their surface - bound t cell receptors ( tcrs ) recognise antigens and thereby detect hazardous organisms or changes inside cells .
tcrs recognize short peptide fragments ( p ) that are bound to major histocompatibility complexes ( mhc ) .
different interactions of tcr / peptide - mhc ( tcr / pmhc ) are believed to be the basis for distinctive stimuli that lead to and trigger different fates of the t cell , for example , t cell development , thymic selection , lineage commitment , differentiation into effector cells , or memory t cell responses to foreign antigens .
mhcs are surface - bound proteins and their role is to present peptide fragments to tcrs , be they self- or alloantigens . to achieve this ,
this cleft comprises two -helices and five subjacent lateral -strands forming a sheet or floor of the cleft .
tcrs engage peptide mhcs in a diagonal arrangement that seems to be a common mode of interaction across tcrs [ 2 , 3 ] .
the two mhc -helices interact with the tcr complementarity determining regions ( cdr ) 1 and 2 , while the mhc - bound peptide interacts with cdr3 , although cdr1 has also been shown to interact with the terminal parts of the peptide .
most of the sequence variability of tcrs is found within cdrs ; these regions are also referred to as hypervariable regions .
the two mhc -helices are of particular interest as they represent stable secondary structural domains interacting with tcrs .
adhesion and signaling proteins together with a set of tcr / pmhc complexes form a junction between t cell and an antigen - presenting cell that is referred to as an immunological synapse .
it serves as a platform for assembly and segregation of signaling complexes , which cooperatively decide the outcome of t cell activation and effector function .
new findings show that this supramolecular complex forms too late to be relevant for initial tcr signaling that happens within seconds after pmhc engagement , with the tcr initiating a tyrosine phosphorylation cascade .
such an early signal may be sufficient to trigger effector function like killer t cell cytolysis of target cells .
stimulation of proliferation , however , requires engagement and signaling for many minutes or even hours [ 58 ] .
it is important to note that -tcr itself has no signaling motif but associates with homo- and heterodimeric cluster of differentiation 3 ( cd3 ) subunits ,  ,  , and  in a noncovalent way .
these subunits contain immunoreceptor tyrosine - based activation motifs ( itams ) that can be phosphorylated and initiate downstream signaling of tcr activation .
protein structures as found in protein databases ( e.g. , protein data bank ( pdb ) ) show a static image .
in contrast to that , in molecular dynamics ( md ) simulations , the protein explores many conformations and allows one to capture its dynamics .
computational immunoinformatics is a well - established , rapidly evolving field . in previous papers [ 10 , 11 ]
we presented a first evaluation of three tcr / pmhc systems that differ only slightly in the mhc amino acid sequence .
macdonald et al .   determined binding characteristics and immunogenicity of mhc alleles hla - b44:02 , hla - b44:03 , and hla - b44:05 in complex with the abcd3 self - peptide ( eeylqafty ) and lc13 tcr .
hla - b44:02 and b44:05 trigger an immune response when bound to the lc13 tcr , while hla - b44:02 does not despite extensive amino acid sequence homology .
determined the x - ray structure of the ternary complex hla - b44:05 , abcd3 nonapeptide , and lc13 tcr that is accessible on the protein database ( pdb , http://www.pdb.org/ , pdb - id : 3kps ) .
due to extensive sequence identity we were able to use in silico site - directed mutagenesis to obtain 3d structures of the missing tcr / pmhc complexes .
md simulations in the nanosecond range could possibly show short - lived changes in dynamic behaviour , conformation , propagation of forces , or early activation signals [ 13 , 14 ] .
the aim of the present paper is to put forward adequate tools for identifying and monitoring of conformational changes with possible functional relevance in mhc -helices , and in particular to monitor geometric characteristics of the mhc peptide - binding groove . here ,
we present ( i ) a robust and parsimonious method to find an approximation of a protein 's -helical axis , ( ii ) an evaluation of polynomial degree adequacy in describing bending or hinge movements of particular -helical axes , and ( iii ) application of polynomial fitting of -helical axis to monitor -helical conformations along md simulations in different tcr / pmhc immunological complexes .
our previous studies established the use of polynomials to model -helices in mhc molecules and monitor their dynamic behaviour . to mathematically describe the structural dynamics of mhc -helices at the tcr / pmhc binding interface we first identified those helical regions which were stable and therein those which are close to the protein - protein interface .
then we extracted the -helical axis and finally determined a polynomial that approximates this axis in a least - squares sense .
various methods to extract a helix axis have been developed , including rotational fitting , using c atoms as control points of b - splines   or fitting to a helix .
we used a fragment - fitting method , based on previous work , to locate the axis of -helices as follows : an ideal , linear -helix fragment comprising four c atoms is superimposed on successive pieces of mhc -helices in a least - squares sense . along the axis of the fitted helical fragment ,
we adopt points as estimates of the mhc -helix axis and fit a polynomial through these points . from the polynomial , geometric parameters can be derived to monitor conformational changes .
polynomials fitted to the -helical axis can , in principle , be of any degree .
however , polynomials of higher order tend to oscillate , adding noise to geometrical quantities computed thereof .
we therefore evaluated polynomials of different degrees for their ability to reproduce bending and hinge motions of an -helix with minimum relative error . between two adjacent -helices ,
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 .
dynamics in the shape of the protein - protein interface might modulate the tcr / pmhc binding affinity .
the x - ray structure of tcr / pmhc b4405 ( number 3 in table 1 ) was taken from the pdb ( pdb - id : 3kps ) . structures b4402 and b4403 were engineered by means of in silico site - directed mutagenesis   using b4405 as a structural template .
we introduced(i)mutation y116d to the mhc molecule to get lc13/abcd3/hla - b44:02 complex ( b4402),(ii)mutations y116d and d156l to the mhc molecule to get lc13/abcd3/hla - b44:03 complex ( b4403).see figure 1(b ) for a 3d representation of amino acid positions y116 and d156 .
the program automatically browses a rotamer library and selects an amino acid rotamer minimizing a scoring function in order to fit the new amino acid in its surrounding and avoid steric clashes with other atoms .
all mhc molecules simulated have an amino acid sequence identity of more than 99% and stay in very similar 3d fold .
mhc molecules appeared stable during all our md simulations as seen in rmsd plots in figure 15 .
mutation y116d to the mhc molecule to get lc13/abcd3/hla - b44:02 complex ( b4402 ) , mutations y116d and d156l to the mhc molecule to get lc13/abcd3/hla - b44:03 complex ( b4403 ) .
the full amino acid sequence of hla - b alleles is accessible from the hla library ( https://www.ebi.ac.uk/ipd/imgt/hla/ ) and a description of its topology is accessible from uniprot ( http://www.uniprot.org/uniprot/q95365 ) .
not surprisingly , a sequence comparison showed that a transmembrane helix ( 24 amino acids ) and cytoplasmic tail ( 30 amino acids ) are missing from the mhc x - ray structure as plasma membrane structures and flexible protein parts are hard to determine using x - ray crystallography .
first , water molecules were added to the protein structure , immersing it in an artificial water bath of rectangular form and allowing a minimum distance of 2  nm between the protein and the box boundaries .
second , water molecules were replaced by sodium and chloride ions to yield a salt concentration of 0.15  mol / l and neutralize the protein net charge .
third , the energy of the solvated system was minimized using a steepest descent method and then the system was gradually heated up to 310  k during 100  ps position restraint md simulation .
finally , md production runs were done with the lincs constraint algorithm acting on all bonds using the gromos 53a6 force field .
hydrogen and fast improper dihedral motions were removed , allowing for an integration step of 5  fs .
van der waals and coulomb interactions were computed using a cut - off of 1.4  nm . long - range coulomb interactions were computed by ewald summation .
velocity rescale temperature coupling was set to 310  k and berendsen isotropic pressure coupling was set to 1 bar .
the selection of the force field and md parameters for pmhcs were evaluated by omasits et al .   and set accordingly . as outlined in the introduction ,
we are interested in monitoring c atoms that form stable -helices and are in close contact with the tcr . from md simulation data we calculated the following:(i)the relative presence of -helical structure in the protein complexes over the simulation time
: we used the dssp algorithm   as implemented in gromacs to identify secondary structural elements of the protein complex over simulation time .
-helices were considered stable if the ratio(1)t-helixtsim0.5with t-helix being the time the main chain ( backbone atoms plus carbonyl oxygen atoms ) meets the dssp criterion for an -helix and tsim being the total simulation time .
the cut - off value of 0.5 is justified by the distinctly bimodal distribution ( see figure 11(a)).(ii)the relative presence of close contacts at the protein - protein interface : c atoms that are less than 1.4  nm apart ( i.e. , the cut - off for electrostatic interactions in our md simulations ) are defined as being in close contact .
atom - atom contacts between tcr and mhc are defined stable if the ratio(2)tcontacttsim0.5tcontact is the time during which two atoms are in close contact , and tsim is the simulation time .
again , the cut - off value of 0.5 is justified by the distinctly bimodal distribution ( see figure 11(b ) ) . to get the residue - wise relative contact time , we averaged the atom - wise relative contact time ( defined in ( 2 ) ) over all atoms per residue.the resulting sets of amino acid residues defined by procedures ( i ) and ( ii )
were intersected ( workflow , see figure 2 ; results , see figure 3 ) before applying further methods described in sections 2.4 and 2.5 .
dynamically stable helices ( green atomic surface in figure 3 ) as defined by procedure ( i ) overlap well with the atoms in close contact of the tcr ( red atomic surface in figure 3 ) especially with the cut - off set to 1.4  nm .
note that at the end of the mhc 's peptide - binding pocket both g - alpha1 and g - alpha2 exhibit a kink followed by a short -helix .
these short -helices are present in the crystal structure , but we found them being unstable during md simulations as they fold and unfold .
we do not consider these helices in our analysis , as they are not in close contact with the tcr .
the relative presence of -helical structure in the protein complexes over the simulation time : we used the dssp algorithm   as implemented in gromacs to identify secondary structural elements of the protein complex over simulation time .
-helices were considered stable if the ratio(1)t-helixtsim0.5with t-helix being the time the main chain ( backbone atoms plus carbonyl oxygen atoms ) meets the dssp criterion for an -helix and tsim being the total simulation time .
the cut - off value of 0.5 is justified by the distinctly bimodal distribution ( see figure 11(a ) ) .
the relative presence of close contacts at the protein - protein interface : c atoms that are less than 1.4  nm apart ( i.e. , the cut - off for electrostatic interactions in our md simulations ) are defined as being in close contact .
atom - atom contacts between tcr and mhc are defined stable if the ratio(2)tcontacttsim0.5tcontact is the time during which two atoms are in close contact , and tsim is the simulation time .
again , the cut - off value of 0.5 is justified by the distinctly bimodal distribution ( see figure 11(b ) ) . to get the residue - wise relative contact time
, we averaged the atom - wise relative contact time ( defined in ( 2 ) ) over all atoms per residue . in order to mathematically describe and quantify -helical geometry and movements , polynomials c(u ) of degree m
were fitted to the -helices , where c represents a vector of 3d coordinates and u is the curve parameter .
prior to fitting we extracted c atom coordinates of those amino acids , which fulfil the criterion of stable -helices and close contacts as described in section 2.3 . according to the structural definition of -helices   a model of one ideal -helical turn ,
that is , a fragment consisting of four c atoms , is constructed , with its axis coinciding with x - axis .
the coordinates of its kth c were assigned as(3)xyz = pk1rcosk1rsink1k=1,  ,4with pitch p = 0.15  nm ( advance from one c to the next ) , helix radius r = 0.23  nm , and  = 100  /180 . along the axis of this helical fragment
we consider three axis points , the initial ( 0,0 , 0 ) , the intermediate ( 1.5  p , 0,0 ) , and the final ( 3.0  p , 0,0 ) . out of an -helix with n  c atoms we pick moving groups of four successive c atoms each , to which we fit the fragment model defined above in a least - squares sense . along x -
axis of the fitted fragment model we adopt points as estimates of the axis of the -helix , see figure 4 . from the very first fitted fragment ( c atoms 1 ,  ,
4 ) we adopt two points as points of the helix : the transformed initial axis point as a1 and the transformed intermediate point as a2 . from fragments fitted
subsequently we adopt only the respective intermediate points , and from the last fragment ( fitted to c atoms n  3 , 
n ) , we again adopt 2 points , its  intermediate  as an2 and its final point as an1 .
thus , n  1 points ( a1 , a2 ,  , an1 ) represent the axis of the -helix .
an1 ) representing the helical axis are fitted by polynomials c(u ) of degree m in a least - squares sense .
separate regression functions fx , fy , and fz are computed for x- , y- , and z - coordinates:(4)cu =
fxufyufzu;for example , for x - coordinate,(5)cxufxu = px , mum+px , m1um1++px,1u+p0.the total number of parameters for this model is nparameter = 3  ( m + 1 ) .
when fitting the curve , parameter u is evaluated at discrete values u = 1,2.5,3.5 ,  , n  1.5 , n of pitch , corresponding to the positions of estimated points along the helical axis .
after the regression has been performed , c(u ) in ( 5 ) may be evaluated for arbitrary , continuous values 1  u 
both g - alpha1 and g - alpha2 helices were modelled in the same way , yielding models c1(u ) and c2(u ) with equal polynomial degrees k. it is well known that ( with equidistant data points ) interpolating polynomials of too high a degree may exhibit severe oscillations near the ends of the interpolation interval .
( approximating polynomials need not pass through data points but rather approximate the shape of their functional dependence in a least - square sense . )
for each polynomial , the curve parameter ranges within 1  u  n , with possibly different values ( n1 , n2 ) for each helix .
we consider l = min  ( n1 , n2 ) equidistant values of u , yielding l reference points on each helix model .
connecting corresponding reference points by straight lines yields rulings , x(u ) = c2(u )  c1(u ) , which span a ruled surface ( see figure 5 ) .
( the rule for defining  corresponding  points has to be adopted in a reasonable way but finally remains to some degree arbitrary . ) 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 .
respective graphs provide well - defined estimates of changes in width of the intrahelical gap ( binding cleft ) as a function of both helical position and time .
likewise , median , quartiles , and extreme values of interhelical distances for each u and of a can be obtained .
the ruled surface between both polynomial helix models may bend and wind in various ways .
describing all aspects would call for a comprehensive mathematical treatment in terms of differential geometry , from which we refrain . instead , we restrict ourselves to describe something like the  torsion of the interhelical surface  in a simple , intuitive way ; see figure 5 .
 it may apply to ( curved ) lines as well as to surfaces in 3d space .
we use both terms in parallel with equal meaning . ) to this end we note that the observer may slide across a surface along various paths and may , in general , observe different values of surface torsion along each of the paths .
we notice that torsion in strict mathematical terms is a local characteristic of the surface , and even more , it depends upon the path one takes to inspect the surface . for the sake of simplicity
we deliberately adopt the centre line between both helix models:(6)mu=12c1u+c2u , and let t(u ) be the unit tangent vector of m(u ) . of note ,
the centre line intersects with each of the rulings . while proceeding along the centre line we inspect the directional change of rulings .
this provides us with a particular characterization of the deformation of the interhelical surface . to quantify this intuitive description ,
 it is based on the change in direction between successive rulings   and is determined by the derivative x(u ) = c1(u ) + c2(u ) .
( in strict mathematical terms ( of differential geometry ) this quantity is called  parameter of distribution ,  which we think is a very nonintuitive label , prone to be mixed up with probability distributions . hence , we prefer the more intuitive term  surface torsion  to quantify the winding of a surface while proceeding along a given path ( in our case the centre line ) , not that surface torsion is generally different in different directions , even in the very same point of a surface . )
what we call surface torsion is given by ( 7)u = xxtx2.the integral value(8)t=path  muuduquantifies the relative tilt of the surface between start and endpoint of the path .
note that (u ) > 0 indicates right - handed surface torsion , (u ) < 0 indicates left - handed surface torsion , and (u ) = 0 indicates that the surface at this point is developable and rulings are parallel .
(  developable  means that a sheet of paper could be bended to exactly match the surface ; the surface could be  flattened .
 ) in summary , the definition of the intrahelical surface depends on a reasonable selection of rulings , x , and a path , m , which are both arbitrary to some extent . despite these shortcomings in a strict mathematical sense , the concept of torsion , as introduced here , mirrors some essential features in describing the interplay between the shape of -helices and their relative orientation in forming the mhc binding cleft
. derivatives of each polynomial helix model can be obtained analytically for each value of the parameter u , yielding the vectors(9)cu = dcdu , cu = d2cdu2.curvature  is a scalar quantity being per definition positive in euclidian 3d space and is obtained via : ( 10)u = cucucu3 .
so far we have described how an -helical axis is reconstructed by our newly proposed fragment - fitting method and fitted by a polynomial of a certain degree . from this polynomial
we wanted to find an optimal degree for these polynomials in order to retrieve conformational deformations with minimal errors . to do so we modelled two different motions such that mhc -helices could perform : a bending motion and a hinge motion ( see figure 6 ) . to model a bending motion we created a curved helix backbone model with known curvature ,
as described in detail in appendix a. following polynomial representation of the backbone , the curvature integral can be obtained analytically , yielding the reference , bendingdu ( see ( a.2 ) in the appendix ) . to test our method ,
the curved helix model was subjected to the fragment - fitting method and the resulting helix axis was fitted by a polynomial in a least - squares sense as described in sections 2.4 and 2.5 .
the polynomial was evaluated at 100 equidistant points and curvature and the curvature integral were calculated numerically , yielding bendingdu ( see ( 10 ) ) . as a quality criterion for regaining the correct curvature integral we used the relative error ( 11)bending=bendingdetectordubendingreferencedu1and evaluated it for polynomial degrees 1 to 8 .
subsequently , the helix was split into two parts : one c atom was selected and the remaining part of the helix rotated around the selected c atom to model a hinge motion ( see figure 6(c ) ) . the position to split the helix ( number of c selected ) was varied from c atom 5 to 15 .
the aim was to examine -helix hinges with two legs unequal in length . from this series of -helical models ( different positions of the hinge and varying hinge angles , hinge )
a polynomial of kth order was fitted to the traced -helical axis in a least - squares sense and the curvature integral calculated numerically , yielding hingedu .
the relative error ( 12)hinge=hingedetectorduhinge1was obtained for polynomials of degrees 1 to 8 , with 0  hinge  /2 being the angle between the two parts of the helix .
to monitor deformations of -helices in md simulations we propose an approximation of the -helical axis using a fragment - fitting method ( section 2.4 ) , fitting the resulting axis by a polynomial in least - squares sense ( section 2.5 ) , and derive geometric quantities from it ( section 2.6 ) . to find an optimal degree of polynomials we applied our method by applying it to modelled -helical bending and hinge motions ( figures 6(a ) and 6(b ) ) and evaluated the polynomial degree that retrieves a certain -helical motion with minimal relative error .
for the hinge motion we noticed that relative errors increase with polynomials of higher order . for the bending motion ,
choosing sixth- or eighth - degree polynomials increases the degree of freedom while at the same time failing to add adequate improvement in relative error .
generally , relative errors in the detection of hinge motions are larger than those for bending motions .
the second - order polynomials reproduced the measured quantities of bending and hinge motions with low relative error and hence were adopted for monitoring mhc -helices in all subsequent analyses .
we admit that second - order polynomials may fail to model -helical axes in full detail , but it seems appropriate for capturing trends in helical motions and shape during an md simulation with minimal relative error .
we applied the geometric analysis to model mhc -helices of tcr / pmhc complexes b4402 , b4403 , and b4405 . by inspecting the curvature integral for mhc -helices , g - alpha1 and g - alpha2
, three interesting cases could be identified showing changes along the 250  ns md simulation ( figure 7 , panels in upper row ) .
phases during which the curvature integral changes either abruptly ( ( b ) and ( c ) ) or gradually ( a ) are highlighted in orange and red . helix conformations (  bundles  ) corresponding to these phases are shown in panels in the middle row . to check if changes in the curvature integral actually indicate a conformational change we calculated rmsd matrices of conformations within and between orange and red helix bundles ( panels in lower row ) . in all three cases ,
rmsd between phases is distinctly larger ( shifted to the right ) as compared to rmsd within phases .
a shift towards higher rmsd values indicates a conformational change and is seen in all three cases ( see table 2 for median rmsd values ) .
these case studies demonstrate that changes in conformation of -helices during md simulations can be monitored using second - degree polynomials fitted to helical axes .
the mhc peptide - binding groove comprises two -helices that interact with the tcr . using second - order polynomials
, we computed ( i ) the integral of the curvature of individual mhc -helices , ( ii ) the area of interhelical surface , and ( iii ) the surface torsion along the imaginary centre line derived from both polynomials that model mhc -helices for single time steps in md simulations .
items ( ii ) and ( iii ) are geometric properties of the ruled surface .
they are used to quantify the geometric relation between the two helices , for example , their relative orientation , and thus capture important aspects of the geometry of the peptide - binding groove , as illustrated in figure 5 .
considering the curvature integral we obtain a measure of the overall bending of the whole curve .
curvatures of polynomials modelling single helices were integrated and monitored over time as seen in figure 8 .
g - alpha1 of b4403 and b4405 each undergoes abrupt fluctuations in helical conformation , reflected in the curvature integral , but is stable in the phases in - between .
we notice that the curvature integral for g - alpha2 is generally higher than that for g - alpha1 , reflecting the kink near its n - terminal end
g - alpha2 of b4403 and b4405 shows only minor fluctuations and no abrupt changes in the curvature integral indicating that the hinge angle stays stable . for b4402 ,
the hinge angle decreases in the first half of the md simulation and remains stable thereafter .
it also changes when both helices bend in opposite directions as these amount to a distension of the surface .
whenever helices bend in similar ways in the same direction , interhelical area will be relatively unaffected .
depending on the complexity in shape of the helical axis , second - order polynomials might not adapt to the axis ' path accurately .
figure 9 shows an increasing trend of interhelical area for complexes b4403 and b4405 , while a declining trend of interhelical area is seen for b4402 .
these changes occur as the helices of the peptide - binding groove move closer together or further apart , respectively .
this is also reflected in the distance of -helical centres of masses ( data not shown ) .
however , inspecting figure 5 clearly demonstrates that the actual shape of an interhelical surface can not be characterized by a single quantity such as the area of interhelical surface .
this parameter describes the change in angle between subsequent tangent planes along a given path , in our case the centre line .
high values of surface torsion , regardless of being positive or negative , describe a rapid change in the angle .
a pair of helices , each being somehow deformed and both being in varying positions to each other , gives rise to an interhelical surface with a plethora of possible shapes .
for describing these shapes geometrically , surface torsion is an important concept , lending itself to quantify the twist in the surface along a prescribed path .
for b4402 , surface torsion along the centre line ( see figure 5(a ) ) is left - handed most of the time , reaching a minimum ( ca .
data not shown ) are seen only in rare cases and near both surface termini .
the ruled surface of the peptide - binding cleft at the n - terminus of g - alpha1 appears to be more stable for b4405 and b4403 than for b4402 .
it is stable for b4405 , but trends are observed for the other two complexes : b4402 shows a drop in the surface torsion integral during the first 60  ns followed by a gradual rise .
b4403 is rather stable during the first 6070  ns and then shows a gradual decrease in surface torsion .
we also looked at the shape complementarity ( sc ) ( lawrence and colman introduced a method to measure how well the surfaces of two proteins at their interface match .
the parameter output by this method is called shape complementarity ( sc ) . ) of the protein - protein interface surface and backbone c rmsd .
we found sc to be stable for all three tcr / pmhc systems ( see figure 14 ) .
as sc is unaffected by deformations of the binding cleft ( as reflected by surface torsion , see above ) we conclude that tcrs follow the conformational changes of the mhc surface .
rmsds of b4402 and b4403 reach a low plateau after a few nanoseconds and remain stable thereafter .
rmsd of b4405 shows an increasing trend over 250  ns simulation time , rising from approximately 0.3  nm to 0.6  nm ; see figure 15 .
the mechanism of tcr activation is still controversially debated and several models have been proposed [ 3238 ] . a recent work of dustin and depoil   summarizes new insights into the function of the t cell synapse .
the authors grouped t cell synapse into three interactive layers including interactions of receptors , a signaling layer , and a cytoskeleton layer , all contributing to tcr activation , regulation , and fine - tuning of signaling and responses .
conformational changes of the tcr complex have been demonstrated to be relevant for signaling of the tcr .
association of cd3 proteins to the tcr / pmhc complex is necessary to transmit the activation signal to intracellular signaling molecules .
evidence suggests that tcr conformational changes are required for full activation , but there are certain signaling pathways that can also be activated in the absence of conformational changes .
the three tcr / pmhc complexes analysed in this work differ only by one or two amino acids in the mhc molecule .
hla - b44:05 ( b4405 ) and hla - b44:02 ( b4402 ) mhc types trigger lc13 tcr activation in the presence of the abcd3 self - peptide .
surprisingly hla - b44:03 ( b4403 ) does not trigger tcr activation . to characterise the dynamics of the mhc antigen binding cleft , we applied ( similar to the methods reported by christopher et al . ) a fragment - fitting method to model stable -helical regions that are in close proximity ( 1.4  nm ) to the tcr and monitored their geometric parameters . however , it is known that geometric quantities derived from polynomial approximations may vary substantially depending on the polynomial degree chosen . to select an appropriate polynomial degree
, we tested the ability of polynomials with different degrees to retrieve predefined parameters of helical motions and deformation . in a simplified model
, we tested the ability of the fragment - fitting method to reproduce the curvature integral of helical bending and hinge motions .
we found that second - order polynomials are best suited to model these -helical motions with low relative error .
the curvature integral derived from polynomials of individual -helices can be related to conformational changes of -helices . between the two mhc -helices a ruled surface can be spanned , of which we computed the area as an estimate for the size of the peptide - binding cleft .
we also calculated the surface torsion along an imaginary centre line characterizing the orientation of both -helices relative to each other .
however , we were not able to find correlations between immunogenicity and certain patterns in the -helical movements .
a limitation of the current analysis is that the ruled surface between the mhc -helices that we use to model the mhc surface presented to the tcr does not consider the shape and dynamics of the peptide that lies between the two helices .
we can not assume that phase space has been sampled comprehensibly for these large molecules .
stepwise fluctuations in measured variables are visible ( see figure 7(b ) , upper row , and figure 15 ) .
also , even with highly optimized simulation performance of 15  ns / day on 1024 computing cores of the ibm bluegene , statistics to discriminate between different simulated systems is not feasible .
enhanced sampling techniques and adequate collective variables might be useful to identify adequate collective variables for such systems .
signaling may involve a series of other proteins of the immunological synapse and interactions that are not considered in these simplified tcr / pmhc models .
interactions between tcr and pmhc take place between two cells that are in close contact to each other .
it has been shown that plasma membrane lipids affect the activity of signaling networks   and some models of tcr / pmhc interaction propose involvement of the plasma membrane . in this work ,
we applied the quantification method to three large tcr / pmhc complexes , due to their size being accessible by md simulation studies only since recently .
we saw that mhc -helices undergo rapid changes in conformation by either bending motion or hinge motion .
surface torsion used for characterizing the mhc surface presented to the tcr is stable in b4405 , which is the most immunogenic complex .
we speculate that rapid changes in helical conformation are part of the intrinsic dynamics of mhcs when engaging with tcrs .
though we were not able to find a clear correlation between immunogenicity and certain patterns in the -helical movements , we could demonstrate that single amino acid polymorphisms in the mhc seem to have a subtle influence on the helixes ' shape dynamics and that it would be interesting to apply the same method in the case of peptide polymorphism .
in summary , the present work demonstrates the feasibility and reliability of deriving shape parameters from simulation data .
next , the influence of the detected small conformational changes on the microscopic dynamics will be investigated to clarify their relation to the biological functions of the complexes of interest .
conclusions regarding functional differences between tcr / pmhc systems characterized by a single - residue polymorphism certainly require advanced sampling techniques in order to sample the conformational phase space appropriately for molecules of this size .
future studies might investigate if the small conformational changes in mhc -helices transmit forces to the tcr .
 mhc -helices form the antigen - binding cleft and are of particular interest for immunological reactions . to monitor these helices in molecular dynamics simulations , we applied a parsimonious fragment - fitting method to trace the axes of the -helices . 
 each resulting axis was fitted by polynomials in a least - squares sense and the curvature integral was computed . to find the appropriate polynomial degree , 
 the method was tested on two artificially modelled helices , one performing a bending movement and another a hinge movement . 
 we found that second - order polynomials retrieve predefined parameters of helical motion with minimal relative error . from md simulations we selected those parts of -helices that were stable and also close to the tcr / mhc interface . 
 we monitored the curvature integral , generated a ruled surface between the two mhc -helices , and computed interhelical area and surface torsion , as they changed over time . 
 we found that mhc -helices undergo rapid but small changes in conformation . 
 the curvature integral of helices proved to be a sensitive measure , which was closely related to changes in shape over time as confirmed by rmsd analysis . 
 we speculate that small changes in the conformation of individual mhc -helices are part of the intrinsic dynamics induced by engagement with the tcr . 
1. Introduction
2. Methods
3. Results
4. Discussion