Variational Methods for Biomolecular Modeling

Structure, function and dynamics of many biomolecular systems can be characterized by the energetic variational principle and the corresponding systems of partial differential equations (PDEs). This principle allows us to focus on the identification of essential energetic components, the optimal parametrization of energies, and the efficient computational implementation of energy variation or minimization. Given the fact that complex biomolecular systems are structurally non-uniform and their interactions occur through contact interfaces, their free energies are associated with various interfaces as well, such as solute-solvent interface, molecular binding interface, lipid domain interface, and membrane surfaces. This fact motivates the inclusion of interface geometry, particular its curvatures, to the parametrization of free energies. Applications of such interface geometry based energetic variational principles are illustrated through three concrete topics: the multiscale modeling of biomolecular electrostatics and solvation that includes the curvature energy of the molecular surface, the formation of microdomains on lipid membrane due to the geometric and molecular mechanics at the lipid interface, and the mean curvature driven protein localization on membrane surfaces. By further implicitly representing the interface using a phase field function over the entire domain, one can simulate the dynamics of the interface and the corresponding energy variation by evolving the phase field function, achieving significant reduction of the number of degrees of freedom and computational complexity. Strategies for improving the efficiency of computational implementations and for extending applications to coarse-graining or multiscale molecular simulations are outlined.Comment: 36 page

In quest of a systematic framework for unifying and defining nanoscience

This article proposes a systematic framework for unifying and defining nanoscience based on historic first principles and step logic that led to a “central paradigm” (i.e., unifying framework) for traditional elemental/small-molecule chemistry. As such, a Nanomaterials classification roadmap is proposed, which divides all nanomatter into Category I: discrete, well-defined and Category II: statistical, undefined nanoparticles. We consider only Category I, well-defined nanoparticles which are >90% monodisperse as a function of Critical Nanoscale Design Parameters (CNDPs) defined according to: (a) size, (b) shape, (c) surface chemistry, (d) flexibility, and (e) elemental composition. Classified as either hard (H) (i.e., inorganic-based) or soft (S) (i.e., organic-based) categories, these nanoparticles were found to manifest pervasive atom mimicry features that included: (1) a dominance of zero-dimensional (0D) core–shell nanoarchitectures, (2) the ability to self-assemble or chemically bond as discrete, quantized nanounits, and (3) exhibited well-defined nanoscale valencies and stoichiometries reminiscent of atom-based elements. These discrete nanoparticle categories are referred to as hard or soft particle nanoelements. Many examples describing chemical bonding/assembly of these nanoelements have been reported in the literature. We refer to these hard:hard (H-n:H-n), soft:soft (S-n:S-n), or hard:soft (H-n:S-n) nanoelement combinations as nanocompounds. Due to their quantized features, many nanoelement and nanocompound categories are reported to exhibit well-defined nanoperiodic property patterns. These periodic property patterns are dependent on their quantized nanofeatures (CNDPs) and dramatically influence intrinsic physicochemical properties (i.e., melting points, reactivity/self-assembly, sterics, and nanoencapsulation), as well as important functional/performance properties (i.e., magnetic, photonic, electronic, and toxicologic properties). We propose this perspective as a modest first step toward more clearly defining synthetic nanochemistry as well as providing a systematic framework for unifying nanoscience. With further progress, one should anticipate the evolution of future nanoperiodic table(s) suitable for predicting important risk/benefit boundaries in the field of nanoscience

Coupling nonpolar and polar solvation free energies in implicit solvent models

Recent studies on the solvation of atomistic and nanoscale solutes indicate that a strong coupling exists between the hydrophobic, dispersion, and electrostatic contributions to the solvation free energy, a facet not considered in current implicit solvent models. We suggest a theoretical formalism which accounts for coupling by minimizing the Gibbs free energy of the solvent with respect to a solvent volume exclusion function. The resulting differential equation is similar to the Laplace-Young equation for the geometrical description of capillary interfaces but is extended to microscopic scales by explicitly considering curvature corrections as well as dispersion and electrostatic contributions. Unlike existing implicit solvent approaches, the solvent accessible surface is an output of our model. The presented formalism is illustrated on spherically or cylindrically symmetrical systems of neutral or charged solutes on different length scales. The results are in agreement with computer simulations and, most importantly, demonstrate that our method captures the strong sensitivity of solvent expulsion and dewetting to the particular form of the solvent-solute interactions

Coupling hydrophobicity, dispersion, and electrostatics in continuum solvent models

An implicit solvent model is presented that couples hydrophobic, dispersion, and electrostatic solvation energies by minimizing the system Gibbs free energy with respect to the solvent volume exclusion function. The solvent accessible surface is the output of the theory. The method is illustrated with the solvation of simple solutes on different length scales and captures the sensitivity of hydration to the particular form of the solute-solvent interactions in agreement with recent computer simulations

Proton movement and coupling in the POT family of peptide transporters

POT transporters represent an evolutionarily well-conserved family of proton-coupled transport systems in biology. An unusual feature of the family is their ability to couple the transport of chemically diverse ligands to an inwardly directed proton electrochemical gradient. For example, in mammals, fungi, and bacteria they are predominantly peptide transporters, whereas in plants the family has diverged to recognize nitrate, plant defense compounds, and hormones. Although recent structural and biochemical studies have identified conserved sites of proton binding, the mechanism through which transport is coupled to proton movement remains enigmatic. Here we show that different POT transporters operate through distinct proton-coupled mechanisms through changes in the extracellular gate. A high-resolution crystal structure reveals the presence of ordered water molecules within the peptide binding site. Multiscale molecular dynamics simulations confirm proton transport occurs through these waters via Grotthuss shuttling and reveal that proton binding to the extracellular side of the transporter facilitates a reorientation from an inward- to outward-facing state. Together these results demonstrate that within the POT family multiple mechanisms of proton coupling have likely evolved in conjunction with variation of the extracellular gate

Proton movement and coupling in the POT family of peptide transporters

POT transporters represent an evolutionarily well-conserved family of proton-coupled transport systems in biology. An unusual feature of the family is their ability to couple the transport of chemically diverse ligands to an inwardly directed proton electrochemical gradient. For example, in mammals, fungi, and bacteria they are predominantly peptide transporters, whereas in plants the family has diverged to recognize nitrate, plant defense compounds, and hormones. Although recent structural and biochemical studies have identified conserved sites of proton binding, the mechanism through which transport is coupled to proton movement remains enigmatic. Here we show that different POT transporters operate through distinct proton-coupled mechanisms through changes in the extracellular gate. A high-resolution crystal structure reveals the presence of ordered water molecules within the peptide binding site. Multiscale molecular dynamics simulations confirm proton transport occurs through these waters via Grotthuss shuttling and reveal that proton binding to the extracellular side of the transporter facilitates a reorientation from an inward- to outward-facing state. Together these results demonstrate that within the POT family multiple mechanisms of proton coupling have likely evolved in conjunction with variation of the extracellular gate

The Accuracy of Generalized Born Forces

Generalized Born (GB) models play an important role in biomolecular dynamics simulations. The issue of the accuracy of solvation forces computed according to these models has not been investigated in much detail to the best of our knowledge. Optimal atomic radii have been derived for most used force fields in order to reproduce molecular dynamics results under the Poisson\u2013Boltzmann (PB) continuum model. In principle, if optimized radii are used, the PB model reproduces well MD simulations. For this reason in this contribution, we consider the most used GB model implementations in publicly available molecular dynamics simulation softwares and assess the accuracy of GB solvation forces compared to the reference PB ones. The results show that current implementations that compute forces by pairwise summations correlate very well with PB forces if judicious choice of model parameters is performed. The latter can be obtained by linear fitting of solvation self-energies computed by the most accurate GBR6 GB model

Residual structures, conformational fluctuations, and electrostatic interactions in the synergistic folding of two intrinsically disordered proteins

To understand the interplay of residual structures and conformational fluctuations in the interaction of intrinsically disordered proteins (IDPs), we first combined implicit solvent and replica exchange sampling to calculate atomistic disordered ensembles of the nuclear co-activator binding domain (NCBD) of transcription coactivator CBP and the activation domain of the p160 steroid receptor coactivator ACTR. The calculated ensembles are in quantitative agreement with NMRderived residue helicity and recapitulate the experimental observation that, while free ACTR largely lacks residual secondary structures, free NCBD is a molten globule with a helical content similar to that in the folded complex. Detailed conformational analysis reveals that free NCBD has an inherent ability to substantially sample all the helix configurations that have been previously observed either unbound or in complexes. Intriguingly, further high-temperature unbinding and unfolding simulations in implicit and explicit solvents emphasize the importance of conformational fluctuations in synergistic folding of NCBD with ACTR. A balance between preformed elements and conformational fluctuations appears necessary to allow NCBD to interact with different targets and fold into alternative conformations. Together with previous topology-based modeling and existing experimental data, the current simulations strongly support an ‘‘extended conformational selection’’ synergistic folding mechanism that involves a key intermediate state stabilized by interaction between the C-terminal helices of NCBD and ACTR. In addition, the atomistic simulations reveal the role of long-range as well as short-range electrostatic interactions in cooperating with readily fluctuating residual structures, which might enhance the encounter rate and promote efficient folding upon encounter for facile binding and folding interactions of IDPs. Thus, the current study not only provides a consistent mechanistic understanding of the NCBD/ACTR interaction, but also helps establish a multi-scale molecular modeling framework for understanding the structure, interaction, and regulation of IDPs in general