**1. Introduction**

Processivity is a kinetic phenomenon widespread among enzymes that act on polymeric substrates, such as DNA, RNA, polysaccharides, and proteins [1]. Once committed, processive enzymes engage in multiple rounds of modification instead of releasing their substrate after modifying it once. Served by different sliding mechanism(s), very effective enzymatic modifiers arose in evolution that can carry out hundreds or thousands of elementary steps upon a single engagement with the substrate [1]. Processivity occurs in: (i) synthesis (e.g., DNA by DNA polymerase [2], RNA by RNA polymerase, and protein by the ribosome [3]); (ii) degradation (e.g., DNA by DNAse [4], RNA by RNAse [5], polysaccharides by glycohydrolases [6] or proteins by the proteasome [7,8]); (iii) structural modification (e.g., DNA by helicase [9]); (iv) chemical modification (e.g., ubiquitination of proteins by ubiquitin ligases [10,11]); or (v) cargo transport (e.g., movement by mechanochemical motors kinesin, dynein and myosin [12–15] along actin and tubulin tracks).

A compilation of domain-linker-domain (DLD)-type monomeric processive enzymes is taken from the comprehensive list given in Supplementary Table S1. Important parameters including the length of predicted disordered linker, mean linker length of orthologous proteins (see Table S2 for species), κ value describing charge distribution, and the level of processivity (such as the length of processive move, the number of steps taken or the number of elementary substrate units covered, if determined at all), are given.

Given the extreme diversity of substrates upon which these processive enzymes act and also the variability of the chemical/mechanochemical changes they make, it is of little surprise that the molecular details of processivity are rather diverse, yet they are based on combinations of two basic designs principles. The classic and amply studied mechanism relies on structural confinement by circular/cylindrical or asymmetric binding domains or subunits of the enzymes. The former occurs, for example, when the PCNA subunit of DNA polymerase encircles the template DNA (Figure 1A) to ensure that the enzyme adds a practically unlimited number of nucleotides [16,17] to the growing DNA polymer. A closely related solution is used by HIV reverse transcriptase [18], which has an asymmetric binding domain that strongly favors sliding along the RNA substrate over dissociating from it (Figure 1B). A completely different mechanism has evolved in mechanochemical motors, such as kinesin and dynein, which move along polymeric protein tracks of tubulin [15]. These dimeric proteins have long coiled-coil stalks and ATPase binding domains, which undergo conformational changes that result in a strong preference for rebinding following dissociation due to a proximity effect, i.e., spatial confinement (Figure 1C). The region connecting the dimerization domain with the binding domain may even undergo transitions between ordered and disordered states [19]. The latter class of processive motors suggests that the presence of two binding elements (motifs or domains) connected by long, conformationally adaptable/flexible linker region(s) appears to be a key element of processivity, which combines deterministic and probabilistic elements of binding [20].

Here we generalize this concept by observing and analyzing that proteins in which binding domains are connected by a disordered linker may show probabilistic bias for re-binding over dissociation from their substrate, due to which they possess processive capacity. As structural disorder of proteins (intrinsically disordered protein/region, IDP/IDR) is widespread in eukaryotic proteomes [21,22], this may be a frequently applied mechanism. IDPs/IDRs often engage in protein-protein interactions [23,24] but their function may also directly stem from the disordered state, termed entropic-chain functions [25]. Binding and entropic-chain functions can actually be combined because often part of the IDP remains disordered even in the bound state, a phenomenon termed fuzziness [26]. Of particular relevance to the observed processivity is that binding motifs embedded in disordered regions, due to the arising "proximity effect" or "optimal effective concentration" around binding sites, may feature facilitated binding, which is central to the concepts of: (i) acceleration of binding by "fly casting" [27], (ii) reduction of binding dimensionality by the "monkey-bar" mechanism [28], and (iii) "ultrasensitive" binding by repetitive binding motifs in signaling proteins [29,30].

By statistical-physical modeling and bioinformatics analysis we show that this kinetic proximity effect is also a widespread inherent property of many monomeric processive enzymes that are capable of multiple rounds of modification of their polymeric substrate. These enzymes, such as a variety of glycohydrolases (e.g., cellulases) [6,31,32], Ribonuclease H1 (RNAse-H1) [5] and matrix metalloproteinase-9 (MMP-9) [33], need no ATP energy for processivity, which makes it a robust and widespread mechanism in the proteome. Here we have selected 12 such monomeric (ATP-independent) processive enzymes from the literature and provide a comprehensive analysis of their physical and structural properties. We show that once engaged with their substrate, their structural organization kinetically biases binding of their free binding domain over dissociation of both its domains, resulting in multiple successive binding events without ever fully releasing the polymeric partner (Figure 1D). We suggest that this type of processivity represents a unique type of "entropic chain" function enabled by the structural disorder of their linker region [25,34], which may be a general mechanism that arises in a broad range of biological contexts.

**Figure 1.** Basic mechanisms of processivity. The figure illustrates the two basic types (and four subtypes) of the mechanism of processivity. The classical mechanism based on structural confinement are represented by folded proteins that either (**A**) completely surround their partner by an oligomeric structure of toroidal shape, such as PCNA (PDB: 1AXC) [16,17], or (**B**) use an asymmetric binding domain to restrict its dissociation, such as in HIV reverse transcriptase (PDB: 1REV) [18]. Basically, different mechanisms are based on spatial confinement allowed by two binding motifs connected by a long, adaptable or flexible linker, as appears in (**C**) the ATP-dependent dimeric mechanochemical motors kinesin-1 and dynein (adapted from [20]), or (**D**) monomeric processive enzymes of domain-disordered linker-domain arrangement. These types of enzymes analyzed here in detail (for cases, see Table 1) bind their polymeric substrate via two binding domains, termed "bound" or "tethered" (T) for the one that anchors the enzyme to the substrate and "unbound" or "free" (F) for the one that is in search for substrate "target" binding sites), connected by a structurally disordered linker. We show by statistical-kinetic modeling that binding via the tethering domain kinetically favors binding via the free domain (a) over full dissociation of the protein (d), which may then result in processive diffusional moves (c) or directed movements driven by energy-dependent binding and/or modification of the substrate (e,f).


**Table 1.** ATP-independent monomeric domain-linker-domain (DLD)-type processive enzymes.

\* no sufficient number of orthologous proteins.

#### **2. Results**

### *2.1. The Classical Mechanisms of Processivity*

For rationalizing the diverse mechanisms of processivity, we suggest that they fall into two broad mechanistic categories (cf. Table S1). The structural underpinning of the mechanism is straightforward when the enzyme uses structural confinement to make dissociation from the substrate highly unfavorable [1]. Complete confinement may result from ring-shaped oligomeric structures (e.g., PCNA [16,17] (Figure 1A)), whereas asymmetric structures of a single polypeptide chain can also either fully (e.g., exonuclease I [1]) or partially (e.g., HIV reverse transcriptase [18] (Figure 1B)) enclose the substrate. These mechanisms can be interpreted in terms of a preferred 1D sliding of the substrate (template) within the well-defined structural element of the enzyme.

Processivity of a completely different structural rationale can be observed in motor enzymes that use chemical energy for unidirectional movement along cytoskeletal tracks [12,13]. These motors usually have a dimeric structure, with their dimerization region and ATPase domains connected to their substrate-binding domains by long and extended structures (stalk) (Figure 1C). Large-scale conformational changes elicited by ATP hydrolysis in the ATPase domain(s) propagate to these binding domains, which result in a preference for the re-binding to the substrate track vs. full dissociation [14,15]. In these mechanisms, passive diffusional moves and energy-driven directional steps are combined, i.e., they represent a combination of confining the sequence of events by structural and spatial means. As outlined in the next paragraph, confinement by the limitation of search space by a disordered linker connecting binding domains (Figure 1D) can also account for processivity of enzymes, which appears to be widely applied in biology.
