scholarly journals Local Energy Estimates for the Fractional Laplacian

2021 ◽  
Vol 59 (4) ◽  
pp. 1918-1947
Author(s):  
Juan Pablo Borthagaray ◽  
Dmitriy Leykekhman ◽  
Ricardo H. Nochetto
Keyword(s):  
Fractional Laplacian ◽  
Energy Estimates ◽  
Local Energy ◽  
Local Energy Estimates

LOCAL ENERGY ESTIMATES FOR THE MAXWELL–LANDAU–LIFSHITZ SYSTEM AND APPLICATIONS

2005 ◽  
Vol 02 (03) ◽  
pp. 565-594 ◽  
Author(s):  
JEAN STARYNKÉVITCH
Keyword(s):  
Stability Result ◽  
Energy Estimates ◽  
Linear Wave ◽  
Local Energy ◽  
Linear Wave Equation ◽  
Lifshitz Equation ◽  
Convergence Results ◽  
Damping Parameter ◽  
Local Energy Estimates ◽  
Stationary Approximation

We study the Maxwell–Landau–Lifshitz system without exchange energy. First, we prove an [Formula: see text] estimate for the linear wave equation and apply this local energy estimate to obtain a bound on the curl of the electromagnetic field, uniformly in time and locally in space. Next, we prove strong convergence results, when the time t tends to ∞ or when the speed of light tends to ∞ (which corresponds to the quasi-stationary approximation). Finally, we establish a stability result with respect to the damping parameter of the Landau–Lifshitz equation.

Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials

2018 ◽  
Vol 149 (03) ◽  
pp. 617-653 ◽  
Author(s):  
Miao Du ◽  
Lixin Tian ◽  
Jun Wang ◽  
Fubao Zhang
Keyword(s):  
Fractional Laplacian ◽  
Energy Estimates ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Fractional Schrödinger Equations ◽  
Embedding Constant ◽  
Normalized Solutions ◽  
Concentration Behavior ◽  
Nonlocal Nature

AbstractIn this paper, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrödinger equations. Comparingwith the Schrödinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo–Nirenberg–Sobolev inequality can be expressed by exact form, which improves the results of [17, 18]. By doing this, we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.

SAMPL7 TrimerTrip Host-Guest Binding Poses and Binding Affinities from Spherical-Coordinates-Biased Simulations

2020 ◽  
Author(s):  
Zhaoxi Sun
Keyword(s):  
Free Energy ◽  
Computational Modelling ◽  
Free Energy Calculations ◽  
Energy Estimates ◽  
Binding Affinities ◽  
Spherical Coordinates ◽  
Collective Variables ◽  
Guest Binding ◽  
Starting Point ◽  
Sampl Challenge

Host-guest binding remains a major challenge in modern computational modelling. The newest 7<sup>th</sup> statistical assessment of the modeling of proteins and ligands (SAMPL) challenge contains a new series of host-guest systems. The TrimerTrip host binds to 16 structurally diverse guests. Previously, we have successfully employed the spherical coordinates as the collective variables coupled with the enhanced sampling technique metadynamics to enhance the sampling of the binding/unbinding event, search for possible binding poses and predict the binding affinities in all three host-guest binding cases of the 6<sup>th</sup> SAMPL challenge. In this work, we employed the same protocol to investigate the TrimerTrip host in the SAMPL7 challenge. As no binding pose is provided by the SAMPL7 host, our simulations initiate from randomly selected configurations and are proceeded long enough to obtain converged free energy estimates and search for possible binding poses. The predicted binding affinities are in good agreement with the experimental reference, and the obtained binding poses serve as a nice starting point for end-point or alchemical free energy calculations.

D3R Grand Challenge 4: Ligand Similarity and MM-GBSA-Based Pose Prediction and Affinity Ranking for BACE-1 Inhibitors

2019 ◽  
Author(s):  
Sukanya Sasmal ◽  
Léa El Khoury ◽  
David Mobley
Keyword(s):  
Binding Mode ◽  
Energy Estimates ◽  
Poor Correlation ◽  
Grand Challenge ◽  
Pose Prediction ◽  
Design Data ◽  
Data Resource ◽  
Affinity Ranking ◽  
Ligand Similarity ◽  
Bace 1

The Drug Design Data Resource (D3R) Grand Challenges present an opportunity to assess, in the context of a blind predictive challenge, the accuracy and the limits of tools and methodologies designed to help guide pharmaceutical drug discovery projects. Here, we report the results of our participation in the D3R Grand Challenge 4, which focused on predicting the binding poses and affinity ranking for compounds targeting the beta-amyloid precursor protein (BACE-1). Our ligand similarity-based protocol using HYBRID (OpenEye Scientific Software) successfully identified poses close to the native binding mode for most of the ligands with less than 2 A RMSD accuracy. Furthermore, we compared the performance of our HYBRID-based approach to that of AutoDock Vina and Dock 6 and found that HYBRID performed better here for pose prediction. We also conducted end-point free energy estimates on protein-ligand complexes using molecular mechanics combined with generalized Born surface area method (MM-GBSA). We found that the binding affinity ranking based on MM-GBSA scores have poor correlation with the experimental values. Finally, the main lessons from our participation in D3R Grand Challenge 4 suggest that: i) the generation of the macrocycles conformers is a key step for successful pose prediction, ii) the protonation states of the BACE-1 binding site should be treated carefully, iii) the MM-GBSA method could not discriminate well between different predicted binding poses, and iv) the MM-GBSA method does not perform well at predicting protein-ligand binding affinities here.

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