Eigencages: Learning a Latent Space of Porous Cage Molecules

Abstract

Porous organic cage molecules harbor nanosized cavities that can selectively adsorb gas molecules, lending them applications in separations and sensing. The geometry of the cavity strongly influences their adsorptive selectivity. For comparing cages and predicting their adsorption properties, we embed/encode a set of 74 porous organic cage molecules into a low-dimensional, latent "cage space" on the basis of their intrinsic porosity. We first computationally scan each cage to generate a three-dimensional (3D) image of its porosity. Leveraging the singular value decomposition, in an unsupervised manner, we then learn across all cages an approximate, lower-dimensional subspace in which the 3D porosity images congregate. The "eigencages" are the set of orthogonal, characteristic 3D porosity images that span this lower-dimensional subspace, ordered in terms of importance. A latent representation/encoding of each cage follows by approximately expressing it as a combination of the eigencages. We show that the learned encoding captures salient features of the cavities of porous cages and is predictive of properties of the cages that arise from cavity shape. Our methods could be applied to learn latent representations of cavities within other classes of porous materials and of shapes of molecules in general.

Figure 1

Structures of all aligned 74 porous organic cage molecules^, comprising the training set in this study, ordered by cage molecule diameter computed from pywindow. Note the diversity of cavities.

Figure 2

Example 3D porosity images. (a–f) The molecular structure of a cage (name and experimental reference^,− in subcaption) is shown along with a contour (0.5) of the 3D porosity image (orange). Bounding box shows [−20 Å, 20 Å]³ dimension of the snapshot, consistent for all cages.

Figure 3

Full singular value decomposition of A in eq 1. Relevant rows/columns are colored and labeled.

Figure 4

Low rank approximant A_ν ≈ A in eq 6. Compare to Figure 3. Eigencage k is row k of V_ν^T. The latent representation of cage i is row i of U_νΣ_ν.

Figure 5

Visualizing the eigencages. (a) Contour surfaces of the average 3D porosity image. Values increase from light to dark green. (b–g) Contour surfaces of the first six eigencages. Blue: low (negative), Gray: intermediate, red: high (positive).

Figure 6

Reconstructing cage B25 with its latent representation. (a) Exact 3D void space image of B25. (b–j) Reconstructions using latent representations of varying dimensions ν. These are contours (0.5) of eq 8 with varying ν (views from an angle to clearly see the cavity).

Figure 7

Latent representations of cages U_νΣ_ν embedded into 2D by t-SNE.^, Salient clusters are highlighted. See Figure S6.1 for the remaining clusters.

Figure 8

Latent representations of cages U_νΣ_ν embedded into 2D by t-SNE.^, The color of points represents the simulated Xe/Kr selectivity of an isolated cage molecule in an empty box at 298 K. Points nearby in the latent cage space are likely to exhibit similar Xe/Kr selectivities. Cages marked with ‘X’ have windows too narrow for xenon to percolate into the cavity.

Figure 9

Walking through latent space from cage NC2 (a) to DC1 (f). Panels (b–e) are fictitious cage cavities generated by walking along a line between the latent representation of NC2 and DC1.

Figure 10

Flexible cages explore an area in latent space. (a) Snapshots of the structure of an isolated, empty cage CC5 undergoing thermal fluctuations during a molecular dynamics simulation. (b) The gray points are the ν = 2D latent representations of the (rigid) cages learned by SVD, rows of U₂Σ₂. The 3D porosity images of 400 snapshots of cages CC2, CC3, CC4, and CC5 (colored) during molecular dynamics simulations are projected onto this 2D latent space. To precisely define the axes, the 3D porosity image of cage k in the scatter plot is best-approximated in terms of a linear combination of the first two eigencages as . Each fluctuating cage explores a latent region determined by the set of cavity shapes explored while undergoing thermal fluctuations. MC6, an outlier, is omitted to avoid stretched axes.