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Abstract We introduce and study the spatial replicator equation with higher order interactions and both infinite (spatially homogeneous) populations and finite (spatially inhomogeneous) populations. We show that in the special case of three strategies (rock–paper–scissors) higher order interaction terms allow travelling waves to emerge in non-declining finite populations. We show that these travelling waves arise from diffusion stabilisation of an unstable interior equilibrium point that is present in the aspatial dynamics. Based on these observations and prior results, we offer two conjectures whose proofs would fully generalise our results to all odd cyclic games, both with and without higher order interactions, assuming a spatial replicator dynamic. Intriguingly, these generalisations for $N\text{⩾}5$strategies seem to require declining populations, as we show in our discussion. 

Interactions among the underlying agents of a complex system are not only limited to dyads but can also occur in larger groups. Currently, no generic model has been developed to capture high-order interactions (HOI), which, along with pairwise interactions, portray a detailed landscape of complex systems. Here, we integrate evolutionary game theory and behavioral ecology into a unified statistical mechanics framework, allowing all agents (modeled as nodes) and their bidirectional, signed, and weighted interactions at various orders (modeled as links or hyperlinks) to be coded into hypernetworks. Such hypernetworks can distinguish between how pairwise interactions modulate a third agent (active HOI) and how the altered state of each agent in turn governs interactions between other agents (passive HOI). The simultaneous occurrence of active and passive HOI can drive complex systems to evolve at multiple time and space scales. We apply the model to reconstruct a hypernetwork of hexa-species microbial communities, and by dissecting the topological architecture of the hypernetwork using GLMY homology theory, we find distinct roles of pairwise interactions and HOI in shaping community behavior and dynamics. The statistical relevance of the hypernetwork model is validated using a series of in vitro mono-, co-, and tricultural experiments based on three bacterial species. 

Biodegradation of insoluble biomass such as cellulose via carbohydrase enzymes is an effective approach to break down plant cell walls and extract valuable materials therein. Yet, the high cost and poor reusability of enzymes are practical concerns. We recently proved that immobilizing multiple digestive enzymes on metal–organic materials (MOMs) allows enzymes to be reused via gravimetric separation, improving the cost efficiency of cereal biomass degradation [ACS Appl. Mater. Interfaces 2021, 13, 36, 43085–43093]. However, this strategy cannot be adapted for enzymes whose substrates or products are insoluble (e.g., cellulose crystals). Recently, we described an alternative approach based on magnetic metal–organic frameworks (MOFs) using model enzymes/substrates [ACS Appl. Mater. Interfaces 2020, 12, 37, 41794–41801]. Here, we aim to prove the effectiveness of combining these two strategies in cellulose degradation. We immobilized multiple carbohydrase enzymes that cooperate in cellulose degradation via cocrystallization with Ca2+, a carboxylate ligand (BDC) in the absence and presence of magnetic nanoparticles (MNPs). We then compared the separation efficiency and enzyme reusability of the resultant multienzyme@Ca–BDC and multienzyme@MNP-Ca–BDC composites via gravimetric and magnetic separation, respectively, and found that, although both composites were effective in cellulose degradation in the first round, the multienzyme@MNP-Ca–BDC composites displayed significantly enhanced reusability. This work provides the first experimental demonstration of using magnetic solid supports to immobilize multiple carbohydrase enzymes simultaneously and degrade cellulose and promotes green/sustainable chemistry in three ways: (1) reusing the enzymes saves energy/sources to prepare them, (2) the synthetic conditions are “green” without generating unwanted wastes, and (3) using our composites to degrade cellulose is the first step of extracting valuable materials from sustainable biomasses such as plants whose growth does not rely on nonregeneratable resources. 

Confining proteins in synthetic nanoscale spatial compartments has offered a cell-free avenue to understand enzyme structure–function relationships and complex cellular processes near the physiological conditions, an important branch of fundamental protein biophysics studies. Enzyme confinement has also provided advancement in biocatalysis by offering enhanced enzyme reusability, cost-efficiency, and substrate selectivity in certain cases for research and industrial applications. However, the primary research efforts in this area have been focused on the development of novel confinement materials and investigating protein adsorption/interaction with various surfaces, leaving a fundamental knowledge gap, namely, the lack of understanding of the confined enzymes (note that enzyme adsorption to or interactions with surfaces differs from enzyme confinement as the latter offers an enhanced extent of restriction to enzyme movement and/or conformational flexibility). In particular, there is limited understanding of enzymes' structure, dynamics, translocation (into biological pores), folding, and aggregation in extreme cases upon confinement, and how confinement properties such as the size, shape, and rigidity affect these details. The first barrier to bridge this gap is the difficulty in “penetrating” the “shielding” of the confinement walls experimentally; confinement could also lead to high heterogeneity and dynamics in the entrapped enzymes, challenging most protein-probing experimental techniques. The complexity is raised by the variety in the possible confinement environments that enzymes may encounter in nature or on lab benches, which can be categorized to rigid confinement with regular shapes, rigid restriction without regular shapes, and flexible/dynamic confinement which also introduces crowding effects. Thus, to bridge such a knowledge gap, it is critical to combine advanced materials and cutting-edge techniques to re-create the various confinement conditions and understand enzymes therein. We have spearheaded in this challenging area by creating various confinement conditions to restrict enzymes while exploring experimental techniques to understand enzyme behaviors upon confinement at the molecular/residue level. This review is to summarize our key findings on the molecular level details of enzymes confined in (i) rigid compartments with regular shapes based on pre-formed, mesoporous nanoparticles and Metal–Organic Frameworks/Covalent-Organic Frameworks (MOFs/COFs), (ii) rigid confinement with irregular crystal defects with shapes close to the outline of the confined enzymes via co-crystallization of enzymes with certain metal ions and ligands in the aqueous phase (biomineralization), and (iii) flexible, dynamic confinement created by protein-friendly polymeric materials and assemblies. Under each case, we will focus our discussion on (a) the way to load enzymes into the confined spaces, (b) the structural basis of the function and behavior of enzymes within each compartment environments, and (c) technical advances of our methodology to probe the needed structural information. The purposes are to depict the chemical physics details of enzymes at the challenging interface of natural molecules and synthetic compartment materials, guide the selection of enzyme confinement platforms for various applications, and generate excitement in the community on combining cutting-edge technologies and synthetic materials to better understand enzyme performance in biophysics, biocatalysis, and biomedical applications. 

Enzyme immobilization offers a number of advantages that improve biocatalysis; however, finding a proper way to immobilize enzymes is often a challenging task. Implanting enzymes in metal–organic frameworks (MOFs) via co-crystallization, also known as biomineralization, provides enhanced reusability and stability with minimal perturbation and substrate selectivity to the enzyme. Currently, there are limited metal–ligand combinations with a proper protocol guiding the experimental procedures. We have recently explored 10 combinations that allow custom immobilization of enzymes according to enzyme stability and activity in different metals/ligands. Here, as a follow-up of that work, we present a protocol for how to carry out custom immobilization of enzymes using the available combinations of metal ions and ligands. Detailed procedures to prepare metal ions, ligands, and enzymes for their co-crystallization, together with characterization and assessment, are discussed. Precautions for each experimental step and result analysis are highlighted as well. This protocol is important for enzyme immobilization in various research and industrial fields.