Interaction Energy Components Research Articles

RLpMIEC: High-Affinity Peptide Generation Targeting Major Histocompatibility Complex-I Guided and Interpreted by Interaction Spectrum-Navigated Reinforcement Learning.

Major histocompatibility complex (MHC) plays a vital role in presenting epitopes (short peptides from pathogenic proteins) to T-cell receptors (TCRs) to trigger the subsequent immune responses. Vaccine design targeting MHC generally aims to find epitopes with a high binding affinity for MHC presentation. Nevertheless, to find novel epitopes usually requires high-throughput screening of bulk peptide database, which is time-consuming, labor-intensive, more unaffordable, and very expensive. Excitingly, the past several years have witnessed the great success of artificial intelligence (AI) in various fields, such as natural language processing (NLP, e.g., GPT-4), protein structure prediction and engineering (e.g., AlphaFold2), and so on. Therefore, herein, we propose a deep reinforcement-learning (RL)-based generative algorithm, RLpMIEC, to quantitatively design peptide targeting MHC-I systems. Specifically, RLpMIEC combines the energetic spectrum (namely, the molecular interaction energy component, MIEC) based on the peptide-MHC interaction and the sequence information to generate peptides with strong binding affinity and precise MIEC spectra to accelerate the discovery of candidate peptide vaccines. RLpMIEC performs well in all the generative capability evaluations and can generate peptides with strong binding affinities and precise MIECs and, moreover, with high interpretability, demonstrating its powerful capability in participation for accelerating peptide-based vaccine development.

Unravelling pH/pKa influence on pH-responsive drug carriers: Insights from ibuprofen-silica interactions and comparative analysis with carbon nanotubes, sulfasalazine, and alendronate

This study employs density functional theory to explore the interaction between ibuprofen (IBU) and silica, emphasizing the influence of the trimethylsilyl (TMS) functional group for designing pH-responsive drug carriers. The surface (S) and drug (D) molecules' neutral (0) or deprotonated (−1) states were taken into consideration during the investigation. The likelihood of these states was determined based on the pKa values and the desired pH conditions. To calculate the pH-dependent interaction energy (EintpH), four different situations have been identified: S0D0, S0D−1, S−1D0, and S−1D−1.The electrostatic component of interaction energy aligns favorably with its theoretical value in both the Debye-Hückel and Grahame models. The investigation has gathered first-hand experimental data on the drug loading and release of pH-responsive mesoporous silica nanoparticles. Effective drug loading was observed in the acidic environment of the stomach (pH 2–5), followed by a release in the slightly basic to neutral pH of the small intestine (pH 7.4), These findings align with existing literature. The results revealed horizontal drug adherence on silica surfaces, improving binding capabilities. Comparisons were made with combinations involving carboxylated carbon nanotubes and ibuprofen, silica, and sulfasalazine, and silica and alendronate, exploring drug loading/release dynamics associated with positive/negative interaction energies. The investigation, supported by experimental data, contributes valuable insights into pH-responsive mesoporous silica nanoparticles, offering new design possibilities for drug carriers.

Benchmarking Swaths of Intermolecular Interaction Components with Symmetry-Adapted Perturbation Theory.

A benchmark database for interaction energy components of various noncovalent interactions (NCIs) along their dissociation curve is one of the essential needs in theoretical chemistry, especially for the development of force fields and machine-learning methods. We utilize DFT-SAPT or SAPT(DFT) as one of the most accurate methods to generate an extensive stock of the energy components, including dispersion energies extrapolated to the complete basis set limit (CBS). Precise analyses of the created data, and benchmarking the total interaction energies against the best available CCSD(T)/CBS values, reveal different aspects of the methodology and the nature of NCIs. For example, error cancellation effects between the S2 approximation and nonexact xc-potentials occur, and large charge transfer energies in some systems, including heavy atoms, can explain the lower accuracy of DFT-SAPT. This method is perfect for neutral complexes containing light nonmetals, while other systems with heavier atoms should be treated carefully. In the last part, a representative data set for all NCIs is extracted from the original data.

Approximating Electrostatic Potential of Molecules with Point Charges Mimicking the Electron Pairs

The electrostatic component used in the traditional force fields significantly impacts their accuracy in modelling the noncovalent interactions peculiar to biomolecular systems, including hydrogen bonding. In this contribution, we present a physical model for approximating the electrostatic potential of a molecule (MEP) based on the first-principle decomposition of its charge density distribution into the localized components. In contrast to conventional schemes, which typically use atom-centered charges to approximate MEP, the proposed approach locates such charges in the positions selected so as to mimic the anisotropy of the electron density distributions related to the electron pairs of atoms or covalent bonds. This peculiarity leads to a more accurate representation of the overall electrostatic potential, as verified by applying the proposed model to approximate the electrostatic component of the intermolecular interaction energy in 145 noncovalently bound molecular complexes from GMTKN55 database. This benchmark showed the root-mean-square difference between the true and approximated values of the electrostatic component of 2.7 kcal/mol, which is 2.2 times lower as compared to the traditional RESP charges method used as a baseline.

Soil organic matter facilitates the transport of microplastic by reducing surface hydrophobicity

Groundwater contamination by microplastic (MP) causes increasing concern about the critical factors that determine MP transport mechanisms in the vadose zone. Strong water repellency of native microplastic surfaces and respective attractive hydrophobic interactions with components of the soil matrix limits its mobility. However, adsorption of dissolved organic matter (DOM) onto microplastic surfaces can potentially make them more hydrophilic, enhancing their mobility. Consequently, the main objective of this study was to systematically determine the effects of DOM on both the surface properties and the mobility of polystyrene microplastic (PSP), and to use the extended DLVO theory (XDLVO) to predict PSP transport behavior with natural DOM for the first time. Breakthrough experiments were conducted in columns filled with quartz sand, through which PSP (1 µm) were percolated at either 0 or 5 mg/L dissolved organic carbon (DOC) derived from beech (Fagus sylvatica) litter at ionic strengths ranging from 0.5 to 2.5 mM CaCl2, since the cation background also affects wettability and mobility. Using sessile drop contact angle and zeta potential measurements, XDLVO interaction energy components including Lewis acid-base energies were approximated. Breakthrough curve analysis confirmed that DOM coming from natural sources strongly enhanced the mobility of PSP in the quartz sand matrix, in particular at higher ionic strength. At 2.5 mM CaCl2, the presence of DOM increased PSP breakthrough from 1 % to 60 %. Consequently, in the absence of DOM, laser microscope images showed much more pronounced formation of microplastic clusters on quartz grain surfaces than at co-percolation of PSP with DOM. This behavior could be clearly predicted with the XLDVO approach, which suggests at 0 mg/L DOC a deep primary energy minimum between individual PSP and between PSP and quartz surfaces. In contrast, repulsive hydrophilic interactions between the surfaces were predicted at 5 mg/L DOC. We conclude that even low concentrations of DOM, as usually found in subsoil, can significantly enhance the mobility of microplastic in soils via rendering the surfaces more hydrophilic. This implies potentially serious repercussions for the spread of this contaminant in the environment.

TB-IECS: an accurate machine learning-based scoring function for virtual screening

Machine learning-based scoring functions (MLSFs) have shown potential for improving virtual screening capabilities over classical scoring functions (SFs). Due to the high computational cost in the process of feature generation, the numbers of descriptors used in MLSFs and the characterization of protein–ligand interactions are always limited, which may affect the overall accuracy and efficiency. Here, we propose a new SF called TB-IECS (theory-based interaction energy component score), which combines energy terms from Smina and NNScore version 2, and utilizes the eXtreme Gradient Boosting (XGBoost) algorithm for model training. In this study, the energy terms decomposed from 15 traditional SFs were firstly categorized based on their formulas and physicochemical principles, and 324 feature combinations were generated accordingly. Five best feature combinations were selected for further evaluation of the model performance in regard to the selection of feature vectors with various length, interaction types and ML algorithms. The virtual screening power of TB-IECS was assessed on the datasets of DUD-E and LIT-PCBA, as well as seven target-specific datasets from the ChemDiv database. The results showed that TB-IECS outperformed classical SFs including Glide SP and Dock, and effectively balanced the efficiency and accuracy for practical virtual screening.

Interplay between different metal–ligand binding modes in tin(II) complexes with pyridine

In this work, the π-complexation interaction between simple tin(II) molecules SnX2 (X = H, F, Cl, Br, I) and the π-cloud of pyridine (py) in a series of their model complexes SnX2·py was characterized using a variety of quantum chemical methods. Moreover, the coexistence of this interaction and the well-known σ-complexation interaction between the lone pair on the N atom of py and another SnX2 molecule was studied in the corresponding series of model complexes SnX2·py·SnX2. The interplay between the two metal–ligand binding modes was established in terms of structural changes, interaction energies, interaction energy components and topological electron density parameters. The interaction between SnX2 and py in their SnX2·py π-complexes is weak and its strength is further diminished by the σ-complexation of another SnX2 molecule, forming the SnX2·py·SnX2 complexes. The weakening is in fact mutual yet the strength of the σ-complexation interaction is reduced only marginally.

Integrating network pharmacology and experimental validation to decipher the mechanism of action of Jingfang Granule in the treatment of viral myocarditis.

This study investigated the mechanisms of Jingfang Granule (JFG) in viral myocarditis (VMC) treatment via network pharmacology-based approach combined with molecular docking and validated the results through in vitro and in vivo experiments. The chemical composition of JFG and its therapeutic targets was queried in Traditional Chinese Medicine Systems Pharmacology (TCMSP) database. The targets related to VMC were retrieved from the disease database, and the overlapping targets were screened. Based on the STRING database, a protein-protein interaction network was constructed. Cytoscape software was used to construct the "component-target-disease" interaction network for visualization. GO and KEGG pathway enrichment analyses were performed using Metascape data. Molecular docking was performed using PyMoL2.3.0 and AutoDock Vina software programs. The target genes were further verified in vitro and in vivo. JFG contains 88 active components. The main biological targets of JFG in VMC include quercetin, luteolin, and kaempferol. The molecular docking results showed that the three key targets showed strong binding properties with both the height components of the molecular docking interaction energies. The results of experimental verification results showed that JFG may be used to treat VMC mainly by down-regulating inflammatory factors TNF-α and NF-κB and inhibiting myocardial apoptosis. The results support the network pharmacological data. JFG reduces myocardial inflammation and myocardial cell apoptosis in VMC and protects myocardial tissue.

Origins of Offset-Stacking in Porous Frameworks

Parallel-displaced π-stacking in the benzene dimer and larger polycyclic aromatic hydrocarbons is driven by competition between dispersion and exchange-repulsion interactions. The present work examines whether the same is true in porous frameworks that exhibit stacking interactions, including the [18]annulene dimer, porphyrin dimer, and several models of the covalent organic framework known as COF-1. Interaction energies and their components are computed using extended symmetry-adapted perturbation theory along two-dimensional scans representing slip-stacking. As in the polycyclic aromatic hydrocarbons studied previously, we find that the van der Waals interaction potential (defined as the sum of dispersion and Pauli repulsion) drives the system into a slip-stacked geometry. Electrostatics is a relatively small component of the total interaction energy. In the case of COF-1, the van der Waals potential drives the conformational preference whether or not a solvent molecule intercalates into the framework, although the presence of the guest (mesitylene) molecule substantially limits the low-energy slip-stacking configurations that are available. Even when the COF-1 pore is empty, a modest lateral offset of ≲1.5 Å is preferred, which is small compared to the pore size.

Effect of the QM Size, Basis Set, and Polarization on QM/MM Interaction Energy Decomposition Analysis.

Herein, an Energy Decomposition Analysis (EDA) scheme extended to the framework of QM/MM calculations in the context of electrostatic embeddings (QM/MM-EDA) including atomic charges and dipoles is applied to assess the effect of the QM region size on the convergence of the different interaction energy components, namely, electrostatic, Pauli, and polarization, for cationic, anionic, and neutral systems interacting with a strong polar environment (water). Significant improvements are found when the bulk solvent environment is described by a MM potential in the EDA scheme as compared to pure QM calculations that neglect bulk solvation. The predominant electrostatic interaction requires sizable QM regions. The results reported here show that it is necessary to include a surprisingly large number of water molecules in the QM region to obtain converged values for this energy term, contrary to most cluster models often employed in the literature. Both the improvement of the QM wave function by means of a larger basis set and the introduction of polarization into the MM region through a polarizable force field do not translate to a faster convergence with the QM region size, but they lead to better results for the different interaction energy components. The results obtained in this work provide insight into the effect of each energy component on the convergence of the solute-solvent interaction energy with the QM region size. This information can be used to improve the MM FFs and embedding schemes employed in QM/MM calculations of solvated systems.

Decoding the infrared spectra changes upon formation of molecular complexes: the case of halogen bonding in pyridine⋯perfluorohaloarene complexes.

A recently developed computational scheme is employed to interpret changes in the infrared spectra of halogen-bonded systems in terms of intermolecular interaction energy components (electrostatic, exchange, induction, dispersion) taking pyridine⋯perfluorohaloarene complexes as examples. For all complexes, we find a strong linear correlation between the different terms of the interaction-induced changes of the IR band associated with an intermolecular halogen bond stretching mode and the corresponding terms of the interaction energy, which implies that the interaction components play similar roles in both properties. This is not true for other vibrational modes localized in one of the monomers studied here, for which the corresponding interaction-induced changes in IR bands may present a completely different decomposition than the interaction energy.

Structure, Quantum Chemical and In Silico Molecular Docking Analysis of some Di-Ortho-Substituted Halogenated Biphenyls

The structures of three di-ortho-substituted halogenated biphenyls have been revisited for their optimized geometry and other quantum chemical investigations. The X-ray data, in conjunction with quantum chemical investigations, reveals some interesting results. The Hirshfeld surface analysis helps visualize various intermolecular interactions and the energy frameworks dwell further on the dominant interaction energy component for each structure. To study the inhibitory behaviour of each biphenyl against Cytochrome-P450-14alpha-sterol demethylase fungal enzyme (PDB code: 1EA1), the results of molecular docking studies suggest that the di-ortho-substituted halogenated biphenyls may be regarded as effective and efficient antifungal drugs.

Choice of DLVO approximation method for quantifying the affinity between latex particles and membranes

The Derjaguin-Landau-Verwey-Overbeek (DLVO) model, as well as the extended model (XDLVO), is popularly employed to quantify the interfacial interactions underlying membrane fouling. However, disagreements between membrane fouling extent and the interaction energies derived from DLVO or XDLVO models have been reported, which suggests gaps in the understanding. This study demonstrates that the DLVO approximation methods for predicting the interfacial foulant-membrane interaction are sensitive to the boundary condition assumptions (e.g., constant charge versus constant potential). In particular, while both the Poisson-Boltzmann (P–B) and linear superposition approximation (LSA) equations can quantify the electrostatics (EL) interaction energy component, the former assumes constant potential, while the latter additionally considers constant charge scenarios. The relative accuracy of these two equations were evaluated here. For dead-end filtration tests, flux decline trends, OCT analysis results and fouling model parameters were obtained. Regarding fouling of pristine PCTE membrane by latex particles of opposite charge signs, both the P–B and LSA equations contribute to predict relative fouling extents correctly. However, for the case of the BPEI-coated membrane, the P–B equation failed whereas the LSA equation gave good agreements. For cross-flow filtration tests in organic solvents, LSA also out-performed the P–B equation in providing more accurate predictions of membrane fouling. The results here highlight the shortcomings in the commonly used P–B equation and are expected to be potentially valuable in the development of better approximations for quantifying interfacial interaction energies.

An account of noncovalent interactions in homoleptic palladium(II) and platinum(II) complexes within the DFT framework: A correlation between geometries, energy components of symmetry-adapted perturbation theory and NCI descriptors

The present work addresses the underlying nature of weak noncovalent interactions (NCIs) in the self-assembled dimers of two square planar palladium(II) and platinum(II) complexes trans-[Pd(Hida)2] (1) and trans-[Pt(Hida)2] (2) (Hida = monoprotonated iminodiacetate) within the framework of density functional theory (DFT) in gas phase. Initial geometries of the dimers in different spatial orientations were extracted from the X-ray crystal structures, reported earlier, and optimized with three dispersion-corrected functionals that are frequently used to explore NCIs. The BP86-D3, M062X-D3 and ωB97X-D3 functionals have been used to test their performances over the present systems. The SARC-ZORA-TZVP and ZORA-def2-TZVP basis sets were applied for the metals and the remaining elements, respectively. The optimizations resulted in equilibrium geometries where the monomers are self-assembled through NCIs to form dimers in a cyclic fashion. This type of structural pattern is absent in the crystal structures of both 1 and 2. Physical components of interaction energies were investigated by symmetry-adapted perturbation theory (SAPT). The UV-Vis absorption spectra of the dimers are described by time-dependent density functional theory (TD-DFT). Global reactivity parameters for the dimers have been computed within the framework of conceptual density functional theory (CDFT). Detailed investigations on NCIs were performed for all dimer geometries. Simulated IR and 1H NMR spectra, charge transfer, QTAIM, NCI-RGD, IGM, ETS-NOCV and ELF studies confirmed the presence of intermolecular hydrogen bonds (HBs) and weak van der Waals interactions. Energies of the hydrogen bonds and associated orbital interaction energies were computed by QTAIM and ETS-NOCV methods, respectively.

Interaction between the Human OX2 Orexin Receptor and Suvorexant and Some of Its Analogues: SAPT (DFT) Interaction Energy Decomposition Analysis.

In this study, the interaction energy (Eint) of suvorexant (as an orexin receptor antagonist) and some of its analogues with the important residues of the human OX2 orexin receptor, determined by molecular docking, is calculated using the symmetry-adapted perturbation theory-density functional theory (SAPT (DFT)) method. Also, the important residues with the dominant interaction with each ligand are determined based on the obtained SAPT (DFT) interaction energies. To analyze the interaction of the receptor with each ligand, the decomposition of Eint to its constituent components including electrostatic (Eele), exchange (Eex), induction (Eind), and exchange-induction (Eex-ind), dispersion (Edisp), and exchange-dispersion (Eex-disp) is performed. The change of interaction energy components with the replacement of the benzoxazole part of suvorexant by pyrimidine containing different functional groups, thieno pyrimidine, and furo pyrimidine is also investigated, separately. It is found that the change in Eint, due to these replacements, is controlled more by the variation of the electrostatic interaction energy component of Eint than by the other interaction energy components. A linear correlation (R2 = 0.91) is found for the variation of Eint versus experimental ligand-binding affinities. Also, the existence of the linear correlation for the variation of the interaction energy components with experimental ligand-binding affinities is investigated. The variation of the electrostatic component versus experimental ligand-binding affinities shows a more linear correlation compared to the other interaction energy components.

High-throughput biochemical profiling reveals functional adaptation of a bacterial Argonaute

Argonautes are nucleic acid-guided proteins that perform numerous cellular functions across all domains of life. Little is known about how distinct evolutionary pressures have shaped each Argonaute's biophysical properties. We applied high-throughput biochemistry to characterize how Thermus thermophilus Argonaute (TtAgo), a DNA-guided DNA endonuclease, finds, binds, and cleaves its targets. We found that TtAgo uses biophysical adaptations similar to those of eukaryotic Argonautes for rapid association but requires more extensive complementarity to achieve high-affinity target binding. Using these data, we constructed models for TtAgo association rates and equilibrium binding affinities that estimate the nucleic acid- and protein-mediated components of the target interaction energies. Finally, we showed that TtAgo cleavage rates vary widely based on the DNA guide, suggesting that only a subset of guides cleaves targets on physiologically relevant timescales.

Comprehensive Basis-Set Testing of Extended Symmetry-Adapted Perturbation Theory and Assessment of Mixed-Basis Combinations to Reduce Cost.

Hybrid or "extended" symmetry-adapted perturbation theory (XSAPT) replaces traditional SAPT's treatment of dispersion with better performing alternatives while at the same time extending two-body (dimer) SAPT to a many-body treatment of polarization using a self-consistent charge embedding procedure. The present work presents a systematic study of how XSAPT interaction energies and energy components converge with respect to the choice of Gaussian basis set. Errors can be reduced in a systematic way using correlation-consistent basis sets, with aug-cc-pVTZ results converged within <0.1 kcal/mol. Similar (if slightly less systematic) behavior is obtained using Karlsruhe basis sets at much lower cost, and we introduce new versions with limited augmentation that are even more efficient. Pople-style basis sets, which are more efficient still, often afford good results if a large number of polarization functions are included. The dispersion models used in XSAPT afford much faster basis-set convergence as compared to the perturbative description of dispersion in conventional SAPT, meaning that "compromise" basis sets (such as jun-cc-pVDZ) are no longer required and benchmark-quality results can be obtained using triple-ζ basis sets. The use of diffuse functions proves to be essential, especially for the description of hydrogen bonds. The "δ(Hartree-Fock)" correction for high-order induction can be performed in double-ζ basis sets without significant loss of accuracy, leading to a mixed-basis approach that offers 4× speedup over the existing (cubic scaling) XSAPT approach.

The Bifurcated σ-Hole···σ-Hole Stacking Interactions.

The bifurcated σ-hole···σ-hole stacking interactions between organosulfur molecules, which are key components of organic optical and electronic materials, were investigated by using a combined method of the Cambridge Structural Database search and quantum chemical calculation. Due to the geometric constraints, the binding energy of one bifurcated σ-hole···σ-hole stacking interaction is in general smaller than the sum of the binding energies of two free monofurcated σ-hole···σ-hole stacking interactions. The bifurcated σ-hole···σ-hole stacking interactions are still of the dispersion-dominated noncovalent interactions. However, in contrast to the linear monofurcated σ-hole···σ-hole stacking interaction, the contribution of the electrostatic energy to the total attractive interaction energy increases significantly and the dispersion component of the total attractive interaction energy decreases significantly for the bifurcated σ-hole···σ-hole stacking interaction. Another important finding of this study is that the low-cost spin-component scaled zeroth-order symmetry-adapted perturbation theory performs perfectly in the study of the bifurcated σ-hole···σ-hole stacking interactions. This work will provide valuable information for the design and synthesis of novel organic optical and electronic materials.

Decomposition of the interaction energy of several flavonoids with Escherichia coli DNA Gyr using the SAPT (DFT) method: The relation between the interaction energy components, ligand structure, and biological activity

In this study, the SAPT (DFT) method is used to determine the components of the electronic interaction energies (electrostatic (Eele), exchange (Eex), induction (Eind), exchange-induction (Eex-ind), dispersion (Edisp), and exchange-dispersion (Eex-disp)) between the several selected flavonoids and the important residues of the active site of Escherichia coli DNA Gyr determined by molecular docking. A significant linear correlation between the calculated SAPT (DFT) interaction energies of flavonoids and their experimental pIC50 values is found, which is not observed for the free binding energies (ΔGb) of flavonoids obtained from molecular docking. The variation of the interaction energy components of flavonoids with their structural differences is investigated to find the relation between the flavonoids structures and their biological activity based on the interaction energy components.

Graphene Oxide–(Ferrocenylmethyl) Dimethylammonium Nitrate Composites as Catalysts for Ammonium Perchlorate Thermolysis

Graphene oxide has attracted attention due to its excellent catalytic properties, and it is expected to be used in the field of burning rate catalysis. To investigate the synergistic effect of graphene oxide and ferrocene derivatives, graphene oxide–(ferrocenylmethyl) dimethylammonium nitrate composites (GO–FcMANO3) were prepared and characterized by X-ray photoelectron spectroscopy (XPS), Raman analysis, X-ray diffraction (XRD) and scanning electron microscopy (SEM), and their thermal stability and catalytic effect on ammonium perchlorate (AP) were studied by thermogravimetry–differential scanning calorimetry (TG–DSC) techniques. Based on interaction region indicator (IRI) analysis, electrostatic potential (ESP) and energy decomposition analysis on the basis of forcefield (EDA-FF), the dispersion effect is the dominant component of interaction energy, and the contribution of electrostatic attraction is small. The frontier molecular orbitals illustrate that a higher highest occupied molecular orbital (HOMO) energy makes GO–FcMANO3 more susceptible to oxidization and is more conducive to catalyzing AP decomposition. The TG data illustrated that the presence of GO in the composites increased the thermal stability of FcMANO3 during AP decomposition. The catalytic effect of GO–FcMANO3 for AP decomposition manifested in two ways: it advanced the peak temperature in the second stage of AP decomposition and reduced the exothermic temperature width between two stages; it also increased the heat released during AP decomposition. Combined kinetic analysis and the Friedman method provided more detailed information about the catalytic behavior of composites for AP thermolysis, and GO–FcMANO3 composites decreased the Ea of the second stage of AP decomposition. The introduction of GO influenced the decomposition physical model of FcMANO3 for AP catalysis. The first stage of AP decomposition transformed from random nucleation and two-dimensional growth of nuclei model (A2) to random nucleation and three-dimensional growth of nuclei model (A3), and the second stage changed from the phase boundary-controlled reaction (R2) to random nucleation and two-dimensional growth of nuclei model (A2).