Nonpolar cavity

First we verify that the water molecules in the nonpolar cavity are properly equilibrated. By comparing the distributions of insertion energies from simulation structures with

*N* water molecules in the cavity to the distributions of removal energies with

*N*+1 water molecules, we can directly test whether the different simulations are consistent with each other, and whether the energies are indeed Boltzmann distributed. shows the distributions of water insertion and removal energies for

*N*=0 and 1, respectively. We find that

*p*_{rem}_{,1}(Δ

*U*) agrees very well with

, as required by

eq 2, over virtually the entire range of overlap (with

*p*_{ins}_{,0}(Δ

*U*) varying over more than six orders of magnitude). We find similar agreement for the pairs of insertion and removal distributions. As shown in , the logarithm of the ratio of insertion and removal probability distributions is a linear function of the energy Δ

*U*, with the required slope of β. From the intercept of the linear fits with fixed slope β, we extract the excess chemical potentials

entering into

eq 1 for the ratio of the occupancy probabilities.

By combining the results for *N*=0, 1, 2, 3, and 4, followed by normalization, we obtain the occupancy probabilities *P*(*N*). From the *P*(*N*), we calculate the water transfer free energies. shows the resulting free energies Δ*A*_{n} of transferring *N* water molecules from bulk into the nonpolar cavity of IL-1β, defined as Δ*A*_{N} = −*k*_{B}T ln *P*(*N*) / *P*(0), ignoring a small *pV* term.^{41} The results for the different combinations of protein force fields and water models are consistent, with only modest force-field associated differences in the transfer free energy. All simulations show that filling the nonpolar cavity of IL-1β with water is thermodynamically unfavorable. The highest free energy penalty of about 5-6 *k*_{B}T is encountered by the first water molecule entering the cavity. This penalty is large despite the fact that the first water molecule is relatively free to move around in the cavity and can even form transient interactions with a structural water molecule that contribute to the low-energy shoulders in the distribution shown in . Subsequent water molecules face lower costs of about 2 *k*_{B}T each, reflecting the fact that they can form hydrogen bonds with the water molecules already in the cavity. Overall, transferring water molecules into the nonpolar cavity is uphill in free energy over the entire range of occupancies considered. We have also performed simulations with *N*=5, and found this filling state to be unstable. Therefore we conclude that according to all simulation models, the nonpolar cavity of IL-1β is predicted to be empty.

Is this due to unfavorable energy or entropy of transfer into the nonpolar cavity? To answer this question we estimate the energy of transfer from the difference in the energy required to remove all *N* water molecules collectively from the cavity (*N*=1 to 4) and the energy of *N* water molecules in the bulk phase:^{21}

The first average is over the removal energy of the entire *N*-water cluster from the cavity, and the second average is over the removal energy of a single water molecule from the bulk phase. This relation is correct in the absence of reorganization energies, which is a reasonable assumption for nonpolar cavities. From the difference between the transfer energy and the transfer free energy we obtain the transfer entropy, Δ*S*_{N} = (Δ*U*_{N} − Δ*A*_{N}) / *T*. Our results are displayed in which show that although filling with 1 and 2 water molecules is energetically unfavorable, it becomes energetically favorable when the occupancy *N* is increased to 3 and 4, likely due to increased hydrogen bonding between the water molecules in the cavity.^{28} The favorable energy at the higher cavity occupancy is outweighed by the unfavorable entropy and overall the free energy of transfer is always positive, as shown in . A four-molecule water cluster is actually observed in our simulations () and by others, but in our study, unlike that of Somani et al.,^{22} this cluster is thermodynamically unstable at room temperature. A structurally similar four-molecule water cluster in the larger nonpolar cavity of tetrabrachion is also unstable at 298 K (see Table 2 of ^{ref 21}), but much less so, by about 1 *k*_{B}T instead of >10 *k*_{B}T here. Although the energies of transfer per water molecule are comparable (about -5.7 vs -5.2 kJ/mol here), the corresponding entropy of transfer into the nonpolar cavity within tetrabrachion is less negative (-2.31 *k*_{B} vs. -5.15 *k*_{B}) because the greater cavity size results in larger translational and rotational entropy of the water cluster. This comparison illuminates the entropic influence of cavity size on the free energy of water transfer into a nonpolar cavity and illustrates the fine balance between energy and entropy in determining the conditions of thermodynamic stability of water in nonpolar cavities.

Comparison to X-ray crystallography

Based on a careful examination of previous crystal structures, and a refinement of a new crystal structure with experimentally determined phases^{20} Matthews and collaborators confirmed their earlier conclusion of an empty cavity,^{42} and also provided a rationale for the apparent electron density seen in an earlier study that used low-resolution data in the refinement.^{19} Our simulation finding of an empty nonpolar cavity is fully consistent with the crystallographic studies of Matthews and collaborators.^{6}^{,}^{20}

To extend this quantitative comparison to experiment, and to check that the simulation methodology is correct, we also studied the polar cavities of IL-1β. The crystal structure determined by Quillin et al.^{20} (PDB code 2NVH) identifies four polar cavities, designated as 1 to 4, respectively ( and Table 4 of ^{ref 20}), two cavities containing two water molecules each, and two containing one water molecule. As shown in , our calculated transfer free energies of water into the polar cavities of the NMR structure are consistent with occupancies determined in the X-ray studies of Quillin et al.,^{20} except that in cavity 4, where the most probable water occupancy number varies between one and two in two calculations based on simulations of 5 ns length, possibly due to small changes in protein conformation. The energies of transfer per water molecule (-22 to -90 kJ/mol in the different polar cavities, ignoring possible reorganization energies associated with changes in the protein structure in different hydration states) are an order of magnitude larger than they are for transfer into the nonpolar cavity, and so are the entropies of transfer (Δ*S/Nk*_{B} = -7 to -29), but here the compensation between energy and entropy works in favor of filling by one or two water molecules. In our simulations, the predicted hydration of both the polar and nonpolar cavities in IL-1β is thus fully consistent with the X-ray crystal data of Quillin et al.^{20}

Comparison to NMR

To compare our simulations to the NMR hydration study,

^{18} we have calculated effective distances between the methyl protons of the residues lining the nonpolar cavity, and the water protons, including water in the polar cavities and in the surrounding solvent. Even with the nonpolar cavity being empty, we find that ~90% of the effective proton-proton distances

*r*_{eff,}_{i} defined in

eq 3 are at or below 5 Å (). We have performed a similar analysis also on five different crystal structures with varying degrees of hydration. With proton positions not resolved, we used the distances between the methyl carbon and water oxygen atoms. For the MD simulation, the results obtained under this assumption are very similar to those for actual proton-proton distances (). We find that the effective distances to water in the crystal largely follow those seen in the simulations, and that ~60% of these distances are at or below 5 Å. For the NMR structure,

^{24} the effective distances are considerably larger, reflecting the fact that it contains only six water molecules buried in polar cavities, but no surface water. These results suggest that most, if not all, of the observed NOEs

^{18} from water protons to methyl groups lining the cavity signals can be explained by interactions with nearby buried and surface water molecules, and likewise a significant fraction of the ROEs, which have a shorter range of less than ~4 Å. We note, however, that if we use the ensembles with

*N*=1 or 2 water molecules in the nonpolar cavity, instead of the structures with an empty cavity, the effective distance

*r*_{eff,}_{i} between water and the methyl groups

*i* lining the cavity, as defined in

eq 3, drops by about 0.5-1 Å, corresponding to a ~4-fold increase in predicted NOE intensities.

Matthews et al.^{42} pointed out that the methyl carbons of Val-58 showed interactions with water, despite being buried. This was explained by Ernst et al.^{27} by flexibility in the protein surface allowing water access. Indeed, in our simulations we find effective proton-proton distances of about 4 and 5 Å between the Val-58 methyl group and water, again consistent with the observed NMR water-methyl NOEs.^{18}

Comparison to other simulation studies

The thermodynamics of transfer of water from the bulk phase into the nonpolar cavity of IL-1β has been studied by Zhang and Hermans^{8} and by Somani et al.^{22} and, more recently, by Oikawa and Yonetani^{23} using molecular dynamics simulations. The methods used are different from our scheme based on the evaluation of the semigrand partition function for cavity water. From calculations of the energy of transfer, Zhang and Hermans^{8} concluded that the nonpolar cavity of IL-1β is insufficiently polar to contain water. Through studies of buried water in several other proteins, they suggest that a minimum of about -50 kJ/mol for the interaction energy of cavity water with the protein is necessary to drive a water molecule into a cavity, in rough agreement with our calculations for polar cavities. They maintain that the hydrogen bond network between water molecules in the cavity cannot compete with the network in bulk water, and that the contribution to the free energy of transfer from this source is small. In contrast, we found here and previously for tetrabrachion^{26} that water-water interactions can be important factors driving the filling of large nonpolar cavities. Zhang and Hermans^{8} also assume that the entropy of transfer is small, even for polar cavities, which is not supported by our rough estimates for the polar cavities of IL-1β, and the free energy of transfer of water into the cavity is therefore dominated by the corresponding energy. Following these arguments, all nonpolar cavities in proteins, including IL-1β, would be empty, in contrast to the experimental^{43} and computational findings for tetrabrachion.^{26}

In contrast to Zhang and Hermans

^{8}, Somani et al.

^{22} predict that the cavity of IL-1β is filled by water. Somani et al.

^{22} studied the transfer of one to four water molecules (N= 1 to 4) into the nonpolar cavity of the 9ILB

^{19} structure of IL-1β, and found that up to 3 water molecules are thermodynamically unstable in the cavity, but that the addition of a fourth shifts the transfer free energy from unfavorable (positive Δ

*A*_{n}) states to a favorable one (negative Δ

*A*_{n}). Their analysis is based on the assumption that the binding energies of cavity water are Gaussian. We find that in the nonpolar cavity of IL-1β that a single Gaussian is a rather poor approximation (), in particular in the high-energy tail most relevant for the transfer free energy.

^{32} For a more accurate estimate, a multistate Gaussian description would be needed.

^{44} We also find some differences between the binding-energy distributions displayed in Figure 8 of

^{ref 22} and ours, especially for

*N*=1 (single occupancy). Somani et al.

^{22} calculated an “excess chemical potential” μ

^{ex} of cavity water from the average binding energy

*u*_{N} of a single water molecule in a cavity with

*N* water molecules, and the variance

of

*u*,

. The first term in this expression is an ad-hoc correction factor corresponding to the free energy of transferring a hard sphere from an ideal gas into water, with a diameter of 2.7 Å, roughly matching the size of a water molecule. For the transfer free energy Δ

*A*_{n}, in our notation, they then use

. This expression for the transfer free energy of cavity water ignores the fact that the transfer is sequential, and that the Gaussian formula should thus be applied for the steps from 0 to 1, from 1 to 2, etc.; and it also assumes a constant “entropic” term of 16.7 kJ/mol that entirely ignores the size of the cavity. Whereas a Gaussian approximation can under some circumstances be used to approximate the Boltzmann average in

eq 1 and

2, −

*kT* ln

exp[−

*β*(

*U*_{N+1}−

*U*_{N})]

_{N} ≈

*U*_{N+1} −

*U*_{N} −

*βσ*^{2}/2, it has to be applied sequentially (i.e., for N=0→1, 1→2, etc.) and with the volume-dependent prefactor given in

eq 1. We therefore conclude that the finding of a 4-water filled cavity in IL-1β is likely based on incorrect assumptions.

Very recently, Oikawa and Yonetani^{23} evaluated the transfer free energy of TIP3P water into the nonpolar cavity of IL-1β, represented by both rigid and unrestrained models for the 9ILB^{19} structure of this protein. By using the method of Roux et al.^{9} and thermodynamic integration they calculated the free energy difference between a cavity filled with *N* water molecules and an empty one. A harmonic restraint on cavity water to prevent expulsion from the cavity was used, similar to the flat-bottomed potential used in some of our simulations. The free energies of transfer are positive (unfavorable) and in qualitative agreement with our calculations for the unrestrained 6I1B structure of the protein. Flexibility in the protein exposes polar groups and embedded water near the cavity walls, thereby increasing cavity polarity and transient hydrogen bond formation. Consistent with this interpretation, we also obtain conformations with low water removal energies, even in the singly occupied nonpolar cavity ( and ). These interactions reduce the free energies of transfer, but the differences are small for *N*=1. At higher occupancies (*N*≥2), local expansion of the cavity is a likely additional effect that explains the more favorable transfer free energies compared to a rigid protein. But despite these contributions, their main conclusion is that the transfer free energy of water into the nonpolar cavity of IL-1β is unfavorable for all reported occupancies from *N*=1 to 4. Oikawa and Yonetani do not report data for the transfer of one and two water molecules (*N*=1 and 2) into the cavity of their unrestrained model of the 9ILB^{19} structure of IL-1β since the water molecules apparently left the cavity during the MD simulations despite the restraining potential employed, unlike cavity water in the rigid model (see Figure 4 of ^{ref 23}, but note a possible inconsistency in the caption, with the figure suggesting that water escapes from *N*=2 and 3, not *N*=3 and 4).We have obtained data for all occupancies from *N*=1 to 4, by either returning ejected water back into the cavity followed by re-equilibration, or by applying a restraining potential at the edges, as stated earlier. The free energies of transfer of one and four water molecules into the nonpolar cavity of the unrestrained model (9ILB^{19}; see Figure 4 of ^{ref 23}) are ~16.7 kJ/mol (~6.7 *k*_{B}T) and ~46 kJ/mol (~18.4 *k*_{B}T) respectively, and are larger by about 2 *k*_{B}T per cavity water molecule than our calculations for the unrestrained 6I1B structure (see ). These differences are more likely due to differences in theoretical methodology rather than differences between the crystal (9ILB^{19}) and NMR (6I1B^{24}) structures of IL-1β. In particular, the use of thermodynamic integration to “grow in” water makes it difficult for a gradually inserted water molecule to exchange position with already present water molecules, and sample the entire cavity. As a result, one would expect a bias toward more unfavorable transfer free energies, which would explain the differences to our work. Nevertheless, the main conclusion by Oikawa and Yonetani^{23} of an empty cavity in IL-1β is fully consistent with our findings.

The thermodynamic instability of water in the nonpolar cavity in IL-1β can be traced to its relatively small volume. Already for the transfer of a single water molecule into the empty cavity, we estimate a slightly unfavorable (negative) entropy (). Likely reasons are, first, that the water-accessible volume of the cavity is only about twice the partial molar volume of bulk water; and, second, that the water molecule in the cavity can make transient interactions with one of the structural water molecules, as evidenced by the low-energy shoulders in the removal energy distribution Both effects are enhanced as additional water molecules form hydrogen-bonded clusters that fill up the small cavity volume. The observed decrease in the relative entropy of transfer () with occupancy is thus the result of a reduction in translational entropy, as the remaining available volume shrinks with increasing occupancy, and of a reduction in both translational and rotational entropy, as water molecules become increasingly localized through hydrogen bonding and cluster formation in the densely filled cavity. A similar decrease in the entropy was observed for the nonpolar cavity in tetrabrachion, where we also tested the approximation

eq 4 used to calculate the transfer energies and entropies.

^{21} The somewhat more positive (favorable) entropies in tetrabrachion are a result of its significantly larger cavity size compared to IL-1β.

Dunitz^{45} has provided rough bounds on the entropy of water molecules filling a cavity, -3.5 < Δ*s*_{n}/Nk_{B} < 0. However, these numbers are based on heuristic arguments and should be taken with care. For one, the estimated bounds apply to cavities that are actually filled at equilibrium under ambient conditions, whereas here we probe the transfer into a cavity that turns out to be mostly empty at equilibrium based on our calculations. Furthermore, the entropy of transfer will depend sensitively on a number of factors, in particular the size, polarity, and filling state of the cavity. Enthalpy-entropy compensation arises from trade-offs between favorable energies of interaction with the protein, or with water molecules already in the cavity, and associated losses in entropy. Consistent with our simulation results for tetrabrachion^{21} and for IL-1β, one would in particular expect the entropy per water molecule to go down as additional water molecules enter the cavity and form new hydrogen bonds that restrict their translational and rotational movement.

The energies of transfer from the bulk phase into the nonpolar cavity of IL-1β are unfavorable (positive) for single and double occupancy (). With the entropies of transfer being negative, single and double occupancy states are therefore thermodynamically unfavorable. Although the energies of transfer of three and four molecules from the bulk phase into the cavity are favorable (negative), they are not large enough to surmount the unfavorable entropies of transfer. In contrast, for the nonpolar cavity of tetrabrachion,^{21} comparable energies of transfer proved sufficient to induce filling because the entropic penalty was smaller for the cavity with a significantly larger volume compared to the one in IL-1β.