Correlations in Charged Multipore Systems: Implications for Enhancing Selectivity and Permeability in Nanoporous Membranes

Nanoporous membranes have emerged as powerful tools for diverse applications, including gas separation and water desalination. Achieving high permeability for desired molecules alongside exceptional rejection of other species presents a significant design challenge. One potential strategy involves optimizing the chemistry and geometry of isolated nanopores to enhance permeability and selectivity while maximizing their density within a membrane. However, the impact of the pore proximity on membrane performance remains an open question. Through path sampling simulations of model graphitic membranes with multiple subnanometer pores, we reveal that nanoscale proximity between pores detrimentally affects water permeability and salt rejection. Specifically, counterion transport is decelerated, while co-ion transport is accelerated, due to direct interactions among water molecules, salt ions, and the dipoles within neighboring pores. Notably, the observed ionic transport time scales significantly deviate from established theories such as the access resistance model but are well explained using the simple phenomenological model that we develop in this work. We use this model to prescreen and optimize pore arrangements that elicit minimal correlations at a target pore density. These findings deepen our understanding of multipore systems, informing the rational design of nanoporous membranes for enhanced separation processes such as water desalination. They also shed light on the physiology of biological cells that employ ion channel proteins to modulate ion transport and reversal potentials.

Ideal conductor/dielectric model (ICDM): A generalized technique to correct for finite-size effects in molecular simulations of hindered ion transport

Molecular simulations serve as indispensable tools for investigating the kinetics and elucidating the mechanism of hindered ion transport across nanoporous membranes. In particular, recent advancements in advanced sampling techniques have made it possible to access translocation timescales spanning several orders of magnitude. In our prior study [Shoemaker et al., J. Chem. Theory Comput. 18, 7142 (2022)], we identified significant finite size artifacts in simulations of pressure-driven hindered ion transport through nanoporous graphitic membranes. We introduced the ideal conductor model, which effectively corrects for such artifacts by assuming the feed to be an ideal conductor. In the present work, we introduce the ideal conductor dielectric model (Icdm), a generalization of our earlier model, which accounts for the dielectric properties of both the membrane and the filtrate. Using the Icdm model substantially enhances the agreement among corrected free energy profiles obtained from systems of varying sizes, with notable improvements observed in regions proximate to the pore exit. Moreover, the model has the capability to consider secondary ion passage events, including the transport of a co-ion subsequent to the traversal of a counter-ion, a feature that is absent in our original model. We also investigate the sensitivity of the new model to various implementation details. The Icdm model offers a universally applicable framework for addressing finite size artifacts in molecular simulations of ion transport. It stands as a significant advancement in our quest to use molecular simulations to comprehensively understand and manipulate ion transport processes through nanoporous membranes.

Ideal Conductor Model: An Analytical Finite-Size Correction for Nonequilibrium Molecular Dynamics Simulations of Ion Transport through Nanoporous Membranes

Modulating ion transport through nanoporous membranes is critical to many important chemical and biological separation processes. The corresponding transport timescales, however, are often too long to capture accurately using conventional molecular dynamics (MD). Recently, path sampling techniques, such as forward-flux sampling (FFS), have emerged as attractive alternatives for efficiently and accurately estimating arbitrarily long ionic passage times. Here, we use non-equilibrium MD and FFS to explore how the kinetics and mechanisms of pressure-driven chloride transport through a nanoporous graphitic membrane are affected by its lateral dimensions. We not only find ionic passage times and free energy barriers to decrease dramatically upon increasing the membrane surface area but also observe an abrupt and discontinuous change in the locus of the transition state. These strong finite size effects arise due to the cumulative effect of the periodic images of the leading ion entering the pore on the distribution of the induced excess charge at the membrane surface in the feed. By assuming that the feed is an ideal conductor, we analytically derive a finite size correction term that can be computed from the information obtained from a single simulation and successfully use it to obtain corrected free energy profiles with no dependence on the system size. We then estimate ionic passage times in the thermodynamic limit by assuming an Eyring-type dependence of rates on barriers with a size-independent prefactor. This approach constitutes a universal framework for removing finite size artifacts in molecular simulations of ion transport through nanoporous membranes and biological channel proteins.

Speeding molecular recognition by using the folding funnel: the fly-casting mechanism

Protein folding and binding are kindred processes. Many proteins in the cell are unfolded, so folding and function are coupled. This paper investigates how binding kinetics is influenced by the folding of a protein. We find that a relatively unstructured protein molecule can have a greater capture radius for a specific binding site than the folded state with its restricted conformational freedom. In this scenario of binding, the unfolded state binds weakly at a relatively large distance followed by folding as the protein approaches the binding site: the "fly-casting mechanism." We illustrate this scenario with the hypothetical kinetics of binding a single repressor molecule to a DNA site and find that the binding rate can be significantly enhanced over the rate of binding of a fully folded protein.

Exploring structures in protein folding funnels with free energy functionals: the denatured ensemble

We discuss the formulation of free energy functionals that describe the formation of structure in partially folded proteins. These free energy functionals take into account the inhomogeneous nature of contact energies, chain entropy and cooperative contributions reflecting the many body character of some folding forces like hydrophobicity, but do not directly account for non-native contacts because they assume the validity of the minimal frustration principle. We show how the free energy functionals can be used to interpret experiments on partially folded proteins that probe the fractional occupancy of specific local structures. In particular, we study the hydrogen protection factors in lysozyme studied in transient experiments by Gladwin and Evans and by Nash and Jonas using equilibrium pressure denaturation and the NMR order parameters measured by Dobson and Kim for the homologous protein alpha-lactalbumin.

Exploring structures in protein folding funnels with free energy functionals: the transition state ensemble

We use free energy functionals that account for the partial ordering of residues in the transition state ensemble to characterize the free energy surfaces for fast folding proteins. We concentrate on chymotrypsin inhibitor and lambda-repressor. We show how the explicit cooperativity that can arise from many body forces, such as side-chain ordering or hydrophobic surface burial, determines the crossover from folding with a large delocalized nucleus and the specific small classical nucleus of the type envisioned in nucleation growth scenarios. We compare the structural correlations present in the transition state ensemble obtained from free energy functionals with those inferred from experiment using extrathermodynamic free energy relations for folding time obtained via protein engineering kinetics experiments. We also use the free energy functionals to examine both the size of barriers and multidimensional representations of the free energy profiles in order to address the question of appropriate reaction coordinates for folding.

Structural correlations in protein folding funnels

While the overall energy landscape of a foldable protein can be described by means of a few parameters characterizing its statistical topography, specific energetic terms subtly bias the representative structures giving rise to residue pair correlations as in a liquid. We use a free energy functional incorporating an inhomogeneous pair contact energy along with a contact formation entropy and a cooperativity contribution to determine residue-specific contact probabilities in the denatured state and the transition state ensemble. The predicted "hot residues" for the theoretical transition state ensemble reasonably agree with experiment for chymotrypsin inhibitor 2, and generally a strong correlation exists with the measured kinetic effects of mutating residues not involved in highly solvent-exposed regions.