Biomedical Engineering Reference
In-Depth Information
on membranes of other RBCs. The Lennard-Jones potential is defined as
σ
LJ
r
6
12
σ
LJ
r
U
LJ
(
r
)=
4
ε
−
,
(10.27)
where
σ
LJ
are energy and length characteristic parameters, respectively. These
interactions are repulsive and vanish beyond
r
ε
and
2
1
/
6
σ
LJ
. In addition, specular re-
flections of RBC vertices on surfaces of other RBCs are necessary due to coarseness
of the triangular network, which represents the RBC membrane.
>
10.2.3.5 RBC adhesion interactions
Adhesion of Pf-RBCs to coated surfaces is mediated by the interactions between
receptors and ligands which are the adhesion sites distributed on a cell and a surface,
respectively. A potential bond between a receptor and a ligand may be formed only if
the receptor is close enough to the free ligand, which is characterized by the reactive
distance
d
on
. A ligand is called free if it is not bound to any receptors. During the
time a receptor is within the distance
d
on
to a free ligand, a bond can be formed with
on-rate
k
on
. Reversely, existing bonds are ruptured with off-rate
k
off
or if their length
exceeds the rupture distance
d
off
. The rates
k
on
and
k
off
are defined as follows
k
on
exp
k
off
exp
σ
2
2
(
−
)
(
−
)
−
σ
l
l
0
l
l
0
on
off
k
on
=
,
k
off
=
,
(10.28)
2
k
B
T
2
k
B
T
where
k
on
and
k
off
are the reaction rates at the distance
l
l
0
between a receptor and
a ligand with the equilibrium spring length
l
0
defined below. The effective on and
off strengths
=
σ
off
define a decrease or an increase of the corresponding rates
within the interaction lengths
d
on
and
d
off
,and
k
B
T
is the unit of energy. The force
exerted on the receptors and ligands by an existing bond is given by
σ
on
and
F
(
l
)=
k
s
(
l
−
l
0
)
,
(10.29)
where
k
s
is the spring constant. The probabilities of bond formation and dissociation
are defined as
P
on
=
t
is the
time step in simulations. This adhesion model is a slight modification of the well-
known adhesive dynamics model developed by Hammer and Apte [54] primarily for
leukocytes. In their model
1
−
exp
(
−
k
on
Δ
t
)
and
P
off
=
1
−
exp
(
−
k
off
Δ
t
)
,where
Δ
σ
on
=
σ
ts
and
σ
off
=
k
s
−
σ
ts
,where
σ
ts
is the transition
state spring constant.
During the course of a simulation the receptor/ligand interactions are considered
every time step. First, all existing bonds between receptors and ligands are checked
for a potential dissociation according to the probability
P
off
. A bond is ruptured if
ξ
<
P
off
and left unchanged otherwise, where
ξ
is a random variable uniformly dis-
tributed on
. If a bond is ruptured the corresponding ligand is available for new
binding. Second, all free ligands are examined for possible bond formations. For
each free ligand we loop over the receptors within the distance
d
on
, and bond forma-
tion is attempted for each found receptor according to the probability
P
on
. This loop
[
0
,
1
]
