Basic Principles in Modeling Adaptive Regulation and Immunodominance - Mathematical Methods and Models in Biomedicine - page 34

encompassing view of immune interactions; however, its complexity obfuscates the
key feedback loop that drives the expansion and contraction of a primary T cell
response and the development of immunodominance.
To gain insight into regulatory mechanisms, we will begin by presenting a
simplified model that elucidates the key feedback loop, while still capturing the
qualitative behavior of the extended model from [ 16 ]. The primary simplification is
that the helper and killer subpopulations are considered as a collective population
of effector T cells, since the expansion and contraction of helper and killer T cells
occur roughly in parallel [ 9 ]. In this manner, the simplified model focuses on the
negative feedback between the effector and regulatory T cell populations. Using
the simplified model, we will discuss insights that are difficult to obtain using the
extended model.
The structure of this chapter is as follows. In Sect. 2 we present our mathematical
models of adaptive regulatory T cell-mediated contraction. First, in Sect. 2.1 we
present the simplified model of adaptive regulation, a model that does not separate
helper and killer T cells. This model is taken from [ 17 ]. The model is extended to
include helper T cells in Sect. 2.2 . Mathematical models of immunodominance are
described in Sect. 3 . The models of immunodominance are based on the models of
adaptive regulation and follow the same pattern of presentation: we start in Sect. 3.1
with the basic model of immunodominance that does not include helper T cells. This
model is the original model we proposed, a model that has not yet been published.
We then continue in Sect. 3.2 with the extended model that includes helper T
cells. This model was published in [ 16 ]. Numerical results are given in Sect. 4 .We
show some results for the adaptive regulation model, results that demonstrate the
robustness of the system to small perturbations in the precursor frequencies. We then
continue in Sect. 4.2 with simulations of both immunodominance models, focusing
on results that were obtained with the simple model. A discussion and concluding
remarks are provided in Sect. 5 .

2 Mathematical Model of Adaptive Regulatory

T cell-Mediated Contraction

In this section we overview our models for adaptive regulatory T cell-mediated
contraction. We start in Sect. 2.1 with the basic adaptive regulation model we
introduced in [ 17 ]. This model is based on the hypothesis that primary response
may be governed by a feedback control system involving adaptive regulatory cells
(iTregs) rather than by intrinsic, intracellular feedback mechanisms.
In Sect. 2.2 we extend the adaptive regulation model of Sect. 2.1 to a more
comprehensive model that includes helper T cells and positive growth signals. While
this model better adheres to the biology when compared with the basic model in
Sect. 2.1 , the basic principle that enables that contraction of the immune response
remains the same: adaptive regulatory cells provide the required negative feedback
to generate the desired contraction.