## 1 Introduction  
Biological polymers, e.g., polypeptide (protein) and polysaccharide, are vital materials and show specific biological functions in living systems [1; 2]. They are mainly composed of regular primary structures, which lead to the architecture of controlled higher-order assemblies that contribute to the exhibit of their specific functions [3]. For example, amylose, which is an abundant polysaccharide present in starch, is comprising a left-handed helical stereo-arrangement, derived from an \(\alpha\)(1\(\rightarrow\)4)-linked repeating glucose (Glc) unit structure, leading to the construction of a water-insoluble double helical assembly [4; 5]. In addition, natural hybrid systems from multiple kinds of biological polymer are known, as an example of saccharide-peptide polymeric conjugates, such as peptidoglycanans and glycoproteins [6; 7; 8]. Therefore, artificial saccharide-peptide polymeric conjugates can be considered to be new biobased functional ingredients, that practically exhibit a potential for applications in the fields of tissue engineering and biomedicine. For example, we synthesized an artificial saccharide-peptide polymeric conjugate by combining amylose and an abundant polypeptide, that is, poly(\(\gamma\)-glutamic acid) (PGA), by enzymatic grafting approach [9]. As PGA shows a water-soluble, anionic, and biodegradable properties, it has been employed for multifunctional applications in food, water treatment, healthcare, pharmaceuticals, and other fields [10; 11]. The enzymatic approach is well accepted as an efficient tool for synthesizing polysaccharides with highly controlled regio- and stereo-arrangements by simple operations [12; 13; 14; 15; 16]. Glucan phosphorylase (GP) is the representative enzyme to be employed for the synthesis of well-defined polysaccharides. GP catalyzes enzymatic polymerization of \(\alpha\)-D-glucose 1-phosphate (Glc-1-P) as a monomer in the presence of a maltooligosaccharide as a primer to precisely synthesize \(\alpha\)(1\(\rightarrow\)4)-glucan (amylose) [17, 18, 19, 20, 21, 22, 23]. Initiation of the polymerization occurs from the nonreducing end of the maltololigosaccharide primer, and then, consecutive transfer of the Glc residues occurs from the monomer to the propagating nonreducing end with the release of inorganic phosphate (Pi) by the GP catalysis, based on primary reaction as follows; [\(\alpha\)(1\(\rightarrow\)4)-Glc]\({}_{\text{n}}\) + Glc-1-P \(\rightarrow\) [\(\alpha\)(1\(\rightarrow\)4)-Glc]\({}_{\text{n+1}}\) + Pi. Therefore, GP-catalyzed polymerization has been used to precisely fabricate various types of amylose-containing functional materials [15, 16, 23, 24, 25]. In the previous studies, amylose-grafted polymeric materials have been fabricated by GP-catalyzed polymerization combined with appropriately designed chemical reactions (chemoenzymatic method) [26, 27, 28]. For the synthesis, first, maltooligosaccharides are covalently linked on the main-chain polymers to yield polymeric primers. Then, GP-catalyzed polymerization is carried out from the nonreducing ends of the primer chains on the main-chains to yield the designed graft materials, such as amylose-grafted poly(L-lysine), poly(\(\alpha\)-L-glutamic acid), cellulose, chitin/chitosan, carboxylmethyl cellulose (CMC), xanthan gum, and alginate [29, 30, 31, 32, 33, 34, 35, 36, 37]. As abovementioned, we also synthesized the amylose-grafted PGA by the chemoenzymatic method, which formed network structures composed of cross-linking points from amylosic double helixes among the PGA chains, giving rise to hydrogels [9].