2.3.4. Modelling Cellulase, a Processive Enzyme

Based on all the foregoing analyses, it appears compelling that the DLD arrangement makes enzymes processive. This seems a general phenomenon, which can be demonstrated by low-resolution statistical-kinetic modelling (Figure 2). Here we proceed to show that by incorporating structural details, i.e., atomistic structural models of the domains, into the model and considering domain-linker interactions (Figure 7), we can quantitatively describe the mechanistic and kinetic behavior of one of the most-studied DLD processive enzymes, that of bacterial cellulase (*Trichoderma. reesei* Cel7A, cf. Table 1). Cel7A has two domains of different size, a larger catalytic domain (CD) that confines the linear cellulose substrate, i.e., in itself tends to be processive, and a smaller cellulose binding domain (also termed motif, CBM) attached with a disordered linker of 33 amino acids in length (Figure 7A). The enzyme is processive, typically carrying out about 20–100 cleavage events before dissociating form its substrate. By modeling all parameters of: (i) linker length and flexibility, (ii) catalytic parameters of the enzymatic domain (for the range of kinetic parameters within the Cel7A family, cf. Table S3) and binding parameters of the free (binding) domain, (iii) structural hindrance arising from the actual structures of the domains and domain-linker interaction, and (iv) distance of cellulose binding sites, we show that average binding time of the CBM domain (Figure 7B) undergoes a minimum at a linker length range that is very close to the observed linker lengths in cellulases (Table 1). Furthermore, binding of the linker to the CD has an effect on the behavior of the system (Figure 7B, cf. blue region in color scheme) as it restricts the freedom of movement of the domains, making it less processive. Since all the known cellulase linkers are highly flexible and contain little or no secondary structural elements, changing the Kuhn-segment length is not applicable in this system. The level of processivity that can be approximated as the ratio of the time of binding of CBM to the time of the catalytic reaction (for the CD of cellulase, Table S3, measured with rather artificial substrates) is on the order of 10–100, which agrees with the values reported (Table 1).

**Figure 7.** Cellulase: a model processive enzyme. (**A**) Model of the Cel7A cellulase based on the structure PDB 8cel for the catalytic domain (CD) and PDB 2mwk for the cellulose-binding domain (CBM). The CD is purple with the cellulose tunnel shown in transparent blue. One glycosylation of the CD is visible in dark red. Further elements marked are the two catalytic amino acids (red and blue stick-and-ball), the linker region (blue with orange mannose glycosylation), the CBM (dark green), and the cellulose sheet (pale green) of which one fibril (yellow-green) is being processed. The sequence and glycosylation is based on UniProt P62694. (**B**) Statistical kinetic modelling considering geometry (size) and binding of the linker to CD shows binding times characteristic of this system. The green area represents typical catalytic times for Cel7A cellulase family (Table S3), whereas the red area marks typical linker region lengths (Figure S2). The four curves correspond to various values of the linker region's partial binding to the CD, which results in it emerging from the CD at different points (see color mark). If we consider the beginning of the CD domain as the origin of the coordinate system and the cellulose filament moves along the X axis, and assume no binding between the linker and the CD, then free end of the linker region reaches −4.2 nm (red in color scale). When the largest portion of the linker is bound to the CD, the starting point of the free linker end is at zero (blue in color scale). Yellow and light blue colors represent intermediate back-binding cases, with −2.5 and −1.5 nm starting points, respectively.
