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 pK_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 2ⁿ charge microstates and use a criterion based on model compound pK_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 interconversions 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.

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.

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.

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.

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.

Figure 5:. Summary of results from q-canonical sampling for the E4K4 and NTL9_12–23 systems.

(a) Surface plot showing the per-residue alpha helicity, calculated using the DSSP algorithm , 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).

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.

Figure 7:. Comparison of results obtained using q-canonical sampling (green), unshifted pKa 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.