Basic Principles in Modeling Adaptive Regulation and Immunodominance - Mathematical Methods and Models in Biomedicine - page 36

2. In the lymph node, APCs activate naıve T cells that enter a minimal developmen-
tal program of m cell divisions.
3. T cells that have completed the minimal developmental program become effector
cells that keep dividing in an antigen-dependent manner as long as they are not
suppressed by iTregs.
4. Effector cells differentiate into iTregs at a constant rate.
5. The iTregs suppress effector cells upon interaction.

For convenience, we group the entire T cell population into one unit consisting
of both CD4+ and CD8+ T cells. This assumption simplifies the model and focuses
on the feedback loop between effector cells and iTregs. This simplification does not
capture the heterogeneous roles of CD4+ and CD8+ T cells in driving and regulating
the overall T cell response. In particular, CD4+ T cells are the primary secreters
of the cytokine interleukin-2 (IL-2), which drives T cell proliferation. In addition,
nonregulatory CD4+ T cells are the major, if not only, source of iTregs generated
in the periphery [ 24 ]. On the other hand, CD8+ T cells proliferate more rapidly and
extensively than CD4+ T cells and also exhibit cytotoxic activity [ 9 ]. To capture
these differences, we develop a more extensive model that includes separate CD4+
and CD8+ subpopulations in Sect. 2.2 .
In addition, we assume that iTregs do not undergo further proliferation after
differentiating from effector T cells. As with the previous assumption, this sim-
plification also allows the model to focus on the feedback loop between effector
cells and iTregs without incorporating an additional positive stimulation of iTreg
via APCs. We also remove this simplification in the extended model of Sect. 2.2 .
The T cell dynamics in the model are based on the concept of antigen-
independent T cell proliferation and contraction. Various experiments have shown
that the during a primary CD8+ T cell response, T cell kinetics are determined
early on (after approximately 24 h of stimulation) [ 20 ], T cell expansion and
differentiation are antigen-independent after initial exposure (approximately 20 h
of stimulation) [ 30 ], and T cells divide at least 7-10 times after stimulation even
if antigen is removed [ 13 ]. Similar results have been found for CD4+ T cells
[ 33 ]. These results along with other related studies have led to the notion of
antigen-independent T cell program . The main principle is that following initial
stimulation, the primary T cell response is governed by an independent program
that is insensitive to the nature and duration of subsequent antigen stimulation.
The implication is that T cells somehow regulate themselves during a primary
response without feedback from the antigen source. Since in this chapter we only
consider immunodominance during a primary T cell response, we model T cell
dynamics from the perspective of an antigen-independent, self-regulating process.
Other examples of mathematical models of antigen-independent primary T cell
response dynamics can be found in Antia et al. and Wodarz et al. [ 1 , 32 ].
The mathematical model corresponding to Fig. 1 is formulated as the following
system of delayed differential equations (DDEs):

A 0 (

t )= s A − d 0 A 0 ( t ) − a ( t ) A 0 ( t ) , (1)