This document presents a mechanistic kinetic model for enzymatic hydrolysis of cellulosic biomass. The model incorporates: 1) Distinct modes of action for endoglucanase I (EGI) and cellobiohydrolase I (CBHI), which randomly cleave interior chains and processively cleave from chain ends, respectively. 2) A distribution of insoluble cellulose chain lengths and accounting for the evolution of enzyme-accessible chains over time. 3) Population balance equations to track the spectrum of chain lengths as hydrolysis progresses, providing a computationally efficient approach. This framework allows investigation of substrate structure effects, enzyme inhibition, and cooperative enzyme action on conversion processes.

1. ARTICLE
A Mechanistic Model for Enzymatic
Saccharification of Cellulose Using
Continuous Distribution Kinetics I:
Depolymerization by EGI and CBHI
Andrew J. Griggs, Jonathan J. Stickel, James J. Lischeske
National Renewable Energy Laboratory, National Bioenergy Center, 1617 Cole Boulevard,
Golden, Colorado 80401; telephone: 303-384-6867; fax: 303-384-6877;
e-mail: jonathan.stickel@nrel.gov
Received 5 May 2011; revision received 16 September 2011; accepted 26 September 2011
Published online 27 October 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/bit.23355
and chemical transformations that occur in several distinct
ABSTRACT: A mechanistically based kinetic model for the processing steps (e.g., pretreatment, enzymatic hydrolysis,
enzymatic hydrolysis of cellulosic biomass has been devel- and fermentation). In order to design reaction vessels
oped that incorporates the distinct modes of action of
and optimize process parameters, it is helpful to have a
cellulases on insoluble cellulose polymer chains. Cellulose
depolymerization by an endoglucanase (endoglucanase I, quantitative understanding of the kinetics of these
EGI) and an exoglucanase (cellobiohydrolase I, CBHI) is physical and chemical transformations. Due to the complex
modeled using population-balance equations, which pro- chemistry and material properties of biomass, our under-
vide a kinetic description of the evolution of a polydisperse standing of reaction kinetics for the degradation of
distribution of chain lengths. The cellulose substrate is
biomass is far from complete. Empirical models, based on
assumed to have enzyme-accessible chains and inaccessible
interior chains. EGI is assumed to randomly cleave insoluble experimental results, have often been used to study
cellulose chains. For CBHI, distinct steps for adsorption, enzymatic hydrolysis kinetics, as reviewed in Bansal et al.
complexation, processive hydrolysis, and desorption are (2009). These models are typically limited to the parameter
included in the mechanistic description. Population-balance space for which the experiments were performed, and
models that employ continuous distributions track the
often provide only limited insight into the underlying
evolution of the spectrum of chain lengths, and do not
require solving equations for all chemical species present in mechanisms of enzymatic hydrolysis.
the reacting mixture, resulting in computationally efficient Mechanistically, or functionally, based models for
simulations. The theoretical and mathematical development enzymatic hydrolysis (Zhang and Lynd, 2004), offer several
needed to describe the hydrolysis of insoluble cellulose advantages to semi-empirical or empirical kinetic models.
chains embedded in a solid particle by EGI and CBHI is
By considering the dynamic interplay of the chemical and
given in this article (Part I). Results for the time evolution of
the distribution of chain sizes are provided for independent physical phenomena occurring during enzymatic hydrolysis,
and combined enzyme hydrolysis. A companion article (Part mechanistic models can provide a deeper understanding,
II) incorporates this modeling framework to study cellulose improve predictive capabilities, and ultimately provide
conversion processes, specifically, solution kinetics, enzyme more directed and rational approaches for process design
inhibition, and cooperative enzymatic action.
and optimization.
Biotechnol. Bioeng. 2012;109: 665–675.
Previous models for enzymatic hydrolysis have made
ß 2011 Wiley Periodicals, Inc.
simplifying assumptions for the cellulosic substrate.
KEYWORDS: cellulose; enzymatic hydrolysis; kinetics
Often, the substrate is described by a single bulk cellulose
model; polymer distribution; substrate structure
concentration, sometimes with varying reactivities, such as
inert and susceptible fractions (Fan and Lee, 1983; Kadam
et al., 2004; Zheng et al., 2009). Such simplifications limit the
Introduction ability of these models to explain many important enzyme–
substrate interactions, such as the relationship between
The biochemical conversion of lignocellulosic biomass to
enzyme adsorption and hydrolysis yield, which may be
liquid transportation fuels involves a multitude of physical
needed to further understand the recalcitrant nature of
Correspondence to: J.J. Stickel cellulosic biomass, identify rate-controlling steps, and
Additional supporting information may be found in the online version of this article. improve cellulose-conversion technologies.
ß 2011 Wiley Periodicals, Inc. Biotechnology and Bioengineering, Vol. 109, No. 3, March, 2012 665

2. Cellulose is an insoluble polymer having varied degrees of chains of varied length. We propose that a meaningful
polymerization, composed of repeating units of cellobiose, mechanistic model should include descriptions of (1) the
which is a glucose dimer having b-(1,4)-glucosidic bonds distinct modes of action of the cellulase enzymes (EGI and
(Zhang and Lynd, 2004). Naturally occurring cellulosic CBHI), (2) the occurrence of insoluble and soluble
substrates often have a wide distribution of chain lengths. substrates, (3) the distribution of chain lengths for the
Previous experimental studies have examined the size insoluble cellulose substrate, and (4) the time evolution
distributions of cellulose chains during enzymatic hydrolysis of the substrate accessibility to cellulases. In this article, we
and shown the susceptibility of the cellulose to enzyme present the model structure to implement these core
hydrolysis can depend on the chain size (Kleman-Leyer features. Population-balance equations are used to describe
et al., 1996; Srisodsuk et al., 1998). So-called ‘‘depolymeri- the transformation of a continuous distribution of cellulose
zation models’’ consider the distribution of chain sizes for chains lengths. The model framework is modular and
polymeric substrates and incorporate various modes of extendable so that features appropriate to a particular
depolymerization by enzymes acting on isolated chains system of study can be readily implemented, including
(Converse and Optekar, 1993; Okazaki and Mooyoung, lignocellulosic biomass substrates. In a companion article
1978; Watanabe and Kawai, 2006; Watanabe et al., 2007; (Griggs et al., 2011)) (Part II), solution kinetics and product
Zhang and Lynd, 2006). These models do not account for inhibition are considered simultaneously with the hetero-
the time evolution of accessible cellulose chains during geneous catalyzed depolymerization of cellulose.
saccharification, but are useful for studying the early stages
of enzyme hydrolysis.
The availability or accessibility of the cellulose substrate to Model Formulation
the enzymes at a given point in hydrolysis also requires careful
consideration. The amount of enzyme-accessible cellulose Cellulose Depolymerization
differs from the total cellulose in a reaction mixture due to the
supra-molecular organization of cellulose chains. The We treat the cellulose substrate as ‘‘populations’’ of varied
evolution of enzyme-accessible cellulose during the course chain lengths. Let P(x) represent an insoluble cellulose chain
of hydrolysis depends on the spatial organization of cellulose comprised of x anhydroglucose units. Assuming x varies
chains for a given cellulose substrate, and therefore, the initial continuously (Aris and Gavalas, 1966), the population
spatial organization of cellulose chains, and the morphologi- distribution of enzyme-accessible cellulose chains at some
cal changes to the cellulose substrate during the course of time, t, is represented as pðx; tÞ, so that pðx; tÞdx is the
enzyme hydrolysis may be key rate-determining factors. number of cellulose chains per volume (molar concentra-
Until recently, few enzymatic hydrolysis studies have tion) having lengths between x and x þ dx. Useful metrics
considered both the size distribution of the cellulose can be obtained from the absolute moments of the
polymers and the embedding of cellulose in a solid distribution:
substrate. Hydrolytic-time evolution of solid substrate Z 1
morphology and enzymatic chain fragmentation were pðnÞ ðt Þ ¼ xn p ðx; t Þdx (1)
both considered in the works of Zhou et al. (2009a; b), 0
which examined short- and long-time conversion of
cellulose. Zhou et al. (2010) used a time-scale analysis The zeroth moment, p(0)(t), is the molar concentration
based on the kinetic model given in Zhou et al. (2009a; b) to of cellulose chains, and the first moment, p(1)(t), gives
study the synergistic behavior of exo- and endo-acting the total concentration of monomeric glucan comprising
cellulases and identified the embedding of cellulose chains in the cellulose chains. Other commonly used metrics are the
a solid particle as a rate-limiting factor for conversion. number-averaged chain length, xN ¼ pð1Þ =pð0Þ , mass-aver-
Levine et al. (2010) described the cellulose polymers as aged chain length, xM ¼ pð2Þ =pð1Þ , and polydispersity index,
individual discrete species, essentially tracking the concen- Ipd ¼ xM =xN . Soluble sugars are distinguished from insolu-
tration for each chain length. Although analytical solutions ble cellulose, and are denoted by Q(xj), for species having j
are possible for some reaction systems, numerical methods glucan units. This distinction arises from the treatment of
are often required to integrate the differential equations and cellulose depolymerization as a heterogeneous catalysis
track the concentration evolution of the reacting species. For reaction. Molar concentrations of soluble sugars are given by
the kinetics of reacting polymers with a distribution of qj(t). In particular, q1(t) is the concentration of glucose
discrete chain lengths, it is possible to form a set of ODEs, (Q(x1)), and q2(t) is the concentration of cellobiose (Q(x2)).
one for each length of polymer, and to solve the system The enzyme mixtures used for biomass deconstruction
numerically (Kostoglou, 2000; Zhang and Lynd, 2006). typically include a variety of processive and non-processive
However, the number of ODEs can become extremely large cellulases. Although the mechanisms of these various
(O(103)) and may be inefficient to solve directly, even when cellulases are not fully understood, experimental evidence
the initial distribution is monodisperse. indicates different enzymes are responsible for facilitating
In this work, we develop a mechanistically based kinetics the hydrolysis of b-(1,4)-glycosidic bonds from the chain
model for the depolymerization of embedded solid cellulose interior and chain ends. Rather than attempting to model
666 Biotechnology and Bioengineering, Vol. 109, No. 3, March, 2012

3. the action of all of the different cellulases, we instead processive action of CBHI is
describe two distinct hydrolysis mechanisms, chain-interior
and chain-end scission. Endoglucanase I (EGI, Cel7B) and rpB h ðx; t Þ ¼ kCBH ½pB ðx þ x2 ; t Þ À pB ðx; tÞŠ;
CBH
h
cellobiohydrolase I (CBHI, Cel7A, an exoglucanase) from (4)
Trichoderma reesei (Srisodsuk et al., 1998) have been well x ! x" :
characterized, and are thought to perform primarily internal
and chain-end hydrolysis, respectively. We assume cellulose Here, we assume processive hydrolysis by CBHI continues
depolymerization occurs at the solid–liquid interface by unhindered along the cellulose chain length until x < xe.
purely endo- (EGI) and exo-acting (CBHI) mechanisms. Although the lower limit, xe, has yet to be experimentally
Enzymatic hydrolysis reactions do not always result in determined, we assume procession stops and CBHI desorbs
depletion of substrate. Rather, generation, transformation, when x < x3. In a recent study, ‘‘processivity’’ values, which
and loss of insoluble substrate (solubilization) may occur. represent the number of processive hydrolysis events that
We assume EGI and CBHI may adsorb anywhere on the occur prior to desorption of CBHI, were determined to be
cellulose surface, and surface-adsorbed EGI and CBHI act to much smaller than the average cellulose chain length
depolymerize the insoluble cellulose at sites specific to their (Kurasin and Valjamae, 2011). However, the processivity
mode of action. The adsorption of enzymes onto cellulose is values have yet to be verified by experimental comparisons
described below in section ‘‘Enzyme Adsorption’’. CBHI to cellulose DP distributions. For the reported processivity
is considered to perform strictly chain-end scission in a values in Kurasin and Valjamae (2011), one would expect an
processive manner from the reducing end of a cellulose appreciable decrease in the average degree of polymeriza-
chain (Igarashi et al., 2009). The chemical-reaction scheme tion, which has not been observed for CBHI hydrolysis
for surface-adsorbed CBHI can be written as: (Kleman-Leyer et al., 1996; Srisodsuk et al., 1998). Our
assumption of complete processivity substantially simplifies
kCBH È
f É the modeling equations. Similarly, although there is some
ECBH À
S ! ECBH P ðxÞ
B (2a)
limited experimental evidence that oligomers of cellulose
as large as seven glucan units may be liberated during
hydrolysis (Zhang and Lynd, 2004), the rate of soluble
È É kCBH È CBH É
ECBH P ðxÞ À ! EB P ðx À x2 Þ þ Q ðx2 Þ
h
B (2b) oligomer production and the relation to component
cellulase enzymes is substrate dependent and still unclear.
For simplicity, we consider only cellobiose and glucose to be
È É kCBH soluble.
ECBH P ðx4 Þ À ECBH þ 2Q ðx2 Þ
! B
h
B (2c) From Equations (2b–d), the rates of generation of soluble
species (cellobiose and glucose) due to processive hydrolysis
are given by
È É kCBH
ECBH P ðx3 Þ À ECBH þ Q ðx2 Þ þ Q ðx1 Þ
!
h
B (2d) Z 1 !
rq2 h ðt Þ ¼ kCBH
CBH
h pB ðx; t Þ dx þ x1 pB ðx4 ; t Þ ; (5)
x"
Surface-adsorbed CBHI, ECBH ,
S becomes catalytically active
following surface diffusion, complexation with a reducing
end, and threading of a chain into its catalytic domain, as and
described by Equation (2a). The distribution of CBHI-
È É
threaded (‘‘bound’’) cellulose chains, ECBH PðxÞ , is rq1 h ðt Þ ¼ kCBH x1 pB ðx3 ; t Þ:
CBH
h (6)
B
denoted as pB ðx; tÞ, and is treated separately from
the population pðx; tÞ. Although experimental studies The first term on the right-hand side (RHS) of Equation (5)
have yet to precisely identify the rate CBHI finds and gives the generation of cellobiose due to scission of chain
threads cellulose chains, we hypothesize that the time ends from all larger threaded chains. The second term on the
required to find a reducing end is proportional to the RHS of Equation (5) and the only term of Equation (6) gives
chain length, so that the rate coefficient has the form the generation of additional cellobiose or glucose from
kfCBH ðxÞ ¼ ^CBH =x. The rate of generation of pB ðx; tÞ due to
kf scission at the end of the chain, depending on whether the
threading of CBHI is then initial chain contained an even or odd number of glucan
units.
^ CBH
kCBH ES EG
Surface-adsorbed EGI, ES , is considered to perform
f
r pB f
CBH
ðx; t Þ ¼ Àrp f
CBH
ðx; t Þ ¼ p ðx; t Þ: (3) strictly random-chain scission. Complexation and hydroly-
x
sis are assumed to occur in a single concerted step,
according to
Population-balance equations have been developed to
describe the transformation given in Equation (2b). The kEG
EEG þ P ðxÞ À P ðx À yÞ þ P ðy Þ þ EEG ;
!
h
time rate of change rate of the population pB(x) due to the S (7)
Griggs et al.: Mechanistic Model for Enzymatic Saccharification, Part I 667
Biotechnology and Bioengineering

4. which shows cleavage of a cellulose polymer of length x, Soluble species may also result from scission of threaded
resulting in two smaller cellulose chains of lengths y and xÀy chains:
EG
and desorption of ES . A population-balance equation can
be used to express the rate of change of pðx; tÞ due to EGI ^ Z
kEG ES ðt Þ 1
EG
action: rq2 :pB ðt Þ ¼ rq1 :pB ðt Þ ¼
EG EG h
pB ðy; t Þ dy: (13)
xN x"
Z 1
EG
rp ðx; t Þ ¼ 2 kEG ðy Þ ES ðt Þ p ðy; t Þ Vðx; yÞ dy
h
EG
x
À kEG ðxÞ ES ðt Þ p ðx; t Þ:
h
EG
(8)
Cellulose Structure
The second term on the RHS accounts for the loss of a chain The kinetic model development in the previous section
of size x. The first term on the RHS accounts for addition applies to the enzyme-accessible cellulose population. Often,
of chains of size x from larger chains of size y. The multiple 2 only a portion of the cellulose substrate is accessible to
comes from the fact that two fragments result from enzymes at a given time. For instance, crystalline cellulose
the scission of a larger chain. The term Vðx; yÞ is known in an aqueous environment exists as either aggregated or
as the ‘‘breakage kernel’’ and gives the probability of discrete insoluble particles (Zhang and Lynd, 2004).
obtaining a chain of size x from a chain of size y (Sterling and Depending on the biomass source and previous processing,
McCoy, 2001). For random-chain scission, Vðx; yÞ ¼ 1=y. these particles may have a number of different shapes and
We propose that the rate coefficient be, kh ðxÞ ¼ ^EG x=xN ,
EG
kh sizes. Microscopically, cellulose chains are commonly
which accounts for more frequent action by EGI on longer oriented together into microfibrils, which may collectively
chains. Substituting these into Equation (8) gives be arranged as macrofibrils. Depending on the processing
used to obtain cellulose from plants, cellulose particles often
^ Z 1 ! consist of groupings of macro- and microfibrils and may be
kEG ES ðt Þ
EG
EG
rp ðx; t Þ ¼ h
2 p ðy; t Þ dy À xp ðx; t Þ ; porous, having both interior and exterior accessible surfaces
xN x
(Hong et al., 2007). Depolymerization and solubilization of
x ! x" ; surface-accessible cellulose leads to a decrease in particle
(9) mass and generation of additional surface-accessible
cellulose by exposing the underlying chains.
where we also include a lower chain-size limit for EGI Here, we assume a population of monodisperse cylindri-
hydrolysis. Because this lower limit is not known precisely, cal particles, comprised of cellulose chains of varied length,
we assume that the limit is the same for EGI as for CBHI, which is intended to represent the micro- or macrofibrils.
specifically, xe, so that random-chain scission by EGI does Experimental findings suggest microfibrils and macrofibrils
not occur for chains smaller than xe. Production of soluble of cellulose are approximately hexagonal in shape across the
glucose and cellobiose may result from random-chain radial dimension (Ding and Himmel, 2006), which is much
scission by EGI. The rate of cellobiose and glucose formation smaller than the particle length, making a cylindrical
by EGI is given by: approximation reasonable. The model could be augmented
to consider polydisperse distributions of particle sizes,
^ Z but would require simultaneous tracking of multiple size
kEG ES ðt Þ 1
EG
rq2 ðt Þ ¼ rq1 ðt Þ ¼ 2
EG EG h
p ðy; t Þ dy: (10) distributions (of particles and of chain length), adding
xN x" significantly to the complexity of the model. The mono-
disperse cylinders have a radius of R and length of L, as
EGI may also act on the CBHI-threaded-chain popula- illustrated in Figure 1. We assume that L ) R, so that only R
tion, pB ðx; tÞ. For this case, random-chain scission results in is a function of time (t) and L is constant. Additionally, we
one threaded and unthreaded chain fragment, so that: denote the thickness of the accessible layer of cellulose as R0,
which may be regarded as the thickness of a single cellulose
^ Z 1 ! chain or roughly the diameter of glucose. Clearly, many
kEG ES ðt Þ
EG
EG
rpB ðx; t Þ ¼ h
pB ðy; t Þ dy À xpB ðx; t Þ ; other geometrical representations for insoluble particles
xN x could be chosen, but our choice of a cylinder is sufficient to
x ! x" ; describe the evolution of accessible and inaccessible sub-
strate with conversion. Porous particles would require
(11)
further consideration, including whether the pores are
large enough to permit entry by enzymes. We do not
and
consider porous particles here.
^ Z A detailed derivation for the rate of loss and generation
kEG ES ðt Þ 1
EG
of surface-accessible cellulose is given in the supporting
rp:pB ðx; t Þ ¼
EG h
pB ðy; t Þ dy; x ! x" : (12)
xN x information. The important relationships that are needed to
668 Biotechnology and Bioengineering, Vol. 109, No. 3, March, 2012

5. Here, E denotes either freely suspended EGI or CBHI, and Es
denotes surface adsorbed enzyme. A detailed derivation of
the relationships between free and adsorbed enzymes is
given in the supporting information. The concentrations of
surface-adsorbed CBHI and EGI are given by
ð0Þ
ET À pB
CBH
ES ¼
CBH
(18)
K CBH
1 þ dð1Þ
pS
and
Figure 1. Schematic of a cylinder comprised of cellulose chains. EG
ET
ES ¼
EG
EG ; (19)
Kd
1 þ ð1Þ
pS
complete the kinetics model are reproduced here. The
population distribution of surface-accessible cellulose is
denoted as pS ðx; tÞ ¼ pðx; tÞ þ pB ðx; tÞ and the total cellulose respectively. In this article we have neglected product
population is denoted as pT ðx; tÞ ¼ pS ðx; tÞ þ pi ðx; tÞ, inhibition and enzyme crowding. We extend the above
where pi ðx; tÞ is the population distribution of inaccessible relationships to include these features and discuss their
cellulose. Surface-accessible cellulose is exposed at a rate, impact in Part II.
rexp, and consumed at a rate, rloss, due to enzymatic
hydrolysis, so that: Numerical Methods
dpS ðx; t Þ Solution methods for population-balance models have often
¼ rloss ðx; t Þ þ rexp ðx; t Þ: (14)
dt used the method of moments, which offers considerable
computational efficiency (Sterling and McCoy, 2001).
The total rate of the loss of cellulose due to enzymatic However, the method of moments provides information
reaction, rloss, is the sum of the rate terms given above that limited to the time evolution of the moments of a
lead to the loss of insoluble polymer: distributed system, not the full distribution. The method
of moments approach was found to be inadequate for our
rloss ðx; t Þ ¼ rp ðx; t Þ þ rpB ðx; t Þ þ rp:pB ðx; t Þ
EG EG EG
mechanistic description of CBHI, because the concentration
of the cut-off species xe must be known precisely (Kostoglou,
þ rpB h ðx; t Þ:
CBH
(15)
2000; Stickel and Griggs, 2010). Specifically, the concentra-
tion of CBHI-threaded cellotetraose (i.e., 2-cellobiose) is
The rate of exposure is derived to be needed to properly describe CBHI desorption following
hydrolysis of a chain and the depletion of a chain from the
R À R0 pi ðxÞ ð1Þ population.
rexp ðx; t Þ ¼ À r ðt Þ;
ð1Þ loss
R ! R0 ; (16)
R p In this work, we map the continuous distribution to fixed
i
discrete grid points and solve the system of rate equations
and the rate of exposure is zero once the radius falls below using numerical methods. Rate equations with finite integral
R0. We assume that the cellulose is distributed uniformly terms over x are evaluated using cumulative-trapezoidal
throughout the particle. However, it would be straightfor- integration, which provides sufficient numerical accuracy.
ward to have the population be a function of radius; e.g., Special consideration must be taken for the evaluation of
the population near the surface may have an average chain Equation (4), which describes the processive chain-end
length that is shorter than the population near the center of scission by CBHI. Calculation of pB ðx þ x2 Þ À pB ðxÞ at
the particle. each grid point x ¼ xi could be accomplished by using
interpolation to determine the value for pB ðxi þ x2 Þ, but
when the grid spacing is much larger than x2, computation-
Enzyme Adsorption ally expensive quadratic or cubic interpolation would be
necessary to obtain reasonable accuracy. Instead, a Taylor-
Prior to hydrolysis, solution-phase CBHI and EGI adsorbs series expansion can be used to approximate the difference
onto the insoluble cellulose surface according to term (Adrover et al., 2003; Stickel and Griggs, 2010):
þPðxÞ
E ÀÀ Es
)*
ÀÀ (17) @pB x2 @2 pB
2 À 3Á
E pB ðx þ x2 Þ À pB ðxÞ % x2 þ þ O x2 : (20)
Kd
@x 2 @x2
Griggs et al.: Mechanistic Model for Enzymatic Saccharification, Part I 669
Biotechnology and Bioengineering

6. Truncating after the second term allows us to rewrite distribution, so that:
Equation (4) as
ð 0Þ
pT; in aÀ1 Àx=b
! pT; in ðxÞ ¼ a x e
@pB ðx; t Þ x2 @2 pB ðx; t Þ
2
b G ða Þ
rpB h ðx; t Þ ¼ kCBH x2
CBH
þ ;
h
@x 2 @x2 (21) ð0Þ
¼ pT;in exp ½ða À 1Þ ln x À x=b À a ln b À ln G ðaÞŠ
x x : (28)
ð0Þ
Central-finite-difference methods were used to evaluate the where a, b, and pT;in determine the mean, width, and total
derivative terms, with boundary conditions given by size of the distribution. The expanded second expression is
more amenable to numerical evaluation. Of this initial total
population, the enzyme-accessible portion is
@p ðx; t Þ
lim p ðx; t Þ ¼ lim ¼ 0: (22)
x!1 x!1 @x R0
pS; in ðxÞ ¼ pin ðxÞ ¼ 2 p T; in ðxÞ: (29)
Rin
For relatively broad distributions, which is often the case
for many naturally occurring plant celluloses, the second The initial population of threaded chains was set to zero for
derivative term in Equation (21) may be neglected (Stickel all x, as was the concentrations for cellobiose and glucose.
and Griggs, 2010). The initial radius of the insoluble-cellulose cylindrical
The set of modeling equations to be solved is the system of particles was related to the specified number and length of
ordinary differential equations: the particles and the initial mass of cellulose.
For the purposes of comparing the total population
distribution with that determined by experiment, pT(x) can
dp ðx; t Þ
¼ rp ðx; t Þ þ rp:pB ðx; t Þ þ rp f ðx; t Þ
EG EG CBH be calculated by
dt
þ rexp ðx; t Þ; (23) pi ðxÞ ð 1Þ
pT ðx; t Þ ¼ ð 1Þ
pi ðt Þ þ p ðx; t Þ þ pB ðx; t Þ; (30)
pi
dpB ðx; t Þ where the normalized distribution pi ðxÞ=pi will be
ð 1Þ
¼ rpB ðx; t Þ þ rpB f ðx; t Þ
EG CBH
dt constant in time according to the initial conditions, as
þ rpB h ðx; t Þ;
CBH
(24) long as the initial distribution of cellulose lengths do not
change with radius, and the total mass of inaccessible
ð1Þ
cellulose is pi ðtÞ ¼ nrpðR ðtÞ À R0 Þ2 L.
dq2 ðt Þ
¼ rq2 ðt Þ þ rq2 :pB ðt Þ þ rq2 h ðt Þ;
EG EG CBH
(25)
dt
Results and Discussion
dq1 ðt Þ Independent Action of EGI
¼ rq1 ðt Þ þ rq1 :pB ðt Þ þ rq1 h ðt Þ;
EG EG CBH
(26)
dt
For all the results presented in this article, total enzyme
loading was fixed to approximately 65 mg of enzyme per
and gram of cellulose, an amount appropriate for demonstrating
changes to the cellulose population with digestion. In this
ð1Þ
dR r ðt Þ section, we will demonstrate how random-chain scission
¼ loss : (27) by EGI changes a population of cellulose chains. An initial
dt 2nprRL
population of cellulose polymers, a Gamma distribution
with an average degree of polymerization of xN ¼ 500, with a
Equations (23) and (24) were evaluated at grid points, x ¼ xi, relatively narrow range of molecular weights (b ¼ 10), was
where x xe. The total number of equations to solve was chosen for the results presented in this section. Figure 2
therefore 2Nx þ 3, where Nx is the number of grid points. shows the time evolution of the mass-weighted distribution
Typical values for Nx were 500 to 1,000. Time integration of of cellulose chains during hydrolysis by EGI, assuming all of
Equations (23–27) was performed using an adaptive-time- the cellulose chains are enzyme accessible. Over time, the
step ODE solver. mean of the distribution shifts progressively toward lower
Any arbitrary distribution may be used to describe the values of xN, indicating shorter chains are being generated
initial population of cellulose. In this work, the initial from longer chains. Further fragmentation of these shorter
cellulose population was generated using a Gamma chains can also be observed throughout the course of
670 Biotechnology and Bioengineering, Vol. 109, No. 3, March, 2012

7. markedly different than those presented in Figure 2, for the
same enzyme and cellulose loading. The extent of random-
chain scission, characterized by reduction in chain length
and production of smaller chains, for the total population is
much less pronounced when cellulose structure is taken into
account. Figure 3b shows results for the surface-accessible
(hydrolyzable) population. However, the extent of random-
chain scission here is still less than that observed from
Figure 2, owing to a reduced availability of cellulose
substrate for enzymatic adsorption and subsequent hydro-
lytic bond cleavage.
Independent Action of CBHI
Separate mechanistic steps for adsorption, surface diffusion
to a reducing-chain end and formation of a catalytically
Figure 2. Mass-weighted cellulose chain size distribution for various times active complex, processive hydrolysis, and desorption were
during EGI hydrolysis of amorphous cellulose. [Color figure can be seen in the online
version of this article, available at http://wileyonlinelibrary.com/bit]
included for CBHI. Following adsorption of CBHI on the
insoluble cellulose surface via the cellulose binding domain,
the rate of formation of catalytically active CBHI depends
hydrolysis. At time, t ¼ 2,000, almost all of the chains from on the availability of reducing ends and the rate at which
the initial distribution have undergone at least one scission, available chain ends are found. The rate parameter
which is expected due to all chains being susceptible to associated with the induction time between adsorption
cleaving by EGI. and complexation, kfCBH , was assigned a value of 2.0. To our
For the results presented in Figure 2, it was assumed that knowledge, this parameter has yet to be determined
all of the cellulose chains were enzyme-accessible. We now experimentally. However, a recent high-speed AFM study
demonstrate the effect of our ‘‘structure’’ model on the of CBHI enzymes acting on insoluble cellulose suggests that
cellulose substrate, where only the chains on the outer this rate parameter may be measurable for a given cellulose
surface are labile to random-chain scission by EGI. Figure 3 substrate (Igarashi et al., 2009). Following complexation
presents results for the (mass-weighted) total cellulose with a reducing chain end, the threaded CBHI enzyme
population (xpT ðx; tÞ) and the surface-accessible popula- processes along the chain, hydrolyzing cellulose by
tion, using our structure model, with R0/R ¼ 0.2 (the successively cleaving cellobiose units from the chain ends.
‘‘amorphous’’ case discussed above was obtained by setting The rate parameter associated with this hydrolysis was set to
R0/R ¼ 1.0). The changes to the total cellulose population a value of kh ¼ 1:0 (b-glucosidic bonds cleaved/time). It
CBH
due to random-chain scission by EGI, shown in Figure 3a are is reasonable to assume that kfCBH is greater than or equal to
a b
Figure 3. Mass-weighted cellulose chain distribution for various times during EGI hydrolysis of cellulose, incorporating cellulose structure. Figure 3(a) shows the total
cellulose population, while (b) shows the enzyme-accessible (surface) cellulose population. [Color figure can be seen in the online version of this article, available at http://
wileyonlinelibrary.com/bit]
Griggs et al.: Mechanistic Model for Enzymatic Saccharification, Part I 671
Biotechnology and Bioengineering

8. the hydrolysis rate constant, especially for crystalline to progressively lower values of x with time, due to
cellulose substrates due to the energy required for processive hydrolysis, eventually reaching a quasi-steady
decrystallization during processive hydrolysis. distribution at longer times.
Figure 4 shows mass-weighted distributions for the total Following complete hydrolysis of a cellulose chain,
cellulose (Fig. 4a) and enzyme-accessible population the CBHI enzyme desorbs, and the cycle of adsorption,
(Fig. 4b) for various times during hydrolysis by CBHI, recognition, and hydrolysis repeats, as illustrated in Figure 5,
using the same initial distribution and enzyme loading value which shows the dynamic evolution of the portion of
described in the previous section. In contrast with the CBHI that is catalytically active, surface adsorbed, or free
observed trends for EGI, the distribution does not exhibit (in solution). Initially, the relative amount of free enzyme
a pronounced shift to lower degrees of polymerization. decreases due to surface adsorption, while surface-adsorbed
Rather, the area under the distribution curve decreases enzyme subsequently threads reducing chain ends, becom-
progressively during hydrolysis, as shown in Figure 4a, due ing catalytically active. An increase in the relative amount
to mass loss as cellulose is converted to soluble cellobiose. of free enzyme can be attributed to desorption of CBHI
The enzyme-accessible population of cellulose chains, following processive hydrolysis, which is accompanied
shown in Figure 4b, exhibits a slight increase in chains by a decrease in the number of CBHI-threaded chains.
having a small degree of polymerization, because CBHI- The oscillations of CBHI concentrations over time shown
threaded-chains are included with the enzyme-accessible in Figure 5 are consistent with our mechanistic description
cellulose population. Figure 4c shows results for the of CBHI hydrolysis, for a relatively narrow distribution
evolution of the CBHI-threaded chain population (num- of cellulose chain lengths. For wider distributions, such
ber-weighted). The distribution of CBHI-threaded chains as those for naturally occurring celluloses, such pronounced
first increases in area (from zero at t ¼ 0) as catalytically fluctuations in CBHI concentrations are unlikely to
active complexes form. The distribution subsequently shifts occur.
a b
c
Figure 4. Cellulose-chain-size distributions for various times during CBHI hydrolysis of cellulose. Parts (a) and (b) show the mass-weighted total cellulose population and
enzyme-accessible (surface) cellulose population, respectively. Part (c) depicts the number-weighted distribution of CBHI-threaded chains. [Color figure can be seen in the online
version of this article, available at http://wileyonlinelibrary.com/bit]
672 Biotechnology and Bioengineering, Vol. 109, No. 3, March, 2012

9. a
b
Figure 5. The dynamic evolution of the distribution of CBHI enzyme, relative to
CBH CBH
total loading (ET ), in the forms of surface-adsorbed (ES ), catalytically active
ð0Þ
(chain-threaded, pB ), and free solution (ECBH) enzyme. [Color figure can be seen in
the online version of this article, available at http://wileyonlinelibrary.com/bit]
Combined Action of CBHI and EGI
In this section, we consider both CBHI and EGI acting
simultaneously to depolymerize cellulose. Figure 6 shows
the time evolution of the mass-weighted cellulose chain-size
distributions for various CBHI to EGI loading ratios, using
an in initial distribution having an average degree of
polymerization of xN ¼ 500. Here, the total enzyme loading c
was kept constant and the proportion of each enzyme was
varied. As can be observed in Figure 6, the distribution
becomes bimodal at intermediate conversions, due to the
accumulation of small chain fragments generated by EGI,
which are eventually solubilized by CBHI.
Conclusions
Chain-end scission by CBHI is captured utilizing a
population-balance approach to describe the depolymeri-
zation of cellulose chains. Separate mechanistic steps
for adsorption, complexation with reducing ends to form
a catalytically active enzyme, processive hydrolysis, and Figure 6. Mass-weighted cellulose chain size distributions for various times
during the course of hydrolysis for combined CBHI and EGI action for various CBHI:EGI
desorption were included. Random-chain scission of ratios. The total enzyme loading was fixed, and the proportion of each enzyme was
insoluble chains by EGI is also handled using population- varied for CBHI:EGI: (a) 2:1, (b) 1:1, and (c) 1:2. [Color figure can be seen in the online
balance equations. The evolution of enzyme accessible version of this article, available at http://wileyonlinelibrary.com/bit]
cellulose is represented by cellulase-mediated erosion of a
cylindrical particle comprised of cellulose chains. Solutions
for the system of ODEs describing the fragmentation and cellulose chain lengths can be experimentally measured
solubilization of chains at the particle surface and the using various techniques (Kleman-Leyer et al., 1996;
evolution of particle size are easily obtained. By examining Srisodsuk et al., 1998) and compared to our model results.
the evolution of the distribution of cellulose chain lengths In a companion article (Part II), we incorporate the model
during enzymatic hydrolysis, in conjunction with soluble framework described here to study cellulose conversion
sugar production, an improved understanding of cellulose processes, including solution kinetics, product inhibition,
depolymerization can be achieved. The distribution of and cooperative enzyme hydrolysis.
Griggs et al.: Mechanistic Model for Enzymatic Saccharification, Part I 673
Biotechnology and Bioengineering

10. Contributions to an earlier version of the kinetics model were made by
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complexed/adsorbed with substrate or inhibitor) of Chemical Engineering, Massachusetts Institute of Technology.
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