Biology Reference
In-Depth Information
Table 1
Nomenc
lature of the symmetry instability model
Symbol
Meaning
b
Angular coordinate of the microtubule contact with the boundary
Δ Distance between the centrosome and the cell center
Δ
e
Equilibrium distance between the centrosome and the cell center
q
Angle of the tangent to the microtubule
q
0
Angle at which the microtubule is clamped at the centrosome
EI Microtubule flexural rigidity
F
Force exerted by all microtubules on the centrosome
f
Vertical component of the force exerted by a microtubule on the centrosome
L
Microtubule length
M
Moment of the microtubule-boundary contact force
N
Number of microtubules in the cell
n
i
Fraction of microtubules in stable (
i
= 1) and metastable (
i
= 2) forms
P
Contact force from the cell boundary on the distal microtubule end
p
=
N
/(2π) Density of microtubules per unit of
q
0
in a flat cell
p
=
N
/(4π) Density of microtubules per unit of the solid angle Ω in a three-dimensional cell
R
Cell radius
s
Axial coordinate in a microtubule
v
Moment arm of the microtubule-boundary contact force
concerned. Table
1
lists the model parameters. The following equations specify the
model for the single microtubule. They consist of the standard equilibrium beam
equation, and of the boundary conditions of clamping on the centrosome and fric-
tionless contact with the cell boundary:
d
d
d
d
2
2
s
xs
() sin(),
=
q
s
s
zs
()
=
cos(), d
q
ss
=
d
x
+
d
z
d
d
Ms
()
,
q
()
s
=
M
(()
s
=
vs vs
(),()
= −
xs
( )cos
q
+
zs
()sin
q
s
EI
x
()
000
=
,()
z
=
Dq q
,
()
0
=
,(
x
LLR
)
=
sin
q
,()
zL
=
R
cos
q
0
Since
xL
()
=
xL
(, ,
Δ
qb
,, ,()
Pz LzL
=
(
,, ,,)
Δ
qb
P
0
0
the contact conditions, which are the characteristic equations for the nonlinear
boundary problem, specify the unknown parameters
P
and
b
as functions
PPL
=
(, ,
Δ
q bb
,
=
(
L
,, )
Δ
q
0
0
In the three-dimensional case there is only one stable solution, and, correspond-
ingly, one (
P
,
b
) pair. In the flat (two-dimensional) case, the nonlinearity leads to
