The kinetic theory of enzymes that modify insoluble substrates is still underdeveloped, despite the prevalence of this type of reaction both in vivo and industrial applications. Here, we present a steady-state kinetic approach to investigate inhibition occurring at the solid–liquid interface. We propose to conduct experiments under enzyme excess (E0 ≫ S0), i.e. the opposite limit compared with the conventional Michaelis–Menten framework. This inverse condition is practical for insoluble substrates and elucidates how the inhibitor reduces enzyme activity through binding to the substrate. We claim that this type of inhibition is common for interfacial enzyme reactions because substrate accessibility is low, and we show that it can be analyzed by experiments and rate equations that are analogous to the conventional approach, except that the roles of enzyme and substrate have been swapped. To illustrate the approach, we investigated the major cellulases from Trichoderma reesei (Cel6A and Cel7A) acting on insoluble cellulose. As model inhibitors, we used catalytically inactive variants of Cel6A and Cel7A. We made so-called inverse Michaelis–Menten curves at different concentrations of inhibitors and found that a new rate equation accounted well for the data. In most cases, we found a mixed type of surface-site inhibition mechanism, and this probably reflected that the inhibitor both competed with the enzyme for the productive binding-sites (competitive inhibition) and hampered the processive movement on the surface (uncompetitive inhibition). These results give new insights into the complex interplay of Cel7A and Cel6A on cellulose and the approach may be applicable to other heterogeneous enzyme reactions.

## Introduction

Enzyme inhibition makes up a cornerstone within enzymology. The field has earned its position not only because inhibition has an important regulatory role in living cells, but also because inhibition kinetics have given fundamental insights into both enzyme mechanism and active-site architecture [1]. On an applied level, many drugs function as enzyme inhibitors [2], and industrial enzymes have been engineered to alleviate inhibition [3,4]. In light of this broad significance, it is not surprising that reversible enzyme inhibition is an essential part of any biochemistry textbook, where it is common to introduce both mechanism (e.g. competitive, uncompetitive, non-competitive or mixed) and a framework of rate equations to analyze the phenomenon [5,6]. The textbook treatment typically assumes that the inhibitor acts on the enzyme and thus reduces catalytic efficiency and/or active-site accessibility, but any molecule that lowers the rate of an enzyme-catalyzed reaction can be considered an inhibitor [7]. This could be a molecule that interacts with the substrate instead of the enzyme. However, this latter type of inhibition has only been sporadically investigated for bulk reactions [7,8], and this probably reflects that it is rare or negligible in these systems.

In the current work, we focus on the inhibition of enzyme reactions occurring at the interface of an insoluble substrate. These so-called heterogeneous enzyme reactions are common both in nature [9] and industrial applications of enzymes [10], but the theoretical framework for their kinetic characterization remains insufficient. Conventional (bulk) kinetic studies are set up to meet the quasi-steady-state assumption, and this implies a large excess of substrate compared with the enzyme [11]. This condition also facilitates inhibition studies, as the reaction rate will be proportional to the total enzyme concentration, and hence sensitive to inhibitors that bind to the enzyme (E) or enzyme–substrate (ES) complex (c.f. Figure 1A). While substrate excess is easily achieved for reactions in bulk, this is not necessarily the case for enzymes that act at a solid–liquid interface [9]. Specifically, the condition of substrate excess S0 ≫ E0, may not be achievable under the desired experimental conditions, and this precludes the application of the classical kinetic approach to inhibition. As an alternative, we propose to study surface-site inhibition of heterogeneous enzyme reactions in the opposite limit of enzyme excess, where E0 ≫ S0 (c.f. Figure 1B). We show that under these conditions, the reaction rate is governed by equations that are analogous to the well-known treatment for substrate excess, except that the roles of enzyme and substrate have been interchanged. The approach is also opposite in the sense that it elucidates the kinetic effects of the inhibitor's binding to the substrate rather than to the enzyme. Binding of surface-site inhibitors to the substrate surface may either reduce enzyme accessibility to the substrate or lower the reactivity of already formed ES-complexes, and we argue that these effects parallel, respectively, competitive and uncompetitive inhibition in bulk reactions.

### Enzymes can be subjected to inhibition both in the bulk solution and on the substrate surface.

Figure 1.
Enzymes can be subjected to inhibition both in the bulk solution and on the substrate surface.

Illustration of conventional (bulk) enzyme inhibition (A) and surface-site inhibition at the interface of an insoluble substrate (B). In conventional enzyme inhibition, the inhibitor (blue) acts by reducing the activity or accessibility of the enzyme (green) to the substrate (black). In surface-site inhibition, the inhibitor molecule (blue) acts by interacting with the substrate surface (gray tiles) and by lowering the reactivity or accessibility of the substrate, rather than the enzyme (green).

Figure 1.
Enzymes can be subjected to inhibition both in the bulk solution and on the substrate surface.

Illustration of conventional (bulk) enzyme inhibition (A) and surface-site inhibition at the interface of an insoluble substrate (B). In conventional enzyme inhibition, the inhibitor (blue) acts by reducing the activity or accessibility of the enzyme (green) to the substrate (black). In surface-site inhibition, the inhibitor molecule (blue) acts by interacting with the substrate surface (gray tiles) and by lowering the reactivity or accessibility of the substrate, rather than the enzyme (green).

In this study, we used cellulases from the model fungus Trichoderma reesei to elucidate surface-site inhibition. These enzymes act at the cellulose–water interface and have received extensive research interest due to their importance in natural carbon cycling [12], and emerging biorefineries [13]. Under industrial conditions, cellulolytic enzymes are faced with a heterogeneous environment containing multiple components that may bind to the surface and inhibit the enzymatic reaction. Such components include lignin [14], hemicellulose [15,16], surfactants [17] or the enzyme itself [18,19]. To mimic this situation in a simplified way, we studied the kinetics of the two major processive cellobiohydrolases of T. reesei, Cel7A and Cel6A on microcrystalline cellulose (Avicel). We used catalytically inactive variants of the same enzymes (Cel7AE212Q and Cel6AE212Q) as surface-site inhibitors. The inactive variants had similar binding properties as the wild type enzymes, but no activity, and hence acted as surface-site inhibitors with known affinity. We made so-called inverse Michaelis–Menten curves (reaction rate vs. E0) in the presence of different inhibitor concentrations and analyzed the results with respect to the surface-site inhibition equation. This enabled us to determine the strength and type of surface-site inhibition. In most cases, the surface-site inhibition found was mixed type with both competitive and uncompetitive contributions, and we discuss molecular underpinnings of these mechanisms under conditions of enzyme excess. The new approach makes it possible to get both mechanistic and quantitative insights into surface reactions using simple experiments and well-known principles of data treatment.

## Materials and methods

All experiments were conducted in a standard 50 mM sodium acetate buffer, pH 5.0.

### Enzymes and inhibitors

We used the two cellobiohydrolases Cel7A (Uniprot P62694) and Cel6A (Uniprot P07987) from the filamentous fungus T. reesei. The surface-site inhibitors were inactive variants of the same enzymes. In Cel7A, we mutated the catalytic nucleophile Glu212 to a Gln (E212Q), while the catalytic acid Asp221 was mutated to an Asn (D221N) in Cel6A. These two variants, termed Cel7AE212Q and Cel6AD221N, with isosteric point mutations, have previously been shown to be catalytically deficient while their structural fold is preserved [20,21]. All four enzymes were expressed heterologously in Aspergillus oryzae and purified as previously described [22,23]. The purified enzymes showed a single band on sodium dodecyl sulfate–polyacrylamide gel electrophoresis (SDS–PAGE, Supplementary Figure S1) and enzyme concentrations were determined by absorbance at 280 nm using theoretical molar extinction coefficients (Gasteiger et al. 2005 [24]) of 97 790 M−1 cm−1 for Cel6 and Cel6AD221N and 86 760 M−1 cm−1 for Cel7A and Cel7AE212Q.

Binding isotherms were determined for all four enzymes at 25C using microcrystalline cellulose (Avicel PH101, Sigma–Aldrich) as substrate. Before use, Avicel was washed seven times with deionized water and then twice in the standard buffer to remove traces of soluble sugars. Adsorption experiments were conducted at two different (fixed) substrate loads (10 g/l and 20 g/l), while the enzyme concentration was varied from 0 to 9 µM. The reaction was started by mixing the enzyme and substrate in a 96-well microtiter plate (655101 Greiner Bio-One, Germany) with a final reaction volume of 250 µl. After 60 min incubation in a thermomixer operating at 1100 rpm, the substrate was pelleted by centrifugation (3 min, 2000× g) and 100 μl of the supernatant was transferred to a black 96-well microtiter plate (655079Greiner Bio-one, Germany). Next, the concentration of free enzyme was quantified by measuring the intrinsic protein fluorescence emission at 345 nm after excitation at 280 nm. The measurements were done using a plate reader (SpectraMax i3, Molecular Devices, Wals, Austria). An enzyme standard curve ranging from 0 to 9 μM was included in all experiments.

As a control, the adsorption reversibility of the inactive enzymes Cel7AE212Q and Cel6AD221N was tested on Avicel. The basic idea behind the experiment is to measure the release of the enzyme upon dilution. If adsorption is reversible, the ‘ascending’ adsorption isotherm obtained at increasing enzyme concentration should be superimposed by the ‘descending’ isotherm obtained by diluting samples. This is discussed in detail elsewhere [25]. In the current study, we used the following procedure. Binding isotherms (ascending) were constructed in 96-well microtiter plates using a fixed substrate load of 15 g/l and an enzyme concentration range from 0.05 to 2 µM. The total volume of each sample was 250 µl. The plates were incubated at 25°C in thermomixers operating at 1100 rpm. After 30 min the shaking was switched off and the plates were left in the thermomixer to let the substrate particles sediment. After 30 min, 100 µl of the clear supernatant was transferred to a black 96-well microtiter plate and the concentration of free enzyme was measured as described above. For the descending binding isotherm, 100 µl of fresh buffer was added onto the initial plate and it was re-incubated for 30 min at 1100 rpm. The plate was again left without shaking for 30 min to let the particles sediment and a new sample of 100 µl was withdrawn from the supernatant and quantified for its concentration of free enzymes as described above. Standard curves of the enzyme without substrate were tested in the same way and included in all the measurements. All adsorption measurements were done in triplicates.

### Kinetics

Reaction rates were measured using Avicel as substrate and a temperature of 25°C. The load of Avicel was 2 g/l and we measured the rate at increasing concentrations of active enzyme either with or without inhibitor (Cel7AE212Q or Cel6AD221N). We studied the following concentration ranges. For Cel7A/Cel7AE212Q the active enzyme concentration (E0) ranged from 0.1 to 9 µM and we used six inhibitor concentrations (I0) from 0.25 to 8 µM of Cel7AE212Q. For Cel6A/Cel6AD221N the active enzyme concentration (E0) ranged from 0.1 to 5.8 µM of Cel6A, and we used five inhibitor concentrations (I0) from 0.14 to 2.2 µM. For Cel7A/Cel6AD221N the active enzyme concentration (E0) ranged from 0.1 to 6.2 µM of Cel7A, with five inhibitor concentrations from 0.1 to 2.2 µM of Cel6AD221N. For Cel6A/Cel7AE212Q the active enzyme concentration (E0) ranged from 0.1 to 8.8 µM of Cel6A, and we included six inhibitor concentrations (I0) from 0.1 to 4 µM of Cel7AE212Q. Before starting the reactions, 240 µl of Avicel suspension and 30 µl of inhibitor were equilibrated in microtiter plates for 30 min in thermomixers operating at 1100 rpm. Then, 30 µl of active enzyme was added for a final reaction volume of 300 µl. The enzyme–substrate contact time was 1 h. To stop the reaction, the plates were centrifuged for 3 min at 2000×g and 60 µl of the supernatant was retrieved, and the concentration of soluble reducing ends was measured using the para-hydroxybenzoic acid hydrazide (PAHBAH) method as described elsewhere [26]. As a control, the possible residual activity of the catalytic deficient enzymes (the surface-site inhibitors) was also measured in comparison with the wild type enzymes. This assay was done as described above but only at one fixed substrate load (2 g/l) and enzyme concentration (5 µM).

## Kinetic analysis

To derive a general rate law for surface-site inhibition we adapt the conventional kinetic theory of enzyme inhibition to conditions of enzyme excess (see Figure 2). In the conventional description, linear, reversible inhibition is described by the microkinetic scheme shown in Figure 2A. In this scheme, the rate of the enzyme-catalyzed reaction (v = ES*kcat) can be lowered in two distinct ways. Either the inhibitor (I) can bind to the free enzyme (E) and lower the available active-sites that the substrate (S) can bind to, or it can act on the ES-complex and reduce the population of productive complexes. In the general case where the inhibitor has an affinity for both E and ES, the inhibition mechanism is called mixed. Typically, three other types of inhibition mechanisms are defined, and they may all be considered special cases of mixed inhibition. If the inhibitor has a negligible affinity for the ES complex, that is, if α ≫ 1 (see Figure 2), the inhibition mechanism is known as competitive inhibition. The other limiting case is when the inhibitor does not bind (or binds very poorly) to the free enzyme (α ≪ 1) is called uncompetitive inhibition. The last type of inhibition is the special (and more rare [7]) case where the inhibitor binds to E and ES with the same affinity (α = 1). This mechanism is called non-competitive inhibition.

### Microkinetic schemes for conventional bulk inhibition and surface-site inhibition.

Figure 2.
Microkinetic schemes for conventional bulk inhibition and surface-site inhibition.

(A) Conventional inhibition scheme. The free enzyme (E) or the enzyme–substrate complex (ES) is inhibited by an inhibitor (I) which lowers the product (P) formation rate. (B) Surfaces-site inhibition scheme. The free substrate (S) or the enzyme–substrate complex is inhibited through the binding of an inhibitor. Note that the substrate in the inhibition scheme for surface-site inhibition is preserved (appear together with the product on the right side of scheme B). As argued in the Discussion and elsewhere [27] this phenomenon is unique for solid substrates where only a miniscule fraction of the total substrate is accessible for enzyme attack.

Figure 2.
Microkinetic schemes for conventional bulk inhibition and surface-site inhibition.

(A) Conventional inhibition scheme. The free enzyme (E) or the enzyme–substrate complex (ES) is inhibited by an inhibitor (I) which lowers the product (P) formation rate. (B) Surfaces-site inhibition scheme. The free substrate (S) or the enzyme–substrate complex is inhibited through the binding of an inhibitor. Note that the substrate in the inhibition scheme for surface-site inhibition is preserved (appear together with the product on the right side of scheme B). As argued in the Discussion and elsewhere [27] this phenomenon is unique for solid substrates where only a miniscule fraction of the total substrate is accessible for enzyme attack.

We can construct an equivalent scheme for surface-site inhibition with the only difference that the substrate and enzyme swap places (Figure 2B). This swap also pertains to the underlying experiments, which must be conducted with an excess of the enzyme (E0 ≫ S0). The conventional interpretation of enzyme inhibition can easily be adapted to the inversed situation of enzyme excess. Thus, we can think of the surface-site competitive inhibition as the case where the inhibitor binds to the substrate and lowers the apparent population of surface-sites that the enzyme can productively bind to (also known as attack sites), while uncompetitive inhibition is the case where the inhibitor reduces the population of productive enzyme–substrate complexes.

A general rate law for surface-site inhibition (eqn 1) can be derived analogously to the conventional rate law for linear mixed inhibition [7] (see Supplementary Section S3 for the derivation of eqn 1).
$Vss=E0⋅invVmaxKM1+I0Ki+E01+I0αKi$
1
The experimental input to eqn (1) comes from plots of the initial rate as a function of the enzyme concentration at selected (fixed) inhibitor concentrations (see Figure 4). The maximal rate (saturation rate)without inhibitor is $invVmax$ and KM is the so-called inverse Michaelis–Menten constant (enzyme concentration for half-saturation) [28]. Finally, Ki is the competitive inhibition constant and α is the constant specified in Figure 2. The variables in eqn (1) are the initial molar concentration of enzyme (E0) and inhibitor (I0).
The maximal rate ($invVmax$) reflects a situation where all productive sites at the surface are occupied by enzyme and is given by the product of the molar concentration of substrate (S0) and the turnover frequency (kcat) of the enzyme–substrate complex: $invVmax=S0kcat$. The molar concentration, S0, is typically not known for an insoluble substrate but one way to overcome this problem is to introduce a parameter for the density of attack sites, $kinΓmax$ in units of mol/g. This parameter quantifies sites on the surface with which the enzyme can form a productive complex and hence enables conversion from the (known) mass load (in g/l) of the substrate into an apparent molar concentration. If $massS0$ is the load of the substrate (in g/l), we may define an apparent S0 (in mol/l) by
$S0=massS0kinΓmax$
2
Using eqn (2), we can express the maximal rate under this condition as
$invVmax=massS0kinΓmaxkcat$
3
As seen from eqn (3), the inverse maximal rate ($invVmax$) depends not only on the turnover frequency of the enzyme (kcat) but also on the density of productive surface-sites ($kinΓmax$). Since not all accessible surface-sites can be used to form a productive enzyme–substrate complex, we use the superscript kin to distinguish the density of productive surface-sites ($kinΓmax$) from the total density of adsorption sites (Γmax, see Table 1). The latter parameter is generally found from binding isotherms (as shown in Figure 3), while the former parameter must be derived from kinetic measurements [28–30].
Table 1.
Langmuir adsorption parameters for the investigated enzymes at 25°C, derived from Figure 3
ProteinΓmax (µmol/g)Kd (µM)
Cel7A 0.27 ± 0.01 0.46 ± 0.06
Cel7AE212Q 0.29 ± 0.01 0.79 ± 0.08
Cel6A 0.22 ± 0.01 1.13 ± 0.14
Cel6AD221N 0.23 ± 0.01 1.39 ± 0.11
ProteinΓmax (µmol/g)Kd (µM)
Cel7A 0.27 ± 0.01 0.46 ± 0.06
Cel7AE212Q 0.29 ± 0.01 0.79 ± 0.08
Cel6A 0.22 ± 0.01 1.13 ± 0.14
Cel6AD221N 0.23 ± 0.01 1.39 ± 0.11

Standard errors are included.

Table 2.
Kinetic parameters and surface-site inhibition mechanism for Cel7A and Cel6A with different surface-site inhibitor combinations (Cel7AE212Q or Cel6AD221N) shown in Figure 4
Enzyme/inhibitor$invVmax/massS0$ (nmol g−1 s−1)$invKM$ (µM)Ki (µM)αInhibition mechanism
Cel7A/Cel7AE212Q 46.8 ± 1.1 1.69 ± 0.1 0.74 ± 0.08 5.11 ± 1.37 Mixed
Cel6A/Cel6AD221N 12.9 ± 0.5 0.44 ± 0.07 0.24 ± 0.06 7.58 ± 3.66 Mixed
Cel6A/Cel7AE212Q 19.5 ± 0.5 0.46 ± 0.04 3.1 ± 1.7 0.38 ± 0.23 Mixed
Cel7A/Cel6AD221N 43.7 ± 1.3 1.43 ± 0.11 0.47 ± 0.04 — Competitive
Enzyme/inhibitor$invVmax/massS0$ (nmol g−1 s−1)$invKM$ (µM)Ki (µM)αInhibition mechanism
Cel7A/Cel7AE212Q 46.8 ± 1.1 1.69 ± 0.1 0.74 ± 0.08 5.11 ± 1.37 Mixed
Cel6A/Cel6AD221N 12.9 ± 0.5 0.44 ± 0.07 0.24 ± 0.06 7.58 ± 3.66 Mixed
Cel6A/Cel7AE212Q 19.5 ± 0.5 0.46 ± 0.04 3.1 ± 1.7 0.38 ± 0.23 Mixed
Cel7A/Cel6AD221N 43.7 ± 1.3 1.43 ± 0.11 0.47 ± 0.04 — Competitive

The inhibition model that best describes the data is also indicated. Standard errors are included.

### Binding isotherms of the four investigated enzymes using Avicel as substrate.

Figure 3.
Binding isotherms of the four investigated enzymes using Avicel as substrate.

Symbols are experimental data while solid lines are the best-fit to a one-site Langmuir binding isotherm model. All experiments were performed at 25C and reported standard deviations are for triplicate measurements.

Figure 3.
Binding isotherms of the four investigated enzymes using Avicel as substrate.

Symbols are experimental data while solid lines are the best-fit to a one-site Langmuir binding isotherm model. All experiments were performed at 25C and reported standard deviations are for triplicate measurements.

### Surface-site inhibition curves for Cel7A and Cel6A in combination with different surface-site inhibitors.

Figure 4.
Surface-site inhibition curves for Cel7A and Cel6A in combination with different surface-site inhibitors.

The graphs show reaction rates for the hydrolysis of Avicel as a function of the enzyme concentration at different surface-site inhibitor concentrations (I0), according to the legend in the top-left corner. The steady-state rates for all inhibition experiments were normalized with the (fixed) substrate load ($massS0$) used in the assay. (A) Inhibition of Cel7A by Cel7AE212Q. (B) Inhibition of Cel7A by Cel6AD221N. (C) Inhibition of Cel6A by Cel6AD221N. (D) Inhibition of Cel6A by Cel7AE212Q. Symbols are experimental data points and lines are the best-fit results from the global, non-linear regression analysis to eqn. (1). Regression analysis was done in Origin Pro v. 2019 (OriginLab Corporation, Northampton, MA, U.S.A.). For inhibition of Cel7A by Cel6AD221N with I0 = 2 µM, we were unable to measure the data point at 6 µM E0 due to an experimental mistake.

Figure 4.
Surface-site inhibition curves for Cel7A and Cel6A in combination with different surface-site inhibitors.

The graphs show reaction rates for the hydrolysis of Avicel as a function of the enzyme concentration at different surface-site inhibitor concentrations (I0), according to the legend in the top-left corner. The steady-state rates for all inhibition experiments were normalized with the (fixed) substrate load ($massS0$) used in the assay. (A) Inhibition of Cel7A by Cel7AE212Q. (B) Inhibition of Cel7A by Cel6AD221N. (C) Inhibition of Cel6A by Cel6AD221N. (D) Inhibition of Cel6A by Cel7AE212Q. Symbols are experimental data points and lines are the best-fit results from the global, non-linear regression analysis to eqn. (1). Regression analysis was done in Origin Pro v. 2019 (OriginLab Corporation, Northampton, MA, U.S.A.). For inhibition of Cel7A by Cel6AD221N with I0 = 2 µM, we were unable to measure the data point at 6 µM E0 due to an experimental mistake.

## Results

### Adsorption and activity of catalytically deficient enzymes

To test whether the binding to cellulose was different for the inactive mutants, we made binding isotherms for all four enzymes investigated using Avicel as substrate. From the result in Figure 3, it can be seen that the inactive variants showed nearly identical binding compared with their respective wild type enzyme. Notably, the enzyme Cel6A had slightly weaker binding compared with Cel7A which has also been reported previously [31]. To quantify the adsorption data, we fitted a simple Langmuir binding isotherms to the experimental data. As seen from the solid lines in Figure 3, the Langmuir isotherms accounted well for the data, and the derived parameters are given in Table 1. It appears that Cel6A had 2.5 times higher Kd and a 20% lower saturation coverage (Γmax) compared with Cel7A. The parameters for Cel6A and Cel6AD221N where identical, while a small increase in Kd was observed for Cel7AE212Q when compared with Cel7A.

Binding reversibility and residual activity were also tested for the two inactive variants to confirm that the enzymes bound reversibly without degrading the substrate. By dilution-induced adsorption measurements, it was found that the two variants had a similar descending and ascending binding isotherms (see Supplementary Figure S1). This is consistent with a reversible adsorption mechanism for both enzymes. In accordance with Ståhlberg et al. [21], Koivula et al. [32], the two variants Cel7AE212Q and Cel6AD221N where catalytic deficient and their activity could not be detected under the current experimental conditions.

### Surface-site inhibition kinetics

We carried out four inhibition experiments using pair combinations of active and inactive enzymes and microcrystalline cellulose (Avicel PH-101) as substrate. Quasi-steady-state rates (vss) were derived from the release of soluble products after 1 h of contact time using a constant, low substrate load ($massS0$) and different enzyme (E0) and inhibitor (I0) concentrations. Such type of measurements have recently been found to give a good estimate of the steady-state rates [33]. We measured both the inhibition of wild type enzymes with their respective inactive variants (Cel7A/Cel7AE212Q and Cel6A/Cel6AD221N) and the enzyme/inhibitor combination were the wild type enzymes were mixed (Cel6A/Cel7AE212Q and Cel7A/Cel6AD221N). We followed a conventional approach for inhibition studies in the sense that we made saturation curves at several inhibitor concentrations for each enzyme/inhibitor pair (see Figure 4). The surface-site inhibition mechanisms for each inhibition experiment was identified using non-linear regression analysis (Origin Pro v. 2019, OriginLab Corporation, Northampton, MA, U.S.A.). Specifically, we impose the following constraints to eqn (1): I0/αKi = 0 (competitive surface-site inhibition), I0/Ki = 0 (uncompetitive surface-site inhibition), α =1 (non-competitive surface-site inhibition) and no constraints (mixed surface-site inhibition). Next, we made separate fits to the experimental data using the four simplified equations (see Supplementary Table S1). All fit was done using global non-linear regressions analysis and the inhibition mechanism was chosen from them equation that best fitted the experimental data. The latter decision was based on the Akaike information criterion (AIC, see Supplementary Data), and the result (the most probable mechanism) and the derived kinetic parameters are listed in Table 2. We note in passing, that this procedure of non-linear regression for each surface-site inhibition mechanism followed by rational assessment (AIC) of the model's ability to account for the data has the same purpose as earlier practices based on linearized equations such as Lineweaver Burk- or Hanes plots. Three of the four investigated enzyme/inhibitor pairs, Cel7A/Cel7AE212Q, Cel6A/Cel6AD221N, and Cel6A/Cel7AE212Q, showed mixed inhibition, while Cel7A/Cel6AD221N showed competitive inhibition.

We may use the α-values to further interpret the cases of mixed surface-site inhibition. Thus, the two homogeneous pairs (Cel7A/Cel7AE212Q, Cel6A/Cel6AD221N) had α > 1. This implies that while the mechanism is mixed, with I binding to both S and ES (Figure 2B), the competitive component (binding to S) dominates. Conversely, α < 1 for the Cel6A/Cel7AE212Q pair, and this suggests the dominance of the uncompetitive element (I binding to ES).

## Discussion

Steady-state kinetics with high enzyme to substrate ratios (E0 ≫ S0) are generally not feasible for bulk reactions because the substrate will be depleted instantaneously. However, the approach may be useful for enzyme reactions that involve insoluble substrates, where only a minuscule fraction of the total substrate is accessible to the enzymes. In the current case, the substrate is the glycosidic bond that links the glucopyranose units in the insoluble cellulose, and the accessible fraction, Θaccess, may be roughly estimated from the adsorption capacity of the enzyme. Specifically, Θaccess ∼ ΓmaxMglu, where Mglu is the molar mass of glucopyranose (162 g/mol). Insertion of Γmax from cellulases adsorption measurements (Table 1) suggests that Θaccess is in the order of 10−5 for the current studied Avicel substrate. It follows that even if the surface is saturated with an enzyme, the apparent molar concentration of hydolyzable sites on the surface, S0, can remain constant in short experiments, because new sites emerge as the upper layer of the particle is removed [27]. This assumption is captured in the upper line of the scheme in Figure 2B, where both substrate and enzyme recurs (together with the product) after the reaction. Recently, we provided evidence for this type of stationarity when Cel6A and Cel7A hydrolyzed cellulose under conditions of the excess enzyme [33]. The current study had identical experimental conditions, ensuring a low substrate conversion even at the highest enzyme concentrations (∼4% for Cel7A and ∼1% for Cel6A). Under the condition of enzyme excess (E0 ≫ S0), the reaction rate can be expressed by an inverse Michaelis–Menten equation [9,27,28,34–36]. This equation describes how the rate increases towards a constant maximum (described by the $invVmax$ parameter) as the surface of an insoluble substrate is gradually saturated with enzymes. It follows, that $invVmax$ not only depends on the turnover frequency of the enzyme (kcat), but also on the density of productive surface-sites ($kinΓmax$) (see eqn 3). This has the interesting consequence that the ability of a cellulase to attack broad range of different surface-sites (i.e. different conformations of cellulose strands on the heterogeneous surface) is just as important for $invVmax$ as the turnover frequency. It also underscores a particular sensitivity to inhibitors that hampers access to sites for example through binding to the substrate rather than the enzyme (c.f. Figure 1).

The schemes in Figure 2 illustrate how the classical interpretation of enzyme inhibition can be easily adopted to the surface-site condition, where the substrate is the limiting reactant. A rate equation for this scheme (eqn 1) can be readily derived as shown in Supplementary Section S3. As in the conventional case, the derivation rests on the assumptions that the substrate concentration remains nearly constant, and that both inhibitor and enzyme bind reversibly to the substrate. The former assumption was discussed above, and the latter was confirmed here (Supplementary Figure S2) and in earlier studies [25]. Figure 2B specifies two distinct ways for a surface-inhibitor to lower the rate of an interfacial enzyme reaction. Either it can compete with the enzyme for a surface-site (competitive inhibition through the formation of a SI-complex), or it can reduce the population of productive ES-complexes through the formation of non-reactive ESI complexes (uncompetitive inhibition). To illustrate the feasibility of the method, we used the cellulases Cel7A and Cel6A as an example of interfacial enzymes and their inactive versions as surface-site inhibitors. Our results with different enzyme–inhibitor combinations mostly showed mixed type surface-site inhibition (Table 2), and this means that the formation of both SI and ESI complexes influenced the overall reaction rate. In the following, we discuss molecular origins and importance of, respectively, the competitive- and uncompetitive components for the cellulases.

The competitive component was expected for self-inhibition experiments (Cel7A/Cel7AE212Q and Cel6A/Cel6AD221N), because the binding properties of both surface-site inhibitor and enzyme were essentially identical (Figure 3). Hence, the catalytically inactive enzyme will compete with the wild type enzyme for the productive binding-sites (forming SI-complexes). This interpretation is illustrated in Figure 5A. Interestingly, the competitive surface-site inhibition constant, Ki, for Cel7A/Cel7AE212Q (Table 2) almost perfectly matched the dissociation constant, Kd, (Table 1) from the binding isotherms (0.74 µM and 0.79 µM for Ki and Kd, respectively). On the other hand, for Cel6A/Cel6AD221N, Ki is about five-fold lower than Kd (0.24 µM and 1.39 µM for Ki and Kd, respectively). This difference may be related to populations of, respectively, complexed and non-complexed enzyme adsorbed on the substrate surface. Both Cel7A and Cel6A are modular enzymes with two domains (see Figure 5); a catalytic domain that binds a single cellulose chain and a carbohydrate-binding domain (CBM) that binds to the cellulose surface [37]. Biochemical studies have suggested that the majority of adsorbed Cel7A had a ligand in the catalytic domain, while a significant population of Cel6A was adsorbed unproductively through the CBM [28,31]. These biochemical results are in line with the current data inasmuch as adsorption and competitive inhibition have similar strength for Cel7A (all adsorbed molecules in the complex) whereas Kd is larger than Ki for Cel6A (some adsorbed molecules not in the complex).

### Molecular interpretation of different surface-site inhibition mechanisms.

Figure 5.
Molecular interpretation of different surface-site inhibition mechanisms.

(A) Competitive surface-site inhibition (formation of the SI-complex in Figure 2B) arises from the competition of sites on the surface between the surface-site inhibitor (inactive enzyme) and the interfacial enzyme (active enzyme). (B) Uncompetitive surface-site inhibition (formation of the ESI complex in Figure 2B) is suggested to reflect ‘traffic jams’, where the immobile inhibitor block the processive movement of a productive enzyme–substrate complex. Mixed surface-site inhibition will be a combination of the competitive and uncompetitive components while non-competitive surface-site inhibition will be the special case of mixed surface-site inhibition where the uncompetitive and competitive component have similar binding strength.

Figure 5.
Molecular interpretation of different surface-site inhibition mechanisms.

(A) Competitive surface-site inhibition (formation of the SI-complex in Figure 2B) arises from the competition of sites on the surface between the surface-site inhibitor (inactive enzyme) and the interfacial enzyme (active enzyme). (B) Uncompetitive surface-site inhibition (formation of the ESI complex in Figure 2B) is suggested to reflect ‘traffic jams’, where the immobile inhibitor block the processive movement of a productive enzyme–substrate complex. Mixed surface-site inhibition will be a combination of the competitive and uncompetitive components while non-competitive surface-site inhibition will be the special case of mixed surface-site inhibition where the uncompetitive and competitive component have similar binding strength.

The enzyme–inhibitor combinations with different cellulases, Cel6A/Cel7AE212Q and Cel7A/Cel6AD221N, also showed competition for sites (Table 2). This is less intuitive as Cel7A and Cel6A are mostly exo-acting hydrolytic enzymes with preference for opposite reducing ends of the cellulose strand [38–40]. However, the result may reflect the occurrence of attack sites that are on different cellulose strands but in close proximity on the surface. Since the catalytic domain of Cel7A covers about 48 cellobiose lattice sites on a cellulose surface [41], the enzyme could sterically hinder association to nearby sites independent of the enzymé site specificity.

Beside the competitive component of surface-site inhibition, all but one of the investigated systems had an uncompetitive component, which gave rise to an overall surface-site mixed type inhibition mechanism (Table 2). To interpret the origins of the uncompetitive component (i.e. the nature of the ESI complex in Figure 2B), we note that both Cel6A and Cel7A are processive enzymes. This means the hydrolytic reaction involves a one-dimensional movement along a cellulose strand [37], and this makes the enzymes vulnerable to any obstacle on the surface that hinder this movement. If indeed such movement is prevented, the enzyme will be trapped unproductively (in a complexed state) behind the obstacle until either enzyme or obstacle dissociate. We propose that the ESI complex represents this situation, where bound surface-site inhibitor keeps the wild type enzyme–substrate complex from further processive movement. This interpretation is in accord with biochemical measurements for both Cel7A and Cel6A [42–45], and supported by molecular imaging [46,47]. In addition, different imaging studies have confirmed that inactive variants of Cel7A and Cel6A remain immobile on the surface [46,48,49]. This understanding of the ESI complex, which is illustrated in Figure 5B, is also in line with a high-speed AFM study, which showed that immobile Cel7A molecules gave raise to ‘traffic jams’ on the surface at high enzyme concentrations [47].

Table 2 shows that α > 1 in the two self-inhibition experiments (Cel7A/Cel7AE212Q and Cel6A/Cel6AD221N). This implies that the surface-site competitive component is stronger than the uncompetitive component (c.f. in Figure 2B), and underscores the importance of site competition when the surface-site inhibitor and enzyme have the same specificity. For the mixed Cel6A/Cel7AE212Q system, we found α < 1, which signified the dominance of the surface-site uncompetitive component in this system. This may rely on the stronger substrate binding [37] and longer residence times [42] of Cel7A compared with Cel6A, as these traits will make Cel7AE212Q a particularly efficient obstacle for processive movement and hence a strong uncompetitive inhibitor.

Inhibition kinetics along the lines proposed here might be useful in mechanistic studies of interfacial enzyme reactions. One example is the analysis of enzyme synergy, which is important for the two cellobiohydrolases studied here, as well as other cellulases. Typically, the degree of synergy (DS) is assessed as the activity ratio of an enzyme mixture and the mono-component enzymes acting separately. For cellulases, this effect is quite significant with DS values commonly about 2–3 and in some cases as high as 10 [50–53]. The current results emphasized that measured synergies encompass both positive contribution of enzyme cooperation and a negative contribution from surface-site inhibition. Moreover, the approach delineated a way to calculate the inhibitory contribution, and hence resolve both elements for a better understanding. In a related study, Jeoh et al. [18] showed that cellulase mixtures exhibit binding cooperativity at high temperature (50°C), but binding inhibition at low temperature (5°C). Since low temperature reduces the rate of processive movement and increases binding [26,54,55], it will lead to lower mobility and higher density of enzymes on the surface. This, in turn, collectively promotes surface-site inhibition. Conversely, at high temperatures, the opposite changes might cause the observed shift towards binding cooperativity. Finally, we note that, for practical reasons, we used inactive enzymes as surface inhibitors in this work. We anticipate that the principles illustrated here might also be useful for other molecules that bind to the surface of cellulose and hence hampers its hydrolysis. These include lignin [14], hemicellulose [16,56] and competing enzymes [18,57]. With regards to lignin, it has been debated whether it inhibits through binding to the substrate [58–61] or enzyme [17], and we propose that this question could be further elucidated through a combined study of conventional and surface-site inhibition kinetics.

## Conclusions

In this study, we developed a new steady-state kinetic approach to elucidate enzyme inhibition at the solid–liquid interface. We used inactive variants of two of the best-studied fungal cellulases, Cel7A and Cel6A, to test the validity and applicability of the approach on an insoluble cellulosic substrate. We demonstrated that the method is applicable for surface-site inhibition of Cel7A and Cel6A, and that both qualitative (inhibition mechanism) and quantitative (inhibition constants) information can be obtained. The results suggested that both competition for attack sites on the substrate surface and obstruction of processive movement contributes to surface-site inhibition, and that the former was particularly important for self-inhibition.

We anticipate that this new kinetic approach will be applicable to different types of interfacial enzyme reactions, and that it will enable mechanistic and applied studies based on activity measurements in much the same way as inhibition analyses under substrate excess have been used for bulk reactions. One potential application is monitoring surface properties of complex insoluble substrates using specific inhibitors akin to the ones used here, and another is the study of synergistic mechanisms.

## Conflict of Interest

K.B., T.H.S., A.M.C. work for Novozymes, a major enzyme-producing company.

## Funding

This work was supported by the Innovation Fund Denmark [grant number: 5150-00020B], the Novo Nordisk Foundation [grant number: NNF15OC0016606 and NNFSA170028392] and the Independent Research Fund Denmark [grant number: 8022-00165B].

## Author Contributions

J.K., C.S.d.C., K.B. and P.W. conceived and designed the study. J.K. mathematically derived the rate equations and provided the initial theoretical framework. C.S.d.C. performed experiments, analyzed the data and interpreted the results along with J.K. T.H.S., A.M.C. carried out cloning and expression of the enzymes. S.F.H. performed experiments of Cel6A/Cel6AD221N. S.F.B. optimized the purification procedure for Cel6A and Cel6AD221N. P.W., J.K., C.S.d.C. wrote the manuscript. K.B. provided the enzymes used in the study. All authors reviewed and approved the manuscript.

## Abbreviations

• AIC

Akaike information criterion

•
• PAHBAH

4-hydroxybenzoic acid hydrazide

•
• SDS–PAGE

sodium dodecyl sulfate–polyacrylamide gel electrophoresis

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## Author notes

*

Both authors contributed equally to this work.