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.

Major histocompatibility proteins share a common overall structure or peptide binding groove. Two binding groove domains, on the same chain for major histocompatibility class I or on two different chains for major histocompatibility class II, contribute to that structure that consists of two α-helices (“wall”) and a sheet of eight anti-parallel beta strands (“floor”). Apart from the peptide presented in the groove, the major histocompatibility α-helices play a central role for the interaction with the T cell receptor. This study presents a generalized mathematical approach for the characterization of these helices. We employed polynomials of degree 1 to 7 and splines with 1 to 2 nodes based on polynomials of degree 1 to 7 on the α-helices projected on their principal components. We evaluated all models with a corrected Akaike Information Criterion to determine which model represents the α-helices in the best way without overfitting the data. This method is applicable for both the stationary and the dynamic characterization of α-helices. By deriving differential geometric parameters from these models one obtains a reliable method to characterize and compare α-helices for a broad range of applications.Program summaryProgram title: MH2c (MH helix curves)Catalogue identifier: AELX_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AELX_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 327 565No. of bytes in distributed program, including test data, etc.: 17 433 656Distribution format: tar.gzProgramming language: MatlabComputer: Personal computer architecturesOperating system: Windows, Linux, Mac (all systems on which Matlab can be installed)RAM: Depends on the trajectory size, min. 1 GB (Matlab)Classification: 2.1, 4.9, 4.14External routines: Curve Fitting Toolbox and Statistic Toolbox of MatlabNature of problem: Major histocompatibility (MH) proteins share a similar overall structure. However, identical MH alleles which present different peptides differ by subtle conformational alterations. One hypothesis is that such conformational differences could be another level of T cell regulation. By this software package we present a reliable and systematic way to compare different MH structures to each other.Solution method: We tested several fitting approaches on all available experimental crystal structures of MH to obtain an overall picture of how to describe MH helices. For this purpose we transformed all complexes into the same space and applied splines and polynomials of several degrees to them. To draw a general conclusion which method fits them best we employed the “corrected Akaike Information Criterion”. The software is applicable for all kinds of helices of biomolecules.Running time: Depends on the data, for a single stationary structure the runtime should not exceed a few seconds.

The authors have notified us of errors in the article. The corrections are given below. We apologize for any inconvenience this may have caused.In the initial software release, the calculation of the curvature and the torsion was done inappropriately due to a missing term resulting from a performed transformation. The software bug is corrected in the current release, which is available as Matlab source code from http://www.meduniwien. ac.at/msi/md/sourceCodes/diffParams/diffParams.htm.This bug affects the test cases shown in the article: All curvature and torsion values depicted for the test cases are incorrect, that is, the depicted values are characteristic properties of the curves, but they do not coincide with the definition of curvature and torsion.The correct curvature and torsion values for the test cases shown in the article can be easily reproduced using the new software release. In addition, for those readers who do not have a Matlab licence, a PDF containing the corrected test cases is available together with the source code.In contrast, the construction of the curves, the ruled surface and its area, the distribution parameter, and the conical curvature were implemented correctly.