Rate of Threading a Cellulose Chain into the Binding Tunnel of a

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Rate of Threading a Cellulose Chain Into the Binding Tunnel of a Cellulase Nicolaj Cruys-Bagger, Kadri Alasepp, Morten Andersen, Johnny T. Ottesen, Kim Borch, and Peter Westh J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b01877 • Publication Date (Web): 01 Jun 2016 Downloaded from http://pubs.acs.org on June 6, 2016

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The Journal of Physical Chemistry

Rate of Threading a Cellulose Chain Into the Binding Tunnel of a Cellulase

Nicolaj Cruys-Bagger,1,2 Kadri Alasepp,1 Morten Andersen,1 Johnny Ottesen,1 Kim Borch2 and Peter Westh,1,3 1: Dept. of Science and Environment, Roskilde University. 1 Universitetsvej, DK-4000, Roskilde Denmark 2: Novozymes A/S, Krogshøjvej 36, DK-2880, Denmark. 3: Corresponding Author. [email protected], tel +45 4674 2879.

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Abstract. The industrially important cellulase, Cel7A, hydrolyzes crystalline cellulose by a complex processive mechanism in which the enzyme slides along the cellulose surface with one strand of the polymeric substrate channeled through its catalytic tunnel. Each processive run must start by threading the tunnel with a cellulose strand and end by the opposite, de-threading process. Different evidence has suggested that threading or de-threading may be rate limiting for the overall enzyme reaction. To directly elucidate the rates of threading and de-threading, we analyzed experimental data with respect to a two-step model that distinguishes enzyme in respectively free-, associated nonthreaded- and threaded states. This approach enabled estimation of rate constants for both steps in both directions. The results showed that Cel7A utilizes a “tapping” mode of attack, where it associates unproductively with the cellulose surface many times before it eventually finds a location, where it gets threaded. Moreover, it was concluded that at quasi steady state, de-threading was the main determinant of the overall hydrolytic rate under most conditions. An exception from this was at very low enzyme/substrate ratio, where other steps also influenced the overall dynamics. These results will be helpful in identifying rate limiting steps for cellulases and in turn targets for rational design of faster enzymes.


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Introduction Lignocellulosic biomass is the most abundant renewable biological resource and this makes it a promising feedstock for large-scale production of sustainable alternatives to fossil fuels and some petrochemical materials. One key challenge in this emerging industry is the enzymatic conversion of polysachharides in the biomass (mainly cellulose) to fermentable sugars. This so-called saccharification is a highly complex process, which requires a mixture of different enzymes to work together in a synergistic way.1 The most abundant type of cellulolytic enzymes both in the secretome

of

cellulose

degrading

fungi

and

commercial

saccharification

cocktails

is

cellobiohydrolases (CBHs). This group of enzymes utilizes an intriguing and highly complex catalytic mechanism, in which multiple hydrolytic events are completed as the enzyme moves along the insoluble substrate with a single cellulose strand threaded through the binding tunnel. This is known as processive hydrolysis, and on a structural level it relies at least in part on binding of multiple glucopyranose moieties to subsites in the long binding tunnel that essentially spans the whole enzyme. In this way, the enzyme and substrate remain associated after product (typically cellobiose) has been released and the hydrolytic cycle can be repeated without dissociation. As for other enzymes, mechanistic and kinetic descriptions of CBH-activity are usually based on a reaction scheme that identifies a number of intermediate species. If these intermediates are appropriately defined, their interconversion may be treated as elementary reactions, and hence used for kinetic modeling of the enzymatic process. Several insightful attempts have been made to define key enzyme-substrate complexes that occur along the processive reaction path,2-4 but in most cases it has proven difficult (if not impossible) to study these intermediates in experiments. In the current work we have tried to do so by zooming in on one part of the enzymatic process, namely the formation of the enzyme-substrate complex (henceforth called the Michaelis complex). Specifically, we have studied association of the CBH, Cel7A from Trichoderma reesei and bacterial cellulose. This association process is generally described by a mechanism where at least two steps separate the free, aqueous enzyme from the Michaelis complex, and this interpretation has its roots in the two3 ACS Paragon Plus Environment


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domain architecture of the enzyme. Thus, the catalytic domain (CD) of T. reesei Cel7A is connected to a carbohydrate binding domain (CBM) through a flexible linker peptide, and this has led to the putative association mechanism illustrated in Fig. 1.

Figure 1. A suggested two-step picture for the formation of the catalytically active Michaelis complex of Cel7A. The scheme distinguishes enzyme in the aqueous phase (E), enzyme-substrate complex with unoccupied active site (ESassoc) and the Michaelis complex with filled (“threaded”) active site (ESthread).

In Fig. 1, Cel7A first adsorbs through the CBM to the surface of the cellulose particle. We call this state, where the binding tunnel in the CD remains unoccupied, associated enzyme. Subsequently, a part of the cellulose strand re-locates from the cellulose particle to the binding tunnel and hence make the Michaelis complex. We will call this threaded enzyme. We note in passing that the term threading of Cel7A has sometimes been used for the initial transfer of the very end of the strand into the mouth (first subsite) of the binding tunnel, but we will use the term less specifically for filling the whole tunnel as illustrated in Fig. 1. The main focus of this work is to assess the rate of the threading (and de-threading) step in Fig. 1. The kinetics of this step is of particular interest because several recent studies have suggested that threading may be slow and thus limit the overall rate of hydrolysis of crystalline cellulose.5-7 It appears intuitive that this step could be time-consuming as it involves the transfer of about ten glucopyranose rings from a rather stable condition in the cellulose crystal8 to the binding tunnel of the enzyme, but the kinetics of threading cellulases remains poorly elucidated. In contrast to these studies, which suggested on-rate control of cellulase kinetics, other works have concluded that the off-rate was rate limiting.9-10 Still other investigations have found that the inner catalytic cycle may be rate limiting under some conditions.2 In the light of this controversy,


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it appears that attempts to quantify the kinetics of threading and de-threading of cellulases could provide important input for both a general understanding of CBH mechanism and identification of rate limiting steps and thus targets for engineering of CBHs with better industrial performance. Recently, experimental methods that distinguish the three enzyme states in Fig. 1 have been much improved. The common principle in these techniques is to measure the hydrolysis of a soluble reporter substrate11-12 in the absence and presence of cellulose13-15. The basic underlying idea is that both the free- and associated enzyme populations are active against the soluble substrate, while the threaded population is not. This means that the activity against the soluble substrate (in samples with both cellulose and soluble substrate) provides a measure of the sum of free- and associated populations. If this information is combined with conventional adsorption measurements, which distinguishes free enzyme from the sum of the two adsorbed states in Fig. 1, the concentration of all three enzymes states can be quantified.13-15 Using this approach, it has previously been reported that at loads of 10-100 mg enzyme/g cellulose Cel7A is predominantly in the threaded form whereas more dilute systems have shown comparable populations of threaded and associated Cel7A.2, 13, 16 Other related enzymes including the processive endoclucanase Cel9A17 as well as a processive chitinase, ChiA 16 were predominantly in the associated (non-threaded) form. In the current work we report the temporal development of the concentration of these states and show how simple kinetic models applied to such experimental data may be used to elucidate the dynamics of adsorption, threading and de-threading.

Experimental methods Materials. Unless otherwise stated, all chemicals were of HPLC grade (> 99% purity) and purchased from Sigma-Aldrich (St. Louis, MO, USA). All solutions were prepared with 50 mM sodium acetate buffer adjusted to pH 5.0. The retaining cellobiohydrolase, Cel7A, from T. reesei and the βglucosidase (BG), Cel3A, from Aspergillus fumigatus were cloned, expressed and purified as described elsewhere.18-19 The expression host was Aspergillus oryzae for both enzymes. Enzyme 5 ACS Paragon Plus Environment


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concentrations were determined from absorbance measurements at 280 nm using an extinction coefficients calculated from the primary structure.20 Bacterial cellulose (BC) from Acetobacter xylinum was prepared from commercially available Nata de Coco (coconut gel in syrup, Monika®, Fitrite Incorporated, Novaliches Quezon City, Philippines) using a previously described protocol.21 Measurement of free active sites. The concentration of free active sites (i.e. the sum of the free and associated enzyme populations as defined in Fig. 1) was measured as described by Jalak and Valjämäe.13 The main principle is to measure the activity of Cel7A against a soluble “reporter” substrate in the presence of different loads of insoluble cellulose. In this work the reporter substrate was 4-methylumbelliferyl-β–D-lactobioside (MUL), which is cleaved by Cel7A to produce lactose and the fluorescent product 4-methylumbelliferone. First, a standard curve for the conversion of MUL as a function of the Cel7A concentration is made without any cellulose. Subsequently, the reduction in MUL conversion induced by the addition of cellulose is quantified. This activity loss (i.e. inhibition of MUL hydrolysis) brought about by cellulose is interpreted as a measure of threaded enzyme, simply because in contrast to free and associated enzyme, the threaded species cannot accommodate a MUL molecule in the active site. More details regarding this principle and its implementation can be found in the aforementioned work by Jalak and Valjämäe. While conceptually simple, the approach is challenged by some practical issues. In particular, Cel7A’s activity on the reporter substrate is inhibited by both cellobiose released from the hydrolysis of cellulose and lactose produced by the hydrolysis of MUL. The former problem, inhibition by cellobiose, can be solved by adding BG, as this enzyme quickly converts cellobiose to glucose, which has no inhibitory effects in the concentration range used here.22-23 Hence, 1 µM BG was added to the sample in all free active site measurements. As shown in Fig. S-2 of the Supporting Information, this concentration of BG was sufficient to keep the concentration of cellobiose below 0.5 µM, which is two or three orders of magnitude below reported inhibition constants for Cel7A.24-25 Hence, it is safe to neglect inhibition by cellobiose. The use of BG, however, was not unproblematic as it has a small but measurable activity against MUL. All in all, this meant that corrections for both inhibition of Cel7A by lactose and BG activity against MUL 6 ACS Paragon Plus Environment

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must be implemented. Both of these effects, however, can be assessed quite precisely and procedures of how to correct for this are described in the Supporting Information (Figs S-4 to S-6). Kinetic measurements. All experiments were conducted at 25°C. A 100 ml suspension of BC containing 10 µM MUL was stirred at 500 rpm in a jacketed beaker. The adsorption process was started by the addition of a mixture Cel7A and BG and 100µl subsets of the sample were subsequently retrieved at defined time points and diluted ten times with 0.1M NaOH to stop the reaction. Released MU in the subsets was measured on a Shimadzu spectrofluorophotometer (λex/λem = 360/450 nm) and the amount of Cel7A with a free active site was quantified from a standard curve generated from the incubation of 10 μM MUL and 0.5 μM Cel7A (no cellulose). Free enzyme was quantified in similar measurements without BG and MUL. In these latter runs, 1 ml aliquots of the reaction mixture were withdrawn with a syringe at defined times and quickly filtered through a Q-max PES low-protein binding filter (Frisenette ApS, Knebel, Denmark). The intrinsic enzyme

fluorescence

(λex/λem =

280/345

nm)

was

determined

with

a

Shimadzu

Spectrofluorophotometer and converted to the concentration of free enzyme on the basis of a separate standard curve. Separate control experiments confirmed that loss of Cel7A resulting from the filtration was negligible (see Fig. S-8 in Supporting Information). The overall time resolution of this (batch) method was about 5 s, and this put some limitations on the systems that could be kinetically characterized. In particular, we were restricted to quite low concentrations of enzyme and substrate as adsorption in more concentrated systems was so fast that the concentration of free enzyme, [E], nearly fell to its equilibrium value within the dead time of the method. Binding isotherm. The equilibrium adsorption of Cel7A on BC was measured as described previously.26 These experiments used a BC load of 0.25 g/l, which was the same as the kinetic measurements. Cellulose conversion. Soluble sugars were quantified by high-performance anion-exchange chromatography with pulsed amperometric detection (HPAEC-PAD) on an ICS-5000 ion 7 ACS Paragon Plus Environment


chromatograph as described elsewhere.27 The concentrations of CBHI, BG and BC substrate were the same as in the free active site measurements.

Results and modeling. Preliminary measurements showed that an enzyme concentration of 0.50µM and a substrate load of 0.25 g BC/l were optimal for time-resolved measurements (see Fig. S-1 in Supporting Information). Representative data for these loads, showing the temporal development of free-, associated- and threaded enzyme states are illustrated in Fig. 2. The concentration of free enzyme, [E], was measured directly in the retrieved, filtered subsets, while concentrations of associated-, [ESassoc], and threaded, [ESthread], enzyme were calculated from the MUL measurements as shown in the Supporting Information.

0,5

Free enzyme, E Threaded enz., ESthread

Concentration (µM)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Associated enz., ESassoc

0,4

0,3

0,2

0,1

0,0 0

200

400

600

800

Time (s)

Figure 2. Experimental data for the temporal development of the three enzyme species, free, associated and threaded defined in Fig. 1. The initial conditions were 0.50µM Cel7A and 0.25 g BC/l.

The equilibrium adsorption of Cel7A on 0.25 g/l BC was measured in separate experiments. We first calculated enzyme coverage of BC (Γ, in µmol Cel7A /g cellulose) from the measured free enzyme concentrations [E] as


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Γ=

E0 − [ E ]

(1)

S0

where E0 and S0 are respectively the (known) total concentration of enzyme and the load of substrate (in g/l). In Fig. 3 the coverage was plotted a function of the concentration of free enzyme, [E]. As in earlier works on Cel7A adsorption,28-29 we found that a Langmuir isotherm based on two sets of independent sites accounted satisfactory for the experimental data, while a simple Langmuir isotherm did not fit the results. Hence, the line if Fig. 3 is the best fit to the experimental data of the expression

Γ = 1 Γ max

1

[E] + 2Γ max K + [E]

2

d

[E ] K + [E]

(2)

d

In eq. (2), Γmax and Kd are the saturation coverage and the dissociation constant for each set of sites respectively. The maximum likelihood parameters for the fit in Fig. 3 were 1Γmax= 0.39 ±0.08 µmol/g and 1Kd=0.008 ±0.003µM for the strong set of sites. The analogous values for the set of weaker adsorption sites were 2Γmax= 1.83 ±0.09 µmol/g and 2Kd=0.31 ±0.07 µM, and these results compare well to recent adsorption data for the same system.29 In the current context, the most important result in Fig. 3 is the total density of binding sites, Γmax + Γmax = 2.22±0.17 µmol/g BC. 1

2



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Figure 3. Equilibrium adsorption data and best fit eq. (2).

To analyze the kinetic data in Fig. 2, we tested two microkinetic reaction schemes for the formation of the Michaelis complex of Cel7A. Model 1: The simplest mechanism that describes the reaction in Fig. 1 is a sequence of two reactions. The first is the combination of free enzyme and substrate to form associated enzyme with unoccupied active site (ESassoc) and the second is the transformation of associated enzyme to threaded enzyme (ESthread) as a piece of cellulose strand is translocated into the binding tunnel. This can be expressed in the following reaction scheme. ka kb →  → ESthread E + S ← ESassoc ← k− a

k− b

Scheme 1 The results in Fig. 2 show that the process approaches (quasi) equilibrium, and we therefore define reactions in both directions in Scheme (1). While conceptually straight forward, the kinetic description of Scheme 1 is somewhat obfuscated by different concentration units for respectively enzyme (molar) and substrate (g/l). To circumvent complications related to this, we converted the 10 ACS Paragon Plus Environment

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mass concentration of substrate to a molar concentration of binding sites on the basis of the binding isotherm in Fig. 3. Here, we found a total capacity of 2.22±0.17 µmol/g BC, and it follows that the molar concentration of sites in 0.25 g BC/l is 0.55µM. This idea of converting substrate load to molar units has been commonly used in cellulase research (see 30-31 for reviews). We will adhere to this approach in the following and use the conventional square brackets to indicate molar concentrations. There are four chemical species in Scheme (1) and hence, we may write four rate equations of the usual type

[ E ] ' ( t ) = −ka [ E ] ( t ) [ S ] ( t ) + k− a [ ESassoc ] ( t ) [ S ] ' ( t ) = −ka [ E ] ( t ) [ S ] ( t ) + k− a [ ESassoc ] ( t ) [ ESassoc ] ' ( t ) = ka [ E ] ( t ) [ S ] ( t ) − ( k− a + kb ) [ ESassoc ] ( t ) + k−b [ ESthread ] ( t ) [ ESthread ] ' ( t ) = kb [ ESassoc ] ( t ) − k−b [ ESthread ] ( t )

(3)

In eq. (3) the prime sign identifies the derivative with respect to time. The initial concentrations of enzyme, [E]0, and substrate, [S]0, before any complex is formed are both known. There are two conserved quantities in the system of differential equations which is seen by noticing

[ E ] ' ( t ) + [ ESassoc ] ' ( t ) + [ ESthread ] ' ( t ) = 0 and [ S ] ' ( t ) + [ ESassoc ] ' ( t ) + [ ESthread ] ' ( t ) = 0 . Following integration and insertion of the initial conditions, this may be expressed

[ E ]0 = [ E ] ( t ) + [ ESassoc ] ( t ) + [ ESthread ] ( t ) [ S ]0 = [ S ] ( t ) + [ ESassoc ] ( t ) + [ ESthread ] ( t )

(4)

Combining Eqs. (3) and (4) shows that the temporal development of all concentrations, can be described by two ordinary differential equations (ODEs) and two conservation equations. If we choose ODEs for E’(t) and ES’thread(t) we may write



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[ E ] ' ( t ) = −ka [ E ] ( t ) [ S ] ( t ) + k− a [ ESassoc ] ( t ) [ ESthread ] ' ( t ) = kb [ ESassoc ] ( t ) − k−b [ ESthread ] ( t ) [ ESassoc ] ( t ) = [ E ]0 − [ E ] ( t ) − [ ESthread ] ( t ) [ S ] ( t ) = [ S ]0 − [ ESassoc ] ( t ) − [ ESthread ] ( t )

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(5)

Hence, the free variables in the model, [ E ] ( t ) and [ ESthread ] ( t ) , match the independent variables in the experiments, and this is suitable in order to get to a unique best fit between model and experiment. The initial (molar) concentrations in the experiment in Fig. 2 were [E]0=0.50µM, [S]0=0.55µM and [ESassoc]0=[ESassoc]0=0. We used these conditions for t=0 and performed numeric analysis of the data in Fig. 2 with respect to the kinetic description in eq. (5). For concrete values of rate constants, the time evolution of all species can be computed rapidly using integrated solvers of Matlab (MathWorks, Natick, MA). For each experimental data point we picked the corresponding data point in the model and squared the distance between model and experiment. The squared distance of all data points was then added to give a total error of the model compared to the data. About half a million different parameter combinations were tested and the one with the smallest total error between model and experiment was selected. This procedure was then iterated with a gradually finer grid around optimal parameter values, and this approach had the advantage of rarely getting trapped in local minima. The experimental data suggested that the different states of enzyme had equilibrated after some 400 s (see Fig 2 above and Fig. 1-S in the Supporting Information). Hence, the average concentration of each species from 600 to 900 s was used as the equilibrium value (i.e. concentrations representing equilibration of both reactions in Scheme 1). The ratios of these equilibrium concentrations for each step in Fig. 1 are equal to the ratios of the corresponding rate constants (both ratios are equal to the equilibrium constant), and insertion of this in eq. (5) reduces the number of free parameters from four to two (we used k-a and k-b). This resulted in a significant reduction of computational requirements and also enabled us to make contour plots to illustrate the total error between model and experiment. Contour plots for the


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analysis with respect to Model 1 are shown in Fig. 4B, and clearly illustrate that there is only one well defined minimum. This means that there is a unique best set of parameters, and the area defined by the contour-lines further reveal the sensitivity of the parameters. In particular they identify what parameter combinations would be possible if we allowed a certain level of total error. These combinations are all the parameter sets contained in the closed contour marked by the chosen error level (see legend of Fig. 4).

Results of the numeric analysis based on Model 1 are shown in Fig. 4A, and it appears that the model accounts well for the experimental data. Numerical values of the four rate constants from Scheme 1 corresponding to the best fit are listed in Tab. 1 along with the associated equilibrium constants for each step calculated at the ratios of the forward- and backwards rate constants.

A

0,5

Free enzyme, E

0,4

B

0,08

Threaded enz. ESthread Associated enz., ESassoc

0,06

-1

k-b (s )

Concentration (µM)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0,3

0,04

0,2

0,02

0,1

0,0 0

200

400

600

800

0,00 0,00

0,05

0,10

0,15

0,20

0,25

k-a (s-1)

Time (s)

Figure 4. Results from the numeric regression based on Model 1. Panel A shows experimental results (symbols) and the deconvolution of these data with respect to eq. (5). Panel B is a contour plot showing the parameter space of k-a and k-b for error levels of respectively 1%, 2%, 4%, 10%, 20% and 50%. The cross represents the best set of parameters (Tab. 1). Small separation of contour lines around the cross signifies a well-defined minimum and hence a unique best set of parameters.

Model 2. In Model (1), the threading process (i.e. the forward reaction of the second step in Fig. 1) is described as a first order (isomerization type) reaction. It could be argued, however, that this is overly simplified description because threading relies on the encounter of two species; associated enzyme and an attack point on the cellulose surface. As such, it may be better represented by a 13 ACS Paragon Plus Environment


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second order process. To capture this in a reaction scheme, we partitioned the total population of surface sites into two groups. These were “association sites”, where the enzyme can adsorb, but a reducing end is not readily available and “attack sites”, where the enzymes can become threaded. If we denote (unoccupied) association- and attack sites respectively Sα and Sβ, the microkinetic scheme for model 2 may be written ka → E + Sα ← ESassoc k− a

kb → ESassoc + S β ← ESthread + Sα k− b

Scheme 2. The distinction of association- and attack sites for Cel7A was recently discussed extensively by Jalak and Väljamäe.29 These workers reported a total binding capacity of 2.57±0.27µmol/g for the BC/Cel7A system, in excellent agreement with the results in Fig. 3, and further succeeded in partitioning this total population into association sites and attack sites. Application of the Sα/Sβ-ratio reported by Jalak and Valjamae gave [Sα]=0.31µM and [Sβ]=0.24µM for the 0.25 g/l BC suspension used in Fig. 2. Returning to Scheme 2, there are five chemical species and hence five ODEs. In addition we may write conservation equations for enzyme, attack sites and association sites. As a result, the kinetics of Scheme (2) may be described by two ODEs and three conservation equations (see Supporting Information for a detailed derivation)

[ E ] ' ( t ) = −ka [ E ] ( t ) [ Sα ] ( t ) + k− a [ ESassoc ] ( t ) [ ESthread ] ' ( t ) = kb [ ESassoc ] ( t )  Sβ  ( t ) − k−b [ ESthread ] ( t ) [ Sα ] ( t ) [ ESassoc ] ( t ) = [ E ]0 − [ E ] ( t ) − [ ESthread ] ( t ) [ Sα ] ( t ) = [ Sα ]0 − [ ESassoc ] ( t )  S β  ( t ) =  S β  − [ ESthread ] ( t ) 0

(6)


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We note that like Model 1, Model 2 can be kinetically described by only two free variables. In eq. (6) we chose [ E ] ( t ) and [ ESthread ] ( t ) , and all free variables are then matched against experimental observations. There are four rate constants, but two of these can be computed from the known experimental values at equilibrium (Tab.1). At t=0 the values of all species are known ([Sα]0=0.31µM, [Sβ]0=0.24µM, [E] 0=0.50µM and [ESassoc]0=[ESthread] 0=0). These initial values were used in numerical analyses of eq. (6) with respect to the experimental results in Fig. 2. Results with best fits of the model and contour plots are shown in the Supporting Information (Fig. S-7). Overall, the results showed that the ability of Model 2 to account for the experiments was similar to Model 1. Thus, Model 1 showed a least square error of 0.20 while it was 0.18 for model 2. The two models both have four parameters where two of these are determined from the equilibrium concentrations of E and ESthread. Best fit parameters for the rate constants derived from Model 2 and the associated equilibrium constants are listed in Tab. 1. One of the limitations of the current models is that they do not account for a possible direct formation of threaded complex without going through the associated intermediate. To test a possible importance of this simplification we tried an extended scheme in which Model 2 was supplemented with the c  → ESassoc . However, we obtained poor resolution of the direct threading reaction E + Sβ ←

k

k− c

(four) free parameters and concluded that this approach was overparameterized with respect to the current experimental information. Application of more complex models like this might be useful, but it will probably require experimental methods with higher time resolution so that a broader range of enzyme and substrate concentrations can be characterized. ka

k-a

kb

k-b

K1

K2

Model 1

0.097 µM-1s-1

0.067 s-1

0.029 s-1

0.013 s-1

1.4 µM-1

2.2

Model 2

0.33 µM-1s-1

0.19 s-1

0.18 µM-1s-1

0.016 µM-1s-1

1.7 µM-1

11 µM-1

Table 1. Rate constants ka, k-a, kb, k-b, derived from numerical analysis of the data in Fig. 2 using respectively Model 1 and Model 2. The equilibrium constants, K1 and K2, are calculated as the ratios of rate constants for each step.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Discussion Many enzymes that modify polymeric substrates such as DNA and polysaccharides utilize a processive mechanism, where repeated catalytic events are conducted as the polymeric substrate slides through a tunnel-shaped protein (or protein assembly) without dissociation.32-35 A related sliding mechanism is also seen for transport of polymers across biological membranes,36 and processive movement of a synthetic toroidal host along a polymer has been described and used as a mimic of processive enzymes.37 In all these cases, the process must commence by threading of the polymer strand through the tunnel or pore. For the processive cellulases, this threading process has been discussed extensively, particularly with respect to its possible role as rate limiting step (RLS) for the overall hydrolytic reaction5-7, 10, 38. Thus, it appears reasonable to expect that the transfer of a piece of cellulose strand from a highly stable crystal lattice to the enzyme binding tunnel may involve a high energy barrier. If indeed so, threading will be slow, but direct evidence to support this interpretation remains sparse, and in the current work we address this through kinetic modeling of the threading process. Before analyzing the results, we notice that the two model approaches tested here are simplified in the sense that they neglect effects of the hydrolytic reaction and only account for the distribution of enzyme between the states defined in Fig. 1. This simplification is necessary to obtain manageable kinetic descriptions such as eqs. (5) and (6), and it rests on the assumption that the physical properties of the substrate do not change significantly over the (short) experiments used here. This assumption is supported by the low (

Rate of Threading a Cellulose Chain into the Binding Tunnel of a

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