Investigating Carboxysome Morphology Dynamics with a Rotationally Invariant Variational Autoencoder

Carboxysomes are a class of bacterial microcompartments that form proteinaceous organelles within the cytoplasm of cyanobacteria and play a central role in photosynthetic metabolism by defining a cellular microenvironment permissive to $CO_2$ fixation. Critical aspects of the assembly of the carboxysomes remain relatively unknown, especially with regard to the dynamics of this microcompartment. We have recently expressed an exogenous protease as a way of gaining control over endogenous protein levels, including carboxysomal components, in the model cyanobacterium \textit{Synechococcous elongatus} PCC 7942. By utilizing this system, proteins that compose the carboxysome can be tuned in real-time as a method to examine carboxysome dynamics. Yet, analysis of subtle changes in carboxysome morphology with microscopy remains a low-throughput and subjective process. Here we use deep learning techniques, specifically a Rotationally Invariant Variational Autoencoder (rVAE), to analyze the fluorescence microscopy images and quantitatively evaluate how carboxysome shell remodelling impacts trends in the morphology of the microcompartment over time. We find that rVAEs are able to assist in the quantitative evaluation of changes in carboxysome location, shape, and size over time. We propose that rVAEs may be a useful tool to accelerate the analysis of carboxysome assembly and dynamics in response to genetic or environmental perturbation, and may be more generally useful to probe regulatory processes involving a broader array of bacterial microcompartments.

Deep Bayesian local crystallography

The advent of high-resolution electron and scanning probe microscopy imaging has opened the floodgates for acquiring atomically resolved images of bulk materials, 2D materials, and surfaces. This plethora of data contains an immense volume of information on materials structures, structural distortions, and physical functionalities. Harnessing this knowledge regarding local physical phenomena necessitates the development of the mathematical frameworks for extraction of relevant information. However, the analysis of atomically resolved images is often based on the adaptation of concepts from macroscopic physics, notably translational and point group symmetries and symmetry lowering phenomena. Here, we explore the bottom-up definition of structural units and symmetry in atomically resolved data using a Bayesian framework. We demonstrate the need for a Bayesian definition of symmetry using a simple toy model and demonstrate how this definition can be extended to the experimental data using deep learning networks in a Bayesian setting, namely rotationally invariant variational autoencoders.

A general method for rotational averages

The theory of nonlinear spectroscopy on randomly oriented molecules leads to the problem of averaging molecular quantities over random rotation. We solve this problem for arbitrary tensor rank by deriving a closed-form expression for the rotationally invariant tensor of averaged direction cosine products. From it, we obtain some useful new facts about this tensor. Our results serve to speed the inherently lengthy calculations of nonlinear optics.

Geometric Interpretation and General Classification of Three-Dimensional Polarization States through the Intrinsic Stokes Parameters

In contrast with what happens for two-dimensional polarization states, defined as those whose electric field fluctuates in a fixed plane, which can readily be represented by means of the Poincaré sphere, the complete description of general three-dimensional polarization states involves nine measurable parameters, called the generalized Stokes parameters, so that the generalized Poincaré object takes the complicated form of an eight-dimensional quadric hypersurface. In this work, the geometric representation of general polarization states, described by means of a simple polarization object constituted by the combination of an ellipsoid and a vector, is interpreted in terms of the intrinsic Stokes parameters, which allows for a complete and systematic classification of polarization states in terms of meaningful rotationally invariant descriptors.