Computational and Modeling Strategies for Cell Motility - Computational Modeling of Biological Systems - page 280

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p 2C 0 2 # d x

Z

1 C

2 2

k B T b
k a

1

f mb D

C M 1 .A./ A..t 0 /// 2 C M 2 .V./ V..t 0 /// 2 : (58)

This will be the free energy density used in our cell model.
We denote v as the velocity of the mixture, and p the hydrostatic pressure. We
denote by 1 the mass density of the fluid outside the membrane and inside the core
and by 2 the mass density of the mixture in the cytoplasm. We assume the material
is incompressible in both domains, i.e., 1 and 2 are constants. The density of the
mixture is defined as

1
2 .1 / C

2
2 .1 C /:

D (59)

From mass conservation, we have

2 1
1 C 2

d
dt :

r v D (60)

This is true when

d
dt C r

1
:

v D (61)

d
dt D

@
@t C

Here, r is the material derivative and is the chemical potential of

the material system.
If we use

v

d
dt D

1

(62)

to transport , the continuity equation should be

2 1

d
dt :

r D

v (63)

The balance of linear momentum is governed by

d v
dt Dr . p I

C / C F e ;

D 1 C 2 ; (64)

where 1 is the stress tensor for the fluid outside the membrane and inside the core,
2 is the stress tensor inside the cytoplasmic region, and F e is the external force
exerted on the complex fluid.