Quantifying Error Correction through a Rule-Based Model of Strand Escape from an [ n]-Rung Ladder

Article abstract of DOI:10.1021/jacs.9b08958

The rational design of 3D structures (MOFs, COFs, etc.) is presently limited by our understanding of how the molecular constituents assemble. The common approach of using reversible interactions (covalent or noncovalent) becomes challenging, especially when the target is made from multivalent building blocks and/or under conditions of slow exchange, as kinetic traps and nonequilibrium product distributions are possible. Modeling the time course of the assembly process is difficult because the reaction networks include many possible pathways and intermediates. Here we show that rule-based kinetic simulations efficiently model dynamic reactions involving multivalent building blocks. We studied "strand escape from an [n]-rung ladder" as an example of a dynamic process characterized by a complex reaction network. The strand escape problem is important in that it predicts the time a dynamic system needs to backtrack from errors involving [n]-misconnections. We quantify the time needed for error correction as a function of the dissociation rate coefficient, strand valency, and seed species. We discuss the simulation results in relation to a simple probabilistic framework that captures the power law dependence on the strand's valency, and the inverse relationship to the rung-opening rate coefficient. The model also tests the synthetic utility of a one-rung (i.e., hairpin) seed species, which, at intermediate times, bifurcates to a long-lived, fully formed [n]-rung ladder and a pair of separated strands. Rule-based models thus give guidance to the planning of a dynamic covalent synthesis by predicting time to maximum yield of persistent intermediates for a particular set of rate coefficients and valency.

Full text of DOI:10.1021/jacs.9b08958

Article
pubs.acs.org/JACS
Cite This: J. Am. Chem. Soc. XXXX, XXX, XXX−XXX
Quantifying Error Correction through a Rule-Based Model of Strand
Escape from an [n]‑Rung Ladder
Morgan M. Cencer,† Andrew J. Greenlee,† and Jeffrey S. Moore*,‡,†
†Department of Chemistry, University of Illinois at Urbana−Champaign, Urbana, Illinois 61801, United States
‡Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana−Champaign, Urbana, Illinois 61801,
United States
S
* Supporting Information
ABSTRACT: The rational design of 3D structures (MOFs,
COFs, etc.) is presently limited by our understanding of how
the molecular constituents assemble. The common approach of
using reversible interactions (covalent or noncovalent)
becomes challenging, especially when the target is made from
multivalent building blocks and/or under conditions of slow
exchange, as kinetic traps and nonequilibrium product
distributions are possible. Modeling the time course of the
assembly process is difficult because the reaction networks
include many possible pathways and intermediates. Here we
show that rule-based kinetic simulations efficiently model
dynamic reactions involving multivalent building blocks. We
studied “strand escape from an [n]-rung ladder” as an example
of a dynamic process characterized by a complex reaction network. The strand escape problem is important in that it predicts
the time a dynamic system needs to backtrack from errors involving [n]-misconnections. We quantify the time needed for error
correction as a function of the dissociation rate coefficient, strand valency, and seed species. We discuss the simulation results in
relation to a simple probabilistic framework that captures the power law dependence on the strand’s valency, and the inverse
relationship to the rung-opening rate coefficient. The model also tests the synthetic utility of a one-rung (i.e., hairpin) seed
species, which, at intermediate times, bifurcates to a long-lived, fully formed [n]-rung ladder and a pair of separated strands.
Rule-based models thus give guidance to the planning of a dynamic covalent synthesis by predicting time to maximum yield of
persistent intermediates for a particular set of rate coefficients and valency.
bonding connectivity does not adhere to that of the target.
Error correction is accomplished either by removing an
improperly placed constituent from the assembling structure
or by repositioning it into its proper location.17,18 Error
correction is a well-known and generally applicable concept of
dynamic, multivalent chemistries.19−21 Removal of the
misplaced constituent is slower the more bonds that must
break and the lower the dissociation rate of the individual
bonding interactions.12 Quantifying how strongly valency,
affinity, and bond dissociation rates impact error correction is
desired for the rational planning of a dynamic synthesis.22−24
To the best of our knowledge, there is no quantitative
description of error correction in the literature. Here we
develop a model for the specific case where error correction
involves complete removal of an improperly placed constitu-
ent.
INTRODUCTION
■
Dynamic synthesis targets ordered structures made from
constituents that are joined through multiple, reversible
bonding sites (i.e., the constituents are multivalent with
dynamic bonds). In many cases, thermodynamic modeling
accurately predicts the major products since equilibrium is
readily achieved.1−3 As increasingly complex structures are
targeted,4 persistent intermediates arise as kinetic traps,5−7
stymieing the synthesis.8,9 Kinetic trapping occurs when the
rate of constituent exchange is slower than the time scale to
perform the overall reaction in the laboratory.10 Since dynamic
chemistries cover a wide kinetic range (Figure 1), the inability
to converge to a thermodynamic minimum is more frequently
encountered in systems with low dissociation rates.11 Addi-
tionally, kinetic traps are also common with high-valency
constituents,12−15 owing to the low probability of simulta-
neously dissociating all of the bonding interactions (i.e., high
avidity).15 In extreme cases, escaping kinetic traps becomes a
rate limiting step to error correction for a dynamic covalent
synthesis.16
Rule-based models (RBMs) handle the construction of
complex reaction networks algorithmically, making it possible
to simulate the time-evolution of dynamic multivalent
Error correction is that part of the synthesis pathway that
reverses the placement of a constituent whose pattern of
Received: August 19, 2019
© XXXX American Chemical Society
A
DOI: 10.1021/jacs.9b08958
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number of reactions and intermediates increases exponentially
(see Table 2 for a specific example). Kinetic models of
Table 2. Characteristics of Reaction Networks for Strand
Escape from [n]-Rung Ladders
a
b
c
d
[n]
no. of species
no. of reactions
no. of rules
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
4
8
16
32
6
21
60
155
378
889
2040
4599
10 230
4
6
8
10
12
14
16
18
20
64
128
256
512
1024
Figure 1. Literature values comparing the dissociation rate
coefficients for a variety of reversible reactions (kd (obs)). Literature
references, assumptions, and methods to compare the rate coefficients
are provided in the Supporting Information.
[10]
a
[n] is the number of rungs in a double-stranded ladder; more
generally, [n] describes the valency of the constituents. Species are
b
defined as ladder (starting reactant), plus all unique intermediates,
plus the free strand (product). Reactions are the elementary steps,
c
reactions.25 RBMs employ simplifying assumptions that allow a
small set of rules to define the reactive chemistry of a system.26
Rules represent reactions that change the bonds (e.g., bond
formation) or state of a reactant (e.g., catalyst deactivation).27
Once the rules capture the desired attributes of the system, the
reaction network is generated and the time evolution is
stochastically simulated starting from an initial species (i.e., the
seed species). RBMs have been broadly used in biochemistry
to handle the complexity of biochemical networks and
pathways28−39 (Table 1). To the best of our knowledge,
i.e., a single rung-opening or rung-closing step. Species are connected
by these single elementary reactions. Strands are irreversibly separated
d
once the last remaining rung opens. The number of rules required to
generate the complete set of reactions listed in the previous column.
dynamic multivalent reactions must incorporate the many
intermediates and pathways relevant to the problem at hand.40
Time course simulations are then able to generate non-
equilibrium product distributions and reveal possible kinetic
traps.
Table 1. Applications of Rule-Based Models
In this paper we address the general class of reactions that
we refer to as “strand escape from an [n]-rung ladder”. The
fundamental significance of this reaction class is that it
generalizes the dissociation of a pair of multivalent
constituents, valency being equated to the number of rungs,
[n]. The specific chemistry does not matter here, as this model
is applicable to any type of dynamic interaction. The kinetics of
strand escape predict the time required for error correction
(i.e., constituent removal) in which backtracking requires the
simultaneous disconnection of [n] interactions. Table 2 shows
that the number of species and number of elementary reactions
associated with the reaction network increase exponentially
with the number of rungs ([n]) (see SI). By employing a
simplifying set of assumptions (vide infra), the number of rules
required to construct the network only depends linearly on [n]
(Table 2). The RBM approach to model construction enables
the simulation of complex reaction networks making it possible
to systematically probe factors such as valency, affinity, and
bond dissociation rates of dynamic multivalent processes.
topic
system modeled
reference
Kozer, N., et al.26
cellular signaling
oligomerization and
phosphorylation of EGFR
protein−protein interactions
Stites, E. C.,
et al.27
interactions between αVβ3
integrin and VEGFR2
FcεRI signaling network
Montgomery, R.,
et al.28
Chylek, L. A.,
et al.29
Hat, B., et al.30
cell fate decision-making in the
p53 system
intracellular competition for
calcium
Antunes, G.,
et al.31
Dolan, D., et al.32
DNA repair and tiles DNA damage repair by
nonhomologous end joining
DNA tile self-assembly
Amarioarei, A.,
et al.33
viral capsid assembly dynamic pathways for T1 capsids Hagan, M.,
et al.34
Schwartz, R.,
et al.35
icosahedral viral capsids
other
thermochemical routes to
biomass conversion
biome mercury exchange
Rangarajan, S.,
MODEL CONSTRUCTION
et al.36
■
We developed strand escape from an [n]-rung ladder as a
model to quantify the time needed for error correction. While
the model refers to [n]-rung ladders, the ladders are equivalent
to a set of misconnections between a pair of constituents. For
example, escape from a 5-rung ladder is equivalent to the
largest possible number of misconnections between two 6-rung
strands. It also corresponds to an error in a periodic lattice that
requires 5 simultaneous disconnections in order to remove the
misplaced constituent (Figure 2, [5]-rung error). We have
focused on the disassembly process (i.e., strand escape), rather
than the assembly process (i.e., ladder formation), to specify a
precise mode by which error correction is achieved (i.e., the
Hartman, J.,
et al.37
this work
dynamic multivalent strand escape from an [n]-rung
chemistry ladder
RBMs have not previously been used to simulate process that
are relevant to dynamic chemistry. As shown below, we have
found that RBMs are readily adapted to dynamic multivalent
chemistry to construct the reaction network and simulate the
time course of the system as it progresses toward equilibrium.
Multivalent reaction networks are combinatorially com-
plex.11 As the valency of the building block increases, the
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Figure 2. Example of errors in COF formation with [5] through [1] misconnections demonstrating that this work can be expanded to 2D and 3D
structures. The misconnected strand in highlighted in red. The light and dark gray shading of the rest of the COF is only provided to guide the eye.
Figure 3. (a) Mechanistic assumptions governing model construction include dynamic rung exchange, strand escape and absence of intermolecular
reactions (i.e., infinite dilution case), geometric specificity, and rung equivalence. (b) The network of reactions shows the reversible and irreversible
steps occurring in strand escape from the [4]-rung ladder. The network converges on two nodesthe fully bonded ladder and the free strands. The
network also shows the number of species (16) and reactions (60). As shown in Table 2, the number of intermediates and reactions grows
exponentially with increasing [n]. (c) The in-model definitions of free strand and ladder, and the rules used in the model for rung exchange and
strand escape. Full text of the model can be found in the SI.
constituent removal mode). Our model is useful for (1) testing
if this mode of error correction is the rate limiting step in a
dynamic synthesis, and (2) predicting the occurrence of
kinetically trapped structures.
Our RBM for [n]-rung ladder disassembly is based on the
assumptions described in Figure 3a. The Supporting
Information provides the details of how these assumptions
were implemented into BioNetGen,41 an RBM simulator.
Using only eight rules (Figure 3c), the complete reaction
network for strand escape from the [4]-rung ladder was
algorithmically generated (Figure 3b). Reaction arrows are
drawn between any two species related by the execution of a
single elementary step (i.e., rung association (right to left) or
rung dissociation (left to right)). Through a sequence of steps,
the [4]-rung ladder is transformed to a pair of free strands, the
shortest pathway involving four steps and three intermediates
C
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Figure 4. (a) The reaction network for escape from a [3]-rung ladder. (b) The distribution of products for a reaction where the seed species is the
three rung ladder (shaded black in (a)), which leads to a rise in two and one rung species as the rungs dynamically open and close. Eventually the
free strands separate. In this reaction kd = ka = 0.01 s−1, (c) shows the same reaction as (b) except that kd = 0.01 s−1, while ka = 0.1 s−1, leading to
kinetic trapping of both the ladder species and the two-rung species. The difference between a dynamic intermediate and a kinetically trapped
species is the lifetime of the species as compared to the reaction time. (d) The distribution of products over a reaction where kd = ka = 0.01 s−1 and
the seed species is a hairpin type structure (shaded green in (a)). (e) The same reaction as (d) except that kd = 0.01 s−1, while ka = 0.1 s−1, leading
to a steady state concentration of ladder, and 2-rung species. (f) The ratio between ladder and free strand from part (e). It peaks at a ratio of 6, at
38 s.
(Figure 3b). All the other reaction networks for ladders with
[2]- to [10]-rungs were similarly generated, as the assumptions
are general for any length ladder. Each rung is assumed to
independently undergo reversible opening (rung dissociation)
and closing (rung association). The rates of these intra-
molecular reactions are governed by first order rate coefficients
kd and ka, respectively. Cooperativity in rung binding was not
considered; in this work the rate coefficients for opening and
closing bonds are assumed to be independent of local
environment. Cooperativity (either positive or negative) is
possible by adding rules that define state dependent rates (e.g.,
the rate depends on the bonding state of adjacent rungs).
Strand escape occurs irreversibly when the final rung
dissociates. The infinite dilution case allows the model to
focus solely and specifically on the escape step of error
correction. The rate coefficient for this step is also assumed to
be kd. Since the model assumes that strand escape takes place
under conditions of infinite dilution, all intermolecular
interactions are prohibited. For simplicity, we only consider
the case where the set of rungs on one strand bond to a
contiguous set of rungs on the opposite strand without rung
crossover. The resulting reaction network (Figure 3b)
converges at two places corresponding to (1) the fully
connected [4]-rung ladder and (2) the pair of separated
strands. As will become apparent, these points of convergence
are the major species that accumulate transiently and as
equilibrium is approached. We surmise that convergent points
on reaction networks are generally either the final equilibrium
product(s) or they are intermediates that become kinetically
trapped (i.e., long-lived, persistent intermediates). Beyond [4]-
rungs, these multivalent reaction networks are difficult to
visualize, which in general, makes predicting reaction outcomes
and identifying the location of traps more challenging. Using
these models, we demonstrate the utility of RBMs for
validating synthetic design starting from an advanced
intermediate and quantify the time required for error
correction (i.e., constituent removal).
MODELING THE TIME COURSE OF STRAND
ESCAPE
■
Once the networks are generated, the time course of the
overall reaction progress is repeatedly simulated using sets of
values for ka and kd. Figure 4 shows simulation results for two
different kinetic values starting from the fully formed [3]-rung
ladder. When the rate coefficient for rung breaking is equal to
the rate coefficient for rung making, the [3]-rung ladder
(shaded black in Figure 4a) gradually disappears, passing
through the expected [2]-rung and, [1]-rung intermediates on
its way to free the strands (Figure 4b, kd = ka). When the
kinetic coefficients favor rung formation (i.e., kd < ka) there are
distinctive changes in the time dependent concentration
profiles (Figure 4c). Noteworthy are the occurrence of broad
plateaus that persist for a significant period of time but
eventually fade as the free strands are generated. These
plateaus are signatures of kinetic traps. Kinetic traps are
persistent, long-lived intermediates, where the “long” is defined
by the time scale of the experiment.
The formation of a kinetic trap is more noticeable when
starting from an advanced intermediate. We simulated escape
from a [3]-rung ladder where the starting (i.e., seed) species
was a “hairpin” (shaded green in Figure 4a). When the rate
coefficient for rung breaking is equal to the rate coefficient for
D
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rung making (ka = kd), all the intermediates on the reaction
network are still reached, including a transient appearance of
the fully bonded [3]-rung ladder (Figure 4d). When the kinetic
coefficients favor rung formation (i.e., kd < ka) the ladder
species becomes kinetically trapped for a significant period of
time (Figure 4e). These trapped species and their time
dependence are of particular interest to the experimental
chemist. For instance, the [3]-rung ladder in Figure 4e might
represent a defect in an assembly process whose “error-
correction mechanism” requires backtracking to the free
strands. To gauge the likelihood of a successful dynamic
multivalent synthesis, the chemist would need to know the
time to undo the defect (i.e., error correction via constituent
removal). In another instance, the [3]-rung ladder in Figure 4e
might represent the desired kinetic product for an assembly
process that begins from the hairpin. Synthetic planning will
benefit from knowing the time point that maximizes the yield
of desired product (i.e., the [3]-rung ladder) in addition to the
time point that maximizes the ratio (Figure 4f) of the desired
product relative to irreversible losses (i.e., free strands). In the
reaction shown in Figure 4e, the largest ratio for fully bonded
ladder to free strand is found at 38 s (Figure 4f), but the
maximum yield of ladder is found at 61 s. Which of these
maxima is the best time to stop a reaction depends on the ease
of separation of the mixture and the potential to isolate and
subsequently resubject intermediates to the dynamic con-
ditions in order to recover a second crop of desired product.
To expand on the synthetic example, we systematically
varied the rate coefficient ratio for rung dissociation (kd) and
rung formation (ka) starting from the hairpin species leading to
ladders having [2]- to [5]-rungs. By varying the rate coefficient
ratio systematically for various length ladders, we are able to
analyze the effect of kinetic coefficients and the effect of
valency on reaction progression and kinetic trapping. The
ladder lifetime (defined in Figure 5a) decreases as KD increases
(KD = kd/ka). For KD < 1, the ladder lifetime depends strongly
on valency (Figure 5b) while the maximum yield exhibits only
a slight dependence on valency (Figure 5c). Interestingly, for a
given value of KD, the time to reach maximum ladder yield is
markedly dependent on valency (Figure 5d). These results
show that a thorough understanding of the time evolution of a
system is necessary to capture the desired nonequilibrium
product distribution.
Figure 5. Different behavior is seen with different rate coefficients. In
all cases shown here the seed species was a hairpin (i.e., one rung is
closed on one end of the strands while the remaining rungs are open,
shown in (a) in schematic form). Note that KD = kd/ka. For KD < 1, ka
was held constant at 0.01 s−1 while kd was varied. For KD > 1, kd was
held constant at 0.01 s−1 while ka was varied. (a) The concentration of
[3]-ladder for three different KD when the seed species is a hairpin.
When ka is greater than or equal to kd there is a distinct time domain
during which the [n]-rung ladder formation exhibits steady state
behavior, known as the ladder lifetime. Ladder lifetime is defined as
the full width at half-maximum height (fwhm) for the ladder
concentration peak. (b) The effects of KD and rung number on the
ladder lifetime when the starting species is a hairpin. When ka is
greater than kd, ladder lifetimes increase, with the number of rungs
having a strong effect on the amount of increase. When ka is less than
kd, all ladder lifetimes converge toward zero with increasing kd. (c)
The effects of KD and rung number on the maximum percent yield of
ladder when the starting species is a hairpin. Increasing the number of
rungs decreases overall yield, although the change between [n] rungs
and [n + 1] rungs decreases with increasing [n]. (d) The effects of
rates and rung number on the time at which maximum ladder yield is
reached when the starting species is a hairpin. Time at maximum
ladder concentration decreases with large kd due to limited formation
of ladder. The same trend is seen with large ka due to rapid formation
of ladder and minimal separation into free strand.
using a dynamic multivalent synthesis involving a building
block having a particular valency, the kinetic coefficient needed
for all species to remain in the dynamic pool may limit the
choice of viable dynamic chemistries according to the ranges
specified in Figure 1.
QUANTIFYING THE TIME REQUIRED FOR ERROR
■
CORRECTION
To expand on the error-correction model for a dynamic
multivalent synthesis that targets an equilibrium product, we
systematically examined how strand escape times of 1 h and 1
day depend on rung dissociation rate coefficient and valency
(Figure 6). The plot in Figure 6 is for a case that is likely to
result in kinetic traps (i.e., where the rung exchange
equilibrium favors rung formation (KD = 0.1)) (results for
KD equal to 1 and 10 are shown in SI Figure S4). Times of 1 h
and 1 day were selected to represent error-correction times
that are of significance to a dynamic multivalent synthesis.
Times that are shorter than 1 h are characteristic of [n]-rung
species that remain in the “dynamic pool” on a reasonable
laboratory time frame. In contrast, times longer than 1 day are
characteristic of [n]-rung species that are in the “static pool” on
a reasonable laboratory time frame. Error correction is seen to
become unlikely for high-valent systems or systems with small
dissociation rate coefficients. To achieve an equilibrium target
The relationship between number of rungs, ladder escape
half-life, and the rung exchange rate coefficients are reasoned
with a simple probabilistic framework (eq 1, full derivation in
the SI) that predicts useful relationships between the valency
of the species ([n]), the kinetic parameters (ka and kd), and the
strand escape half-life (te1s/c2p, i.e., the time at which half of the
strands have escaped the ladder architecture).
[n]
i
j
j
j
j
j
y
z
z
z
z
z
1 + K
KD
ln(2)
kd
escp
D
t
=
×
1/2
(1)
k
{
escp
1/2
This equation predicts a power law relationship between t
and the valency of the species, consistent with the results
plotted in Figure 6. The equation also predicts an inverse
relationship between ladder escape half-life and kd, which we
have confirmed in simulation (SI Figures S2 and S3). The
escp
dependency of t on [n] and kd is evident in Figure 6. Half-
1/2
E
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Article
enough to allow backtracking as needed to correct errors. By
combining experimentally determined kinetic parameters and
RBMs, we believe that chemists will be able to rationally design
increasingly complicated synthetic targets. While this pub-
lication has focused on 1D [n]-rung ladder misconnection
types, it is possible to extend this approach to more complex
structures such as 2D materials and 3D MOFs and COFs. We
are currently working to combine experimentally derived
kinetic coefficients and RBMs to rationally design advanced
constructs. Future models will include intermolecular reactions
to describe the complete synthetic pathway, including the off-
target pathways (i.e., error formation and reversal).
ASSOCIATED CONTENT
■
S
* Supporting Information
The Supporting Information is available free of charge at
https://pubs.acs.org/doi/10.1021/jacs.9b08958.
Simulation files and information, additional graphs, full
derivation of the probabilistic framework presented, and
explanation of the data in Figure 1 (PDF)
Figure 6. Error correction profile for KD = 0.1 (an example that
generates kinetic traps). The model predicts the dissociation rate
escp
1/2
coefficient (kd) needed to achieve specific ladder escape half-lives t
based on the number of rungs in the ladder. The dissociation rate
AUTHOR INFORMATION
escp
■
coefficient (kd) needed to achieve a t of 1 h or less is shaded green
1/2
escp
1/2
while the dissociation rate coefficient (kd) needed to achieve a t of
Corresponding Author
*jsmoore@illinois.edu
ORCID
Morgan M. Cencer: 0000-0003-2806-8317
Andrew J. Greenlee: 0000-0002-9085-6309
Jeffrey S. Moore: 0000-0001-5841-6269
1 day or more is shaded red. Additional information is in SI Figures
escp
1/2
S3 and S4. Any species with a t of an hour is considered within the
dynamic pool and will lead to equilibrium DC synthesis. Any species
escp
1/2
with a t of a day (or more) is outside of the dynamic pool and will
lead to nonequilibrium DC synthesis and kinetic trapping. Interesting
behavior and trapping are also found in the region between those two
times.
Notes
The authors declare no competing financial interest.
lives are highly dependent on both the valency and the rate
coefficients. Accounting for the impact of both valency and
kinetics is critical for rationally designing a dynamic multi-
valent synthesis.
ACKNOWLEDGMENTS
■
This material is based upon work supported by the National
Science Foundation under Grant No. CHE 16-10328 and
Grant No. CHE 19-04180. We are deeply grateful to Dr. Chris
Pattillo, Summer Laffoon, and Professor Corrin Clarkson for
valuable discussions.
Imine ladder formation provides a qualitative example of
how these findings are applied to an experimental system. In
Figure 1, we see that the fastest imine metathesis has a kd of
about 10−3 s−1. With this kd we predict that three rung ladders
are likely to experience some trapping as their escape half-life is
longer than 1 h, and that four or more rungs are likely to see
significant kinetic trapping as escape half-lives for these species
are 1 day or longer (Figure 6). Experimentally, it has been
found that imine ladders of four or more rungs are indeed
kinetically trapped.42 Raising the reaction temperature notably
increased the amount of strand scrambling. By raising the
temperature, the value of kd shifts to a larger value, returning
misconnected strands back into the dynamic pool. Exper-
imental determination of bond dissociation rates can thus
establish the likelihood that a system will form kinetic traps.
Using our model along with experimentally obtained kinetic
details will help guide the researcher about the conditions for
which error correction is rapid, thus avoiding lengthy
optimization studies.
REFERENCES
■
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constituent, seed species, and kinetic parameters of the system
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