T cells circulate in blood and the lymphatic system, continually engag
ing cells through transient non-specific adhesion. In a normally funct
ioning immune system, these interactions permit sufficient time for T-
cell receptors (TCRs) to sample major histocompatibility complex (MHC)
-peptide complexes for the presence of foreign antigen, with detection
of the latter to some extent being triggered by a longer dwell time o
f the receptor on the complex. Precisely how this incremental stabilit
y, which may be relatively small, leads to activation is unclear, but
it appears to be related to diffusion-mediated formation of ternary co
mplex dimers. The formation of stable dimers can explain the high sens
itivity of the response, but leaves a number of questions unaddressed,
including the following: i) How can high sensitivity be reconciled wi
th high specificity, and how can a short TCR dwell time be reconciled
with a comparably short time for ternary complex pair formation? ii) W
hat is the nature of the early signals on the plasma membrane that det
ermine alternative responses e.g. proliferation at one extreme and apo
ptosis at the other? iii) What are the cell-surface correlates of biph
asic dose response functions i.e. of responses that peak as a function
of dose and then descend? This paper has two loosely coupled goals. O
ne is to review and assess the mathematical and computational methods
available for analyzing reactions with and between mobile membrane-bou
nd receptors. These methods range from phenomenological to mechanistic
, the latter being based on the details of atomic structure. The other
is to apply these methods to address biological questions, such as th
ose raised above, part of whose answer may lie in the kinetic competit
ion between alternative reaction paths.