Graph Theoretical Description of Phase Transitions in Complex Multiscale Phases with Supramolecular Assemblies.

Phase transitions are typically quantified using order parameters, such as crystal lattice distances and radial distribution functions, which can identify subtle changes in crystalline materials or high-contrast phases with large structural differences. However, the identification of phases with high complexity, multiscale organization and of complex patterns during the structural fluctuations preceding phase transitions, which are essential for understanding the system pathways between phases, is challenging for those traditional analyses. Here, it is shown that for two model systems- thermotropic liquid crystals and a lyotropic water/surfactant mixtures-graph theoretical (GT) descriptors can successfully identify complex phases combining molecular and nanoscale levels of organization that are hard to characterize with traditional methodologies. Furthermore, the GT descriptors also reveal the pathways between the different phases. Specifically, centrality parameters and node-based fractal dimension quantify the system behavior preceding the transitions, capturing fluctuation-induced breakup of aggregates and their long-range cooperative interactions. GT parameterization can be generalized for a wide range of chemical systems and be instrumental for the growth mechanisms of complex nanostructures.

Spanning Network Gels from Nanoparticles and Graph Theoretical Analysis of Their Structure and Properties.

Gels self-assembled from colloidal nanoparticles (NPs) translate the size-dependent properties of nanostructures to materials with macroscale volumes. Large spanning networks of NP chains provide high interconnectivity within the material necessary for a wide range of properties from conductivity to viscoelasticity. However, a great challenge for nanoscale engineering of such gels lies in being able to accurately and quantitatively describe their complex non-crystalline structure that combines order and disorder. The quantitative relationships between the mesoscale structural and material properties of nanostructured gels are currently unknown. Here, it is shown that lead telluride NPs spontaneously self-assemble into a spanning network hydrogel. By applying graph theory (GT), a method for quantifying the complex structure of the NP gels is established using a topological descriptor of average nodal connectivity that is found to correlate with the gel's mechanical and charge transport properties. GT descriptions make possible the design of non-crystalline porous materials from a variety of nanoscale components for photonics, catalysis, adsorption, and thermoelectrics.

Unifying structural descriptors for biological and bioinspired nanoscale complexes.

Biomimetic nanoparticles are known to serve as nanoscale adjuvants, enzyme mimics and amyloid fibrillation inhibitors. Their further development requires better understanding of their interactions with proteins. The abundant knowledge about protein-protein interactions can serve as a guide for designing protein-nanoparticle assemblies, but the chemical and biological inputs used in computational packages for protein-protein interactions are not applicable to inorganic nanoparticles. Analysing chemical, geometrical and graph-theoretical descriptors for protein complexes, we found that geometrical and graph-theoretical descriptors are uniformly applicable to biological and inorganic nanostructures and can predict interaction sites in protein pairs with accuracy >80% and classification probability ~90%. We extended the machine-learning algorithms trained on protein-protein interactions to inorganic nanoparticles and found a nearly exact match between experimental and predicted interaction sites with proteins. These findings can be extended to other organic and inorganic nanoparticles to predict their assemblies with biomolecules and other chemical structures forming lock-and-key complexes.

Generator based approach to analyze mutations in genomic datasets.

In contrast to the conventional approach of directly comparing genomic sequences using sequence alignment tools, we propose a computational approach that performs comparisons between sequence generators. These sequence generators are learned via a data-driven approach that empirically computes the state machine generating the genomic sequence of interest. As the state machine based generator of the sequence is independent of the sequence length, it provides us with an efficient method to compute the statistical distance between large sets of genomic sequences. Moreover, our technique provides a fast and efficient method to cluster large datasets of genomic sequences, characterize their temporal and spatial evolution in a continuous manner, get insights into the locality sensitive information about the sequences without any need for alignment. Furthermore, we show that the technique can be used to detect local regions with mutation activity, which can then be applied to aid alignment techniques for the fast discovery of mutations. To demonstrate the efficacy of our technique on real genomic data, we cluster different strains of SARS-CoV-2 viral sequences, characterize their evolution and identify regions of the viral sequence with mutations.

Deciphering the generating rules and functionalities of complex networks.

Network theory helps us understand, analyze, model, and design various complex systems. Complex networks encode the complex topology and structural interactions of various systems in nature. To mine the multiscale coupling, heterogeneity, and complexity of natural and technological systems, we need expressive and rigorous mathematical tools that can help us understand the growth, topology, dynamics, multiscale structures, and functionalities of complex networks and their interrelationships. Towards this end, we construct the node-based fractal dimension (NFD) and the node-based multifractal analysis (NMFA) framework to reveal the generating rules and quantify the scale-dependent topology and multifractal features of a dynamic complex network. We propose novel indicators for measuring the degree of complexity, heterogeneity, and asymmetry of network structures, as well as the structure distance between networks. This formalism provides new insights on learning the energy and phase transitions in the networked systems and can help us understand the multiple generating mechanisms governing the network evolution.

Network science characteristics of brain-derived neuronal cultures deciphered from quantitative phase imaging data.

Understanding the mechanisms by which neurons create or suppress connections to enable communication in brain-derived neuronal cultures can inform how learning, cognition and creative behavior emerge. While prior studies have shown that neuronal cultures possess self-organizing criticality properties, we further demonstrate that in vitro brain-derived neuronal cultures exhibit a self-optimization phenomenon. More precisely, we analyze the multiscale neural growth data obtained from label-free quantitative microscopic imaging experiments and reconstruct the in vitro neuronal culture networks (microscale) and neuronal culture cluster networks (mesoscale). We investigate the structure and evolution of neuronal culture networks and neuronal culture cluster networks by estimating the importance of each network node and their information flow. By analyzing the degree-, closeness-, and betweenness-centrality, the node-to-node degree distribution (informing on neuronal interconnection phenomena), the clustering coefficient/transitivity (assessing the "small-world" properties), and the multifractal spectrum, we demonstrate that murine neurons exhibit self-optimizing behavior over time with topological characteristics distinct from existing complex network models. The time-evolving interconnection among murine neurons optimizes the network information flow, network robustness, and self-organization degree. These findings have complex implications for modeling neuronal cultures and potentially on how to design biological inspired artificial intelligence.

Biomorphic structural batteries for robotics.

Batteries with conformal shape and multiple functionalities could provide new degrees of freedom in the design of robotic devices. For example, the ability to provide both load bearing and energy storage can increase the payload and extend the operational range for robots. However, realizing these kinds of structural power devices requires the development of materials with suitable mechanical and ion transport properties. Here, we report biomimetic aramid nanofibers-based composites with cartilage-like nanoscale morphology that display an unusual combination of mechanical and ion transport properties. Ion-conducting membranes from these aramid nanofiber composites enable pliable zinc-air batteries with cyclic performance exceeding 100 hours that can also serve as protective covers in various robots including soft and flexible miniaturized robots. The unique properties of the aramid ion conductors are attributed to the percolating network architecture of nanofibers with high connectivity and strong nanoscale filaments designed using a graph theory of composite architecture when the continuous aramid filaments are denoted as edges and intersections are denoted as nodes. The total capacity of these body-integrated structural batteries is 72 times greater compared with a stand-alone Li-ion battery with the same volume. These materials and their graph theory description enable a new generation of robotic devices, body prosthetics, and flexible and soft robotics with nature-inspired distributed energy storage.