Modeling the activity burst in the initial phase of cellulose hydrolysis by the processive cellobiohydrolase Cel7A

Abstract The hydrolysis of cellulose by processive cellulases, such as exocellulase TrCel7A from Trichoderma reesei, is typically characterized by an initial burst of high activity followed by a slowdown, often leading to incomplete hydrolysis of the substrate. The origins of these limitations to cellulose hydrolysis are not yet fully understood. Here, we propose a new model for the initial phase of cellulose hydrolysis by processive cellulases, incorporating a bound but inactive enzyme state. The model, based on ordinary differential equations, accurately reproduces the activity burst and the subsequent slowdown of the cellulose hydrolysis and describes the experimental data equally well or better than the previously suggested model. We also derive steady‐state expressions that can be used to describe the pseudo‐steady state reached after the initial activity burst. Importantly, we show that the new model predicts the existence of an optimal enzyme‐substrate affinity at which the pseudo‐steady state hydrolysis rate is maximized. The model further allows the calculation of glucose production rate from the first cut in the processive run and reproduces the second activity burst commonly observed upon new enzyme addition. These results are expected to be applicable also to other processive enzymes.

unbound state are cyclic arrangements between the states: E → EC → (EC) b → E or E → (EC) b → EC → E. These configurations are rather restrictive, and when used to fit the experimental data did not lead to good data fits. It is therefore worth considering models with four rates.
There are in total 15 different combinations of four non-zero rates, six of which can be directly excluded because they do not allow either attachment to or detachment from the substrate or access to one of the two bound states, or they lead to the accumulation of all enzyme in one state. The remaining nine four-rate models can be categorized into three classes (A, B, C), each containing 3 models, depending on the state in which the enzyme is upon binding to the substrate (Fig. S1).
The enzyme can bind to be immediately active (EC), with the blocked state (EC) b entered later (class A), or the enzyme can bind to become either active or blocked (class B), or the enzyme binds to be in the blocked state (EC) b (which could be interpreted as, for example, a state searching for the active site, that is, the cellulose chain end), only to enter the active (hydrolyzing) state later (class C).
A detailed analysis of the four-rate models in classes A, B and C shows that the models in class C cannot produce an activity maximum similar to the one observed experimentally. All six models in classes A and B can produce a clear activity burst for some range of kinetic parameters.
The quality of fits to the experimental data varies among the models; the choice of a 'favourable' model is however complicated by the equivalence of some models, as described in more detail in the following section.

Model equivalence
A closer inspection of the four-rate models reveals that the solutions of some of these are mathematically equivalent to each other, with respect to the type of data used in this article. By equivalence we mean here that one model can be converted into another by transforming the set of rates k i of one model to the set of rates k i of the other model while preserving Eq. 2, which defines the problem and its solution. As a result, the solutions of the two models, y i (t) and y i (t) are equal, up to a scaling factor. A direct consequence of this fact is that one cannot distinguish between these models when describing the experimental data. An exception to this indistinguishability is the situation when one of the two equivalent models can be excluded because it contains non-physical parameter values, for example, negative rate constants.
A search for parameter transformations among the nine models in Fig. S1 preserving the form of Eq. 2 shows that there are three groups of equivalent models. These are described in more detail below. In all cases we set y T = 1 for simplicity and without a loss of generality; the concentrations y 1 and y 2 have then the meaning of fractions of enzyme molecules in the two states. The concentrations of the free species y 1 (t) are equal in both models: y 1 (t) = y 1 (t) because of the initial condition y 1 (0) = y 1 (0) = 1 (all enzyme molecules are free at the beginning of the experiment). The concentrations of the bound active species EC can be related by a scaling factor β: y 2 (t) = βy 2 (t). This scaling degree of freedom is present because experimentally we do not obtain the absolute concentration y 2 (t) but only the rate of cellobiose formation, which is proportional to y 2 (t). In order to eliminate this model equivalence, more experimental data would be necessary. The three groups of equivalent models are: (Ab, Ba), (Aa, Bb, Bc) and (Ca, Cb).
In the following, several examples of transformations between equivalent models are given.

Ab -Ba
The model Ab can be converted into the model Ba by the following transformation (the parameters of Ba are marked with a prime): This means that if y 1 (t) and y 2 (t) are the solutions of the model Ab (Eq. 2) with the parameters k 1 , k 2 , k 3 , k 4 , then y 1 (t) = y 1 (t) and y 2 (t) = βy 2 (t) are the solutions of the model Ba with the parameters k 1 , k 2 , k 4 , k 6 . As long as both sets of parameters are physically meaningful, we cannot distinguish between the two models based on the unscaled y 2 (t) data alone (the experimentally determined time evolution of the rate of cellobiose production). Similar reasoning applies to the following model groups.

Aa -Bb -Bc
The model Aa can be converted into the model Bb by the following transformation (the parameters of Bb are marked with a prime): The transformation is physically meaningful only if k 2 > k 5 because β must be positive. The model Aa can be converted into the model Bc by the following transformation (the parameters of Bc are marked with a prime): Again, the transformation is physically meaningful only if k 2 < k 5 because β must be positive.
This condition is complementary to the condition for the transformation Aa → Bb shown above; the model Aa is therefore meaningfully equivalent to either Bb or Bc but not to both models at the same time.
The model Bb can be converted into the model Bc by the following transformation (the parameters of Bc are marked with a prime): The transformation is physically meaningful only if k 2 < k 5 because β must be positive.

Ca -Cb
The model Ca can be converted into the model Cb by the following transformation (the parameters of Cb are marked with a prime): The transformation is physically meaningful only if k 2 < k 5 because β must be positive.
Ac Bc Cc

Processivity
We use the definition of processivity as the average number n of hydrolysis steps during one processive run, that is, while the enzyme is in the active bound state EC (intrinsic processivity): where p i is the probability, that i units will be hydrolyzed during one processive run. The probability p i depends on the duration t of the processive run and can be calculated by integrating over all possible durations of the processive run: where p(i, t) is the probability of i hydrolysis steps during a processive run of length t, and p T (t) is the probability density that the processive run has a duration t. The probability p(i, t) is a Poisson distribution of i with the mean value k 7 t, where k 7 is the rate constant of hydrolysis: The probability p T (t)dt of leaving the active bound state EC after time t can be expressed as the product of the probability that the enzyme does not leave the EC state during the time interval (0, t): e −k2t and the probability that it leaves the state EC during the interval (t, t + dt): k 2 dt, where k 2 is the rate constant of leaving the EC state: Combination of the above equations yields: In the models Aa and Ab, where the active bound state EC can be left either by detachment (rate constant k 2 ) or by entering the blocked state (rate constant k 3 ), k 2 in eq. S14 is replaced by the sum of these two rate constants k 2 + k 3 , resulting in the processivity n = k 7 /(k 2 + k 3 ).
Using the above definition, the processivity in the published model (Praestgaard et al., 2011) would be k 2 /k 3 (meaning k cat /k off ) if the chain length were not restricted to n = 150 units. With this restriction, the processivity is n = k 2 /k 3 (1 − (k 2 /(k 2 + k 3 )) n ).

Experimental details
The data fitted in Fig. 5 and S2 were taken from ref. (Praestgaard et al., 2011). The enzyme concentration in all measurements was 50 nM. The substrate was reconstituted amorphous cellulose (RAC) prepared from cellulose Sigmacell 20. The substrate concentrations, expressed as the concentration of reducing ends, ranged from 1.5 µM to 110.9 µM. The hydrolytic activity of the enzyme was measured by a calorimetric method; the cellobiose produced by hydrolysis of cellulose was further converted to glucose and D-glucono-δ-lactone by added β-glucosidase and glucose oxidase, thus siginificantly amplifying the heat signal. The measured heat flow (in J·s −1 ) was converted to the rate of cellobiose production. The response time of the calorimeter was approximately 15 s. The measurements were performed at 25 • C. For more details see the original publication (Praestgaard et al., 2011).
The data fitted in Fig. 6 were taken from ref.