## 1 Introduction  
Folded polypeptides work not only in monomeric states but also in assembled states, including homo- and heterooligomers [1, 2, 3]. In the course of enzyme evolution, symmetrical protein homooligomers with polygonal and polyhedral shapes emerged from their monomeric ancestors, presumably due to the advantages in their biological properties over the respective monomeric states, including enzymatic activities and structural stability [1, 2, 3]. Interfaces between monomer units play a key role in the assembly of protein oligomers. The artificial design of protein oligomers with symmetrical shapes has been explored with the use of protein-protein interaction interfaces extracted from naturally occurring protein oligomers, generating the field of protein nanostructure research [4, 5, 6]. Recent challenges in this field include de novo design of artificial protein-protein interaction interfaces that can serve as modular structural units for versatile protein nanostructure design [7, 8]. Among the various forms of symmetrical protein homooligomers, symmetrical protein homodimers can be regarded as the simplest molecular parts for the rational design and assembly of protein structures [9, 10, 11]. RNA is a biopolymer that can fold into defined 3D structures with biological functions. Structured RNAs are known to perform many types of protein-like functions, including roles as catalysts and receptors [12, 13, 14]. Large and complex 3D RNA structures frequently formed through noncovalent assembly of structural modules, which can fold locally within a single polynucleotide chain [15]. Intramolecular assembly of RNA structural modules is supported and maintained by RNA-RNA interaction interfaces [15; 16]. Their association is usually strong enough to reconstitute the catalytic activity in a multimolecular format, in which separately prepared modular RNAs are assembled noncovalently by RNA-RNA interaction interfaces [17; 18; 19; 20; 21; 22; 23; 24; 25]. We recently engineered an RNA-RNA interaction interface of the _Tetralumena_ group I ribozyme to design a prototype of ribozyme modules (abbreviated as RzM-0, Figure 1) with self-dimerizing capability [26; 27]. The RNA-RNA interface in the RzM-0 homodimer is supported by two sets of tertiary interactions between the \(\Delta\)P5 core ribozyme module and P5 abc module, which are connected by flexible linker elements (P5-P5a linker) in the parent ribozyme and form strong intramolecular associations (WT group I ribozyme, Figure 1). We rationally replaced the flexible P5-P5a with a rigid duplex to prevent intramolecular \(\Delta\)P5-P5 abc assembly (Figure S1C), resulting in homodimerization of the resulting variant (RzM-0 in Figure 1) in a head-to-head manner by forming RzM:RzM interface **0:0**[26; 27]. A pair of symmetrical \(\Delta\)P5-P5 abc interfaces yielding RzM:RzM homodimer was then engineered to asymmetrical recognition interfaces, such as RzM:RzM interface **1:1'**, yielding RzM:RzM heterodimers (Figure 1). We then connected a pair of RzM monomers in a tail-to-tail manner at their P6b region, enabling the resulting covalent dimer (termed unit ribozyme dimer and abbreviated as Urd) to homooligomerize in a directional manner (Figure 1) [28]. Unit ribozyme dimers (Urds) homooligomerized to form an open chain assembly (Figure 1). The extents of oligomerization were programmable by designing the specificity of the assembly interfaces between two ribozyme modules (RzMs) in Urds [28]. We also expected that the parent Urds could be modified to produce closed forms of symmetrical oligomers if we used a bent P6b-P6b linker. In this study, we constructed and examined a new series of Urds, which were designed to form closed trimers with a triangular shape. Figure 1: Design of closed oligomers based on engineered group I ribozyme dimers. Scheme of modular redesign to generate unit ribozyme dimers (kUrds) and their oligomerization. White arrows indicate structural redesign to prepare RzM RNAs and their noncovalent dimerization as reported by [PERSON] et al. in 2016 [26].