Characterization of the pH-Dependent Stabilizing Forces of Insulin’s Native and Fibril States

Abstract

Insulin fibrillation poses a major challenge for protein therapeutics, yet the molecular origins of its strong pH dependence remain incompletely understood. Here we employ constant-pH molecular dynamics simulations to quantify residue-specific protonation behavior in native monomeric insulin and in fibrils of increasing size. While the native monomer exhibits largely uncoupled titration with modest pK _a perturbations, conversion to the fibrillar fold induces systematic upshifts of acidic residues driven by partial desolvation and reduced dielectric screening. Upon self-association, strong electrostatic coupling emerges between neighboring glutamate residues aligned along the fibril axis, giving rise to cooperative and multistep titration behavior. In fibril assemblies, central residues consistently deprotonate first, despite residing in partially solvent-shielded environments within the fibril core, indicating that collective electrostatic frustration overrides simple solvent accessibility effects. This cooperative charge redistribution is accompanied by a progressive loss of intermonomer contacts, revealing pH-dependent weakening of fibril cohesion. Together, our results demonstrate that insulin fibrillation is governed by collective electrostatics that arises only after conformational rearrangement and assembly, providing an atomistic framework for understanding pH-regulated amyloid stability.

Introduction

Insulin, a peptide hormone found in all vertebrates, regulates glucose homeostasis through activation of the insulin receptor. The mature form of insulin is produced from proinsulin via removal of the connecting peptide (C-peptide), yielding two separate peptide chains: the A chain (21 amino acids) and the B chain (30 amino acids). The two chains are linked by two interchain disulfide bonds and an additional intrachain disulfide bond within the A chain. , Impaired insulin production or reduced cellular sensitivity leads to diabetes mellitus, necessitating lifelong therapeutic insulin administration. ,

A major challenge in the use of protein-based therapeutics lies in maintaining proteins’ structural and functional integrity during storage, transport, and administration. Experimental studies have shown that insulin can undergo an α-to-β structural transition and form amyloid fibrils, especially under acidic conditions or at neutral pH in response to mechanical or thermal stress. − These fibrils are pharmacologically inactive and highly stable, reducing therapeutic efficacy. Moreover, fibrillar aggregates have been associated with cytotoxicity and local inflammatory responses at the injection sites. ,

Recent structural insights into insulin’s monomeric and fibrillar , states provide a valuable basis for mechanistic investigation. These distinct conformational states are stabilized by different solution conditions, highlighting pH as a central factor in shaping insulin’s free energy landscape and aggregation propensity. Because early stages of aggregation remain difficult to capture experimentally, molecular simulations offer a powerful complementary approach for elucidating how pH governs the transition toward fibril formation at atomic resolution.

The pH sensitivity of proteins arises from the presence of titratable groups. Most commonly these include Asp, Glu, Lys, Arg, His, Tyr, and free Cys, as well as the terminal amino and carboxyl groups. The protonation states of these groups vary with the local environment, modulating net charge, hydrogen bonding, salt bridge formation, and the overall electrostatic landscape. These electrostatic changes can propagate throughout the protein structure, influencing solubility, conformational stability, and intermolecular interactions. In the case of insulin, where aggregation pathways are tightly linked to conformational flexibility and inter-residue interactions, pH-induced shifts in protonation can perturb critical contacts and potentially (dis)­favor aggregation-prone states.

A common way to quantify pH effects is through the determination of pK _a shifts. The shifts represent deviations from intrinsic solution values that arise due to environmental perturbations such as desolvation, hydrogen bonding, or Coulombic interactions. Particularly in buried or partially buried sites, these shifts can serve as sensitive indicators of conformational or energetic changes. , Computational methods capable of estimating these shifts have become essential tools for dissecting pH-dependent processes in proteins and can offer insight into the atomistic underpinnings of pH-driven aggregation. −

In this study, we employ constant-pH molecular dynamics (CpHMD) simulations using a hybrid explicit/implicit solvent protocol to determine pK _a values of individual ionizable residues in both the native monomeric and fibrillar states of insulin. Rather than simulating the full aggregation process, we directly compare the protonation behavior of these two states to evaluate how fibril formation alters the local electrostatic environment. By identifying residue-specific pK _a shifts and patterns of charge redistribution, we aim to uncover the molecular determinants of insulin’s pH sensitivity and clarify the electrostatic mechanisms that may stabilize fibrillar states in insulin and other amyloid-forming proteins. Because aggregation is most relevant in the low to neutral pH range, our discussion here focuses on the acidic glutamate residues that dominate protonation changes in this regime.

Methodology

pH-Replica Exchange Molecular Dynamics

All-atom discrete-state constant-pH replica exchange molecular dynamics (pH-REMD) simulations were performed using Amber24 , and AmberTools24. Two structural classes were investigated: native monomeric insulin and fibrillar assemblies.

Native insulin simulations employed a monomeric structure containing a B30 Thr → Ala substitution (PDB ID 3I40). The fibril state simulations were based on a truncated insulin type II protofilament 2 structure (PDB ID 8SBD), in which both A- and B-chain termini are truncated due to being disordered and thus not modeled in the cryo-EM derived structure. The exposed termini resulting from these truncations were capped with acetyl and N-methylamide groups to avoid introducing artificial terminal charges. From this reference fibril structure, three systems were constructed: an isolated fibril monomer (1-mer), a trimeric fibril (3-mer), and a heptameric fibril (7-mer), generated by replication along the fibril axis preserving the fibril structure. The systems are visualized in Figure .

1.

Overview of the insulin systems studied: native monomeric insulin, monomeric fibril state, trimeric fibril, and heptameric fibril assemblies. The titratable residues present are illustrated in licorice, with the three disulfide bridges visualized as yellow bonds.

All systems’ initial configurations were prepared using tleap and solvated in periodic rectangular boxes with a minimum distance of 20 Å between any protein atom and box faces. This resulted in 9960 water molecules for the native monomer, 13,782 for the 1-mer, 15,950 for the 3-mer, and 20,664 for the 7-mer. Explicit water was represented using the TIP3P model.

Proteins were modeled using the AMBER constant-pH force field derived from ff10, with modified representations of carboxylate residues. Each protonatable carboxyl oxygen is represented by two hydrogen sites, one in the syn and one in the anti position, constrained 180° apart by an improper torsion. This dual-site configuration prevents unphysical rotations of “deprotonated” sites and ensures sampling ergodicity by allowing the system to transition from either rotameric orientation. By decoupling the protonation state change from the physical rotation of the O–H bond, we circumvent the kinetic and thermodynamic barriers associated with the disparate populations of these rotor states. This is particularly relevant as the energy landscape of the deprotonated site often favors a specific orientation (e.g., in vacuum-like conditions) that may differ from the thermally accessible states in the active, solvated environment. Consequently, this approach ensures that the protonation sampling remains unbiased by the local geometry of the deprotonated state.

Energy minimization was performed in two stages: an initial 1000 steps of steepest descent followed by up to 99,000 steps of conjugate gradient minimization, or until the convergence was obtained. During energy minimization, positional restraints were applied to all protein backbone atoms using a harmonic potential with a force constant of 10 kcal mol^–1 Å^–2, with the reference positions defined by the initial coordinates of the structure. Following minimization, velocities were initialized from a Maxwell–Boltzmann distribution at 10 K, and the systems were heated to 298.15 K over 300 ps using a linear temperature ramp. A further 100 ps of equilibration at constant temperature followed. Temperature regulation was performed using a Langevin integrator , with a collision frequency of 1 ps^–1. Subsequently, pressure equilibration was conducted in the isothermal–isobaric (NPT) ensemble for 200 ps using a Monte Carlo barostat , targeting 1 atm, with isotropic volume scaling attempted every 0.2 ps.

For all fibrillar systems (1-mer, 3-mer, and 7-mer), harmonic positional restraints (10 kcal mol^–1 Å^–2) were maintained on heavy atoms throughout equilibration and production to preserve structural integrity of the fibril structure throughout the simulations.

Production simulations were performed in the semigrand canonical ensemble of protons (Δμ_H⁺ VT), − where the side chains of Glu, His, Lys, and Tyr residues were allowed to dynamically switch protonation states. In the discrete constant-pH formalism employed here, protonation changes are represented using the “proton” sites for which the protonated and deprotonated states possess identical masses, holonomic constraints, and bonded interactions. When deprotonated, the “proton” site retains only local covalent interactions and its nonbonded interactions (electrostatic and van der Waals) are turned off. Protonation moves were attempted every 0.2 ps using Markov chain Monte Carlo sampling. During these moves, explicit solvent was temporarily replaced by a generalized Born implicit solvent model (Onufriev, Bashford, Case solvation model , ) with a Debye–Hückel salt concentration of 0.1 M. , Upon acceptance of a protonation change, 0.2 ps of solvent relaxation was performed after reinstating the explicit solvent. Because protonation states fluctuate, the total charge of the simulation box varies. No counterions were located explicitly in the MD box, and the apparent non-neutrality was handled by the uniform neutralizing background of the Ewald method. ,

The Metropolis acceptance criterion employed a transfer free energy framework in which each titratable site is referenced to a model compound with known solution pK _a values (see SI sections S1 and S3). , The reference compound defines the intrinsic proton affinity of a given residue type, and the effect of the protein environment is captured as a perturbation to this reference through a thermodynamic cycle. The effective chemical potential of protons at each pH is determined relative to these reference states, allowing for Monte Carlo sampling within the semigrand ensemble. The reference pK _a values employed in this study were the default values in AMBER, corresponding to experimentally measured values in solution. −

Enhanced sampling was achieved through pH replica exchange ,, with exchanges attempted every 0.6 ps between adjacent replicas. For the folded monomeric insulin, the pH range was 2.0–13.5, and for the fibrillar insulin systems (1-mer, 3-mer, and 7-mer) the pH range was 0.0–13.5. All systems employed a uniform spacing of 0.5 pH units (see SI section S2 for the acceptance criterion).

For all systems, each replica was simulated for 240 ns. Trajectory frames were saved every 50 ps for subsequent statistical analysis. All simulations employed a 2 fs time step with bonds involving hydrogen atoms constrained using the SHAKE algorithm. Long-range electrostatics were computed using the smooth particle mesh Ewald (PME) method with a real-space cutoff of 1.0 nm. PME grid dimensions and interpolation were handled using AMBER’s default settings, which include fourth-order B-spline interpolation and automatically selected grid sizes to ensure accurate evaluation of electrostatic interactions. A nonbonded cutoff of 1.0 nm was also used for Lennard-Jones interactions.

Rather than inferring the pK _a values from single-pH simulations, deprotonated fractions were computed across the full pH ladder to construct complete titration curves. For sites exhibiting an apparent sigmoidal titration behavior, pK _a values and Hill coefficients were obtained by fitting to the Hill-modified Henderson–Hasselbalch equation: −

$$y_{\mathrm{b}}=\frac{1}{1+10^{n(\mathrm{p}{K}_{\mathrm{a}}-\mathrm{p}\mathrm{H})}}$$ 1

where y _b is the deprotonated fraction, n is the Hill coefficient, and pK _a is the negative base-10 logarithm of the acid dissociation constant.

In the present work, the Hill coefficient is treated as an empirical measure of titration hypersensitivity, reflecting the degree to which the protonation response is broadened (n < 1) or sharpened (n > 1) relative to an ideal sigmoidal titration curve. If a titration curve is well-described by eq , the corresponding free-energy difference between protonated and deprotonated states is necessarily linear in pH, with a slope proportional to n. Importantly, however, this linearity follows directly from the assumed functional form and does not uniquely specify the microscopic origin of the effect. Deviations from n = 1 may arise from electrostatic interactions, conformational redistribution between protonation states, or other forms of coupling between protonation and the underlying structural ensemble. ,

For coupled (multitransition) titration curves, a formal treatment based on multiprotic acid models can, in principle, be used to extract site-unresolved and protonation-state-conditional pK _a values from protonation microstate populations, as described by Ullmann. However, because the number of residue-specific and residue-residue coupling parameters increases exponentially with the number of titratable sites, such models rapidly become highly parameter-rich. To avoid overparameterization of the titration model and the associated ambiguities in uniquely resolving protonation-state-conditional constants, we instead focus on directly analyzing deprotonated fractions and emergent cooperative features of the titration curves. This approach is sufficient to characterize the collective protonation behavior observed here without imposing a particular microscopic (state-conditional) decomposition.

All structural properties were analyzed using MDTraj (v. 1.10). Solvent-accessible surface areas (SASA) were calculated using the Shrake–Rupley algorithm using a probe radius of 1.4 Å, 960 sphere points per atom, and element-based atomic radii of 1.20 Å for H, 1.70 Å for C, 1.55 Å for N, 1.52 Å for O, and 1.80 Å for S.

Statistical uncertainties for all reported quantities, including both structural and protonation data, were determined by block averaging, partitioning each trajectory into 20 equally spaced blocks.

Results and Discussion

Independent Titrations in Monomeric Native Insulin

To establish an electrostatic baseline for understanding how pH modulates insulin fibrillation, we first characterize the native monomeric state. Figure shows the titration curves of all titratable residues obtained from the CpHMD simulations. Across all residues, the titration curves are monophasic, exhibiting a single sigmoidal transition over the investigated pH range. Accordingly, each site can be described by an effective single-transition model, and the simulated deprotonated fractions were fitted to the Hill-modified Henderson–Hasselbalch equation to obtain apparent pK _a values and Hill coefficients (summarized in Table S1). While the simulations yield complete titration profiles for all ionizable groups, the discussion below focuses on residues whose deprotonation occur in the low to neutral pH range, conditions most relevant to insulin aggregation under both physiological and formulation-relevant scenarios. Histidines are included because their intrinsic pK _a values lie near physiological pH, where insulin aggregation can also be induced by thermal or mechanical stress.

2.

Titration curves of all titratable residues in human natively folded monomeric insulin from constant-pH molecular dynamics simulations. Deprotonation fractions are shown as a function of pH for 11 titratable residues grouped by residue type: (A) glutamate (Glu A:4, A:17, B:13, B:21), (B) tyrosine (Tyr A:14, A:19, B:16, B:26), (C) histidine (His B:5, B:10), and (D) lysine (Lys B:29). Circles represent raw deprotonation data from simulation, while lines indicate nonlinear least-squares fits to the Hill-modified Henderson–Hasselbalch equation (eq ). The fitted curves provide estimates of apparent pK _a values and Hill coefficients, reflecting site-specific titration behavior and possible cooperativity.

For the glutamic acid residues (Figure A), the obtained titration curves group into two distinct categories based on their apparent pK _a values. Glu A:4 and Glu B:13 exhibit pK _a values of 4.44 and 4.28, respectively, close to that of the reference compound. By contrast, Glu A:17 and Glu B:21 show elevated pK _a values of 5.13 and 5.08, corresponding to upshifts of 0.5–0.6 units. This shift equates to a free energy difference of approximately 3.4 kJ mol^–1, comparable to the system’s thermal energy at physiological temperature.

In the case of Glu B:21, the elevated pK _a appears to be driven by the dipole moment of the B-chain α-helix. This interpretation is supported by the distance distribution between the carboxylate carbon of Glu B:21 and the backbone nitrogen of Cys B:19, which marks the C-terminal end of the B-chain α-helix (Figure ). At pH 2, where Glu B:21 is protonated, the distance distribution shows three peaks around 0.5, 0.9, and 1.05 nm (Figure A). The shortest-distance peak corresponds to conformations where the side chain carboxylic acid lies near the negative end of the helix dipole. The two longer-distance peaks reflect solvent-exposed states (Figure B). In the protonated (neutral) state, Glu B:21 interacts favorably with both the helix dipole and the solvent, with a preference for the latter. Upon deprotonation (pH 10), the short-distance peak shifts to ∼ 0.6 nm and diminishes in intensity, while the solvent-exposed peak becomes dominant suggesting electrostatic repulsion from the negatively polarized end of the α-helix. This shift in conformational preference destabilizes the deprotonated state, thereby elevating the apparent pK _a.

3.

Distance distribution and structural depiction of Glu B:21’s interaction with the B-chain α-helix. (A) Probability density of the distance between the carboxylate carbon (C_δ) of Glu B:21 and the backbone nitrogen of Cys B:19 at pH 2 (blue) and pH 10 (orange). The shaded areas indicate the 95% confidence intervals of the probability distribution obtained from block averaging. (B) Structural representation of Glu B:21 and the helix dipole, highlighting spatial context.

As a structural comparison to Glu B:21, we also examine the distance distribution of Glu A:17, located in the vicinity of the helix. Unlike Glu B:21, its distribution is predominantly single-peaked at 0.7 nm (Figure S3). At pH 2, a minor shoulder at shorter distances is observed, but this population remains weak relative to the dominant state. Overall, Glu A:17 does not exhibit the bimodal distribution characteristic of Glu B:21. The bimodal distribution of Glu B:21 reflects the coexistence of two distinct conformational states, corresponding to helix-associated and solvent-exposed interaction modes.

Helix dipoles are well-known to influence local electrostatics in proteins. , However, their effective magnitude depends sensitively on orientation, solvent exposure, and dielectric screening. In the present case, the B-chain α-helix is solvent-exposed at both termini, suggesting that water screening may attenuate the net dipole effect. Because protonation state sampling was carried out using an implicit solvent model, the precise magnitude of the observed pK _a upshift may therefore reflect the effective dielectric screening inherent to that model.

In the case of Glu A:17, the upshifted pK _a appears to arise from reduced solvent exposure and partial hydrophobic burial. To quantify this effect, we computed both the mean solvent-accessible surface area (SASA) and the full SASA distributions for the four Glu residues. The average values are shown in Figure A, while the corresponding probability distributions are provided in Figure S4.

4.

Solvent accessibility of glutamate residues varies with pH. (A) Average SASA for Glu A:4, Glu A:17, Glu B:13, and Glu B:21 at pH 2 (blue) and pH 10 (orange). Error bars show 95% confidence intervals obtained via block averaging. Decoupled protons are retained at both pH values to isolate environmental effects. (B) Representative protein structure highlighting the positions of the four Glu residues. All residues are well separated, minimizing spatial overlap in SASA calculations.

Across all residues, SASA increases modestly upon deprotonation, consistent with the unfavorable energetics of charged groups in low-dielectric environments and the conformational relaxation following proton removal. Comparing residues, the SASA follows the order Glu A:17 < Glu A:4 ≈ Glu B:13 ≪ Glu B:21, and this trend holds at both pH values. Although Glu A:17 has the lowest solvent exposure among the four, the absolute SASA difference relative to Glu A:4 and Glu B:13 is modest, suggesting that reduced SASA alone does not fully account for its upshifted pK _a. To resolve this discrepancy, we examined the local structural environment (Figure B). Glu A:4, Glu B:13, and Glu B:21 are all located within α-helical regions. A common property of helices at protein–water interfaces is amphipathicity, in which polar side chains are oriented toward the solvent and hydrophobic side chains are buried in the protein interior. Glu A:4 and Glu B:13 occupy the solvent-exposed face of their respective helices, and even when partially buried, their neighboring contacts are largely polar in nature, minimizing any penalty to the deprotonated state. By contrast, Glu A:17 is positioned near the hydrophobic core at the edge of a compact helical segment, resulting in partial burial of its side chain in a nonpolar environment. Such burial disfavors formation of the charged carboxylate, thereby destabilizing the deprotonated state and elevating the apparent pK _a.

To assess whether the changes in solvent exposure and inter-residue distances reflect underlying structural adaptation, we examined the χ₁ side chain dihedral distributions of the glutamate residues (Figure S5). Although the global backbone remains stable (See Figure S6), modest pH-dependent shifts in rotamer populations are observed, consistent with localized conformational rearrangements that reduce electrostatic strain upon deprotonation. Collectively, these results indicate that protonation changes are accommodated through subtle side chain reorganization without disruption of the overall insulin fold.

The two histidine residues, His B:5 and His B:10, exhibit titration behavior largely consistent with the reference compound, with apparent pK _a values of 6.45 and 6.64, respectively (Figure C and Table S1). This indicates that their protonation equilibria are only modestly perturbed by the local protein environment under the conditions studied. The corresponding Hill coefficients are close to unity (0.92–0.99), consistent with well-defined single-transition behavior.

Overall, titration of the native insulin monomer is characterized by predominantly single-transition behavior and only modest pK _a perturbations. The protonation equilibria therefore remain close to reference values across the investigated pH range, without pronounced cooperative features or strong shifts enforcing specific protonation states at physiological pH. Consequently, any marked pH dependence of aggregation is unlikely to be encoded in the isolated monomer and is instead expected to emerge upon structural rearrangement into fibril-like conformations, where intermolecular packing and hydrophobic burial reshape the underlying protonation energetics. In the following section, we quantify how these conformational changes alter pK _a values and promote cooperative proton binding in the monomeric fibril state of insulin.

Isolated Monomeric Fibril State (1-mer): Local Protonation Changes Induced by Conformational Change

The transition from the native monomer to the fibril conformation entails substantial backbone rearrangement, primarily an α-to-β transition, accompanied by redistribution of solvent exposure across the protein. These structural changes alone may influence protonation equilibria, even in the absence of intermolecular packing effects present in larger fibrils. To isolate electrostatic consequences arising purely from conformational change, we first analyze the titration behavior of a single insulin monomer restrained in its fibril fold, i.e., a fibril 1-mer.

The titration curves for the monomeric (1-mer) fibril state of insulin are shown in Figure . The corresponding fitted pK _a values and Hill coefficients are summarized in Table S2. All glutamate residues exhibit upshifted apparent pK _a values relative to the reference compound, with values of 5.34, 5.24, and 5.10 for Glu A:17, Glu B:13, and Glu B:21, respectively. Similarly, all tyrosine residues display elevated pK _a values (10.26, 10.65, 11.30, and 10.65 for Tyr A:14, Tyr A:19, Tyr B:16, and Tyr B:26). The histidine residues show pK _a values of 6.61 and 6.90 for His B:5 and His B:10, respectively, falling within the range of reference values for the δ and ε nitrogens of histidine.

5.

Titration curves of titratable residues the monomeric (1-mer) fibril state of insulin from constant-pH molecular dynamics simulations. Deprotonation fractions are shown as a function of pH for 9 titratable residues grouped by residue type: (A) glutamate (Glu A:17, B:13, B:21), (B) tyrosine (Tyr A:14, A:19, B:16, B:26), and (C) histidine (His B:5, B:10). The absence of Glu A:4 and Lys B:29 reflects unresolved disordered termini in the cryo-EM structure. Circles represent raw deprotonation data from simulation, while lines indicate nonlinear least-squares fits to the Hill-modified Henderson–Hasselbalch equation (eq ). The fitted curves provide estimates of apparent pK _a values and Hill coefficients, reflecting site-specific titration behavior and possible cooperativity.

At first glance, it is surprising that the fibrillar monomer exhibits systematic upshifts in Glu and Tyr pK _a values, as many of these residues remain nominally solvent exposed and isolated acidic groups would normally imply downshifted pK _a values. However, solvent exposure in a folded protein does not imply equilibration with bulk water. Even in the monomeric fibril state, titratable groups reside within a structured peptide environment, where proximity to the backbone, partial burial by neighboring side chains, and restricted hydration collectively reduce the effective local dielectric constant relative to bulk solution. In contrast to reference compounds, which are fully solvated and experience isotropic dielectric screening, protein-bound residues sample anisotropic and dynamically heterogeneous environments. In bulk solution, the charged state is strongly stabilized relative to the neutral form. Within the fibrillar environment, however, partial desolvation and restricted polarization reduce this differential stabilization. As a result, the free-energy contrast between charged and neutral states is diminished, leading to reduced stabilization of the ionized form and a relative shift toward protonation.

This effect is further amplified by restraining the backbone in the fibrillar conformation, which limits conformational relaxation and solvent reorganization following protonation changes. Together, partial desolvation, altered backbone proximity, and reduced structural flexibility destabilize the ionized states of acidic and phenolic residues, leading to systematic pK _a upshifts already at the level of a single fibrillar monomer. To evaluate whether these shifts are accompanied by local structural adaptation, we analyzed the χ₁ side chain dihedral distributions and solvent-accessible surface area (SASA) distributions of the glutamate residues (Figures S7–S9). In contrast to the native monomer, where subtle pH-dependent rotamer redistributions were observed, the fibrillar 1-mer exhibits markedly constrained side chain behavior. Glu A:17 and Glu B:21 display nearly identical χ₁ distributions at pH 2 and pH 10, indicating limited capacity for local rearrangement upon deprotonation. Only Glu B:13 shows a modest redistribution of rotamer populations, consistent with its orientation toward the fibril interior and potential side chain interactions. The SASA distributions similarly show only minor pH-dependent shifts. Together, these observations suggest that the fibrillar backbone conformation restricts local conformational relaxation, thereby limiting structural compensation upon ionization.

Thus, substantial pK _a shifts arise from conformational rearrangement alone, prior to any intermolecular packing. However, in this monomeric fibril state of insulin each titratable group remains electrostatically independent, and no cooperative behavior is observed. The monomeric fibril state therefore provides a well-defined reference state that isolates conformational electrostatics from interchain interactions.

Having established this baseline, we next examine how protonation equilibria are modified upon self-association, where neighboring monomers introduce electrostatic coupling, solvent exclusion, and charge invasion.

Trimeric Fibril (3-mer): Self-Association Induces Cooperative Glu Titration

We now turn our attention to a fibril consisting of three truncated insulin monomers. Relative to the isolated fibril 1-mer, this assembly introduces electrostatic coupling arising from self-association, as well as solvent exclusion and charge invasion, i.e., the presence of nearby charged species in the vicinity of titrating groups. Due to the parallel β-sheet packing of insulin monomers within the fibril, equivalent charged residues are brought into close spatial proximity along the fibril axis, providing a structural basis for strong intermonomer coupling. To describe positions of insulin monomers along the fibril, we adopt a monomer-centered indexing scheme. The central monomer is assigned index 0. For reproducibility, we define the viewing direction along the fibril axis with the N-terminal of the central insulin monomer facing the viewer and the C-terminal pointing away. Insulin monomers located to the right of the central insulin monomer (from this viewpoint) are assigned negative indices (−1, −2, ...), and those to the left are assigned positive indices (+1, +2, ...). The magnitude of the index indicates the insulin monomer’s position from the central insulin monomer, with larger absolute values corresponding to monomers farther away along the fibril axis. This convention provides a consistent reference for labeling monomers and reporting positional trends along potentially asymmetric fibrils.

Complete titration curves for all titratable residues in the 3-mer fibril are provided in Figure S10. To more directly assess the effect of self-association, Figure S11 further isolates the central Glu and His residues, whose protonation behavior is expected to be most sensitive to intermonomer electrostatic coupling at low and neutral pH. Here, we focus on the Glu residues and explicitly compare the central monomer with the neighboring monomers, as these exhibit the clearest signatures of cooperative titration. The titration curves for the Glu residues in the insulin 3-mer fibril are shown in Figure . Residues are grouped by sequence position: A:17 (Figure A), B:13 (B), and B:21 (C), with each panel displaying the behavior of the central monomer together with its neighboring monomers along the fibril axis. In all three cases, the central Glu residue exhibits a biphasic titration curve, despite possessing only a single titratable proton. This behavior indicates strong coupling between equivalent Glu residues along the fibril axis, consistent with their close spatial proximity imposed by the parallel β-sheet architecture. Such multistep transitions are characteristic of interacting titrating groups and coupled protonation equilibria, rather than independent Henderson–Hasselbalch behavior. Accordingly, the curves in Figure were not fitted to Hill-modified Henderson–Hasselbalch equations.

6.

Titration curves of Glu residues in the human 3-mer insulin fibril from constant-pH molecular dynamics simulations. Deprotonation fractions are shown as a function of pH for 9 Glu residues, grouped by equivalent positions in the insulin sequence: (A) A:17, (B) B:13, and (C) B:21. Curves are shown for monomers along the fibril, with indices indicating position relative to the central monomer: 0 refers to the central monomer, negative indices (−1) indicate monomers on one side, and positive indices (+1) indicate monomers on the other side. Circles represent raw deprotonation data from simulation, and lines are drawn between points for visual guidance.

A common feature across all sites is that the central Glu residue titrates at lower pH than the corresponding residues on terminal monomers. This ordering reflects cooperative electrostatic interactions between Glu residues on adjacent monomers along the fibril axis: the central residue interacts with titratable sites on both sides, making its initial deprotonation energetically more favorable, whereas residues on terminal monomers interact with only a single adjacent site and therefore titrate at higher pH.

Residues A:17 and B:21 (Figure A,C), which are located on the fibril exterior, display comparatively smooth and monotonic titration curves for the residues on terminal monomers, while retaining clear coupling to the central residue. Their behavior is characteristic of solvent-exposed acidic residues, with apparent pK _a shifts dominated by inter-residue electrostatic coupling rather than burial or persistent hydrogen-bonding interactions.

In contrast, B:13 (Figure B), which faces the fibril interior, exhibits markedly more complex titration behavior. Most notably, the central B:13 residue titrates first, despite its interior-facing location. This is unexpected, as burial and reduced solvation would normally disfavor charge formation in a low-dielectric environment, thereby shifting titration to higher pH and favoring deprotonation of the corresponding residues on terminal monomers. The observed ordering therefore indicates that inter-residue electrostatic coupling within the 3-mer outweighs simple desolvation effects at this site.

Notably, Glu B:13 is located in proximity to Thr A:8 and Ser A:9, both polar residues capable of stabilizing charged groups. This local arrangement creates a comparatively polar microenvironment within the fibril interior, likely mitigating the energetic penalty associated with charge formation and partially offsetting desolvation effects.

Furthermore, the central B:13 residue displays nonmonotonic titration behavior, including partial reprotonation over a narrow pH range (approximately 4.5–6). This behavior reflects redistribution among multiple charge microstates rather than a simple two-state transition. At low pH, the system predominantly occupies the fully protonated configuration. Upon increasing pH, the central residue initially deprotonates, yielding a singly charged state. However, as pH increases further and the corresponding residues on terminal monomers begin to ionize, the system preferentially populates configurations in which those terminal residues are deprotonated while the central residue remains protonated, thereby avoiding the higher electrostatic penalty associated with three simultaneously deprotonated residues. This shift in microstate populations transiently stabilizes the protonated state of the central residue, giving rise to the observed nonmonotonic titration curve.

Together, these results demonstrate that Glu residues within the 3-mer fibril are strongly coupled across monomers, with the central residue consistently acting as the initial ionization site. The interior-facing B:13 position shows particularly pronounced multistate behavior arising from a delicate balance between desolvation penalties and inter-residue electrostatic frustration. Compared to the fibril 1-mer, the 3-mer thus represents the first level of assembly at which cooperative electrostatics emerge as a dominant determinant of protonation behavior.

Because spontaneous fibril dissociation occurs on time scales far beyond those accessible to classical molecular dynamics, we instead sought a structural proxy for fibril stability under varying pH. Throughout the simulations, harmonic restraints were applied to the fibril backbone to preserve overall integrity while allowing reliable sampling of protonation equilibria. Consequently, direct monomer separation is suppressed, necessitating an indirect measure of destabilization.

To this end, we quantified intermonomer side chain–side chain contacts between neighboring insulin units as a function of pH. Contacts were defined using a 0.45 nm heavy-atom distance cutoff and evaluated between adjacent monomers along the fibril axis. Figure shows the resulting normalized total intermonomer contacts, together with a decomposition into contacts involving glutamate side chains, separated into Glu–Glu and Glu–other interactions.

7.

pH-dependent relative changes in intermonomer side chain contacts in the insulin 3-mer fibril. Pairwise side chain–side chain contacts between neighboring monomers were quantified using a heavy-atom distance cutoff of 0.45 nm. Contacts were decomposed into Glu–Glu, Glu–other, and other–other contributions, where “other” denotes all nonglutamate residues; the total number of intermonomer contacts is shown for reference. Averages were computed over trajectory frames using block averaging. For each contact class (including the total), values are normalized to the corresponding mean at pH 0.0, such that unity represents the fully protonated reference state. Error bars denote 95% confidence intervals estimated from 20 blocks. This representation emphasizes relative pH-induced changes in interfacial packing rather than absolute contact probabilities.

At low pH, where glutamate residues are predominantly protonated, interfacial packing is maximal and the number of contacts remains close to that at the fully protonated reference state. In this regime, electrostatic repulsion between neighboring glutamate residues is minimal, allowing stable intermonomer packing along the fibril axis. At intermediate pH (∼3–6), a modest increase in total contacts is observed, accompanied by a pronounced rise in Glu–Glu contacts. This behavior reflects partial deprotonation of interfacial glutamate residues and the emergence of correlated electrostatic interactions across the neighboring monomers. The resulting network of charged side chains appears to stabilize the packing arrangement between neighboring monomers. At higher pH, the trend reverses. As glutamate residues become increasingly deprotonated, Glu–Glu contacts decrease sharply, coinciding with a pronounced drop in the total number of intermonomer contacts. This loss reflects the buildup of electrostatic repulsion between negatively charged side chains aligned along the fibril axis. Glu–other contacts also decline with pH, though more gradually, while other–other contacts remain largely unchanged until very high pH. This indicates that the primary structural perturbation originates from the glutamate-mediated intermonomer network rather than from a global disruption of packing.

Notably, although harmonic restraints prevent macroscopic dissociation, the progressive loss of interfacial side chain contacts provides a sensitive indicator of pH-induced weakening of fibril cohesion. Together, these observations support a mechanistic picture in which increasing negative charge along the fibril axis progressively disfavors intermonomer packing, thereby predisposing the assembly toward destabilization at higher pH.

Strongly coupled protonation equilibria are well-established in proteins, particularly when ionizable residues are positioned in close spatial proximity. Examples include salt bridges and catalytic carboxylate pairs in glycoside hydrolases, where two residues act cooperatively as nucleophile and general acid/base. , In such systems, protonation states are energetically interdependent and cannot be described by independent Henderson–Hasselbalch behavior. Notably, coupled titrations can give rise not only to broadened or biphasic transitions but also to nonmonotonic titration curves with apparent reprotonation of a monitored site. A well-characterized example is diethylenetriaminepentaacetate (DTPA), which contains five interacting titratable groups and exhibits apparent reprotonation behavior when individual nuclei are monitored by NMR spectroscopy. ,, In this case, the observed titration curve of a given site reflects proton redistribution among multiple coupled charge microstates rather than a simple two-state transition.

The behavior of Glu B:13 in the 3-mer fibril is directly analogous. As pH increases, the system initially favors deprotonation of the central residue, consistent with stabilization of the singly charged configuration. At intermediate pH, however, population shifts toward states in which the corresponding residues on the terminal monomers are ionized while the central site becomes transiently reprotonated, thereby reducing electrostatic repulsion within the cluster. Such nonmonotonic behavior is therefore a hallmark of strongly coupled ionizable networks. Importantly, this ordering is governed primarily by interaction topology rather than solvent exposure alone. The central residue interacts electrostatically with two neighboring monomers, whereas each terminal residue interacts with only one, preferentially stabilizing central-first ionization. For the interior-facing B:13 residue, this cooperative stabilization competes with desolvation penalties that would otherwise disfavor charge formation. Local polar residues Thr A:8 and Ser A:9 likely mitigate this penalty, positioning B:13 near a delicate energetic balance that gives rise to both early ionization and transient reprotonation. Taken together, these findings indicate that the observed central-first titration and reprotonation behavior are natural consequences of cooperative electrostatic coupling within the 3-mer fibril.

Although electrostatic coupling formally emerges already in a dimer, the 3-mer fibril represents the smallest assembly containing a well-defined interior monomer that experiences interactions from neighboring chains. However, longer-range electrostatic effects and cumulative charge frustration along the fibril axis remain inherently limited by the finite extent of the assembly. To examine how these collective effects evolve in more extended fibrillar environments, we therefore proceed to investigate a larger fibril system.

Heptameric Fibril (7-mer): Amplified Electrostatic Frustration and Dehydration in an Extended Assembly

The 7-mer fibril provides a representation of an extended fibrillar environment in which a central insulin monomer experiences electrostatic interactions from multiple neighboring insulin monomers along the fibril axis. In this assembly, the two outermost monomers are separated by approximately 28 Å, corresponding to nearly 3 times the Debye length in 0.1 M NaCl solution. In contrast to the 3-mer system, where coupling is largely restricted to nearest neighbors, the 7-mer allows cumulative charge frustration and longer-range electrostatic effects to emerge. This enables direct assessment of how protonation equilibria and interfacial stability evolve as the fibril approaches a bulk-like regime.

The complete titration curves for Glu, His, and Tyr residues in the heptameric fibril are shown in Figure S12, central residues are summarized in Figure S13, and Glu residues are summarized in Figure S14. Consistent with the 3-mer system, extensive coupling between proximal titratable residues is observed. Several residues exhibit broadened or multiphasic transitions, reflecting state-dependent effective pK _a values arising from electrostatic interactions along the fibril axis. Notably, the most centrally positioned residues do not always display the strongest apparent coupling, indicating that geometric centrality alone does not determine electrostatic frustration in the 7-mer assembly.

In principle, the seven Glu residues along the fibril axis constitute a network of coupled protonation equilibria. In the absence of strict symmetry, a complete site-resolved description would require 2⁷ – 1 = 127 independent conditional equilibrium constants to account for all possible protonation microstates, each defined with respect to a specific protonation state of the remaining sites. While such a model is physically well-defined, it is practically intractable. We therefore adopt an alternative approach: rather than extracting protonation-state-conditional pK _a values, we focus on characterizing the collective process by which protons are stripped from the fibril and how this progression relates to electrostatic frustration and structural stability.

To visualize this collective protonation process, we constructed residue–pH heatmaps of the Glu residues along the fibril axis (Figure ). These maps provide a spatially resolved representation of how protonation states evolve as a function of pH and allow direct comparison between symmetry-related positions within the assembly.

8.

Residue–pH heatmaps of protonation states in the insulin fibril for (A) Glu A:17, (B) Glu B:13, and (C) Glu B:21. The color scale indicates the degree of deprotonation (0 = fully protonated, 1 = fully deprotonated) across the simulated pH range. Monomer-centered indices denote position along the fibril, where 0 corresponds to the central monomer and ±n indicates monomers located n units away on either side of the center.

The trends identified in the 3-mer become further accentuated in the 7-mer assembly. First, Glu B:13 located in the central region of the fibril titrates at lower pH than the corresponding central residues of Glu A:17 and Glu B:21. Thus, proton release from B:13 in the fibril core precedes that of other Glu residues that are similarly buried within the extended assembly. This ordering reinforces the conclusion that burial alone does not dictate the titration sequence. Instead, electrostatic interactions within the densely packed core dominate over desolvation effects, driving early ionization of B:13 despite its limited solvent exposure.

Second, the heatmaps reveal the emergence of spatially correlated protonation patterns along the fibril axis. As the pH increases into the partially deprotonated regime, proton release does not occur uniformly across neighboring monomers, but instead gives rise to alternating protonation patterns in which protonated and deprotonated states preferentially occupy spatially separated positions within the fibril. Such behavior reflects electrostatic frustration between nearby like charges, whereby deprotonation of one residue stabilizes proton retention in adjacent monomers, leading to intrinsically nonindependent and cooperative titration behavior. The persistence of these alternating patterns across extended regions of the fibril further suggests that protonation-state correlations can propagate collectively through the fibril assembly.

Comparison across the three Glu residues further highlights residue-specific electrostatic roles within the fibril. Whereas Glu A:17 and Glu B:21 display pronounced alternating protonation patterns and spatially distributed deprotonation within the fibril core, Glu B:13 exhibits earlier and more localized loss of protonation in the central monomers. This suggests that B:13 may play a distinct role in mediating electrostatic frustration within the interior of the assembly, potentially acting as an initial trigger for charge redistribution along the fibril axis.

Taken together, these observations indicate that extension from the 3-mer to the 7-mer amplifies electrostatic coupling and promotes collective proton release within the fibril core. Rather than behaving as independent titratable sites perturbed only by local burial, the Glu residues participate in a cooperative network in which protonation states are dynamically redistributed to minimize repulsive interactions. The resulting proton-stripping process is spatially organized and residue-specific.

In the following, we examine how this progressive charge redistribution correlates with structural metrics of fibril stability, allowing direct assessment of how electrostatic frustration contributes to destabilization of the extended fibril. The 3-mer analysis presented above demonstrated that increasing deprotonation leads to a redistribution of interfacial contacts followed by a marked loss of stabilizing interactions at alkaline pH. Here, we extend this analysis to the 7-mer to assess how fibril length modulates this response (Figure S15).

Decomposition of intermolecular side chain contacts into Glu–Glu and Glu–Other contributions reveals a qualitatively similar three-regime behavior in both systems. At low pH, where Glu residues remain predominantly protonated, total interfacial contacts are maximal and the assemblies exhibit compact packing. Upon partial deprotonation (pH ≈ 4–6), Glu–Glu contacts increase transiently while Glu–Other interactions decrease, yet total contacts remain nearly constant. This indicates electrostatic reorganization rather than immediate structural destabilization, and suggests cooperative buffering of emerging charge heterogeneity. At high pH, where Glu residues are largely deprotonated, Glu–Glu interactions decrease substantially. In the 3-mer, this coincides with a pronounced reduction in total contacts, consistent with electrostatic repulsion driving interfacial weakening. In contrast, the 7-mer exhibits a more gradual decline in total contacts, indicating that extended fibrils distribute electrostatic stress across multiple interfaces and are therefore more resistant to charge-induced destabilization. These results support a size-dependent stabilization mechanism in which larger assemblies partially buffer electrostatic frustration, while smaller oligomers are more susceptible to repulsive destabilization under alkaline conditions.

Despite increased burial and cumulative packing in the 7-mer, Glu B:13, whose side chain is considerably less solvent-exposed than the other glutamate residues, continues to titrate prior to more solvent-exposed sites. At first glance, this behavior appears counterintuitive, as one might anticipate that increasing burial within a hydrophobic core would penalize charge formation and shift deprotonation to higher pH relative to more solvent-exposed sites. However, experimental studies from the García-Moreno laboratory have demonstrated that the effective microscopic dielectric properties of protein interiors can be substantially higher than predicted by simple continuum models. , In particular, internal ionizable residues can be stabilized by local structural reorganization, water penetration, and polar side chain rearrangements, leading to unexpectedly favorable charge formation even within nominally buried environments. These findings underscore that solvent accessibility alone is not a reliable predictor of ionization behavior in complex protein assemblies. The persistence of early B:13 ionization in the 7-mer is therefore consistent with a locally polarizable interior capable of accommodating charge, rather than a strictly low-dielectric, rigid core. The presence of nearby Thr A:8 and Ser A:9 residues may further contribute to this stabilization. Notably, experimental substitutions of these residues to more hydrophobic amino acids significantly alter the pH-dependent aggregation behavior of insulin, ,, supporting the notion that local polarity within the fibril interior plays a central role in modulating electrostatic stability.

Taken together, the 7-mer results demonstrate that extended fibrillar assemblies do not simply behave as rigid, low-dielectric cores. Instead, collective electrostatic interactions and locally polar microenvironments enable interior ionizable residues to participate actively in protonation equilibria while simultaneously generating size-dependent electrostatic frustration. These findings provide a mechanistic bridge between protonation dynamics and the structural stability of amyloid assemblies, and motivate the broader conclusions drawn below regarding electrostatic control of fibril formation and destabilization.

Concluding Remarks

In this work, we used constant-pH molecular dynamics to dissect how protonation equilibria are reshaped as insulin transitions from its native monomeric fold into progressively larger fibrillar assemblies. While the native state displayed largely independent titrations with only modest pK _a perturbations, fibril formation fundamentally reorganizes the electrostatic landscape. Even at the level of a single fibril monomer, conformational rearrangement promotes systematic stabilization of protonated states through partial desolvation and reduced dielectric screening. Upon self-association, independent titration is replaced by strong inter-residue coupling along the fibril axis, yielding heterogeneous and cooperative protonation behavior in which equivalent residues experience distinct effective pK _a shifts depending on their position within the assembly, resulting in multistep titration profiles inaccessible to single-site models.

A central finding is that proton release initiates in the fibril core, despite its apparent hydrophobic character, demonstrating that collective electrostatic frustration outweighs simple burial effects. At low pH, near-complete protonation of acidic residues minimizes repulsive interactions and stabilizes compact interfacial packing within the fibril. As the pH increases into the partially deprotonated regime, structural integrity is preserved through redistribution of charge along the fibril axis, giving rise to alternating protonation patterns that buffer electrostatic frustration while maintaining overall intermolecular contacts. At high pH, however, widespread deprotonation overwhelms this cooperative buffering capacity, leading to progressive weakening of intermonomer contacts and size-dependent destabilization driven by electrostatic repulsion.

Together, these results establish cooperative electrostatics as a key determinant of pH-dependent fibril stability in insulin. Rather than being governed solely by burial or hydrogen bonding, stability emerges from a balance between protonation equilibria, collective charge interactions, and structural adaptability within the fibril.

More broadly, the emergent and cooperative protonation behavior observed here is unlikely to be unique to insulin fibrils, but may represent a general feature of biomolecular and soft-matter systems in which protonation-state fluctuations are strongly coupled to conformational organization and intermolecular interactions. The constant-pH molecular dynamics framework employed here may therefore be extended to investigate other systems in which protonation equilibria influence stability, binding, biological function, or pH-responsive material behavior. In particular, analysis of equilibrium protonation and conformational ensembles across pH conditions may provide insight into pH-dependent functional behavior, including enzymatic activity and optimal pH, through identification of the thermodynamically accessible states and their coupling to known or proposed functional mechanisms. Although the present framework does not directly provide kinetic information, because the stochastic protonation-state Monte Carlo moves disrupt physically meaningful dynamics, the equilibrium populations of the protonation states are often closely linked to catalytic accessibility, functional activity, and emergent pH responsiveness, potentially enabling prediction of qualitative or relative trends in pH-dependent behavior and informing the engineering of pH-responsive polymers and related materials.

In the specific case of insulin amyloid formation, our findings highlight how aggregation can qualitatively reprogram protonation behavior, creating emergent properties absent in the native protein. From an applied perspective, this framework offers a mechanistic basis for understanding why acidic conditions stabilize insulin fibrils and suggests that targeting collective charge interactions, rather than individual residues, may be a viable strategy for improving formulation stability and mitigating aggregation in protein therapeutics. More generally, the present results demonstrate how protonation equilibria can become collectively reorganized in extended molecular assemblies through the interplay between electrostatic frustration, conformational adaptability, and intermolecular organization. Such behavior may be relevant not only to protein aggregation, but also to other biomolecular and soft-matter systems exhibiting cooperative pH-responsive behavior. Within the specific context of insulin, future application of this framework to native insulin oligomerization and zinc-stabilized storage assemblies may provide complementary insight into how protonation equilibria regulate the balance between functional self-association and pathological aggregation.

All input files, analysis scripts, and Jupyter notebooks required to reproduce the simulations and analyses reported in this study are publicly available on Zenodo at DOI: 10.5281/zenodo.19027365 Processed data used to generate the figures, including protonation-state sampling results and derived structural properties, are also provided in human-readable format within the repository. Full molecular dynamics trajectory files are not included due to storage constraints. However, all simulation inputs and analysis workflows are provided to enable reproducibility of the reported results within the same computational framework (Amber/AmberTools).

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsphyschemau.6c00045.

• Derivations of the constant-pH and pH-replica exchange Monte Carlo acceptance criteria; simulation protocols; residue-specific pK _a values and Hill coefficients; titration curves; solvent-accessible surface area distributions; χ₁ dihedral distributions; structural sampling analyses; distance distributions; and pH-dependent intermolecular contact analyses for monomeric and fibrillar insulin systems (PDF)

The study was designed and conceptualized by S.H.-H. with input from N.M. Simulations and analysis were performed by S.H.-H. The results were discussed and interpreted by both authors. The manuscript was written by S.H.-H. and revised by N.M.

The authors declare no competing financial interest.