Computational and Modeling Strategies for Cell Motility - Computational Modeling of Biological Systems - page 272

3.2 Active Polar Gel Model

In a series of papers, Joanny, Prost, Kruse, Julicher et al. [ 56 , 57 , 63 , 64 , 92 ] studied
active gels pertinent to cytoskeletal dynamics. We discuss one of their generic
models below. We denote the domain occupied by the gel by , the number density
of monomers in the gel by , an average velocity transporting the gel by v .The
transport equation for is given by

@
@t Cr . v / D k d ı.S/ C k p ı.S/;

(28)

where k p is the rate of polymerization and h d is the rate of depolymerization at
the gel surface defined by the level surface . The polymerization and

depolymerization in this model are assumed to only take place at the gel surface.
Let a be the number density of diffusing free monomers and the diffusive flux j a of
free monomers. The transport equation for a is

f x j S D 0 g

@ a
@t Cr

j a D k d ı.S/ k p ı.S/: (29)

Note that the total number of monomers is conserved

@
@t . C a / Cr . v C

j a / D 0:

(30)

Active processes are mediated by molecular motors. Let c .b/ be the concentration
of bound motors and c .m/ the concentration of the free diffusing motors. The con-
servation equations for the motors are given by

@c .m/
@t Cr

k on c .m/ n ; j .m/ D k pff c .b/

@c .b/

@t Cr v c .b/ Cr

C k on c .m/ n ; j .b/ D k pff c .b/

(31)

where k on and k off denote the attachment and detachment rate, respectively, and j .b/
and j .m/ are the flux of free motors and the bounded ones relative to the gel motion.
In the timescales considered in their model, the momentum balance is replaced
by a force balance equation

r total … I C

f ext D 0; (32)

where I is the identity matrix, f ext is the external force, total

denotes the total stress

tensor, and … is the pressure.