Extending pressure-matching to inhomogeneous systems via local-density potentials.

Bottom-up coarse-grained models describe the intermolecular structure of all-atom (AA) models with desirable accuracy and efficiency. Unfortunately, structure-based models for liquids tend to dramatically overestimate the thermodynamic pressure and, consequently, tend to vaporize under ambient conditions. By employing a volume potential to introduce additional cohesion, self-consistent pressure-matching provides a simple and robust method for accurately reproducing the pressure equation of state (EoS) for homogeneous fluids, while still preserving an accurate description of intermolecular structure. Because they depend upon the global density, though, volume potentials cannot be directly employed for inhomogeneous systems, such as liquid-vapor interfaces. In the present work, we demonstrate that volume potentials can be readily adapted as potentials of the local density. The resulting local-density potentials provide an accurate description of the structure, pressure EoS, and local density fluctuations of an AA model for liquid methanol. Moreover, we demonstrate that very slight modifications to these local-density potentials allow for a quantitative description of either local or global density fluctuations. Most importantly, we demonstrate that the resulting potentials, which were parameterized to describe a homogeneous liquid, also generate stable liquid-vapor coexistence. However, further work is necessary to more accurately reproduce the interfacial density profile.

Extended Ensemble Approach to Transferable Potentials for Low-Resolution Coarse-Grained Models of Ionomers.

We develop an extended ensemble method for constructing transferable, low-resolution coarse-grained (CG) models of polyethylene-oxide (PEO)-based ionomer chains with varying composition at multiple temperatures. In particular, we consider ionomer chains consisting of 4 isophthalate groups, which may be neutral or sulfonated, that are linked by 13 PEO repeat units. The CG models represent each isophthalate group with a single CG site and also explicitly represent the diffusing sodium counterions but do not explicitly represent the PEO backbone. We define the extended ensemble as a collection of equilibrium ensembles that are obtained from united atom (UA) simulations at 2 different temperatures for 7 chemically distinct ionomers with varying degrees of sulfonation. We employ a global force-matching method to determine the set of interaction potentials that, when appropriately combined, provide an optimal approximation to the many-body potential of mean force for each system in the extended ensemble. This optimized xn force field employs long-ranged Coulomb potentials with system-specific dielectric constants that systematically decrease with increasing sulfonation and temperature. An empirical exponential model reasonably describes the sensitivity of the dielectric to sulfonation, but we find it more challenging to model the temperature-dependence of the dielectrics. Nevertheless, given appropriate dielectric constants, the transferable xn force field reasonably describes the ion pairing that is observed in the UA simulations as a function of sulfonation and temperature. Remarkably, despite eliminating any explicit description of the PEO backbone, the CG model predicts string-like ion aggregates that appear qualitatively consistent with the ionomer peak observed in X-ray scattering experiments and, moreover, with the temperature dependence of this peak.

Trimethylamine N-oxide stabilizes proteins via a distinct mechanism compared with betaine and glycine.

We report experimental and computational studies investigating the effects of three osmolytes, trimethylamine N-oxide (TMAO), betaine, and glycine, on the hydrophobic collapse of an elastin-like polypeptide (ELP). All three osmolytes stabilize collapsed conformations of the ELP and reduce the lower critical solution temperature (LSCT) linearly with osmolyte concentration. As expected from conventional preferential solvation arguments, betaine and glycine both increase the surface tension at the air-water interface. TMAO, however, reduces the surface tension. Atomically detailed molecular dynamics (MD) simulations suggest that TMAO also slightly accumulates at the polymer-water interface, whereas glycine and betaine are strongly depleted. To investigate alternative mechanisms for osmolyte effects, we performed FTIR experiments that characterized the impact of each cosolvent on the bulk water structure. These experiments showed that TMAO red-shifts the OH stretch of the IR spectrum via a mechanism that was very sensitive to the protonation state of the NO moiety. Glycine also caused a red shift in the OH stretch region, whereas betaine minimally impacted this region. Thus, the effects of osmolytes on the OH spectrum appear uncorrelated with their effects upon hydrophobic collapse. Similarly, MD simulations suggested that TMAO disrupts the water structure to the least extent, whereas glycine exerts the greatest influence on the water structure. These results suggest that TMAO stabilizes collapsed conformations via a mechanism that is distinct from glycine and betaine. In particular, we propose that TMAO stabilizes proteins by acting as a surfactant for the heterogeneous surfaces of folded proteins.

Van der Waals Perspective on Coarse-Graining: Progress toward Solving Representability and Transferability Problems.

Low-resolution coarse-grained (CG) models provide the necessary efficiency for simulating phenomena that are inaccessible to more detailed models. However, in order to realize their considerable promise, CG models must accurately describe the relevant physical forces and provide useful predictions. By formally integrating out the unnecessary details from an all-atom (AA) model, "bottom-up" approaches can, at least in principle, quantitatively reproduce the structural and thermodynamic properties of the AA model that are observable at the CG resolution. In practice, though, bottom-up approaches only approximate this "exact coarse-graining" procedure. The resulting models typically reproduce the intermolecular structure of AA models at a single thermodynamic state point but often describe other state points less accurately and, moreover, tend to provide a poor description of thermodynamic properties. These two limitations have been coined the "transferability" and "representability" problems, respectively. Perhaps, the simplest and most commonly discussed manifestation of the representability problem regards the tendency of structure-based CG models to dramatically overestimate the pressure. Furthermore, when these models are adjusted to reproduce the pressure, they provide a poor description of the compressibility. More generally, it is sometimes suggested that CG models are fundamentally incapable of reproducing both structural and thermodynamic properties. After all, there is no such thing as a "free lunch"; any significant gain in computational efficiency should come at the cost of significant model limitations. At least in the case of structural and thermodynamic properties, though, we optimistically propose that this may be a false dichotomy. Accordingly, we have recently re-examined the "exact coarse-graining" procedure and investigated the intrinsic consequences of representing an AA model in reduced resolution. These studies clarify the origin and inter-relationship of representability and transferability problems. Both arise as consequences of transferring thermodynamic information from the high resolution configuration space and encoding this information into the many-body potential of mean force (PMF), that is, the potential that emerges from an exact coarse-graining procedure. At least in principle, both representability and transferability problems can be resolved by properly addressing this thermodynamic information. In particular, we have demonstrated that "pressure-matching" provides a practical and rigorous means for addressing the density dependence of the PMF. The resulting bottom-up models accurately reproduce the structure, equilibrium density, compressibility, and pressure equation of state for AA models of molecular liquids. Additionally, we have extended this approach to develop transferable potentials that provide similar accuracy for heptane-toluene mixtures. Moreover, these potentials provide predictive accuracy for modeling concentrations that were not considered in their parametrization. More generally, this work suggests a "van der Waals" perspective on coarse-graining, in which conventional structure-based methods accurately describe the configuration dependence of the PMF, while independent variational principles infer the thermodynamic information that is necessary to resolve representability and transferability problems.

Assessing Coupled Protein Folding and Binding Through Temperature-Dependent Isothermal Titration Calorimetry.

Broad interest in the thermodynamic driving forces of coupled macromolecular folding and binding is motivated by the prevalence of disorder-to-order transitions observed when intrinsically disordered proteins (IDPs) bind to their partners. Isothermal titration calorimetry (ITC) is one of the few methods available for completely evaluating the thermodynamic parameters describing a protein-ligand binding event. Significantly, when the effective ΔH° for the coupled folding and binding process is determined by ITC in a temperature series, the constant-pressure heat capacity change (ΔCp) associated with these coupled equilibria is experimentally accessible, offering a unique opportunity to investigate the driving forces behind them. Notably, each of these molecular-scale events is often accompanied by strongly temperature-dependent enthalpy changes, even over the narrow temperature range experimentally accessible for biomolecules, making single temperature determinations of ΔH° less informative than typically assumed. Here, we will document the procedures we have adopted in our laboratory for designing, executing, and globally analyzing temperature-dependent ITC studies of coupled folding and binding in IDP interactions. As a biologically significant example, our recent evaluation of temperature-dependent interactions between the disordered tail of FCP1 and the winged-helix domain from Rap74 will be presented. Emphasis will be placed on the use of publically available analysis programs written in MATLAB that facilitate quantification of the thermodynamic forces governing IDP interactions. Although motivated from the perspective of IDPs, the experimental design principles and data fitting procedures presented here are general to the study of most noncooperative ligand binding equilibria.

Bottom-Up Coarse-Graining of Peptide Ensembles and Helix-Coil Transitions.

This work investigates the capability of bottom-up methods for parametrizing minimal coarse-grained (CG) models of disordered and helical peptides. We consider four high-resolution peptide ensembles that demonstrate varying degrees of complexity. For each high-resolution ensemble, we parametrize a CG model via the multiscale coarse-graining (MS-CG) method, which employs a generalized Yvon-Born-Green (g-YBG) relation to determine potentials directly (i.e., without iteration) from the high-resolution ensemble. The MS-CG method accurately describes high-resolution models that fluctuate about a single conformation. However, given the minimal resolution and simple molecular mechanics potential, the MS-CG method provides a less accurate description for a high-resolution peptide model that samples a disordered ensemble with multiple distinct conformations. We employ an iterative g-YBG method to develop a CG model that more accurately describes the relevant distribution functions and free energy surfaces for this disordered ensemble. Nevertheless, this more accurate model does not reproduce the cooperative helix-coil transition that is sampled by the high resolution model. By comparing the different models, we demonstrate that the errors in the MS-CG model primarily stem from the lack of cooperative interactions afforded by the minimal representation and molecular mechanics potential. This work demonstrates the potential of the MS-CG method for accurately modeling complex biomolecular structures, but also highlights the importance of more complex potentials for modeling cooperative transitions with a minimal CG representation.

Deciphering the glycosylation code.

Asparagine-linked carbohydrates profoundly impact glycoprotein folding, stability, and structure. However, the "glycosylation code" that relates these effects to protein sequence remains unsolved. We report atomically detailed replica exchange molecular dynamics simulations in explicit solvent that systematically investigate the impact of glycosylation upon peptides with the central sequon Pro-Asn-Gly/Ala-Thr-Trp/Ala. These simulations suggest that the effects of glycosylation may be quite sensitive to steric crowding by the side chain immediately following the glycosylation site but less sensitive to stacking interactions with the aromatic Trp residue. In addition, we compare our simulated ensembles with the known structures for full length glycoproteins. These structures corroborate the simulations and also suggest a remarkable consistency between the intraprotein and protein-glycan interactions of natural glycoproteins. Moreover, our analysis highlights the significance of left-handed conformations for compact ß-hairpins at glycosylation sites. In summary, these studies elucidate basic biophysical principles for the glycosylation code.

Investigation of coarse-grained mappings via an iterative generalized Yvon-Born-Green method.

Low resolution coarse-grained (CG) models enable highly efficient simulations of complex systems. The interactions in CG models are often iteratively refined over multiple simulations until they reproduce the one-dimensional (1-D) distribution functions, e.g., radial distribution functions (rdfs), of an all-atom (AA) model. In contrast, the multiscale coarse-graining (MS-CG) method employs a generalized Yvon-Born-Green (g-YBG) relation to determine CG potentials directly (i.e., without iteration) from the correlations observed for the AA model. However, MS-CG models do not necessarily reproduce the 1-D distribution functions of the AA model. Consequently, recent studies have incorporated the g-YBG equation into iterative methods for more accurately reproducing AA rdfs. In this work, we consider a theoretical framework for an iterative g-YBG method. We numerically demonstrate that the method robustly determines accurate models for both hexane and also a more complex molecule, 3-hexylthiophene. By examining the MS-CG and iterative g-YBG models for several distinct CG representations of both molecules, we investigate the approximations of the MS-CG method and their sensitivity to the CG mapping. More generally, we explicitly demonstrate that CG models often reproduce 1-D distribution functions of AA models at the expense of distorting the cross-correlations between the corresponding degrees of freedom. In particular, CG models that accurately reproduce intramolecular 1-D distribution functions may still provide a poor description of the molecular conformations sampled by the AA model. We demonstrate a simple and predictive analysis for determining CG mappings that promote an accurate description of these molecular conformations.

Role of Ordered Proteins in the Folding-Upon-Binding of Intrinsically Disordered Proteins.

In this work, we quantitatively investigate the thermodynamic analogy between the folding of monomeric proteins and the interactions of intrinsically disordered proteins (IDPs). Motivated by the hypothesis that similar hydrophobic forces guide both globular protein folding and also IDP interactions, we present a unified experimental and computational investigation of the coupling between the folding and binding of the intrinsically disordered tail of FCP1 when interacting with the cooperatively folding winged-helix domain of Rap74. Our calorimetric measurements quantitatively demonstrate the significance of hydrophobic interactions for this binding event. Our computational studies indicate that IDPs relieve frustration at the surface of ordered proteins to generate a minimally frustrated complex that is strikingly similar to a globular monomeric protein. In summary, these results not only quantify the thermodynamic forces driving disordered protein interactions but also highlight the role of ordered proteins for IDP function.

Native-based simulations of the binding interaction between RAP74 and the disordered FCP1 peptide.

By dephosphorylating the C-terminal domain (CTD) of RNA polymerase II (Pol II), the Transcription Factor IIF (TFIIF)-associating CTD phosphatase (FCP1) performs an essential function in recycling Pol II for subsequent rounds of transcription. The interaction between FCP1 and TFIIF is mediated by the disordered C-terminal tail of FCP1, which folds to form an α-helix upon binding the RAP74 subunit of TFIIF. The present work reports a structure-based simulation study of this interaction between the folded winged-helix domain of RAP74 and the disordered C-terminal tail of FCP1. The comparison of measured and simulated chemical shifts suggests that the FCP1 peptide samples 40-60% of its native helical structure in the unbound disordered ensemble. Free energy calculations suggest that productive binding begins when RAP74 makes hydrophobic contacts with the C-terminal region of the FCP1 peptide. The FCP1 peptide then folds into an amphipathic helix by zipping up the binding interface. The relative plasticity of FCP1 results in a more cooperative binding mechanism, allows for a greater diversity of pathways leading to the bound complex, and may also eliminate the need for "backtracking" from contacts that form out of sequence.

The role of many-body correlations in determining potentials for coarse-grained models of equilibrium structure.

Coarse-grained (CG) models often employ pair potentials that are parametrized to reproduce radial distribution functions (rdf's) determined for an atomistic model. This implies that the CG model must reproduce the corresponding atomistic mean forces. These mean forces include not only a direct contribution from the corresponding interaction but also correlated contributions from the surrounding environment. The many-body correlations that influence this second contribution present significant challenges for accurately reproducing atomistic distribution functions. This work presents a detailed investigation of these many-body correlations and their significance for determining CG potentials while using liquid heptane as a model system. We employ a transparent geometric framework for directly determining CG potentials that has been previously developed within the context of the multiscale coarse-graining and generalized Yvon-Born-Green methods. In this framework, a metric tensor quantifies the relevant many-body correlations and precisely decomposes atomistic mean forces into contributions from specific interactions, which then determine the CG force field. Numerical investigations reveal that this metric tensor reflects both the CG representation and also subtle correlations between molecular geometry and intermolecular packing, but can be largely interpreted in terms of generic considerations. Our calculations demonstrate that contributions from correlated interactions can significantly impact the pair mean force and, thus, also the CG force field. Finally, an eigenvector analysis investigates the importance of these interactions for reproducing atomistic distribution functions.

Specific and nonspecific effects of glycosylation.

Glycosylation regulates vital cellular processes and dramatically influences protein folding and stability. In particular, experiments have demonstrated that asparagine (N)-linked disaccharides drive a "conformational switch" in a model peptide. The present work investigates this conformational switch via extensive atomically detailed replica exchange molecular dynamics simulations in explicit solvent. To distinguish the effects of specific and nonspecific interactions upon the peptide conformational ensemble, these simulations considered model peptides that were N-linked to a disaccharide and to a steric crowder of the same shape. The simulations are remarkably consistent with experiment and provide detailed insight into the peptide structure ensemble. They suggest that steric crowding by N-linked disaccharides excludes extended conformations, but does not significantly impact the tetrahedral structure of the surrounding solvent or otherwise alter the peptide free energy surface. However, the combination of steric crowding with specific hydrogen bonds and hydrophobic stacking interactions more dramatically impacts the peptide ensemble and stabilizes new structures.

Atomistic simulations reveal structural disorder in the RAP74-FCP1 complex.

We report atomically detailed molecular dynamics simulations characterizing the interaction of the RAP74 winged helix domain with the intrinsically disordered C-terminal of FCP1. The RAP74-FCP1 complex promotes the essential dephosphorylation of RNA polymerase II prior to initiation of transcription. Although disordered in solution, the C-terminal of FCP1 forms an amphipathic helix when bound to RAP74. Our simulations demonstrate that this interaction also reorganizes and stabilizes RAP74. These simulations illuminate the significance of hydrophobic contacts for stabilizing disordered protein complexes, provide new insight into the mechanism of protein binding by winged helix domains, and also reveal "dynamic fuzziness" in the complex as FCP1 retains significant flexibility after binding. In conjunction with our recent NMR experiments identifying residual structure in unbound FCP1, these simulations suggest that FCP1 loses relatively little conformational entropy upon binding and that the associated coupled folding-binding transition may be less sharp than expected.