Multiscale Modeling of Primary Cilia - Multiscale Modeling in Biomechanics and Mechanobiology - page 90

ignore the inertia effects and focus on the Stokes flow regime. In 2012 we advanced
previous models [ 58 , 60 ] by taking into account the large rotation at the cilium base
and a consistent hydrodynamic drag force [ 62 ]. Later, we used the slender-body
theory (SBT) to compute the hydrodynamic interaction more accurately with large
deformation of the cilium under a planar shear flow [ 63 ]. Within the SBT frame-
work we incorporated a rotational stiffness at the axoneme base to model the basal
anchorage, and we obtained good quantitative agreement in the dynamics of cilium
bending under flow with experiments. We further used this model to investigate the
ciliary dynamics with a more complex temporally periodic flow (see Sect. 5.4.2.2 ),
and the bending dynamics of a cilium with a time-varying length that depends on the
balance between axonemal assembly and dissemble rates (see Sect. 5.6 ).

5.3.1 Drag Force Model

Downs et al. [ 62 ] modeled the primary cilium axoneme as an elastic, homogeneous
beam with a uniform cross-section along the centerline. Under load the bending of
such an elastic beam can be described by Euler-Bernoulli beam theory:

d d s = M ( , ʸ)

E I

s

, (5.1)

where is the angle between the tangent and the vertical, M is the bending moment,

s is the arclength along the beam from the tip to the base, E is Young's modulus, and
I is the second moment of inertia. In this formulation they assume an infinitesimal
strain, and make no assumption about the axoneme angle at the free end to allow for
large deflection at equilibrium.
At equilibrium the governing equation for the angle

ʸ

ʸ( s ) is

E I d 2

ʸ
d s 2

d M
d s =

= Q x ( s ) sin ʸ( s ) − Q y ( s ) cos ʸ( s ), (5.2)

where Q x and Q y are the total forces in the x and y directions, respectively. In this
model the main focus is on a unidirectional flow in the y direction, and force-free
condition is assumed in the x -direction. Thus the governing equation for ʸ( s ) is

E I d 2

ʸ
d s 2

=− Q y ( s ) cos (ʸ( s ) − ʸ 0 ), (5.3)

where ʸ 0 , the basal angle of the initial

shape of the nonstressed beam. The cantilevered beam boundary conditions with a
specified rotation

ʸ( s ) − ʸ 0 is the difference in angle from

ʸ base at the fixed end are used: ʸ( L ) = ʸ base and a zero bending

d d s | s = 0 =

moment at the free end, 0 .