$q$-Canonical Monte Carlo Sampling for Modeling the Linkage Between Charge Regulation and Conformational Equilibria of Peptides

Abstract

The overall charge content and the patterning of charged residues have a profound impact on the conformational ensembles adopted by intrinsically disordered proteins. These parameters can be altered by charge regulation, which refers to the effects of post-translational modifications, pH dependent changes to charge, and conformational fluctuations that modify the ${\text{pK}}_{\text{a}}$ values of ionizable residues. Although atomistic simulations have played a prominent role in uncovering the major sequence-ensemble relationships of IDPs, most simulations assume fixed charge states for ionizable residues. This may lead to erroneous estimates for conformational equilibria if they are linked to charge regulation. Here, we report the development of a new method we term $q$-canonical Monte Carlo sampling for modeling the linkage between charge regulation and conformational equilibria. The method, which is designed to be interoperable with the ABSINTH implicit solvation model, operates as follows: For a protein sequence with $n$ ionizable residues, we start with all $2^n$ charge microstates and use a criterion based on model compound ${\text{pK}}_{\text{a}}$ values to prune down to a subset of thermodynamically relevant charge microstates. This subset is then grouped into mesostates, where all microstates that belong to a mesostate have the same net charge. Conformational distributions, drawn from a canonical ensemble, are generated for each of the charge microstates that make up a mesostate using a method we designate as proton walk sampling. This method combines Metropolis Monte Carlo sampling in conformational space with an auxiliary Markov process that enables inter-conversions between charge microstates along a mesostate. Proton walk sampling helps identify the most likely charge microstate per mesostate. We then use thermodynamic integration aided by the multistate Bennett acceptance ratio method to estimate the free energies for converting between mesostates. These free energies are then combined with the per-microstate weights along each mesostate to estimate standard state free energies and pH dependent free energies for all thermodynamically relevant charge microstates. The results provide quantitative estimates of the probabilities and preferred conformations associated with every thermodynamically accessible charge microstate. We showcase the application of $q$-canonical sampling using two model systems. The results establish the soundness of the method and the importance of charge regulation in systems characterized by conformational heterogeneity.

1. Introduction

Intrinsically disordered proteins (IDPs) highlight the functional importance of conformational heterogeneity ^{1, 2}. Studies over the past decade have uncovered relationships between IDP sequences and global as well as local features of conformational ensembles ^3–30. To first order, the sequence-ensemble relationships ^31–33 of IDPs are governed by compositional biases such as the fraction of charged residues (FCR), net charge per residue (NCPR), and mean hydrophobicity ³⁴. In addition, sequence patterning of oppositely charged residues ²⁹ and the patterning of proline and charged residues vis-à-vis other residues ³⁰ can have a profound impact on the overall dimensions, amplitudes of conformational fluctuations, and local conformational preferences of IDPs.

Atomistic simulations based on efficient and accurate implicit solvent models ^{7, 26, 28, 31–33, 35–40} as well as improved descriptions of protein-solvent interactions using explicit models for solvent molecules ^{41, 42} have been deployed to study a wide variety of IDPs ^43–48. These simulations have yielded quantitative descriptions of sequence-ensemble relationships. The overall picture that has emerged may be summarized as follows: IDPs come in distinct sequence flavors; and the sequence-specific interplay between chain solvation vs. intramolecular interactions leads to distinct relationships between IDP sequences and conformational ensembles.

The knowledge base generated to date highlights the importance of charged residues, specifically parameters such as FCR, NCPR, and the patterning of charged residues, as determinants of sequence-ensemble relationships and as drivers of functional consequences of IDPs ^{29, 33, 34, 38, 49–52}. The effects of charged residues can be altered through charge screening, charge renormalization, or charge regulation. Charge screening refers to the effects of solution ions and conformational fluctuations on the strengths of intra- and intermolecular electrostatic interactions among charged residues of IDPs ^{5, 26, 51, 53}. In contrast, charge renormalization refers to alterations or even inversions of charge profiles ^{54, 55} that result from the accumulation of solution ions, specifically multivalent ions, around regions of high charge density. While the effects of charge renormalization have been well established for nucleic acids⁵⁶, this has yet to be demonstrated for IDPs, although such effects might prevail for highly charged polyelectrolytes such as histone tails ^{57, 58}, protamines ²⁶, and acid-rich sequences such as prothymosin $\text{α}$⁵³. It is also likely that charge renormalization contributes to phase separation via complex coacervation of complexing IDPs ^59–61.

Charge regulation refers to changes in the FCR, NCPR, and charge patterning, and these changes may arise through post-translational modifications, due to pH effects such as intracellular pH gradients ^{62, 63}, or through conformation-dependent changes to the protonation states of ionizable residues ³⁴. Due to their conformational heterogeneity, solvent accessibility, and propensity for post-translational modifications, IDPs are likely to be much more susceptible to charge regulation than folded domains. The consequences of charge regulation through post-translational modifications such as Ser / Thr phosphorylation ^{8–10, 30}, Lys acetylation ^64–67, and Arg methylation / citrullination ⁶⁸ are becoming topics of intense scrutiny for experimental studies of IDPs. Simulations of the impact of charge regulation via post-translational modifications are in their infancy, limited mainly by the absence of well-vetted forcefield parameters for modified amino acids.

Recent experiments indicate that charge regulation effects are likely to be important for disorder-to-order and order-to-disorder transitions in IDPs ⁶⁹, the overall charge profiles of IDPs ⁶, and IDP-driven liquid-liquid phase separation ⁷⁰. As an example, we consider data for the intrinsically disordered sidearms of neurofilaments ^{71, 72}. These sequences are polymer brushes comprising of multiple repeats of the hexapeptide KSPAEA. The brush height is governed by the end-to-end distance distribution of these polymers. At extremes of pH, the sidearms are either cationic (low pH) or anionic polyelectrolytes (high pH) that form stiff brushes of maximal brush height ^{71, 72}. Between these extremes, the polymer brushes should, in theory, be symmetric polyampholytes corresponding to lower brush heights. Interestingly, the pH dependence of measured brush heights is suggestive of upshifted ${\text{pK}}_{\text{a}}$ values for some fraction of the Glu residues. An even more striking example of sequence-specific charge regulation in IDPs comes from single molecule nanoscale electrometry measurements on prothymosin $\text{α}$ ⁶. These measurements provide an estimate of the net charge of the system at very low salt concentrations. At pH 7.0, and very low salt concentrations, one would estimate the net charge to be –46 based on the sequence of prothymosin $\text{α}$. In contrast, the electrometry measurements suggest an effective charge of –28.5 ± 1.2. This implies that at least a third of Glu / Asp residues in prothymosin $\text{α}$ have significantly upshifted ${\text{pK}}_{\text{a}}$ values such that at pH 7.0 they are in their protonated (uncharged) states.

Despite their importance, the impacts of pH, sequence, and conformation dependent charge regulation are seldom accounted for in simulations of IDPs. Instead, it is common practice to fix the charges of ionizable groups using the intrinsic ${\text{pK}}_{\text{a}}$ values of blocked amino acids as reference. Accordingly, the amine and guanido groups of Lys and Arg are assigned a charge of +1, the carboxyl groups of Asp and Glu are assigned charges of –1, and the imidazole group of His is typically set to be electroneutral. Critically these charge states are immutable throughout the course of a simulation, regardless of the local environment. The use of fixed charge models, which are based on the ${\text{pK}}_{\text{a}}$ values of blocked amino acids, is a questionable approach ³¹. One way to work around this approximation is to fix the chemical potential of the proton, as in constant pH molecular dynamics ^73–80 or Monte Carlo approaches ^81–83. In these approaches, pH, sequence, and conformation dependent changes to the charge states of ionizable groups are realized via the uptake or release of protons to a proton bath. Successes have been achieved in the deployment of constant pH molecular dynamics methods ^74–76 to calculate ${\text{pK}}_{\text{a}}$ shifts of surface ionizable groups of globular proteins ^{84, 85} and for pH-dependent protein dissociation ^{86, 87}. However for IDPs where over a third of the residues are ionizable, the combinatorics of proton uptake and release options becomes computationally unwieldy. This calls for different approaches that adapt advances in free energy calculations such as Wang-Landau sampling ⁸⁸.

Here, we describe a two-stage method designated as $q$-canonical Monte Carlo sampling that for a fixed temperature (a) samples conformational distributions for all thermodynamically relevant charge microstates (defined below) associated with an IDP sequence and (b) uses the multistate Bennett acceptance ratio (mBAR)⁸⁹ estimator to calculate free energy changes associated with alchemic conversions between distinct pairs of charge microstates. This two-stage sampling allows us to compute pH-dependent weights of different charge microstates and their conformational ensembles. The current version of $q$-canonical Monte Carlo sampling is designed to be interoperable with the ABSINTH implicit solvation paradigm ^{28, 90, 91}, although the tenets of the model are interoperable with any implicit solvation model.

In what follows, we describe the overall methodology and the approaches we have incorporated to make the joint sampling of charge microstates and conformational distributions a tractable proposition. We illustrate the utility of the method by applying it to a pair of archetypal systems comprising of Asp, Glu, and Lys residues.

2. Theory and Computational Details

We introduce the different aspects of $q$-canonical Monte Carlo sampling using peptide models of IDPs that contain multiple ionizable residues. For example, in a sequence such as ace-EEE-nme, where ace and nme respectively denote N-acetyl and N′-methylamide, each of the three Glu residues can either be protonated (E) or deprotonated (e). For a sequence with $n$ ionizable residues, there is a theoretical maximum of $2^n$ distinct charge microstates to consider. For ace-EEE-nme, these charge microstates are EEE, eEE, EeE, EEe, eeE, Eee, eEe, and eee, respectively. The overall “brute-force” strategy underlying $q$-canonical Monte Carlo sampling is straightforward. For each of the $2^n$ conceivable charge microstates, we obtain conformational distributions sampled from the canonical ensemble at temperature $T$. Conformational ensembles for each of the $2^n$ charge microstates are stitched together using free energy calculations. This allows us to compute pH-dependent relative populations for each of the charge microstates and the conformations for each charge microstate.

The overall computational complexity is vastly reduced by: a) Decreasing the number of charge microstates from its theoretical maximum and restricting considerations to the thermodynamically relevant microstates; b) Grouping microstates into mesostates and using a proton walk algorithm to reduce the computational cost of obtaining conformational ensembles to that of a single charge microstate; c) Minimizing the overall number of free energy calculations using maximum likelihood methods to identify the most representative charge microstates and associated conformational ensembles for each mesostate. The details are described in the following sections.

Eliminating forbidden charge microstates:

The effective number of thermodynamically relevant charge microstates can be considerably smaller than $2^n$ for sequences with $n$ ionizable residues. This pruning of charge microstates is readily achieved for sequences with a combination of acidic and basic groups. Consider the case of ace-EKEK-nme. Since $n$ = 4, there are, in theory, 16 distinct charge microstates. Based on the intrinsic ${\text{pK}}_{\text{a}}$ values of Glu and Lys, we stipulate that the protonation of Glu and deprotonation of Lys are unlikely to occur simultaneously. This allows us to disregard eight of the charge microstates (ekek, ekeK, ekEK, ekEk, EKek, EkeK, eKEk, and eKek) and designate them as forbidden microstates. Accordingly, we reduce the effective number of charge microstates to 7 from 16. The ansatz of forbidden charge microstates is particularly effective for sequences that are symmetric polyampholytes, which refers to sequences where the number of groups that can be neutralized by uptake vs. release of a proton is essentially the same. We use a defined protocol to identify forbidden charge microstates, eliminate them, and prune the set of $2^n$ charge microstates to the thermodynamically relevant set of charge microstates per mesostate. The protocol uses model compound ${\text{pK}}_{\text{a}}$ values to compute a reference free energy based on the assumption of additivity and calculates probabilities for each of the microstates along a mesostate. If this probability falls below 10^–3, then the charge microstate in question is ignored. A particular advantage of the forbidden states ansatz is the well-documented observation that polyampholytic sequences make up roughly 70% of the sequence space of IDPs where FCR exceeds 0.25. Less than 5% of this sequence space comprises of polyelectrolytes. Therefore, the forbidden states ansatz leads to substantial reduction in the number of thermodynamically relevant charge microstates.

Grouping charge microstates into mesostates:

Charge microstates can be grouped into mesostates based on their net charge. In the example considered above, eEE, EeE, and EEe make up a mesostate where the charge per microstate is –2; similarly, eeE, Eee, and eEe constitute a mesostate where the charge per microstate is –1; finally, EEE and eee belong to the two mesostates where the charge per microstate is –3 and 0, respectively. Our grouping ensures that all charge microstates within a mesostate will have the same net charge. Accordingly, each mesostate is designated by the label $q_k$ where $q$ denotes the net charge per microstate within the mesostate and $k$ denotes the total number of charge microstates within the mesostate. Grouping of charge microstates into mesostates also allows us to reduce the cost of conformational sampling per charge microstate. If a pair of microstates is within the same mesostate then the free energy changes associated with converting between charge microstates will be independent of pH. We use this to our advantage in a proton walk sampling approach, as described next. This yields relative populations of charge microstates that belong to a specific mesostate.

Proton-walk Monte Carlo sampling:

Consider a $q_k$-mesostate of net charge $q$ that comprises of $k$ charge microstates. The goal is to obtain converged conformational ensembles for each of the $k$ charge microstates. We use Metropolis Monte Carlo methods⁹² to sample conformations from the canonical ensemble for each charge microstate. Sampling is initiated for a randomly chosen charge microstate designated as $i$. In addition to switching between different conformations for a given charge microstate, we also propose switches to the identities of charge microstates. For example, for the –1₃ mesostate of EEE, a switch between charge microstates might involve a switch from microstate eeE to Eee or eEe. Such switches involve swapping the positions of protonated vs. unprotonated Glu residues in the sequence. One approach to switching of charge microstates involves a switch from charge microstate $i$ to charge microstate $j$ while keeping the conformation fixed. This can become computationally inefficient due to inevitable steric clashes involving the atom that is introduced at the site of protonation and the rest of the peptide. To solve this problem, we adapt the previously developed Hamiltonian Switch Metropolis Monte Carlo (HS-MMC or HS for short) ⁹³ whereby we introduce an auxiliary Markov process that samples from an alternative potential function to alleviate problems associated with switching between charge microstates.

The HS method works as follows: we designate the current conformational state of the system of interest as $l$; an auxiliary Markov process is spliced in whereby sampling from the potential of interest is switched to sampling from an alternate, computationally inexpensive potential that also enhances conformational exploration; a series of conformational states are then generated using the alternate potential; these proposed conformations are accepted / rejected using the standard Metropolis criterion; after some number of moves, the system finds itself in a new conformation designated as $m$; the potential is now switched back from the alternative one to the actual potential and the new state $m$ is accepted / rejected according to the criterion ⁹³:

$${\alpha }_{l\to m}=\min \left(\frac{{\pi }_m}{{\pi }_l}\frac{{\pi }_l^{\prime }}{{\pi }_m^{\prime }},1\right);$$ (1)

In Equation (1), ${\text{π}}_m$ and ${\text{π}}_l$ are the Boltzmann weights associated with the potential of interest for conformations $l$ and $m$ whereas ${{\text{π}}^{\prime }}_m$ and ${{\text{π}}^{\prime }}_l$ are Boltzmann weights associated with the alternative potential for conformations $l$ and $m$, respectively. Gelb has shown ⁹⁴ that the structure of the acceptance ratio preserves detailed balance. The HS method allows us to make arbitrary choices for the alternative potential. In our case, we use HS aided proton walks to switch between charge microstates within a mesostate (Figure 1). To enable efficiency in proton walks, we choose an alternative potential – see below – to ensure that the sidechain to be protonated / deprotonated is solvent exposed thus reducing the rejection of proton walk moves because solvent exposed groups have minimal steric overlaps with surrounding peptide atoms. Our use of an implicit solvent model alleviates any concerns about overlaps with solvent atoms.

Figure 1: Schematic of the HS aided proton walk algorithm.

During the transformation, the potential is set such that the two interchanging residues are both uncharged, while having the high free energy of solvation of the charged state, and the Lennard-Jones are set to half of those of the fully grown atom. Spheres represent amino acids, colored in black for uncharged moieties, blue for charged moities, and pink for moieties that are in the alternative potential state used for charge transfer in the auxilary Markov chain.

For the standard ABSINTH potential ^{90, 91}, the total potential energy for conformation $l$ is written as a sum of the direct mean field interaction (DMFI) with the implicit solvent, which is essentially the mean-field estimate of the conformation specific free energy of solvation, the electrostatic (el) interactions among polar and charged groups screened by a conformation specific inhomogeneous dielectric, the Lennard-Jones (LJ) potential to model van der Waals interactions, and any other terms that are incorporated to model stereoelectronic effects ²⁸ and / or geometric constraints on the system. Accordingly, the potential energy for conformation $l$ takes the form:

$$W_{\text{ABSINTH}}^{(l)}=W_{\text{DMFI}}^{(l)}+W_{\text{el}}^{(l)}+W_{\text{LJ}}^{(l)}+W_{\text{other}}^{(l)}$$ (2)

In the alternative potential, we set the electrostatic term to zero and scale the LJ term by a factor of 0.5 using a soft-core potential ⁹⁵ (see Equation (4) below) for residues whose charge microstates are to be altered.

The upshot of the HS aided proton walk algorithm is that the system, which is in conformation $l$ and charge microstate $i$ prior to the introduction of the auxiliary Markov process, can undergo a dual switch to conformation $m$ in charge microstate $j$ (Figure 1). Since all charge microstates along a mesostate have identical numbers of atoms, the switching between charge microstates does not require any special treatment, as would be the case in a grand or semi-grand canonical ensemble. Charge microstates and their associated conformational ensembles that belong to a specific mesostate make up what we refer to as a $q$-canonical ensemble. From a single Markov process, aided by the auxiliary process wherein the switching of charge microstates is realized, one obtains a distribution of conformations associated with each of the charge microstates, and the sampling frequencies identify the most likely conformations and charge microstates for a given mesostate.

Overall, the HS aided proton walk algorithm requires the specification of three sets of parameters: (a) The frequency with which a HS aided proton walk move will be attempted; (b) the average number of steps to be taken using the alternative potential before accepting or rejecting charge / conformational states; and (c) the maximum number of protons that can change position in a single move. The choice of charge microstates for switching the positions of protonated vs. deprotonated residues in a proton walk move can either be chosen at random or from a pre-set list to restrict sampling to relevant states. If the assumptions made to designate forbidden states are correct, then the two methods should yield very similar results, although restricting the states to be used for sampling leads to substantially improved efficiency. For each mesostate, we pick the most likely charge microstate and use this – and its associated conformational ensemble – in free energy calculations as described next. When the number of charge microstates per mesostate is large, there is the formal possibility that several microstates will have similar probabilities that are equivalent and high. In this case, a charge microstate is chosen at random from the most likely set.

In the current setup, the ionizable residues are Asp, Glu, Lys, and His. Extant data suggest that Arg does not become deprotonated, even in seemingly hydrophobic environments such as the interiors of globular proteins ⁹⁶ or well inside lipid bilayers ⁹⁷. A recent potentiometric and NMR investigation sets the model compound ${\text{pK}}_{\text{a}}$ of Arg to be 13.8 ± 0.1 ⁹⁸, which is 1.8 pH units higher than the value of ~12 that is quoted in textbooks. These data suggest that it is reasonable and appropriate to assume that Arg will always be protonated. Histidine has two neutral forms and it is treated as a special case in the proton walk algorithm where the proton is allowed to switch positions within a residue as well as between other His residues along the sequence. The fundamental structure of the algorithm does not change, but the frequency with which proton walks are attempted will increase in accord with the number of His residues in the sequence.

Setup of the free energy calculations:

We shall consider a pair of mesostates adjacent in charge space and designate them as $q_r$ and $\left(q_r{}_{\pm 1}\right)=q_s$. The most likely charge microstate within mesostates $q_r$ and $q_s$ are designated as $q_{ri}$ and $q_{sj}$, respectively. We calculate the free energy change associated with alchemic conversion between $q_{ri}$ and $q_{sj}$. These calculations are performed using modules introduced into the CAMPARI modeling suite (http://campari.sourceforge.net). The transformation between charge microstates from adjacent mesostates is based on thermodynamic integration (TI) and uses three distinct Kirkwood coupling parameters^{99, 100} ${\text{λ}}_{\text{LJ}}$, ${\text{λ}}_{\text{DMFI}}$ and ${\text{λ}}_{\text{el}}$, respectively that are inserted as pre-factors into Equation (2). Each parameter λ takes values between 0 and 1. The transformation involves addition of a proton to a specific site on the peptide of interest. The new site is introduced using four distinct steps, with ${\text{λ}}_{\text{LJ}}$ increasing systematically between 0 and 1 in steps of 0.25. Once ${\text{λ}}_{\text{LJ}}$ = 1, the values for ${\text{λ}}_{\text{DMFI}}$ and ${\text{λ}}_{\text{el}}$, are set to unity in one step. Sampling along the alchemic path between charge microstates is aided by the use of Hamiltonian replica exchange ^{101, 102} between pairs of replicas corresponding to different potentials defined by the coupling parameters. Data gathered along the alchemic transformation are combined with the multistate Bennett Acceptance Ratio (mBAR) method ⁸⁹ as implemented in the mBAR package (https://github.com/choderalab/pymbar) ¹⁰³ to estimate the free energy changes associated with the transformation.

We use TI ¹⁰⁰ to evaluate the free energy change associated with transforming between charge microstates from adjacent mesostates. In order to increase the reliability of the free energy changes assessed using TI, we calculate relative free energy changes referenced to model compounds for which the free energy change associated with protonation / deprotonation are known. Not using absolute free energies allows us to circumvent some of the limitations of standard free energy methods. For example, the use of fixed bond lengths and bond angles in the ABSINTH model ⁹¹ allows us to ignore the cost of changes to bond lengths and bond angles the in the free energy calculations ^{104, 105}. This contribution remains constant and is independent of changes to the rest of the system properties and these contributions are dereferenced by subtraction. Accordingly, if the transformation is the protonation of a Glu / Asp or deprotonation of Lys, then we perform calculations of the free energy change associated with Protonation In a Reference Model-compound (PIRM) and dereference this against experimentally derived free energy changes for the same reference model compound. Accordingly, we estimate the free energy for the conversion between charge microstates $q_{ri}$ and $q_{sj}$ using the relationship:

$$\text{Δ}{F}_{i(r)\leftrightarrow j(s)}=\text{Δ}{F}_{\text{PIRM}}^{\text{exp}}+\left(\text{Δ}{F}_{i(r)\leftrightarrow j(s)}^{\text{mBAR}}-\text{Δ}{F}_{\text{PIRM}}^{\text{mBAR}}\right);$$ (3)

In Equation (3), $\text{Δ}{F}_{\text{PIRM}}^{\text{exp}}$ is the experimentally derived free energy change associated with protonation in the relevant reference model compound and $\text{Δ}{F}_{\text{PIRM}}^{\text{mBAR}}$ is the estimate obtained from the simulations for the same model compound. This approach sets the reference energy scales using experimentally derived values for $\text{Δ}{F}_{\text{PIRM}}$. The approach prescribed in Equation (3) allows us to reduce systematic errors in the free energy calculations that arise due to errors in the ABSINTH forcefield.

The ABSINTH model allows for simulations as a function of salt concentration using explicitly modeled solution ions ^{91, 106}. Additionally, the model is versatile in allowing for the incorporation of temperature dependent reference free energies of solvation for model compounds that constitute the backbone and sidechain moieties ⁷. Therefore, care is taken to ensure that values for $\text{Δ}{F}_{\text{PIRM}}^{\text{exp}}$ are derived by treating the temperature and salt concentration as part of the protein context that will influence changes in pH dependence of the protonation reaction. This is important given clear evidence for salt ¹⁰⁷ and temperature dependence ¹⁰⁸ of the deprotonation reaction of sidechains with ionizable groups. Salt and temperature dependent ${\text{pK}}_{\text{a}}$ values used to calculate $\text{Δ}{F}_{\text{PIRM}}^{\text{exp}}$ are obtained from the data of Platzer et al. ¹⁰⁹.

Details of the paths chosen for TI:

Since the free energy of solvation is by far the most important change in energy among all transformation coordinates, we choose a path for alchemic transformation that allows a drastic reduction in the number of replicas while ensuring smooth transformations between replicas. To avoid singularities linked to having a charge assigned to a dummy atom, changes in LJ parameters are made after charges have been turned off. To further smooth the energy landscape across replicas, we use a soft core potential for the changes in LJ potential ¹¹⁰, using a ${\text{λ}}_{\text{LJ}}$ dependent potential of the form:

$$W_{\text{LJ}}\left(r;{\text{λ}}_{\text{LJ}}\right)=4\text{ε}{\text{λ}}_{\text{LJ}}\left[{\left(0.5\left(1-{\text{λ}}_{\text{LJ}}\right)+{\left(\frac{r}{\sigma }\right)}^6\right)}^{-2}-{\left(0.5\left(1-{\text{λ}}_{\text{LJ}}\right)+{\left(\frac{r}{\sigma }\right)}^6\right)}^{-1}\right];$$ (4)

Here, $\text{ε}$ and $\text{σ}$ are the well depth and hard sphere radii for the interaction pair in question and the overall functional form derives from previous calibrations ^{111, 112}. We then change the free energy of solvation of the appropriate solvation groups, simultaneously, introducing an energy bath that compensates for the difference in reference free energy between the two end states. We arrived at an optimal schedule for changing the Kirkwood coupling parameters that result in relatively high overlaps and low variance for estimates to the changes in free energy ¹¹³, with just six steps (see Table 1). The creation of an atom ex nihilo is the single largest factor that influences rejection rates between adjacent steps along the transformation process. Accordingly, as noted above, we vary ${\text{λ}}_{\text{LJ}}$ in increments of 0.25 whereas all other parameters are varied in one step from 0 to 1.

Table 1:

Values for the Kirkwood coupling parameters used in setting up efficient paths for TI calculations.

Replicas	${\text{λ}}^{\text{Q}\text{off}}$	${\text{λ}}_{\text{LJ}}$	${\text{λ}}^{\text{FOS}}$	${\text{λ}}^{\text{Q}\text{on}}$
0	0	0	0	0
1	1	0.25	0	0
2	1	0.5	0	0
3	1	0.75	0	0
4	1	1	0	0
5	1	1	1	1

Generation of pH dependent conformational ensembles:

To illustrate the method, we introduce a simple system, ace-EE-nme, and the corresponding charge microstates as shown in Figure 2. This system calls for two free energy calculations between adjacent mesostates and one HS aided proton walk simulation along the –1₂ mesostate. For convenience, we set the free energy of the fully protonated state (ee) to be zero. In step (1) we perform HS aided proton walk simulations for the –1₂ mesostate that comprises of charge microstates eE and Ee, which identifies Ee as the most likely charge microstate. We also perform standard Metropolis Monte Carlo simulations for the charge microstates EE and ee. These simulations yield conformational distributions for each of the charge microstates. In step (2), we pick the most likely charge microstate for the –1₂ mesostate (Ee) and estimate the free energy changes associated with the transformation between Ee and EE as well as Ee and ee.

Figure 2: Illustration of $q$-canonical sampling for the ace-EE-nme system.

The schematic lists the four charge microstates, depicts the grouping of charge microstates into mesostates, the use of proton walk sampling to extract weights for different charge microstates within a mesostate, and the use of free energy sampling for estimating the free energies for alchemic transformation between adjacent mesostates.

The standard state free energies associated with each of the relevant charge microstates viz., $F_{\text{Ee}}^0$ and $F_{\text{EE}}^0$ changes are calculated using estimates based on mBAR by setting $F_{\text{ee}}^0=0$ and using the following equations:

$$\begin{array}{c}{F}_{\text{Ee}}^0=F_{\text{ee}}^0+\text{Δ}{F}_{\text{ee}\to \text{Ee}}^{\text{mBAR}}=\text{Δ}{F}_{\text{ee}\to \text{Ee}}^{\text{mBAR}};\\ \text{and}{F}_{\text{EE}}^0=F_{\text{Ee}}^0+\text{Δ}{F}_{\text{Ee}\to \text{EE}}^{\text{mBAR}}=\text{Δ}{F}_{\text{ee}\to \text{Ee}}^{\text{mBAR}}+\text{Δ}{F}_{\text{Ee}\to \text{EE}}^{\text{mBAR}};\end{array}$$ (5)

For an arbitrary pH, we have to also account for the chemical potential of the proton and accordingly, the relevant free energies, in terms of the thermal energy RT become:

$$\begin{array}{r}{F}_{\text{eE}}(\text{pH})=F_{\text{eE}}^0-RT\ln \left({10}^{\text{pH}}\right),\\ \text{and}{F}_{\text{EE}}(\text{pH})=F_{\text{EE}}^0-RT\ln \left({10}^{\text{pH}}\right);\end{array}$$ (6)

For systems with $n$ ionizable residues, the standard state free energies and the pH dependent free energies for each charge microstate are calculated in direct analogy to the approach shown in Equations (5) and (6), respectively. The standard state and pH dependent free energies for each of the thermodynamically relevant charge microstates are calculated using the results obtained from the free energy estimator and the relative weights obtained for microstates along a mesostate using proton walk sampling. For a mesostate $q_k$ of net charge $q$ that comprises of $k$ charge microstates, the calculations analogous to those Equations (5) and (6) yield the standard state and pH-dependent free energies for charge microstate $i$. If $w_{j\left(q_k\right)}$ and $w_{i\left(q_k\right)}$ are the weights obtained from visitation frequencies of charge microstates $j$ and $i$ from mesostate $q_k$, then the relevant free energies for charge microstates $j\ne i$ are obtained using:

$$F_{j\left(q_k\right)}^0=F_{i\left(q_k\right)}^0-RT\ln \left[\frac{w_{j\left(q_k\right)}}{w_{i\left(q_k\right)}}\right],$$ (7)

$$\text{and}{F}_{j\left(q_k\right)}(\text{pH})=F_{i\left(q_k\right)}(\text{pH})-RT\ln \left[\frac{w_{j\left(q_k\right)}}{w_{i\left(q_k\right)}}\right];$$ (8)

Given the information obtained in Equations (7) and (8), the free energies associated with each mesostate can be calculated using information regarding the free energies per microstate. For example, the standard state free energies and pH dependent free energies are calculated as:

$$\begin{array}{c}{F}_{q_k}^0=-RT\ln \left(\sum _{i=1}^k\exp \left[-\frac{F_{i\left(q_k\right)}^0}{RT}\right]\right)\\ \text{and}{F}_{q_k}(\text{pH})=-RT\ln \left(\sum _{i=1}^k\exp \left[\frac{F_{i\left(q_k\right)}(\text{pH})}{RT}\right]\right);\end{array}$$ (9)

In Equation (9), $F_{i\left(q_k\right)}^0$ and $F_{i\left(q_k\right)}(\text{pH})$ are the standard state and pH dependent free energies, respectively for charge microstate $i$ from mesostate $q_k$, while $F_{q_k}^0$ and $F_{q_k}(\text{pH})$ are the standard-state and pH dependent mesostate free energies, respectively. For a system with $n_T$ thermodynamically relevant charge microstates distributed across all possible mesostates, we calculate the pH dependent, charge-microstate-specific Boltzmann probabilities as:

$$p_i(\text{pH})=\frac{w_i}{\sum _{j=1}^{n_T}{w}_j}\text{where}{w}_i=\exp \left(-\frac{F_i(\text{pH})}{RT}\right);$$ (10)

These calculations yield the pH dependent populations for the entire set of thermodynamically relevant charge microstates.

Obtaining estimates of errors:

The statistical error associated with free energy estimates based on mBAR are obtained using the method of Shirts et al. ⁸⁹. Errors in estimates of visitation frequencies for distinct charge microstates along a mesostate are linked to the quality of the proton walk Monte Carlo simulations. These errors are estimated using bootstrapping and a non-parametric resampling of the state probabilities, using 10³ distinct samples. Errors in estimates of standard state free energies for each of the charge microstates are obtained by propagation of the errors linked to each method along the path used for their determination. Because the probabilities are constructed relative to a reference charge microstate, the error grows as we consider charge microstates that are farther away from the reference microstate. To minimize the error for all charge microstates, we construct the free energies and their respective errors starting from both sides of the pH range (i.e., fully basic vs. fully acidic states). Since the charge microstates used for the construction are the same in both directions, the mean is independent of the direction used to retrieve the free energies, and only the estimates of errors will change. As a consequence, we set the error in the estimate of the free energy as the minimum error obtained for each microstate.

Archetypal systems used to demonstrate the $q$-canonical method:

We demonstrate the applicability and working of $q$-canonical sampling use two model peptides. The peptide ace-${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$-nme, designated hereafter as ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$, is a model system that has been used to study the stabilities of so-called Charged Single Alpha Helices (CSAHs) ¹¹⁴. Sequences of CSAHs typically contain nearly perfect repeats of four Glu residues followed by four Lys residues ¹¹⁵. These repeats of blocks of four Glu and four Lys residues can range from being 25 to 200 residues long ¹¹⁶, and they are known to form long, stable alpha helices. The presence of alternating repeats of four Glu and four Lys residues lead to the postulate that alpha helicity in (${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$) repeats are stabilized by salt bridges between deprotonated Glu and protonated Lys residues that are four residues apart along the sequence ¹¹⁶. In this context, it is worth noting the (${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$) repeats are imperfect in that they are often interrupted by substitutions for Glu with Gln, Leu or other polar / non-polar residues ¹¹⁷. The second system NTL9_12–23 is a 12-residue peptide excised from the N-terminal domain of the ribosomal protein L9. Kuhlman et al.¹¹⁸ used this as a model system to quantify the pH dependence of alpha helicity in peptides that fold autonomously into structures they adopt in folded states. The sequence of NTL9_12–23 is: ace-KGKKGEIKNVAD-nme.

Details of the simulation setup for each of the systems:

The simulation temperature was set to 298 K. Four independent simulations were performed for the HS aided protonation walk and free energy calculations based on mBAR. Results from the four independent simulations were pooled for joint analysis and errors were estimated using bootstrap analysis methods as described above. The peptides were enclosed in spherical droplets (70 Å for ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ and 75 Å for NTL9_12–23). Solution ions ¹⁰⁶ including neutralizing counterions and ions to mimic NaCl concentrations of 10 mM and 100 mM, for ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ and the NTL9_12–23 peptide, respectively were modeled explicitly as has been the case in all ABSINTH-based simulations reported to date.

Each HS aided protonation walk simulation comprises of 5×10⁹ independent steps. On average, a HS step that switches to an auxiliary Markov chain was attempted once every 2.5×10⁴ steps and each auxiliary process involved sampling for 10 steps using the alternative potential. This combination results in an acceptance of ca. 4% of the proposed transitions within the auxiliary process. The identities of charge microstates and associated conformations were recorded once every 10⁴ steps. Alchemic transformation between charge microstates across adjacent mesostates uses 5×10⁹ independent steps of Monte Carlo sampling along the TI path. Hamiltonian replica exchange was attempted once every 10⁴ steps and the native and foreign energies of mBAR were evaluated once every 10⁴ steps.

Results

For the ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ system, there are nine possible mesostates corresponding to net charges per microstate that range from –4 to +4 (Figure 3). All eight residues are ionizable; accordingly the theoretical maximum for the number of charge microstates is 2⁸ = 256. However, once we deploy the forbidden microstates ansatz, only ~12% of the conceivable charge microstates are thermodynamically relevant. This reduces the number of relevant charge microstates to 31 from 256 (Figure 3). Similarly, for NTL9_12–23 a theoretical maximum of 2⁶ = 64 charge microstates is reduced to 19 thermodynamically relevant charge microstates.

Figure 3: Demonstration of the maximum likelihood minimal transformation approach to select the free energy transformation path.

The color of the background of the boxes is representative of the ratio of the population of the corresponding charged microstate compared to that of the most populated microstate in the corresponding mesostate. Red arrows represent the free energy transformation chosen, and black arrows states that are on the same layer. Black lower-case letters represent the uncharged states of the corresponding amino acids.

For each of the 31 and 19 thermodynamically relevant charge microstates of ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ and NTL9_12–23, respectively we perform HS aided proton walk simulations to obtain conformational ensembles for each of the charge microstates. We use a maximum likelihood approach to identify the charge microstate that is most representative of a particular mesostate as shown in Figure 3 for ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$. Once the most likely charge microstates for each mesostate have been identified, we compute the free energy changes for alchemic transformations between mesostates. The path used for free energy calculations for ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ is also shown in Figure 3.

From the $q$-canonical simulations that combine HS aided proton walks and free energy calculations we are able to compute the pH-dependent probabilities associated with each of the thermodynamically relevant charge microstates. These results, obtained for the ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ and NTL9_12–23 systems are summarized in Figure 4. Panels (a) and (b) show the pH-dependence of charge microstate and mesostate probabilities for the ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ whereas panels (c) and (d) shows the corresponding profiles for the NTL9_12–23 system.

Figure 4: Probability distributions for all charge microstates and mesostates obtained from $q$-canonical sampling.

(a) Results for the 31 thermodynamically relevant charge microstates of ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$; (b) Results shown in panel (a) synthesized in terms of the mesostates for ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$; (c) Results for the 19 thermodynamically relevant charge microstates of NTL9_12–23; (d) Results shown in panel (c) synthesized in terms of the mesostates for NTL9_12–23. The envelopes for mesostate distributions quantify accumulated error in our estimates of the mesotstate statistics.

For ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$, the single dominant charge microstate in the pH range of 7–9 is the one where all Glu residues are unprotonated and all Lys residues are protonated. Below a pH of 7.0, we record contributions from charge microstates corresponding to the +1₄ and +2₆ mesostates and likewise, above pH 9.0 the charge microstates from the -1₄ and −2₆ mesostates start to make significant contributions. The pH dependence of the per-residue fractional helicity profiles are shown in panel (a) in Figure 5 whereas panel (b) shows the pH dependence of the ensemble-averaged radii of gyration (${\text{R}}_{\text{g}}$) and standard deviations. The average ${\text{R}}_{\text{g}}$ values change by ~0.5 Å between a pH range of 2.0 and 10.0 and decreases more substantially for pH values that are above 10.0. The net charge of the system, calculated as a weighted average over the contributions of the spectrum of charge microstates shows that the net charge is zero only within a narrow pH range of 7.0 – 9.0, falling below zero above 9.0 and becoming positive below a pH of 7.0.

Figure 5: Summary of results from $q$-canonical sampling for the ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ and NTL9_12–23 systems.

(a) Surface plot showing the per-residue alpha helicity, calculated using the DSSP algorithm ¹⁵⁷, for each of the eight residues in ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ as a function of pH. (b) Ensemble-averaged radius of gyration (blue curve) as a function of pH and standard deviations of the pH-dependent distributions for radii of gyration (pink envelope) for ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ system. (c) Mean net charge (blue curve) as a function of pH and standard deviation for the pH-dependent net charge distributions for the ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ system. Panels (d), (e), and (f) are results for the NTL9_12–23 system and are equivalent to panels (a), (b), and (c).

Results for the NTL9_12–23 system are also summarized in Figure 4 (panels (c) and (d)) and Figure 5 (panels (d) – (f)). The wild-type sequence belongs to the +2₁ mesostate. In the pH range between 5.0 and 8.5, the NTL9_12–23 system has a negligible preference for forming alpha helices or any other regular secondary structures. This is consistent with observations from the studies of Kuhlman et al.¹¹⁸ Below a pH of 6.0 there is a discernible increase in alpha helicity, especially within the C-terminal half of the peptide. A similar increase in helicity, mostly through the middle of the peptide is observed for pH values above 8.5. These preferences can be traced to the contributions of the +3₂ mesostate in the pH range between 3.0 and 6.0 and the +1₄, 0₆, and -1₄ mesostates in the pH range 8.0–11.0. These effects, which are indicative of smooth transitions in helical preferences of NTL9₂₃ well away from the model compound ${\text{pK}}_{\text{a}}$ values of Asp, Glu, and Lys are suggestive of upshifted ${\text{pK}}_{\text{a}}$ values for Glu 17 and Asp 23 combined with downshifted ${\text{pK}}_{\text{a}}$ for Lys residues, although the magnitudes shifts are dependent on the specific Lys residue.

Apparent ${\text{pK}}_{\text{a}}$ values of ionizable residues depend on sequence context:

In Figure 6 we show our quantification of the probability of deprotonating different Asp, Glu, and Lys residues within the two systems. In panel (a) the vertical lines shown in green and black correspond, respectively to the model compound ${\text{pK}}_{\text{a}}$ values of Glu and Lys. The apparent ${\text{pK}}_{\text{a}}$ values of the different Glu and Lys residues within ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ are estimated as the pH value at which the probability of observing the residue in a deprotonated state is precisely 0.5. Values for the apparent ${\text{pK}}_{\text{a}}$ values calculated using the $q$-canonical simulations are shown in Table 2. The apparent ${\text{pK}}_{\text{a}}$ value of E1 is downshifted by 0.2 pH units with respect to the model compound value; conversely, when compared to the model compound ${\text{pK}}_{\text{a}}$ value of Glu, the apparent ${\text{pK}}_{\text{a}}$ values of E2, E3, and E4 are upshifted by 0.67, 0.73, and 0.57 pH units, respectively. The apparent ${\text{pK}}_{\text{a}}$ values of the four Lys residues are upshifted by 0.4 pH units with respect to the model compound ${\text{pK}}_{\text{a}}$ values.

Figure 6: Probability of deprotonating ionizable residues as a function of pH.

(a) Results for the eight ionizable residues within ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$. The green and black vertical lines intersect the abscissa at model compound ${\text{pK}}_{\text{a}}$ values for Glu and Lys, respectively. The horizontal dashed line intersects the ordinate at the value of 0.5. The intersection of this horizontal dashed line with the residue-specific “titration curves” is used to estimate the apparent ${\text{pK}}_{\text{a}}$ value for the residue in question. The curves are colored according to the residues as shown in the legend. (b) Results for the six ionizable residues within NTL9_12–23. The vertical lines shown in blue, green, and cyan intersect the abscissa at pH values that correspond to the model compound ${\text{pK}}_{\text{a}}$ values for Asp, Glu, and Lys, respectively.

Table 2:

Apparent ${\text{pK}}_{\text{a}}$ values calculated using $q$-canonical sampling for ionizable residues within ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ and NTL9_12–23.

E 4 K 4	Residue	Apparent pK a	NTL9 12–23	Residue	Apparent pK a
	E1	4.11		K12	10.00
	E2	4.98		K14	9.74
	E3	5.04		K15	9.80
	E4	4.88		E17	4.45
	K5	10.77		K19	10.00
	K6	10.72		D23	4.34
	K7	10.70
	K8	10.72

For NTL9_12–23 the apparent ${\text{pK}}_{\text{a}}$ values for Glu17 and Asp23 are 4.45 and 4.35, respectively. Kuhlman et al.¹¹⁸ used pH titrations and chemical shift measurements to estimate the ${\text{pK}}_{\text{a}}$ values of Glu17 and Asp23 to be 4.11 ± 0.17 and 4.11 ± 0.11, respectively. The values obtained using $q$-canonical sampling (Table 2) are in accord with the estimates from experiments. The key point being that the apparent ${\text{pK}}_{\text{a}}$ values of both residues are upshifted with respect to their model compound values, with the shift being more pronounced for Asp23, in accord with the experiments of Kuhlman et al.¹¹⁸ These results highlight the importance of local sequence contexts on the ${\text{pK}}_{\text{a}}$ values of ionizable residues. The $q$-canonical simulations also help quantify the apparent ${\text{pK}}_{\text{a}}$ values for Lys residues as summarized in Table 2.

Comparison of q-canonical sampling to results obtained using unshifted ${\text{pK}}_{\text{a}}$ values and fixed charge models:

As noted in the introduction, fixed charge models specify charges for ionizable residues using the ${\text{pK}}_{\text{a}}$ values of model compounds. We performed simulations using fixed charge models and compared the results to those obtained using the $q$-canonical approach (Figure 7). In the fixed charge simulations, the charges of ionizable residues are immutable. For example, Glu is assumed to be deprotonated for all pH values above its model compound ${\text{pK}}_{\text{a}}$ and protonated for all pH values below the model compound ${\text{pK}}_{\text{a}}$. Accordingly, for a sequence such as ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$, only the ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$, ${\text{e}}_{\text{4}}{\text{K}}_{\text{4}}$, and ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ charge microstates contribute to the calculation of the pH dependent profiles. We can allow for charge state fluctuations while assuming unshifted ${\text{pK}}_{\text{a}}$ values by weighting the contributions of all thermodynamically relevant charge microstates by model compound ${\text{pK}}_{\text{a}}$ values. Comparisons are summarized in terms of the structural quantifies namely, the fractional helicity, calculated in terms of ensemble-averaged DSSP-H values – panels (a) and (b) – for ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ and NTL9_12–23 and the ensemble-averaged ${\text{R}}_{\text{g}}$ values – panels (c) and (d). These comparisons show that pronounced deviations from the $q$-canonical results come from the fixed charge simulations thus highlighting the errors associated with quenching charge state fluctuations. Accounting for these fluctuations vastly improves the calculated pH dependent profiles vis-à-vis the $q$-canonical results and this is true even if we assume that the ${\text{pK}}_{\text{a}}$ values are unshifted with respect to the model compounds. However, the impact of shifted ${\text{pK}}_{\text{a}}$ values is made clear in the quantitative comparisons of the results from $q$-canonical sampling and those obtained using unshifted ${\text{pK}}_{\text{a}}$ values (Figure 7).

Figure 7: Comparison of results obtained using $q$-canonical sampling (green), unshifted ${\text{pK}}_{\text{a}}$ values that nevertheless use all of the thermodynamically relevant charge microstates (green), and fixed charge models (black).

The top row shows how the ensemble-averaged helical propensities, quantified as probabilities, vary with pH for the ${\text{E}}_{\text{4}}{\text{K}}_{\text{4}}$ system (left) vs. the NTL9_12–23 system (right). The bottom row shows a similar comparative analysis for the ensemble-averaged radii of gyration.

Discussion

We have described a new method, which we designate as $q$-canonical Monte Carlo sampling, to model the effects of pH, sequence, and conformation dependent charge regulation in peptides and IDPs. We applied $q$-canonical Monte Carlo sampling to two systems and show how the method yields a complete pH dependent description of populations for charge microstates and their conformations. The results highlight the holistic picture one obtains for the diversity of charge microstates that contribute to conformational distributions. They also highlight the highly averaged descriptions one obtains using fixed charge models that do not allow for charge regulation. Strictly speaking, the results one obtains using fixed charge models will be accurate if and only if sequence context and conformational changes do not influence ${\text{pK}}_{\text{a}}$ values of ionizable groups. Results obtained using fixed charge models are also likely to be reliable if a single mesostate dominates over a range of pH values. This is likely to be the case for proteins characterized by minimal conformational heterogeneity ^119–121. In contrast, systems such as IDPs will require full consideration of contributions from all thermodynamically relevant charge microstates, and $q$-canonical sampling enables this sort of sampling.

Although we have demonstrated the deployment of $q$-canonical sampling using the ABSINTH implicit solvation and forcefield paradigm, there is nothing about its design that prevents its interoperability with other implicit solvation models. However, some of the distinct advantages of ABSINTH are likely to be lost in making $q$-canonical sampling interoperable with other classes of implicit solvation models, especially those that come under the rubric of Poisson ^{122, 123} or generalized Born ⁸² approaches.

Ongoing work:

Effects of charge regulation are likely to become more pronounced in longer sequences where long-range interactions between non-nearest neighbor residues can alter charge states. Extant data already highlight the effects of charge regulation in sequences where the local charge density is high, as would be the case with the striking example of prothymosin $\text{α}$ ⁶. Long linear clusters of charged residues ³⁴ and the modulating effects of long-range interactions ²⁹ are rather common in intrinsically disordered proteins. Therefore, we expect that $q$-canonical sampling applied to a host of recently studied systems will reveal the contributions from charge regulation.

Although we have focused our narrative on the application of $q$-canonical sampling to the effects of charge regulation in IDPs, it is noteworthy that accounts of ${\text{pK}}_{\text{a}}$ shifts have been well documented for folded proteins, especially variants of staphylococcal nuclease ^{96, 98, 119–121, 124–152}. The design of $q$-canonical sampling does not come with a formal restriction of being applicable to IDPs alone. The absence of this restriction is also true for ABSINTH / ABSINTH-C models. These are physics-based implicit solvation models that are interoperable with standard molecular mechanics forcefields. Accordingly, the model itself can be applied in conjunction with $q$-canonical sampling to modeling the effects of ${\text{pK}}_{\text{a}}$ shifts in folded proteins. This might require the use of hybrid Monte Carlo and torsional molecular dynamics methods that have been designed to work with the ABSINTH-style models ¹⁵³. Estimators of enthalpies and entropies designed to be interoperable with the mBAR method ⁹⁵ will help in uncovering the contributions of entropy to charge regulation effects. Grouping of microstates into mesostates, and the general approach of decoupling proton rearrangements and proton release should be usable in conjunction free energy methods for the study of charge regulation effects in folded proteins.

Conclusions

We have presented a detailed description of the $q$-canonical sampling methodology and applied it to a set of short peptides, with the longest sequence being NTL9_12–23. Our demonstration of the initial version of $q$-canonical sampling was based on the default ABSINTH implicit solvation model and forcefield paradigm with one essential distinction: we used the bond length and bond angle parameters derived from the CHARMM forcefield, which is different from the standard practice of using the Engh and Huber ¹⁵⁴ values for bond lengths and bond angles. Importantly, recent efforts, based on the development of a quantitative touchstone for statistics of backbone dihedral angles for all twenty residues ¹⁵⁵, Choi and Pappu developed an optimized version of ABSINTH, and referred to as ABSINTH-C ¹⁵⁶, that vastly improves the description of local conformational equilibria. As a follow up study to the developments of ABSINTH-C and $q$-canonical sampling for modeling charge regulation effects we are combining the results of the two efforts to perform a large-scale calibration of how the two major improvements to ABSINTH-based simulations alter our descriptions of conformational equilibria for a variety of IDP systems.