Accuracy and tractability of a kriging model of intramolecular polarizable multipolar electrostatics and its application to histidine

Abstract

We propose a generic method to model polarization in the context of high-rank multipolar electrostatics. This method involves the machine learning technique kriging, here used to capture the response of an atomic multipole moment of a given atom to a change in the positions of the atoms surrounding this atom. The atoms are malleable boxes with sharp boundaries, they do not overlap and exhaust space. The method is applied to histidine where it is able to predict atomic multipole moments (up to hexadecapole) for unseen configurations, after training on 600 geometries distorted using normal modes of each of its 24 local energy minima at B3LYP/apc-1 level. The quality of the predictions is assessed by calculating the Coulomb energy between an atom for which the moments have been predicted and the surrounding atoms (having exact moments). Only interactions between atoms separated by three or more bonds ("1, 4 and higher" interactions) are included in this energy error. This energy is compared with that of a central atom with exact multipole moments interacting with the same environment. The resulting energy discrepancies are summed for 328 atom-atom interactions, for each of the 29 atoms of histidine being a central atom in turn. For 80% of the 539 test configurations (outside the training set), this summed energy deviates by less than 1 kcal mol(-1).

Transferable kriging machine learning models for the multipolar electrostatics of helical deca-alanine

“…The kriging method is briefly outlined here but has been discussed in greater detail in a previous publication of group [5] and is based on the treatment of Jones et al [62,63]. Kriging maps an output's response to any given input.…”

“…The quantity a i is the ith element of the vector a = R −1 (y − 1μ) where R is a matrix of error correlations between training points, and 1 is a column vector of ones. The error from the global term is determined by the distance between the new input point (x * ) and a known input point (x i ), each scaled by the magnitude [5] ϕ. The sum of these errors gives the appropriate deviation from the background term and results…”

“…It has been suggested that p h can be fixed at 2 for most cases, and this has held true for past publications, but we continue to optimise this hyperparameter. For that purpose we use particle swarm optimisation [64] against a fitness function described before [5]. This optimisation technique is also used for each θ h .…”

“…The electrostatic energy often defies simple computational solutions such as the still-ubiquitous static atom-centred point charges because atomic electron densities are anisotropic and polarise significantly due to changes in molecular geometry. In the past, we have reported on a different and new approach that predicts highrank multipolar and fully polarised electrostatic interaction energies in water clusters [1], ethanol [2], alanine [3], serine [4], N-methylacetamide and histidine [5], aromatic amino acids [6], hydrogen-bonded dimers [7], and atomic kinetic energies of methanol, N-methylacetamide, glycine, and triglycine [8]. These studies feature a developing force field, QCTFF, which predicts electrostatic multipole moments using machine learning, fully taking into account polarisation.…”

Accurate prediction of the energetics of weakly bound complexes using the machine learning method kriging

“…More precisely, FFLUX needs to be trained by a sufficient number of relevant geometries such that it can interpolate a property of a given atom of interest between the data learnt. The selected [12] machine learning method is Kriging [13], which has been tested successfully on a variety of systems, including ethanol [14], (peptide-capped) alanine [15], the microhydrated sodium ion [15], N-methylacetamide (NMA) and histidine [16], the four aromatic (peptidecapped) amino acids [17], all naturally occurring amino acids [18], helical deca-alanines [19,20], water clusters [21], cholesterol [22] and carbohydrates [23]. This collective work shows an existing proof-of-concept that kriging models generate sufficiently accurate atomic property models, and they do this directly from the coordinates of the surrounding atoms.…”