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\begin{document}

\title{A Kinetic Framework for Modeling Oleochemical Biosynthesis in E. coli}



\author[1]{Jackson Peoples}%
\author[1]{Sophia Ruppe}%
\author[1]{Kathryn Mains}%
\author[1]{Elia C. Cipriano}%
\author[1]{Jerome Fox}%
\affil[1]{University of Colorado Boulder}%


\vspace{-1em}



  
  \date{May 27, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
Microorganisms build fatty acids with biocatalytic assembly lines, or
fatty acid synthases (FASs), that can be repurposed to produce a broad
set of fuels and chemicals. Despite their versatility, the product
profiles of FAS-based pathways are challenging to adjust without
experimental iteration, and off-target products are common. This study
uses a detailed kinetic model of the \emph{E. coli} FAS as a foundation
to model nine oleochemical pathways. These models provide good fits to
experimental data and help explain unexpected results from \emph{in
vivo} studies. An analysis of pathways for alkanes and fatty acid ethyl
esters, for example, suggests that reductions in titer caused by enzyme
overexpression can result from shifts in pools of metabolic
intermediates that are incompatible with the substrate specificities of
downstream enzymes. In general, different engineering objectives (i.e.,
production, unsaturated fraction, and average chain length) show
experimentally consistent sensitivities to pathway enzymes, and
model-based compositional analyses indicate simple shifts in enzyme
concentrations can alter the product profiles of pathways with
promiscuous components. The study concludes by integrating all models
into a graphical user interface. The models supplied by this work
provide a versatile kinetic framework for studying oleochemical pathways
in different biochemical contexts.%
\end{abstract}%



\sloppy


A Kinetic Framework for Modeling Oleochemical Biosynthesis in \emph{E.
coli}

Jackson Peoples\textsuperscript{a}, Sophia Ruppe\textsuperscript{a},
Kathryn Mains\textsuperscript{a}, Elia C. Cipriano\textsuperscript{a},
and Jerome M. Fox\textsuperscript{a*}

\textsuperscript{a}Department of Chemical and Biological Engineering,
University of Colorado, Boulder,

3415 Colorado Avenue, Boulder, CO, 80303

*To whom correspondence should be addressed. E-mail:
jerome.fox@colorado.edu

\textbf{Abstract}

Microorganisms build fatty acids with biocatalytic assembly lines, or
fatty acid synthases (FASs), that can be repurposed to produce a broad
set of fuels and chemicals. Despite their versatility, the product
profiles of FAS-based pathways are challenging to adjust without
experimental iteration, and off-target products are common. This study
uses a detailed kinetic model of the \emph{E. coli} FAS as a foundation
to model nine oleochemical pathways. These models provide good fits to
experimental data and help explain unexpected results from \emph{in
vivo} studies. An analysis of pathways for alkanes and fatty acid ethyl
esters, for example, suggests that reductions in titer caused by enzyme
overexpression can result from shifts in pools of metabolic
intermediates that are incompatible with the substrate specificities of
downstream enzymes. In general, different engineering objectives (i.e.,
production, unsaturated fraction, and average chain length) show
experimentally consistent sensitivities to pathway enzymes, and
model-based compositional analyses indicate simple shifts in enzyme
concentrations can alter the product profiles of pathways with
promiscuous components. The study concludes by integrating all models
into a graphical user interface. The models supplied by this work
provide a versatile kinetic framework for studying oleochemical pathways
in different biochemical contexts.

\textbf{1 \textbar{} INTRODUCTION}

Microorganisms can produce a diverse set of oleochemicals from renewable
feedstocks and are centrally important to the emerging green bioeconomy
(Akinsemolu, 2018; Kumar and Kumar, 2017). Microbially produced
alcohols, esters, olefines, alkanes, ketones, and polyesters are
promising transportation fuels (Bao et al., 2016; Choi and Lee, 2013;
Teo et al., 2015), herbicides (Kim et al., 2015), lubricants (Rui et
al., 2015), flavors (Lian and Zhao, 2015), fragrances (Goh et al., 2012;
Sherkhanov et al., 2016), and polymer additives (Bowen et al., 2016;
Clomburg et al., 2015). In engineered microbes, oleochemical
biosynthesis typically begins with a fatty acid synthase (FAS), which
builds acyl-ACPs of varying lengths, and concludes with the enzymatic
conversion of acyl-ACPs to target products ((Sarria et al., 2017; White
et al., 2005); Fig. 1). Despite the versatility of FAS-based systems,
their activities are challenging to tune---a reflection of their
sensitivity to the concentrations, kinetics, and substrate specificities
of multiple enzymes (Mains et al., 2022; Ruppe and Fox, 2018; Tan et
al., 2018; Xu et al., 2013). This challenge is illustrated by ongoing
efforts to engineer oleochemical pathways with well-defined product
profiles (Blitzblau et al., 2021; Haushalter et al., 2014; Hern\selectlanguage{ngerman}ández
Lozada et al., 2018; Hernández Lozada et al., 2020); pathway
optimization typically requires iteration, and off-target products are
common (Grisewood et al., 2017; Sarria et al., 2018).

Mathematical models can facilitate the systematic analysis of cellular
metabolism and have guided the design of oleochemical-producing strains.
Genome-scale models use the stoichiometries of metabolic networks to
predict flux under different growth conditions (Machado and Herrgård,
2015; Zhang and Hua, 2016); when paired with flux balance analysis
(FBA), they can reveal genetic adjustments (e.g., gene deletion) likely
to re-route flux toward specific products (Agren et al., 2013; Ravi and
Gunawan, 2021; Yoshikawa et al., 2017).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Figure 1.} Oleochemical biosynthesis in \emph{E. coli} . The
fatty acid synthase (FAS) of\emph{E. coli} (orange) can supply
precursors for a broad set of oleochemicals. Engineered pathways tend to
exploit either (i) acyl-ACPs, which are important intermediates in fatty
acid synthesis, or (ii) free fatty acids, which are released by
thioesterase-catalyzed hydrolysis of acyl-ACPs. This study focuses on a
subset of oleochemicals (boxes) and enzymatic steps (solid lines).
Colors denote enzymatic steps unique to specific products (see
Nomenclature for full enzyme names). Pathways not included in this study
appear as dotted lines; enzymes that are not included appear in gray.

Unfortunately, these models typically rely on fixed metabolite
concentrations and steady-state assumptions that limit their ability to
capture cellular dynamics. Kinetic models derived from reduced
genome-scale models can reproduce some dynamic responses to cellular
perturbations (e.g., a glucose spike; (van Rosmalen et al., 2021;
Stanford et al., 2013)). These models, however, typically lack the
resolution required to predict how experimentally relevant changes in
the concentrations, kinetics, or substrate specificities of multiple
enzymes affect the outputs of complex metabolic processes.

In recent work, we developed a mechanistic kinetic model of the type II
FAS of \emph{E. coli} and used it to determine how different FAS
components work together to control fatty acid synthesis. Type II FASs,
which consist of discrete monofunctional subunits, are common in
bacteria and plant plastids. We modeled the \emph{E. coli} FAS as a
well-mixed reaction with fixed enzyme concentrations, a constraint
supported by the tendency of engineered strains to overproduce free
fatty acids in stationary phase (Cho and Cronan, 1995). Our model
explained a variety of perplexing results from the literature (e.g., the
inhibitory effects of enzyme overexpression) and provided new strategies
for engineering FAS systems. Recently, we used it to explore the effects
of enzyme concentration on fatty acid synthesis. Using our model
alongside \emph{in vitro} and \emph{in vivo} experiments, we showed that
simple changes in enzyme concentration can enhance the titers of
specific chain lengths by as much as 125-fold (Mains et al., 2022). Our
analysis illustrates how kinetic models can guide metabolic engineering.

In this study, we expanded our kinetic model by adding pathways for a
diverse set of oleochemicals. These pathways pose a challenge because
they include enzymes that are poorly characterized, relative to FAS
components. In adding them, we focused on three goals: (i) to develop a
framework for incorporating new biosynthetic steps, including those
catalyzed by poorly characterized enzymes, (ii) to identify the most
influential steps within each pathway, and (iii) to build a graphical
user interface (gui) that facilitates model implementation. With these
tasks, we sought to build a versatile set of tools for designing
oleochemical-producing microbes.

\textbf{2 \textbar{} MATERIALS AND METHODS}

\textbf{2.1 \textbar{} Solution and Optimization of the Kinetic Model}

Our models consist of systems of rate equations and mass balances. We
solved them by using the MATLAB solver ode15s with relative and absolute
error tolerances of 10\textsuperscript{-6} and a vectorization step to
reduce solve time. We parameterized the initial model by using
parameters reported in experimental studies of purified enzymes or prior
modeling analyses, and we adjusted these parameters by using the MATLAB
function fminsearch (a derivative-free minimization function) to carry
out the fitting routines described below.

We optimized our newly refined model of fatty acid biosynthesis by using
a series of objective functions. To incorporate the contributions of
FabF and FabZ to unsaturated fatty acid synthesis, we added reactions
identical to those used for FabB and FabA, respectively (Tables S1-S2),
and we adjusted values of k\textsubscript{cat} for FabA, FabB, FabZ, and
FabF by using Obj\textsubscript{1}, where

Obj\textsubscript{1} = SSE\textsubscript{1} (Eq. 1)

SSE\textsubscript{1} is the sum of squared errors between measured and
predicted unsaturated fractions generated by experimentally
reconstituted FASs lacking each enzyme (i.e., we normalized unsaturated
fractions by the unsaturated fraction generated by the complete FAS;
Tables S3-S4; (Ruppe et al., 2020)). To incorporate the contributions of
FabF and FabB to initiation, we used optimization routines that include
SSE\textsubscript{2}, SSE\textsubscript{3}, and SSE\textsubscript{4},
the respective sums of squared errors

Obj\textsubscript{2} =
SSE\textsubscript{2}\selectlanguage{ngerman}·SSE\textsubscript{3}·SSE\textsubscript{4} (Eq. 2)
Obj\textsubscript{3} =
(1+SSE\textsubscript{2})·(1+SSE\textsubscript{3})·(1+SSE\textsubscript{4})
(Eq. 3) Obj\textsubscript{4} =
(1+SSE\textsubscript{2})·(1+SSE\textsubscript{3})\textsuperscript{2}·(1+SSE\textsubscript{4})
(Eq. 4)

for the time course, product distribution, and initial rates of fatty
acid synthesis between modeled and reconstituted FASs. Our optimization
process included three steps carried out in series: (i) We removed the
production of C\textsubscript{20} fatty acids, which are not observed
experimentally, and used Obj\textsubscript{2} to fit the 18 scaling
parameters described in our prior work (Table S5; (Mains et al., 2022)).
(ii) Our initial fit caused an overaccumulation of FabB·acetyl-CoA (Fig.
S2). To address this issue, we replaced scaling parameters for the
dissociation of FabF·acetyl-ACP and the reversal of FabB activation by
acetyl-CoA with parameters that scale FabF binding to malonyl-ACP and
FabB binding to acetyl-CoA. We used Obj\textsubscript{3} to optimize
these two parameters and two that scale FabF- and FabB-catalyzed
decarboxylation of malonyl-ACP (Table S1). (iii) We improved the fit for
the product profile by using Obj\textsubscript{4} to fit the same four
parameters. Figure S1 shows the results of our optimization. Hereafter,
we refer to this model as our ``base model''. We used it to build the
oleochemical-specific models, which used similar optimization routines
(SI Methods)

\textbf{2.2 \textbar{} Sensitivity analysis}

We used the Morris Method to examine the sensitivity of different
objectives to various model parameters (e.g., enzyme concentrations or
kinetic constants; (Morris, 1991)). This method calculates a set of
derivatives (i.e., the change in an objective with respect to a
specified parameter) by sampling randomly within a range of parameter
values. Briefly, we used the SAFE toolbox with a radial method and Latin
hypercube sampling (r = 1,000 trajectories), and verified the
convergence of the elementary effects by calculating the first
derivative of each parameter's normalized elementary effects with
respect to the number of model evaluations (i.e.,
\textbar{}dEE/dn\textbar{} \textless{} 0.0002) before reporting them
(Pianosi et al., 2015). We examined kinetic parameters associated with
methyl ketone biosynthesis (Fig. S6) by using parameter ranges of
0.01-100 s\textsuperscript{-1}, and we explored the sensitivity of
different biochemical objectives to enzyme concentrations (Fig. 4) by
using 0.01-100 \selectlanguage{greek}μ\selectlanguage{english}M for all enzymes except those in the alkane pathway;
for this low-producing pathway, the narrower range of 1-100 \selectlanguage{greek}μ\selectlanguage{english}M enzyme
facilitated convergence.

\textbf{2.3 \textbar{} Analysis of competing objectives}

We constructed ``phase diagrams'' that depict tradeoffs between average
chain length and total production for each alcohol pathway by optimizing
enzyme concentrations to maximize or minimize average chain length at
different production levels. Briefly, to maximize average chain length,
we minimized Obj\textsubscript{5}, where L\textsubscript{norm} is the
normalized average chain length (i.e., average

Obj\textsubscript{5} = (1-L\textsubscript{norm})\textsuperscript{2} +
(Prod\textsubscript{target}- Prod)\textsuperscript{2} +
(Enz\textsubscript{target}- Enz)\textsuperscript{4} (Eq. 5)

Obj\textsubscript{6} = (4/18-L\textsubscript{norm})\textsuperscript{2} +
(Prod\textsubscript{target}- Prod)\textsuperscript{2} +
(Enz\textsubscript{target}- Enz)\textsuperscript{4} (Eq. 6)

chain length divided by 18, the maximum chain length),
Prod\textsubscript{target} is the desired alcohol concentration
(palmitic acid equivalents), Prod is the actual alcohol concentration,
Enz\textsubscript{target} is the desired total enzyme concentration, and
Enz is the actual enzyme concentration. To minimize average chain
length, we minimized Obj\textsubscript{6}, where 4/18 is the minimum
achievable chain length. For all models, we constrained total enzyme
concentration (Enz\textsubscript{target}) by supplementing the total
concentration in our base model with 10 \selectlanguage{greek}μ\selectlanguage{english}M of each pathway-specific
enzyme: 38 \selectlanguage{greek}μ\selectlanguage{english}M for a carboxylic acid reductase (CAR) model, 38 or 48 \selectlanguage{greek}μ\selectlanguage{english}M
for two acyl-CoA reductase (ACR2 and ACR1) models, and 18 or 108 \selectlanguage{greek}μ\selectlanguage{english}M for
an acyl-ACP reductase (ATR) model. During each optimization, the
concentrations of specific enzymes changed; we retained only results for
which differences in both (i) Prod\textsubscript{target} and Prod and
(ii) Enz\textsubscript{target} and Enz differed by less than 5\%.

\textbf{2.4 \textbar{} Graphical user interface}

We integrated our final models into a graphical user interface (gui) by
using MATLAB App Designer. This software package facilitates the design
of interactive interfaces with drag-and-drop tools that link to a MATLAB
script. We coded the functionality of each added feature (e.g., drop
down menus or text boxes) by using the development window, and we
adjusted each model slightly to enable integration with the back-end
code for the resulting gui.

\textbf{3 \textbar{} RESULTS AND DISCUSSION}

\textbf{3.1 \textbar{} A Refined Model for Fatty Acid Synthesis}

In prior work, we developed a detailed kinetic model of the FAS
of\emph{E. coli} by modeling the nine enzymes necessary to convert
malonyl-CoA and acetyl-CoA to free fatty acids (Fig. 1) (Mains et al.,
2022; Ruppe et al., 2020; Ruppe and Fox, 2018). The model uses rate
equations based on detailed reaction mechanisms reported in the
literature and includes separate association and dissociation steps for
all heteromeric complexes. In this study, we refined our model by
incorporating commonly ignored secondary activities of FabZ
(\selectlanguage{greek}β-\selectlanguage{english}hydroxyacyl-ACP dehydratase), FabB (\selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP synthase I), and
FabF (\selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP synthase II).

We began by focusing on unsaturated fatty acid synthesis. Previous
studies indicate that FabF and FabZ can produce unsaturated fatty acids,
although less efficiently than FabA (\selectlanguage{greek}β-\selectlanguage{english}hydroxy-decanoyl-ACP dehydratase)
and FabB (Cronan et al., 1969; Feng and Cronan, 2009; Silbert and
Vagelos, 1967). We incorporated this set of secondary activities by
adding rate equations describing (i) the allylic rearrangement
of\emph{trans} -dec-2-enoyl-ACP to \emph{cis} -dec-3-enoyl ACP by FabZ
and (ii) the condensation of \emph{cis} -dec-3-enoyl-ACP with
malonyl-ACP by FabF. The updated model captured the relative differences
in unsaturated fractions between reconstituted FASs lacking FabA, FabB,
FabZ, or FabF but underpredicted concentrations of unsaturated products
(Tables S3 and S4). This finding is consistent with our observation that
\emph{in vitro} systems tend to produce larger unsaturated fractions
than\emph{in vivo} systems (Mains et al., 2022; Ruppe et al., 2020),
which we used to optimize the original model (and which might reasonably
bias the final model toward smaller unsaturated fractions)

Next, we incorporated the contributions of FabF and FabB to acyl chain
initiation. Despite their primary role in acyl-ACP elongation, FabF and
FabB can also generate \selectlanguage{greek}β-\selectlanguage{english}ketobutyryl-ACP, an intermediate that initiates
fatty acid synthesis (i.e., it is the primary product of \selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP
synthase III, or FabH). To add this activity, we introduced reactions
for (i) FabB- and FabF-catalyzed decarboxylation of malonyl-ACP, (ii)
FabB-catalyzed condensation of malonyl-ACP with acetyl-ACP or acetyl-CoA
to form \selectlanguage{greek}β-\selectlanguage{english}ketobutyryl-ACP, and (iii) FabF- catalyzed condensation of
malonyl-ACP with acetyl-ACP to form \selectlanguage{greek}β-\selectlanguage{english}ketobutyryl-ACP (Table S1; (Mains
et al., 2022)). We fit this newly expanded model to initial rates and
product profiles of experimentally reconstituted FASs (Fig. S1). The
final optimized model recreated an unusual C\textsubscript{18}-rich
product profile to which it was not fit (Fig. S3)---an indication that
it adequately captures the contributions of different core FAS enzymes
to fatty acid synthesis.

\textbf{3.2 \textbar{} Biosynthesis of oleochemicals}

We expanded our model to include various oleochemicals by adding
Michaelis-Menten parameters for new enzymes (SI Methods). These
parameters, which are commonly reported in the literature, are
reasonable when enzyme-substrate complexes maintain steady-state
concentrations during oleochemical production---a reasonable assumption
in stationary phase, where protein concentrations are approximately
constant, and rates of cell formation and death are approximately equal
(Li et al., 2014; Pletnev et al., 2015). With these parameters in hand,
we assembled models for the biosynthesis of alcohols, alkanes, methyl
ketones, fatty acid methyl esters (FAMEs), and fatty acid ethyl esters
(FAEEs) by carrying out two steps: (i) We added estimated values of
k\textsubscript{cat} and K\textsubscript{m} for the requisite enzymes
(Fig. 1), and (ii) we adjusted enzyme concentrations and kinetic
parameters by fitting our model to the product profiles of engineered
strains (Figs. 2 and 3). We note: In all fits, we scaled titers reported
from experimental cultures to titers expected for \emph{in vitro}systems
at 12 minutes (SI Methods). Our models provided good fits to
experimental data with two notable exceptions: Tow models for methyl
ketone biosynthesis underpredicted the production of either (i)
2-nonanone by a pathway containing \emph{Umbellularia
californica}UcFatB1 (BTE), a thioesterase specific for short chains
(strain 5 in Fig. 3A), or (ii) palmitic acid by TesA, a thioesterase
specific for medium chains that is native to \emph{E. coli} (Fig. 3B).
The underprediction of 2-nonanone matches the narrow product profile
that we used to parameterize BTE; the associated study may have
overlooked minority products. For palmitic acid production by TesA, the
large experimental concentration may result from contamination by
membrane lipids, which are hydrolyzed in common extraction procedures
(Grisewood et al., 2017). Importantly, we recreated the experimental
profile by reducing k\textsubscript{cat} for fatty acid-CoA ligase
(FadD) on palmitic acid by four-fold (Fig. S8), but this adjustment is
inconsistent with the reported substrate specificity of FadD (i.e., it
requires the k\textsubscript{cat} for C\textsubscript{16} acyl-CoAs to
be lower than for C\textsubscript{14} and C\textsubscript{18} substrates
(Arora et al., 2005)). In the absence of a strong experimental
justification for re-parameterizing the enzymes in question---that is, a
rationale for using one experimental dataset over another---we left
their kinetic parameters unchanged.

Our modeling analysis allowed us to classify oleochemical pathways by
the availability of kinetic data and the required fitting procedure. We
created three groups: (i) Pathways with\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Figure 2.} Biosynthesis of fatty alcohols and alkanes. (A) We
optimized a model that generates alcohols with a carboxylic acid
reductase (CAR) by fitting the product profile of a strain of \emph{E.
coli} overexpressing four enzymes: a thioesterase (TE), a CAR, an
aldehyde reductase (AHR), and a 4'-phosphopantetheinyl transferase (Sfp;
(Kalim Akhtara et al., 2013)). (B) We optimized our model for alkane
biosynthesis by fitting the product profile and titer of a strain
overexpressing both an acyl-ACP reductase (AAR) and an aldehyde
deformylating oxygenase (ADO; (Song et al., 2016)). Table S9 describes
the compositions of the modeled pathways.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure 3.} Biosynthesis of methyl ketones and FAEEs. (A-B) We
optimized our first set of models for methyl ketone biosynthesis with
simultaneous fits to two datasets: (A) the product profiles and titers
of five strains that overexpress a thioesterase (TE), a fatty-acid-CoA
ligase (FadD), an acyl-CoA dehydrogenase (FadE), an oxidoreductase
(FadB), and an acyl-CoA thioesterase (FadM; (Yan et al., 2020)), and (B)
the ratio of C\textsubscript{12} fatty acids to C11 methyl ketones for a
sixth strain with similar enzymes overexpressed (Goh et al., 2012).
(C-D) We optimized models that generate fatty acid methyl esters (FAMEs)
and fatty acid ethyl esters (FAEEs) by fitting the product profiles and
titers of strains that overexpress (C) an O-methyltransferase (MT;
(Sherkhanov et al., 2016)) or (D) both a wax ester synthase (WS) and a
TE, (Steen et al., 2010)). Table S9 describes the compositions of the
modeled pathways.

well-characterized enzymes (e.g., an alcohol pathway that uses
carboxylic acid reductase, or

CAR, and a thiolase-based methyl ketone pathway discussed later). For
these pathways, we fit experimental data by adjusting either a single
kinetic parameter or multiple enzyme concentrations (but not both). (ii)
Pathways with partially characterized enzymes (e.g., the alkane, FAME,
and FAEE pathways). Here, we optimized a single substrate-specific
k\textsubscript{cat} and either (a) multiple enzyme concentrations or
(b) an overall k\textsubscript{cat} vector (i.e., a single factor
multiplied by all substrate-specific k\textsubscript{cat}s). (iii)
Pathways with poorly characterized enzymes (e.g., the \selectlanguage{greek}β-\selectlanguage{english}oxidation-based
methyl ketone pathway and several alcohol pathways discussed later).
Here, we optimized multiple enzyme-specific kinetic parameters. For the
methyl ketone pathway, which is poorly characterized, we used a
sensitivity analysis to identify the most influential kinetic parameters
before fitting. Overall, the reasonable fits afforded by our models
indicate that Michaelis-Menten parameters provide an adequate means of
modeling the activities of enzymes from different oleochemical pathways
(Figs. 2-3).

We parameterized enzymes consistently across models. For each enzyme, we
adjusted kinetic parameters only once, and we maintained consistent
concentrations between pathways. For example, when optimizing pathway
that generates alcohols via a CAR and an aldehyde reductase (AHR), we
changed the concentrations of FabA, FabZ, FabB, and FabF---all core FAS
enzymes---and retained these concentrations for all models; this
constraint is consistent with the native FAS expression levels exploited
in most engineered strains. For heterologously expressed enzymes, by
contrast, we occasionally adjusted concentrations. TesA, in particular,
is common in engineered strains, but its overexpression relies on
different promoters and ribosome binding sites (Barrick et al., 1994;
Lozano Terol et al., 2021); variability in its concentration between
strains is reasonable. In general, the consistent parameterization of
enzymes in different pathways suggests that our models can capture the
activities of important pathway enzymes in different biochemical
contexts.

\textbf{3.3 \textbar{} A sensitivity analysis of different models.}

Enzyme concentration is an important design consideration for engineered
pathways, where it can be controlled with carefully selected promoters,
ribosome binding sites, and dynamic transcriptional systems (Meyer et
al., 2019; Salis et al., 2009; Zhao et al., 2018). We used a global
sensitivity analysis to determine how changes in enzyme concentrations
influence three important engineering objectives---the total production,
unsaturated fraction, and average length of oleochemical products (Fig.
4). Our results indicate that total production is most sensitive to the
concentrations of acyl-ACP thioesterase and downstream
oleochemical-specific enzymes, while product profile (i.e., unsaturated
fraction and average chain length) is most sensitive to concentrations
of acyl-ACP thioesterase and core FAS enzymes. These sensitivities are
consistent with the experimentally observed improvements in titer (and
constant product profiles) that result from the overexpression of
oleochemical-specific enzymes (Song et al., 2016; Yan et al., 2020).

The specific effects revealed by our analysis are consistent
with\emph{in vivo} studies. We will briefly comment on a few examples:
Starting with the alcohol pathway, we found that concentrations of TesA,
CAR, and AHR have a pronounced influence total production, while TesA
and FabF affect average chain length. The original study of this pathway
did not examine each enzyme in isolation---a difficult feat for an
experimental analysis---but it found that the incorporation of a highly
active AHR could increase alcohol production (Kalim Akhtara et al.,
2013); additionally, shifts in fatty acid compositions produced
corresponding shifts in alcohol\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure 4} : Sensitivity analyses of oleochemical pathways. We
used a global sensitivity analysis to examine the influence of enzyme
concentrations on three distinct biochemical objectives: total
production, unsaturated fraction, and average length of oleochemical
products. In our models, total production is most strongly influenced by
the concentrations of either (i) the thioesterase and a single
product-specific enzyme or (ii) two product-specific enzymes, while
unsaturated fraction and average chain length are most sensitive to the
concentrations of core FAS enzymes.

profiles. For the methyl ketone pathway, our results indicate that
thioesterase concentration has an outsized influence on both total
production and chain length, but also suggest that FadD and
\selectlanguage{greek}β-\selectlanguage{english}ketoacyl-CoA thioesterase (FadM) can affect both objectives. This
finding is consistent with \emph{in vivo} data showing that the
thioesterase, FadD, and FadM need overlapping substrate specificities to
maximize the production of specific chain lengths. Importantly,
several\emph{in vivo} studies have showed that overexpression of
acyl-coenzyme A dehydrogenase (FadE) can improve titers of methyl
ketones, but our sensitivity analysis does not reproduce this influence;
this discrepancy suggests that FadE may be overactive in our model (Yan
et al., 2020). We conclude our brief comparison of general effects by
noting that in both model and experiment, the concentrations of final
enzymes in the FAME and FAEE pathways (i.e., the O-methyltransferase
(MT) and the wax synthase (WS), respectively) strongly influence total
production, perhaps a result of their low activities, relative to other
enzymes (Sherkhanov et al., 2016; Steen et al., 2010).

We used our models to investigate several intriguing effects
from\emph{in vivo} studies. An experimental study of the alkane pathway
examined different expression levels of acyl-ACP reductase (AAR) and
(ii) an aldehyde deformylating oxygenase (ADO) by using alternative
plasmids, promoters, and pseudo-operon configurations (Song et al.,
2016). Shifts in the expression of both enzymes had a prominent
influence on total production, but not product profile. We recreated
this effect by changing the AAR:ADO ratio in our model; production
peaked at an intermediate ratio, while average chain length stayed
constant (Figs. 5A-5B). This behavior suggests that AAR and ADO catalyze
rate-limiting steps and must be overexpressed at similar levels to
improve titer. Their inability to affect product profile, in turn,
probably reflects their narrow substrate specificities (i.e.,
C\textsubscript{16} and C\textsubscript{18}). Intriguingly, the same
experimental study also showed that the overexpression of FabH and FabB
can reduce alkane production. Our model\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure 5} . Concentration effects in alkane and FAEE pathways.
The FAS supplemented with (A-C) an acyl-ACP reductase (AAR) and an
aldehyde deformylating oxygenase (ADO) or (D-F) an acyl-ACP thioesterase
specific for medium chains (TesA), a fatty-acid-CoA ligase (FadD), and a
wax synthase (WS). (A) Alkane production peaks at a high AAR:ADO ratio,
which does not alter (B) average chain length. (C) Overexpression of
\selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP synthase III (FabH) or \selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP synthase I (FabB)
decrease alkane production \emph{in vivo} (black) and in our model
(i.e., 1 \selectlanguage{greek}μ\selectlanguage{english}M to 100 \selectlanguage{greek}μ\selectlanguage{english}M; white with black hatching). (D) Increasing
concentrations of WS increase FAEE production, while TesA decreases it;
FadD has no effect. (E) High concentrations of TesA increase the
production of short-chain acyl-CoAs at the expense of long-chain
acyl-CoAs, the preferred substrates of WS. (F) FAEE production decreases
as a result. In A-F, the concentrations of pathway enzymes not mentioned
are unchanged from Figs. 2B and 3B.

recreates this effect and suggests that it results from a reduction in
appropriately sized acyl-ACPs (Fig. 5C). In brief, FabH overexpression
increases short-chain acyl-ACPs, at the expense of long chains, while
FabB overexpression sequesters long-chain acyl-ACPs in FabBw\selectlanguage{english}ACP
complexes (Fig. S10). Both effects reduce the pool of substrates for
AAR.

Next, we turned to the FAEE pathway. In an early study of this pathway,
TesA overexpression reduced FAEE titers (Steen et al., 2010). We
speculated that this effect, which is recreated in our model (Fig. 5D),
might result from a mismatch between (i) the pool of available acyl-ACPs
and (ii) the substrate specificity of WS. Our model allowed us to
explore this mismatch. As expected, high concentrations of TesA
increased the production of short-chain acyl-ACPs at the expense of long
chains, which are the preferred substrates of WS (Figs. 5E-5F). Our
focused analyses of alkane and FAEE production illustrates the
importance of kinetic models that capture changes in concentrations of
length-specific metabolic intermediates.

\textbf{3.4 \textbar{} A comparison of competing objectives in
alternative alcohol pathways.}

Our prior work on fatty acid biosynthesis indicates that coordinated
shifts in enzyme concentration can tune average chain length---a
critical design objective of engineered systems---without altering total
production (Mains et al., 2022). We sought to evaluate this engineering
strategy in four pathways for alcohol biosynthesis, where control over
length remains difficult. We began by supplementing our first alcohol
pathway, which relies on FadD, a CAR, and an aldehyde reductases (AHR),
with three additional pathways that extend the core FAS with 1-4
enzymes: (i) an acyl-ACP reductase (ATR), (ii) a TesA, FadD, and an
acyl-CoA reductase (ACR2), or (iii) TesA, FadD, a different acyl-CoA
reductase (ACR1), and an aldehyde reductase (AHR). As before, we added
new enzymes to our base model and optimized poorly characterized kinetic
parameters with fits to experimental data (Fig. S5). With our four
alternative models in hand, we examined the range of average chain
lengths accessible with simple shifts in enzyme concentration.
Intriguingly, for all pathways, the breadth of chain lengths decreases
as total production increases, and the highest production levels
culminate in a single average chain length (Fig. 6). This tradeoff is
sharpest for the ATR and ACR1 pathways (Figs. 6B and 6D), where critical
enzymes---namely ATR and ACR1---have very narrow substrate
specificities. The CAR and ACR2 pathways, by contrast, enable broad
control until very high production levels, with the CAR pathway
providing the greatest flexibility (Fig. 6A and 6C). The control
afforded by this pathway reflects the broad substrate specificities of
all enzymes downstream of the FAS.

An analysis of enzyme compositions at high and low average chain lengths
reveals the mechanisms of control (Dataset S1). For the CAR, ACR1, and
ACR2 pathways, high concentrations of TesA and low concentrations of
FabF and FabB facilitate the production of short-chain alcohols, while
the opposite shifts encourage long-chain products. This trend is
consistent with influence of TesA, FabF, and FabB on fatty acid
biosynthesis (Mains et al., 2022). Results also suggest that a reduction
in FabA concentration and increase in FabZ can promote the production of
short-chain alcohols---perhaps, the result of the preference of FabZ for
short substrates---but this effect is less pronounced (i.e., changes in
enzyme concentrations are smaller than those of TesA, FabF, and FabB).
For the ATR pathway, which has no thioesterase, chain length appeared
most sensitive to the concentrations of FabB and FabA (with trends
consistent with those exhibited by the other pathways).

For each pathway, we used a total enzyme concentration informed by the
overexpression strategy described in an experimental study. The ATR and
ACR1 enzymes, however, are not particularly active, and high enzyme
concentrations could blunt the tradeoff between chain\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

\textbf{Figure 6.} A comparison of different pathways for alcohol
biosynthesis. (A-D) The range of average chain lengths afforded by
alternative alcohol pathways: FAS supplemented with (A) TesA, CAR, and
AHR, (B) acyl-ACP reductase (ATR), (C) acyl-ACP thioesterase specific
for medium chains (TesA), a fatty-acid-CoA ligase (FadD), and an
acyl-CoA reductase (ACR2), and (D) TesA, FadD, a differentacyl-CoA
reductase (ACR1), and an aldehyde reductase (AHR). At each production
level, the shaded region denotes the average chain lengths achievable by
changing enzyme concentrations; the range of lengths narrows as
production increases. Dataset S1 contains the source data and enzyme
compositions for plots A-D.

length and production. We explored this effect by repeating the ATR
analysis with a much higher enzyme concentration
(\textasciitilde{}ninefold). This adjustment broadened the range of
accessible chain lengths at intermediate production levels. We note,
however, that our model does not account for the potential for ATR to
bind holo-ACP, an interaction that can inhibit fatty acid synthesis and
may limit the usefulness of ATR overexpression.

A sensitivity analysis of all pathways examined in this work suggests
that, in general, product profiles are most sensitive to concentrations
of acyl-ACP thioesterases and \selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP synthases (Fig. S11). This
sensitivity, however, does not imply that all pathways are compatible
with concentration-based control strategies; the narrow substrate
specificities of ATR and ACR1, for example, are limiting. Rather, our
focused analysis of the alcohol pathways suggest that coordinated
changes in enzyme concentration provide a versatile means of tuning the
product profiles of pathways with promiscuous enzymes downstream of the
FAS.

\textbf{3.5 \textbar{} A graphical user interface for modeling
oleochemical biosynthesis}

To facilitate the rapid implementation of our models, we integrated them
into a MATLAB-based graphical user interface (gui; Fig. 7). This gui
permits user-defined changes in enzyme, substrate, and cofactor
concentrations, and allows the user to specify alternative versions of
enzymes explored in this study (e.g., different thioesterases). Its
plotting feature can generate final product profiles at user-defined
time points or time courses for specific metabolites (e.g., acyl-ACPs).
Motivated by our analysis of different alcohol pathways, we added a
thiolase-based pathway for methyl ketone production (Fig. S9). The final
gui includes nine pathways, each parameterized with a mixture of\emph{in
vitro} and \emph{in vivo} studies.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{Figure 7} . A graphical user interface. (A) A MATLAB-compatible
graphical user interface (gui) facilitates modeling analyses of
different (B) oleochemical pathways and (C) metabolites. Note: The
12-minute time point provides a convenient means of exploring the
influence of enzyme compositions on final product profiles; we used this
time point in the analyses of Figs. 2-6.

\textbf{4 \textbar{} CONCLUSION}

Oleochemical-producing microbes with arbitrary, user-defined product
profiles remain an elusive goal of metabolic engineering (Marella et
al., 2018; Sharma and Yazdani, 2021; Yan and Pfleger, 2019). The
activities and substrate specificities of critical enzymes are difficult
to predict (or change), and their integration into new pathways
typically requires multiple rounds of iteration; titer and product
profile, in particular, tend to be nonintuitively coupled (Greenhalgh et
al., 2021; Grisewood et al., 2017; Sarria et al., 2018). In this study,
we developed a set of kinetic models that facilitates the design and
analysis of oleochemical pathways. These models provide good fits to
experimental data and can capture the effects of previously reported
pathway modifications. The final models enable mechanistic studies of
unexpected phenomena (e.g., the inhibition of FAEE biosynthesis by TesA
overexpression) and facilitate large, system-wide analyses that are
experimentally intractable. Our sensitivity analysis, for example,
allowed us to probe the influence of isolated shifts in protein
concentration within a broad set of pathways. Findings indicate that
product profiles are most sensitive to concentrations of FabB, FabF, and
acyl-ACP thioesterase (when present) and indicate that coordinated
shifts in these concentrations can adjust the product profiles of
pathways with promiscuous enzymes downstream of the FAS.
Oleochemical-specific enzymes, in turn, primarily influence total
production, an indication that their overexpression can improve titer
without altering final product profiles. These observations provide
guidance for optimizing enzyme expression levels in oleochemical
pathways.

The incorporation of different oleochemical pathways into our FAS model
allowed us to refine general rules for characterizing pathway enzymes.
Results indicate that substrate-specific Michaelis-Menten parameters are
sufficient to capture the activities of a broad set of enzymes. These
parameters rely on steady-state assumptions that are reasonable during
oleochemical production in stationary phase. Importantly, our consistent
parameterization of enzymes between pathways suggests that our models
can capture their activities in different biochemical contexts.
Enzyme-specific kinetic parameters are most accurately measured
with\emph{in vitro} kinetic assays (e.g., the CAR-based alcohol
pathway), but our results indicate that \emph{in vivo} product profiles
that differ in the identity or expression level of individual enzymes
can also support reasonable estimates (e.g., alternative thioesterases).
Finally, protein expression levels, which can vary between strains (Monk
et al., 2016), are important for accurate and consistent kinetic models.
The regular use of proteomics to measure protein concentrations in
engineered strains could improve model accuracy and help explain
strain-to-strain variability in oleochemical pathways.

The paper concludes by developing a gui to facilitate the rapid
implementation of our models. This gui is a useful tool for exploring
the influence of various pathway modifications on final product profiles
and could guide experimental studies (e.g., promoter selection or
configuration). It complements the complete model files, which enable
detailed analyses of oleochemical production (e.g., plots of various
metabolites or analyses of tradeoffs between objectives) with minor
changes in code or short optimization routines. Together, the gui and
its core models provide a versatile kinetic framework for studying
oleochemical biosynthesis.

\textbf{ACKNOWLEDGEMENTS}

This work was supported by the United States Army Research Office (J.P.,
S.R., K.M. E.C.P., and J.M.F., award W911NF-18-1-0159). J.P.
acknowledges support from the National Science Foundation (J.P., award
1804897) for a summer research experience that supported his initial
work on this project. K.M. acknowledges fellowship support from the
Department of Defense (DoD) through the National Defense Science \&
Engineering Graduate (NDSEG). E.C.C. acknowledges fellowship support
from the NIH/CU Molecular Biophysics Program and NIH Biophysics Training
Grant T32 GM-065103.

\textbf{DATA AVAILABILITY STATEMENT}

Code associated with this work is available online at
https://github.com/jmfoxai/versatilemodel1. All other data is available
from the corresponding author upon reasonable request.

\textbf{SUPPORTING INFORMATION}

Additional supporting information may be found in the online version of
the article at the publisher's website.

\textbf{ORCID}

Jackson Peoples: https://orcid.org/0000-0003-3261-582X

Sophia Ruppe: https://orcid.org/0000-0003-0743-8796

Kathryn Mains: https://orcid.org/0000-0002-3623-1383

Elia C. Cipriano: https://orcid.org/0000-0001-5460-4094

Jerome M. Fox: https://orcid.org/0000-0002-3739-1899

\textbf{NOMENCLATURE}

FAS, fatty acid synthase; FAME, fatty acid methyl ester; FAEE, fatty
acid ethyl ester; ACP, acyl carrier protein; CoA, coenzyme-A; FabD,
Malonyl CoA-ACP transacylase; CO\textsubscript{2}, carbon dioxide;
NADPH, reduced nicotinamide adenine dinucleotide phosphate;
H\textsuperscript{+}, hydrogen ion; NADP\textsuperscript{+},
nicotinamide adenine dinucleotide phosphate; FabA,
\selectlanguage{greek}β-\selectlanguage{english}hydroxy-decanoyl-ACP dehydratase; FabZ, \selectlanguage{greek}β-\selectlanguage{english}hydroxyacyl-ACP dehydratase;
NADH, reduced nicotinamide adenine dinucleotide; NAD\textsuperscript{+},
nicotinamide adenine dinucleotide; FabI, enoyl-ACP reductase; FabF,
\selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP synthase II; FabB, \selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP synthase I; FabH;
\selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP synthase III; TE, thioesterase; TesA, thioesterase I;
H\textsubscript{2}O, water; SAM, S-adenosyl-L-methionine; SAH,
S-adenosyl-L-homocysteine; DmJHAMT or MT, \emph{Drosophila
melanogaster}juvenile hormone acid O-methyltransferase; ATP, adenosine
triphosphate; AMP, adenosine monophosphate; ADP, adenosine diphosphate;
P\textsubscript{i}, phosphate; PP\textsubscript{i}, pyrophosphate; CAR,
carboxylic acid reductase; ACR, acyl-ACP/acyl-CoA reductase; ATR,
acyl-ACP reductase; ADH, alcohol dehydrogenase; AHR, aldehyde reductase;
ADO, aldehyde deformylating oxygenase; O\textsubscript{2}, oxygen; FadD,
fatty acid-CoA ligase; WS, wax ester synthase; EtOH, ethanol; FadE,
acyl-CoA dehydrogenase; FAD, flavin adenine dinucleotide;
FADH\textsubscript{2}, reduced flavin adenine dinucleotide; FadB,
3-hydroxyacyl-CoA dehydrogenase; FadM, \selectlanguage{greek}β-\selectlanguage{english}ketoacyl-CoA thioesterase; MKS,
methyl ketone synthase; PhaG, (R)-3-hydroxydecanoyl-ACP:CoA
transacylase; FabG, \selectlanguage{greek}β-\selectlanguage{english}ketoacyl-ACP reductase; PhaJ, (R)-enoyl-CoA
hydratase; Lig, ligase; PhaC, PHA synthase; PHA, polyhydroxyalkanoate;
FadA, 3-keto-acyl-CoA thiolase.

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