Differential geometric analysis of alterations in MH α‐helices

Abstract

Antigen presenting cells present processed peptides via their major histocompatibility (MH) complex to the T cell receptors (TRs) of T cells. If a peptide is immunogenic, a signaling cascade can be triggered within the T cell. However, the binding of different peptides and/or different TRs to MH is also known to influence the spatial arrangement of the MH α-helices which could itself be an additional level of T cell regulation. In this study, we introduce a new methodology based on differential geometric parameters to describe MH deformations in a detailed and comparable way. For this purpose, we represent MH α-helices by curves. On the basis of these curves, we calculate in a first step the curvature and torsion to describe each α-helix independently. In a second step, we calculate the distribution parameter and the conical curvature of the ruled surface to describe the relative orientation of the two α-helices. On the basis of four different test sets, we show how these differential geometric parameters can be used to describe changes in the spatial arrangement of the MH α-helices for different biological challenges. In the first test set, we illustrate on the basis of all available crystal structures for (TR)/pMH complexes how the binding of TRs influences the MH helices. In the second test set, we show a cross evaluation of different MH alleles with the same peptide and the same MH allele with different peptides. In the third test set, we present the spatial effects of different TRs on the same peptide/MH complex. In the fourth test set, we illustrate how a severe conformational change in an α-helix can be described quantitatively. Taken together, we provide a novel structural methodology to numerically describe subtle and severe alterations in MH α-helices for a broad range of applications. © 2013 Wiley Periodicals, Inc.

Exploring peptide/MHC detachment processes using hierarchical natural move Monte Carlo

“…Consequently, most structural analysis has been carried out on bound pMHC ( Hischenhuber et al. , 2012 , 2013 ) and TCR/pMHC structures ( Dunbar et al. , 2014 ; Knapp et al.…”

Geometry Dynamics ofα-Helices in Different Class I Major Histocompatibility Complexes

“…From rulings we estimate distances | X ( u )| between polynomials across the cleft. The resulting polygon mesh was triangulated and interhelical area A was calculated, as previously outlined [ 18 , 28 ]. Each of these quantities may be monitored over time, for example, A ( t ); see Figure 9 .…”

“…Between two adjacent α -helices, as found in MHC proteins, the polynomials serve to span a ruled surface. This interhelical surface lends itself to derive several quantitative characteristics of shape: (a) total area [ 18 , 19 ], (b) a profile of interhelical distances along the binding cleft, and (c) heuristic “centre line of the cleft” which may be constructed, along which the surface torsion, that is, a twist or screw of the interhelical surface, can be computed. The latter characterizes the positions and bending of helices relative to each other and defines the geometrical shape of the peptide-binding cleft that is ligated to the TCR.…”

Geometric Analysis of Alloreactive HLAα-Helices

“…Each helix is represented by a spline, and the interhelical area is represented by a surface defined by “rulings” (i.e., straight lines) spanned between corresponding points (opposite to each other) on these splines [9]. We use M rulings (1200 ≤ M ≤ 1500) parameterized by a common parameter u .…”

“…From distances between splines $\begin{array}{c}d\left(u_i\right)=||{\mathbf{c}}_{\mathrm{2}}\left(u_i\right)-{\mathbf{c}}_{\mathrm{1}}\left(u_i\right)||\hspace{1em}\mathrm{1}\le i\le M\end{array}$ and distances between rulings, the total intrahelical area, A , is computed as outlined in [9]. Likewise, median, quartiles, and extreme values (boxplots) of d ( u i ) over time are calculated for each i ; see Section 3.1.…”