Probabilistic Analysis and Simulation of Stochastic Models of Immune Signaling

This dissertation presents a comprehensive exploration of T-cell activation and antigen recognition, focusing on the mechanisms of kinetic proofreading (KP) and the stochastic nature of T-cell receptor (TCR) activation. We introduce two models: the First Receptor Activation Model (FRAM) and the Signal Activation Model (SAM), both of which demonstrate the exceptional discriminatory capabilities of a kinetic proofreading mechanism. In this thesis, we evaluate the T-cell as a classifier with the task of identifying the condition of an antigen presenting cell (APC), where the APC can be in one of two states, agonist positive or agonist negative. Agonist positive APCs display a small agonist, or foreign, antigen population along with a large self antigen population. Agonist negative APCs only display a large self antigen population. The T-cell classifies an APC as being agonist positive when a T-cell forms a contact with an opposing APC and the T-cell activates prior to the contact expiration. Otherwise, the T-cell classifies the APC as being agonist negative.

In the FRAM we describe T-cell activation as the event when any one receptor is triggered by an antigen ligand before the cellular contact expires. We demonstrate that within the context of the FRAM, kinetic proofreading is more than sufficient to describe any arbitrarily high accuracy observed in biological T-cells, as long as there is sufficient time and/or energy. We show that this is true even in a challenging classification environment, where self and agonist antigen differ only slightly in dissociation rate to the TCR, and encountering an agonist positive APC is rare. We find that a KP model of first receptor activation can overcome this challenge of similarity, and rarity, by having a large number of KP steps. However, we also demonstrate that there is a potential cost, which is that a high APC classification accuracy under challenging environmental conditions requires a large number of reactions and/or a long T-cell/APC contact duration.

The SAM, on the other hand, takes into account the pathways activated downstream of the calcium signal. The KP model is simulated using the Gillespie algorithm, and TCR activation signals are collected throughout the duration of the T-cell/APC contact. We use an autoencoder neural network to demonstrate that even when the steady state TCR activation (calcium) signal is achieved within a few seconds, extending the duration of the cellular contact diversifies agonist positive and agonist negative signals. We further explore the diversity in TCR activation signals by utilizing a recurrent neural network that computes a belief state, which is the model's belief that a given TCR activation signal is a result of interactions with an agonist positive APC. Utilizing this belief state, we describe T-cell activation in the SAM in a recurrent computation framework, where past calcium signal activity influences the T-cell response given the current intracellular calcium signal. Like with the autoencoder, we show that extending the duration of the cellular contact can significantly improve the belief representations, i.e., higher valued beliefs for agonist positive contacts and lower valued beliefs for agonist negative contacts. Furthermore, we demonstrate that when there is sufficient duration of T-cell/APC contact, increasing the number of kinetic proofreading steps improves the SAM's sensitivity to small populations of agonists.

Together, our findings suggest that given the conditions of the FRAM and SAM, kinetic proofreading may be sufficient to accurately distinguish agonist positive and agonist negative cells, even in challenging environments. This suggests a more comprehensive understanding of kinetic proofreading requires us to abandon the steady-state assumption and take into account the dynamic and stochastic nature of the process. More so, our models demonstrate that the interplay between other T-cell features beyond the KP mechanism, such as the duration of the cellular contact or the TCR activation signal, may contribute to the process of antigen discrimination in unexpected ways. Our hope is that our models serve as a basis for a piece-by-piece understanding of antigen discrimination in T-cells, where in the future, more biological realism and complexity can be introduced.