A new topological descriptor for water network structure
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Abstract
Bulk water molecular dynamics simulations based on a series of atomistic water potentials (TIP3P, TIP4P/Ew, SPC/E and OPC) are compared using new techniques from the field of topological data analysis. The topological invariants (the different degrees of homology) derived from each simulation frame are used to create a series of persistence diagrams from the atomic positions. These are averaged over the simulation time using the persistence image formalism, before being normalised by their total magnitude (the L1 norm) to ensure a size independent descriptor (L1NPI). We demonstrate that the L1NPI formalism is suitable for the analysis of systems where the number of molecules varies by at least a factor of 10. Using standard machine learning techniques, a basic linear SVM, it is shown that differences in water models are able to be isolated to different degrees of homology. In particular, whereas first degree homology is able to distinguish between all atomistic potentials studied, OPC is the only potential that differs in its second degree homology. The L1 normalised persistence images are then used in the comparison of a series of Stillinger–Weber potential simulations to the atomistic potentials and the effects of changing the strength of three-body interactions on the structures is easily evident in L1NPI space, with a reduction in variance of structures as interaction strength increases being the most obvious result. Furthermore, there is a clear tracking in L1NPI space of the λ parameter. The L1NPI formalism presents a useful new technique for the analysis of water and other materials. It is approximately size-independent, and has been shown to contain information as to real structures in the system. We finally present a perspective on the use of L1NPIs and other persistent homology techniques as a descriptor for water solubility.
Keywords
Persistent homology Water networks Topological data analysisIntroduction
The water network problem
Understanding the structure and dynamics of water networks is an important task in a wide variety of fields. This is due to the anomalous behaviour of water, such as the well-known density maximum. Further, these anomalies have been shown to play important roles in physical, chemical, and biological processes [1, 2]. There have therefore been many studies of simulated water systems, often looking at radial distribution functions [3] or spatial distribution functions [4]. In particular, the tetrahedral nature of local water has been investigated [5, 6].
This has led to a plethora of computational techniques for understanding water network structure. In general, these can be split into categories such as coordination number studies [3, 7, 8, 9] and graph-theoretical studies [10, 11, 12, 13]. Both of these categories have drawbacks, namely the difficulty in interpreting data beyond nearest neighbours, and the requirement for a connectivity heuristic respectively.
Mathematical techniques drawn from topology look highly suitable to make progress in the analysis of connectivity. In particular, persistent homology is a recent development in mathematics, in the field of topological data analysis, and creates a multiscale representation of an arbitrary point cloud [14]. This is achieved by converting this point cloud into a filtration of topological structures, and observing how topological invariants change in this filtration. Persistence has found many uses in chemistry, mainly in proteins [15, 16, 17, 18, 19, 20, 21, 22], but also as a small molecule descriptor [23, 24] or a descriptor for the analysis of crystal structures and other materials [25, 26, 27, 28, 29, 30]. Furthermore, persistent homology has recently been applied to understanding water networks [31] however these methods did not take into account the dynamic nature of such systems.
In this work, we develop the ideas discussed in [31] and by the use of persistence images [32] are able to develop what we term l_{1}-normalised persistence images (L1NPIs) which take into account the dynamic nature of the molecular dynamics simulations. These descriptors are size-agnostic, meaning they can be used between systems with vastly different numbers of water molecules, and are well-suited for machine learning techniques. We apply this technique to a range of atomistic water models and a coarse-grained Stillinger–Weber (SW) potential [33], and using this technique are able to not only distinguish between these models, but relate these differences to the underlying water network. We lastly present a perspective as to how this technique can be used to understand the solute–solvent interaction, as well as potential challenges and pitfalls.
Theory
Rather than present the fundamentals of persistent homology (see references [34, 35, 36, 37] for introductions to the field), we will instead present a ‘greatest hits', where we will aim to give the reader a basic understanding, while paying little attention to the man behind the curtain.
Persistent homology
Definition 1
For every pair of points in (x, y) in S, if d(x, y)<, we draw a line between x and y. If every pair in a triplet (quartet, etc.) is connected, we draw the triangle (tetrahedron, etc.) between them.
Description of different Betti numbers \(\beta_{n}\) and their associated values for a sphere and torus
\(\beta_{n}\) | Description | Sphere | Torus |
---|---|---|---|
0 | Connected components | 1 | 1 |
1 | (Non-contractible) loops | 0 | 2 |
2 | Voids | 1 | 1 |
The final ingredient of persistent homology is the ‘persistence’. One may ask—What is the best value of to define a VR complex on a data set? Persistent homology answers: all of them. By considering how the topology of the VR complex changes as we go through a range of δ, we hope to gain understanding as the structure of the underlying set of points. Any hole born at \(t\) must be filled in by some \(t '\). Therefore, we represent the persistent homology of a set of points by considering when topological features are born and when they die, in a persistence diagram.
Persistence images
For this work, we will be trying to understand simulated water networks through the lens of persistent homology. Rather than comparing descriptors computed from single frames of simulation, which would be susceptible to noise, we would like to use a notion of average persistence. However, the persistence diagram is not well-suited to such a task (for more details we direct the reader to [40], particularly Fig. 3 therein). Therefore, there have been many attempts to construct vector representations of persistence diagrams that can have statistical techniques applied to them, including persistence landscapes [40, 41], kernel embeddings [42], and persistence images [32]. In this work, we use the persistence image, which transforms a single persistence diagram into a literal grayscale image. Furthermore, calculating the average of a set of images is as simple as finding the average value for each pixel. Lastly, persistence images are relatively simple to interpret, as they look similar to the persistence diagram.
- 1.
Select a single degree of homology
- 2.
Transform each point of this degree from \(\left( {b,d} \right)\) to \(\left( {b,p} \right),\) where \(p = d - b\)
- 3.For each point \(\left( {b,p} \right),\) define the function:$$g\left( {x,y} \right) = \frac{1}{{2\pi\sigma^{2} }}\exp \left( {\frac{{ - \left( {\left( {x - b} \right)^{2} + \left( {y - p} \right)^{2} } \right)}}{{2\pi\sigma^{2} }}} \right)$$
- 4.
Multiply each function \(g\left( {x,y} \right)\) by \(\phi \left( {x,y} \right)\), where \(\phi\left( {x,0} \right) = 0.\) This is done for stability reasons, and is discussed in more detail in [32]
- 5.
Integrate \(g\left( {x,y} \right)\phi\left( {x,y} \right)\), over a collection of pixels
- 6.
The persistence image I is this discretisation of \(g\left( {x,y} \right)\phi\left( {x,y} \right)\)
Simulation details
A brief summary is given here for the molecular dynamics simulations and for more information about simulation details, please refer to the Additional file 1.
Atomistic potentials
The parameters of the various water models used in this study, and their physical meaning Open image in new window
Model | \(q/e\) | l/Å | z/Å | \(\theta_{LJ} /\) ° | σ_{LJ}/Å | \(\epsilon_{LJ}/{\text{kJmol}}^{- 1}\) | \(n_{atom}\) |
---|---|---|---|---|---|---|---|
TIP3P | 0.4170 | 0.9572 | N/A | 104.52 | 3.15061 | 0.636 | 4287 |
TIP4P/Ew | 0.5242 | 0.9572 | 0.1250 | 104.52 | 3.16345 | 0.681 | 4254 |
SPC/E | 0.4238 | 1.0000 | N/A | 109.47 | 3.16600 | 0.890 | 4287 |
OPC | 0.6971 | 0.8724 | 0.1594 | 103.60 | 3.16655 | 0.89036 | 4302 |
SW | – | – | – | – | – | – | 512 |
The Stillinger–Weber potential
In this work, we use the parameters A = 7.049556277, B = 0.6022245584, p = 4, q = 0, \(\cos \theta_{0} = \frac{1}{3}\), γ = 1.2, and a = 1.8. All simulations of the SW potential were performed at the ambient temperature and pressure corresponding to the melting temperature at that particular λ.
Persistent homology procedure for water simulations
Persistent homology
Given a single frame of a simulation, we use the locations of the oxygen atoms as our point cloud. This leads to a substantially quicker computation time, as we are reducing the number of points in our system by 2/3. This decision also makes sense from a theoretical perspective, namely that it is the tetrahedral nature of the oxygen lattice which is of interest, and including the hydrogen atoms as equal in the persistent homology would likely ‘wash out' this information, and instead simply capture the persistent of densely sampled Euclidean 3-space. We note that this procedure is much simpler than the element specific persistent homology of Cang and Wei [16] and the multiparameter persistence of PHoS developed by Keller, Lesnick and Willke [48]. However, the relative simplicity of our systems compared to the drug-like biomolecules used in their work allows us to use such a simple procedure. Furthermore, our procedure naturally extends to the coarse-grained SW potential. For each degree of homology separately, we calculate the persistent homology for every frame of simulation, before converting each persistence diagram into a persistence image.
L1-normalised persistence images
Comparison to other techniques
The radial distribution function (RDF) is a standard tool when analysing simulations of materials, such as the water networks discussed in this work. The RDF describes the relative density of water molecules as a function of distance, and allows the discovery of solvation shells. The RDF has previously been used to compare different water models such as in [3], where it was shown that the slight differences in Lennard-Jones and Coulombic terms led to pronounced changes in density of second-nearest neighbours. An extension of the RDF, the spatial distribution function (SDF) was developed, which does not integrate out the angular distribution in the manner of the RDF. The SDF, when applied to SPC/E water, led to the discussion of two different motifs, a temperature independent tetrahedral water, and a non-tetrahedral structure that appeared to vary with temperature [4].
Persistent homology is a more complex and rich tool for analysing these structures. Rather than simply studying the relative positions of pairs of water molecules, the simplicial complex required in persistent homology contains information about groups of water molecules. For example, the presence of the triangle abc in the simplicial complex requires all pairs ab, ac, bc, to be within a particular distance of each other. This leads to information that can be related to the RDF—the nearest neighbour distance can be estimated using zeroth-degree homology—but also information that is not so easily extracted from either the RDF or SDF—such as the presence of rings of water structures.
Results and discussion
Comparison of persistence images and L1NPIs
Comparison of atomistic potentials
Comparison of series of Stillinger–Weber potentials
It is also interesting to note that the distribution of points in L1NPI space narrows as λ increases. This suggests a reduction in the variance of the persistent homology. As λ increases, the relative strength of the three-body interaction defined in the SW potential increases. This leads to a reduction in the variance of next-nearest neighbour distances, which is then reflected in the persistence.
Comparison of atomistic and Stillinger–Weber potentials
We note that in first degree homology, the atomistic potential simulations are closest to the value of λ = 23.15. This is the value of λ that is considered to lead to water-like structures, as it reproduces the density profile of water on a range of temperatures. Interestingly, this is not the case in second degree homology, where the L1NPI descriptor suggests that the atomistic systems are more similar to λ = 23.95, with OPC being closest to λ = 23.15. This separation of degrees of homology is a useful property of the L1NPI analysis, where we are able to say that although the atomistic structures have the same ‘loops’ of the SW structures, they do not match the ‘holes’, with OPC being the closest.
We finally return to the size-independent nature of our L1NPI descriptor. Whereas the atomistic potentials have in excess of 4000 water molecules, the simulations of the SW potential have 512. However, the L1NPI descriptor can be used to compare such systems, irrespective of system size. We do note that there is one consequence of size in the L1NPI formalism, which can be seen in the PCA images. Namely, systems with more molecules lead to a tighter distribution of points in L1NPI space. Considering a single frame of a simulation, we note that more molecules will lead to more points in the persistence diagram, which will become more ‘filled in’. This implies that the individual persistence diagrams (and therefore images and L1NPIs) will be more similar to each other in simulations with a larger number of particles. Therefore, the L1NPI descriptor is not entirely size-independent, although it is far more size independent than other persistence representations.
Conclusion
We have derived a new descriptor for water network structure, using topological data analysis. By applying persistent homology, the study of holes in data, to the point cloud defined by oxygen atom coordinates, we are able to gain insight as to what distinguishes various structures created by different intermolecular potentials. Whereas more commonly used techniques, such as persistence landscapes [40] are unable to be used on systems of widely varying sizes, we have shown that our technique, the L_{1}-normalised persistence image (L1NPI) is relatively size-independent.
We first applied the L1NPI formalism to four commonly used atomistic potentials: TIP3P, TIP4P/Ew, SPC/E and OPC. We were able to determine that first degree homology (i.e. loops) were enough to distinguish between these potentials, even with a relatively simple linear support vector machine. In contrast, second degree homology (holes) was only able to distinguish between OPC and the other models. We consider this to be a consequence OPC’s rather unique parameterisation technique. We are also able to show that TIP4P/Ew and SPC/E are more similar than the other atomistic models, purely based on their proximity in L1NPI space.
We then investigated a series of Stillinger–Weber potentials. By tuning the parameter λ, the relative strength of the three-body interaction can be altered. The L1NPI formalism showed that differences in structure caused by changing λ are much more pronounced than those found in the atomistic potentials. Furthermore, we were able to relate properties such as nearest neighbour distances to observations in L1NPI space.
We finally compared the atomistic systems to the Stillinger–Weber potential series. We noticed that in first degree homology, the atomistic structures are closest to the widely accepted value of λ = 23.15. In contrast, second degree homology suggests that the structures are closer to slightly higher values of λ, with OPC being closest to 23.15. Furthermore, by comparing systems of widely different sizes (512 vs. 4000 water molecules), we show that the L1NPI formalism is size-independent.
It would be interesting to study generalisations of the persistence image to other means, rather than simply the L1 norm, as a method of future work. The use of generalised mean-based descriptors is well established, such as in [50, 51], and we feel that different means could be able to account for other discrepancies than system size.
We conclude by discussing the application of the L1NPI formalism to the solubility problem, Although it is widely accepted that there is a need to produce better models (as evidenced by the ‘Solubility Challenge’ [52, 53]) models are still unable to accurately predict water solubility [53]. We feel that a large amount of research is invested in producing models with more complex designs. This, coupled with the lack of high-quality solubility data, leads to overfitted models, as well as poor interpretability. However, the L1NPI formalism could be applied to solute–solvent systems, and in particular differences in L1NPIs could be related to perturbations of the water network. We plan to expand upon this problem further in a future publication.
Notes
Acknowledgements
The authors thank Jacek Brodzki, Mariam Pirashvili and Francisco Belchi for helpful discussions. L.S. thanks Dr Francis Longford for helpful discussions. The authors also acknowledge the use of the IRIDIS High Performance Computing Facility in the completion of this work.
Authors’ contributions
JGF designed the study with contributions from LS and JR Simulations of atomistic potentials were performed by LS, and the Stillinger–Weber potential simulations were performed by JR Analysis pipeline was designed and implemented by LS Results were analyzed by all authors. The manuscript was written by LS with contributions from all authors. All authors read and approved the final manuscript.
Funding
This research was supported by EPSRC Grants EP/N014189/1 Joining the dots: from data to insight, and L.S. thanks the EPSRC for the studentship support through the Centre for Doctoral Training in Theory and Modelling in Chemical Sciences EP/L015722/1. J.R. acknowledges support from the European Research Council Grant DLV-759187 and the Royal Society University Research Fellowship.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Supplementary material
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