Pairwise Approximation Research Articles

Multibody interactions between protein inclusions in the pointlike curvature model for tense and tensionless membranes.

The pointlike curvature constraint (PCC) model and the disk detachment angle (DDA) model for the deformation-mediated interaction of conical integral protein inclusions in biomembranes are compared in the small deformation regime. Given the radius of membrane proteins, which is comparable to the membrane thickness, it is not obvious which of the two models should be considered the most adequate. For two proteins in a tensionless membranes, the PCC and DDA models coincide at the leading-order in their separation but differ at the next order. Yet, for distances larger than twice the proteins diameter, the difference is less than . Like the DDA model, the PCC model includes all multibody interactions in a non-approximate way. The asymptotic many-body energy of triangular and square protein clusters is exactly the same in both models. Pentagonal clusters, however, behave differently; they have a vanishing energy in the PCC model, while they have a non-vanishing weaker asymptotic power law in the DDA model. We quantify the importance of multibody interactions in small polygonal clusters of three, four and five inclusions with identical or opposite curvatures in tensionless or tense membranes. We find that the pairwise approximation is almost always very poor. At short separation, the three-body interaction is not sufficient to account for the full many-body interaction. This is confirmed by equilibrium Monte Carlo simulations of up to ten inclusions.

The spindle approximation of network epidemiological modeling

Understanding the dynamics of spreading and diffusion on networks is of critical importance for a variety of processes in real life. However, predicting the temporal evolution of diffusion on networks remains challenging as the process is shaped by network topology, spreading non-linearities, and heterogeneous adaptation behavior. In this study, we propose the ‘spindle vector’, a new network topological feature, which shapes nodes according to the distance from the root node. The spindle vector captures the relative order of nodes in diffusion propagation, thus allowing us to approximate the spatiotemporal evolution of diffusion dynamics on networks. The approximation simplifies the detailed connections of node pairs by only focusing on the nodal count within individual layers and the interlayer connections, seeking a compromise between efficiency and complexity. Through experiments on various networks, we show that our method outperforms the state-of-the-art on BA networks with an average improvement of 38.6% on the mean absolute error. Additionally, the predictive accuracy of our method exhibits a notable convergence with the pairwise approximation approach with the increasing presence of quadrangles and pentagons in WS networks. The new metric provides a general and computationally efficient approach to predict network diffusion problems and is of potential for a large range of network applications.

Pairwise Additivity and Three-Body Contributions for Density Functional Theory-Based Protein-Ligand Interaction Energies.

The prediction of protein-ligand binding energies is crucial in computer-assisted drug design. This property can be calculated in a straightforward fashion as the difference in the energies between a binding site-ligand complex and the separated binding site and ligand. Often, though, there is value in knowing how different amino acid residues in the protein binding site interact with the ligand. In this case, the interaction energy can be calculated as the sum of pairwise energies between each amino acid residue in the binding site and the ligand, and the sum of these energies is often equated with the total interaction energy. The validity of this pairwise additivity approximation can be assessed by experimental evidence, such as double-mutant cycles. In this work, we test the pairwise additivity approximation on the sulfotransferase-l-DOPA complex for 16 density functional theory (DFT) methods with varying degrees of exact (Hartree-Fock) exchange. Several "families" of functionals are studied, including BLYP, B3LYP, and CAM-B3LYP, as well as M06L, M06, and M062X. We also calculate the three-body contributions to interaction energy for the same DFT methods and assess when they are significant. We find that the amount of exact exchange or other nonlocal contributions has a direct influence on how closely the sum of pairwise energies approximates the total interaction energy. We also find that three-body interactions can be significant and that their significance can be predicted with good accuracy.

Magnetostriction in the magneto-sensitive elastomers with inhomogeneously magnetized particles: Pairwise interaction approximation

We analyze the magnetostriction effect occurring in the magneto-sensitive elastomers (MSEs) containing inhomogeneously magnetized particles. As it was shown before, the expression for the interaction potential between two magnetic spheres, that accounts for their mutual inhomogeneous magnetization, can be obtained from the Laplace equation. We use this potential in the approximation formula form to construct magnetic energy of the sample in terms of the pairwise interactions of the particles. We show that this form of magnetic energy leads to the same demagnetizing factor as predicted by the continuum mechanics, confirming that only dipole–dipole magnetic interactions are important on a large scale. As the next step, we examine the role played by the particles arrangement on the magnetostriction effect. We consider different spatial distributions of the magnetic particles: a uniform one, as well as several lattice-type distributions (SC, BCC, HCP and FCC arrangements). We show that the particles arrangement affects significantly the magnetostriction effect if the separation between them became comparable with the particles dimensions. We also show that, typically, this contribution to the magnetostriction effect is of the opposite sign to the one related with the initial elastomer shape. Finally, we calculate the magnetostriction effect using the same interaction potential but expressed in a form of a series expansion, qualitatively confirming the above findings.

High-accuracy approximation of evolutionary pairwise games on complex networks

Previous studies have shown that the topological properties of a complex network, such as heterogeneity and average degree, affect the evolutionary game dynamics on it. However, traditional numerical simulations are usually time-consuming and demand a lot of computational resources. In this paper, we propose the method of dynamical approximate master equations (DAMEs) to accurately predict the evolutionary outcomes on complex networks. We demonstrate that the accuracy of DAMEs supersedes previous standard pairwise approximation methods, and DAMEs require far fewer computational resources than traditional numerical simulations. We apply the DAMES to investigate prisoner’s dilemma and snowdrift game on regular and scale-free networks, demonstrating its effectiveness. Overall, our method facilitates the investigation of evolutionary dynamics on a broad range of complex networks, and provides new insights into the puzzle of cooperation.

Impact of human contact patterns on epidemic spreading in time-varying networks.

Human contact behaviors involve both dormant and active processes. The dormant (active) process goes from the disappearance (creation) to the creation (disappearance) of an edge. The dormant (active) time is the elapsed time since the edge became dormant (active). Many empirical studies have revealed that dormant and active times in human contact behaviors tend to show a long-tailed distribution. Previous researches focused on the impact of the dormant process on spreading dynamics. However, the epidemic spreading happens on the active process. This raises the question of how the active process affects epidemic spreading in complex networks. Here, we propose a novel time-varying network model in which the distributions of both the dormant time and active time of edges are adjustable. We develop a pairwise approximation method to describe the spreading dynamical processes in the time-varying networks. Through extensive numerical simulations, we find that the epidemic threshold is proportional to the mean dormant time and inversely proportional to the mean active time. The attack rate decreases with the increase of mean dormant time and increases with the increase of mean active time. It is worth noting that the epidemic threshold and the attack rate (e.g., the infected density in the steady state) are independent of the heterogeneities of the dormant time distribution and the active time distribution. Increasing the heterogeneity of the dormant time distribution accelerates epidemic spreading while increasing the heterogeneity of the active time distribution slows it down.

Epidemic Dynamics of Two-Pathogen Spreading for Pairwise Models

In the real world, pathogens do not exist in isolation. The transmission of one pathogen may be affected by the presence of other pathogens, and certain pathogens generate multiple strains with different spreading features. Hence, the behavior of multi-pathogen transmission has attracted much attention in epidemiological research. In this paper, we use the pairwise approximation method to formulate two-pathogen models capturing cross-immunity, super-infection, and co-infection phenomena, in which each pathogen follows a susceptible-infected-susceptible (SIS) mechanism. For each model, we calculate the basic reproduction number and analyze the stability of equilibria, and discuss the differences from the mean-field approach. We demonstrate that simulations are in good agreement with the analytical results.

Fuzzy Information Discrimination Measures and Their Application to Low Dimensional Embedding Construction in the UMAP Algorithm.

Dimensionality reduction techniques are often used by researchers in order to make high dimensional data easier to interpret visually, as data visualization is only possible in low dimensional spaces. Recent research in nonlinear dimensionality reduction introduced many effective algorithms, including t-distributed stochastic neighbor embedding (t-SNE), uniform manifold approximation and projection (UMAP), dimensionality reduction technique based on triplet constraints (TriMAP), and pairwise controlled manifold approximation (PaCMAP), aimed to preserve both the local and global structure of high dimensional data while reducing the dimensionality. The UMAP algorithm has found its application in bioinformatics, genetics, genomics, and has been widely used to improve the accuracy of other machine learning algorithms. In this research, we compare the performance of different fuzzy information discrimination measures used as loss functions in the UMAP algorithm while constructing low dimensional embeddings. In order to achieve this, we derive the gradients of the considered losses analytically and employ the Adam algorithm during the loss function optimization process. From the conducted experimental studies we conclude that the use of either the logarithmic fuzzy cross entropy loss without reduced repulsion or the symmetric logarithmic fuzzy cross entropy loss with sufficiently large neighbor count leads to better global structure preservation of the original multidimensional data when compared to the loss function used in the original UMAP algorithm implementation.

Efficient Hit-to-Lead Searching of Kinase Inhibitor Chemical Space via Computational Fragment Merging.

In early-stage drug discovery, the hit-to-lead optimization (or "hit expansion") stage entails starting from a newly identified active compound and improving its potency or other properties. Traditionally, this process relies on synthesizing and evaluating a series of analogues to build up structure-activity relationships. Here, we describe a computational strategy focused on kinase inhibitors, intended to expedite the process of identifying analogues with improved potency. Our protocol begins from an inhibitor of the target kinase and generalizes the synthetic route used to access it. By searching for commercially available replacements for the individual building blocks used to make the parent inhibitor, we compile an enumerated library of compounds that can be accessed using the same chemical transformations; these huge libraries can exceed many millions─or billions─of compounds. Because the resulting libraries are much too large for explicit virtual screening, we instead consider alternate approaches to identify the top-scoring compounds. We find that contributions from individual substituents are well described by a pairwise additivity approximation, provided that the corresponding fragments position their shared core in precisely the same way relative to the binding site. This key insight allows us to determine which fragments are suitable for merging into single new compounds and which are not. Further, the use of pairwise approximation allows interaction energies to be assigned to each compound in the library without the need for any further structure-based modeling: interaction energies instead can be reliably estimated from the energies of the component fragments, and the reduced computational requirements allow for flexible energy minimizations that allow the kinase to respond to each substitution. We demonstrate this protocol using libraries built from six representative kinase inhibitors drawn from the literature, which target five different kinases: CDK9, CHK1, CDK2, EGFRT790M, and ACK1. In each example, the enumerated library includes additional analogues reported by the original study to have activity, and these analogues are successfully prioritized within the library. We envision that the insights from this work can facilitate the rapid assembly and screening of increasingly large libraries for focused hit-to-lead optimization. To enable adoption of these methods and to encourage further analyses, we disseminate the computational tools needed to deploy this protocol.

Probabilistic threshold analysis by pairwise stochastic approximation for decision-making under uncertainty

The concept of probabilistic parameter threshold analysis has recently been introduced as a way of probabilistic sensitivity analysis for decision-making under uncertainty, in particular, for health economic evaluations which compare two or more alternative treatments with consideration of uncertainty on outcomes and costs. In this paper we formulate the probabilistic threshold analysis as a root-finding problem involving the conditional expectations, and propose a pairwise stochastic approximation algorithm to search for the threshold value below and above which the choice of conditionally optimal decision options changes. Numerical experiments for both a simple synthetic testcase and a chemotherapy Markov model illustrate the effectiveness of our proposed algorithm, without any need for accurate estimation or approximation of conditional expectations which the existing approaches rely upon. Moreover we introduce a new measure called decision switching probability for probabilistic sensitivity analysis in this paper.

Rigorous treatment of pairwise and many-body electrostatic interactions among dielectric spheres at the Debye\u2013H\xfcckel level

Electrostatic interactions among colloidal particles are often described using the venerable (two-particle) Derjaguin–Landau–Verwey–Overbeek (DLVO) approximation and its various modifications. However, until the recent development of a many-body theory exact at the Debye–Hückel level (Yu in Phys Rev E 102:052404, 2020), it was difficult to assess the errors of such approximations and impossible to assess the role of many-body effects. By applying the exact Debye–Hückel level theory, we quantify the errors inherent to DLVO and the additional errors associated with replacing many-particle interactions by the sum of pairwise interactions (even when the latter are calculated exactly). In particular, we show that: (1) the DLVO approximation does not provide sufficient accuracy at shorter distances, especially when there is an asymmetry in charges and/or sizes of interacting dielectric spheres; (2) the pairwise approximation leads to significant errors at shorter distances and at large and moderate Debye lengths and also gets worse with increasing asymmetry in the size of the spheres or magnitude or placement of the charges. We also demonstrate that asymmetric dielectric screening, i.e., the enhanced repulsion between charged dielectric bodies immersed in media with high dielectric constant, is preserved in the presence of free ions in the medium.Graphic abstract

A Quantum Chemical Topology Picture of Intermolecular Electrostatic Interactions and Charge Penetration Energy.

Based on the Interacting Quantum Atoms approach, we present herein a conceptual and theoretical framework of short-range electrostatic interactions, whose accurate description is still a challenging problem in molecular modeling. For all the noncovalent complexes in the S66 database, the fragment-based and atomic decomposition of the electrostatic binding energies is performed using both the charge density of the dimers and the unrelaxed densities of the monomers. This energy decomposition together with dispersion corrections gives rise to a pairwise approximation to the total binding energy. It also provides energetic descriptors at varying distance that directly address the atomic and molecular electrostatic interactions as described by point-charge or multipole-based potentials. Additionally, we propose a consistent definition of the charge penetration energy within quantum chemical topology, which is mainly characterized in terms of the intramolecular electrostatic energy. Finally, we discuss some practical implications of our results for the design and validation of electrostatic potentials.

An unrestricted approach for the accurate calculation of the intermolecular potential of (O2)4: Implications for the solid epsilon phase.

Oxygen in its elemental form shows a variety of magnetic properties in its condensed phases; in particular, the epsilon solid phase loses its magnetism. These phenomena reflect the nature of the intermolecular forces present in the solid and the changes that arise with variations in pressure and temperature. In this study, we use intermolecular potentials obtained with unrestricted ab initio methods to model the singlet state of the oxygen tetramer [(O2)4], which is the unit cell, consistent with the non-magnetic character of this phase. To do this, we perform an analysis of the coupled-uncoupled representations of the spin operator together with a pairwise approximation and the Heisenberg Hamiltonian. We start from unrestricted potentials for the dimer calculated at a high level as well as different density functional theory (DFT) functionals and then apply a finite model to predict the properties of the epsilon phase. The results obtained in this way reproduce well the experimental data in the entire pressure range below 60 GPa. Additionally, we show the importance of calculating the singlet state of the tetramer as opposed to previous DFT periodic calculations, where the unrestricted description leads to a mixture of spin states and a poor comparison with the experiment. This point is crucial in the recent discussion about the coexistence of two epsilon phases: one where the identity of each O2 with spin S = 1 is retained within the tetramer unit vs another at higher pressures where the tetramer behaves as a single unit with a closed-shell character.

Pair approximation model for the vaccination game: predicting the dynamic process of epidemic spread and individual actions against contagion.

We successfully establish a theoretical framework of pairwise approximation for the vaccination game in which both the dynamic process of epidemic spread and individual actions in helping prevent social behaviours are quantitatively evaluated. In contrast with mean-field approximation, our model captures higher-order effects from neighbours by using an underlying network that shows how the disease spreads and how individual decisions evolve over time. This model considers not only imperfect vaccination but also intermediate protective measures other than vaccines. Our analytical predictions are validated by multi-agent simulation results that estimate random regular graphs at varying degrees.

Improving pairwise approximations for network models with susceptible-infected-susceptible dynamics

Network models of disease spread play an important role in elucidating the impact of long-lasting infectious contacts on the dynamics of epidemics. Moment-closure approximation is a common method of generating low-dimensional deterministic models of epidemics on networks, which has found particular success for diseases with susceptible-infected-recovered (SIR) dynamics. However, the effect of network structure is arguably more important for sexually transmitted infections, where epidemiologically relevant contacts are comparatively rare and longstanding, and which are in general modelled via the susceptible-infected-susceptible (SIS)-paradigm. In this paper, we introduce an improvement to the standard pairwise approximation for network models with SIS-dynamics for two different network structures: the isolated open triple (three connected individuals in a line) and the k-regular network. This improvement is achieved by tracking the rate of change of errors between triple values and their standard pairwise approximation. For the isolated open triple, this improved pairwise model is exact, while for k-regular networks a closure is made at the level of triples to obtain a closed set of equations. This improved pairwise approximation provides an insight into the errors introduced by the standard pairwise approximation, and more closely matches both higher-order moment-closure approximations and explicit stochastic simulations with only a modest increase in dimensionality to the standard pairwise approximation.

Remote Sensing Image Ship Detection under Complex Sea Conditions Based on Deep Semantic Segmentation

Under complex sea conditions, ship detection from remote sensing images is easily affected by sea clutter, thin clouds, and islands, resulting in unreliable detection results. In this paper, an end-to-end convolution neural network method is introduced that combines a deep convolution neural network with a fully connected conditional random field. Based on the Resnet architecture, the remote sensing image is roughly segmented using a deep convolution neural network as the input. Using the Gaussian pairwise potential method and mean field approximation theorem, a conditional random field is established as the output of the recurrent neural network, thus achieving end-to-end connection. We compared the proposed method with other state-of-the-art methods on the dataset established by Google Earth and NWPU-RESISC45. Experiments show that the target detection accuracy of the proposed method and the ability of capturing fine details of images are improved. The mean intersection over union is 83.2% compared with other models, which indicates obvious advantages. The proposed method is fast enough to meet the needs for ship detection in remote sensing images.

A Linear-Scaling Method for Noncovalent Interactions: An Efficient Combination of Absolutely Localized Molecular Orbitals and a Local Random Phase Approximation Approach.

A novel method for the accurate and efficient calculation of interaction energies in weakly bound complexes composed of a large number of molecules is presented. The new ALMO+RPAd method circumvents the prohibitive scaling of coupled cluster singles and doubles while still providing similar accuracy across a diverse range of weakly bound chemical systems. Linear-scaling procedures for the Fock build are given utilizing absolutely localized molecular orbitals (ALMOs), resulting in the a priori exclusion of basis set superposition errors. A bespoke data structure and algorithm using density fitting are described, leading to linear scaling for the storage and computation of the two-electron integrals. Electron correlation is included through a new, linear-scaling pairwise local random phase approximation approach, including exchange interactions, and decomposed into purely dispersive excitations (RPAxd). Collectively, these allow meaningful decomposition of the interaction energy into physically distinct contributions: electrostatic, polarization, charge transfer, and dispersion. Comparison with symmetry-adapted perturbation theory shows good qualitative agreement. Tests on various dimers and the S66 benchmark set demonstrate results within 0.5 kcal mol-1 of coupled cluster singles and doubles results. On a large cluster of water molecules, we achieve calculations involving over 3500 orbital and 12,000 auxiliary basis functions in under 10 min on a single CPU core.

Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks

The global behaviour of the compact pairwise approximation of SIS epidemic propagation on networks is studied. It is shown that the system can be reduced to two equations enabling us to carry out a detailed study of the dynamic properties of the solutions. It is proved that transcritical bifurcation occurs in the system at \begin{document}$ \tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n} $\end{document} , where \begin{document}$ \tau $\end{document} and \begin{document}$ \gamma $\end{document} are infection and recovery rates, respectively, \begin{document}$ n $\end{document} is the average degree of the network and \begin{document}$ \langle n^{2}\rangle $\end{document} is the second moment of the degree distribution. For subcritical values of \begin{document}$ \tau $\end{document} the disease-free steady state is stable, while for supercritical values a unique stable endemic equilibrium appears. We also prove that for subcritical values of \begin{document}$ \tau $\end{document} the disease-free steady state is globally stable under certain assumptions on the graph that cover a wide class of networks.

Accurate and Efficient ab Initio Calculations for Supramolecular Complexes: Symmetry-Adapted Perturbation Theory with Many-Body Dispersion.

Symmetry-adapted perturbation theory (SAPT) provides a chemically meaningful energy decomposition scheme for nonbonded interactions that is useful for interpretive purposes. Although formally a dimer theory, we have previously introduced an "extended" version (XSAPT) that incorporates many-body polarization via self-consistent charge embedding. Here, we extend the XSAPT methodology to include nonadditive dispersion, using a modified form of the many-body dispersion (MBD) method of Tkatchenko and co-workers. Dispersion interactions beyond the pairwise atom-atom approximation improve total interaction energies even in small systems, and for large π-stacked complexes these corrections can amount to several kilocalories per mole. The XSAPT+MBD method introduced here achieves errors of ≲1 kcal/mol (as compared to high-level ab initio benchmarks) for the L7 data set of large dispersion-bound complexes and ≲4 kcal/mol (as compared to experiment) for the S30L data set of host-guest complexes. This is superior to the best contemporary density functional methods for noncovalent interactions, at comparable or lower cost. XSAPT+MBD represents a promising method for application to supramolecular assemblies, including protein-ligand binding.

The activity coefficient within the solvation thermodynamics formalism: Fundamentals and applications to neutral solutes

The solvation thermodynamics is a molecular based statistical mechanics description of the solvation process, which provides a proper distinction from other processes that uniquely and unambiguously defines the solvation of any species in any environment. Because the activity coefficient involves the behavior of a given species in different environments, it is then related to the solvation process, which should be described within the solvation thermodynamics formalism. Indeed, the activity coefficient of a species i within an α phase, γi,αo,ρ, is expressed in terms of the binding energy of i to the α phase, Bi, α, and to the dilute-ideal phase, Bi, o. In addition to this molecular description, the activity coefficient can also be related to the Gibbs solvation energies of these two phases within the solvation thermodynamics formalism, which can provide, for instance, unique values of the activity coefficients from the experimental Gibbs solvation energies. The relevance and advantages of using the binding energy (or the coupling work of the solute with its surroundings) as the main molecular quantity for evaluating the activity coefficient is explored in detail. For instance, several cancellations occur when evaluating the binding energies, especially those related to the non-additive corrections (many-body effects), which can more promptly justify the use of the pairwise approximation. In addition, by partitioning the potential energy into two or more contributions, the activity coefficient can also be partitioned into these contributions, yielding exact expressions that are suitable for approximations and interpretations. In fact, a direct connection with perturbation theories can be established and new partitions can be proposed to explore recent approaches based on embedded quantum calculations of the potential energy in condensed phases. It has been shown that the approximation 〈e−βBi, α〉 ≅ e−β〈Bi, α〉, with β−1 = kBT, can be valid even when the condition 〈Bi, α〉 ≪ kBT is not satisfied, as long as the fluctuations, 〈(Bi, α − 〈Bi, α〉)2〉, and the higher order deviations of the binding energy, 〈(Bi, α − 〈Bi, α〉)n〉, (n ≥ 3), are negligible. This small fluctuations approximation (SFA) can rationalize and justify the good results obtained for systems with large binding energies. The SFA can also provide a new interpretation and insights for the activity coefficient. The partition of the potential energy into hard sphere (cavity) and attractive (non-cavity) contributions and the SFA were employed to obtain approximate expressions that allowed to explore the general behavior of the activity coefficient of neutral systems. For instance, the activity coefficient of strongly interacting large molecules (hard-spheres interacting by Lennard-Jones potential in an implicit solvent) tends to decrease with the increasing of the concentration of the solute, which is accentuated by the increase of the dielectric constant of the medium. In addition, the cavity contribution (modeled by hard-sphere mixtures of different sizes) to the activity coefficient is influenced by the relative sizes of the molecules and by the pure liquid densities. This contribution dominates the behavior of the activity coefficient with the concentration.