During the process of T1D pathogenesis, culminating in beta-cell destruction and overt diabetes, islet autoantibodies manufactured by the immune system and directed against one or more host self-proteins serve as reliable surrogate predictive markers of disease onset. The measurement of islet autoantibodies is now a clear prerequisite in screening for individuals at risk of developing insulin requirement. The presence of two or more of these autoantibodies to islet autoantigens (such as insulin and/or GAD65 or IA-2, or insulin or ICA) can be used as entry criterion for interventional trials 
. However, the design of these trials should be based on the understanding that about 30% or more of relatives of type 1 diabetics might not develop an insulin requirement within 15 years. Therefore, the prediction of the rate of progression to clinical disease is still difficult. Mathematical modeling may help determine the key risk factors controlling the timing of disease onset in high-risk individuals for early diagnosis and early enrollment in preventative therapies. In this paper, we have taken a first step by investigating two processes, T-cell avidity and T-cell killing efficacy, that underlie both autoantibody levels and the risk of progression. Such modeling can be a useful adjunct to experimental studies because the frequency of autoreactive CD4
T cells in the peripheral circulation is very low, and the methods for detecting these T cells are laborious, requiring expansion of CD4
T cells with antigen for 10 days.
We first used a one-clone model to show that not only avidity but also killing efficacy of T cells plays a role in determining the timing of disease onset. These two notions determine the ability of T cells in responding to stimulation (via the level of expression of peptides on APCs) and their ability to induce apoptosis into beta cells. Both of these will have to be taken into account in evaluating the predictive power of autoantibodies reactive to the same epitopes as the T cells. Four different regimes associated with the time course of autoantibody level were identified, depending on the parameter regime specified by the reciprocal of avidity,
, and killing efficacy,
(). In one particular parameter regime (labeled (i), with small
and high-avidity), we found that high-risk subjects may exhibit a high level of autoantibodies from the start of the (weak) autoimmune attack and throughout their whole lives but never develop T1D, a feature consistent with conventional autoantibodies that are hypothesized to be associated with less avid T-cell clones. The three remaining outcomes, on the other hand, were all associated with subjects who eventually developed autoimmune diabetes, but exhibited various behaviours in the time course of their autoantibodies. The first outcome (exhibited by the parameter regime (ii) just right of the critical threshold, with intermediate
values) expressed elevated level of autoantibodies from the start of the autoimmune attack and remained elevated throughout, a feature consistent with experimental observations associated with novel autoantibodies (such as IA-2 
). In the second outcome in regime (iii) (also corresponding to intermediate
values but with slightly higher
), autoantibodies were elevated until disease onset only, then decayed to undetectable level. Such an outcome is similar to what has been observed with Insulin autoantibody (IAA) in NOD mice 
. Finally, the last outcome in regime (iv) (corresponding to low
values) exhibited an absence of detectable autoantibodies throughout, which does not commonly occur in high-risk subjects.
The timing of disease onset, defined here as the time it takes to reach the 30% critical threshold, was also evaluated for each of the four cases (regimes) discussed above. The time range for case (ii), which corresponded to high-risk subjects expressing elevated level of autoantibodies throughout, was 7–15 years, depending on the values of
, but more so on
. Such a range is consistent with what has been observed experimentally when comparing the predictability of conventional and novel autoantibodies.
We expanded the model to include several competing clones of T cells with different avidities and autoantigenic specificities to shed further light on the role of T-cell intra- and cross-clonal competition on disease progression. The model consisted of two autoreactive T- and B-cell clones specific to two different autoantigens. Each T-cell clone included high- and low-avidity subclones. Using this more complex model, we showed that the maturation of average avidity within each clone determined the level of beta-cell destruction and the level of autoantibodies manufactured by the immune system. For example, in the middle row of , the low-avidity sub-clones came to dominate, resulting in a sub-clinical outcome, whereas in the bottom row, the average avidity of the more avid clone increased (see ), resulting in clinical disease. In the latter case, reducing the avidity of the less harmful Auto-Ag
-reactive subclone, caused greater loss of beta cells. This was due to a reduction in the competition level within the same clone, leading to an increase in the size of the other competing subclone that was reactive to the same autoantigen. Such complex dynamics is better understood by studying model responses to variations in the avidity ratio within each clone and investigating the impact of such variations in inducing their corresponding autoantibodies. That work will appear in an upcoming manuscript.
In our analysis, we have shown that in most cases, the level of autoantibodies declines to its basal level shortly after the decline of beta cells to levels below the 30% critical threshold needed to prevent the outbreak of clinical symptoms of T1D. The basal level in our model is due to the secretion of immunoglobulin by the inept B cells. As mentioned earlier, this decline in the level of islet-specific autoantibodies is consistent with some of the results obtained from NOD mice in 
. Experimental evidence in humans, however, suggests that the level of circulating islet-specific autoantibodies remains elevated beyond disease onset, an outcome that is exhibited by our model in a limited parameter regime within the
-plane. Other possible factors that may account for this type of behaviour have been neglected by our model for simplicity, such as considering a separate pool of memory T cells with long half-life; beta-cell neogenesis/replication; and the possibility of B cells secreting immunoglobulin more efficiently than assumed here, each of which could keep the level of autoantibodies elevated even after the cessation of beta-cell destruction.
Another simplifying assumption in the model is the linear nature of the effect of autoantigen on the transformation of B cells into plasma cells (Eqn. 1b). This is a first degree approximation to Michaelis-Menten type of kinetics like the formulation used for peptide-dependent T-cell activation. This simplifying assumption made the reduced one-clone model easier to analyze. In future efforts, we propose to relax this assumption with the aim of obtaining better correlation between T-cell avidity and autoantibody affinity in predicting disease onset.
There are other possible alternatives for the immunodominance of certain epitope-specific autoantibodies that have not been addressed in the formalism presented here. Examples include: (a) The level of expression of dominant autoantigenic epitopes (such as, GAD65 and aa1-256) on DCs and beta cells could be higher than those less dominant autoantigenic peptides, leading to higher chances of T- and B-cell activation 
. In other words, epitope dominance may be regulated by the level of expression of autoantigens on DCs and beta cells, rather than by T-cell avidity; (b) ER stress, which may lead to protein misfolding or unfolding in beta cells, could be another factor that determines the dominance of these autoantigenic peptides and their corresponding autoreactive autoantibodies and T cells 
. That is, such proteins are more inclined to be misfolded/unfolded than others, making them more susceptible to degradation and thus rendering them more immunodominant. Testing such hypotheses will require building models that take into account the various intracellular pathways responsible for such behaviour in beta cells and DCs.
Even though our models presented here (with their limitations) have focused on a subset of factors within a highly complex immunological system, we were successful in testing the hypothesis that there is a correlation between the avidity of islet-specific autoreactive T cells and the risk of developing T1D determined by circulating autoantibodies reactive to the same autoantigenic peptides. That correlation, however, also depends on the killing efficacy of the T cells. Modeling T-cell avidity maturation may shed light on the mechanisms by which benign self-reactive T cells develop into a pathological autoreactive T-cell population during T1D progression, which is potentially of great use given the technical challenges in collecting longitudinal peripheral blood lymphocytes (PBL) to quantify autoreactive T-cell avidities. The models presented here provide a qualitative and quantitative analysis of this correlation and explain the reason for the discrepancy in the timing of disease onset between rapid and slow progressors.
Because many of the parameters of the model are not tightly constrained by experiment, it is important to assess the sensitivity of the model to variation in those values. We have done this in two ways. First, we used the reduced one-clone model, whose repertoire of responses can be fully explored qualitatively by phase-plane analysis. We then expanded this module to build the two-clone model and explored in the range of behaviors possible under variation of avidity and killing efficacy, two key parameters identified from the phase-plane analysis. A more complete, formal sensitivity analysis 
of the parameters used in the models presented here, to investigate the impact of various parameters on the general behaviour of the model would be an important future step toward making these models more reliable quantitatively.
Such models may contribute to the development of tools to measure the level of risk associated with each epitope-specific autoantibody and thus prove helpful in diagnosing the disease before irreversible loss of beta cells. That may help us understand the disease process and more accurately identify high-risk individuals at an early stage and enroll them in therapies that can either block the disease or suppress the immune-mediated attack.