Biomedical Engineering Reference
In-Depth Information
Let
p
be the polarity vector describing the polar direction of the monomer. The
time rate change of the system free energy is given by
Z
d
x
h
total
P
P
a
a
i
;
P
P
F
D
c
.b/
.b/
c
.m/
.m/
Wr
v
C
h
P
C
r
(33)
@
where
h
is the molecular field,
P
p
is the corotational
derivative of
p
, is the chemical force conjugate to the ATP production rate
r which determines the number of ATP molecules hydrolyzed per unit time and
unit volume. The dot denotes time derivative, ;
a
;
.b/
;
.m/
are the chemical
potentials corresponding to ;
a
;c
.b/
;c
.m/
, respectively. The total stress is given by
D
@t
p
C
v
r
p
C
1
2
.
ph
total
D
C
hp
/;
(34)
where is the symmetric part of the stress. The symmetric stress tensor ,
P
,and
the ATP consumption rate r can be decomposed into reactive part and dissipative
part, respectively,
D
r
C
d
;
P
r
P
d
;
D
C
P
r
D
r
r
C
r
d
;
(35)
where the superscripts r denote the reactive response and d the dissipative response.
The constitutive equations for the dissipative response are given by
1
2
D
2
Dt
2
d
1
2
.
ph
h
/
I
;
D
Dt
D
2
D
C
hp
/
CQ
1
.
p
1
2
D
2
Dt
2
P
d
1
2
D
2
Dt
2
h
1
C
1
p
D
C
D
Dt
.
1
r
v
p
CQ
1
Tr.
r
v
/
p
/;
r
d
r
.b/
;
D
ƒ
C
1
p
h
C
p
(36)
where
1
;
Q
1
;ƒ;
1
are model parameters and is the relaxation time. The fluxes
for the monomers and motor molecules, which do not have reactive parts, are
given by
j
.a/
D
D
.a/
r
.a/
C
.a/
p
;
j
.m/
D
D
.m/
r
c
.m/
C
.m/
p
;
j
.b/
D
D
.b/
r
c
.b/
C
.b/
p
;
(37)
where D
.i/
are the diffusion coefficients,
.i/
are coupling parameters.
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