# Alexander, Wahl, 2010

#### Model Status

This CellML model represents system 1 (equations 1a-1d) in the original publication. The model runs in both COR and OpenCell to replicate the published results (figure 2) with R0=0.8. The units have been checked and they are consistent.

#### Model Structure

ABSTRACT: The class of immunosuppressive lymphocytes known as regulatory T cells (Tregs) has been identified as a key component in preventing autoimmune diseases. Although Tregs have been incorporated previously in mathematical models of autoimmunity, we take a novel approach which emphasizes the importance of professional antigen presenting cells (pAPCs). We examine three possible mechanisms of Treg action (each in isolation) through ordinary differential equation (ODE) models. The immune response against a particular autoantigen is suppressed both by Tregs specific for that antigen and by Tregs of arbitrary specificities, through their action on either maturing or already mature pAPCs or on autoreactive effector T cells. In this deterministic approach, we find that qualitative long-term behaviour is predicted by the basic reproductive ratio R(0) for each system. When R(0) is less tHAN 1, only the trivial equilibrium exists and is stable; when R(0) is greater than 1, this equilibrium loses its stability and a stable non-trivial equilibrium appears. We interpret the absence of self-damaging populations at the trivial equilibrium to imply a state of self-tolerance, and their presence at the non-trivial equilibrium to imply a state of chronic autoimmunity. Irrespective of mechanism, our model predicts that Tregs specific for the autoantigen in question play no role in the system's qualitative long-term behaviour, but have quantitative effects that could potentially reduce an autoimmune response to sub-clinical levels. Our results also suggest an important role for Tregs of arbitrary specificities in modulating the qualitative outcome. A stochastic treatment of the same model demonstrates that the probability of developing a chronic autoimmune response increases with the initial exposure to self antigen or autoreactive effector T cells. The three different mechanisms we consider, while leading to a number of similar predictions, also exhibit key differences in both transient dynamics (ODE approach) and the probability of chronic autoimmunity (stochastic approach).

Flow chart illustrating interactions among populations and flow in/out of compartments in System 1. Populations to be modelled explicitly are in black boxes; those that are considered only implicitly or as intermediaries are in grey boxes. Movement in or out of a compartment is indicated by a black arrow; activating influences are indicated with green arrows; suppressive influences are indicated with red arrows; and other interactions or effects are indicated with blue arrows. |

The original paper reference is cited below:

Self-tolerance and Autoimmunity in a Regulatory T Cell Model, Alexander HK and Wahl LM, 2010, *Bulletin of Mathematical Biology*. PubMed ID: 20195912