171 Chapter 12 Mathematical Modeling of Synaptic Patterns Anastasios Siokis, Philippe A. Robert, and Michael Meyer-Hermann Abstract During antigen recognition by T cells, a specific spatial structure is formed at the contact face to an antigen-presenting cell (APC), called an immunological synapse (IS). The IS supports bidirectional signal- ing and release of effector molecules and is widely studied both biologically and numerically, in order to understand the process of T cell activation and signaling. This specialized structure harbors a central area (central supramolecular activation cluster, cSMAC) populated by T cell receptor-peptide-major histocom- patibility complex (TCR-pMHC) interactions, hedged by a peripheral ring (peripheral supramolecular activation cluster, pSMAC) of integrin lymphocyte function associated-1 interactions with its immuno- globulin superfamily ligand intercellular adhesion molecule-1 (LFA-1-ICAM-1). These two regions form the “bull’s eye” pattern characteristic of the mature IS. In theoretical studies, different modeling architectures, including partial differential equations (PDE) and agent-based models, have been developed with the purpose to answer mechanistic questions about the IS dynamics. In this chapter, we explain possible physiological mechanisms that lead to the formation of ISs and technical issues that may occur in the course of development of agent-based models. Key words Immunological synapse, Patterns, Agent-based modeling, Partial differential equations (PDEs), Mechanics, Computational biology 1 Introduction The study of the IS initially began at the end of the 1990s and begin- ning of the 2000s [1], and the description of this phenomenon has undergone continuous improvement with advances in microscopy and molecular methods [2–4]. While biologists focused on details of cytoskeletal and vesicular transport as candidate mechanisms, physi- cists envisioned important contributions of membrane bending, and mathematicians took unbiased approaches to the discovery of mini- mal forces acting between molecules that could lead to these patterns. In this chapter, we will focus on the mathematical approaches to build minimal systems, explaining in depth the agent-based approach. Cosima T. Baldari and Michael L. Dustin (eds.), The Immune Synapse: Methods and Protocols, Methods in Molecular Biology, vol. 1584, DOI 10.1007/978-1-4939-6881-7_12, © Springer Science+Business Media LLC 2017 Inquiries regarding this work may be addressed to Anastasios Siokis <Anastasios.Siokis@helmholtz-hzi.de> or Michael Meyer-Hermann <mmh@theoretical-biology.de>.