model by

Riedel-Kruse et al. [6] and on seminal work by Brokaw [5,8]. The interested reader is pointed to these references for

further details. We idealize the 3D axoneme by its planar projection in the plane of motion, described as an elastic

structure of width a, length L, and centerline xf(s). In this projection, microtubules from the opposite sides of the

axoneme are represented by two polar ﬁlaments x±=xf±aˆ

n/2 clamped at the base at s= 0 and connected to one

another by passive nexin crosslinkers as well as dynein motors, which exert shear forces ±fm(s)ˆ

t(s) per unit length.

These forces result in a sliding displacement ∆(s, t) = a(φ(s, t)−φ(0, t)) between the two ﬁlaments. The sliding

force density can be expressed as

fm(s, t) = ρ(n+F++n−F−)−K∆(s, t),(9)

where ρis the line density of motors, n±(s, t) is the fraction of bound motors on x±,F±is the force exerted by an

individual dynein, and Kis the stiﬀness of nexin links modeled as linear springs. The force exerted by the motors

follows a linear force- velocity relation F±=±f0(1 ∓˙

∆(s, t)/v0), where f0is the stall force of dynein, ˙

∆ is the sliding

velocity, and v0is a characteristic velocity scale. The inability of the microtubules to freely slide apart means that

the sliding forces give rise to an active bending moment

M(s, t) = ˆ

ez"Bφs(s, t)−aZL

fm(s0, t) ds0#,(10)

where Bis the bending rigidity of the ﬂagellum. Moment balance in the out-of-plane direction from equation (4)

yields

Bφss +afm+N= 0.(11)

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摘要：

AchemomechanicalmodelofspermlocomotionrevealstwomodesofswimmingChenjiLi,1BratoChakrabarti,2PedroCastilla,1AchalMahajan,1andDavidSaintillan1,1DepartmentofMechanicalandAerospaceEngineering,UniversityofCaliforniaSanDiego,LaJolla,CA92093,USA2CenterforComputationalBiology,FlatironInstitute,NewYork,NY100...

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A chemomechanical model of sperm locomotion reveals two modes of swimming Chenji Li1Brato Chakrabarti2Pedro Castilla1Achal Mahajan1and David Saintillan1 1Department of Mechanical and Aerospace Engineering.pdf

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A chemomechanical model of sperm locomotion reveals two modes of swimming Chenji Li1Brato Chakrabarti2Pedro Castilla1Achal Mahajan1and David Saintillan1 1Department of Mechanical and Aerospace Engineering

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