Entropy

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15 pages, 4545 KiB

Open AccessArticle

QUBO Problem Formulation of Fragment-Based Protein–Ligand Flexible Docking

by Keisuke Yanagisawa, Takuya Fujie, Kazuki Takabatake and Yutaka Akiyama

Entropy 2024, 26(5), 397; https://doi.org/10.3390/e26050397 - 30 Apr 2024

Cited by 1 | Viewed by 2222

Protein–ligand docking plays a significant role in structure-based drug discovery. This methodology aims to estimate the binding mode and binding free energy between the drug-targeted protein and candidate chemical compounds, utilizing protein tertiary structure information. Reformulation of this docking as a quadratic unconstrained [...] Read more.

Protein–ligand docking plays a significant role in structure-based drug discovery. This methodology aims to estimate the binding mode and binding free energy between the drug-targeted protein and candidate chemical compounds, utilizing protein tertiary structure information. Reformulation of this docking as a quadratic unconstrained binary optimization (QUBO) problem to obtain solutions via quantum annealing has been attempted. However, previous studies did not consider the internal degrees of freedom of the compound that is mandatory and essential. In this study, we formulated fragment-based protein–ligand flexible docking, considering the internal degrees of freedom of the compound by focusing on fragments (rigid chemical substructures of compounds) as a QUBO problem. We introduced four factors essential for fragment–based docking in the Hamiltonian: (1) interaction energy between the target protein and each fragment, (2) clashes between fragments, (3) covalent bonds between fragments, and (4) the constraint that each fragment of the compound is selected for a single placement. We also implemented a proof-of-concept system and conducted redocking for the protein–compound complex structure of Aldose reductase (a drug target protein) using SQBM+, which is a simulated quantum annealer. The predicted binding pose reconstructed from the best solution was near-native (RMSD = 1.26 Å), which can be further improved (RMSD = 0.27 Å) using conventional energy minimization. The results indicate the validity of our QUBO problem formulation. Full article

(This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications II)

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8 pages, 286 KiB

Open AccessArticle

Ising Ladder with Four-Spin Plaquette Interaction in a Transverse Magnetic Field

by Maria Eugenia S. Nunes, Francisco Welington S. Lima and Joao A. Plascak

Entropy 2023, 25(12), 1665; https://doi.org/10.3390/e25121665 - 16 Dec 2023

Cited by 1 | Viewed by 1405

Abstract

The spin-1/2 quantum transverse Ising model, defined on a ladder structure, with nearest-neighbor and four-spin interaction on a plaquette, was studied by using exact diagonalization on finite ladders together with finite-size-scaling procedures. The quantum phase transition between the ferromagnetic and paramagnetic phases has [...] Read more.

The spin-1/2 quantum transverse Ising model, defined on a ladder structure, with nearest-neighbor and four-spin interaction on a plaquette, was studied by using exact diagonalization on finite ladders together with finite-size-scaling procedures. The quantum phase transition between the ferromagnetic and paramagnetic phases has then been obtained by extrapolating the data to the thermodynamic limit. The critical transverse field decreases as the antiferromagnetic four-spin interaction increases and reaches a multicritical point. However, the exact diagonalization approach was not able to capture the essence of the dimer phase beyond the multicritical transition. Full article

(This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications II)

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25 pages, 15573 KiB

Open AccessArticle

The Capabilities of Boltzmann Machines to Detect and Reconstruct Ising System’s Configurations from a Given Temperature

by Mauricio A. Valle

Entropy 2023, 25(12), 1649; https://doi.org/10.3390/e25121649 - 12 Dec 2023

Viewed by 1387

Abstract

The restricted Boltzmann machine (RBM) is a generative neural network that can learn in an unsupervised way. This machine has been proven to help understand complex systems, using its ability to generate samples of the system with the same observed distribution. In this [...] Read more.

The restricted Boltzmann machine (RBM) is a generative neural network that can learn in an unsupervised way. This machine has been proven to help understand complex systems, using its ability to generate samples of the system with the same observed distribution. In this work, an Ising system is simulated, creating configurations via Monte Carlo sampling and then using them to train RBMs at different temperatures. Then, 1. the ability of the machine to reconstruct system configurations and 2. its ability to be used as a detector of configurations at specific temperatures are evaluated. The results indicate that the RBM reconstructs configurations following a distribution similar to the original one, but only when the system is in a disordered phase. In an ordered phase, the RBM faces levels of irreproducibility of the configurations in the presence of bimodality, even when the physical observables agree with the theoretical ones. On the other hand, independent of the phase of the system, the information embodied in the neural network weights is sufficient to discriminate whether the configurations come from a given temperature well. The learned representations of the RBM can discriminate system configurations at different temperatures, promising interesting applications in real systems that could help recognize crossover phenomena. Full article

(This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications II)

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13 pages, 500 KiB

Open AccessFeature PaperArticle

Heat-Bath and Metropolis Dynamics in Ising-like Models on Directed Regular Random Graphs

by Adam Lipowski, António L. Ferreira and Dorota Lipowska

Entropy 2023, 25(12), 1615; https://doi.org/10.3390/e25121615 - 2 Dec 2023

Viewed by 1594

Abstract

Using a single-site mean-field approximation (MFA) and Monte Carlo simulations, we examine Ising-like models on directed regular random graphs. The models are directed-network implementations of the Ising model, Ising model with absorbing states, and majority voter models. When these nonequilibrium models are driven [...] Read more.

Using a single-site mean-field approximation (MFA) and Monte Carlo simulations, we examine Ising-like models on directed regular random graphs. The models are directed-network implementations of the Ising model, Ising model with absorbing states, and majority voter models. When these nonequilibrium models are driven by the heat-bath dynamics, their stationary characteristics, such as magnetization, are correctly reproduced by MFA as confirmed by Monte Carlo simulations. It turns out that MFA reproduces the same result as the generating functional analysis that is expected to provide the exact description of such models. We argue that on directed regular random graphs, the neighbors of a given vertex are typically uncorrelated, and that is why MFA for models with heat-bath dynamics provides their exact description. For models with Metropolis dynamics, certain additional correlations become relevant, and MFA, which neglects these correlations, is less accurate. Models with heat-bath dynamics undergo continuous phase transition, and at the critical point, the power-law time decay of the order parameter exhibits the behavior of the Ising mean-field universality class. Analogous phase transitions for models with Metropolis dynamics are discontinuous. Full article

(This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications II)

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29 pages, 5081 KiB

Open AccessArticle

Characteristics of an Ising-like Model with Ferromagnetic and Antiferromagnetic Interactions

by Boris Kryzhanovsky, Vladislav Egorov and Leonid Litinskii

Entropy 2023, 25(10), 1428; https://doi.org/10.3390/e25101428 - 9 Oct 2023

Viewed by 1118

Abstract

In the framework of mean field approximation, we consider a spin system consisting of two interacting sub-ensembles. The intra-ensemble interactions are ferromagnetic, while the inter-ensemble interactions are antiferromagnetic. We define the effective number of the nearest neighbors and show that if the two [...] Read more.

In the framework of mean field approximation, we consider a spin system consisting of two interacting sub-ensembles. The intra-ensemble interactions are ferromagnetic, while the inter-ensemble interactions are antiferromagnetic. We define the effective number of the nearest neighbors and show that if the two sub-ensembles have the same effective number of the nearest neighbors, the classical form of critical exponents (

α = 0

,

β = 1 / 2

,

γ = γ^{'} = 1

,

δ = 3

) gives way to the non-classical form (

α = 0

,

β = 3 / 2

,

γ = γ^{'} = 0

,

δ = 1

), and the scaling function changes simultaneously. We demonstrate that this system allows for two second-order phase transitions and two first-order phase transitions. We observe that an external magnetic field does not destroy the phase transitions but only shifts their critical points, allowing for control of the system’s parameters. We discuss the regime when the magnetization as a function of the magnetic field develops a low-magnetization plateau and show that the height of this plateau abruptly rises to the value of one when the magnetic field reaches a critical value. Our analytical results are supported by a Monte Carlo simulation of a three-dimensional layered model. Full article

(This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications II)

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11 pages, 804 KiB

Open AccessArticle

Relevant Analytic Spontaneous Magnetization Relation for the Face-Centered-Cubic Ising Lattice

by Başer Tambaş

Entropy 2023, 25(2), 197; https://doi.org/10.3390/e25020197 - 19 Jan 2023

Viewed by 2155

Abstract

The relevant approximate spontaneous magnetization relations for the simple-cubic and body-centered-cubic Ising lattices have recently been obtained analytically by a novel approach that conflates the Callen–Suzuki identity with a heuristic odd-spin correlation magnetization relation. By exploiting this approach, we study an approximate analytic [...] Read more.

The relevant approximate spontaneous magnetization relations for the simple-cubic and body-centered-cubic Ising lattices have recently been obtained analytically by a novel approach that conflates the Callen–Suzuki identity with a heuristic odd-spin correlation magnetization relation. By exploiting this approach, we study an approximate analytic spontaneous magnetization expression for the face-centered-cubic Ising lattice. We report that the results of the analytic relation obtained in this work are nearly consistent with those derived from the Monte Carlo simulation. Full article

(This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications II)

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11 pages, 307 KiB

Open AccessArticle

What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution

by Gandhimohan M. Viswanathan, Marco Aurelio G. Portillo, Ernesto P. Raposo and Marcos G. E. da Luz

Entropy 2022, 24(11), 1665; https://doi.org/10.3390/e24111665 - 15 Nov 2022

Cited by 5 | Viewed by 2553

Abstract

An exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly [...] Read more.

An exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two-dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function Z, but all such attempts have failed due to unavoidable conceptual or mathematical obstructions. Given the importance of this simple yet paradigmatic model, here we set out clear-cut criteria for any claimed exact expression for Z to be minimally plausible. Specifically, we present six necessary—but not sufficient—conditions that Z must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken “solutions”, unveiling their shortcomings. Full article

(This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications II)

7 pages, 253 KiB

Open AccessArticle

Generalized Solution of Inverse Problem for Ising Connection Matrix on d-Dimensional Hypercubic Lattice

by Boris Kryzhanovsky and Leonid Litinskii

Entropy 2022, 24(10), 1424; https://doi.org/10.3390/e24101424 - 6 Oct 2022

Viewed by 1174

Abstract

We analyze a connection matrix of a

d

-dimensional Ising system and solve the inverse problem, restoring the constants of interaction between spins, based on the known spectrum of its eigenvalues. When the boundary conditions are periodic, we can account for interactions between [...] Read more.

We analyze a connection matrix of a

d

-dimensional Ising system and solve the inverse problem, restoring the constants of interaction between spins, based on the known spectrum of its eigenvalues. When the boundary conditions are periodic, we can account for interactions between spins that are arbitrarily far. In the case of the free boundary conditions, we have to restrict ourselves with interactions between the given spin and the spins of the first

d

coordination spheres. Full article

(This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications II)

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