- Research article
- Open Access
Are different stoichiometries feasible for complexes between lymphotoxin-alpha and tumor necrosis factor receptor 1?
© Mascarenhas and Kästner; licensee BioMed Central Ltd. 2012
- Received: 26 August 2011
- Accepted: 8 May 2012
- Published: 8 May 2012
Tumor necrosis factors, TNF and lymphotoxin-α (LT), are cytokines that bind to two receptors, TNFR1 and TNFR2 (TNF-receptor 1 and 2) to trigger their signaling cascades. The exact mechanism of ligand-induced receptor activation is still unclear. It is generally assumed that three receptors bind to the homotrimeric ligand to trigger a signaling event. Recent evidence, though, has raised doubts if the ligand:receptor stoichiometry should indeed be 3:3 for ligand-induced cellular response. We used molecular dynamics simulations, elastic network models, as well as MM/PBSA to analyze this question.
Applying MM/PBSA methodology to different stoichiometric complexes of human LT-(TNFR1)n=1,2,3 the free energy of binding in these complexes has been estimated by single-trajectory and separate-trajectory methods. Simulation studies rationalized the favorable binding energy in the LT-(TNFR1)1 complex, as evaluated from single-trajectory analysis to be an outcome of the interaction of cysteine-rich domain 4 (CRD4) and the ligand. Elastic network models (ENMs) help to associate the difference in the global fluctuation of the receptors in these complexes. Functionally relevant transformation associated with these complexes reveal the difference in the dynamics of the receptor when free and in complex with LT.
MM/PBSA predicts complexes with a ligand-receptor molar ratio of 3:1 and 3:2 to be energetically favorable. The high affinity associated with LT-(TNFR1)1 is due to the interaction between the CRD4 domain with LT. The global dynamics ascertained from ENMs have highlighted the differential dynamics of the receptor in different states.
- Elastic network model (ENM)
- Tumor necrosis factor (TNF)
Protein-protein interactions are critical for signaling events within a cell. An investigation on the precise recognition of ligands by their respective receptors is an active field of research, since breakdown of such specific recognition is the root cause of several diseases and infections. One of the rational motives to understand such phenomena is to develop antibodies and small-molecule inhibitors that modulate the outcome of such interactions. One such system that generated immense attention owing to its central role in inflammatory effect, immunological response, but also in several autoimmune diseases and several pathogeneses is the tumor necrosis factor (TNF) [1, 2]. Two TNF ligands, namely TNF-α (or TNF) and TNF-β (or lymphotoxin-α, LT) have been extensively studied to methodologically dissect cellular signaling and diseases related to their malfunction [3–6]. It is now well recognized that several cellular responses are directly dictated by TNFs and about 20 homologous cytokines have been identified .
The TNFRs exhibit distinct functional roles and diverse signaling capabilities. While TNFR1 is expressed in all tissues, TNFR2 is expressed particularly in immune cells and other specialized cell types like endothelial and neuronal tissue [8, 15]. TNFR1 primarily invokes cytotoxic activities of the cell whereas TNFR2 functions as a receptor for T-cell signaling and for mediating host infections . In contrast to TNFR2, TNFR1 contains a death domain in its cytoplasmic region and upon ligand binding is capable of activating the apoptotic pathway . Experimental evidence also suggests only TNFR2 to exhibit differential binding to soluble and membrane bound TNF-α . Although the two receptors share good homology in their extracellular domain, their cytoplasmic regions show significant differences in their sequences. Due to their central role in cellular signaling, several diseases are directly linked to the TNF family of ligands and receptors . Animal models of diseases have predicted the predominant role of TNFR1 in several pathogeneses and adverse causes of enhanced inflammation . In contrast, TNFR2 has been demonstrated to be involved in defects related to cell-mediated immunological response . Anti-TNF antibodies and engineered soluble TNFRs have been developed for the treatment of rheumatoid arthritis and other diseases [21–23].
Interaction of TNF with its receptors imparts a conformational change in the receptors that triggers a cellular response. But the precise mechanism of receptor activation by their ligand is still under debate. The first proposal for receptor activation, known as the ligand trimerization model, emphasizes the ligand to recruit the receptors to form the final complex with a ligand-receptor molar ratio of 3:3 . Recent evidence has, however, raised serious questions on this mechanism of activation of the receptors. One school of thought envisages receptors as dimers or trimers in the absence of ligand and propose a pre-ligand assembly domain (PLAD) formed by the association of two or three receptors at the membrane distal CRD1 domain, prior to ligand binding [25, 26]. Cross-linking experiments with TNFR1-Fas and TNFR2-Fas also suggest the formation of homodimers in the absence of a ligand . These homo-dimers/trimers of receptors then build up to form cluster-aggregates on the cell surface upon ligand binding [28, 29]. The CRD1 domain has also been shown to be important for stabilizing the CRD2 domain for efficient ligand binding . The recent crystal structure of the CD40-CD154 complex in a 2:3 molar ratio has further hinted that the stoichiometry of TNF-TNFR complexes may not always be 3:3 . Also, recent work has indicated that the formation of a trimer-monomer complex of a ligand trimer-receptor monomer complex of the TNF family member TRAIL is quite stable and may be the first step in the formation of the complex . In this work we investigate the free energy of binding of LT-TNFR1 complexes. The objective of this work is to shed light on the way in which the receptors bind to the ligand and to estimate the free energy of binding involved in complex formation.
Molecular dynamics (MD) simulations have been carried out on the three stoichiometric complexes of LT with TNFR1, represented as LT-(TNFR1)1/2/3 along with their individual binding partners, monomeric TNFR1 (mTNFR1), the dimeric receptor ((TNFR1)2) and trimeric lymphotoxin (LT), each for 35 ns. The residues 28–171 of each chain of human LT and the residues 15–153 of human TNFR1 were included in our model. The receptor is made up of four cystein-rich domains: CRD1 (residues 15–53), CRD2 (54–97), CRD3 (98–138), and CRD4 (139–153). For analysis purposes and to have similar number of residues to those observed in the dimeric receptor (TNFR1)2 (PDB: 1NCF), only the residues 15–150 from TNFR1 were used in the MM/PBSA calculations and other analysis. Free energy of binding has been computed by using MM/PBSA methodology in single-trajectory and separate-trajectory methods. The components of free energies, gas-phase energies, and solvation free energies have been averaged over 1001 snapshots from MD trajectories.
Naismith and Sprang  have classified the structure of the receptor into two major types of sub-domains, namely A1 and B2 modules, based on the structural topology and on disulfide bridges. They denoted the receptor to be made up of three A1 and B2 sub-domains each, in the following order: A1 (residues 15–29), B2 (30–52), A1 (55–70), B2 (73–96), A1 (98–114), B2 (117–137), and A1 (139–153). These authors also relate the structure of TNFR1 to be similar to a spiral, where the B-modules correspond to the plates and the A-modules to the bolts about which they pivot. The dynamic cross-correlation matrix (DCCM) extracted for the receptors explicate the relations between these domains. The correlation patterns of mTNFR1, LT-(TNFR1)2, and LT-(TNFR1)3 are rather similar, but differ strongly from the pattern of LT-(TNFR1)1 (see Additional file 1: Figure S1). In the former, the B2 module of CRD2 and the A1 module of CRD3 (residues 73–96 and 98–114, respectively) are highly correlated. Both these modules are also anti-correlated to the B2 module (residues 30–52) of the CRD1 domain, to the CRD4 domain, and to some extent to the A1 module (residues 15–29) of CRD1. In contrast to LT-(TNFR1)2 and LT-(TNFR1)3, highly correlated fluctuations observed in mTNFR1 might be an artifact of its high flexibility since it is present in an unbound form. In mTNFR1, LT-(TNFR1)2, and LT-(TNFR1)3, the B2 module of CRD2 and the A1 module from CRD3 are highly correlated. Thus it can be argued that these sub-domains form a stable motif across these complexes. The loss of correlations in the dimer (TNFR1)2 is not surprising considering the interaction in this complex happens mainly via the CRD1 and CRD4 domains. However, the significant silencing of correlations in LT-(TNFR1)1 further supports a unique nature of the interaction between the LT and TNFR1 in LT-(TNFR1)1.
Residues involved in binding
A good estimation of the polar interaction between two molecules can be made from estimating the hydrogen-bonding interaction between them. The hydrogen bonding interaction between the receptors in (TNFR1)2 and the receptor-ligand complexes in LT-(TNFR1)n were measured purely based on the following geometric constraints using VMD . A distance cutoff of 0.35 nm between the donor and acceptor with an angle cutoff of 60° in the angle donor-hydrogen-acceptor were defined to count for a successful hydrogen bonding interaction. The average number of hydrogen bonding interactions (per interface) over the trajectories was 25.5 for (TNFR1)2 and 31.4, 25.7, and 27.7 for LT-(TNFR1)1, LT-(TNFR1)2, and LT-(TNFR1)3, respectively. Hence, the interaction between the receptor and the ligand is strongest in the LT-(TNFR1)1 complex while it is pretty similar within all other complexes.
Complex structures and receptor motions
Results from the elastic network model
One of the major drawbacks of MD simulations is that a system needs to be simulated for long time scales to arrive at a meaningful interpretation of functionally relevant motions. This naturally requires computational time ranging from weeks to months for systems like the one studied here. In order to overcome such time-consuming calculations several coarse-grained computational methods have been developed. One such model that has received wide popularity is the elastic network model (ENM). Several studies have shown the low-frequency normal modes obtained from ENM to capture the conformational transition of several biomolecules which have been summed up nicely in the following reviews [34, 35]. Hence, ENM is considered a powerful tool to establish the large-scale motions of proteins. One factor that dictates the outcome of the ENM is the spring constant for the interacting atoms. Several groups have explored distinct ways to rationalize their choice of force constants [36, 37]. In this work, as discussed in Methods, we defined a set of three force constants depending on the nature of the bonds. It is to be noted that the ENM was constructed based on the X-ray structures and is, thus, independent of the results of the MD simulations.
Free energy of binding
In this work the formation of LT-(TNFR1)3 was split into the following three fundamental steps in accordance with the trimerization model.
Our motive for applying the MM/PBSA method on this system was to shed light on the stability of LT-(TNFR1)n complexes of different stoichometry. Though precise estimations of binding free energies for protein-protein complexes are tough, results from MM/PBSA are known to correlate well with experimental binding free energies . The precise mechanism for the activation of TNFR1 has been subject to immense debate. The previous belief of a 3:3 molar ratio of the ligand-receptor has been hugely influenced by the first crystallographic structure of the LT-(TNFR1)3 complex. In the recent past, however, evidence and arguments have been presented that question if indeed that should be the case. Recently Reis et al.  showed for the TRAIL-DR5 system, a system similar to LT-TNFR1, that the affinity of DR5 for TRAIL is strongest for the binding of the first receptor molecule compared to the binding of second and third, suggesting a ligand-receptor molar ratio of 3:1. Another family of TNF-receptor systems, the CD154-CD40, crystallizes in the molar ratio 3:2 . Hence, it is worth to analyze if such 3:1 and 3:2 stoichiometric complexes are stable and plausible for LT-(TNFR1)n. The major advantage of the MM/PBSA method is its ability to determine free energies with relatively low computational expense coupled with the advantage of breaking down the free energy components into different energy terms obtained from molecular mechanics and solvation. Nevertheless, the MM/PBSA analysis presented here should more be understood as providing qualitative insight rather than quantitative numbers.
Results from single-trajectory simulations (SITA)
For the calculation of free energy components of the binding energy from MD simulations, one needs to extract the coordinates of the individual binding partners as well as the complex. It is possible to obtain the coordinates of the individual proteins from a single simulation of the complex, which is referred as the single-trajectory approach (SITA). Alternatively, when individual MD runs have been performed on the individual binding partners and their complex separately, we refer to them as separate-trajectory approach (SETA). One major advantage associated with SITA is the reduction in the computational requirement since only a single simulation of the complex needs to be performed. But this approach is valid only if the binding partners do not undergo major conformational and dynamic changes upon complex formation. In the present system, the receptors exhibit huge fluctuations and domain movements as discussed above.
Binding energies (in kJ/mol) obtained from single-trajectory analysis
−671.6 ± 1.2
−739.5 ± 2.2
−600.7 ± 1.7
−670.4 ± 1.9
−668.7 ± 3.5
−676.5 ± 4.3
−567.4 ± 4.6
−662.8 ± 4.2
709.3 ± 3.3
700.9 ± 4.0
589.9 ± 2.0
680.4 ± 1.8
−60.7 ± 0.1
−73.4 ± 0.2
−59.3 ± 0.1
−66.7 ± 0.2
−691.6 ± 5.0
−788.5 ± 6.3
−637.5 ± 6.4
−719.4 ± 6.2
Results from separate-trajectory simulations (SETA)
Binding energies (in kJ/mol) obtained from SETA
94.4 ± 4.1
−44.3 ± 4.6
110.0 ± 4.9
−202.2 ± 5.0
−21.4 ± 6.3
−169.7 ± 6.9
−434.2 ± 12.5
−183.0 ± 13.4
−257.6 ± 13.5
461.6 ± 11.0
143.3 ± 11.7
267.0 ± 11.8
−36.4 ± 0.3
−13.6 ± 0.3
−36.0 ± 0.2
425.2 ± 11.0
129.7 ± 11.7
231.1 ± 11.8
−116.8 ± 17.9
−119.0 ± 19.4
−86.2 ± 19.8
The free energy of binding, ∆Ggas+solv, estimated from SETA for LT-(TNFR1)1, LT-(TNFR1)2 and LT-(TNFR1)3, is −116.8, −119.0 and −86.2 kJ/mol, respectively. The electrostatic interaction between receptor and ligands is quite high for these complexes. However, the total electrostatic interaction (∆Helect + ∆Gpolar), which is the sum of the contribution of electrostatic interaction between the binding partners and the solvation energy, gives a true picture of the electrostatic interaction between the proteins in the complex. The values for the steps 1 to 3 in this investigation are 27.3, −39.7, and 9.3 kJ/mol. Hence, binding of the second receptor to LT is electrostatically favorable in contrast to binding of first and third receptors. The non-polar interaction between the receptor and the ligand is negative; for the binding of second receptor the value is comparatively less pronounced. All this suggests that the binding of second receptor imparts a significant change to LT-(TNFR1)1.
A range of forces and constraints are at play when two proteins interact to form a complex. The conformational freedom of the individual binding partners varies between the complex and their free form. A parameter that reflects conformational restrain is the change in the internal energy. When this parameter is positive it indicates the binding partners have to be conformationally constrained to form the complex while a negative value indicates that the conformational restrains on the individual binding partners have been relaxed. For steps 1 and 3 the ∆Hint values are positive while for step 2 it is negative, indicating that binding of two receptors to the ligand is favored. The association of two capable binding partners occurs invariably at the cost of entropy. Entropic changes are hard to estimate in MM/PBSA. However, in our case in each of the three steps the receptor from free solution binds to the ligand. The major contribution to entropy arises then from the loss in entropy of the receptor from its state free in solution to the state bound to the ligand. Since we mainly compare the free energies of the different stoichiometric complexes, the entropy contribution arising from this step should then be comparable and cancels in the differences. For this reason we have ignored entropic contributions in the free energy calculations. The free energy of binding (ΔGgas+solv) values obtained from our study suggest a stoichiometric ratio 3:1 and 3:2 are of similar stability and are little higher in comparison to a 3:3 complex, suggesting such complexes are energetically feasible.
In this work, we tried to judge using MM/PBSA methodology if LT-(TNFR1)n=1,2,3 complexes can form with a ligand-receptor molar ratio of 3:1 and 3:2. The exact mechanism of receptor activation is still unknown. In accordance with the ligand trimerization model, the free energy of binding involved in the sequential binding of the receptors has been estimated. Using an ENM the global fluctuations that are associated with these complexes have been investigated. The results from MD simulations of the three stoichiometric complexes of the receptor with LT reveal that the CRD4 domain is attached to LT and stabilized in LT-(TNFR1)1 while it exhibits extensive fluctuations in the other two complexes. The low-frequency normal modes as observed from ENM analysis display highly symmetric motions of the three CRD4 domains in the LT-(TNFR1)3 complex. A direct impact of these motions on the cytoplasmic domains can be postulated. It has been recently proposed that six Fas intracellular death domains come in close proximity for inducing the formation of the oligomeric complex of Fas molecules . Motions as observed in our ENM might be necessary and appropriate for aligning the intracellular domains in a systematic fashion and with right steric requirements to activate the signaling cascade. Such domain motions in tandem can change between an activated state, where they bring the intracellular domains to correct proximity, and a moderately active or inactivate state, where such proximity between the intracellular domains is either partially or completely lost.
In accordance with TRAIL-DR5, we observed that LT-(TNFR1)1 complex is stable, which is proposed to arise from the binding of the CRD4 domain to LT. We observed that only in the LT-(TNFR1)1 complex, the CRD4 domain binds to LT. Three residues, namely Phe144, Arg146, and Glu147 have been determined to be crucial for such an interaction. If such an interaction leads to a stable LT-(TNFR1)1 complex, as predicted from our MM/PBSA studies, it opens the debate if the LT-(TNFR1)1 complex represents an inactive state of the receptor. While LT-(TNFR1)1 would be inactivate the binding of subsequent receptors could lead to its activation. Our results from ENM indicate quite similar domain motions of the receptors in LT-(TNFR1)2 and LT-(TNFR1)3 which differ from those in LT-(TNFR1)1.
The results of the free energy of binding, ΔGgas+solv, estimated from MM/PBSA from single-trajectory analysis reveal the LT-(TNFR1)1 complex to be the most stable among LT-(TNFR1)n=1,2,3 while that from separate-trajectory analysis suggest LT-(TNFR1)n=1/2 to be equally stable. Although both methods utilize the same complex trajectory as input, only in SETA does one include the coordinates of ligand and protein in their unbound form from independent simulations. Since the receptors undergo a huge conformational change upon complex formation, the results from SETA should be more trustworthy. Several factors have a direct effect on the results of MM/PBSA, which include the force-field used, simulation time, charge models, solute dielectric constant and the surface boundary [41–43]. While the actual numbers obtained may be too high, their relative magnitude is expected to be more reliable. Estimation of free energies of binding for a protein-protein complex is a tedious task. Two major bottlenecks need to be overcome in such simulations, sufficient sampling and accurate estimation of entropy. The energy values obtained from this study are from 20 ns of data which we believe are a good compromise between size of the system and the number of simulations that needs to be undertaken coupled with the corresponding computational cost. The objective of this investigation was to get a hint whether ligand binding in a sequential fashion, as in steps 1, 2, and 3 leading to the final 3:3 complex strengthens or weakens the protein-protein interaction. The absolute numbers might not be that relevant but their relative values aid in better understanding of the interaction of the protein-protein complexes in different stoichiometry. In that sense our free energy results suggests both LT-(TNFR1)1 and LT-(TNFR1)2 to be more stable than LT-(TNFR1)3.
The present study provides new insight into the LT-(TNFR1)n complexes. The CRD4 domain of the receptor in the LT-(TNFR1)1 complex was observed to bind to LT. With the aid of ENM models the functional motions exhibited by LT-TNFR1 complexes have been portrayed. Our analysis suggests the CRD4 to exhibit a kind of zig-zag motion in LT-(TNFR1)2 and LT-(TNFR1)3 but to be well immobilized in LT-(TNFR1)1. The low-frequency normal modes derived from ENM analysis also support the CRD4 domain to be involved in highly fluctuating motions. Our free energy results based on MM/PBSA calculations on single-trajectory and separate-trajectory support the proposal of stable LT-(TNFR1)1 and LT-(TNFR1)2 complexes.
We model the interactions of LT with TNFR1. The starting structure for our simulation was the crystallographic structure of the human LT-(TNFR1)3 complex (PDB ID: 1TNR)  and the receptor dimer (TNFR1)2 (PDB ID: 1NCF) . Other structures for the simulations, LT, TNFR1, LT-(TNFR1)1, and LT-(TNFR1)2 were extracted from the trimeric complex structure of LT-(TNFR1)3. All simulations were performed with GROMACS (ver. 4.0.7)  using the Gromos 43a2 (united-atom) force-field . The proteins were placed such that a minimum distance of 0.7 nm is ensured between any sides of the dodecahedral unit cell and protein atoms. Proteins were then solvated in water modeled as simple point charge (SPC) . To preserve electro-neutrality Na+ or Cl– ions where added when necessary. The whole setup was energy minimized, first with steepest decent followed by conjugate gradient. As the first step in molecular dynamics, random velocities were generated at 300 K. Keeping the heavy atoms of the protein restrained with a force constant of 1000 kJ mol-1 nm-2, the solvent molecules were allowed to equilibrate for 30 ps. The restraints were then removed and system was further allowed to equilibrate at 300 K for another 1 ns. The simulation was then extended to 35 ns of MD run. The temperature was maintained at 300 K using the Berendsen thermostat  with a coupling constant of 0.1 ps. Protein and solvent were independently coupled to the reference temperature. In the first equilibration phase a Berendsen barostat was used while in the subsequent MD run Parrinello–Rahman  pressure coupling (with a coupling constant of 1 ps) was applied. Short-range non-bonded interactions were cut off at 1.2 nm. Electrostatic interactions above this range were evaluated using PME. The pair list was updated every 5 steps. All bonds were constrained using the LINCS algorithm  permitting an integration time step of 2 fs.
Estimation of the free energy of binding by MM/PBSA
In the above expressions Hbond, Hangle, and Hdihedral are the contributions to internal energy (Hint) obtained from the components of potential energy of the force field. The energy terms HvdW and Helect are van der Waals and electrostatic interaction energy, respectively. Helect was computed using the coulomb module of the APBS software . The HvdW energies were computed fully, i.e., without either periodic boundary conditions or cutoff using GROMACS. The free energy contribution from solvation (Gsolv) is estimated from the polar (Gpolar) and apolar (Gapolar) contributions to the solvation. The energy terms are averaged over 1001 equally spaced snapshots extracted from the last 20 ns of the molecular dynamics trajectory. VMD version 1.9 was used for visualization and for the hydrogen bond analysis .
The electrostatic component of the solvation free energy Gpolar, resulting from the Poisson-Boltzmann equation, was calculated with the program APBS. In this study the PARSE parameters were used . The interior relative dielectric constants of the protein and the solvent dielectric were set to 2 and 78.54, respectively. The van der Waals surface was used for the dielectric boundary. 225 grid points in each direction and a grid spacing of 0.5 Å were used for all calculations. No counterions were included for the calculation. The non-polar contributions to solvation were estimated from the solvent accessible surface area (SASA), , where γ = 0.0227 kJ mol-1Å-2, b = 3.85112 kJ mol-1. The SASA of the solute molecules were calculated using APBS. The objective of the study was to compare the free energies of the different stoichiometric complex. Since estimating entropy contribution to binding in a protein-protein complex is a challenging task, especially for a protein of this size we have ignored entropy contributions to free energy.
Elastic Network Model (ENM)
Coarse-grained elastic network models (ENM) have gained enormous attention in the past decade to study the intrinsic motions of a protein . Two commonly used ENM models are the Gaussian network model (GNM)  and the anisotropic network model (ANM) . In this work we used an ANM to extract the functionally relevant motions exhibited by the protein. In a conventional ANM analysis, only the Cα atoms are considered and connections between them are defined based on a cutoff. In this study we have incorporated few extensions to this general approach. (a) In addition to the Cα atoms the side chains of the residues were also included in this model at a coarse-grained level. Hence, for every residue (except for glycine) two nodes have been defined, one at the Cα and the other at the center of mass of the heavy atoms of the side chain. A similar strategy has been previously adopted on the chaperonin GroEL . For the residues ASP, ASN, ARG, LYS, GLN, and GLU, which have their interaction center primarily at the terminus of the side chain, we used the Cγ, Cγ, Cζ, Nζ, Cδ, and Cδ positions, respectively, instead of their side chain center. (b) Several types of force constants have been assigned between atoms depending on their bonding criteria or distance. A side chain node is attached to its Cα with a force constant of 10 kcal mol-1 Å-2. For atoms that are within 0.4 nm, between 0.4 and 0.8 nm, and between 0.8 and 1.2 nm, a force constant of 3, 2, and 1 kcal mol-1 Å-2, respectively, was assigned. Interactions between atoms further than 1.2 nm apart were ignored. (c) The receptors in this study possess several disulfide bonds, which act as major forces that render stability to the receptor. Hence, including these interactions in ENMs was considered imperative. To identify these disulfide bridges we used the DSSP program . A force constant of 10 kcal mol-1 Å-2 was assigned between the side chain nodes of the two cysteine residues to mimic the disulfide bridge. (d) Secondary structural information was included by raising the force constant between Cα atoms in the backbone of α-helices and β-sheets to 6 kcal mol-1 Å-2. Structural elements were identified using the program STRIDE . We calculated the Hessian matrix with the 'pdbmat' program  and the matrix was diagonalized using Prody  to calculate the eigenvalues and eigenvectors and to perform further analysis related to ENM.
Prof. Peter Scheurich and Prof. Frank Allgöwer, as well as Jan Hasenauer are acknowledged for helpful discussions. The authors thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart.
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