Somatic genetics of CDR3 control TCR V-domain rotational probability effecting germline CDR2 scanning of polymorphic MHC

The mechanism which adapts the T-cell antigen receptor (TCR) within a given major histocompatibility complex (MHC; HLA, in humans) genotype is essential for protection against pathogens. Historically attributed to relative affinity, genetically vast TCRs are surprisingly focused towards a micromolar affinity for their respective peptide (p) plus MHC (pMHC) ligands. Thus, the somatic diversity of the TCR with respect to MHC restriction, and (ultimately) to pathogens, remains enigmatic. Here, we derive a triple integral equation (from fixed geometry) for any given V-domain in TCR bound to pMHC. We examine solved complexes involving HLA-DR and HLA-DQ, where genetic linkage to the TCR is most profound. Certain V-beta domains displayed rare geometry within this panel—specifying a very low (highly-restricted) rotational probability/volumetric density (dV). Remarkably, hydrogen (H)-bond charge-relays distinguished these structures from the others; suggesting that CDR3 binding chemistry dictates CDR2 contacts on the respective MHC-II alpha-helix.

Some Further Triple Integral Equation Solutions

A solution of a triad of integral equations involving Bessel functions is given. This, like earlier ones, is in the form of a pair of Fredholm integral equations, which may be solved by iteration in certain cases. In spite of a slightly more general formulation of the problem, the kernels of these equations are simpler than those given in earlier solutions. Certain extensions are considered and a formal solution given. Application is made to the problem of incompressible inviscid flow normal to an annular disc, and to the flow due to the slow rotation of such a disc in a viscous fluid.