Mariano Nicolas Cruz Bournazou, Tilman Barz, D. B. Nickel, Diana
Carolina Lopez Cárdenas, Florian Glauche, Andreas Knepper, Peter
Neubauer
Online optimal experimental re-design in robotic
parallel fed-batch cultivation facilities
Open Access via institutional repository of Technische Universität Berlin
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Journal article | Accepted version
(i. e. final author-created version that incorporates referee comments and is the version accepted for
publication; also known as: Author’s Accepted Manuscript (AAM), Final Draft, Postprint)
This version is available at
https://doi.org/10.14279/depositonce-15644
Citation details
This is the peer reviewed version of the following article:
Cruz Bournazou, M. N., Barz, T., Nickel, D. B., Lopez Cárdenas, D. C., Glauche, F., Knepper, A., & Neubauer,
P. (2016). Online optimal experimental re-design in robotic parallel fed-batch cultivation facilities. In
Biotechnology and Bioengineering (Vol. 114, Issue 3, pp. 610–619). Wiley. https://doi.org/10.1002/bit.26192,
which has been published in final form at https://doi.org/10.1002/bit.26192. This article may be used for
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For Peer Review
1
Online Optimal Experimental Re-Design in Robotic Parallel Fed-
Batch Cultivation Facilities for Validation of Macro-Kinetic
Growth Models using E. coli as an Example
M.N. Cruz Bournazou, T.Barz, D. Nickel, D. Lopez Cárdenas, F. Glauche, A. Knepper, P.
Neubauer
corresponding author:
Dr. M. Nicolas Cruz Bournazou
Laboratory of Bioprocess Engineering
Department of Biotechnology
Technische Universität Berlin (TU Berlin)
Ackerstr. 76, ACK24
D-13355 Berlin, Germany
Tel. direct: +49 30 314 72527
Secretary: +49 30 314 72573
Fax: +49 30 314 27577
www.bioprocess.tu-berlin.de
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Abstract
We present an integrated framework for the online optimal experimental re-design of
parallel nonlinear dynamic processes that aims to precisely estimate the parameter set of
macro kinetic growth models with minimal experimental effort. This provides a systematic
solution for rapid validation of a specific model to new strains, mutants, or products. In
biosciences, this is especially important as model identification is a long and laborious
process which is continuing to limit the use of mathematical modeling in this field.
The strength of this approach is demonstrated by fitting a macro-kinetic differential equation
model for Escherichia coli fed-batch processes after six hours of cultivation. The system
includes two fully-automated liquid handling robots; one containing eight mini-bioreactors
and another used for automated at-line analyses, which allows for the immediate use of the
available data in the modeling environment. As a result, the experiment can be continually
re-designed while the cultivations are running using the information generated by periodical
parameter estimations.
The advantages of an online re-computation of the optimal experiment are proven by a
fifty-fold lower average variation coefficient on the parameter estimates compared to the
sequential method (4.83% instead of 235.86%). The success obtained in such a complex
system is a further step towards a more efficient computer aided bioprocess development.
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Introduction
Today, mathematical modeling in biotechnology is not hampered by computer capacity or
by insufficient understanding of the microbial systems, but rather by the lack of fast, cheap,
and informative experiments – especially in the case of dynamic processes such as
cultivations describing industrial conditions. When properly validated, simplified
mathematical models are capable of describing the dynamics of complex systems (Cruz
Bournazou et al. 2012; Kokkalis et al. 2014; Legmann et al. 2009; Rosen et al. 2006; Zavrel
et al. 2008). In literature, we can find models that successfully describe biological systems
at all scales (Brunk et al. 2012; Neubauer and Junne 2010), including models for
optimization of dynamic processes (Cruz Bournazou et al. 2013; Delgado San Martín et al.
2014; Hidalgo-Bastida et al. 2012; Lu et al. 2013; Oddone et al. 2007; Venkata Mohan et al.
2005) and processes with mixtures of microbial populations with low quality data (Junker
and Wang 2006; Su et al. 2005). However, contrary to other industries where processes are
typically predicted using existing equations and literature values (thermodynamics, chemical
reactions, combustion, etc.), model building in biotechnology is inevitably coupled with
iterative and laborious experimental validation for each process-product pair (Ataman and
Hatzimanikatis 2015; Medema et al. 2012). Regardless of its level of complexity, a model
needs to be constantly re-fitted against observations in order to adapt its outputs to
variations in the environment or in the microorganism itself (e.g mutations). Therefore,
creating an experimental facility that can generate the data required to fit a specific model
with high cost-time efficiency is certainly relevant for research and industry. This implies
that the experimental set-ups should: i) run fast and cheap, ii) emulate process relevant
conditions, and iii) consider the evolution of the system over time.
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These requirements have pushed the development in automation, miniaturization, and
parallelization of experimental facilities developing automatic Liquid Handling Stations
(LHS) for High Throughput Screening (HTS) and High Throughput Bioprocess Design
(HTBD), (Long et al. 2014; Puskeiler et al. 2005; Schäpper et al. 2009) as well as sensor
and Process Analytical Technologies (PAT) for a better and less invasive insight into
biological systems (Neubauer et al. 2013). The rapid growth of these technologies brings
additional challenges, such as the correct handling and analysis of very large data sets
(Kozak 2014; Shockley 2015) and the design of extremely complex experiments. The
design of multiple parallel experiments can be increased by using methods for design of
experiments (Glauche et al. 2016; Lutz et al. 1996) and neural networks (Glassey et al.
1994) among others. Nevertheless, we need to go beyond “endpoint” or “steady state”
experiments to uncover the dynamics of the process and predict its evolution over time. This
means that the design of the dynamic experiment cannot be done exclusively with statistical
tools. This time dependent behavior is better described by nonlinear differential equations.
Methods for Optimal Experimental Design (OED), also called Model Based Design of
Experiments (MBDoE), which design the experiments based on nonlinear differential
equation systems, have been developed (Franceschini and Macchietto 2007a; Körkel et al.
2004) and specific applications in biotechnology exist (Baltes et al. 1994; Takors et al.
1997). In order to maximize the precision of the parameter estimates, the experimenter
needs to find the optimal combination of input variables (e.g. feeding strategy) and sampling
setup (type and time) within the feasible region. In addition, each sampling consists of a
number of “actions” (pipetting, mixing, incubating, measuring, etc.) and requires different
“resources” (1-, 8-, or 96-chanel pipette, shaker, photometer, flow cytometer, reaction
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vessels, plates, etc.) which have to be optimally coordinated in an efficient schedule.
Unfortunately, tools to plan and perform dynamic experiments exploiting full LHS capacity
are not available. Despite various publications using OED in biotechnology (Franceschini
and Macchietto 2007b; Galvanin et al. 2011; Gernaey et al. 2002; Kreutz and Timmer 2009;
López Cárdenas et al. 2013; Skanda and Lebiedz 2010) and design of experiments applied
in HTS, to our knowledge, there are no publications dealing with the online design of
nonlinear dynamic parallel experiments as they are needed for HTBD.
Furthermore, in order to minimize the experimental effort and increase the robustness of the
experiment, the model should be re-fitted after each sample and the experiment re-designed
as data is being generated. This requires an efficient solution of a fairly large nonlinear and
possibly ill-conditioned optimization program.
Works dealing with online experimental re-design
(Galvanin et al. 2012; Stigter et al. 2006;
Zhu and Huang 2011) show that the information generated in each experiment is
significantly higher compared to experiments that are planned in sequence (offline). It may
appear that the extension to parallel experiments is fairly straight forward from the
theoretical point of view (Cruz Bournazou et al. 2014a); nevertheless, former studies are
applied on simple systems and its application to a real set of parallel cultivations presents
many challenges. This includes the complexity of the biological system, the operation of the
experimental facilities, the typically long delays and the limited information content of the
analyses, and the scheduling of all actions considering resource availability (Mayer and
Raisch 2004). Finally, a robust and cheap computation of both optimization programs,
namely the Parameter Estimation (PE) and the Experimental Design (ED) problem, is
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critical in order to handle data and compute the optimal strategy at the speed dictated by the
running experiment.
Problem statement
The goal of this work is to achieve an online computation of optimal parallel dynamic
experiments together with its automatic implementation in a LHS for fed-batch cultivations
of the bacterium E. coli (Figure 1) using a Sliding Window Optimal experimental Re-
Design (SWORD). We regularize the singularity problem by an appropriate selection of the
parameter subset to assure a well-conditioned parameter estimation even with reduced
experimental information, and use a moving horizon approach to reduce the computational
burden of the optimization. Ultimately, it is possible to re-design the running experiment
after each measurement and take full advantage of the present state of information to plan
the following steps.
Materials and Methods
The most relevant aspects of the experiment are presented here, for a more detailed
description of the experimental setting, the reader is referred to the Appendix in the
supplementary material and to (Nickel et al. submitted). The experiments were carried out
using two different LHSs. The mini-bioreactor system bioREACTOR 48 (2mag AG,
Munich, Germany) was integrated in a TECAN Freedom Evo LHS (Tecan, Crailsheim,
Germany) (Figure 2) to automate all liquid handling steps. General process data
(temperature, stirrer speed, pH, dissolved oxygen) was stored in the iLab-Bio database
(infoteam, Bubenreuth, Germany). For pH control of the vessels, a LabVIEW (National
Instruments, Munich, Germany) based pH controller was used. Worklists were generated of
titrant volumes for the Tecan Gemini software. Samples were analyzed on a Hamilton
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Microlab Star platform (Hamilton Robotics, Bonaduz, Switzerland), which is equipped with
a Biotek Synergy MX microplate reader (BioTek Instruments GmbH, Bad Friedrichshall.
Germany). The data was stored in the iLab-Bio database for further processing.
Bacterial strain and cultivation medium
The experiments were performed using E. coli W3110 (DSM No. 5911) stored at – 80 °C in
LB medium containing 20% glycerol. Cultivations were carried out in EnPresso media
(BioSilta Ltd., Cambridge, UK): the seed cultures were grown in EnPresso B medium and
the main culture was performed in EnPresso B defined medium. Unless otherwise stated,
media were prepared according to the manufacturer’s instructions.
Seed and main cultures
The seed culture was performed in a 125 mL UltraYield Flask (Thompson Instrument
Company, Oceanside, USA) containing 25 mL Enpresso B, which was inoculated with 24
µL of cryo culture. For glucose release, 1.5 U L
-1
Reagent A (BioSilta) were added. The
shake flask was covered with AirOtop enhanced shake flask seals (Thompson Instrument
Company) and incubated at 30°C and 220 rpm in a shaking incubator with 50 mm offset
until the mid-log phase was reached (4 – 6 h). Then the main culture was prepared in 100
mL of EnPresso B defined medium containing 10 g L
-1
glucose with an initial OD
620
of
0.01. For the initial batch phase, pre-calibrated baffled 500 mL sensor flasks for online pH
and DO measurements were used (PreSens GmbH, Regensburg, Germany). The flasks were
incubated overnight at 30°C and 200 rpm with 25 mm offset. At 20 h, the 11 mL of culture
were transferred into the 8 mini-bioreactors.
The cultures were incubated at 30°C with an aeration rate of 0.1 L min
-1
and a stirrer speed
of 2600 rpm until they reach glucose limitation. This was indicated by a sudden increase of
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DO. Then 6 UL
-1
of Reagent A were added to initiate the glucose limited fed-batch phase.
The pH was adjusted to 7.0 before starting the SWORD experiment by the addition of
6.25 % (v/v) ammonia solution via the Freedom Evo LHS.
SWORD experiment
Deck layout of Freedom Evo LHS
For addition of Reagent A, a 3000 U L
–1
Reagent A solution was placed on a cooled plate
carrier at 4 °C. The EnPresso B defined medium, the 40 gL
-1
acetate additive solution and
the glucose additive solution were pipetted individually into 24 deep well plates (Ritter,
Schwabmünchen, Germany). Samples for analysis were collected into a V-bottom plate. The
pH was adjusted by adding 6.25 % (v/v) ammonia solution prepared in a sterile container.
Sterilization of the metallic pipetting tips was done in a washing station using 70 % (v/v)
ethanol (EtOH).
Sequential arrangement of additions during SWORD
The SWORD experiment was divided into cycles of 20 minutes with a total experimental
time of 6 h. One cycle comprised 5 subroutines each of which lasted 4 minutes (Figure 3).
During each 4-minute subroutine, all steel needles were firstly sterilized. Then, either a
MATLAB work list for the addition of a specific pulse was loaded into the Gemini
environment and executed or the LHS took 300 µL samples from each mini reactor into the
V-bottomed plate.
Analytics
Sampling was done every 20 minutes. The samples were pipetted into plates containing
anhydrous NaOH and mixed thoroughly to a final NaOH concentration of 0.1 M.
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Afterwards, 20 µL of sample were used for optical density measurements. The remaining
sample volumes were centrifuged at 3000 rpm and 4 °C for 10 min to separate the cells.
Duplicates of 75 µL supernatant were pipetted for glucose and acetate analyses. Samples
from four successive time points were cooled at 4 °C and analyzed together.
The OD
620
samples were diluted 15 fold in flat bottom plates and were measured manually
with a PHOmo microplate reader (Anthos, Krefeld, Germany). Samples were shaken inside
the microplate reader before the measurement. The OD
620
values were corrected against the
blank, correlated to OD
600
in a 1 cm cuvette by multiplication with the factor 2.62. The cell
dry weight (CDW) was approximated by dividing the OD values by 3.3 as obtained from
dry weight calibrations. OD-measurements were carried out every 20 min.
The extracellular glucose concentration was measured with the enzymatic Glucose
Hexokinase FS kit (DiaSys Diagnostic Systems, Holzheim, Germany) on the Hamilton LHS
as described previously (Knepper et al. 2014). The extracellular acetic acid concentration
was measured with the enzymatic Enzytec fluid
TM
kit (R-Biopharm, Darmstadt, Germany).
Both enzymatic assays were performed on the Microlab Star LHS.
Software
All numeric computations were performed using MATLAB Release 2013b. Model and
parameter sensitivity equations were integrated using CVODES solver from SundialsTB
Toolbox (Hindmarsh et al. 2005).
The computed parameter sensitivities, normalized with respect to initial parameter guesses
and measurements, were used to accurately calculate the gradients of the PE problem and
the Fisher Information Matrix (FIM). The PE problems were solved by single shooting and
using TOMLABs clsSolve solver with the Wang, Li, Qi Structured MBFGS method and
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user defined gradients (Holmström et al. 2003). The experimental design (ED) problems
were solved using the TOMLAB implementation of SNOPT (Gill et al. 2002). The objective
was computed from the FIM. The gradients were computed with finite differences with four
threads in parallel.
E. coli kinetic model
The macro-kinetic model used in this work describes E. coli substrate limited growth and a
Matlab® version is available online (see Appendix). The validity of the model is
constrained to a narrow operability region, namely i) glucose concentrations lower than 0.2
g/l, ii) at least 20% of dissolved oxygen tension, and iii) overflow phases must be shorter
than 60 seconds. In addition to substrate feeding, the model considers glucose release by the
EnBase system (Panula-Perälä et al. 2008). The model consists of 6 state variables, namely
biomass concentration () in [g/l], substrate concentration () in [g/l], dissolved oxygen
tension () in [% of saturation], acetic acid () in [g/l], liquid volume () in [l], and
glucose release enzyme concentration () in [Units/l]. These are represented by an Ordinary
Differential Equation (ODE) system as given in the appendix in equations A1 to A6. The
structure of the model is based on the work of (Lin et al. 2001). The model was modified to
add recent discoveries in the acetate production mechanism (Valgepea et al. 2010). As a
result, acetate production and consumption are always active and at equilibrium with a net
acetate production of 0.
SWORD
We present a brief introduction to the basics of OED. The reader is referred to (Barz et al.
2012; Cruz Bournazou et al. 2014b; Franceschini and Macchietto 2007a) for further details.
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Optimal experimental design
Let us consider a model consisting of a nonlinear implicit differential equation system that
describes the dynamic behavior of a process (e.g. an E. coli fed-batch cultivation). This
model will contain a set of unknown parameters that can be varied to fit the outputs of the
model against observations of the real process. OED aims at finding experimental setups
such that the statistical uncertainty of estimates of the unknown model parameters is
minimized (Korkel 2002). Generally, the main factors that affect the identifiability of a
model (i.e. the size of its confidence intervals) are: i) the structure of the model, ii) the
quality of the measurements (type, frequency, accuracy), and iii) the design of the
experiment used to generate the data (inputs, conditions, measurmenents, etc.). In other
words, the computed optimal conditions aim to maximize the information content of the
measurements so that the parameters are determined most accurately.
Roughly speaking, the inputs should excite the system so as to let the experimenter observe
its dynamics by sampling in the most sensitive points of the experiment. To find this optimal
setup, mixed second order sensitivities of the least square functional with regard to
parameters and states have to be computed.
Mathematical formulation
A simplified description of the problem is presented, the reader is referred to Appendix for
further details. We consider a twice continuously differentiable nonlinear Ordinary
Differential Equation (ODE) system:
=
,
,
;
=
ℎ
;
=
1
where ∈
,
⊆ℝ is the time, ∈ℝ
#
are dependent state variables, ∈ℝ
$
are the time-varying input or experimental design variables and ∈ℝ
%
the unknown
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parameter vector and initial conditions are given by
. The vector ∈ℝ
&
contains the
predictions of the observed responses. In Eq. (1) we consider a set of ℛ ≔ {*
+
,⋯,*
-
}
parallel experiments, with /
0
being the number of reactors. The robot simultaneously feeds
all mini-reactors at discrete feeding time instances 1
2
≔3
+
,⋯,
4
5, with /
2
being the
total number of injections into one reactor. Each reactor has its corresponding input signals
or experimental design variables
0
∈ℝ
$
, with ∈1
2
and *∈ℛ. For all reactors, the
experimental design variables are collected in the vector
6 ≔7
08+
28+
9
,⋯,
08+
:
4
;
9
<
9
,…,7
-
28+
9
,⋯,
-
:
4
;
9
<
9
∈ℝ
$
⋅
4
⋅
-
with the number of possible individual species to be injected /
?
, the total number of
injections /
2
and the number of reactors /
0
. The robot simultaneously takes samples from
all mini-reactors at discrete measurement time instances 1
@
≔At
+
,⋯,t
C
D
E, with n
@
being
the number of samples taken by the robot from one reactor. The measurements are defined
by G
H
≔I
08+
H
J8+
9
,⋯,
08+
H
K
9
L
9
,…,I
-
H
J8+
9
,⋯,
-
H
K
9
L
9
∈ℝ
&
⋅
K
⋅
-
with the number of predicted responses /
M
, the number of samplings taken by the robot /
J
and the number of reactors /
0
. In the same way, corresponding predicted response variables
are defined as
0
;
0
,∈ℝ
&
, with ∈1
J
and * ∈ℛ and collected in the vector
G
6
,
∈
ℝ
&
⋅
K
⋅
-
2
Note that the predicted responses are obtained from the solution of Eq. (1) for each reactor *
and therefore depend on
0
and .
Model parameters are estimated by maximum likelihood estimation minimizing the
quadratic residual (Bard 1974):
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N
:
=
arg
min
U
1
2
G
6
,
−
G
H
9
Y
H
Z
+
G
6
,
−
G
H
3
with the weighting matrix Y
H
∈ℝ
&
⋅
K
⋅
-
×
&
⋅
K
⋅
-
being the variance-covariance matrix of
the measurement errors in the data. Y
H
is assumed to be a diagonal matrix with the variance
\
]^
of each measurement _ in its diagonal entries. The precision of a maximum-likelihood
estimate
N can be analysed by evaluation of the so called Fisher Information Matrix (FIM)
F6,
N or its inverse (lower bound on the covariance matrix Y
U
6,
N) (Bard 1974). The
design of an optimal experiment for improving parameter precision therefore minimizes
some metric of Y
U
6,
N by optimally choosing the inputs or experiment design variables 6
subject to equality and inequality constraints. In this contribution we apply the so called A-
optimal criterion a
b
to formulate the optimal experiment design objective function:
a
b
,
N
≔
1
/
c
tr
:
d
Z
+
6
,
N
;
4
Online optimal experimental re-design
The real information content of an ED depends on the accuracy of the assumed parameter
values (initial parameter guess). Unfortunately, uncertainties in the initial parameter guess
and initial conditions cause a mismatch between computed and experimental outputs leading
to suboptimal ED. This mismatch can be reduced by an adaptive online ED, where the
experiment is iteratively re-designed as information is generated. This idea has been
discussed in previous publications and is described in detail by (Barz et al. 2012).
In sequential OED, the identifiability of the parameter estimation problem can be studied
and solved offline. In contrast, when performing a recursive fit of the model and a design of
each sampling setup, an efficient solution of nonlinear and possibly ill-conditioned problems
is required. (Barz et al. 2012) propose the addition of a local parameter identifiability
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analysis method to select the parameters such that well-conditioned optimization problems
are guaranteed. Therefore, it is possible to redesign the running experiment in order to take
full advantage of the present state of information and plan the following step (optimal input
, sampling time
]
, sampling setup) after each measurement. This is, to the best of our
knowledge, the first attempt to carry out an online parallel design of dynamic experiments
using real automated cultivations systems.
The F matrix needs to be well conditioned as it is used to approximate the covariance matrix
of the parameters C
θ
by inversion. If this is not the case, e.g. due to insufficient
measurement data or correlations in the parameters, a regularization method is applied to
approximate F by a well-conditioned matrix. The regularization is based on the Subset
Selection (SsS) method which is very effective when applied to ED problems with ill-
conditioned matrices (López Cárdenas et al. 2015). Moreover, the SsS method also proposes
a reduction of the parameter space (and the PE and ED problem) by finding the parameters
which cannot be identified for the existing measurement data (identifiability analysis).
Experimental implementation of SWORD
The duration of the SWORD experiment is 6 h and is divided in 18 cycles of 20 min. A
detailed description of one cycle is given in Figure 3. The initial experimental design is
carried out up to minute 140 ( first re-design is available). The first at-line data is available
after 100 minutes. Once new at-line data is available, i) the identifiability of the parameters
is checked, ii) a PE is solved for max. 20 min, and iii) the parameter estimates obtained are
passed to re-compute the optimal experiment (max. time 20 min). This procedure is repeated
three more times until 300 min where the last re-design is calculated. The experiment run is
depicted in Figure 4.
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Column C reads the real time. Column A shows the tasks performed by the first computer,
namely the PE (green; cell [4,Q]) and SsS and identifiability analysis (dark green; cell
[5,Q])). Column B depicts the performance of the second computer which was responsible
for the ED-SH optimizations of the short horizon (dark red; cell [6,Q])) and DE-LH random
seed global optimizations computing the long horizon (red; cell [7,Q])), and the data that is
available is depicted by bars (black; cell [10,Q]). The time points of the samples and designs
can be seen in row 1 (columns D to V). For example: at 100 min (cell [7,C]) the data from
sampling at 0 and 20 min is stored as depicted in the bar in cells [7,D- E] and the comp1
initiates the parameter estimation (cell [7,A]). We can also see the three optimization
horizons (orange; cell [8,Q]), which were implemented at 140 min, 240 min, and 340 and
finally the results of the long term horizon (blue; cell [9,Q]), which were available at 0 min,
220 min and 320 min. The run of the experiment is depicted at the example of biomass for
reactor 1 in Figure 5.
Results and discussion
SWORD was able to fit the model despite the fact that the enzymatic assay for acetate
quantification was inaccurate, proving the advantages of an online re-design approach
Figure 6. The program could obtain reliable parameter estimates by re-designing the
experiment taking the unexpectedly high variance in acetate into account. Further
screenshots of the monitoring station are presented in the Appendix to show the
development of the experiment and the inputs selected over time. The reactor replicates
proved to be statistically equal according to the hypothesis test and confidence intervals
(Montgomery and Runger 2010) carried out on the cell dry weight data over the complete
experiment, see Figure 7.
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In Figure 8, the normal probability density function of the whole parameter vector centered
at the normalized parameter estimate
N with variance \
^
are displayed. The size of the
variability of the parameters with large uncertainty can be observed (i.e., Yam and Yofm).
‘qSmax’ has the lowest relative standard deviation \
0
=0.56% whereas ‘Yam’ and
‘Yofm’ have the highest ones with 341% and 822 % respectively.
The identifiability list for this problem can be found in Table 1. It should be stressed that
due to parameter correlations, the ranking between variance, sensitivity and SsS are not
equal. Still parameter with the largest effect on the outputs is also 'qSmax' and the parameter
with the smallest effect on the measurable variables after all orthogonal projections is
'Yofm' showing consistency with the precision results above mentioned. In addition, the
sensitivity matrix is not full-rank but 23, i.e. only 23 parameters are identifiable. The
unidentifiable parameters being 'Yam', and 'Yofm'. Additionally, the sensitivity ranking list
built based on the overall output-parameter sensitivity information is displayed. The metric
used is the sensitivity measure δ=
+
l
m
,n
o
,n
p
∗∑s
]t
^
with s
]t
^
=
uM
v
uU
w
which considers the norm
of the sensitivity of the whole predicted response variables to a change in the parameter
t
.
Table 1. Parameter ranking list according to: parameter variance x
y
, relative standard deviationx
z
and
mean sensitivity {
|
, of the predicted response variables } to a change in the parameter |.
Now let us show the advantages of the re-design of the experiment. We compare the results
obtained in the real experiment against a simulation of the initial design with the final
parameter values. The A-criterion, which represents the average coefficient of variation
(normed standard deviation) of the parameter estimates is almost 50 times lower (235.86%
vs. 4.83%) with SWORD considering all 23 identifiable parameters.
Furthermore, the relative variance σ
r
of the parameters at the estimated vector θ
f
is depicted
in Figure 9 in ascending order (light green bars). A comparison between the parameter
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precision at the initial guess θ
0
and the estimated vector θ
f
shows the superiority of the
estimated parameter vector θ
f
over θ
0
. The identifiability increased from only 13 parameters
with a variance lower than 10% with the original design (dark brown bars) to 23 due to the
continues re-design of the inputs. This proves that the final experiment cannot be computed
beforehand since, due to wrong initial estimates, the predicted outputs are in reality
suboptimal.
Conclusions
The online re-design of optimal experiments for 4x2 parallel fed-batch cultivations is
possible using a moving horizon approach which reduces the experimental effort compared
to sequential designs. In this case study, one experimental run generated sufficient
information to fit a macro-kinetic model of E. coli W3110, significantly reducing time and
costs of model validation. This is achieved by re-designing the optimal strategy of the
parallel cultivations using the existing data as it is generated during the experiment.
The results show that the SWORD program can overcome complications related with
ill-posed optimization problems and highly nonlinear designs in parallel bioreactors. This is
a first step towards new model based tools that can fully exploit the advantages of LHS and
speed up the development from screening to production in bioprocesses. Faster, cheaper,
and more efficient experiments will encourage the use of mechanistic models, allowing for
the application of computer-aided tools in design, monitoring, control, and optimization of
bioprocesses.
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Acknowledgments
The authors acknowledge financial support by the German Federal Ministry of Education
and Research (BMBF) within the Framework Concept ”Research for Tomorrow’s
Production”
(project no. 02PJ1150, AUTOBIO project) managed by the Project Management
Agency Karlsruhe (PTKA).
We would like to thank
Sven Olof Enfors for fruitful discussions during the development of
the model, Tiffany Wong, and students Michael Heisser, Lorenz Thielert, Sebastian Hans,
and Terrance Wilms.
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Figure 1: simplified flow diagram of the online re-design of experiments procedure
172x39mm (150 x 150 DPI)
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Figure 2 (a) bioREACTOR 48 integrated into the Freedom Evo LHS. (b) Data flow concept of SWORD. On-
line signals from the mini-bioreator system (MBR, I) and measured data from samples (II) is stored in a
database, which can be accessed for the sliding window re-
design (III). The LHS was controlled via MATLAB.
(c): Deck layout of the Freedom Evo LHS: bioREACTOR 48 (1), Additives: Reagent A (2), EnPresso B defined
(8), 40 g L-1 sodium acetate (9), glucose (10); sample plate (11).The steel needles were washed in
deionised water (3), twice in EtOH (4, 6) and in sterile deionised water (7) pH adjustment was done with
6,25 % ammonia (5).
152x101mm (150 x 150 DPI)
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Figure 3: Scheduling of the SWORD experiment. Cycles of 20 min were repeated for 6h. During these
intervals, in cycles of 4 min EnPresso B defined medium, Reagent A, acetate and glucose were added to the
cultivations. Afterwards, 300 µL samples were taken from each reactor. Each addition routine consisted of
several subroutines (right).
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Figure 4: Time schedule of the experiment. Following time column C, the development of the experimental
setup is to be seen (starting at the top). The activity of the computers (computer1 PE in column A, and
computer2 ED in column B). Availability of at line measurement (biomass, glucose, and acetate) are
represented by black bars in relation to measurement number and sampling time, row 1.
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Figure 5: Evolution of SWORD experiment at the example of cell dry weight of the firs reactor (from plot A
to D). Red dashed and blue lines represent the simulation results before and after parameter estimation
respectively. The bars represent the horizon of available online and at-
line data (black), the time required to
compute the parameter estimation (blue), the time for ED optimization (red), and horizon length of the
experimental re-design (green).
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Figure 6: Screenshot of the monitoring station. Data collected over the complete experiment. Continuous
lines represent the simulations. The parameters are shown in the left side of the Monitor. Active parameters
(considered in the PE) are
marked with “#” on the left side. Measurements vs. simulations of all four reactor
pairs (first reactor blue second reactor red) are depicted in a 5x4 format. Column-wise beginning from the
upper graph, cell dry weight (X) in grams per liter is first plotted followed by glucose concentration (Gl) in
grams per liter, dissolved oxygen tension (DOT) in % of saturation, and acetate (Ac) in grams per liter in
the fourth graph. In the semi-log graph at the bottom of each column, the volumes in micro liters added
every 4 min (or extracted for sampling) are depicted with bars. Following the order of addition in each cycle,
medium additions are represented by green, enzyme by red, acetate by magenta, glucose by blue and
samples (fixed at 300 µl) by cyan bars. While this screenshot was taken, the additions between time 140
and 240 min were re-designed with the new parameter values.
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Figure 7: Graphical representation of the 95% confidence intervals (shaded area) of the difference in the
means µ_1-
µ_2 (solid line) for 4 experimental conditions (2 reactor replicates) over the whole experimental
run. µ_1 and µ_2 represent the means of the dry biomass of each reactor in a replicate. A confidence
interval containing the zero-value suggests no difference in the means at the evaluated experimental point.
The percentages of accepted measurements are: A=89%, B=79%, C=68%, and D=95%.
120x96mm (150 x 150 DPI)
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Parameter σ
σσ
σ
2
σ
σσ
σ
r
δ
δδ
δ
Ranking
Parameter
σ
σσ
σ
2
σ
σσ
σ
r
δ
δδ
δ
Ranking
Variance
Sensitivity
SsS
Variance
Sensitivity
SsS
'qSmax' 3.13E-05 1% 19.543
1 1 1 'Kaq' 1.02E-03
3% 7.272
14 19 16
'qAmax' 3.97E-04 2% 16.100
2 16 10
'Ks' 1.50E-03
4% 6.933
15 21 17
'KLa-4' 5.01E-04 2% 11.309
3 5 4 'Ke' 2.48E-03
5% 6.889
16 17 19
'KLa-3' 5.01E-04 2% 9.825
4 7 5 'Yem' 6.05E-03
8% 5.334
17 3 3
'KLa-1' 5.02E-04 2% 9.427
5 12 9 'Kap' 7.24E-03
9% 4.220
18 18 18
'KLa-2' 5.03E-04 2% 9.358
6 14 14
'Yaof' 9.73E-03
10% 3.610
19 15 15
'KLa-7' 5.04E-04 2% 9.236
7 10 7 'Ksq' 1.16E-02
11% 1.461
20 23 22
'KLa-8' 5.04E-04 2% 9.229
8 11 8 'Yosresp' 1.26E-02
11% 1.213
21 6 20
'KLa-5' 5.07E-04 2% 9.229
9 9 6 'Ko' 1.74E-02
13% 0.709
22 22 21
'KLa-6' 5.08E-04 2% 8.997
10 13 11
'qm' 7.90E-02
28% 0.626
23 20 23
'pAmax' 5.62E-04 2% 8.932
11 4 13
'Yam' 1.16E+01
341% 0.218
24 24 24
'Yaresp' 7.11E-04 3% 8.629
12 2 2 'Yofm' 6.75E+01
822% 0.111
25 25 25
'GRmax' 8.14E-04 3% 8.029
13 8 12
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Figure 8: Estimator analysis through normal probability density functions of the parameters.
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Figure 9: Estimator analysis with parameter precision ranking in terms of relative standard deviation.
Comparison between the variance obtained with the original design (brown bars) and with 3 re-designs
(green bars).
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