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.
To ascertain the extent of deviation of the structure from their initial crystallographic conformation and to demonstrate proper equilibration, the time-dependence of the root-mean square deviation (RMSD) of Cα atoms was calculated with reference to the starting X-ray structure, see Figure . Since the terminal residues belonging to CRD3 and CRD4 domains, residues 134–153 of TNFR1, show high temperature factors and fluctuate readily in MD simulations, we omitted them in the RMSD calculations. All systems but mTNFR1 are well equilibrated within the simulation time scale. LT-(TNFR1)1 exhibits only minimal conformational transformation from its starting structure since its RMSD remains within 0.3 nm indicating it to be the most rigid among the studied protein-protein complexes. LT-(TNFR1)2 also follows a similar trend as exhibited by LT-(TNFR1)1. However, in the last 10 ns of the simulation there seems to be a slight increase in its RMSD value. On the other hand the LT-(TNFR1)3 complex exhibits a higher RMSD value and seems to equilibrate around 0.4 nm. The monomeric mTNFR1 displays a higher RMSD, which is expected since it is simulated in an unbound form and lacks any kind of external stabilization. The dimeric complex (TNFR1)2 also deviates from its initial structure by 0.35 nm, similar to LT-(TNFR1)3, and equilibrates after about 10 ns of simulation.
The root mean square deviation (RMSD) of the Cα atoms of the receptor as a function of time.
To assess the dynamics associated with the receptor in ligand-unbound form, in dimeric form, and as complexes of different stoichiometries, the root-mean-square fluctuations (RMSF) of the Cα-atoms of the receptors were calculated, see Figure . The mTNFR1, as expected, exhibits relatively high peaks in the RMSF plots along its whole length, indicating its strongly fluctuating character. The dimer (TNFR1)2
is well stabilized by comparison, especially the residues in the CRD1 and CRD4 domains since these are the regions the two receptors interact with one another. Interestingly there seems to be immense stabilization even of the CRD2 and CRD3 domains in (TNFR1)2
. Hence, the interaction in the CRD1 and CRD4 domains restrain the fluctuations of the CRD2 and CRD3 domains. These CRD2 and CRD3 domains of the receptor are the regions that are primarily involved in ligand binding in the LT-(TNFR1)n
complexes. Residues 77–81 and 107–114 of TNFR1 reside at the cleft formed by the interface between the LT monomers [12
]. The fluctuations in these binding regions are, hence, well constrained in the dimer as well as in the complex compared to the monomeric form of the receptor. Further down we will discuss that Trp-107TNFR1
is one of the well-buried residues in the complex being sandwiched between the interface of the two monomeric ligands. The CRD4 had been claimed to be highly disordered [12
] correspondingly huge fluctuations are observed for this region in our study. The later part of CRD3 and the whole CRD4 domain showed high levels of mobility in mTNFR1, LT-(TNFR1)2
, but notably not in LT-(TNFR1)1
. In fact in LT-(TNFR1)1
the residues in these regions display a similar profile to that observed for (TNFR1)2
. Hence the immense stability attained by the CRD4 domains in LT-(TNFR1)1
indicates a different behavior compared to the LT-(TNFR1)2
complexes which will be rationalized in more detail below.
The root mean square fluctuations (RMSF) of the Cα atoms of the receptor. Regions with low RMSF indicate high stability.
Naismith and Sprang [32
] 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
, 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 qualitative measure of the underlying strength of interaction between two biomolecules can be gained by measuring the buried surface area of the complex. The buried-surface area of individual residues of the receptors in different complexes averaged over the trajectory is shown in Figure . In the LT-(TNFR1)n complexes, residues in the CRD2 and CRD3 domains are buried while in the (TNFR1)2 complex residues in CRD1 and CRD4 are buried. It is not surprising that residues which are highly buried in the complexes with LT (in the domains CRD2 and CRD3) are highly exposed in (TNFR1)2. This gives valid proof in support of the argument that in the parallel form of the dimer the binding site domains are exposed to the solvent and can bind the approaching LT-ligands without any major structural change. In all LT-(TNFR1)n complexes residues Leu71, Cys72, Arg77, Lys78, and Glu79 of CRD2 are strongly buried. In CRD3, residue Trp107TNFR1 is well buried and lies almost exactly at the interface between the chains of LT (at the membrane-proximal part of LT). In the LT-(TNFR1)1 complex two residues in CRD4, namely Phe144TNFR1 and Arg146TNFR1, are well buried indicating a strong interaction between the CRD4 domain and the ligand. In the (TNFR1)2 dimer, residues Gln17TNFR1, Lys19TNFR1, and His34TNFR1 in the CRD1 region and residue Phe144TNFR1 of the CRD4 domain are well buried.
The buried surface area of the individual residues of the receptor in various complexes.
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 [33
]. 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
, 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
complex crystallizes as a trimer and hence one would expect it to be the most stable complex with minimal fluctuating character. On the contrary, the analysis performed so far highlights the immense stability of LT-(TNFR1)1
among the LT-(TNFR1)n
. This is interesting considering one recent investigation of the stability of various stoichiometric complexes of TRAIL-(DR5)n
concluded the TRAIL-(DR5)1
complex to be more stable than the corresponding dimeric and trimeric complexes [31
]. Figure shows the starting and the final structures from MD of LT-(TNFR1)1
. It was found that the CRD4 domain of TNFR1 in LT-(TNFR1)1
bends towards the homotrimeric LT to form a stable interaction with its residues.
Starting structure (left) and final simulated structure (right) of LT-(TNFR1)1. LT is shown in red, the receptor CRDs 1 to 4 are colored yellow, green, purple, and blue. The figure demonstrates the attachment of the CRD4 domain to the LT.
In order to investigate the residues that contribute to the interaction between CRD3 and CRD4 of TNFR1 and LT in the LT-(TNFR1)1 complex, the hydrogen bonding interactions between these were investigated. In about 94% of the snapshots a hydrogen bond between the side chain of Tyr86LT and that of Glu147TNFR1 was observed. Residue Arg146TNFR1 was also observed to interact with several residues of LT through hydrogen bonding interactions: Leu125LT-chA, Gln126LT-chA, Glu127LT-chA, and Tyr122LT-chB where chA corresponds to one chain of LT, chB to the neighboring chain, see Figure . Considering that Arg146TNFR1 and Glu147TNFR1 are stronger buried in LT-(TNFR1)1 than in LT-(TNFR1)2 and LT-(TNFR1)3 it is not surprising that these are involved in strong hydrogen bonding with the ligand. These two hydrogen bonding interactions remain fairly strong throughout the simulation and contribute significantly to the binding of CRD4 of TNFR1 to LT in the LT-(TNFR1)1 complex. The bending of the CRD4 domain is further stabilized by a strong internal hydrogen bond between Glu147TNFR1 and the backbone of Asn116TNFR1. Initially the distance between the two atoms Glu147-CδTNFR1 and the peptide-H of Asn116TNFR1 is 1.4 nm. It decreases to about 0.25 nm during the last 20 ns of the simulation indicating the strong interaction between the two residues. Arg146TNFR1 and Glu127LT-chA form a salt bridge. The distance between the two side-chain atoms Cδ and Cζ, of Arg146TNFR1 and Glu127LT-chA are plotted as function of time in Figure . It is obvious that the distances remain within the range of a strong salt-bridge.
The major interaction sites of CRD4 with LT in the complex LT-(TNFR1)1. Residues of TNFR1 (left, backbone colored orange) are drawn as sticks, while those of LT (right, backbone colored blue) are represented as ball-stick models.
Distance between the peptide-H of Asn116TNFR1and Cδ of Glu147TNFR1as well as Cδ of Glu127LTand Cζ of Arg146TNFR1during the simulation. These hydrogen bonds stabilize the CRD4 at the membrane-proximal part of LT.
The CRD4 domain has been observed to fluctuate readily in these complexes. Hence, to ascertain the motion of this domain, the distance between the center of LT and the center of the CRD4 domains were monitored for the whole 20 ns of one set of production runs (15–35 ns) (Figure ). The distances between them fluctuates strongly in all complexes except LT-(TNFR1)1
. In LT-(TNFR1)2
as well as LT-(TNFR1)3
the domains move independently from each other: while one CRD4 of LT-(TNFR1)2
remains at about 2.8 nm distance from LT, the other moves significantly further away, to about 3.5 nm. The same holds true for LT-(TNFR1)3
. In LT-(TNFR1)1
, however, the CRD4 is immobilized close to LT, at distances between 2.0 and 1.7 nm. To confirm this result, we ran additional simulations. Within 35 ns each, the CRD4 domains did not attach to LT in LT-(TNFR1)2
. In two of the three additional simulations carried out for LT-(TNFR1)1
, the CRD4 domain attached to LT after 14 ns and 24 ns, respectively, confirming our proposal of the attachment of CRD4 to LT. Only in one simulation did it not attach within this simulation time. We assume that longer simulations would clarify this issue. This explains the distinct role of LT-(TNFR1)1
compared to the other complexes already found by the alternative analysis tools. The inability of the CRD4 domain to bind to LT in LT-(TNFR1)2
can be explained on the basis of their structure. The three LT monomers are arranged in a triangular cone with their narrow ends pointing at the membrane. The width of LT at the top site is about 50 Å while it is only 30 Å at the membrane proximal region [12
]. Hence the space at the bottom of the LT is quite small (Figure ). The space on the LT is probably only adequate to accommodate one CRD4 domain. In LT-(TNFR1)2
it may be that the competition between the CRD4 domains prevents either of them to bind.
Figure 8 The distance between the CRD4 domain of the receptor (TNFR1) and the center of LT during the simulation in the three different complex stoichiometries. It can clearly be seen that CRD4 attaches to LT in LT-(TNFR1)1 while it remains unbound in LT-(TNFR1) (more ...)
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
]. 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
]. 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.
The low-frequency normal modes obtained from ENMs have been shown to be functionally relevant motions of the protein in many cases [38
]. As a first method of validation of the results obtained from the ENM, one usually compares the fluctuating nature of the individual residues as ascertained from the ENM to that obtained from the experimental temperature factors (Figure ). Here, we additionally compare it to the fluctuations found in MD. The rapidly fluctuating segments of the protein as obtained from the MD simulation agree well with those obtained from the ENM. The fluctuations appearing at the CRD1 domains are reduced for (TNFR1)2
compared to LT-(TNFR1)3
. This is expected since CRD1 is the main interaction site in (TNFR1)2
. The CRD4 domains on the other hand fluctuate in a similar fashion for the (TNFR1)2
complexes in experiment (temperature factors in the crystal structures). But results from ENM paint a completely different picture. The CRD4 domains are the most fluctuating domains of the receptors, corroborating with our results from MD simulations (Figure ). One major difference between MD and ENM, though, is that the CRD4 domain of the receptor in LT-(TNFR1)1
, which is the least fluctuating LT-(TNFR1)n
complex in the MD simulations, also displays huge fluctuating behavior in ENM. This can clearly be rationalized by the fact that CRD4 is immobilized and bound to LT in LT-(TNFR1)1
in the MD simulations while it is unbound in the ENM at the crystal structure.
Figure 9 Comparison of the B factors (temperature factors) as obtained from crystallographic structures (1TNR[]and 1NCF[], respectively) to those obtained from an ENM.
Having compared the fluctuations of the receptors in different states, we now compare the conformational changes of the receptor in different states as estimated from the ENM normal modes. This was accomplished by comparing the overlap of the first 10 vibrational eigenmodes (Figure ). The term “overlap” here refers to the dot product of the two eigenvectors. A high degree of overlap of the eigenmodes indicates that both proteins explore a similar conformational space. The RMSD between residues 15–150 of the receptors in LT-(TNFR1)3 (PDB: 1TNR) and (TNFR1)2 (PDB ID: 1NCF) is just 1.65 Å, indicating little geometrical difference between these states. Hence one is tempted to speculate that the receptors in the two forms would exhibit similar conformational changes. In contradiction, comparison of the eigenmodes shows that the global fluctuations of the receptors are well distinguished in the two forms (Figure ). Also between the two ligand-unbound forms of the receptor, namely, mTNFR1 and (TNFR1)2, no significant overlap of the eigenmodes was found. This shows that the global fluctuations of the receptors do not resemble each other in the monomeric and the dimeric forms. The corresponding overlap between LT-(TNFR1)3 and mTNFR1 is also weak, indicating the ligand-binding to cause completely different receptor-motions than in both ligand-unbound forms. Comparison of the overlap between LT-(TNFR1)3 and LT-(TNFR1)2,1 indicates how ligand-binding transforms the receptor to a diverse range of conformational transformations. The binding of just one receptor to the ligand improves the overlap observed for the eigenmodes of the receptors to LT-(TNFR1)3. The binding of the second receptor improves this correlation even more. A very high degree of overlap is observed for the first two eigenmodes of LT-(TNFR1)2 and LT-(TNFR1)3. The third eigenmode from LT-(TNFR1)3 overlaps with the the forth eigenmode of LT-(TNFR1)2 and vice versa (Figure ). Since the lowest-frequency normal modes are the most significant functional motions of the protein, the high degree of overlap between LT-(TNFR1)2 and LT-(TNFR1)3 shows that the dominant motions of the receptors are similar in these two complexes. These results give an overall perspective of the difference in receptor’s functional motions upon ligand binding.
Overlap between the first 10 eigenmodes of the receptor in different complexed or monomeric states. These data are obtained from an ENM. Strong overlap indicates similar receptor dynamics.
A correlation analysis clearly shows the difference between the two ligand-free forms of the receptors (Figure ). The anti-correlated regions observed for mTNFR1 are somewhat diminished for (TNFR1)2. For example the anti-correlations observed between residues 15–34, 37–54 and 75–114 are significantly reduced. Also the anti-correlated motions between the residues 15–34 and 124–138, visible for mTNFR1, are completely lost in (TNFR1)2. Another interesting aspect to be extracted from this comparison is that the strong correlations observed within residues 74–114 in mTNFR1 is lost in (TNFR1)2. However, the correlations within residues 54–114 in (TNFR1)2 resemble those of the LT-(TNFR1)n complexes. Thus, the dynamic motions of the CRD2 and CRD3 domains, which form the ligand-binding domains of TNFR1, fluctuate in a similar fashion in (TNFR1)2 and LT-(TNFR1)n. This suggests that the CRD2 and CRD3 domains in (TNFR1)2 are optimally aligned and fluctuate in manner that the ligands can easily identify them. From these conclusions it can be speculated that ligands prefer to bind to the receptor in its dimeric form rather than to monomers. The receptors in their ligand-bound forms resemble each other very closely except that the anti-correlations of the CRD2,3 domains with CRD3,4 are somewhat diminished in LT-(TNFR1)2 and more so in LT-(TNFR1)3. Also in ligand-bound forms of the receptor, a strong correlation seems to prevail within the CRD1 domain as well as within the second-half of CRD3 and CRD4 domains. Such correlations are more strongly observed for LT-(TNFR1)2,3 than for LT-(TNFR1)1.
The correlation matrix calculated from the 25 lowest-frequency normal modes of the ENM. Top row: mTNFR1, (TNFR1)2 , LT-(TNFR1)1. Bottom row: LT-(TNFR1)2, LT-(TNFR1)3.
The most dominant functional motion as extracted from the lowest-frequency normal mode of the elastic network model (ENM) is shown as arrows in Figure . The lengths of the arrows are proportional to the magnitude of the fluctuation of the residues. The vectors are scaled to result in an RMSD of 2 Å between the elongations in both directions. In all complexes the strongest fluctuations are exhibited by residues of the CRD4 domain. The ligand hardly contributes to this motion. The first mode extracted from LT-(TNFR1)1 is predominantly a hinge-bending motion. The overall motion drives the CRD4 domain towards the membrane-proximal center of LT. Notably, the CRD4 domain exhibits a similar bending motion in the MD. Hence, the results from the ENM are in good agreement with the results obtained from MD. The dominant motions of LT-(TNFR1)2 and LT-(TNFR1)3 are also concentrated at the CRD4 domains, but their direction is perpendicular to the direction of motion observed for CRD4 in the LT-(TNFR1)1 complex.
Figure 12 The dominant functional motions as extracted from the lowest frequency normal modes obtained from the ENM. Only the Cα atoms of the receptors (purple) and LT (yellow) are shown. The arrows indicate the dominant motion. They are colored in the (more ...)
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 [39
]. 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. [31
] 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 [30
]. 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.
The long-range electrostatic interactions hugely influence protein-protein complexes. Hence accurate estimation of these influences is eminent in any free energy method. One usually estimates the accuracy of these calculations based on the trade-off between the gas-phase electrostatic interaction (
) and the polar contribution to solvation (
) obtained from Poissan-Boltzmann (PB) calculations. The total electrostatic interaction (
) is a compromise between the electrostatic energy between the individual protein in the complex and the cost associated with desolvation of the respective proteins. Hence a positive value indicates the cost of desolvation is higher than the electrostatic interaction between the binding partners to form the complex. In all the complexes considered in the present study, it seems that electrostatic interaction disfavors protein-protein binding. For example, from the SITA an electrostatic interaction of −668.7 kJ/mol in (TNFR1)2
is lost due to a higher polar solvation energy of 709.3 kJ/mol, resulting in an unfavorable total electrostatic interaction of 40.6 kJ/mol, see Table . In the same manner an unfavorable total electrostatic interaction (
) of 24.4 kJ/mol, 22.4 kJ/mol, and 17.6 kJ/mol is found for LT-(TNFR1)1
, and LT-(TNFR1)3
, respectively. Hence, from SITA one can conclude that the polar contributions to the free energy only disfavors interaction between the two binding partners. The apolar contribution to solvation free energy (
) on the other hand is favorable across these complexes. The van der Waals interaction between the proteins (
) is very high in all these complexes indicating that such non-polar interactions contribute majorly to complex stability. Therefore it is safe to conclude that the total non-polar components contribute favorably towards binding free energies while the overall electrostatic term disfavors complex formation.
Binding energies (in kJ/mol) obtained from single-trajectory analysis
Results from separate-trajectory simulations (SETA)
We have additionally used the SETA method to calculate the total free energy of binding in these complexes. There are two ways one can extract the energy components of TNFR1, either from TNFR1 or from (TNFR1)2. (TNFR1)2 was chosen since crystal structures suggest receptors to exist as dimers in the absence of ligand. Table shows the components of the free energy obtained from separate-trajectory simulations. In all complex formations except that of LT-(TNFR1)2 the internal energy obtained from the force field is negative, i.e., binding of two receptors to LT is favorable while binding of one and three receptors to LT is not. This can be explained by the unfavorable conformational strain caused by the bending of the CRD4 of TNFR1 towards LT in LT-(TNFR1)1. In LT-(TNFR1)2, the binding of the second receptor relaxes the strain since both CRD4 domains of TNFR1 are unbound again.
Binding energies (in kJ/mol) obtained from SETA
The free energy of binding,
, estimated from SETA for LT-(TNFR1)1
, 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 (
), 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
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.