Likely over-stabilization of charge-charge interactions in CHARMM36m(w): a case for a99SB-disp water

Abstract

Recent years have witnessed drastic improvement in general-purpose explicit solvent protein force fields, partially driven by the need to study intrinsically disordered proteins (IDPs). Yet, the state-of-the-art force fields such as CHARMM36m (c36m) and a99SB-disp still provide different performance in simulating disordered proteins states, where c36m has a bias towards over-compaction for large IDPs. Here, we examine the performance of c36m and a99SB-disp in describing the stabilities of a set of 46 amino acid backbone and side chain pairs in various configurations. The free energy results show that the c36m systematically predicts stronger interactions compared to a99SB-disp, by an average of 0.2 kcal/mol for nonpolar pairs, 0.6 kcal/mol for polar pairs, and 0.8 kcal/mol for salt-bridges. The most severe overstabilization in c36m is observed for charged pairs involving the Arg and Glu side chains, by up to 2.9 kcal/mol. Importantly, the systematic overstabilization of c36m is only marginally alleviated by c36mw, an ad hoc patch to c36m that increases the dispersion interactions between TIP3P hydrogens and protein atoms. Guided by free energy decomposition, we evaluated if revising the charges alone could alleviate the severe over-stabilization of salt-bridges of c36m(w) vs a99SB-disp. The results suggested that direct modification of protein-water interactions is also necessary. Towards this end, we proposed a tentative modification to c36m, referred to as c36mrb-disp, which combines modified Arg side chain charges, retuned backbone hydrogen bonding strength, and the a99SB-disp water model. The modified force field successfully reproduces the secondary structures of several intrinsically disordered peptides and proteins including (AAQAA)₃, GB1p, and p53 transactivation domain, while maintaining the stability of a set of folded proteins. This work provides a set of useful systems for benchmarking and optimizing protein force fields and highlights the importance of balancing protein-protein and protein-water electrostatic interactions for accurately describing both folded and disordered proteins.

Graphical Abstract

Introduction

Intrinsically disordered proteins (IDPs) represent a unique class of functional proteins that remain fully or partially unstructured in the unbound state under physiological conditions^1–8. Unlike the well-folded global proteins, IDPs are generally enriched with charged and hydrophilic residues but lack hydrophobic residues, rendering them structural heterogeneous and more exposed to the solvent^9–13. The inherent thermodynamic instability of IDP conformations enables them to respond sensitively to a wide range of stimuli such as post-translations and binding of ligands and other macromolecules, positioning them as central players in cellular regulation and signaling^14–19. Dysregulation, mis-signaling and aggregation of IDPs are linked to a spectrum of human diseases, including diabetes, amyloidosis, and cancer^20–25. Therefore, understanding the intricate interplay between IDP dynamics and function is critical to mechanistic understanding of IDP-related biological processes at the molecular level.

Studying the conformation and dynamics of IDP poses a significant challenge when relying on traditional ensemble-based experimental techniques like NMR, SAXS, and FRET^{18, 26, 27}. The ensemble average properties alone are insufficient to resolve detailed conformational fluctuations^28–30. This limitation impedes the resolution of the heterogeneity and transient structural features inherent to IDPs. Therefore, molecular dynamics (MD) simulations using transferable energy functions can be a particularly powerful tool for resolving the heterogeneity and transient structural features of disordered protein states. To date, many empirical protein force fields have been extensively optimized to improve the balance of describing both folded and disordered proteins³¹, including CHARMM36 (c36)³², CHARMM36m(w) (c36m and c36mw)³³, a99SB-disp³⁴, ff99SB-ILDN³⁵, and ff99SB-ILDN-TIP4P-D³⁶. This was achieved generally by balancing the protein-protein and protein-water interactions as well as backbone conformational propensities to reproduce experimental measurements on a range of local and global structural properties. For instance, c36mw re-optimizes the protein-water interactions by increasing the Lennard-Jones parameters of the water hydrogen, which not only improves the solvation free energies of amino acid sidechains but also reduces the over-compaction of IDPs without substantially destabilizing structured proteins^{33, 37}. Besides, a new water model TIP4P-D enhances the dispersion interactions of both water-water and protein-water interactions, facilitating sampling of expanded disordered states and significantly improving the descriptions of folded and unfolded states³⁶.

At present, c36m(w) and a99SB-disp represents two types of most widely used protein force fields that are capable of generating realistic conformational ensembles of a series of peptides with helix or sheet conformations, including Aβ₄₀, p53-TAD (p53 N-terminal transaction domain), RS, Histain5, PolyQ, hIAAP, and GB1p^{33, 34}. Yet, significant inconsistencies have been observed in the c36m and a99SB-disp force fields for various proteins^{38, 39}. For example, c36m force fields were found to favor the overly collapsed states for the nuclear coactivator binding domain⁴⁰, the phosphorylated disordered peptides^{41, 42}, and aggregates of ubiquitin proteins⁴³. The a99SB-disp force field was found to be limited for describing protein–protein complexes⁴⁴ and predict overly weak intermolecular interactions⁴³.

In this work, we carefully examine and compare the inconsistencies of three protein force fields (c36m, c36mw and a99SB-disp) in describing the protein-protein and protein-water interactions. First, we calculated the free energy profiles of a large set of backbone and sidechain moieties in representative configurations. The results reveal a surprising similarity in the stabilities predicted by these force fields despite vary different protein charges and water models. However, there is a modest but systematic tendency of c36m or c36mw for favoring protein-protein interactions, which is likely the origin for their tendency to generate more compact protein ensembles compared to a99SB-disp. The apparent over-stabilization is particularly severe for salt-bridges. We examine if the over-stabilization can be alleviated by increasing polarity of the charged sidechains in c36m to similar levels in a99SB-disp. The results show that increasing polarization alone is insufficient to reproduce a99SB-disp stabilities. Instead, the use of the same a99SB-disp water model can significantly reduce the difference. As such, we propose a tentative combination of modification of c36m Arg side chain charges and a99SB-disp water and optimize the backbone hydrogen bonding strengths and torsion profiles to describe the conformational properties of both folded and unfolded proteins. Our study not only provides valuable insights for optimizing and benchmarking protein force fields but also yields a useful modification to c36m for more accurate simulation of folded and unfolded proteins.

Methods

Model systems for evaluating protein-protein and protein-water interactions

A set of model compounds representing major amino acid side chains were used in the current work, together with the alanine dipeptide (alad), to represent backbone carbonyl (bco), and a modified version with one of the peptide plane deleted (alam), to represent backbone amide (b) with minimal steric hinderance (Figure S1). These model compounds were described using the c36m standard parameters for proteins. We considered the pair-wise interactions between these model compounds in various configurations designed to reflect those frequently observed in protein structures, such as hydrophobic, pi-pi stacking, backbone/sidechain hydrogen-bonds, and salt-bridge electrostatic interactions (Figure S2 & S3). Besides these protein-protein interactions, we directly evaluate the interactions of charged amino acids (Glu, Arg, and Lys) with different water models (TIP3P* and a99SB-disp), to dissect the contributions of protein-water interactions to the free energy of protein-protein interactions.

Free energy profiles, stabilities, and decomposition

The free energy profiles were calculated using free energy perturbation (FEP) for each dimer in fixed configurations in three force fields, namely, c36m(w) and a99SB-disp. During FEP, we ran multiple simulations at distances ranging from 1.2 to 13.5 Å with a step of 0.1 Å. The distance was defined using the pair of atoms listed in Figures S2 and S3. Two half step perturbations were performed around each window, and the free energy difference between two perturbation end points (F_ab) can approximated as the mean of individual reduced potential energy differences between two states (f_ab),

$$F_{ab}\approx \frac{1}{N}{\sum }_{n=1}^N{f}_{ab}\left({\mathbf{\text{x}}}_n\right),$$ (1)

where $f_{ab}=\beta \left(E_a-E_b\right)$, E_a and E_b are the potential energy of state a and b, respectively, β is the inverse of k_BT, x_n is one configuration from equilibrium sampling of the current window, and N is the number of samples collected. The overall potential of mean force (PMF) profile was then simply obtained by adding multiple states. The length of sampling was 10 ns per window for all pairs, which is adequate to achieve convergence below 0.1 kcal/mol (Figure S4). We defined the stability of one test system as the free energy difference between the first valley and the average PMF values along distances more than 11.0 Å, summarized in Figure S5. The overall free energy profile can be decomposed into contributions from protein-protein and protein-water interactions,

$$f_{ab}\left(\mathbf{\text{x}}\right)=f_{ab}^{\text{pp}}\left(\mathbf{\text{x}}\right)+f_{ab}^{\text{pw}}\left(\mathbf{\text{x}}\right),$$ (2)

where the $f_{ab}^{\text{ww}}\left(\mathbf{\text{x}}\right)$ is zero because the same water configuration is used for both states a and b. Further decomposition can be trivially made to estimate contributions of different energy components, such as van de Waals (vdW) and electrostatic interactions.

Force field optimization by tuning backbone torsion profile and hydrogen bond strength

We fist adjusted the charges of Arginine side chain, to reduce the stronger stability of “rrsa” pair (Table S1). Once the pair-wise interactions between various side chain and backbone moieties are properly parameterized, achieving a balanced protein force field generally requires additional tuning of peptide backbone torsion profile and hydrogen bond strengths^{32, 34, 45}. The calibration was guided by conformational properties of an appropriate set of folded and disordered model proteins. In this work, we used a set of small and disordered model peptides, including (AAQAA)₃, which is widely used in protein force field optimizations with residue helicity well characterized by NMR (~20% at 300 K)⁴⁶, and β-hairpins GB1p (GEWTYD DATK TFTVTE) and its mutational variant GB1m1 (GEWTYD DATK TATVTE), whose residual structure propensities are distinct due to the single F-to-A mutant (~30% folded for GB1p and ~6% folded for Gb1m1 at 278 K as characterized by NMR chemical shift analysis⁴⁷). The use of both helical peptide and β-hairpin allows one to calibrate the force field’s secondary structure propensities. The use of both GB1p and GB1m1 allows one to balance backbone vs side chain interactions in modulating conformational equilibrium. The sheet propensities between the GB1p and GB1m1 series were characterized by calculating the population of native hydrogen bonds as done previously⁴⁸. The backbone hydrogen bonding strength was tuned by directly adjusting the strength of the Lennard-Jones interaction between the backbone carbonyl oxygen (O) and polar hydrogen (H), which is increased to 0.52 kcal/mol from the original c36m(w) value of 0.0743 kcal/mol (Table S3). The peptide backbone torsion propensity is controlled by the CMAP dihedral cross-term^{49, 50}. It turned out that no adjustment to the original c36m backbone CMAP was needed for achieving the balance between helical and β-sheet propensities.

Disordered and ordered protein benchmarks

The performance of the new force field, which contains adjusted charges for Arg residues, a99SB-disp water model, and optimized backbone parameters, herein referred to as c36mrb-disp, was evaluated using a set of disordered and ordered proteins. Specifically, the 61-residue p53-TAD was used to examine the model’s ability to capture nontrivial local and long-range structural features within the disordered ensemble. Protein p53-TAD contains multiple residual helical regions and transient long-range intrachain contacts and has been well studied by multiple experimental techniques including NMR, SAXS, and smFRET³⁹. The folded protein sets include a 46-residue 3-helix bundle (PDB ID: 1BDC)⁵¹, the protein G B1 domain (PDB ID: 3GB1)⁵², a 56-residue protein with mixed helix and sheet structures, the ribosomal protein S6 lacking β2 (PDB ID: 3ZZP)⁵³, a 76-residue protein with mixed helix-sheet structures, and the sheet-dominated West Nile virus (WNV) NS2B/NS3 proteases (PDB ID: 5IDK)⁵⁴. Particularly, the protein G B1 domain contains three experimentally characterized solvent exposure salt bridges, whose stability could be used to evaluate the force field performances^55–57. While the WNV NS2B/NS3 proteases prefer the compact folded conformation in the solvent as indicated by NMR and X-ray studies (Figure S6), the C-terminal hairpin of NS2B displays significant dynamics in the ligand-free state⁵⁴. As such, the NS2B/NS3 protease represents a good test case for evaluating the force field’s ability to maintain the folded structures and capture large-scale dynamics simultaneously.

Computational details

The atomic coordinates of each dimer were first subjected to energy minimization and then fixed in all FEP simulations. Rectangle simulation boxes were used, and the size was set to be large enough to accommodate the largest separation (13.5 Å), so that they do not have interactions with their images. The GROMACS program was used to generate the initial configurations and add the water models for all force fields⁵⁸. The OpenMM program was used to equilibrate the systems and run the production window sampling simulations⁵⁹. The cutoff scheme in the OpenMM program was used to calculate the nonbonded interactions with d_c = 12.0 Å with a default reaction field approximation to capture the effect of atoms beyond the cutoff distance. The Langevin thermostat with a collision frequency of 1.0 ps⁻¹ was used for maintaining temperature at 300 K. The time step of 2 fs was used for all production simulations. Other parameters were default values provided by the v7.4 OpenMM program.

The termini of all model peptides and proteins are neutralized using amine (-NH2) and carboxyl group (-COOH). Temperature replica exchange (T-REX) was used for simulating the conformational ensembles of (AAQAA)₃, GB1p, and GB1m1^{60, 61}. For (AAQAA)₃, 16 replicas were distributed exponentially between 298 and 450 K. For GB1p and GB1m1 peptides, 16 replicas were distributed exponentially between 278 and 400 K. For larger disordered peptide p53-TAD, the replica exchange with solute tempering (REST3)^62–64 were used to allow the use of 16 replicas exponentially distributed between 298 and 450 K. In REST3, only the protein region was subjected to tempering, which was realized by scaling the solute-solute and solute-solvent interactions to achieve various effective temperatures. Specifically, for REST3, scaling of the solute-solvent interaction was optimized to promote more efficient conformational diffusion throughout the temperature range⁶⁴. Replica exchange was attempted every 2 ps. The vdW interactions were cut off at 1.0 nm and the long-range electrostatic interactions were treated using the particle mesh Ewald (PME) method^{65, 66}. T-REX and REST3 simulations for disordered proteins were all initiated with sets of mixed conformations and lasted 2.0 μs per replica. The folding trajectory for p53-TAD from a99SB-disp, c36m, and c36mw force fields were obtained from Ref.⁶⁴ and Ref.³⁹, respectively. For folded proteins, two constant temperature MD simulations starting from the folded structure were carried out at 298 K and lasted for 2.0 μs each. The stability analyses towards the salt bridge for GB1 protein followed the same approach proposed in Ref⁵⁶. The first 200 ns were discarded in the final analysis. For disordered protein simulations with T-REX or REST3 acceleration, the last 1.8 μs trajectory at the target temperature was divided into two 900 ns trajectories for estimating the error bars.

The paramagnetic relaxation enhancement (PRE) effects at four sites (D7, E28, A39, and D61) were analyzed to evaluate the accuracy of transient long-range interactions of p53-TAD predicted by various force fields. The results were directly compared with previously published experimental data⁶⁷. The ratios of peak intensities between oxidized (paramagnetic, I_ox) and reduced (diamagnetic, I_red) resonances were calculated according to $\frac{I_{ox}}{I_{red}}=\frac{R_{2}exp\left(-R_2^{sp}t\right)}{R_2+R_2^{sp}}$, where $R_2^{sp}=K<r^{-6}>\left(4{\tau }_c+\frac{3{\tau }_c}{1+{\omega }_H^2{\tau }_c^2}\right)$^{68, 69} with r being the distance between spin label and residue, K = 1.23×10⁻³² cm⁶s⁻², and <> indicates ensemble averaging. Consistent with experimental conditions, Larmor frequency ω_H is 600 MHz; the correlation time τ_c is 3.3 ns; the intrinsic R₂ relaxation rate is 16 s⁻¹; and the INEPT evolution time t is set to 9.8 ms. The Cα–Cα distances were used to approximate the electron–proton distances, as described previously³⁹. VMD was used to visualize molecular structures⁷⁰.

Results and Discussion

Free energy profiles of peptide backbone and sidechain dimers

Pair-wise interactions between representative backbone and side chain moieties provide an effective assessment on how different force fields describe the balance between various protein-protein and protein-water interactions. The pairs examined in this work (Figures S2 and S3) include the nonpolar-nonpolar, nonpolar-polar, polar-polar, polar-charged, charged-charged pairs. Examples of the PMFs of six selected pairs in c36m, c36mw and a99SB-disp are shown in Figure 1. Not surprisingly, PMFs from all three force fields are highly similar, even though those from a99SB-disp are more distinctive in both the amplitudes and locations of valleys and peaks. Overall, the slight increase in water hydrogen Lennard-Jones parameters in c36mw leads to a small decrease in the dimer stability, by 0.2 kcal/mol on average (Figure 2). Even such a small systematic reduction in dimer interactions is enough to greatly alleviate the over-compaction bias of c36m^{33, 39}. However, compared to a99SB-disp, there is a clear systematic bias in both c36m and c36mw to predict stronger dimer interactions, by an average of 0.2 kcal/mol for nonpolar pairs, 0.6 kcal/mol for polar pairs, and 0.8 kcal/mol for salt-bridges, respectively. The pi-pi stacking interactions involving the Trp side chain are slightly less stable in c36m. Curiously, the stabilization is particularly pronounced for the polar and charged pairs, with five of them > 1.0 kcal/mol more stable in c36m(w) compared to a99SB-disp (Figure 2B). The most severe overstabilization in c36m is observed for charged pairs involving the Arg and Glu side chains, by up to 2.9 kcal/mol. For example, the “rrsa” pair is almost twice as stable in c36m(w) (>5.0 kcal/mol) compared to a99SB-disp (2.4 kcal/mol). It has been shown that the a99SB-disp force field provides better hydration free energies of amino acid side chains than the c36m force fields⁷¹. The observed over-stabilization of charged pairs in c36m could reflect an under-solvation of charged residues, but could also result from an over-estimation of protein interactions. Nonetheless, it is likely that the observed systematic difference in describing the pair-wise interactions between backbone and side chain moieties is a major contribution to the tendency of c36mw in favoring compact conformational states, beside the inherent rigidity and conformational propensities of the peptide backbone, which is consistent with the observed compactness in several protein systems^{40, 72}.

Figure 1. Representative PMF results for dimers in different force fields.

PMF profiles of six representative dimers in c36m, c36mw, and a99SB-disp. These dimers represent nonpolar, pi-pi stacking, hydrogen bonding, and charge-charge interactions.

Figure 2. Pair-wise interactions stabilities in different force fields.

(A) Stabilities of backbone and side chain dimers in c36m, c36mw, and a99SB-disp. See Figures S2 and S3 for molecular configurations and notations. (B) The relative stabilities with respect to a99SB-disp, where both root mean square difference (RMSD) and mean absolute difference (MAD) are shown (in kcal/mol). Five charged pairs show >1.0 kcal/mol differences in calculated stabilities.

Arg side chain interactions: re-tuning the partial charges

We further examine the interactions involving charged residues by first focusing on the Arginine self-stacking interaction, which is predicted by both a99SB-disp and c36mw to be highly stable (Figure 3A). The stability is overestimated by more than 2.5 kcal/mol in c36mw compared to a99SB-disp (Figure 2B). Figure 3B shows that the atomic partial charges of Arg side chain in a99SB-disp are quite different from those in c36m(w), with the former having a substantially more polar guanidinium group. For example, the C2 atom type has much more positive charges in a99SB-disp (1.035e) than in c36mw (q = 0.64e). Decomposition of the total PMF to protein-protein and protein-water contributions reveals that the difference in the stabilities in c36mw vs a99SB-disp is mainly due to the protein-water free energy component (Figure 3C), which can be apparently attributed to increased guanidinium group polarity in a99SB-disp. We further calculated the potential energy between the Arg side chain and a single TIP3P* (in c36mw) or a99SB-disp water molecule, and the results show that the electrostatic interaction makes a dominant contribution to the observed difference in total protein-water contributions (Figure 3D). It is noted that a99SB-disp readjusted the side-chain charges of aspartate, glutamate, and arginine residues to align with the guanidinium acetate association constant³⁴. Interestingly, the apparent over-stabilization of Arg self-stacking in c36mw can be effectively reduced by the above modification of the charges of guanidinium group to mimic those in a99SB-disp (Figure 3A and B). Specifically, we first adhered to the convention used in CHARMM force fields by maintaining neutral partial charges for -CH3 and -CH2 groups, despite this not being the case in a99SB-disp. We then replaced the charges of the guanidinium group with those from the a99SB-disp force field and adjusted the charge of the C2 atom to ensure the overall charge remained +1 (see the Table S1 details). Although the charges could be further optimized, results shown in Figure 3A demonstrated that partial charges within the guanidinium group play a crucial role in balancing protein-protein and protein-water interactions involving the Arg side chain.

Figure 3. Interactions of the Arg side chain.

(A) PMFs of Arginine self-stacking interactions (“rrsa” in Figure S3) in c36mw, a99SB-disp, c36m with modified Arg charges (c36mr), and c36mrb-disp. (B) Partial charges of Arg side chains in c36m (brown), a99SB-disp (green), and c36mrb-disp (red). (C) Difference of in the total free energy the “rrsa” pair between the a99SB-disp and c36mw and its decomposition into the protein-protein and protein-water components. The distance of the total free energy minimum is marked using a green dot on the total free energy difference curve (black trace). (D) Difference between the potential energy between Args and a single water molecule in a99SB-disp and c36mw and its decomposition into the electrostatic and vdW protein-water contributions.

Glu-involved pairs: rebalancing protein-protein and -water electrostatic interactions

We then examined if adjusting partial charges would be effective in rebalancing dimers involving Glu side chains. As shown in Figure 4A, there are significant differences in both the protein-protein and protein-water contributions to the Glu-Lys dimer (“eks”, see Figure S3), although these different contributions cancel each other to achieve a much smaller difference in the total free energy profiles. Further decomposition of the protein-protein interaction into the vdW and electrostatic contributions (Figure 4B) confirms that stronger electrostatic interactions are the source of stronger protein-protein interactions in a99SB-disp. We further calculated the potential energy of Glu side chain interaction with a single TIP3P* or a99SB-disp water molecule, and the results show that electrostatic interactions are also the source of much strong protein-water interactions observed in c36mw (Figure 4C). We note that a99SB-disp and c36mw give similar descriptions of the Lyss-H₂O system for both vdW and electrostatic interactions (see Figure S6), which suggests that the different protein-protein and -water electrostatic interactions can be attributed to the presence of Glutamic acid side chain.

Figure 4. Glu-Lys interaction in c36mw and a99SB-disp.

(A) The difference in the total free energy of interaction of the “eks” pair between the a99SB-disp and c36m and its decomposition into the protein-protein (pp) and protein-water (pw) contributions. The distance of the free energy minimum is marked as a green dot on the total free energy difference curve (black trace). Further decomposition of the differences in the protein-protein free energy (B) and protein-water potential energy (C) in a99SB-disp and c36mw force fields into vdW and electrostatic contributions.

Following a similar strategy used for “rrsa” pair, we then examined whether changing charges would be effective in rebalancing dimers involving Glu side chains. Unfortunately, modifying atomic charges of Glu side chain has only modest impacts on reducing the difference of dimer stabilities. The reason is that tuning atomic charges modify both the protein-protein and protein-water interactions similarly. These two effects largely cancel, and the total interaction profile is rather insensitive. Instead, it seems necessary to directly change the protein-water interaction, by either changing the water model itself or tuning the protein-water vdW parameters as done previously³⁶. We tested a direct combination of the c36m protein force field with the a99SB-disp water model for Glus-involved pairs without changing the protein parameters. It should be noted that the a99SB-disp water model is different in both vdW parameters and partial charges from the TIP3P* water model in c36mw. Surprisingly, the stabilities of all three pairs are greatly reduced to within 0.5 kcal/mol from those in a99SB-disp (Figure 5). Additionally, it provides a reduced stability for the “rrsa” pair (Figure 3A). This indicates that the use of a99SB-disp water model can reduce the stabilities of pairs resulted from c36m force field.

Figure 5. PMF profiles of interactions of three dimers involving the Glutamic acid side chain.

(A) eks, (B) hpe, and (C) he in c36m, c36mw, a99SB-disp, and a36mrb-disp. (D) Summary of the stabilities of the three dimers in four force field options.

Optimization of c36mrb-disp using model peptides

Given the apparent effectiveness of adopting the a99SB-disp water model, we recalculated the stabilities of all backbone and side chain dimers (see Figure S5) using the c36mrb-disp protein parameters (see Tables S1 and S2). The results suggest that using a99SB-disp water model significantly reduces the apparent over-stabilization of c36m(w), including both polar and nonpolar pairs. The overall RMSD/MAD is reduced from 0.7/0.5 kcal/mol (c36mw) and 0.7/0.4 kcal/mol (c36m) to 0.3/0.2 kcal/mol (c36mrb-disp). This is somewhat surprising considering the substantial difference in protein vdW parameters and partial charges in c36m and a99SB-disp.

Encouraged by the apparent transferability of the a99SB-disp water model to c36m, we reoptimized the peptide backbone hydrogen bonding strength, through adjusting the vdW interaction between backbone carbonyl oxygen and polar hydrogen, guided by conformational equilibria of three helical and β-hairpin peptides (see Methods). It should be emphasized that the optimization here was not intended as a comprehensive one for the c36mw force fields; instead, our goal was to evaluate if comparison with a99SB-disp could provide a useful benchmark and if the improvement on dimer stabilities using a99SB-disp could effectively translate into better description of dynamic protein structures. Specifically, we retained the sigma values but adjusted the epsilon value to balance the stability of the alanine dipeptide (“ala2” in Figure S3), which is a representative backbone hydrogen bonding configuration, to make it closer to that observed with the c36mw force field (Figure S7). Results showed that these minor changes (see Table S3) can achieve a balance between helical and coil/sheet propensities, even if the original c36m CMAP can be used. For example, Figure 6 plots the conformational distributions of (AAQAA)₃ at 300 K, as well as those of GB1p and GB1m1 at 278 K. It shows that the c36mrb-disp force field yields slightly higher residual helicity for (AAQAA)₃ compared to a99SB-disp, even though both are still lower compared to either the original c36m force field³³ or NMR⁴⁶ (Figure 6A, black trace). We note that the residue helicity profile of (AAQAA)₃ from c36m is in excellent agreement with NMR at 298 K⁷³. However, this peptide was used to in the optimization of c36m but not for a99SB-disp³⁴. Importantly, c36mrb-disp is able to correctly resolve non-trivial distinction of sheet propensities between the GB1p and GB1m1 peptides, yielding folded populations of ~20% and ~5%, respectively (Figure 6B). This is highly consistent with the NMR results showing that the single F to A mutation reduced the folded population from ~30% for GB1p to ~6% for Gb1m1 at 278 K⁴⁷. This ability indicates a relatively balanced competition of side chain-mediated interactions with backbone hydrogen bonding interactions and secondary structure propensities in c36mrb-disp.

Figure 6. Conformational distributions of model peptides.

(A) Residue helicity profiles for (AAQAA)₃ calculated at 298 K using a99SB-disp and c36mrb-disp force fields in comparison to the NMR result⁴⁶. (B) Probability distributions of the number of hydrogen bond of GB1p and GB1m1 at 278 K calculated using the c36mrb-disp force field. The error bars were estimated via the differences between two 900 ns trajectories divided from the last 1.8 μs trajectories of the REST3 simulation (see Methods).

Evaluation of c36mrb-disp for disordered and ordered proteins

The new c36mrb-disp force field was first evaluated using p53-TAD, a longer IDP with nontrivial local and long-range nontrivial structures (see Methods). The results show that c36mrb-disp is able to reproduce the expansion of the disordered ensemble; the distributions of both radius of gyration and end-to-end distance (Figure 7B and C) compare well to a99SB-disp simulations^{39, 64} and NMR (R_g ~ 2.89 nm for residues 1–70)^{39, 67}. Importantly, increased chain dimension compared to c36m is achieved without compromising the stability of residual structures. As shown in Figure 7A, the c36mrb-disp model has a capability to predict the presence of a residual helix in the AD1 region (residues 15 to 26), where a comparable helical profile was observed in the residue 20 to 26, although little helix population was observed in residues 15 to 20. The predicted ~20% helicity compares well with either a99SB-disp or NMR³⁹. However, the residual helicity near the AD2 region (residues 40 to 52) is apparently underestimated (~1%), compared to ~5% from NMR or ~10% from a99SB-disp, suggesting room for further optimization of the force field. In contrast, c36mw fails to capture substantial residual helicity throughout the sequence, even though it does alleviate the over-compaction of c36m. We note that the disordered ensembles calculated using c36m and c36mw are limited in convergence despite the 2.0 μs sampling time per REST2 replica.

Figure 7. Conformational properties of p53-TAD in different force fields.

(A) Residual helicity, (B) radius of gyration and (C) end-to-end distance distributions of p53-TAD calculated at 298 K using the a99SB-disp (blue), c36mrb-disp (red), c36m (yellow), and c36mw (green) force fields. The vertical dotted lines in panels B and C mark the average values. The error bars were estimated via the differences between the first and second 900 ns segments of the last 1.8 μs trajectory (see Methods).

The ability of c36mrb-disp to describe long-range, transient structural features of p53-TAD is further evaluated by comparing the PRE effects with experimental data⁶⁷. PRE combined with site-specific spin labeling is a powerful NMR technique for detecting transient interactions between the spin label and the rest of the protein up to 35 Å^{74, 75}. Specifically, for p53-TAD, experimental PRE data are available for spin labeling at four sites: D7, E28, A39, and D61, providing a comprehensive benchmark for chain terminal and middle region dynamics⁷⁶. In Figure 8, we compare the PRE results calculated using c36mrb-disp and a99SB-disp with experimental values. It shows that c36mrb-disp is comparable to a99SB-disp in reproducing the experimental PRE profiles, suggesting that the new force field is able to capture the transient long-range interactions of p53-TAD.

Figure 8. Simulated and measured PRE profiles of p53-TAD at four labeling sites.

(A) E28, (B) D7, (C) A39, and (D) D61. Error bars were calculated using the two 900 ns trajectories divided from the equilibrated 1.8 μs REST3 simulations at 298 K for a99SB-disp (blue, trajectories from Ref.⁶⁴) and c36mrb-disp (red), respectively. The NMR experimental data were taken from Ref.⁷⁶.

Finally, we assessed the ability of c36mrb-disp to accurately simulate the structures of folded proteins, using four proteins with different sizes and topologies (see Methods). The results, summarized in Figure 9, show that these folded structures remain stable throughout 2-μs MD simulations. Figure S8 compares the initial and last snapshots of these proteins, confirming the new force field’s ability to maintain both secondary and tertiary structures. The largest conformational fluctuation in WNV NS2B/NS3 protease is mainly attributed to the dynamics of the N-terminal of the NS3 loop and the NS2B C-terminal domain (CTD). This is consistent with findings from previous studies, which highlighted the dynamic nature of the NS2B CTD in controlling and responding to the functional activity of the proteases^77–79. We also assessed the stabilities of three solvent-exposed salt bridges of protein GB1, namely, K12-E23, K39-E35, and K58-D55, that have been examined previously^{56, 57}. As shown in Figure S9, the c36mrb-disp force field effectively captures the formation of these salt bridges, with energy local minima occurring at an N-O distance of approximately 5 Å. These salt bridges are highly dynamic throughout the 2-μs simulations (Figure S10), and their populations are summarized in Table S4. Previous studies^{56, 57} showed that a99SB-disp overestimates the stabilities of salt bridges K12-E23 and K58-D55 (also see Table S4), even though they were shown by NMR to only weakly formed⁵⁵. Unsurprisingly, the stabilities of these salt-bridges are similar in c36mrb-disp (Table S4), as it is parameterized to reproduce pair-wise protein-protein interactions of a99SB-disp (Figure S5). Nonetheless, it appears that K12-E23 is weaker and K39-E35 is slightly stronger in c36mrb-disp, aligned more closely with experimental observations⁵⁵. Overall, c36mrb-disp demonstrates a good balance in maintaining ordered structures and capturing conformational dynamics and it is likely suitable for simulation of both folded and disordered proteins.

Figure 9. Stability of four folded proteins in c36mrb-disp.

For each protein, two independent simulations (Rep 1 and 2) were run, and all backbone heavy atoms were included in the RMSD calculation.

Conclusions

We constructed a set of representative amino acid side chain and backbone pairs to examine the difference of three latest all-atom protein force fields, namely, c36m, c36mw, and a99SB-disp, in describing protein-protein interactions. Free energy calculations show that both the c36m and c36mw force fields yield similar PMF profiles, with c36mw predicting an average of 0.2 kcal/mol reduction in stabilities for all pairs except “he” (Figure S3). Curiously, both c36 and c36mw predicted higher stabilities for most pairs compared to a99SB-disp, in particular for the polar and charged pairs. The apparent systematic over-stabilization of protein-protein interactions could explain the tendency of c36m(w) to predict more compact conformation ensembles of disordered proteins. Free energy decomposition analysis suggested that the free energy difference between the a99SB-disp and c36m force fields was likely attributed to the imbalanced electrostatic interactions of protein-protein and protein-water. We further showed that direct modification of the protein-water interactions, such as by using the a99SB-disp water model with c36m, is necessary to mitigate apparent over-stabilization of pair-wise protein interactions. Towards this end, we optimized the backbone hydrogen bonding strength and torsion profiles, guided by simulation of the conformational equilibria of helical and β-hairpins peptides, to derive a tentative c36mrb-disp force field more accurate simulation of the conformational properties of both disordered and folded proteins. Results showed that the new force field is greatly improved over the original c36m(w) in describing both local and long-range structural features of IDPs, without compromising the ability to simulate well-folded structures. The current work highlights the importance of balancing protein-protein and -water electrostatic interactions as well as a likely need for changing the water model directly for rebalancing the protein force field. While more systematic optimization is necessary to further improve the force field, c36mrb-disp provides a viable working version for simulation of both folded and disordered proteins.

Data and software availability:

PDB structures of all dimers, simulation data of the dimer stabilities, and the topology and parameter files of the c36mrb-disp force field can be found at GitHub: https://github.com/mdlab-um/prot-dimer-pmf.