Poisson-Boltzmann Research Articles

Evaluation and improvement of Gay-Berne interaction potential to simulate 3D DLVO interaction of clay particles

This paper first presents a set of DLVO-based energy-separation functions for a pair of finite uniformly charged square platelets of infinitesimal thickness in three elementary configurations: face-to-face, edge-to-edge, and edge-to-face. The novel dataset was generated by summing the electrostatic interaction energy computed numerically by solving the non-linear 3D Poisson-Boltzmann equation and the van der Waals interaction energy calculated analytically. The dataset aims to inform qualitatively and quantitatively the energy/force separation functions used in the Discrete Element Method (DEM) and Coarse-Grained Molecular Dynamics (CGMD) modelling of clays. The same dataset was then used to calibrate and evaluate two Gay-Berne (GB)-type potentials: i) a DLVO-adapted Gay-Berne potential, where the Born-van der Waals branches of the underlying Lennard-Jones (LJ) potential are replaced with van der Waals-Columbic branches to represent DLVO interactions; ii) the Mie potential, where the exponents of the two energy terms are ‘unlocked’ instead of being set equal to 12 and 6 as per the original LJ potential. It is shown that the orientation parameter, µ, and the anisotropy parameter, ν, need to be different from µ = 2 and ν = 1 as adopted in CGMD clay modelling to capture the progression of the shape of the pair energy-separation function from face-to-face to edge-to-face and edge-to-edge configuration. It is also shown that the MIE potential (with exponents m = 3 and n = 1.5) better captures the slow decay of the electrostatic repulsive energy component of the DLVO potential energy Coulombic branch of the interaction potential compared to the DLVO-adapted GB potential, which embeds the Lennard-Jones (LJ) exponents m = 12 and n = 6.

The essentials of the finite element method to solve differential equations: an illustrative case in physics

Here we review the basic steps of the modern and accurate finite element method in its application to solve ordinary differential equations in physics. To exemplify this well-in-vogue numerical technique, we have chosen the second order Poisson-Boltzmann equation, which is a classic equation of colloid science. Aiming to formulate a viable, but didactic, implementation of the finite element technique, we have combined a linear basis of functions, the Galerkin weighted residuals method, the Swartz-Wendroff approximation and the Picard iteration algorithm. In summary, the finite element method transforms a differential equation into a simpler system of algebraic equations for the coefficients of the approximate solution in terms of a set of basis functions. We describe the full computational realisation of the finite element procedure and, also, we examine the corresponding Poisson-Boltzmann numerical predictions for various representative conditions.

Interrelationship of Electric Double Layer Theory and Microfluidic Microbial Fuel Cells: A Review of Theoretical Foundations and Implications for Performance

Microbial fuel cells and their related microfluidic systems have emerged as promising greener energy alternatives for the exploitation of avenues related to combined power and wastewater treatment operations. Moreover, the potential for their application in biosensing technology is large. However, while the fundamental principles of science that govern the design and operation of microbial fuel cells (MFCs) and microfluidic microbial fuel cells (MMFCs) are similar to those found in colloid science, the literature shows that current research lacks sufficient reference to the electrostatic and electrokinetic aspects, focusing mostly on aspects related to the architecture, design, anodes, microbial growth and metabolism, and electron transfer mechanisms. In this regard, research is yet to consider MFCs and MMFCs in the context of electrostatic and electrokinetic aspects. In this extensive review, we show, for the first time, the interrelationship of MFCs and MMFCs with electric double layer theory. Consequently, we show how the analytical solution to the mean field Poisson–Boltzmann theory relates to these systems. Moreover, we show the interrelationship between MFC and MMFCs’ performance and the electric double layer and the associated electrostatic and electrokinetic phenomena. This extensive review will likely motivate research in this direction.

Concentrated aqueous solutions of multivalent macroions enhance the electrical double layer capacitance and differential capacitance of a planar supercapacitor

The electrostatic properties of the ionic cloud, or electrical double layer, neutralizing a charged electrode are relevant to improve the storage of electrical energy and the efficiency of supercapacitors and/or batteries using coulombic fluids. In this work, we study the microscopic profiles of charged particles, integrated surface charge density, and the mean electrostatic potential associated to a pair of infinite parallel plates uncharged or, else, with opposite charge, in the presence of a salt-free aqueous solution constituted by big positive macroions and small anions, which interact via an explicit electrostatic coulombic potential and are dissolved in an implicit solvent. The particle profiles and the integrated surface charge density associated to Monte Carlo simulations and integral equations theory display an oscillatory or non-monotonic behaviour producing the phenomena of charge inversion and charge reversal. These phenomena are not predicted by the non-linear Poisson-Boltzmann equation, which highlights the importance of ion correlations and ionic volume excluded effects. The phenomenon of surface charge amplification is observed near the point of zero charge, which is yielded by the particle size-asymmetry. For a fixed concentration of macroions, a monotonic increase of the mean electrostatic potential difference between the charged parallel plates is predicted by simulations and theories as a function of the electrodes' charge. For a fixed surface charge density, the capacitance of the parallel plates capacitor increases when the ionic strength of the macroion's solution increases. A concavity inversion and an augment of the differential capacitance is also observed for an electrode near the point of zero charge as function of the macroion's concentration.

Peristalsis of Fractional Second Grade Fluid in the Presence of Electro Osmotic Phenomenon with Heat and Mass Transfer

MRI method is used in magnetic resonance imaging (MRI) to diagnose brain disorders by detecting the changing conditions within the brain. Peristaltic waves created on the flexible walls of the brain cause particles to move, and understanding this fluid flow can benefit the treatment of malignant tissues. In light of this, the electro-osmotic impact on the peristaltic flow of a fractional second grade fluid (ionic solution) in a microchannel with a heat and mass transfer is investigated. The governing equations of the modulated problem can be solved analytically by assuming a low Reynolds number (Re → 0) and a long wavelength (δ << 1). An analysis of the Poisson-Boltzmann equation is considered to examine the electro-kinetic mechanism. Using the Mathematica program, we solved a set of dimensionless equations and displayed the generated graphs. We analyzed relevant factors for their impacts on temperature, velocity, and pressure gradient. Biomedical engineers may find inspiration to create bio-microfluidic devices that aid in the movement of physiological fluids based on the model and its related findings.

Variational implicit solvation with Legendre-transformed Poisson–Boltzmann electrostatics

The variational implicit-solvent model (VISM) is an efficient approach to biomolecular interactions, where electrostatic interactions are crucial. The total VISM free energy of a dielectric boundary (i.e. solute–solvent interface) consists of the interfacial energy, solute–solvent interaction energy and dielectric electrostatic energy. The last part is the maximum value of the classical and concave Poisson–Boltzmann (PB) energy functional of electrostatic potentials, with the maximizer being the equilibrium electrostatic potential governed by the PB equation. For the consistency of energy minimization and computational stability, here we propose alternatively to minimize the convex Legendre-transformed Poisson–Boltzmann (LTPB) electrostatic energy functional of all dielectric displacements constrained by Gauss’ Law in the solute region. Both integrable and discrete solute charge densities are treated, and the duality of the LTPB and PB functionals is established. A penalty method is designed for the constrained minimization of the LTPB functional. In application to biomolecular interactions, we minimize the total VISM free energy iteratively, while in each step of such iteration, minimize the LTPB energy. Convergence of such a min–min algorithm is shown. Our numerical results on the solvation of a single ion indicate that the LTPB performs better than the PB formulation, providing possibilities for efficient biomolecular simulations.

Calculation of solvation force in molecular dynamics simulation by deep-learning method

Electrostatic calculations are generally used in studying the thermodynamics and kinetics of biomolecules in solvent. Generally, this is performed by solving the Poisson-Boltzmann equation on a large grid system, a process known to be time-consuming. In this study, we developed a deep neural network to predict the decomposed solvation free energies and forces of all atoms in a molecule. To train the network, the internal coordinates of the molecule were used as the input data, and the solvation free energies along with transformed atomic forces from the Poisson-Boltzmann equation were used as labels. Both the training and prediction tasks were accelerated on GPU. Formal tests demonstrated that our method can provide reasonable predictions for small molecules when the network is well-trained with its simulation data. This method is suitable for processing lots of snapshots of molecules in a long trajectory. Moreover, we applied this method in the molecular dynamics simulation with enhanced sampling. The calculated free energy landscape closely resembled that obtained from explicit solvent simulations.

Understanding electrostatic interaction on strong cation-exchanger via co-ion valency effects

Empirical adsorption models have been extensively used to design and optimize ion exchange chromatography (IEC) processes for proteins. The equations go 40 years back to the qualitative findings about the electrical double layer (EDL) in ion exchangers and form the basis of the stoichiometric displacement (SD) model widely used in preparative chromatography. While the SD model reduces the experimental effort to find salt-eluting conditions for the separation, knowledge transfer is restricted from one system to another. However, this limitation can be overcome by understanding the physicochemical interaction mechanism between the solid adsorbent and the electrolyte. Via a theoretical and experimental approach, we investigated the physicochemical adsorption mechanism in IEC and developed a methodology to determine it quantitatively by measuring the effective EDL thickness. We performed negative adsorption experiments in high-performance liquid chromatography to measure the excluded volume of co-ions, citrate, or oxalate on strong cation exchange resin. Together with the physical specifications of the column and the deployment of a modified nonlinear Poisson-Boltzmann equation, we identified the effects of the electrolyte composition on the size of the EDL. While it depends on the concentration, valency, and size of the counterion, we derived that the expansion of the EDL is indicated by different valencies of the carboxylate co-ions in trace amounts. Our findings provide a self-consistent theory of the transport phenomena in a solid/fluid system with all parameters specified with the physical properties of the chromatographic process. Further, optimizing the resin design or improving the adsorption and desorption conditions for biomolecules may be facilitated. Altogether, our work may improve material designing and process development and, thereby, help to overcome the concurrent technological and economic bottlenecks of the well-deployed purification step of IEC.

Bioinformatics analysis and molecular dynamics simulations of azoreductases (AzrBmH2) from Bacillus megaterium H2 for the decolorization of commercial dyes

The present study aimed to investigate the decolorization of various commercial dyes by azoreductases (AzrBmH21, AzrBmH22/3, and AzrBmH24/5) through bioinformatics means, comprising molecular docking, molecular dynamics simulation, and molecular mechanics Poisson–Boltzmann surface area (MM-PBSA). Therefore, four commercial dyes, namely acid orange 7, cresol red, methylene blue, and malachite green, were selected as potential targets for degradation by the above said azoreductases derived from Bacillus megaterium H2. The prediction of ligand binding or catalytic sites for AzrBmH21, AzrBmH22/3, and AzrBmH24/5 were performed using a machine learning algorithm based on the Prank Web and DeepSite chemoinformatic tool. This analysis revealed that several amino acids of AzrBmH2 interacted with the tested dyes, indicating the presence of distinct ligand-binding sites for AzrBmH2-dye complexes. Likewise, the binding affinity for AzrBmH21, AzrBmH22/3, and AzrBmH24/5 ranged from − 9.4 to − 5.5 kcal/mol, − 9.2 to − 5.4 kcal/mol, and − 9.0 to − 5.4 kcal/mol, respectively, with each complex stabilized at a minimum of 0–5 hydrogen bonds. MD simulations revealed stable AzrBmH2-dye complexes with RMSD and RMSF values ranging from 0.15 to 0.42 nm and 0.05 to 0.48 nm, respectively, with Rg values between 1.75 and 1.88 nm. MM-PBSA calculations indicated that the AzrBmH2–dye complexes, except for AzrBmH2–malachite green, exhibited the lowest binding energy (− 191.05 ± 7.08 to 314.19 ± 6.88 kcal/mol), with prevalent hydrophobic interactions (− 268.25 ± 12.25 to − 418.92 ± 29.45 kcal/mol) through van der Waals forces. Therefore, this study was able to highlight the potential role of enzymes, specifically azoreductases from Bacillus megaterium H2, in predicting the decolorization of commercial dyes. These findings could contribute to our understanding of the azoreductases’ mechanisms in bioremediation and for biotechnological applications.

Poisson-Boltzmann-based machine learning model for electrostatic analysis

Electrostatics is of paramount importance to chemistry, physics, biology, and medicine. The Poisson-Boltzmann (PB) theory is a primary model for electrostatic analysis. However, it is highly challenging to compute accurate PB electrostatic solvation free energies for macromolecules due to the nonlinearity, dielectric jumps, charge singularity, and geometric complexity associated with the PB equation. The present work introduces a PB-based machine learning (PBML) model for biomolecular electrostatic analysis. Trained with the second-order accurate MIBPB solver, the proposed PBML model is found to be more accurate and faster than several eminent PB solvers in electrostatic analysis. The proposed PBML model can provide highly accurate PB electrostatic solvation free energy of new biomolecules or new conformations generated by molecular dynamics with much reduced computational cost.

Structural dynamics and anti-biofilm screening of novel imidazole derivative to explore their anti-biofilm inhibition mechanism against Pseudomonas Aeruginosa

The biofilm formation is still prevalent mechanism of developing the drug resistance in the Pseudomonas aeruginosa, gram-negative bacteria, known for its major role in nosocomial, ventilator-associated pneumonia (VAP), lung infections and catheter-associated urinary tract infections. As best of our knowledge, current study first time reports the most potent inhibitors of LasR, a transcriptional activator of biofilm and virulence regulating genes in, Pseudomonas aeruginosa LasR, utilizing newly functionalized imidazoles (5a-d), synthesized via 1,3-dipolar cycloaddition using click approach. The synthesized ligands were characterized through Mass Spectrometry and 1H NMR. The binding potency and mode of biding of ligands. Quantum Mechanical(QM) methods were utilized to investigate the electronic basis, HOMO/LUMO and dipole moment of the geometry of the ligands for their binding potency. Dynamics cross correlation matrix (DCCMs) and protein surface analysis were further utilized to explore the structural dynamics of the protein. Free energy of binding of ligands and protein were further estimated using Molecular Mechanical Energies with the Poisson–Boltzmann surface area (MMPBSA) method. Molecular Docking studies revealed significant negative binding energies (5a − 10.33, 5b −10.09, 5c − 10.11, and 5d −8.33 KJ/mol). HOMO/LUMO and potential energy surface map estimation showed the ligands(5a) with lower energy gaps and larger dipole moments had relatively larger binding potency. The significant change in the structural dynamics of LasR protein due to complex formation with newlyfunctionalized imidazoles ligands. Hydrogen bond surface analysis followed by MMPBSA calculations of free energy of binding further complemented the Molecular docking revelations showing the specifically ligand (5a) having the relatively higher energy of binding(-65.22kj/mol). Communicated by Ramaswamy H. Sarma

Optimized parallelization of boundary integral Poisson-Boltzmann solvers

The Poisson-Boltzmann (PB) model governs the electrostatics of solvated biomolecules, i.e., potential, field, energy, and force. These quantities can provide useful information about protein properties, functions, and dynamics. By considering the advantages of current algorithms and computer hardware, we focus on the parallelization of the treecode-accelerated boundary integral (TABI) PB solver using the Message Passing Interface (MPI) on CPUs and the direct-sum boundary integral (DSBI) PB solver using KOKKOS on GPUs. We provide optimization guidance for users when the DSBI solver on GPU or the TABI solver with MPI on CPU should be used depending on the size of the problem. Specifically, when the number of unknowns is smaller than a predetermined threshold, the GPU-accelerated DSBI solver converges rapidly thus has the potential to perform PB model-based molecular dynamics or Monte Carlo simulation. As practical applications, our parallelized boundary integral PB solvers are used to solve electrostatics on selected proteins that play significant roles in the spread, treatment, and prevention of COVID-19 virus diseases. For each selected protein, the simulation produces the electrostatic solvation energy as a global measurement and electrostatic surface potential for local details.

Discovery of novel inhibitor via molecular dynamics simulations against D-alanyl-D-alanine carboxypeptidase of Enterobacter cloacae

Antibiotics resistance by bacterial pathogens is a major concern to public health worldwide resulting in high health care costs and rising mortality. Subtractive proteomics prioritized D-alanyl-D-alanine carboxypeptidas (DacB) enzyme from Enterobacter cloacae ATCC 13047 as a potential candidate for drugs designing to block pathogen cell wall biosynthesis. Virtual screening of an antibacterial library against the target unraveled a hit compound (2-[(1-methylsulfonylpiperidin-3-yl)methyl]-6-(1H-pyrazol-4-yl) pyrazine) showing high affinity and stability with the target. The N-methyl-N-propyl-methanesulfonamide of the compound is seen as a closed affinity towards domain involving strong hydrogen bonds with Ser41, Lys44, Ser285, and Asn287. The 4-methyl-1H-pyrazole is posed towards the open cavity of domain I and II and formed hydrophobic and hydrophilic contacts. The system is highly stable with average carbon-alpha deviations of 1.69 Å over trajectories of 400-ns. Three vital residues projected: Arg437, Arg438 and Leu400 from enzyme pocket via Radial distribution function (RDF) assay, which actively engaged the inhibitor. Further confirmation is done by estimating binding free energies, which confirms the very low delta energy of −7.24 kcal/mol in Generalized Born (GB) method and −7.4363 kcal/mol in Poisson-Boltzmann (PB) method. WaterSwap calculations were performed that revealed the energies highly converged, an agreement on good system stability. Lastly, three DacB mutants were created to investigate the role of functional active residues and a decline in binding affinity of the residues was noticed. These computational results provide a gateway for experimentalists to further confirm their efficacy both in-vitro and in-vivo. Communicated by Ramaswamy H. Sarma

New solution of the non-linear Poisson-Boltzmann differential equation for solid particle dispersions in dissymmetrical electrolytes

A new mathematical solution to the non-linear Poisson-Boltzmann differential equation for solid-liquid dispersions in presence of different dissymmetrical electrolytes was given. The analytical expressions of the surface and charge density of solid particles were given. The variations of electrostatic potential ψ (x) and charge density σ (x) of dispersed particles against the distance x were obtained. For colloidal particles in presence of E(m-n) electrolytes with m≠n with m≥3,n≥3 and for E(2–3) and E(3-2) electrolytes, the mean electrostatic potential as a function of the distance was numerically integrated by Mathematica program version 13.The experimental study of silica suspensions in presence with the following electrolytes NaCl, Na2SO4, CaCl2, Na3PO4, AlCl3, Al2(SO4)3, Ca3(PO4)2, Na4P2O7 and Na5P3O10 led to confirm the theoretical predictions obtained from the analytical solution of Poisson-Boltzmann equation. The results obtained allowed to determine the surface potential as a function of pH of the suspension and the electrostatic potential versus the distance x. The variations of the dissociation coefficient of silica surfaces were determined. An important effect of the anion and cation valences of the dissymmetrical electrolytes on the surface charge density and potential was highlighted.

Effects of joule heating and viscous dissipation on electromagneto-hydrodynamic flow in a microchannel with electroosmotic effect: Enhancement of MEMS cooling

This paper inspects the effect of Joule heating and viscous dissipation due to electric double layer (EDL) and electroosmotic effect on steady fully developed electromagnetohydrodynamic flow in a microchannel. Dimensionless formulations of the Poisson-Boltzmann, momentum, and energy equations are derived for the electric potential, velocity profile, and temperature distribution in the microchannel. Exact solutions for the temperature distributions and velocity profile were obtained using the method of undetermined coefficients. The Debye-Hückel linearization is used to get exact solution for the electric potential. The results showed that Brinkmann number [Formula: see text], Joule heating parameter [Formula: see text], Debye-Hückel parameter [Formula: see text], Hartmann number [Formula: see text], and electric field [Formula: see text] have a substantial impact on flow formation and heat transfer. The complex interaction between joule heating, viscous dissipation, and the EOF effect were accurately captured. The range values for the governing parameter for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] are [Formula: see text], and[Formula: see text] respectively.

EIF4A3 targeted therapeutic intervention in glioblastoma multiforme using phytochemicals from Indian medicinal plants – an integration of phytotherapy into precision onco-medicine

Glioblastoma Multiforme (GBM), an aggressive brain tumor (grade-IV astrocytoma), poses treatment challenges. Poor prognosis results from the rapid growth, highlighting the role of EIF4A3 in regulating non-coding RNAs. EIF4A3 promotes the expression of several non-coding RNAs, viz, Circ matrix metallopeptidase 9 (MMP9), a prominent oncogene, by interacting with the upstream region of the circMMP9 mRNA transcript and acts on cell proliferation, migration, and invasion of GBM. However, research shows that EIF4A3 knockdown inhibits glioblastoma progression and increases apoptosis. In this study, we explored the efficiency of the phytochemicals from plants like Withania somnifera and Castanea sativa with potential anti-glioblastoma effects as obtained from the Indian Medicinal Plants, Phytochemistry and Therapeutics (IMPPAT) database. Consequently, we have performed a virtual screening of the compounds against the protein EIF4A3. We further investigated the efficiency of the shortlisted compounds based on docking scores evaluated using GOLD, AutoDock4.2, LeDock, and binding free energy analyses using Molecular Mechanics Poisson–Boltzmann Surface Area (MMPBSA). Among the phytochemicals studied so far, several Withania-specific compounds from Withania somnifera and a single dietary compound, viz., Thiamine from Castanea sativa, have exhibited comparatively good blood-brain barrier permeability, significant binding affinity towards the EIF4A3, and good ADMET properties. Furthermore, we have verified the interaction stability of the lead molecules with EIF4A3 using MD simulations. Thus, the present study offers an opportunity to develop drug candidates targeting glioblastoma caused by EIF4A3 over-expression, integrating phytotherapy into precision oncology to create tailored and precise natural treatment strategies for cancer. Communicated by Ramaswamy H. Sarma

About the Force of Electrostatic Interaction between Two Spheroidal Macroparticles in the Poisson–Boltzmann Model

About the Force of Electrostatic Interaction between Two Spheroidal Macroparticles in the Poisson–Boltzmann Model

Enhancing the sensitivity of nanopipette biosensors for protein analysis

AbstractBackgroundThis paper compares experimental findings and simulation outcomes of single and multiple protein models moving through a nanopipette biosensor. It provides insights into the factors influencing the process and explores their relevance to proteomics.MethodsNanopipette biosensors were produced by pulling borosilicate glass tubes and treating them with an electron beam. A scanning electron microscope was used to characterize the nanopipettes. The study measured and modeled ionic currents for the elastase‐specific inhibitor protein. Simulation models were developed using the finite element method and Poisson–Boltzmann formalism, considering different protein configurations and translocation scenarios.ResultsThe results showed that the pore current of a nanopipette decreases as the protein approaches the nanopipette. The minimum pore current occurs at the widest part of the protein, and the current increases as the protein progresses through the nanopipette. For multiple protein translocations, the pore current decreases between the widest parts of the first and second proteins, and the lowest current is observed at the broadest part of the second protein. After the third protein, the pore current remains constant. It is also found that the fractional blockade difference, translocation speed, fluctuation in pore current, and dwell time are all affected by the number of proteins translocating through the nanopipette. The fractional blockade difference, the decrease in pore current caused by the protein, increases with the number of proteins while the translocation speed decreases. The fluctuation in pore current and dwell time is also longer for three‐protein translocations than for single‐protein translocations.ConclusionThis study offers valuable insights into biomolecule transport through nanopipettes, enhances our understanding of protein dynamics in restricted environments, and significantly contributes to single‐protein sequencing studies, drug screening, and proteomics.

Aromatase inhibitors identified from Saraca asoca to treat infertility in women with polycystic ovary syndrome via in silico and in vivo studies

Polycystic ovary syndrome (PCOS) is a widely occurring metabolic disorder causing infertility in 70%–80% of the affected women. Saraca asoca, an ancient medicinal herb, has been shown to have therapeutic effects against infertility and hormonal imbalance in women. This study was aimed to identify new aromatase inhibitors from S. asoca as an alternative to the commercially available ones via in silico and in vivo approaches. For this, 10 previously reported flavonoids from S. asoca were chosen and the pharmacodynamic and pharmacokinetic properties were predicted using tools like Autodock Vina, GROMACS, Gaussian and ADMETLab. Of the 10, procyanidin B2 and luteolin showed better interaction with higher binding energy when docked against aromatase (3S79) as compared to the commercial inhibitor letrozole. These two compounds showed higher stability in molecular dynamic simulations performed for 100 ns. Molecular mechanics Poisson–Boltzmann surface analysis indicated that these compounds have binding free energy similar to the commercial inhibitor, highlighting their great affinity for aromatase. Density functional theory analysis revealed that both compounds have a good energy gap, and ADMET prediction exhibited the drug-likeness of the two compounds. A dose-dependent administration of these two compounds on zebrafish revealed that both the compounds, at a lower concentration of 50 µg/ml, significantly reduced the aromatase concentration in the ovarian tissues as compared to the untreated control. Collectively, the in silico and in vivo findings recommend that procyanidin B2 and luteolin could be used as potential aromatase inhibitors for overcoming infertility in PCOS patients with estrogen dominance. Communicated by Ramaswamy H. Sarma

Ion-ion association is lost by linearizing the Poisson-Boltzmann equation when deriving the Debye-Hückel equation.

In this work, we demonstrate how the ion association constant can be attributed to the difference between the full Poisson-Boltzmann equation and its linearized version in very dilute solutions. We follow a pragmatic approach first by deriving an analytical approximated solution to the Poisson-Boltzmann equation, then calculating its respective Helmholtz free energy and activity coefficient, and then finally comparing it to the contribution from the mass action law principle. The final result is the Ebeling association constant. We conclude that electrostatic ion-ion interaction models miss the ion association contribution naturally introduced in higher-order electrostatic theories. We also demonstrate how the negative deviations from the Debye-Hückel limiting law can be physically attributed to the ion association phenomenon.