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Keywords:

  • burst phase;
  • calorimetry;
  • cellulase;
  • kinetic equations;
  • slowdown of cellulolysis

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Results and Discussion
  5. Materials and methods
  6. Acknowledgements
  7. References
  8. Supporting Information

Cellobiohydrolases (exocellulases) hydrolyze cellulose processively, i.e. by sequential cleaving of soluble sugars from one end of a cellulose strand. Their activity generally shows an initial burst, followed by a pronounced slowdown, even when substrate is abundant and product accumulation is negligible. Here, we propose an explicit kinetic model for this behavior, which uses classical burst phase theory as the starting point. The model is tested against calorimetric measurements of the activity of the cellobiohydrolase Cel7A from Trichoderma reesei on amorphous cellulose. A simple version of the model, which can be solved analytically, shows that the burst and slowdown can be explained by the relative rates of the sequential reactions in the hydrolysis process and the occurrence of obstacles for the processive movement along the cellulose strand. More specifically, the maximum enzyme activity reflects a balance between a rapid processive movement, on the one hand, and a slow release of enzyme which is stalled by obstacles, on the other. This model only partially accounts for the experimental data, and we therefore also test a modified version that takes into account random enzyme inactivation. This approach generally accounts well for the initial time course (approximately 1 h) of the hydrolysis. We suggest that the models will be useful in attempts to rationalize the initial kinetics of processive cellulases, and demonstrate their application to some open questions, including the effect of repeated enzyme dosages and the ‘double exponential decay’ in the rate of cellulolysis.

Database The mathematical model described here has been submitted to the Online Cellular Systems Modelling Database and can be accessed at http://jjj.biochem.sun.ac.za/database/Praestgaard/index.html free of charge.


Abbreviations
CBH

cellobiohydrolase

Cel7A

cellobiohydrolase I

ITC

isothermal titration calorimetry

RAC

reconstituted amorphous cellulose

Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Results and Discussion
  5. Materials and methods
  6. Acknowledgements
  7. References
  8. Supporting Information

The enzymatic hydrolysis of cellulose to soluble sugars has attracted increasing interest, because it is a critical step in the conversion of biomass to biofuels. One major challenge for both the fundamental understanding and application of cellulases is that their activity tapers off early in the process, even when the substrate is plentiful. Typically, the rate of hydrolysis decreases by an order of magnitude or more at low cellulose conversion, and experimental analysis has led to quite divergent interpretations of this behavior. One line of evidence has suggested that the slowdown is a result of the heterogeneous nature of the insoluble substrate. Thus, if various structures in the substrate have different susceptibility to enzymatic attack, the slowdown may reflect a phased depletion of the preferred types of substrate [1,2]. Other investigations have emphasized enzyme inactivation as a major cause of the decreasing rates [3]. This inactivation could reflect the formation of nonproductive enzyme–substrate complexes [4–6] or the adsorption of cellulases on noncellulosic components, such as lignin [7,8], although the role of lignin remains controversial [9]. Recently, Bansal et al. [10] have provided a comprehensive review of theories for cellulase kinetics, and it was concluded that no generalization could be made regarding the origin of the slowdown. In particular, so-called ‘restart’ or ‘resuspension’ experiments, in which a substrate is first partially hydrolyzed, then cleared of cellulases and finally exposed to a second enzyme dose, have alternatively suggested that enzyme inactivation and substrate heterogeneity are the main causes of decreasing hydrolysis rates (see refs. [10,11]).

Further analysis of different contributions to the slowdown appears to require a better theoretical framework for the interpretation of the experimental material. In this study, we introduce one approach and test it against experimental data for the cellobiohydrolase Cel7A (formerly CBHI) from Trichoderma reesei. Our starting point is classical burst phase theory for soluble substrates [12], and we extend this framework to account for the characteristics of cellobiohydrolases, such as adsorption onto insoluble substrates, irreversible inactivation and processive action. The latter implies a propensity to complete many catalytic cycles without the dissociation of enzyme and substrate. For cellobiohydrolases, the processive action may involve the successive release of dozens or even hundreds of cellobiose molecules from one strand [13], and some previous reports have suggested a possible link between this and the slowdown in hydrolysis [8,13,14].

Results and Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Results and Discussion
  5. Materials and methods
  6. Acknowledgements
  7. References
  8. Supporting Information

Theory

Burst phase for soluble substrates and nonprocessive enzymes

The concept of a burst phase was introduced more than 50 years ago, when it was demonstrated that an enzyme reaction with two products may show a rapid production of one of the products in the pre-steady-state regime [15,16]. Later work has shown that this is quite common for hydrolytic enzymes with an ordered ‘ping–pong bi–bi’ reaction sequence [12]. At a constant water concentration, this type of hydrolysis may be described by Eqn (1), which does not explicitly include water as a substrate (the process is considered as an ordered uni–bi reaction):

  • image(1)

In an ordered mechanism, the product P1 is always released from the complex before the product P2, and it follows that, if k3 is small (compared with k1S0 and k2), there will be a rapid production of P1 (a burst phase) when E and S are first mixed. Subsequently, at steady state, a large fraction of the enzyme population will be trapped in the EP2 complex, which is only slowly converted to P2 and free E, and the (steady state) rate of P1 production will be lower. The result is a maximum in the rate of production of P1 but not P2 (see Fig. 1). To analyze this maximum, we need an expression for the rate of P1 production: P1′(t). Here, and in the following analyses of the reaction schemes, we first try to derive analytical solutions, as this approach provides rigorous expressions that may help to identify the molecular origin of the burst and slowdown. In cases in which analytical expressions cannot be found, we use numerical treatment of the rate equations. The results based on analytical solutions were also tested by the numerical treatment, and no difference between the two approaches was found. The equation for P1′(t) has previously been solved on the basis of different simplifications, such as merging the first two steps in Eqn (1) [17,18] or using a steady-state approximation for the intermediates [15,19]. The equations may also be solved numerically without resorting to any assumptions, or solved analytically if it is assumed that the change in S is negligible. If the initial substrate concentration S0 is much larger than E0, the assumption of a constant S during the burst is very good, and we have used this approach to derive expressions for both the rates P1′(t) and P2′(t), and the concentrations P1(t) and P2(t) (see Data S1). Figure 1 shows an example of how these functions change in the pre-steady-state regime, when parameters similar to those found below for Cel7A are inserted.

image

Figure 1.  Initial time course of the concentrations P1(t) and P2(t) (A) and the rates P1′(t) and P2′(t) (B) calculated from Eqns <10>–<13> in Data S1. Full and broken lines indicate P1 and P2, respectively, and the dotted line shows the steady-state condition with constant concentrations of the intermediates ES and EP2, and hence constant rates. The intersection π is a measure of the extent of the burst (see text for details). The parameters were S0 = 20 μm, E0 = 0.050 μm, k2 = 0.3 s−1, k1 = 0.002 s−1·μm−1, k−1 = k3 = 0.002 s−1; these values are similar to those found below for Cel7A.

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The initial slopes in Fig. 1A are zero and, after about 100 s, both functions asymptotically reach the steady-state value, where the concentrations of both intermediates ES and EP2, and hence the rates P1′(t) and P2′(t), become independent of time (Fig. 1B). For P2(t), the slope in Fig. 1A never exceeds the steady-state level, but P1(t) shows a much higher intermediate slope that subsequently falls off towards the steady-state level. This behavior is more clearly illustrated by the rate functions in Fig. 1B, and it follows that a method that directly measures the reaction rate (rather than the concentrations) may be particularly useful in the investigation of burst phase kinetics. This is the rationale for using calorimetry in the current work.

Experimental analysis of the burst phase often utilizes the intersection π of the ordinate and the extrapolation of the steady-state condition for P1(t) (dotted line in Fig. 1A). This value is used as a measure of the amount of P1 produced during the burst, i.e. the excess of P1 with respect to the steady-state production rate, and it is therefore a measure of the magnitude of the burst. An expression for π is readily obtained by inserting t = 0 in the (asymptotic) linear expression for P1(t), which results from considering t [RIGHTWARDS ARROW] ∞ (see Data S1). Under the simplification that k−1 = k3, π may be written:

  • image(2)

If Eqn (2) is considered for the special case in which the first two steps in Eqn (1) are much faster than the third step (i.e. k1S0 >> k3 + k−1 and k2 >> k3), it reduces to the important relationship π = E0, which is the basis for so-called substrate titration protocols [20], in which the concentration of active enzyme is derived from experimental assessments of π. The intuitive content of this is that each enzyme molecule quickly releases one P1 molecule, as described by the first two steps in Eqn (1), before it gets caught in a slowly dissociating EP2 complex.

Burst phase for processive enzymes

Kipper et al. [13] studied the hydrolysis of end-labelled cellulose by Cel7A, and found that the release of the first (fluorescence-labelled) cellobiose molecule from each cellulose strand showed a burst behavior, which was qualitatively similar to that shown in Fig. 1. This suggests that this first hydrolytic cycle may be described along the lines of Eqn (1). Unlike the example in Eqn (1), however, Cel7A is a processive enzyme that completes many catalytic cycles before it dissociates from the cellulose strand [13]. This dissociation could occur by random diffusion, but some reports have suggested that processivity may be linked to the occurrence of obstacles and imperfections on the cellulose surface [4,6,14]. These observations may be captured in an extended version of Eqn (1) that takes processivity and obstacles into account. Thus, we consider a cellulose strand Cn, which has no obstacles for the processive movement of Cel7A between the reducing end (the attack point of the enzyme) and the nth cellobiose unit [i.e. there is a ‘check-block’ that prevents processive movement from the nth to the (n + 1)th cellobiose unit]. The processive hydrolysis of this strand may be written as:

inline image

We note that this reaction reduces to Eqn (1) when n = 2 and k−1 = k3. In Eqn (3), the free cellulase (E) first combines with a cellulose strand (Cn) to form an ECn complex. This process, which will also include a possible diffusion on the cellulose surface and the ‘threading’ of the strand into the active site, is governed by the rate constant k1 at a given value of S0. The ECn complex is now allowed to decay in one of two ways. Either the enzyme makes a catalytic cycle in which a cellobiose molecule (C) is released whilst the enzyme remains bound in a slightly shorter ECn−1 complex. Alternatively, the ECn complex dissociates back to its constituents E and Cn. The rate constants for hydrolysis and dissociation are k2 and k3, respectively. This pattern continues so that any enzyme–substrate complex ECni (where i enumerates the number of processive steps) can either dissociate [vertical step in Eqn (3)] or enter the next catalytic cycle [horizontal step to the right in Eqn (3)], which releases one more cellobiose. A typical cellulose strand is hundreds or thousands of glycosyl units long, and it follows that the local environment experienced by the cellulase may be similar for many sequential catalytic steps. Therefore, we use the same rate constants k2 and k3 for consecutive hydrolytic or dissociation steps. This version of the model neglects the fact that the Cn−1, Cn−2, … strands are also substrates (free E is not allowed to associate with these partially hydrolyzed strands). This simplification is acceptable in the early part of the process where Cn >> E0. After n processive steps, the enzyme reaches the ‘check block’, and this necessitates a (slow) desorption from the remaining cellulose strand (designated Cx) before the enzyme can continue cellobiose production from a new Cn strand. In other words, the strand consists of n + x cellobiose units in total, but because of the ‘check block’, only the first n units are available for enzymatic hydrolysis. This interpretation of obstacles and processivity is similar to that recently put forward by Jalak & Valjamae [14].

A kinetic treatment of Eqn (3) requires the specification of the substrate concentration. This is not trivial for an insoluble substrate, but, as the enzyme used here attacks the reducing end of the strand, we use the molar concentration of ends for S0 throughout this work. This problem may be further addressed by introducing noninteger (fractal) kinetic orders that account for the special limitations of the heterogeneous reaction (see refs. [31,32]). For this model, this is readily performed by introducing apparent orders in Eqn (5). However, the current treatment is limited to the simple case in which the kinetic order is equal to the molecularity of the reactions in Eqn (3). This implies that the adsorption of enzyme onto the substrate is described by a kinetic (rather than equilibrium) approach (c.f. Ref. [21]). Based on this and the simplifications mentioned above, the kinetic equations for each step in Eqn (3) were written and solved with respect to the ECni intermediates as shown in Data S1. As cellobiose production in Eqn (3) comes from these ECni complexes, which all decay with the same rate constant k2, the rate of cellobiose production C′(t) follows the equation:

  • image(4)

Using the expressions in Data S1, the sum in Eqn (4) may be written as:

  • image(5)

where inline image is the so-called upper incomplete gamma function [22]. Equations (4) and (5) provide a description of the burst phase for processive enzymes. In the simple case, this approach will eventually reach steady state with constant concentrations of all ECn−i complexes and hence constant C′(t). We emphasize, however, that there are no steady-state assumptions in the derivation of Eqn (5) and, indeed, we use it to elucidate the burst in the pre-steady-state regime. As discussed below, Eqn (3) is found to be too idealized to account for experimental data, and some modifications are introduced. Nevertheless, Eqn (5) is the main result of the current work and is the backbone in the subsequent analyses.

Examination of a processive burst phase as specified by Eqns (4) and (5) reveals some similarity to the simple burst behaviour in Fig. 1. Hence, if we insert the same rate constants as in Fig. 1, and use an obstacle-free path length of n = 100 cellobiose units, the rate of cellobiose production C′(t) (full curve in Fig. 2) exhibits a maximum akin to that observed for P1′(t) in Fig. 1B. However, the occurrence of fast sequential steps in the processive model produces a more pronounced maximum in both duration and amplitude. Figure 2 also illustrates the meaning of the three terms that are summed in Eqn (5). The chain line shows the contribution from the first (simple exponential) term on the right-hand side of Eqn (5), which describes the kinetics devoid of any effect from obstacles (corresponding to n [RIGHTWARDS ARROW] ∞). The broken line is the sum of the last two terms (the terms with gamma functions) and quantifies the (negative) effect on the hydrolysis rate arising from the ‘check blocks’. For the parameters used in Fig. 2, this contribution only becomes important above t ≈ 300 s, and this simply reflects the minimal time required for a significant population of enzyme to bind and perform the 100 processive steps to reach the ‘check block’. After about 600 s, essentially all enzymes have reached their first encounter with a ‘check block’ and we observe an abeyance with reduced C′(t) because a significant (and constant) fraction of the enzyme is unproductively bound in front of a ‘check block’.

image

Figure 2.  The rate of cellobiose production C′(t) (solid curve) calculated according to Eqns (4) and (5) and plotted against time. The rate constants are the same as in Fig. 1 and the initial concentrations were E0 = 0.050 μm and S0 = 5 μm reducing ends. The obstacle-free path n was set to 100 cellobiose units. The chain curve shows the first term in Eqn (5), which signifies the rate of cellobiose production on an ‘obstacle-free’ substrate (i.e. for n [RIGHTWARDS ARROW] ∞). The broken curve, which is the sum of the last two terms in Eqn (5), signifies the inhibitory effect of the obstacles. The two curves sum to the full curve.

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The extent of the processive burst may be assessed from the intersect πprocessive defined in the same way as π for the simple reaction (see Fig. 1A). As shown in Data S1, πprocessive may be written as:

  • image(6)

We note that πprocessive is proportional to E0 and, if we again consider the case in which adsorption and hydrolysis are fast compared with desorption (i.e. k1S0 >> k3 and k2 >> k3), Eqn (6) reduces to πprocessive = nE0. This implies that, under these special conditions, every enzyme rapidly makes one run towards the ‘check block’, and thus produces the number of cellobiose molecules n which are available to hydrolysis in the obstacle-free path.

Modifications of the model

In analogy with the simple case in Eqn (1), the rate C′(t) specified by Eqn (3) runs through a maximum and falls towards a steady-state level (Fig. 2) in which the concentrations of all intermediates ECni and the rate C′(t) are independent of time. This behavior, however, is at odds with countless experimental reports, as well as the current measurements, which suggest that the activity of Cel7A does not reach a constant rate. Instead, the reaction rate continues to decrease. This suggests that, in addition to the burst behavior described in Eqn (3), other mechanisms must be involved in the slowdown. The nature of such inhibitory mechanisms has been discussed extensively and much evidence has pointed towards product inhibition, reduced substrate reactivity or enzyme inactivation (see, for example, refs. [10,11,23] for reviews). In the current work, we observed this continuous slowdown even in experiments with very low substrate conversion (< 1%), where the hydrolysis rates are unlikely to be affected by inhibition or substrate modification (an inference that is experimentally supported in Fig. 9 below). In the coupled calorimetric assay used here, the product (cellobiose) is converted to gluconic acid. The concentration is in the micromolar range, and previous tests have shown that this is not inhibitory to cellulolysis or the coupled reactions (see Ref. [48]). Therefore, the continuous decrease in the rate of hydrolysis was modeled as protein inactivation. To this end, we essentially implemented the conclusions of a recent experimental study by Ma et al. [24] in the model. As with earlier reports [3,14,25–27], Ma et al. discussed unproductively bound cellulases, and found that substrate-associated Cel7A could be separated into two populations of reversibly and irreversibly adsorbed enzyme. The latter population, which grew gradually over time, was found to lose most catalytic activity. This behavior was introduced into the model through a new rate constant k4, which pertains to the conversion of an active enzyme–cellulose complex (ECni) into a complex of cellulose and inactive protein (ICni). In other words, any ECni complex in Eqn (7) is allowed three alternative decay routes, namely hydrolysis (k2), dissociation (k3) or irreversible inactivation (k4). We also introduced a separate rate constant k−1 for the dissociation of substrate and enzyme ECn before the first hydrolytic step. With these modifications, we may write the reaction:

image

Figure 9.  Rate of cellobiose production C′(t) as a function of time for S0 = 70 μm. One aliquot of 50 nm Cel7A was added at t = 0 and a second dose (bringing the total enzyme concentration to 100 nm) was added at t = 3600 s.

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inline image

We were not able to find an analytical solution for C′(t) on the basis of Eqn (7), and we instead used a numerical treatment with the appropriate initial conditions [i.e. all initial concentrations except E(t) and Cn(t) are zero].

One final modification of the model was introduced to examine the effect of ‘polydispersity’ in n. Thus, n as defined in Eqns (3) and (7) is a constant, and this implies that all enzymes must perform exactly n catalytic cycles before running into the ‘check block’. This is evidently a rather coarse simplification and, to consider the effects of this, we also tested an approach which used a distribution of different n values. For example, the substrate was divided into five equal subsets (i.e. each 20% of S0) with n values ranging from 40% to 160% of the average value. We also analyzed different distributions and subsets of different sizes (with a larger fraction close to the average n and less of the longest/shortest strands). In all of these analyses, the rate of cellobiose production from each subset was calculated independently and summed to obtain the total C′(t).

Experimental

Two parameters from the model, namely the substrate and enzyme concentrations (E0 and S0), can be readily varied in experiments, and we therefore firstly compared measurements and modeling in trials in which S0 and E0 were systematically changed. Figure 3A shows a family of calorimetric measurements in which Cel7A was titrated to different initial substrate concentrations (S0 in μm of reducing ends – this unit can be readily converted into a weight concentration using the molar mass of a glycosyl unit and the average chain length for the current substrate, DP = 220 glycosyl units). The concentration of Cel7A was 50 nm in these experiments and the experimental temperature was 25 °C. Figure 3B shows model results for the same values of E0 and S0. Here, we used the model in Eqn (3) [Eqns (4) and (5)] and manually adjusted the kinetic constants and n by trial and error. The parameters in Fig. 3B are k1 = 0.0004 s−1·μm−1, k2 = 0.55 s−1, k3 = 0.0034 s−1 and n = 150. Comparison of the two panels shows that the idealized description of processive hydrolysis in Eqn (3) cannot account for the overall course of the process, but some characteristics, both qualitative and quantitative, are captured by the model. For example, the model accounts well for the diminished burst (i.e. the disappearance of the maximum) at low S0 (below 5–10 μm). In these dilute samples, the rate of cellobiose production C′(t) increases slowly to a level which is essentially constant over the time considered in Fig. 3. At higher S0, a clear maximum in C′(t) signifies a burst phase in both model and experiment. On a quantitative level, comparisons of the maximal rate at the peak of the burst (t = 150 s in Fig. 3C) and after the burst (t = 1400 s in Fig. 3C) showed a reasonable accordance between experiments and model. In addition, the substrate concentration that gives half the maximal rate (5–10 mm) is similar to within experimental scatter (Fig. 3C). Conversely, two features of the experiments do not appear to be captured by Eqn (3). Firstly, the model predicts a sharp termination of the burst phase, which tends to produce a rectangular shape of the C′(t) function at high S0 (Fig. 3B). This is in contrast with the experiments which all show a gradual decrease in C′(t) after the maximum. Secondly, the model suggests a constant C′(t) well within the time frame covered in Fig. 3, but no constancy was observed in the experiments. We return to this after discussing the effect of changing E0.

image

Figure 3.  Comparison of the results from experiment and model [Eqn (3)] for different substrate concentrations (S0 in μm reducing ends). The enzyme concentration E0 was 50 nm. Experimental (A) and model (B) C′(t) results from Eqns (4) and (5) using the parameters k1 = 0.0004 s−1·μm−1, k2 = 0.55 s−1, k3 = 0.003 s−1 and n = 150 cellobiose units. (C) Experimental (circles) and modeled (lines) rates at two time points plotted as a function of S0.

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Figure 4 shows a comparison of the calorimetric measurements and model results for a series in which the enzyme load was varied and S0 was kept constant at 40.8 μm reducing ends. The model calculations were based on the same parameters as in Fig. 3 without any additional fitting, and it appears that C′(t) increases proportionally to E0. This behavior, which was seen in both model and experiment, implies that the turnover number C′(t)/E0 is constant over the studied range of time and concentration, and this, in turn, suggests that the extent of the burst scales with E0. To analyze this further, πprocessive was estimated from the data in Fig. 4. For the model results (Fig. 4B), this is simply done by inserting the kinetic parameters in Eqn (6). For the experimental data, we first numerically integrated the rates in Fig. 4A to obtain the concentration of cellobiose C(t), and then extrapolated linear fits to the data between 1400 and 1600 s to the ordinate as illustrated in the inset of Fig. 5. In analogy with the procedure used for nonprocessive enzymes (Fig. 1A), this intercept between the extrapolation and the C(t) axis was taken as a measure of the experimental πprocessive.

image

Figure 4.  Comparison of experimental and model results for different enzyme concentrations (E0). The substrate concentration was 40.8 μm reducing ends. Experimental (A) and model (B) C′(t) results using the same parameters as in Fig. 3.

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image

Figure 5.  Theoretical (open symbols) and experimental (filled symbols) estimates for the extent of the burst (πprocessive) based on the results in Fig. 4. Theoretical values were obtained by insertion of the kinetic constants from Fig. 3 into Eqn (4), and the experimental values represent extrapolation of the C′(t) function to t = 0 as illustrated in the inset. The extrapolations were based on linear fits to C′(t) from 1400 to 1600 s.

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The proportionality of the theoretical πprocessive and E0 seen in Fig. 5 follows directly from Eqn (6). The slope of the theoretical curve is about 42, suggesting that each enzyme molecule completes 42 catalytic cycles (produces 42 cellobiose molecules) during the burst phase. This is about three times less than the obstacle-free path (n), which is 150 in these calculations, and this discrepancy simply reflects that k1S0 is too small for the simple relationship πprocessive = nE0 to be valid (see Theory section). Thus, low k1 and the concomitant slow ‘on rate’ tend to smear out the burst and, consequently, πprocessive/E0 < n. This is a general weakness of the extrapolation procedure [17,18], also visible in Fig. 1, where the dotted line intersects the ordinate at a value slightly less than E0. It occurs when the rate constants and S0 attain values that make the fractions on the right-hand side of Eqns (2) and (6) smaller than unity (this implies that the criteria for a simple π expression, k1S0 >> k3 + k−1 and k2 >> k3, discussed in the Theory section, are not met [17,18]). More importantly, the experimental data also show proportionality between πprocessive and E0 with a comparable slope (about 65), and this supports the general validity of Eqn (3).

We now return to the two general shortcomings of Eqn (3) which were identified above: (a) the abrupt termination of the modeled burst phase (Fig. 3B), which is evident for high S0 and not seen in the experiments; and (b) the regime with constant C′(t) (see, for example, t > 500 s in Fig. 4B and inset in Fig. 6), which is also absent in the measurements. We suggest that, at least to some extent, (a) is a consequence of the ‘polydispersity’ in n in a real substrate and (b) depends on the random inactivation of the enzyme. As discussed in the Theory section, simplified descriptions of these properties may be included in the model, and these modifications considerably improve the concordance between theory and experiment. To illustrate this, we considered a substrate distribution with five subsets (each 20% of S0) with n = 40, 70, 100, 130 and 160, respectively. We analyzed the initial 1700 s of all trials in Fig. 3 using Eqn (5) and the nonlinear regression routine in Mathematica 7.0. It was found that, above S0 ∼ 15 μm, the parameters derived from each calorimetric experiment were essentially equal, and we conclude that one set of parameters can describe the results in this concentration range. The parameters were k2 = 1.0 ± 0.2 s−1, k3 = 0.0015 ± 0.0003 s−1 and k1S0 = 0.0052 ± 0.001 s−1, and some examples of the results are shown in Fig. 6. Parameter interdependence was evaluated partly by the confidence levels given by Mathematica and partly by ‘grid searches’, which provide an unambiguous measure of parameter dependence [28,29] and hence reveal possible overparameterization. In the latter procedure, the standard deviation of the fit was determined in sequential regressions, where two of the rate constants were allowed to change, whilst the third was inserted as a constant with values slightly above or below the maximum likelihood parameter [28,29]. These analyses showed moderate parameter dependence with 95% confidence intervals of about ±10% (slightly asymmetric with larger margins upwards). This limited parameter interdependence is also illustrated in the correlation matrix in Data S1, which shows that all correlation coefficients are below 0.7, and we conclude that it is realistic to extract three rate constants from the experimental data. The parameters from this regression analysis may be compared with recent work [30], which used an extensive analysis of reducing ends in both soluble and insoluble fractions to estimate apparent first-order rate constants for processive hydrolysis and enzyme–substrate disassociation, respectively. Values for the system investigated in Fig. 6 (i.e. T. reesei Cel7A and amorphous cellulose) were 1.8 ± 0.5 s−1 (hydrolysis) and 0.0032 ± 0.0006 s−1 (dissociation) at 30 °C [30]. The concordance of these values, which were derived by a completely different approach, and k2 and k3 from Fig. 6 provides strong support of the molecular picture in Eqn (3). With respect to the ‘on rate’, it is interesting to note that a constant value of k1 provided very poor concordance between theory and experiment (not shown), whereas constant k1S0 gave satisfactory agreement (Fig. 6). This suggests that the initiation of hydrolysis (adsorption to the insoluble substrate and ‘threading’ of the cellulase) exhibits apparent first-order kinetics. This may reflect the reduced dimensionality or fractal kinetics, which has previously been proposed for cellulase activity on insoluble substrates [31,32], and it appears that the current approach holds some potential for systematic investigations of this phenomenon.

image

Figure 6.  Experimental data (symbols) and model results (lines) based on Eqn (3). In this case, the substrate was treated as a mixture with different obstacle-free path lengths. Specifically, S0 was divided into five subsets with n = 40, 70, 100, 130 and 160. The nonlinear regression was based on the data for the first 1700 s. The inset shows an enlarged picture of the course after 1700 s and illustrates that, for the simple model [Eqn (3)], the experimental values fall below the model beyond the time frame considered in the regression.

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The model could not account for the measurements at the lowest S0, and this may reflect the fact that the assumption S0 >> E0, used in the derivation of the expression for C′(t), becomes unacceptable. Thus, the concentration of reducing ends S0 : E0 ranges from 30 to 2200 in this work (for S0 = 15 μm, it is 300). If, however, we use instead the accessible area of amorphous cellulose, which is about 42 m2·g−1 [33], and a footprint of 24 nm2 for Cel7A [34], we find an S0 : E0 area ratio (total available substrate area divided by monolayer coverage area of the whole enzyme population) which is an order of magnitude smaller (3–240). These latter numbers are rough approximations as the average area of randomly adsorbed enzymes will be larger than the footprint, and only a certain fraction of the enzyme will be adsorbed in the initial stages. Nevertheless, the analysis suggests that not all reducing ends are available in amorphous cellulose, and hence the deficiencies of the model at substrate concentrations below 15 μm could reflect the fact that the premise S0 >> E0 becomes increasingly unrealistic.

The results in Fig. 6 are for the fixed average and distribution of n mentioned above. We also tried wider or narrower distributions with five subsets, distributions with 10 subsets and distributions with a predominance of n values close to the average (e.g. 5%, 20%, 50%, 20%, 5%, instead of equal amounts of the five subsets). The regression analysis with these different interpretations of n polydispersity gave comparable fits and parameters. In addition, average n values of 100 ± 50 were found to account reasonably for the measurements, and we conclude that detailed information on the obstacle-free path n will require a broader experimental material, particularly investigations of different types of substrate.

We consistently found that the experimental C′(t) fell below the model towards the end of the 1-h experiments (see inset in Fig. 6). For a series of 4-h experiments (not shown), this tendency was even more pronounced. This was interpreted as protein inactivation, as discussed in the Theory section. Numeric analysis with respect to Eqn (7) showed that the inclusion of inactivation and the same polydispersity as in Fig. 6 enabled the model to fit the data reasonably over the studied time frame for S0 above approximately 15 μm. Some examples of this for different S0 are shown in Fig. 7.

image

Figure 7.  Experimental data (full lines) and results from the model in Eqn (7) (broken lines) at different substrate concentrations. The concentration of Cel7A was 50 nm. The parameters were k1S0 = 5.2 × 10−3 s−1, k2 = 1 s−1, k3 = k−1 = 1.2 × 10−3 s−1 and k4 = 2 × 10−4 s−1. The obstacle-free path lengths were 40, 70, 100, 130 and 160, respectively, for the five substrate subsets so that the average n was 100. It appears that inclusion of the inactivation rate constant k4 enables the model to account for 1-h trials.

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The parameters from the analysis in Fig. 7 were k1S0 = (5.2 ± 1.6) × 10−3 s−1, k2 = 1 ± 0.3 s−1, k3 =k−1 = (1.2 ± 0.6) × 10−3 s−1 and k4 = (2 ± 0.7) × 10−4 s−1. The parameter dependence of these fits is illustrated in the correlation matrix in Data S1. It appears that k3 and k4 show some interdependence, with an average correlation coefficient of 0.88, whereas other correlation coefficients are low or very low. This result supports the validity of extracting four parameters from the analysis in Fig. 7. The parameters for k1S0, k2 and k3 are essentially equal to those from the simpler analysis in Fig. 6, and the inactivation constant k4 is about an order of magnitude lower than k3. The rates in Fig. 7 were integrated to give the concentration C(t), and two examples are shown in Fig. 8. In this presentation, the accordance between model and experiment appears to be better, and this underscores the fact that the rate function C′(t) provides a more discriminatory parameter for modeling than does the concentration C(t). Figure 8 also shows that the percentage of cellulose converted during the experiment (right-hand ordinate) ranges from a fraction of a percent for the higher to a few percent for the lower S0 values.

image

Figure 8.  Concentration of cellobiose produced by 50 nm Cel7A at 25 °C plotted as a function of time. These results for S0 = 110.9 μm (filled symbols) and 7.5 μm (open symbols) and for the model in Eqn (7) (lines) were obtained by integration of the data in Fig. 7. The broken and chain lines show the conversion in percent of the initial amount of cellulose.

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The qualitative interpretation of Fig. 7 is that Cel7A produces a burst in hydrolysis when enzymes make their initial ‘rush’ down a cellulose strand towards the first encounter with a ‘check block’, and then enters a second phase with a slow, single-exponential decrease in C′(t) as the enzymes gradually become inactivated. In this latter stage, all enzymes have encountered a ‘check block’ and, in this sense, it corresponds to the constant rate regime in Fig. 2. Unlike in Fig. 2, however, C′(t) is not constant, but decreasing, as dictated by the rate constant of the inactivation process k4. In this interpretation, the extent of inactivation scales with enzyme activity (number of catalytic steps) and not with time. Hence, for any enzyme–substrate complex ECni, the probability of experiencing inactivation when it moves one step to the right in Eqn (7) is inline image. For the parameters in Fig. 7, this translates to about one inactivation for every 5000 hydrolytic steps, which is consistent with the frequency of inactivation (1 : 6000) suggested for a cellobiohydrolase working on soluble cello-oligosaccharides [35]. As the final C(t) is about 40 μm in Fig. 8, and we used E0 = 50 nm, each enzyme has performed about 800 hydrolytic steps in these experiments. With a probability of 2 × 10−4, some inactivation can be observed within the experimental time frame used here, and this is further illustrated in Fig. 11. It is also interesting to note that the probability of hydrolysis of an ECni complex (k2) is about 800 times larger than the probability of disassociation (k3), and hence a processivity of that magnitude would be expected for an ideal, ‘obstacle-free’ cellulose strand.

image

Figure 11.  Time-dependent distribution of enzyme between the four states defined in Eqn (7). The values were calculated at different time points using the kinetic parameters listed in Fig. 7. The total enzyme concentration (E0) was 50 nm and S0 was 37.4 μm (hence corresponding to the middle panel of Fig. 7).

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The notion of two partially overlapping phases of the slowdown is interesting in the light of the experimental observations of a ‘double exponential decay’ reported for the rate of cellulolysis [6,36–38]. In these studies, hydrolysis rates for quite different systems were successfully fitted to empirical expressions of the type C′(t) = Aeαt + Beβt. This behavior has been associated with two-phase substrates (high and low reactivity) [37], but, in the current interpretation, it relies on the properties of the enzyme. The first (rapid) time constant α reflects the gradual termination of the burst as the enzymes encounter their first ‘check block’, and the second (slower) constant β represents inactivation and is related to k4 in Eqn (7). As the extent of the first phase will scale with the amount of protein, this interpretation is congruent with the proportional growth of πprocessive with E0 shown in Fig. 5. This enzyme-based interpretation of the double exponential decay predicts that a second injection of enzyme to a reacting sample would generate a second burst (whereas a second burst in C′(t) would not be expected if the slowdown relied on the depletion of good substrate). Figure 9 shows that a second dosage of Cel7A after 1 h indeed gives a second burst, which is similar to the first, and this further supports the current explanation of the double exponential slowdown.

In the last section, we show two examples of how the analysis of the kinetic parameters may elucidate certain aspects of the activity of Cel7A. First, we consider changes in the ratio k1S0/k3. This reflects the ratio of the ‘on rate’ and ‘off rate’. At a fixed k2, a change in this ratio may be interpreted as a change in the affinity of the enzyme for the substrate. Hence, we can assess relationships of this ‘affinity parameter’ and the hydrolysis rate C′(t). The results of such an analysis using S0 = 25 μm and the simple model [Eqn (3)] are illustrated in Fig. 10. The black curve, which is the same in all three panels, represents the cellobiose production rate C′(t), calculated using the parameters from Fig. 3. Figure 10A illustrates the effects of increased ‘affinity’, inasmuch as k1/k3 is enlarged by factors of two, three and five for the red, green and blue curves, respectively. This was performed by both multiplying the original k1 and dividing the original k3 byinline image, inline image and inline image, respectively. It appears that these changes strongly promote the initial burst, but also decrease the rate later in the process (the curves cross over around t = 300 s). This decrease in C′(t) is mainly a consequence of smaller k3 values (‘off rates’), which make the release of enzymes stuck in front of a ‘check block’ the rate-limiting step [the population of inactive ECx in Eqn (3) increases]. Figure 10B shows the results when the k1/k3 ratio is decreased in an analogous fashion. This reduces C′(t) over the whole time course, and this is mainly because the population of unbound (aqueous) enzyme becomes large when k1 (the ‘on rate’) is diminished. The blue curves in Fig. 10B, C also illustrate how a moderate increase in k3 tends to abolish the burst (maximum) in C′(t) altogether. This is because the inhibitory effect of the ‘check block’, as defined by the broken line in Fig. 2, becomes unimportant when the release rate is increased. Multiplying both k1 and k3 by inline image, inline image and inline image, respectively, will obviously not change the ratio (or ‘affinity’), but will speed up both adsorption and desorption, and hence increase the rate of hydrolysis (Fig. 10C).

image

Figure 10.  Parameter dependence of the rate C′(t) calculated from the simple model [Eqn (3)] using S0 = 25 μm. The black curves are identical in the three panels and were calculated from the parameters listed in Fig. 3. The other curves represent C′(t) when the ratio k1/k3 is increased (A) or decreased (B) by a factor of two (red), three (green) or five (blue), respectively. (C) Ratio k1/k3 is constant, but the values of both k1 and k3 are multiplied by inline image, inline image and inline image, respectively.

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For the model in Eqn (7), the enzyme is distributed between four states: aqueous (E), catalytically active (ECni), stuck at ‘check block’ (ECx) or inactivated (ICni). These enzyme concentrations can be numerically derived from the parameters found in Fig. 7. Figure 11 shows an example of such an analysis for E0 = 50 nm and S0 = 37.4 μm (i.e. corresponding to the middle panel in Fig. 6). It appears that the concentration of free enzyme (E) decreases for about 10 min and then reaches a near-constant (slowly decreasing) level which is about 20% of E0. This calculated course of E(t) is in line with earlier experimental results on different types of substrate [39–43]. In addition, an 80% reduction in free enzyme after about 10 min matches our own adsorption measurements for a mixture of T. reseei cellulases on amorphous cellulose (L. Murphy, unpublished data). The population of catalytically active enzyme is highest (and about 25% of E0) after a few minutes, but decreases at later stages, as a growing fraction of the enzyme becomes stuck in front of a ‘check block’. After about 12 min, this population is well over half of E0 and this transition from active ECni to stuck ECx is the origin of the burst in cellobiose production. As the inactivation of enzyme in Eqn (7) is modeled as an irreversible transition, the concentration of this species grows monotonically. This behavior also appears from Fig. 11, but further analysis of ICni is postponed until calorimetric trials over extended time frames (and hence more precise values of k4) become available.

In summary, we have proposed an explicit model that describes the initial burst and subsequent slowdown in the rate of cellobiose production for processive enzymes such as Cel7A. The focus is on the initial phase of the process, where inhibition from accumulated product and/or the depletion of good attack points on the substrate are of minor importance. We found that a burst and slowdown may indeed occur as a consequence of obstacles to processive movement, on the one hand, and the relative size of rate constants for adsorption, processive hydrolysis and desorption, on the other. This interpretation is analogous to that conventionally used for the description of burst phases in systems with soluble substrates and nonprocessive enzymes. The theory was tested against calorimetric measurements of the hydrolysis of amorphous cellulose by T. reesei Cel7A. No other enzymes or substrates were investigated, and the conclusions thus only pertain directly to this system. We note, however, that, if the origin of the slowdown is linked to low dissociation rates (low k3), as suggested here, an analogous burst behavior should be expected on other substrates, and it appears relevant to conduct such measurements. We found that some experimental hallmarks were reproduced in a simple burst model, where the only cause of the slowdown was a protracted release of enzyme that had reached the obstacle on the cellulose chain. However, to account more precisely for the experimental data, it was necessary to consider enzyme inactivation as well as some heterogeneity in the obstacle-free path length. We implemented the former as an irreversible inactivation step that competed with the production of cellobiose in each hydrolytic cycle. The result was a more complex model which could explain the ‘double exponential decay’ in the rate of cellobiose production which has been reported in several earlier studies. Thus, in this interpretation, the fast component in the double exponential decay reflects the first sweep of each cellulase down a cellulose strand, whereas the slow component is ascribed to random inactivation which is unrelated to the stage of the process. It has recently been stated that ‘processivity is more about disassociation than about the rate of hydrolysis’ [44], and a pronounced improvement in activity has indeed been observed in an enzyme variant with diminished processivity [45]. We suggest that the models presented here may be useful in attempts to elucidate and rationalize such interrelationships of activity and processivity.

Materials and methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Results and Discussion
  5. Materials and methods
  6. Acknowledgements
  7. References
  8. Supporting Information

All mathematical analysis and numerical fitting were performed using the software package Mathematica 7.0 (Wolfram Research, Inc. Champaign, IL, USA).

The substrate in the calorimetric measurements was reconstituted amorphous cellulose (RAC) prepared essentially as described by Zhang et al. [46] Briefly, 0.4 g cellulose (Sigmacell 20) was suspended in 0.6 mL MilliQ-water and placed on ice before adding 8 mL cold 85% phosphoric acid with vigorous stirring. After a few minutes, an additional 2-mL aliquot of phosphoric acid was added. This mixture was incubated for 40 min on ice with continuous stirring. Then, 40 mL of MilliQ-water was slowly added with vigorous stirring. The suspension was transferred to a 50-mL centrifuge tube and centrifuged at 2500 g for 15 min. The cellulose was washed in water and spun down three times, and then resuspended in 50 mL of 0.05 m Na2CO3 to neutralize traces of acid. The carbonate was removed by four washes in water and four in buffer (50 mm sodium acetate, pH 5.00 + 2 mm CaCl2), and the final product was then suspended in 50 mL of acetate buffer. RAC was blended for 5 min in an coaxial mixer.

The number of reducing ends (i.e. attack points for Cel7A) in the produced RAC was determined by the BCA method [47]. The BCA stock reagents A (1.942 g·L−1 disodium-2,2′-bicinchoniate + 54.28 g·L−1 Na2CO3 + 24.2 g·L−1 NaHCO3) and B (1.248 g·L−1 CuSO4.5H2O + 1.262 g·L−1 l-serine) were mixed 1 : 1. RAC was diluted 20 times before mixing 0.75 mL RAC and 0.75 mL BCA (working solution) in a 2-mL Eppendorf tube. After 30 min at 75 °C in a thermomixer, the cellulose was centrifuged down at 9000 g for 5 min, and the absorbance at 560 nm was measured (Shimadzu UV1700, Kyoto, Japan) and quantified against a 0–50-μm cellobiose standard curve.

Trichoderma reesei Cel7A was purified by column chromatography. Desalted concentrated culture broth from a T. reesei strain with deletion of the Cel7A gene was applied in 20 mm Tris, pH 8.5, to a Q-Sepharose Fast Flow column (GE Healthcare Lifesciences, Little Chalfont, UK) and eluted in the same buffer with a gradient to 1 m NaCl. Fractions containing purified Cel7A were identified by SDS/PAGE and pooled. The fraction with Cel7A was mixed with ammonium sulfate to 1 m, and applied to Phenyl Sepharose (GE Healthcare Lifesciences), and eluted in a gradient from 1 to 0 m ammonium sulfate in 20 mm Tris, pH 7.5. Fractions containing purified Cel7A were identified by SDS/PAGE, pooled, concentrated and buffer exchanged to 20 mm Tris, ∼150 mm NaCl, pH 7.5.

The enzymatic activity was measured by the calorimetric method recently described in detail by Murphy et al. [48]. RAC at different concentrations was loaded into the cell of the isothermal titration calorimeter (VP-ITC, Microcal, Piscataway, NJ, USA) at 25 °C and titrated with Cel7A from the syringe. All samples were dissolved in 50 mm sodium acetate with 2 mm calcium chloride, pH 5.00. In addition to the substrate, the calorimetric cell also contained 0.3 mg·mL−1β-glucosidase, 25 GODU·mL−1 glucose oxidase and 25 CIU·mL−1 catalase [48]. As a result, the cellobiose produced by the hydrolysis of RAC is first cleaved into two glucose molecules, and then oxidized to two d-glucono-δ-lactone molecules. This strongly amplifies the heat signal and hence allows measurements at low enzyme dosages such as those used here. The advantages and limitations of the coupled calorimetric assay are discussed elsewhere [48]. The raw result from the calorimetric measurements is the heat flow in J·s−1 (W), and this is readily converted to the rate of cellobiose production (in M·s−1) by division with the molar enthalpy change of the coupled reaction (−360 kJ·mol−1) [48] and the volume of the calorimetric cell (1.42 mL). The response time of the calorimeter is about 15 s and no correction for this was introduced in the analysis.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Results and Discussion
  5. Materials and methods
  6. Acknowledgements
  7. References
  8. Supporting Information

This work was supported by The Danish Council for Strategic Research (grants 09-063210 and 2104-07-0028). Expert experimental assistance from David Osborne and Erik L. Rasmussen is gratefully acknowledged.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Results and Discussion
  5. Materials and methods
  6. Acknowledgements
  7. References
  8. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Results and Discussion
  5. Materials and methods
  6. Acknowledgements
  7. References
  8. Supporting Information

Data S1. Derivations of the expressions for P1(t), P2(t), P1′(t) and P2′(t) used in Fig. 1 and derivations of Eqns (2) and (4)–(6).

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