Abstract. We study the first passage time problem for a diffusing molecule in an enclosed region to hit a small spherical target whose surface contains many small absorbing traps. This study is motivated by two examples of cellular transport. The first is the intracellular process through which proteins transit from the cytosol to the interior of the nucleus through nuclear pore complexes that are distributed on the nuclear surface. The second is the problem of chemoreception, in which cells sense their surroundings through diffusive contact with receptors distributed on the cell exterior. Using a matched asymptotic analysis in terms of small absorbing pore radius, we derive and numerically verify a high order expansion for the capacitance of the structured target which incorporates surface effects and gives explicit information on interpore interaction through a Coulomb-type discrete energy with additional logarithmic dependencies. In the large N dilute surface trap fraction limit, a single homogenized Robin boundary condition ∂nv + κv = 0 is derived in which κ depends on the total absorbing fraction, the characteristic pore scale, and parameters relating to interpore interactions.

The T cell receptor (TCR) initiates the elimination of pathogens and tumors by T cells. To avoid damage to the host, the receptor must be capable of discriminating between wild-type and mutated self and nonself peptide ligands presented by host cells. Exactly how the TCR does this is unknown. In resting T cells, the TCR is largely unphosphorylated due to the dominance of phosphatases over the kinases expressed at the cell surface. However, when agonist peptides are presented to the TCR by major histocompatibility complex proteins expressed by antigen-presenting cells (APCs), very fast receptor triggering, i.e., TCR phosphorylation, occurs. Recent work suggests that this depends on the local exclusion of the phosphatases from regions of contact of the T cells with the APCs. Here, we developed and tested a quantitative treatment of receptor triggering reliant only on TCR dwell time in phosphatase-depleted cell contacts constrained in area by cell topography. Using the model and experimentally derived parameters, we found that ligand discrimination likely depends crucially on individual contacts being ∼200 nm in radius, matching the dimensions of the surface protrusions used by T cells to interrogate their targets. The model not only correctly predicted the relative signaling potencies of known agonists and nonagonists but also achieved this in the absence of kinetic proofreading. Our work provides a simple, quantitative, and predictive molecular framework for understanding why TCR triggering is so selective and fast and reveals that, for some receptors, cell topography likely influences signaling outcomes.

The T-cell receptor (TCR) triggers the elimination of pathogens and tumors by T lymphocytes. In order for this to avoid damage to the host, the receptor has to discriminate between thousands of peptide ligands presented by each host cell. Exactly how the TCR does this is unknown. In resting T-cells, the TCR is largely unphosphorylated due to the dominance of phosphatases over kinases expressed at the cell surface. When agonist peptides are presented to the TCR by major histocompatibility complex (MHC) proteins expressed by antigen-presenting cells (APCs), very fast receptor triggering occurs, leading to TCR phosphorylation. Recent work suggests that this depends on the local exclusion of the phosphatases from regions of contact of the T cells with the APCs. Here, we develop and test a quantitative treatment of receptor triggering reliant only upon TCR dwell-time in phosphatase-depleted cell-cell contacts constrained in area by cell topography. Using the model and experimentally-derived parameters, we find that ligand discrimination is possible but that it depends crucially on individual contacts being 400 nm in diameter or smaller, i.e. the size generated by microvilli. The model not only correctly predicts the relative signaling potencies of known agonists and non-agonists, but achieves this in the absence of conventional, multi-step kinetic proof-reading. Our work provides a simple, quantitative and predictive molecular framework for understanding why TCR triggering is so selective and fast, and reveals that for some receptors, cell topography crucially influences signaling outcomes.Significance statementOne approach to testing biological theories is to determine if they are predictive. A simple, theoretical treatment of TCR triggering suggests that ligand discrimination by the receptor relies on just two physical principles: (1) the time TCRs spend in cell-cell contacts depleted of large tyrosine phosphatases; and (2) constraints on contact size imposed by T cells using finger-like protrusions to interrogate their targets. The theory not only allows agonistic and non-agonistic TCR ligands to be distinguished but predicts the relative signalling potencies of agonists with remarkable accuracy. This suggests that the theory captures the essential features of receptor triggering.

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We consider the mean first passage time (MFPT) of a two-dimensional diffusing particle to a small trap with a distribution of absorbing and reflecting sections. High-order asymptotic formulas for the MFPT and the fundamental eigenvalue of the Laplacian are derived which extend previously obtained results and show how the orientation of the trap affects the mean time to capture. We obtain a simple geometric condition which gives the optimal trap alignment in terms of the gradient of the regular part of a regular part of a Green's function and a certain alignment vector. We find that subdividing the absorbing portions of the trap reduces the mean first passage time of the diffusing particle. In the scenario where the trap undergoes prescribed motion in the domain, the MFPT is seen to be particularly sensitive to the orientation of the trap.