Using Quantum Atomics and Machine Learning to Advance Picotechnology

doi:10.21203/rs.3.rs-4669576/v1

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Using Quantum Atomics and Machine Learning to Advance Picotechnology

https://doi.org/10.21203/rs.3.rs-4669576/v1

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We explore the use of machine learning to predict spectroscopic properties and interaction energies of the carbonyl groups in 225 ketones, aldehydes, imides, and amides. In the combined spirit of Density Functional Theory (DFT) and the Quantum Theory of Atoms in Molecules (QTAIM), but with an eye toward eventually using databases of transferable fragment densities, we limit the training data to small sets of descriptors (from 18 to 48 per molecule) that are based on topological features in the total charge density, ρ, and/or its Laplacian, ∇2ρ. We obtain a mean absolute error under 1% for carbonyl stretching frequencies, and just over 1% for C-13 NMR shifts. Predicting interaction energies with a model nucleophile (fluoride ion) is significantly more challenging. Mean absolute errors just over 3 kcal/mol were obtained for covalent bond formation energies. Similar mean absolute errors were obtained for much weaker van der Waals interaction energies. We also conducted a stress-test to see if our small molecule-based machine learning could predict covalent bond formation energy in a model of the active site of the E. coli enzyme, D-fructose-6-phosphate aldolase.

QTAIM

charge density topology

picotechnology

quantum atomics

machine learning

Nanotechnology is the scientific investigation of matter at the nanometer scale (1 nm = 10^-9 m) – roughly from one to hundreds of nanometers – and engineering systems that take advantage of the properties of matter at that scale. Drug-like organic molecules have dimensions measuring up to just a few nanometers. So, dating back to the mid-1800s, organic chemistry is the oldest form of nanotechnology.

Similarly, but a century or so younger, femtotechnology corresponds to the scientific investigation of matter at the femtometer scale (1 fm = 10^-15 m) and engineering systems that take advantage of the properties of matter at that scale. It is better known as nuclear engineering.

Any future picotechnology should therefore be defined as the scientific investigation of matter at the picometer scale (1 pm = 10^-12 m) – from one to hundreds of picometers - and engineering systems that take advantage of the properties of matter at that scale. In other words, atomic engineering!

Since most atoms have diameters over 100 pm, picotechnology is necessarily concerned with subatomic features (outside the nucleus). Chemists routinely refer to such features conceptually, such as when using arrow-pushing diagrams to show a nonbonding electron pair on one atom filling an unoccupied atomic orbital on another atom, forming a new covalent bond. Technologies, however, are not purely conceptual. They also have physical manifestations. Physical, reproducible, and chemically significant subatomic features have been identified and in the topology of the Laplacian of the charge density, Ñ2r, for atoms from all across the periodic table, and in many forms of matter, by both computational and experimental researchers alike [1-4].

Pioneering chemical informatics work by Alsberg et al. noted that “The (QTAIM) approach is well suited for machine inference as it is topologically based and close to the naïve atom-bond current ILP (inductive logic programming) representation.” [5]. Concurrently, Popelier et al. began developing Quantum Topological Molecular Similarity (QTMS) with charge density topological data (at bond critical points) to predict a variety of empirical chemical parameters, such as Hammett constants for benzoic acids derivatives [6]. These and similar studies motivated us to consider whether other charge density descriptors could prove useful in training neural networks to predict more localized and directional chemical properties, such as mapping van der Waals interaction energies, which are key to drug design and modelling biomolecular interactions.

For our neural network training data, in addition to the bond critical point data that others have used, we also include topological features in Ñ2r, specifically within the valence-shell charge concentration (VSCC) of carbon atoms in carbonyl groups (Fig. 2). Such subatomic topological features have previously been shown to predict Bürgi-Dunitz angles of nucleophilic attack [7], and even changes to that angle upon protonation of the carbonyl oxygen [1].

However, like all other wavefunction-based quantum chemical modelling of chemical reactivity, the topological analysis of localized sites of attack first required time-consuming Coulomb and exchange-correlation integrals spanning the entire molecule. Even using DFT, which omits the latter and most time-consuming integrals, if the carbonyl group of interest is part of a very large protein complex, then most computation time is wasted on computing the charge density around atoms unrelated to the reactive site. QM/MM embedding methods such as ONIOM [8] can further economize by focusing high-level computing effort on the reactive site, but that is not the objective of the current work.

Our goal is to investigate the feasibility of training neural networks to predict carbonyl reactivity using descriptors based on topological properties of subatomic features within the VSCC of carbonyl carbon atoms (measured in picometers) – regardless of how the charge density is computed. It could, in fact, be measured with high-resolution, single-crystal X-ray diffraction [4].

Ultimately, because the charge densities of functional groups can be very transferable between molecules [2] (even when they differ greatly on the periphery), we expect that in the future picotechnology that we envision, the necessary descriptor subatomic topological features will be very rapidly and accurately constructed from fragment databases [9]. Thus, when picotechnology is mature, no quantum chemical calculations, or high-resolution X-ray diffraction experiments will be required to predict chemical reactivity at the reactive site of a molecule.

2.1Quantum Chemistry

The initial Cartesian coordinates of 225 carbonyl compounds, including aldehydes, ketones and amides were obtained using molecular modelling software Spartan’10 [10]. The molecules used in this study vary in size ranging from 7 to 31 atoms. A complete list of compounds included in the training set can be found in Supplementary Materials (Table 1). Full geometry optimization (of default Spartan-generated conformers) and generation of the wavefunction files were carried out using Gaussian09 [11]. All the molecules were optimized at B3LYP/6-31+G* and wavefunctions were obtained at M05-2X/6-311++G** level of theory [12-16]. The wavefunctions obtained from these calculations were used to calculate the topological properties of the molecules such as electron density, Laplacian of electron density, Hessian eigenvalues (of the corresponding Hessian matrix, r or Ñ2r), bond critical points, and critical points in Ñ2r. The topological properties of electron density at bond critical points were determined using AIMAll [17].Whereas the topological properties at critical points of the Laplacian of the charge density were determined using Denprop [18]. The topological data used for descriptors in neural network training were extracted using a Python script to create the data sets. The types of critical points that are located and used as training data in this research are illustrated in Figures 1 and 2.

2.2 Data Sets and Quantum Atomics

Quantum Atomics refers to topological and/or integrated atomic data obtained via, or within the context of the Quantum Theory of Atoms in Molecules (QTAIM). Numerous computational and experimental groups worldwide have been collecting such data for decades, without referring to it as such. It is perhaps a useful term for distinguishing such data from other computational chemistry and crystallographic parameters that are routinely used, but lack the rigorous foundation of QTAIM. A second over-arching goal of this research is to explore the information content “quality” of various types of Quantum Atomic data for specific machine learning goals. For instance, if the goal is to predict chemical reactivity, should training data include topological properties of the charge density, or the Laplacian of the charge density? Although we have not explored such questions here, future research could address whether more efficient and accurate machine learning can be achieved with specific integrated atomic properties that are defined uniquely within QTAIM, such as atomic energy, magnetic susceptibility, electric polarizability, multipole moments, etc…

In this research, a total of nine separate data sets were used: three types topological data (BCP, LCP, combined) trained with three types of physicochemical data (C-13 NMR, C=O frequencies, nucleophile interaction energies) with each data set containing critical point descriptors for all 225 molecules were used to train artificial neural networks (ANNs) to predict C-13 chemical shifts, C=O stretching frequencies, and interaction energies with a model nucleophile (the fluoride ion). Among the nine data sets were three of each of the following: bond critical point (BCPs), Laplacian critical point (LCPs), and combined data sets (with both BCP and LCP data). Each of the previously mentioned data sets contains the following: a class label from experimental C-13 shifts, C=O stretching frequencies, or theoretical interaction energy values. The experimental C-13 chemical shift in ppm and C=O stretching frequency in cm^-1values were collected from spectral database of organic compounds library (SDBS) [19]. All topological data in the training sets can be found in Supplementary Materials (Table 2). An example of the LCP portion of such a data set for 1-butanal is shown in Table 1.

Table 1. Sample Laplacian critical point (LCP) input data for 2-chloro-4-fluorobenzaldehyde (all units au)

Distance from C nucleus	λ₁	λ₂	λ₃	ρ	∇²ρ
1.0282	-1.140	-0.614	10.800	0.141	0.0294
1.0282	-1.140	-0.614	10.800	0.141	0.0294
0.9701	4.300	5.060	22.000	0.292	-1.0100
0.9790	6.480	6.900	18.800	0.302	-1.2300
0.9811	5.230	8.960	18.000	0.431	-1.0700

The interaction energy (ΔE_interaction) between carbonyl compounds and a fluoride ion (F^-1) in the nucleophilic addition reaction was calculated at B3LYP/6-31+G* level with the following approach,

ΔE_interaction= E_CC+F- E_CC- E_F (Eqn. 1)

where E_CC+F, E_CCand E_F are total energies of the carbanion complex, carbonyl compound, and the fluoride ion nucleophile, respectively. Basis set superposition errors (BSSE) were not calculated,as they would have considerably increased computational time, while yielding no significant value. The machine learning here is based on properties of the electron density, which are completely unaffected by BSSE.

2.3 Artificial Neural Network Model

Over the past few decade artificial neural networks (ANNs) have had huge success in machine learning and data mining applications [20]. Recently, Handley and Popelier have pioneered use of QTAIM data (atomic multipole moments) in conjunction with machine learning to model the fluctuating polarizability of water molecules in molecular dynamics simulations [21]. ANNs are powerful tools in advanced computing that analyse information quantitatively by learning from training data. The important properties of ANNs are the learning ability of a network from its environment and improving performance by learning. A learning algorithm is a procedure in which the learning rules are used to adjust the weights. The ANNs consists of input layer, one or more hidden layers and output layer. The input signal propagates layer by layer in the forward direction and these networks are commonly called as multilayer perceptrons (MLP) [22,23]. MLP with back-propagation learning method (BP) is one of the successfully used methods in chemistry and drug design because of its well-defined and explicit set of equations for weight corrections [24]. This is a supervised learning algorithm in which the network is trained with a training data with expected outputs are provided to get the algorithm trained. The learning consists of a forward pass and backward pass. In the forward pass, the input vector is applied to the input layer and these input values are modified by a fixed weight and its effect passes through the network layer by layer. Finally, output produced by the network compared with desired output to calculate the error signal. This error signal is then back propagated in the backward pass to adjust the weights in such a way that the actual output value move closer to the desired output value according to error correction rule [22].

In this study, the machine learning package WEKA 3.6.13 was used for the ANN model development [25]. In our model, we used 1 hidden layer with 9 neurons configuration when it was trained with bond critical point data and 1 hidden layer with 15 neurons configuration when the model was trained with Laplacian critical point data. The predictive ability of the network is determined by validation techniques. We have used the leave-one-out cross-validation technique, in which the whole data set was divided into 225 pieces, 224 pieces used for training and one piece for testing. The mean absolute percent errors were calculated by comparing predicted values with actual values.

As a proof-of-concept exercise, we first attempted to predict routinely measured infrared stretching frequencies of the carbonyl bonds and the ¹³C chemical shift of the carbonyl carbons. Such properties can be accurately approximated by semi-empirical wavefunction-based methods, but their prediction with spectroscopic precision using true ab initio methods is still quite challenging. Development of alternative and highly efficient methodologies for predicting even these properties is still of interest.

Table 2 presents a summary of results using optimized machine learning parameters within WEKA. Default parameters yield results of significantly lower quality. While nothing close to spectroscopic accuracy, the results are promising considering that data for only 3 BCPs and 5 LCPs were used for each compound. Predictions for individual carbonyl types, using default and optimized parameters (learning rate, number of epochs and hidden neurons), as well as the optimized ANN parameters themselves, can be found in Supplementary Materials (Tables 3, 4, and 5, respectively). For instance, the Mean Absolute Percentage Error (MAPE) in predicting C=O stretching frequency for the 100+ ketones only, was 0.41% (better than results for all carbonyl types, Table 2).

It is interesting to note that, as expected, the combined (BCP and LCP) data set yields better predictions, on average. And while there is insignificant difference between the BCP and LCP data sets in predicting C=O stretching frequencies, the topological properties of the Laplacian of the charge density appears to be “better” training data for predicting chemical shifts. This, too, may not be unexpected since the NMR chemical shift of a nucleus depends on shielding over the entire volume of the carbon atom, not just in the s-plane, where all the BCPs are located.

Table 2. MAPE of predicted values of C13 chemical shifts and C=O stretching frequencies for 225 carbonyl compounds

Data set	C13 chemical shift	C=O stretching frequency
Laplacian critical point data	1.335	0.659
Bond critical point data	1.596	0.652
Combined data	1.308	0.641

Here, we report the prediction of both covalent and van der Waals interaction energies using ANNs trained on r and/or Ñ2r critical point descriptors. A nucleophilic addition reaction between a fluoride ion and a carbonyl group was taken as a model chemical reaction for our investigation of machine learning of chemical reactivity. As shown in Figure 3, preliminary investigations of reaction energy profiles, at multiple levels of theory, between a fluoride ion and acetone revealed dual minima; one at a short C-F distance (“covalent” bond formation), and one at a long C-F distance (van der Waals interaction). The interaction energies (Eqn. 1) were calculated for both strong (covalent bond formation) and weak (van der Waals) interactions for our set of 225 carbonyl-containing molecules. Our ANN was then trained on this data, using BCP, LCP, and combined data sets, as before. A summary of Mean Absolute Error (MAE) results for leave-one-out cross-validation machine learning predictions of interaction energies are shown in Table 3.

Table 3. MAE of predicted interaction energies with optimum parameters for training the ANN model

Data set	covalent interaction energy (kcal/mol)	van der Waals interaction energy (kcal/mol)
Laplacian critical point data	2.63	5.06
Bond critical point data	3.39	4.80
Combined data	2.56	4.78

The ANN model yields much better predictions for covalent, as opposed to van der Waals interaction energies. This is an unsurprising result, since machine learning is highly dependent on the quality of the training data, and the B3LYP functional is well-known to poorly reproduce the dispersion energy, which is critical to van der Waals interactions [26]. Since DFT has become a standard workhorse in molecular modelling, future research in machine learning based on charge density descriptors that have been derived from more generally reliable functionals is warranted. With regard to the covalent interaction energy predictions of the optimized ANN model, MAEs are above the desired chemical accuracy that can be achieved easily with inexpensive empirical methods, but they support the idea that there is very high-quality information content in just a small number of charge density descriptors. We note in particular that LCP data is significantly better at predicting covalent interaction energies than the commonly used BCP data. This should not be surprising, because although the topological properties of the charge density led to the development of QTAIM in the first place, it is the topological properties of the Laplacian of the charge density that have been more useful in subsequent [27], as well as ongoing, studies of chemical reactivity based on the charge density [28]. This is perhaps the most salient observation from the current investigation: Not only is the topology of the Laplacian of the charge density more complex than that of the electron density (many more critical points), but the information content at these critical points is higher when the objective is to predict chemical reactivity. Predictions for individual carbonyl types, using default and optimized parameters (learning rate, number of epochs and hidden neurons), as well as the optimized ANN parameters themselves, can be found in Supplementary Materials (Tables 6, 7, and 8, respectively).

The accuracy of our ANN model predictions of the spectroscopic properties and chemical reactivity of the 225 carbonyl-containing compounds studied in this research is not the competitive with standard quantum chemical or even semi-empirical methods. And our ANN model is based on training data that is derived from the aforementioned quantum chemical methods. However, if developed further, the type of ANN model that we have explored will become competitive when the chemical systems become too big to do rapid conventional computational chemistry of any kind. Our ANN model depends only on topological properties of the charge density, which is measurable, and only at the site of interaction. While it is necessary, currently, to have very high-resolution single-crystal X-ray diffraction data [4] to determine the topological properties of the charge density that our ANN model is trained with, progress is being made in Quantum Crystallographic databases [29] that would enable very rapid approximation of the topological properties required to train or apply our ANN model, and for a molecule of any size, as long as the nuclear coordinates are known, either by low-resolution structure determination methods, or by rudimentary classical molecular mechanics.

With a leap of faith into this hopeful future, we tested the accuracy of our small molecule-trained ANN model in a stress-test involving nucleophilic addition to a natural carbonyl-containing substrate of the E. coli enzyme, D-fructose-6-phosphate aldolase (FSA), which catalyzes such a nucleophilic addition, and which has a known molecular structure with a bound substrate (glycerol) [30]. We replaced glycerol with 3-hydroxypropanal (3HP), which nucleophilic addition to is catalyzed naturally by FSA, retaining hydrogen bonding contacts between the hydroxyl group and residues Asn28 and Asp6 (Residues 4 and 5, respectively, in Fig. 4). We then optimized the substrate in a fixed binding pocket. The ∇²ρ = 0.0 isosurface for the Laplacian of the charge density for 3HP in the binding pocket is shown in Figure 4. Many π-holes are evident, such as those in the VSCC of carbons in the aromatic ring of tyrosine (Residue 2 in Fig. 4), which measure about 20 picometers in diameter, or less, if there is a π-hole at all. As expected, the largest π-hole is on the carbonyl carbon of 3HP (near the center of the cluster in Figure 4, bottom), which, after all, is the target of the nucleophilic attack. Its π-hole measures approximately 50 picometers in diameter, and is also shown in a cross-sectional view in Figure 5, both before and after covalent bond formation. These well-defined, observable, and often transferable subatomic features, which are made evident by the Laplacian of the charge density, are clearly related to the chemical properties of the respective atoms. Data associated with these features, such as their sizes, locations, and topological properties, all fall under the label of Quantum Atomics. Their size, in particular, also serves to illustrate and justify at least the prefix of the term picotechnology.

Eqn. 2 was used to calculate interaction energies quantum mechanically, at the same level of theory as before. In addition to 3HP, the fluoride nucleophile was optimized alone in the fixed pocket, as were the combined reactants. All charge density analysis was done as before (with the 225 carbonyl-containing compounds). Only a covalent interaction energy was determined, as there was not enough room in the pocket for a van der Waals complex to form between the fluoride ion and 3HP. The same ANN model that was optimized for the 225 carbonyl-containing compounds was used to predict the covalent interaction energy, using both the BCP and LCP data, as well as the combined data set. The results of our quantum chemical calculations and the ANN model predictions are summarized in Table 4.

ΔE_interaction= E_{pocket+reactants(CC+F)}– E_pocket+CC– E_pocket+F + E_pocket (Eqn. 2)

Table 4. Comparison of covalent ΔE_interaction in the binding pocket of the FSA enzyme (kcal/mol)

DFT	ANN model	ANN model	ANN model
B3LYP/6-31+G*	BCP data	LCP data	Combined data
-22.7	-11.9	-19.7	-14.9

As in the case of using the ANN model to predict interaction energies of the 225 carbonyl-containing compounds, training data based on the topological properties of the Laplacian of the charge density led to more accurate predictions of chemical reactivity. We note that when LCP data is used, the 3.0 kcal/mol absolute error in the prediction of the covalent interaction energy of this large and complex enzyme model is about the same as the Mean Absolute Error in covalent interaction energies for smaller and simpler carbonyl compounds, even though no additional training was done. A very recent study combining charge density analysis with machine learning sought to investigate the inhibition mechanism of cruzain, a cysteine protease that is key to the life cycle of the parasite responsible for Chagas disease [31]. Their charge density-based descriptors were limited to BCP training data. Our research indicates that such studies could be greatly improved by including Laplacian-based descriptors among the training data.

The rapid advancement of machine learning technology is matched by the growing range of applications to which it is being applied. But like any other advanced technology, it is not a one-size-fits-all tool. We already know that the type and quality of training data is as important as the method by which training occurs. The research presented here reinforces this. Additionally, the converse of this principle is that machine learning can also be used as a tool to empirically explore the information content of different properties of matter that are accessible yet not yet fully understood. When it comes to machine learning applications in materials science and engineering, nothing that is measurable is more fundamental than the charge density. If we completely understood the forces shaping the charge distribution in matter, functional development in Density Functional Theory would be a straightforward matter. It is not. The research presented here has offered insight into the types of applications that different properties of the charge density are better suited as training data. Simultaneous with advancements in machine learning, developments in both the theory and practice of Quantum Crystallography [32] make measurement of the charge density and its derivative properties ever more accessible and accurate. Advancement of nanotechnology has ultimately been dependent on engineering molecular structure and reactivity [33], which chemists achieve via substitutions of one type or another - at the atomic or functional group level. Again, this is not straightforward, because there is incomplete understanding of how atomic substitutions determine changes in molecular properties. And that is the long-term goal of this research; to advance a picotechnology by combining the data-mining and property-prediction power of machine learning with the ever-increasing understanding of the charge distribution - with subatomic resolution.

Author Contribution

PJM wrote the main manuscript text and KKD performed the calculations and prepared figures 1-5. All authors reviewed the manuscript.

Acknowledgement

Financial support is acknowledged from the Office of Science, U.S. Department of Energy (DE-SC0005094).

Data Availability

Data is provided within the manuscript or supplementary information files.

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No competing interests reported.

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Using Quantum Atomics and Machine Learning to Advance Picotechnology

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Abstract

Figures

WHAT IS PICOTECHNOLOGY?

COMPUTATIONAL METHODS

2.1Quantum Chemistry

2.2 Data Sets and Quantum Atomics

2.3 Artificial Neural Network Model

Proof-of-concept Spectroscopic Predictions

Nucleophilic Interaction Energy Predictions

stress-test: predicting interaction energies of substrates in an enzyme’s active site

CONCLUSIONS AND OUTLOOK

Declarations

Author Contribution

Acknowledgement

Data Availability

References

Additional Declarations

Supplementary Files

Status:

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