Biomedical Engineering Reference
In-Depth Information
Tabl e 2
Estimates for additional parameters used in the extended model
Param.
Description
Estimate
δ
H
Death/turnover rate of effector CD4+ T cells
0.23
δ
K
Death/turnover rate of effector CD8+ T cells
0.4
H
0
(
)
Initial naıve CD4+ T cell concentration
0
see Scenario 2
K
0
(
0
)
Initial naıve CD8+ T cell concentration
see Scenario 2
δ
0
H
0
Supply rate of naıve CD4+ T cells
(
)
s
H
0
δ
0
K
0
s
K
Supply rate of naıve CD8+ T cells
(
0
)
m
1
# of divisions in minimal CD4+ developmental program
2
m
2
# of divisions in minimal CD8+ developmental program
7
ρ
H
Duration of one T cell division
11/24
ρ
K
Duration of one T cell division
1/3
σ
H
Duration of min developmental program: 1
+(
m
H
−
1
)
ρ
H
1
.
46
σ
K
Duration of min developmental program: 1
+(
m
K
−
1
)
ρ
K
3
r
1
Rate of secretion of positive growth signal by CD4+ cells
100
r
2
Rate of secretion of positive growth signal by CD8+ cells
1
δ
Decay rate of free positive growth signal
5.5
P
r
Rate of differentiation of effector cells into iTregs
0.02
Other parameters are the same as those used in Table
1
for the simplified model
Concentrations are in units of k/
μ
L, and time is measured in days
In this model, we use the same parameters as in Table
1
for the simplified model,
except for those listed in Table
2
. We assume that CD4+ and CD8+ T cells have
halflives of 3 days and 41 h, respectively, yielding death rates of
δ
=
0
.
23 and
H
δ
4/day [
9
]. We assume that CD4+ and CD8+ populations have doubling times
of 11 h and 8 h, respectively, yielding cell division rates of
=
0
.
K
3
day [
9
]. We do not have good estimates of the secretion rates of positive growth
signal by effector T cells, hence we estimate that CD4+ and CD8+ T cells secrete
growth signal at rates
r
1
ρ
=
11
/
24 and
ρ
=
1
/
H
K
1/day, respectively. We assume that free
positive growth signal decays with a halflife of 3 h, yielding an estimate of
=
100 and
r
2
=
δ
=
P
5
5/day. In this model only effector CD4+ T cells can differentiate into iTregs, so
the new estimate of the iTreg differentiation rate,
r
, must be higher than the previous
estimate of
r
.
=
0
.
01/day to maintain similar dynamics. Hence, in this model, we set
r
=
0
.
03/day.
3
Mathematical Models of Immunodominance
In this section we explain how to expand our model for monoclonal T cell responses
to polyclonal responses. We show that the expanded model automatically recreates
elements of the characteristic behavior associated with immunodominance. In this
way, we demonstrate that immunodominance may occur as a natural result of iTreg-
mediated self-regulation of polyclonal T cell responses.
We begin in Sect.
3.1
with deriving a basic immunodominance model
that demonstrated immunodominance as an extension of the basic model of
adaptive regulation. We then continue in Sect.
3.2
with an extended model of
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