Exploring optimal objective functions and additional constraints for

Exploring optimal objective functions and additional
constraints for flux prediction in genome-scale
models
Nguyen Hoang Son
Thesis to obtain the Master of Science Degree in
Biotechnology
Supervisors:
Dr.
Prof.
Rafael Costa
Isabel Maria de Sá Correia Leite de Almeida
Examination Committee
Chairperson:
Prof.
Duarte Miguel de França Teixera dos Prazeres
Supervisor:
Dr.
Rafael Costa
Members of the Committee:
Dr.
Daniel Machado
October 2014
Acknowledgments
Apart from the efforts of myself, the success of any project depends largely on the encouragement
and guidelines of many others. First and foremost, I would like to thank to my supervisor Dr. Rafael
Costa for the valuable guidance and advice. This research project would not have been finished
successfully without his continuously support and assistance. His enthusiastic supervision helped
me in all the time of research and writing of this thesis.
I want take this opportunity to express my sincere gratitude to Prof. Erik Aurell and Prof. Isabel Sá
Correia, the coordinators of euSYSBIO program, for their dedicated and constant support throughout
2 years of my studying. Prof. Isabel Sá Correia is also the co-supervisor of this thesis who helped me
so much to keep the work on progress and I really appreciate that.
I would like to gratefully acknowledge the tremendous encouragement and insightful comments of
Prof. Susana Vinga and her group during the time of research of this thesis. In addition, the author
truly convey thanks to the officers, lab-mates at INESC-ID for providing the useful preferences and
laboratory facilities.
I also wish to express love and gratitude to my beloved families; for their understanding and endless love, through the duration of my studies.
Finally, my thanks and appreciations go to my friends in Lisbon, to the city and people here for
their warm hearts and all the tremendous memories with them for my meaningful time in this beautiful
country that would never be forgotten.
i
Abstract
With the technological developments in the post-genomic era there has been an increasing focus
of metabolic reconstructions for a large number of model organisms, such as, the prokaryotic Escherichia coli, the eukaryote Saccharomyces cerevisiae and human cells. Genome-scale reconstructions are usually stoichiometric and analyzed under steady-state assumption using constraint-based
modelling with Flux Balance Analysis (FBA).
Several applications that use constraint-based models have been particularly successful in computationally assessing metabolic networks, as well as for predicting maximum yield of a certain desired
product and analysis of gene essentially, under environmental and/or genetic perturbations. These
various applications of FBA require not only the stoichiometric information of the network, but also an
appropriate cellular objective function and possible additional physico-chemical constraints to compute the unique/multiple resulting flux distributions of an organism. One approach usually used to
address this optimization problem and to explore the metabolic capabilities of the organism is the
linear programming (LP) framework.
To compute metabolic flux distributions in microbes, the most common objective assumption is
to consider the optimization of the growth rate (“biomass equation”). However, other objectives may
be more accurate in predicting phenotypes. Since objective function selection seems to be, in general, highly dependent on the growth conditions, quality of the constraints and comparison datasets
specific, more investigations are required for better understanding the universality of the objective
function.
In this work, we explore the validity of different cases of optimization criteria and the effect of single
(or combinations) cellular constraints in order to improve the predictive power of intracellular flux distribution. These comparisons were evaluated to compare predicted fluxes to published experimental
13
C-labelling fluxomic datasets using three metabolic models with different conditions and comparison
datasets.
It can be observed that by using different conditions and metabolic models, the fidelity patterns of
FBA can differ considerably. However, despite of the observed variations, several conclusions could
be drawn. First, the maximization of biomass yield achieves one of the best objective function under
all conditions studied. For the batch growth condition the most consistent optimal criteria appears to
be described by maximization of the biomass yield per flux or by the objective of maximization ATP
yield per flux unit. Moreover, under N-limited continuous cultures the criteria minimization of the flux
distribution across the network was determined as the most significant. Secondly, the predictions
iii
obtained by flux balance analysis using additional combined standard constraints are not necessarily
better than those obtained using only the single constraint. Finally, the multi-objective optimization in
FBA illustrate a potential improvement of metabolic flux distribution predictions.
The experimental datasets, metabolic models and ready-to-run MATLAB scripts are freely available at https://github.com/hsnguyen/ObjComparison.
Keywords
Metabolic networks, constraint-based models, flux balance analysis, objective functions, constraints, flux distribution prediction, multi-objective optimization.
iv
Resumo
Com a evolução tecnológia na era pós-genómica tem havido um foco em crescendo de reconstruções
metabólicas para um grande número de organismos, tais como a Escherichia coli, a Sacharomyces
cerevisae e células humanas. As reconstruções à escala genómica são geralmente estequiométricas
e analisadas assumindo estado-estacionário, utilizando modelação baseado em restrição pela análise
de balanço de fluxos (flux balance analysis, FBA).
Várias aplicações que usam modelos baseados em restrições tem sido particularmente bem
sucedidas na computação de redes metabólicas, bem como na previsão do rendimento máximo
de um determinado produto e análise de genes essenciais por perturbações genéticas e ambientais. Estas várias aplicações de FBA requer não apenas informação estequimétrica da rede mas
também uma função objectivo objectivo celular e possiveis restrições fı́sico-quı́micas adicionais para
computar a distribuição de fluxo única/multipla de um organismo. Um método geralmente utilizada
para resolver este problema de optimização e explorar as capacidades metabólicas do organismo é
a programação linear.
Para computar a destribuição de fluxos metabólicos a função objectivo mais comum é considerar
a optimização da taxa de crescimento “equação da biomass”). Contudo, outros objectivos podem
ser mais precisos na previsão de fenótipos. Uma vez que a função objetivo parece ser, em geral,
altamente dependente das condições de crescimento, qualidade de restrições e conjunto de dados
de comparação, mais investigações são necessárias para uma compreensão da universalidade da
função objectivo.
Neste trabalho, foi explorado a validade de diferentes funções de optimização e o efeito de
restrições celulares singluares (ou combinações) para melhorar o poder preditivo da distribuição
de fluxos intracelulares. Estas comparações foram avaliadas para comparar os fluxos previstos com
um conjunto de dados fluxomicos de carbono marcado (C 13 ) em diferentes condições, usando três
modelos metabólicos. Foi possivel observar que usando diferentes condições experimentais e modelos metabólicos, os perfis de fidelidade em FBA podem divergir consideravelmente. No entanto,
apesar das variações observadas, várias conclusões podem ser retiradas. Em primeiro lugar, a
maximização do rendimento da biomassa alcança uma das melhores funções objectivo em todas
as condições estudadas. Em condições de crescimento “batch” o critério óptimo aparenta ser descrito pela maximização do rendimento da biomassa ou pelo maximização do rendimento de ATP
por unidade de fluxo. Além disso, em condições de quimiostato limitado na fonte de amônia o
critério de minimização a distribuição de fluxo através da rede foi determinada como mais signi-
v
ficativa. Em segundo lugar, as previsões obtidas por FBA usando restrições standard combinadas
não são necessáriamente melhores do que as restrições singulares. Finalmente, a optimizção multiobjectivo em FBA ilustra uma potencial melhoria na previsão da distribuição metabólica de fluxos. O
conjunto de dados experimetais, modelos metabólicos e os scripts em MATLAB estão disponı́veis e
de acesso livre em https://github.com/hsnguyen/ObjComparison.
Palavras Chave
Redes metabólicas, modelos baseado em restrições, análise de balanco de fluxos, funções objectivo, restrições, previsão distribuição fluxo, optimização multi-objectivo.
vi
Contents
1 Introduction
1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1 Systems biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.2 Metabolic networks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.3 Genome-scale models and metabolic engineering applications . . . . . . . . . .
4
1.1.4 Cell cultures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2 Fluxomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1 Metabolic Flux Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.2 Constraint-based analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 Flux Balance Analysis
11
2.1 General definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Variations of Flux Balance Analysis (FBA) . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Review of recent works examining objective functions . . . . . . . . . . . . . . . . . . . 17
2.4 The COBRA Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Methods
21
3.1 Metabolic models and experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Cellular constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Objective functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.1 Single objective functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Multi-objective formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.1 Pareto optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4.2 -constraint algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 Calculating distance between predicted and experimental data . . . . . . . . . . . . . . 27
4 Results and Discussion
29
4.1 The impact of objective functions by FBA in different conditions . . . . . . . . . . . . . . 30
4.1.1 Carbon (C)-limited chemostat cultures . . . . . . . . . . . . . . . . . . . . . . . . 30
vii
4.1.2 Ammonium (N)-limited chemostat cultures . . . . . . . . . . . . . . . . . . . . . . 32
4.1.3 Batch cultures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 The impact of objective functions by FBA in different metabolic models . . . . . . . . . . 33
4.3 Evaluation of pairwise constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 Evaluation of multi-objective functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Conclusions and Future Work
5.1 Main conclusions and work summary
39
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Ongoing and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Bibliography
43
Appendix A Experimental Datasets and Matches between experimental and model reactions
49
Appendix B FBA simulation results
55
Appendix C Published abstract and article
75
viii
List of Figures
1.1 Application of systems biology in metabolic engineering approaches. . . . . . . . . . . .
2
1.2 A typical metabolic network of E. coli central carbon metabolism with well-known pathways indicated. Solid arrows represent fluxes to biomass building blocks. Figure from
[1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3 Applications and number of studies of E. coli genome-scale models. This Figure is
taken from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4 An example of isotope tracing mechanism in glycolysis pathway. Isotopic carbon atoms
13
C appeared in glycolytic intermediates clearly or being measured by MS or NMR
spectroscopy (source [3]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.5 Six typical steps of a standard 13 C-Metabolic Flux Analysis (MFA) study (source [4]). . .
7
2.1 An example of a network with its corresponding basic Flux Balance Analysis (taken
from [5]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The solution space is decreased when introducing additional criteria in the formulation
of a typical FBA model (source [6]). Note that the optimal solution usually is non-unique. 14
2.3 An example for working mechanism of Parsimonious enzyme usage FBA. (source [7]). . 17
2.4 COBRA toolbox features overview (source [8]). . . . . . . . . . . . . . . . . . . . . . . . 20
3.1 Map from parameter space into objective function space. The red piece of curve is
noninferior solutions in this case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 A graphic illustration for -constraint algorithm in case of bi-objective problem. Figure
3.2a presents a noninferior solution found by using -constraint algorithm on a single
point. Figure 3.2b shows the Pareto front of the whole problem represented by the
red-highlighted region on the objective function space. . . . . . . . . . . . . . . . . . . . 26
3.3 Workflow presents the summary performed in the thesis.
. . . . . . . . . . . . . . . . . 28
4.1 Predictive fidelities for objective functions using different constraints (a,c,e,g) and corresponding individual flux predictions (b,d,f,h) between predicted and experimental fluxes
for all the objective functions evaluated without additional constraint of four E. coli (C)limited chemostat cultures on the Genome-scale model. Missing lines represent infeasible solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
ix
4.2 Predictive fidelities for objective functions with constraints (a,c) and corresponding individual flux predictions (b,d) between predicted and experimental fluxes for all the
objective functions evaluated without additional constraint of E. coli N-limited chemostat culture conditions on the Core and Genome-scale model. Missing lines represent
infeasible solutions in the simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Predictive fidelities for objective functions with constraints (a,c) and corresponding individual flux predictions (b,d) between predicted and experimental fluxes for all the objective functions evaluated without additional constraint of E. coli batch culture conditions
on the Genome-scale model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Predictive fidelities of Parsimonious enzyme usage FBA (pFBA) simulation for each
objective function in several datasets with three different metabolic models. Values
above 1 are out of range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5 Comparison between pFBA predictive fidelities using single (4.5a) and pairwise (4.5b)
constraints for the dataset from Ishii under the dilution rate of 0.7h−1 . . . . . . . . . . . . 36
B.1 Predictive fidelities for objective functions with single constraints for E. coli Core model.
56
B.2 Predictive fidelities for objective functions with single constraints for E. coli Schuetz
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
B.3 Predictive fidelities for objective functions with single constraints for E. coli iAF1260
Genome-scale model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
B.4 Predictive fidelities for objective functions with pairwise constraints for E. coli Core model. 62
B.5 Predictive fidelities for objective functions with pairwise constraints for E. coli Schuetz
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
B.6 Predictive fidelities for objective functions with pairwise constraints for E. coli iAF1260
Genome-scale model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
B.7 Individual flux predictions between predicted and experimental fluxes for all the objective functions evaluated without additional constraint of the Core model. . . . . . . . . . 68
B.8 Individual flux predictions between predicted and experimental fluxes for all the objective functions evaluated without additional constraint of the Schuetz model. . . . . . . . 70
B.9 Individual flux predictions between predicted and experimental fluxes for all the objective functions evaluated without additional constraint of the Genome-scale model. . . . . 72
B.10 Predictive fidelities of pFBA simulation for each objective function in six remaining
datasets with three different metabolic models. Values above 1 are out of range. . . . . 74
x
List of Tables
2.1 Single objective functions tested in FBA simulation from [9]. . . . . . . . . . . . . . . . . 18
3.1 Experimental datasets for E. coli from different literature references. . . . . . . . . . . . 22
3.2 Cellular constraints used as additional information for FBA in this work. Most of them
are adapted from [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 List of single objective functions tested and related references. . . . . . . . . . . . . . . 24
4.1 Predictive fidelities of pFBA simulation using single and multi-objective functions in the
Core model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Predictive fidelities of pFBA simulation using single and multi-objective functions in the
Schuetz model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Predictive fidelities of pFBA simulation using single and multi-objective functions in the
Genome-scale model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A.1 Experimental flux data for growth of E. coli in different conditions from different sources
(Ishii and Emmerling). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.2 (continue) from Nanchen, Yang, Perrenoud and Holm. . . . . . . . . . . . . . . . . . . . 51
A.3 Mapping from experimental reactions to the corresponding ones in model reconstructions. The “-” symbol indicates opposite direction of the reaction. Note that beside
this common mapping, some dataset may provide information about additional specific
reactions in the model reconstruction also. . . . . . . . . . . . . . . . . . . . . . . . . . . 52
A.4 Specific reaction fluxes used in each objective function. Reactions with ‘ f’ and ‘ b’
suffixes mean forward and backward irreversible reactions respectively which are originated from a reversible reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
xi
Abbreviations
COBRA Constraint-Based Reconstruction and Analysis
LP Linear Programming
MILP Mix-Integer Linear Programming
MFA Metabolic Flux Analysis
NLP Non-Linear Programming
QP Quadratic Programming
MOMA Minimization Of Metabolic Adjustment
FCA Flux Coupling Analysis
FBA Flux Balance Analysis
FVA Flux Variability Analysis
FSA Flux Sensitivity Analysis
OMNI Optimal Metabolic Network Identification
ROOM Regulatory On/Off Minimization
pFBA Parsimonious enzyme usage FBA
xiii
1
Introduction
Contents
1.1
1.2
1.3
1.4
Background . . . . . . . .
Fluxomics . . . . . . . . .
Motivation and Objectives
Structure of the thesis . .
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1
Chapter 1. Introduction
1.1
Background
In this Chapter some background knowledge and general definitions related to the thesis topic is
presented.
1.1.1
Systems biology
System biology is an interdisciplinary field that studies mainly the complex biological interactions
systemically. The ‘systems thinking’, in contrast with the traditional ‘partitions thinking’ of reductionists,
requires incredible workload of analysis and integrations that could only be accomplished thanks to
our innovative computational technologies. The history of the orientation shift from regular molecular
biology to systems biology has been well described in [10]. In which, the authors proposed that the
contemporary holistic methodology is not only evolved from the development of reductive mainstream
molecular biology to deal with bursting informations harnessed by high-throughput technologies. It
results also from the lesser well-known line of works involving nonequilibrium thermodynamics that
initially study novel functional states of multiple molecules interacted simultaneously.
Figure 1.1: Application of systems biology in metabolic engineering approaches.
Systems biologists usually utilize mathematical modelling methods to analyse biological interactions represented by different types of networks such as metabolic pathways, transcriptional regulations, signal transductions, etc to understand behaviour of cell as a whole. Together with the development of high-throughput technologies in molecular biology [11], the extension of computational and
systems biology is becoming increasingly significant. Among cellular activities, metabolism has direct relations with phenotyping mechanisms thus very helpful in diagnosis and treatment of diseases.
Metabolic engineering, genetic therapies and various other biotechnologies also benefit immensely
from systematic researches in metabolic networks [12].
2
1.1. Background
1.1.2
Metabolic networks
A metabolic network could be briefly depicted as the relationships between nodes of biochemical
molecules (metabolites) represented by corresponding hyper edges (reactions) in a biological system.
An edge in the graph is usually free of self-loops and report an irreversible flux (rate) of a reaction.
Thus, a reversible one constitutes two opposite fluxes and usually being indicated by a bi-directional
arrow. Internal reactions are edges that connected two groups of nodes (reactants and productions)
while exchange fluxes contain only one node with the other side lies on the outer environment. An
example for metabolic network of central carbon flow in E. coli is given in Figure 1.2.
Figure 1.2: A typical metabolic network of E. coli central carbon metabolism with well-known pathways indicated.
Solid arrows represent fluxes to biomass building blocks. Figure from [1].
Metabolic networks always play critical roles in numerous studies about organisms since this type
of network theoretically provides us the very underlying reactions controlling all the physico-chemical
aspects of a cell. Although still at starting points, the bursting development in this field keeps bringing
significant achievements to our knowledge about this biological world.
Recently, omics methods have been developing to cover all biological activities in cells. They range
step-by-step from levels of nucleotide sequences (genomics) to transcriptional matters (transcriptomics), then interactions of amino acids (proteomics) and metabolites’ properties (metabolomics)
[13]. However, most of them only offer qualitative in vitro aspects of molecular biology in the cell.
An emerging application of systems biology in metabolism, known as fluxomics, is expected to quantitatively predict the flux values of metabolic reactions in vivo which provide the critical link between
genes, proteins and the observable metabolic phenotypes. To do that, a number of modelling methods
3
Chapter 1. Introduction
have been employed to tackle the problem of complicated kinetic parameters which are used to measure the absolute values of metabolic concentrations in a direct way [14–16]. These methods usually
require the combination of an experimental metabolic flux analysis (Metabolic Flux Analysis (MFA))
with stoichiometric model of the network. Details for MFA and current technology will be described in
the next section.
The metabolism reconstruction of prokaryotic Escherichia coli has been one of the first but most
concerned problem and still being updated day-by-day. This is reasoned by various peer-reviewed
studies related to genome-scale metabolic network reconstruction of this microbial organism in different applications [2]. On this thesis work, only investigations on this bacteria’s metabolism will be
established under various living conditions by different model configurations.
1.1.3
Genome-scale models and metabolic engineering applications
The holistic approach of Systems Biology encourages the integration of more detailed biological
components to achieve comprehensive systems of living organism. The development of genome-
Figure 1.3: Applications and number of studies of E. coli genome-scale models. This Figure is taken from [2].
scale metabolic reconstructions is an example for that phenomenon. From the earlier studying that
utilize networks with only central carbohydrate metabolism [17, 18], successive versions with larger
scales emerged and perpetually being completed [19, 20]. The expansion has been done by adding
4
1.2. Fluxomics
novel subsystems such as fatty acid, alternate carbon metabolism or cell wall synthesis [2]. The
promotion of metabolic reconstructions to the genome scale stems from the boosting availability of
biological information in this post-genomic era and efforts to incorporate all these literature into single
systemic model. A genome-scale model, as being defined in [21], is a mathematical representation
of reconstructed genome-scale metabolism which contains information about metabolites and fluxes,
their stoichiometry, also the genes encoding enzymes of the metabolic network.
Figure 1.3 demonstrates six major applications of E. coli genome-scale models and the amount
of researches in each of those fields. Among them, metabolic engineering is one of the most topic
being concerned with the genome scale models. This field serves as a step required to design
newly cell families with expected characteristics by using mathematical and experimental tools in
metabolic analysis and modification [22]. Therefore, a systemic modelling would shed light to the
problem of complex nature of cellular metabolism and regulation in metabolic engineering. Recently,
unlike traditional methods for genetic modifications such as random mutagenesis and screening, the
contemporary engineering strategy is increasingly applying genome-scale metabolic reconstructions
to predict cellular phenotype from a systems level before in vivo implementation [2].
1.1.4
Cell cultures
Culture in this case is considered as the environmental condition where the cells grow. Studying
biological systems under various cultures is important because organisms living in different conditions
utilize appropriate mechanisms to demonstrate diversed phenotypes, thus give us knowledge about
adaptations and/or evolutions in biological world. There are two typical classes of cultures in research:
batch and chemostat cultures. The former stands for an ideal circumstance in which living things could
grow with a surplus amount of resources. The later represents more interesting and noteworthy cases
that usually happen in real world: certain nutrient is restrainted. This could be achieved by operating
a bioreactor which consists of a continuous flow of the feed (fresh medium) coming in and a outflow of
the effluent (culture liquid) being removed to keep the culture volume constant [23, 24]. The addition
rate of the limiting nutrient is well calibrated in order to control the growth rate of the micro-organism.
When the steady state is reach, the specific growth rate (µ) of the cells is the same as the dilution rate
of the bioreactor (D).
1.2
Fluxomics
Fluxomics is a quantitative representation of metabolic state by estimating the reaction rates of
genome-scale metabolism. As one of ‘omic’ studying field, it has been growing fast with numerous
of approaches to be implemented. Basically, there appeared two classes of modelling methods for
all techniques in fluxomics: stoichiometric (static) and kinetic (dynamic). While kinetic modelling
could be applied to study the potential space of dynamic metabolic fluxes under pertubations away
from steady state, this approach requires the use of complex biochemical reactions with large set of
kinetic parameters which are technically not easy to determine [25]. On the other hand, stoichiometric
5
Chapter 1. Introduction
models utilize the mass balance and other constraints in form of relative simple linear equations
system which is more convenient for further analysis.
Metabolic flux analysis (MFA) [26] and constraint-based analysis [27, 28] are two prominent techniques in fluxomics that take advantage of stoichiometric information of the metabolic network. While
MFA allows the experimental measurement of reaction fluxes in vivo by isotope labeling, the constraintbased analysis perform predictions on reaction rates in silico by using computational modelling. Both
of the analyses have their own pros and cons and integration of those two is becoming the current
trend in fluxomics approaches. MFA technologies are just on the first steps to be applied to largescale metabolic system in a high-throughput fashion [5]. However, the experiment on living cells
usually reflect more accurate values and usually used as references for the results from computer
simulations.
1.2.1
Metabolic Flux Analysis
Metabolic Flux Analysis (MFA) is usually known as an experimental method used in fluxomics to
measure incoming and outgoing rates of metabolites in a biological system. This definition is applied
in the realm of this thesis, although not ubiquitous in the whole system biology community. In some
circumstances, the term is used interchangeable with ‘fluxomics’ definition.
Figure 1.4: An example of isotope tracing mechanism in glycolysis pathway. Isotopic carbon atoms 13 C appeared
in glycolytic intermediates clearly or being measured by MS or NMR spectroscopy (source [3])
.
In isotopic labelling, the reactant undergone the interested reaction is marked by atom with a variation (isotope). The isotopes’ location and pattern in the products (under steady state) are detected
by MS (mass spectrometry) [29] or NMR (nuclear magnetic resonance) technology [30] thanks to
their different characteristics compared to the original atoms. This information could be used to determine the passage the isotopic atom followed in the metabolic pathway and the magnitude of the flux
through each reaction. Figure 1.4 demonstrates a simple example of this technique applied to study
the glycolysis pathway.
13
C-MFA is one of the most popular experimental MFA techniques which functions by using 13 C iso-
tope to measure intracellular metabolic fluxes [31]. Together with the improvement of its accurateness
and reliability on determining in vivo fluxes of metabolic pathways, the studies utilized
6
13
C-MFA has
1.2. Fluxomics
been increasing and reached a high level of maturity [4]. Recently, there are numerous of datasets
on E. coli central carbon fluxes published and ready for used in modelling analysis.
Figure 1.5: Six typical steps of a standard 13 C-MFA study (source [4]).
In this work, dozen datasets of in vivo fluxes in different experiments of
13
C-MFA were used as
references for the predictions which will be given in details later. A standardized work flow of 13 C-MFA
is given in Figure 1.5 and more information about this tracing method could be found in [4].
1.2.2
Constraint-based analysis
Metabolite mass balancing and stoichiometric information could provide the formulation of an linear equation system as a constraint. Theoretically, if this equation could be solved completely then all
reaction rates in the metabolic network are determined values. However, in fact, this ideal condition
is hardly happened because the number of linear equations, or constraints, is usually less than the
number of variables (or fluxes). Mathematically, this linear equation systems occurring in the analysis
is called under-determined or there is a degree of freedom in those flux variables space.
Because of this lack of measurement or constraints, the under-determined network could not be
computed. To tackle the problem, several approached has been suggested [32]: (i) utilizing isotopic
tracing experiment (ii) adding new metabolic constraints, e.g empirical knowledge about uptake rate,
flux boundaries, co-metabolites constraints (NAD(P)H, ATP), etc. (iii) using optimization framework
with suitable objective functions. The first approach (i) has been discussed already in the previous
section. Though this method give extra measurements with high confidence, the cost for experimental
and time setting made it unavailable for large system. Options (ii) and (iii) are highly feasible and still
being applied for in silico studying in fluxomics. They were the reason for constraint-based analysis to
be invented. This method introduced the application of optimization into metabolic network analysis
together with appropriate physico-chemical constraints.
A constraint optimization problem in general, is to optimize a cost function of optimization variables
subject to a set of constraints. The constraints could be represented by equalities or inequalities. The
7
Chapter 1. Introduction
mathematical formulation of an optimization problem is given as follow.
minx f (x)
s.t.
(1.1)
h1 (x) = 0
···
hp (x) = 0
g1 (x) ≤ 0
···
gm (x) ≤ 0
where x ∈ Rn is the optimization variable, f : Rn → R is the cost function; h1 , . . . hp are equality constraints while g1 , . . . , gm are inequality constraints. It is worth noting that every maximization problem
can be converted be minimization problem by changing the sign of f (x).
Usually in constraint-based analysis, the cost function is linear which represent set of biochemical
of interest and also known as objective function. The detail formulation will be given in Chapter 2.
In this thesis, a typical constraint-based modelling technique, Flux Balance Analysis, is used as the
main tool for predicting the flux distribution of E. coli metabolic networks.
1.3
Motivation and Objectives
Inspired by the importance of fluxomics in System Biology and the fast-growing constraint-based
modelling techniques in this field, this thesis work is conducted based on the supports from previous studies. This work considers identifying objective functions and extra constraints that could be
used in order to improve the predictive power of constraint-based models of metabolism, by using
different metabolic networks and experimental data than previous works [9, 33, 34]. Although there
appeared some studies followed this direction, the universality of the objective function of biomass
growth remains an open question. Since objective functions are highly depend on the metabolic models/systems, growth conditions, model parameters and training data in general, more investigation
should be established for better understanding the universality of the objective function in various
conditions. This need has been recognized also by the system biology community [35, 36].
In line with that, the main goal of the current work is to explore different optimality principles and
the effect of single (or combinations) empirical constraints in order to improve the predictive power
of intracellular flux distribution in metabolic networks. This idea has been tested with three E. coli
metabolic subsystems in different cultures and novel comparison datasets, expanding a previous
evaluation [9].
1.4
Structure of the thesis
This thesis is divided into five chapters. In the present Chapter 1, the developed work is contextualized and the aims are mentioned. Chapter 2 provides a background and relevant knowledge about
8
1.4. Structure of the thesis
Flux balance analysis, a common method in constraint-based analysis. The mathematical formulation
of Flux Balance Analysis (FBA) is introduced along with an example and its derived forms in different
applications are described. Finally, the recent works related to objective function studying in FBA is
described. Chapter 3 starts with the description about the objective functions, cellular constraints,
type of model reconstructions examined and the datasets that are used in this work. The MATLAB
implementation is also introduced in this chapter. Chapter 4 presents and discusses the obtained
results of all simulations. Finally, Chapter 5 provides the summary and the conclusion of the thesis,
together with suggestions to extend it in the future. The supporting information are given as Appendix.
Appendix A is for more detail about the used experimental flux datasets and the mapping from those
to the models’ reactions. Appendix B contains all plots generated from simulation results, divided by
three parts: simulations on single constraints, on pairwise constraints and relationship between predicted versus experimental fluxes represented by scatter plots. The publications based on this work
are given in Appendix C.
9
Chapter 1. Introduction
10
2
Flux Balance Analysis
Contents
2.1
2.2
2.3
2.4
General definitions . . . . . . . . . . . . . . . . . . . . .
Variations of FBA . . . . . . . . . . . . . . . . . . . . . .
Review of recent works examining objective functions
The COBRA Toolbox . . . . . . . . . . . . . . . . . . . .
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12
13
17
19
11
Chapter 2. Flux Balance Analysis
2.1
General definitions
The metabolic networks usually contain only stoichiometric information and being analysed under
steady-state assumption using constraint-based modelling with flux balance analysis (FBA) [37]. The
constraints here refer to the different limited conditions that a given biological system must satisfy,
such as physico-chemical, topological and environmental factors [5]. Steady-state assumption forms
the underlined linear constraint that prevent metabolites from dilution due to growth [38].
Figure 2.1: An example of a network with its corresponding basic Flux Balance Analysis (taken from [5]) .
FBA is a well-known method to deal with modelling and analyzing these sets of flux. In which, the
flux balance and other constraints could be manifested as mathematical expressions of a constrained
system that is subjected to an optimization criterion later. The reason for further optimizing step is that
in fact, normally the available constrains are still not sufficient to make the system fully determined.
Thus the desired flux distribution, among subspace of solutions, would be presumed to optimize a
certain cellular objective which is also described as a function in this method. For example, the most
common objective is to consider the maximization of the growth rate (biomass equation) in microbes
to compute metabolic flux distributions [39].
An illustration about the use of FBA to a given system is shown in Fig. 2.1. This example used a
toy model of metabolic network which consists of only 3 metabolites (A, B, C) and 7 fluxes (R1, . . .
12
2.2. Variations of FBA
R7). Among them, R1, R2, and R3 are internal fluxes while the remaining four represent drain (or exchange) reactions. At first, based on the reconstruction of the network, the stoichiometric information
is extracted and stored in form of a node-arc incidence matrix. This stoichiometric matrix, denoted as
S, has rows corresponding to metabolites and columns mapping to all fluxes involved. So that (i, j)
element of this matrix represents the stoichiometric coefficients of metabolite i taking part in reaction
j. The sign of this value depend on the role of the metabolite in the corresponding flux:

if metabolite i is a product in reaction j
 1
−1
if metabolite i is a reactant in reaction j
sign(Sij ) =

0
if metabolite i does not take part in reaction j
For the large-scale metabolic network, S contains mostly zeros among very few of non-zeros elements
and thus it would be treated as sparse matrix with special operations for the sake of computational
cost. The FBA problem is then formulated with the first constraint regarding the steady-state assumption, namely mass balance condition:
Sv =
dC
=0
dt
(2.1)
In which, the flux rates are limited by upper and lower bound: viub ≥ vi ≥ vilb or in short with matrix
notation: v ub ≥ v ≥ v lb .
Another constraints for certain flux could be usually included depend on the specific environmental
or physico-chemical restrictions in order to characterize certain cellular network function. The mass
balance equation together with these constraints form the solution space of all possible flux state
points. Then FBA typically solve an optimization problem on this space, with objective functions
linearly being expressed as:
Z =c×v
where c is coefficient vector that define the weight of each flux in the objective function. The parameter
c can take the form of simple sparse vector with only one non-zero element in case of objective
involving single reaction, such as biomass yield or ATP yield [40]. If multiple fluxes involved, such as
redox potential or ATP producing optimization [33], objective function Z is then a linear combination
of vi with varied coefficients ci . In the example from Figure 2.1, Z = v5 thus we could infer that
c = [0 0 0 0 1 0 0].
In case of more complex functions, mixed-integer or non-linear programming have to be used to
solve the optimization problem [9]. The general mathematical expression for simple FBA problem is:
min/max (Z = f (v))
s.t.
(2.2)
Sv = 0
v ub ≥ v ≥ v lb
2.2
Variations of FBA
Recently, the applications of constraint-based analysis has been expanding perpetually with expect to improve our knowledge about the complicated characteristics of metabolism in living cells.
13
Chapter 2. Flux Balance Analysis
Among them, numerous modelling techniques were remoulded from the idea of flux balance analysis.
FBA has become an important tool for metabolic engineering and systems biology [5, 12, 41]. Next,
more details of selected in silico algorithms will be described.
The original FBA is typically applied to estimate the system’s state in which certain biochemical
productions and/or growth rate (biomass) are optimized. For further comprehend the property of
metabolic network and relationship of reactions, there appeared some other tools such as flux variability analysis (Flux Variability Analysis (FVA)), flux coupling analysis (Flux Coupling Analysis (FCA))
and flux sensitivity analysis (Flux Sensitivity Analysis (FSA)) [28]. While FVA provides information
about the variation in flux distribution when the objective is reached its optimized value due to the
under-determined of the solution space [42], FCA studies the flux change of objective function in response to the pertubations of other fluxes [43] and FCA is to examine correlation between all pairwise
fluxes in a metabolic network [44].
Figure 2.2: The solution space is decreased when introducing additional criteria in the formulation of a typical
FBA model (source [6]). Note that the optimal solution usually is non-unique.
On the other hand, several formulations of FBA had been developed to investigate the flux states of
mutants created by specific genetic modifications. This stems from the fact that the objective used for
wild-type might not accurately represents the behaviour of systems with gene-knockouts. There were
two common frameworks usually proposed for this case, namely Minimization Of Metabolic Adjustment (MOMA) (minimization of metabolic adjustment) and Regulatory On/Off Minimization (ROOM)
(regulatory on/off minimization). Both of two approaches calculated flux state of a mutant by minimizing corresponding distance metric to adjust the solutions from wild-type to mutant case. In MOMA [15],
the adjustment metric is the Euclidean distance between flux state points in mutant solution space
with optimal point in wild-type solution found by FBA beforehand. This could be formally expressed
as a quadratic programming as follow:
min(Lv + v T Qv)
s.t.
(2.3)
Sv = 0
v ub ≥ v ≥ v lb
vj = 0, j ∈ A
here L = −2v wt where v wt is the optimal solution found by FBA in wild-type strain; Q in quadratic part
is simply unit matrix and indices j represent reactions involved when gene is deleted.
On the other hand, ROOM [16] used the number of ‘significant’ flux changes from the wild-type
14
2.2. Variations of FBA
flux distribution as the metric and try to minimize this measure to find the flux state of mutant. The
mixed-integer linear programming is used to solve this problem:
Pm
min( i=1 yi )
s.t.
(2.4)
Sv = 0
v ub ≥ v ≥ v lb
vj = 0, j ∈ A
vi − yi (viub − wiu ) ≤ wiu
vi − yi (vilb − wil ) ≥ wil
yi ∈ {0, 1}
wiu = wi + δ|wi | + wil = wi − δ|wi | − where wiu and wil are used for determine the ’significant’ of flux changes, which are denoted by
Boolean variable yi . The tolerance parameters δ (multiplicative) and (additive) are chosen regarding
the trade-off between resultant fluxes and running time.
Another emerged class of constraint-based analysis takes advantage of multi-objective system.
The theory is to optimize an objective function while other precursor objectives are still achieved.
A typical case study is to enhancing the production rate of a desired biochemical metabolite while
the growth rate is still secured by setting an additional constrain about biomass optimization (primal
problem) in the system. OptKnock [45] is a well-known representative which functions such that bilevel
optimization mechanism:
max vproduct
s.t.
(OptKnock)
(2.5)
max vbiomass (Primal)
s.t.
Sv = 0
v ub ≥ v ≥ v lb
min
vbiomass ≥ vbiomass
vjmin yj ≤ vj ≤ vjmax yj
X
(1 − yj ) ≤ K
j
yi ∈ {0, 1}
min
where vbiomass
is a minimum level of biomass production, vjmin and vjmax are determined by minimizing
and maximizing every reaction flux subject to the constrains in Primal problem ,K is the number of
allowable knock-outs [45].
Using the basic concept of OptKnock framework as starting point, several constraint-based modelling methods has been invented for the purpose of network design. OptStrain, OptGene and OptReg
were among them. OptStrain [46] ultilized a two-step algorithm to determine the maximum yield of the
desired biochemical production and at the same time, the number of required non-native reactions for
15
Chapter 2. Flux Balance Analysis
the system to reach this optimal state had been minimizing. OptGene [47] used genetic programming
to do the optimization work to get the phenotypic objective of interest. OptReg [48] was designed to
investigate the activation/inhibition and elimination reaction set needed to satisfy the level of desired
phenotype.
Another noteworthy algorithm is called optimal metabolic network identification (Optimal Metabolic
Network Identification (OMNI)) [49]. This tool is for identifying set of reactions in which experimental
13
C-based fluxes would come to agreement with predicted values. A bilevel Mix-Integer Linear Pro-
gramming (MILP) optimization is employed: the first step is to generate solution space of fluxes by
FBA with biomass being optimized, and the second optimization explores the set of ‘agreed’ reactions
by minimizing the discrepancy between experimental data and predicted flux distribution.
P
min( j ωi | vjopt − vjexp |)
s.t.
(2.6)
v opt = max vbiomass
Sv = 0
v ub ≥ v ≥ v lb
vdmin yd ≤ vd ≤ vdmax yd
vl = vlexp
opt
min
vbiomass
≥ vbiomass
P
j (1 − yd ) ≤ K
yd ∈ {0, 1}
in which, d, l, j denoted for indices of deleted reactions due to genetic knock-out, experimentally
measured reactions and all of the reactions in the system [28]. The parameter ω is weight vector for
measured fluxes.
Parsimonious enzyme usage FBA is an important derivative of FBA that has been utilizing broadly
in fields related to metabolic analysis of genome-scale constraint-based models [7]. This approach is
originally not only used to study flux distribution of a network as a typical constraint-based modelling
analysis but also be able to classify genes based on condition-specific pathway usage as predicted
in silico. As the output of this method, there are five classes of genes associated with reactions that
(1) are essential for optimal and suboptimal growth, (2) are inside the Parsimonious enzyme usage
FBA (pFBA) optima, (3) are ELE, requiring more enzymatic steps than alternative pathways that
meet the same cellular need, (4) are MLE, requiring a reduction in growth rate if used, or (5) cannot
carry a flux in the given environmental condition/genotype (pFBA no-flux) [7]. An example is given
in Figure 2.3. In which, Gene A, classified as MLE, represents an enzyme that uses a suboptimal
co-factor to catalyze a reaction, thereby reducing the growth rate if used. Gene B, classified as pFBA
no-flux, cannot carry a flux in this example since it is unable to take up or produce a necessary
precursor metabolite. Genes E and F in this example require two different enzymes to catalyze the
same transformation which Gene D can do alone; therefore they are classified as ELE. Gene G is
16
2.3. Review of recent works examining objective functions
essential, since its removal will stop the flux through all pathways. Genes C and D represent the
most efficient (topologically and metabolically) pathway and therefore are part of the pFBA optima [7].
Basically this approach finds a flux distribution with minimum absolute values among the alternative
Figure 2.3: An example for working mechanism of Parsimonious enzyme usage FBA. (source [7]).
optima, assuming that the cell attempts to achieve the selected objective function while allocating the
minimum amount of resources (i.e. minimal enzyme usage). Mathematical formulation is given by:
min
Pj=1
m
(2.7)
virrev,j
s.t. max vbiomass = vbiomass,lb
s.t. Sirrev virrev = 0
0 ≤ virrev,j ≤ vmax
where m is the number of gene-associated irreversible reactions in the network, vbiomass refers to the
growth rate and vbiomass,lb is the lower bound for the biomass rate which is determined by FBA of irreversible network with Sirrev and non-negative steady-state flux virrev . In order to achieve irreversible
network from a certain one, all reversible reactions are split into two irreversible reactions.
2.3
Review of recent works examining objective functions
In numerous researches, maximizing growth rate has been assumed as the most appropriate
objective function for micro-organisms; however it has been found that this may not occur in all cases
[39]. Over the last years, a number of studies have been carried out to test the use of objective function
optimization by FBA with different networks for predicting metabolic phenotypes. This set of studies
can roughly be divided into two ways [35] (i) studies examining hypotheses on presumed cellular
objective functions through comparison to measured fluxes generated by
13
C-labeling techniques
17
Chapter 2. Flux Balance Analysis
[9, 34, 50], and (ii) studies examining optimization techniques to discover or algorithmically predict
biological objective functions from experimental data [51, 52].
As part of the former, an evaluation of objective functions for Escheriachia coli central metabolism
model has been established by Schuetz et al [9]. There are in total 11 objective functions, together
with 8 physico-chemical constraint forming 99 simulations has been evaluated in attempt of finding
the combination that could best predict the
13
C-determined in vivo intracellular fluxes in different
environment conditions. Table 2.1 shows a summary of the different comparative objective functions
for FBA reported in [9].
Table 2.1: Single objective functions tested in FBA simulation from [9].
Objective
function
Description
Max
biomass
Maximization
biomass
[19, 40, 53]
Max ATP
Mathematical
definition
Reported performance
max vvbiomass
glucose
Batch
Good
Chemostat
Very good
Maximization of ATP
yield [40, 54]
vAT P
max vglucose
Good
Very good
Minimization of l2 norm of all fluxes
[55, 56]
min
+/-
Poor
Max ATP per
flux unit
Maximization of ATP
yield per flux unit [57]
AT P
max vP
v2
Very good
Useless
Max BM per
flux unit
Maximization
of
biomass yield per flux
unit.
P 2
max vbiomass
v
Good
Poor
Min glucose
Minimization of glucose consumption.
glucose
min vbiomass
Useless
+/-
Min reaction
step
Minimization of reaction steps
Pn
min i=1 yi2 ,
yi ∈ {0, 1}
Poor
+/-
Max ATP per
reaction step
Maximizatin of ATP
yield per reaction step
min PvnAT Py2 ,
i=1 i
yi ∈ {0, 1}
Useless
Poor
Min
redox
potential
Minimization of redox
potential [33]
min
P
Poor
Poor
Min ATP production
Minimization of ATP
producing fluxes [33]
min
P
vAT P
vglucose
Poor
Poor
Max
ATP
production
Maximization of ATP
producing fluxes [33,
58, 59]
max
P
Useless
Good
Min
P
v2
of
yield
P
vi2
i
i
v
vN ADH
vglucose
vAT P
vglucose
The result shows that in unlimited resource condition (batch culture), the best objective is maximiza18
2.4. The COBRA Toolbox
tion of ATP yield per unit of flux while in chemostat culture with limited nutrient, the linear maximization
of ATP or biomass yield proved to be more suitable objectives; however in the cases tested there was
still significant variation between predicted and experimental fluxes. It also worth noting that in this result the artificial constraints become unimportant if appropriated objective function is chosen in given
condition. As a continuous work, a study based on multi-objective optimization theory is taken place
to understand the principles behind the evolution of flux states [50]. This research pointed out a
combination of three efficiency objectives, namely maximum ATP yield and maximum biomass yield
(as defined in Table 2.1) with minimum sum of absolute fluxes, produced good approximation to the
experimental data. It is worth noting that in this study, the l1 norm (Manhattan norm) of all intracellular
fluxes was used instead of l2 norm (also Euclidean norm) as in the premise. The results from these
two has offer useful hints to the method used in this thesis which will be revealed in the next sections.
Alternatively, others have developed algorithms to systematically identify or predict a relevant objective function using experimental data. ObjFind [51] is an optimization-based framework that created for that purpose. This in silico procedure attempts to solve the coefficients of importance, which
equivalent to vector c in Z, on reaction fluxes while keeping the divergence between resultant and
experimental flux distribution to be as small as possible [51]. A bi-level optimization problem is used
to describe this task: minimize the error between in vivo and in silico fluxes by quadratic programming, subject to the fundamental FBA problem. BOSS is another tool that is claimed to be able to
recapitulate the actual objective including even excluded reaction from reconstruction model [52].
Another idea rising recently is that metabolism should be represented by near-optimal flux distributions, rather a single and fixed solution by introducing a novel objective function. This would form
a region that regulated cellular fluxes are driven into and considered plausible without any further
optimized step. PSEUDO (perturbed solution expected under degenerate optimality) [60] is created
as a tool to predict suboptimal flux distribution utilizing a mathematical formulation of this theory. A
good review for these objective discovery approaches could be found in [35].
2.4
The COBRA Toolbox
The COnstraints Based Reconstruction and Analysis (Constraint-Based Reconstruction and Analysis (COBRA)) toolbox is a set of utilities integrated as form of a software package to provide researchers with easy access to core COBRA methodologies [8]. The openCOBRA project has been
developing based on that idea. Starting with tools for MATLAB, it now has grown to include Python
modules and still on the way of improving itself to deal with the next complex COBRA models.
In this thesis, COBRA toolbox for MATLAB will be employed to deal with the model reconstructions
and optimization framework of FBA. This toolbox offers the processing of SBML files of metabolic
network structure and also several functions that are able to deal with flux analysis tasks. Especially, the MATLAB function optimizeCbModel could be invoked for FBA simulations and its variant
with appropriate additional model settings. It supports optimization on linear programming (Linear
Programming (LP), mix integer linear programming (MILP) and quadratic programming (Quadratic
19
Chapter 2. Flux Balance Analysis
Programming (QP)) with suitable solvers integrated. The functionality for non linear programming (
Non-Linear Programming (NLP)) is currently still on progress. For more details see [8, 14].
Figure 2.4: COBRA toolbox features overview (source [8]).
20
3
Methods
Contents
3.1
3.2
3.3
3.4
3.5
3.6
Metabolic models and experimental data . . . . . . . . . . . . .
Cellular constraints . . . . . . . . . . . . . . . . . . . . . . . . . .
Objective functions . . . . . . . . . . . . . . . . . . . . . . . . . .
Multi-objective formulations . . . . . . . . . . . . . . . . . . . . .
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculating distance between predicted and experimental data
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23
24
26
27
21
Chapter 3. Methods
3.1
Metabolic models and experimental data
In present thesis two published condensed and one genome-scale metabolic reconstruction of
E. coli were selected as the metabolic models for flux analysis. The first condensed genome-scale
metabolic model (Core Model) [6] contains 72 metabolites forming in total a set of 95 chemical reactions which consist of 75 internal and 20 drain fluxes. The second model (Schuetz model) [9] includes
98 reaction and 60 metabolites. The genome-scale model reconstruction iAF1260 [20] was also used
in this work (Genome-scale model). This model contains in total 1668 metabolites and 2382 reactions.
The 12 experimental datasets used to compare with the model predictions consists of measurement of fluxes and extracellular rates in different cultures extracted from 13 C-MFA analyses on E. coli.
These datsets are available at KiMoSys repository [61] and come from various literature sources
[1, 62–66] to diversify the test environments (chemostat aerobic N- and C-limited, batch aerobic), as
described in Table 3.1.
Table 3.1: Experimental datasets for E. coli from different literature references.
Reference
Nanchen et al.[63]
Emmerling et al.[1]
Yang et al.[66]
Ishii et al.[62]
Perrenoud et al.[65]
Holm et al.[64]
Dilution rate
0.09h−1
0.4h−1
0.09h−1
0.09h−1
0.4h−1
0.1h−1
0.55h−1
0.1h−1
0.4h−1
0.7h−1
0.62h−1
0.67h−1
Environment
chemostat, C-limited
chemostat, C-limited
chemostat, N-limited
chemostat, C-limited
chemostat, C-limited
chemostat, C-limited
chemostat, C-limited
chemostat, C-limited
chemostat, C-limited
chemostat, C-limited
batch, aerobe
batch, aerobe
a
mcore/gc
38
38
33
33
33
32
32
47
47
47
36
37
b
mSchuetz
66
66
49
49
49
47
47
69
67
67
57
60
a
b
mcore/gc : number of matched reactions for the E. coli Core and Genome-scale model.
mSchuetz : number of matched reactions for the Schuetz model.
Usually the number of fluxes measured by these
13
C-MFA is less than those of the model re-
constructions. In order to evaluate the performance of FBA simulations, there needed a matching
step from the predicted flux distribution to the reference dataset. The mapping of available measured
fluxes to corresponding reactions in FBA model is given in Appendix A.
3.2
Cellular constraints
One problem of using FBA is that the simulation usually returns predictions which are diverse from
reality. To solve this, biological or physiochemical restrictions are added to limit the solution space of
the model. We systematically tested 6 single (or combined) constraints: P-to-O (P/O) ratio was set to 1
[67] by setting the energy-coupling NADH dehydrogenase I (Nuo) and cytochrome oxidase bo3 equal
to zero [9]; the upper limit bound for the maximal oxygen consumption rate (qO2 max) was set to 15
mmol/gh as experimentally reported for chemostat cultures [62]; the bounds on cellular maintenance
22
3.3. Objective functions
energy (ATPM reaction) as the requirement for growth-independent was left at the default model
value of 8.39 mmol/gh [6]; bound all fluxes (bounds) to maximal 200% of the glucose uptake rate as
observed experimently [68] and 35% of NADPH overproduction compared to the NADPH requirement
for biomass production. Finally the combination of all above constraints is also included. They are
empirical conditions which are synthesized gradually from literature and former researches as could
be seen from Table 3.2
Empirical constraints related to the bounds of fluxes given in the original models were discarded
and at least the relative substrate uptake rate (glucose) was constrained in each case.
Table 3.2: Cellular constraints used as additional information for FBA in this work. Most of them are adapted
from [9].
Constraint
P/O=1
qO2 max ≤ 15
Maintenance constraint
Bound
NADPH
All
3.3
Meaning
P-to-O ratio is set to unity due to known coupling efficiencies and expression levels of respiratory chain
components [9]
Maximum of oxygen uptake is set to 15 mmol/(gh)
as another level of limitation about oxygen utilization.
ATPM reaction flux is set to 8.39mmol/(gh) empirically as the requirement for growth-independent
maintenance of the cell.
Bound all fluxes to maximal 200% of the glucose
uptake rate as observed by experiments.
35% of NADPH overproducted than needed for
biomass production.
Apply all above constraints.
Reference
[67]
[17]
[18, 63]
[1, 68–70]
[63]
Objective functions
This section presents the objective functions tested in the FBA models. There were not only
the single objective function being examined, but the combinations of up to three of them were also
involved at the same time in multi-objective optimization.
3.3.1
Single objective functions
For the experimental datasets selected we evaluated the predictive ability of eight different objective functions: maximization of biomass yield (max BM) and ATP (maintenance reaction) (max ATP),
P
minimization of overall intracellular fluxes l1 -norm (min | v |), maximization of ATP and biomass
P
P
yield per unit flux (max ATP v and max BM v, respectively), minimization of redox potential (min RD), and minimization and maximization of ATP producing fluxes per unit substrate
P
P
(min ATPS and max ATPS, respectively). The objective functions correspond to the most
significant performed in [9, 50] and were mathematical defined as in Table 3.3.
23
Chapter 3. Methods
Table 3.3: List of single objective functions tested and related references.
Objective function
Description
Maximization of biomass yield
Mathematical definition
max vvbiomass
glucose
Max BM
[19, 40, 53]
Max ATP
Maximization of ATP yield.
vAT P
max vglucose
[40, 54]
Min Flux
Minimization of l1 -norm of all fluxes.
min
Max ATP/flux
Maximization of ATP yield per flux unit.
AT P
max vP
v2
P
|vi |
Reference
[50]
[57]
i
Maximization of biomass yield per flux
unit.
P 2
max vbiomass
v
Min Rd
Minimization of redox potential.
min
Min ATPprod
Minimization of ATP producing fluxes.
min
Max ATPprod
Maximization of ATP producing fluxes.
max
Max BM/flux
[9]
i
P
[33]
P
vAT P
vglucose
[33]
P
[33, 58, 59]
vN ADH
vglucose
vAT P
vglucose
For the two nonlinear objective functions max ATP/flux and max BM/flux, the l2 -norm (Euclidean
norm) of the fluxes vector was used in accordance with [9] while min Flux refers to the l1 -norm (Manhattan norm) of the same vector as in [50]. This is because using l1 -norm reported better predictive
results than that of l2 -norm when it comes to minimization-of-all-fluxes objective function [50].
For linear optimization, there are usually non-unique sets of flux values v that give the same optimal value of objective function [71]. To avoid typical degeneracy of FBA solutions, the principle of
parsimonious enzyme usage (pFBA) was used [7]. The detail formation of this constraint-based analysis has been given in Chapter 2. However, the purpose of pFBA application here is not necessarily
to classify gene into five categories. Adapted to this work, the optimization problem is mathematically
described as follows:
min
s.t.
Pn
i=1
| vi |
(3.1)
Z = Zoptima
S.v = 0
v ub ≥ v ≥ v lb
3.4
Multi-objective formulations
Usually in nature, optimization of one single objective function could not represent completely the
behaviour of cell evolution. For example, cells tend to freely maximize growth rate in batch culture
but within resource scarce environment, they also have to take the optimization of food usage into
consideration. For that reason, several objective functions should be put together for studying at
the same time. One possible method to deal with this problem is to use multi-objective or Pareto
optimization instead of the single objective.
24
3.4. Multi-objective formulations
3.4.1
Pareto optimization
A mathematical formulation of Pareto optimization could be described as follow:
min/max(Z1 , Z2 , ..., Zn )
s.t.
(3.2)
Sv = 0
v ub ≥ v ≥ v lb
where Z1 , Z2 , ..., Zn form the set of objective functions that must be targeted simultaneously in this
problem. By doing this, it allows the trade-off between objective functions and since they usually
competitive, there is no unique solution. The solutions of particular problem should be a set of varied
Zi and is known as noninferiority solutions set or simply Pareto front. It is defined as the solutions
that improvement in one objective requires a degradation of another.
Figure 3.1: Map from parameter space into objective function space. The red piece of curve is noninferior
solutions in this case.
Mathematical definition: v ∗ ∈ Ω is a noninferior solution if there does not exist ∆v such that
(v ∗ + ∆v) ∈ Ω and k, l ∈ {1 . . . n} such that
Zk (v ∗ + ∆v) ≤ Zk (v ∗ )
(3.3)
Zl (v ∗ + ∆v) < Zl (v ∗ )
An example of Pareto optimization with only two objective functions of two variables is shown in Figure
3.1.
One method to find Pareto front of an multiple objective functions optimization is the -constraint
algorithm. In this thesis, Pareto optimality is applied in FBA model with up to the three best single
linear objective functions in each growth condition.
3.4.2 -constraint algorithm
The method used in this thesis to calculate the Pareto front of multiple objective system is based
on the idea of -constraint algorithm [72] which is also the method used in [50]. -constraint involves
25
Chapter 3. Methods
(a)
(b)
Figure 3.2: A graphic illustration for -constraint algorithm in case of bi-objective problem. Figure 3.2a presents
a noninferior solution found by using -constraint algorithm on a single point. Figure 3.2b shows the Pareto front
of the whole problem represented by the red-highlighted region on the objective function space.
minimizing a primary objective while expressing the others in the from of inequality constraints:
minv∈Ω Zp (v)
Zi (v) ≤ i
s.t.
i = 1, . . . , n. i 6= p
where Zp is the primary objective to considered among the set of all functions (Z1 , . . . , Zn ). An
example for the case of bi-objective Pareto optimization by using -constraint is given in Figure 3.2a.
In this illustration, Z1 is the primary objective.
In order to adapt this method to solve the cases of interest, a numerical algorithm is employed.
In which, the possible ranges of all non-primary objective functions are divided by nstep = 1000 then
-constraint will be applied to every ∈ Rn−1 retrieved by those partitions.
=
i
=
{1 , . . . p−1 , p+1 , . . . n }
(Zi,ub − Zi,lb )
Zi,lb + k(i)
, k(i) ∈ {1, . . . , nstep}
nstep
i = 1, . . . , n. i 6= p
where Zi,lb and Zi,ub is lower and upper bound of Zi respectively. A graphic illustration is shown in
Figure 3.2b for the Pareto front of two objective functions. Due to the exponential iterations needed
in -constraint algorithm, the number of objective considered in Pareto optimization is limited to three.
The only three single objective being chosen for this simulation are max BM, max ATP and min Flux.
3.5
Implementation
All calculations were implemented in Matlab 2012b (Mathworks Inc. Software) and simulations
were performed using the Constraint-Based Reconstruction and Analysis (COBRA) toolbox (v. 2.0.5)
[8]. In terms of optimization solvers, GLPK [73] was used for linear problems.
26
3.6. Calculating distance between predicted and experimental data
For the two non-linear non-convex objective functions, a numeric approximation method was used.
For example, with the objective function max ATP/flux, firstly the range of ATP flux is calculated then
1000 value points are uniformly selected along the distance. After that, for each point, the ATP reaction
P 2
rate is constrained to this value and the l2 -norm of all fluxes in this system
vi is minimized simultaneously. The maximum of all 1000 ratios between ATP fluxes with their corresponding minimized
flux norms is a good approximation for the objective max ATP/flux. Similar approach was used for
max BM/flux.
Simulations were executed in parallel on a server machine with 8 AMD processors of 2.3GHz
each. The libSBML [74] was the package used for reading SBML model files. Experimental datasets,
metabolic models and ready-to-run matlab scripts are freely available at https://github.com/hsnguyen/
ObjComparison.
3.6
Calculating distance between predicted and experimental data
To evaluate the prediction ability, the definition of predictive fidelity [9] was used. This error is
defined by:
d(vcomp , vexp ) = T W =
Wi,i =
(3.4)
vcomp −vexp
vglucose
1
(
σiexp
P
1
−1
i σiexp )
where d(vcomp , vexp ) is the standardized Euclidean distance between predicted fluxes vcomp and the
experimental in vivo vexp fluxes weighted by their experimental variances σ exp . The set of compared
reactions are given as Appendix A. Smaller predictive fidelity represent a better agreement between
computational predicted and experimental fluxes.
For the original definition of predictive fidelity in [9], there are different Euclidean distances to the
in vivo fluxes due to alternative optima. Consequently the fidelity “range” is defined by the minimum
and maximum Euclidean distances from predicted fluxes and corresponding in vivo fluxes.
min/max d(v, vexp )
s.t.
(3.5)
Z = Zoptima
Sv = 0
v ub ≥ v ≥ v lb
where d(v, vexp ) is the Euclidean distance between fluxes v and the available in vivo vexp . In this
thesis, by applying pFBA in every simulations, the alternative optima phenomenon is efficiently neutralized and the fidelity “range” could be considered as a single erroneous value of the prediction.
A summary of the method applied in this thesis is shown in Figure 3.3. In total, there are three
model reconstructions, each of them would be tested with 8 single objective functions together with
7 single cellular constraints plus 10 pairwise constraints. Thus for the first round there would be
27
Chapter 3. Methods
Figure 3.3: Workflow presents the summary performed in the thesis.
3 × 8 × 17 = 408 simulations needed to be executed. The simulations for the multi-objective models is
computationally expensive due to the application of Pareto optimization problem.
28
4
Results and Discussion
Contents
4.1
4.2
4.3
4.4
The impact of objective functions by FBA in different conditions . . . .
The impact of objective functions by FBA in different metabolic models
Evaluation of pairwise constraints . . . . . . . . . . . . . . . . . . . . . .
Evaluation of multi-objective functions . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
30
33
36
36
29
Chapter 4. Results and Discussion
FBA uses optimization of cellular criteria to find a subset of optimal states in the possibly large
solution space shaped by mass balance and capacity constraints [19]. The cellular objective functions for an organism are strongly dependent on the input data, comparison-data and size/type of the
metabolic models. Thus, across three different metabolic models and using available intracellular flux
measurements for E. coli, we examine the optimal criteria derived in a previous study [9]. Furthermore, the effect of single (or pairs) standard constraints and multiple objective functions on FBA are
also evaluated. The metabolic models used are three stoichiometric models of E. coli metabolism
[6, 9, 20] and the flux distributions for eight different objective functions. The experimental conditions included batch cultures as well as chemostat cultures under aerobic glucose and ammonium
limitation.
The results obtained in the FBA simulations for each case were ordered according to the fidelity
error (Equation 3.4) between predicted and experimental fluxes. Thus the estimation of predictive
result is based on two perspectives: (i) the predictive errors represented by plots for different simulations and (ii) scatter plots of separated flux-by-flux comparisons to see how each experimental flux
distribution matched to the corresponding predictions.
4.1
The impact of objective functions by FBA in different conditions
The following section describes the predictive fidelities on three E. coli metabolic models for various objective and constraint combinations in different experimental conditions. It can be observed
that for simulations using different growth conditions, the corresponding error patterns can differ considerably. The 8 objective functions showed a great difference in the accuracy.
4.1.1
Carbon (C)-limited chemostat cultures
Under nutrient scarcity (chemostat cultures) in glucose-limited conditions, the linear maximization
of the biomass yield (max BM) achieved the best predictive accuracy comparing to the other objective
functions and for the most of constraints (Figure 4.1 and the remaining datasets and models are
given in Appendix B). For the ”none” constraint case, the predicted flux values are plotted against
the experimental flux values (Figure 4.1b, 4.1d, 4.1f and 4.1h). The predictive errors of max BM on
average always reported the lowest values when compared to other objective functions. In fact,
by comparing the predicted fluxes against the experimental flux values (Figure 4.1b, 4.1d, 4.1f and
4.1h), it is clear that for the max BM objective function, there is a correlation which reflects the highly
agreement between in silico to in vivo fluxes. This result agrees with previous works that supported
the use of maximization of biomass as the most important objective function for FBA in continuous
cultures [1, 9, 40].
Another interesting criteria is the parsimony criteria (min
P
| v |). The finding that minimization of
the overall intracellular flux may play a key role, as shown for the genome-scale model (e.g. Figure
4.1c), is in line with a previous work for hybridoma cells in continuous culture [55]. The two nonlinear
30
4.1. The impact of objective functions by FBA in different conditions
Nanchen 0.4h−1, C−limited, Genome−scale model
Nanchen 0.4h−1, C−limited, Genome−scale model
0.8
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
predicted values
0.2
−1
0
1
2
0
−1
−2
−2
−1
0
−1
−1
0
1
2
2
1
0
−1
−2
−2
−1
max ATPprod
1
0
−1
−1
0
1
2
min
max
ATPprod ATPprod
−2
−2
−1
−1
0
1
2
0
−1
−2
−2
predicted values
0
−1
0
1
2
0
−1
0
1
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
1
0
−1
−2
−2
−1
2
2
1
0
−1
−2
−2
−1
0
1
2
observed values
1
0
−1
−2
−2
−1
0
1
2
max ATP
0
−1
−1
0
1
2
−2
−2
−1
0
1
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
max BM/flux
max ATP/flux
min Rd
predicted values
1
0
−1
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
2
1
0
−1
−2
−2
0
−1
2
observed values
2
−2
−2
1
predicted values
−2
−2
min Flux
2
predicted values
predicted values
1
−1
0
1
2
2
2
2
1
0
−1
−2
−2
−1
0
1
2
observed values
2
1
0
−1
−2
−2
−1
0
1
2
observed values
Ishii 0.1h−1, C−limited, Genome−scale model
max ATP
1
0
−1
−1
0
1
2
1
0
−1
−2
−2
1
0.8
1
0
−1
−2
−2
−1
0
1
2
min
Rd
min
max
ATPprod ATPprod
0
−1
−1
0
1
observed values
−2
−2
−1
0
−1
−2
−2
−1
0
1
0
1
2
min Rd
2
2
1
0
−1
−2
−2
−1
0
1
2
observed values
max ATPprod
1
−2
−2
0
−1
observed values
observed values
2
predicted values
max
ATP/flux
predicted values
0.2
2
1
min ATPprod
0.4
1
2
observed values
0.6
0
1
max ATP/flux
predicted values
1.2
−1
2
observed values
max BM/flux
2
predicted values
−2
−2
min Flux
2
predicted values
predicted values
predicted values
max BM
2
observed values
(g)
2
2
observed values
predicted values
Predictive Fidelity
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
max
BM/flux
1
1
observed values
2
Ishii 0.1h−1, C−limited, Genome−scale model
min
Flux
0
0
min Rd
2
(f)
2
max
ATP
−1
observed values
2
(e)
max
BM
2
−2
−2
max ATPprod
1
−1
1
0
−1
observed values
2
−2
−2
0
1
max ATP/flux
1
−1
−1
2
observed values
2
−2
−2
min Flux
predicted values
−2
−2
predicted values
0.2
0
2
2
1
max BM
predicted values
0.4
1.4
1
predicted values
predicted values
0
−1
0.6
1.6
1
Emmerling 0.4h−1, C−limited, Genome−scale model
predicted values
0.8
1.8
0
2
observed values
predicted values
Predictive Fidelity
1
max
BM/flux
0
(d)
1.2
min
Flux
−1
observed values
max ATP
1
predicted values
min
Rd
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
max
ATP
−2
−2
Ishii 0.4h−1, C−limited, Genome−scale model
predicted values
max
ATP/flux
2
max
BM
0
observed values
2
Emmerling 0.4h−1, C−limited, Genome−scale model
0
1
−1
0
(c)
1.4
2
1
max BM
predicted values
0.2
1.6
2
2
−1
min ATPprod
0.4
1.8
1
2
observed values
max
BM/flux
1
observed values
min ATPprod
2
0.6
min
Flux
0
observed values
max BM/flux
0.8
max
ATP
0
min Rd
observed values
1
max
BM
−1
observed values
observed values
predicted values
Predictive Fidelity
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0
−2
−2
max ATP/flux
Ishii 0.4h−1, C−limited, Genome−scale model
1.4
2
0
−1
(b)
2
1.6
1
1
observed values
(a)
1.8
0
2
max BM/flux
1
−2
−2
1
observed values
2
−2
−2
predicted values
−2
−2
0.6
0.4
predicted values
0
−1
min Flux
2
predicted values
1
1
predicted values
1.2
max ATP
2
predicted values
Predictive Fidelity
1.4
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
predicted values
1.6
max BM
predicted values
2
1.8
2
2
1
0
−1
−2
−2
−1
0
1
2
observed values
(h)
Figure 4.1: Predictive fidelities for objective functions using different constraints (a,c,e,g) and corresponding individual flux predictions (b,d,f,h) between predicted and experimental fluxes for all the objective functions evaluated
without additional constraint of four E. coli (C)-limited chemostat cultures on the Genome-scale model. Missing
lines represent infeasible solutions.
31
Chapter 4. Results and Discussion
objectives max BM/flux and max ATP/flux were also proved to be useful for this growth condition
while max ATPprod in all cases revealed large discrepancies between in silico and in vivo fluxes.
Moreover, the effects of each of the single cellular constraints to every cellular objective is also
shown in the Figure 4.1. With the exception of max BM, max ATP and min Flux, there are no significant
changes in the predictive fidelities for all the constraints tested. Our analysis indicated that the single
standard constraints namely none, qO2 , maintenance energy or NADPH reported better predictive
fidelities than the other constraints. Although common properties with the previous work [9], some
relative alterations in this work can be observed for the Schuetz model and Nanchen dataset. For
instance, the behaviour of maximization of ATP per unit flux was consistent and able to work well
instead of being useless without additional constraints (see Appendix B Figure B.2). On the other
hand, the maximization of ATP was not a good choice as expected in continuous cultures. The
probable reason could be the use of the relative flux values instead of the split flux ratios used by
Schuetz et al [9] when comparing computational and experimental results.
4.1.2
Ammonium (N)-limited chemostat cultures
The predictive fidelities for the N-limited chemostat cultures are shown in Figure 4.2 (remaining
datasets are given in Appendix B). For the N-limited chemostat culture, a similar pattern was observed
Emmerling 0.09h−1, N−limited, Core model
Emmerling 0.09h−1, N−limited, Core model
−1
0
1
2
1
0
−1
−2
−2
1
0.8
0
−1
−1
0
1
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
predicted values
0.2
2
0
−1
0
1
−2
−2
−1
2
0.2
(c)
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
predicted values
0.4
max
BM/flux
−2
−2
−1
0
1
2
observed values
1
0
−1
−2
−2
−1
0
1
2
max ATP
1
0
−1
−1
0
1
2
−2
−2
−1
0
1
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
max BM/flux
max ATP/flux
min Rd
predicted values
1
0
−1
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
2
1
0
−1
−2
−2
0
−1
2
observed values
2
−2
−2
1
predicted values
−2
−2
min Flux
2
predicted values
predicted values
2
0.6
min
Flux
0
−1
observed values
predicted values
0.8
max
ATP
2
1
2
max BM
predicted values
1
max
BM
2
Emmerling 0.09h−1, N−limited, Genome−scale model
predicted values
Predictive Fidelity
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0
1
1
2
(b)
2
1.4
0
0
min Rd
−1
observed values
Emmerling 0.09h−1, N−limited, Genome−scale model
1.6
−1
observed values
0
(a)
1.8
−2
−2
max ATPprod
1
−1
0
−1
observed values
2
−2
−2
2
1
min ATPprod
0.4
1
2
observed values
0.6
0
1
max ATP/flux
1
−2
−2
−1
2
observed values
max BM/flux
2
predicted values
−2
−2
predicted values
0
−1
min Flux
2
observed values
1.2
0
predicted values
1
predicted values
Predictive Fidelity
1.4
max ATP
2
predicted values
1.6
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
predicted values
1.8
max BM
predicted values
2
−1
0
1
observed values
2
2
2
2
1
0
−1
−2
−2
−1
0
1
2
observed values
2
1
0
−1
−2
−2
−1
0
1
2
observed values
(d)
Figure 4.2: Predictive fidelities for objective functions with constraints (a,c) and corresponding individual flux
predictions (b,d) between predicted and experimental fluxes for all the objective functions evaluated without
additional constraint of E. coli N-limited chemostat culture conditions on the Core and Genome-scale model.
Missing lines represent infeasible solutions in the simulation.
32
4.2. The impact of objective functions by FBA in different metabolic models
at the simulation results. The max BM was located at the group of top best objective functions but less
accurate than the previous aerobic chemostat cultures. The max ATPprod objective function provided
the worse prediction of flux distribution for the Core and Genome-scale model (Figure 4.2). Instead,
the performance of two non-linear objective and minimization of the net of fluxes were improved to
the same level of max BM. However, when considering scatter plots on the right then max BM objective
function, without any additional constraints, gives a higher prediction and will be most useful.
Overall, as shown in Figure 4.2a,4.2c, the best objective for the ammonium-limited continuous
culture was obtained by minimization of flux distribution or by maximization of the biomass. However
when considering the individual flux prediction as in scatter plots (Figure 4.2b,4.2d), minimization of
flux distribution did not perform significantly as expected.
Unlike the C-limited conditions, some constraints improved the prediction power of FBA compared to “none” scenario. However, the pattern was not consistent through all objectives or model
reconstructions that were tested. For example, in Figure 4.2a, the “maintenance” constraint indeed
improved the predictive fidelity on max BM, but is useless for other objectives (e.g. max ATP).
4.1.3
Batch cultures
The predictive fidelities for the batch cultures in the genome-scale model are shown in Figure 4.3.
For batch cultures the results suggest that the cells “prefer” the maximization of biomass yield per flux
unit. In contrast to a previous work [33], it can be observed that the minimization of redox potential
did not reflect a good objective function for E. coli batch cultures. Furthermore, the minimization and
maximization of ATP producing fluxes were the worst choices. The max BM and max BM/flux objective
were the three best functions for the simulation of E. coli batch cultures.
Additionally, consistent and promising fidelities on the two batch cultures (Holm and Perrenoud
dataset) were observed also for max ATP/flux. A similar result was reported by Schuetz et al. [9].
However, some differences to their conclusions can be observed. First, the max BM/flux was slightly
better than max ATP/flux making this objective function the best choice, rather than the reported
max ATP/flux for the batch culture. Second, in contrast to previous work [9], max ATPprod was considered a bad objective function in batch cultures (Figure 4.3).
Overall, the two nonlinear objectives seem to be reliable objectives in the cases tested, while maximization of biomass and minimization of net flux also presents consistent performances. Additionally,
in accordance with results from Schuetz et al [9], the effect of the constraints appeared to be mostly
insignificant to the choice of the objective function for all the conditions.
4.2
The impact of objective functions by FBA in different metabolic
models
We next examine whether any of these findings are consistent across different metabolic models of
the same system. Here, predictive fidelities were presented to focus the comparison of three diferent
metabolic models (Core model, Schuetz model and iAF1260 Genome-scale model).
33
Chapter 4. Results and Discussion
Holm 0.67h−1, batch, Genome−scale model
Holm 0.67h−1, batch, Genome−scale model
0.8
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
predicted values
0.2
−1
0
1
2
0
−1
−2
−2
−1
−1
−1
0
1
2
2
1
0
−1
−2
−2
−1
max ATPprod
1
0
−1
−1
0
1
2
2
2
2
1
0
−1
−2
−2
−1
0
1
2
observed values
1
0
−1
−2
−2
−1
0
1
2
observed values
Perrenoud 0.65h−1, batch, Genome−scale model
max ATP
1
0
−1
−1
0
1
2
1
0
−1
−2
−2
predicted values
0
−1
−1
0
1
2
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
0
−1
−1
0
1
observed values
2
0
−1
−2
−2
−1
0
−1
−1
0
1
1
2
min Rd
1
−2
−2
0
observed values
2
2
1
0
−1
−2
−2
−1
0
1
2
observed values
max ATPprod
1
−2
−2
1
2
observed values
2
predicted values
0.2
predicted values
0.4
0
1
max ATP/flux
1
−2
−2
−1
2
observed values
2
predicted values
−2
−2
min Flux
2
predicted values
predicted values
predicted values
max BM
2
min ATPprod
(c)
1
2
observed values
max
BM/flux
1
observed values
min ATPprod
2
0.6
min
Flux
0
observed values
max BM/flux
0.8
max
ATP
0
min Rd
observed values
1
max
BM
−1
observed values
observed values
predicted values
Predictive Fidelity
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0
−2
−2
max ATP/flux
0
Perrenoud 0.65h−1, batch, Genome−scale model
1.4
2
0
−1
(b)
2
1.6
1
1
observed values
(a)
1.8
0
2
max BM/flux
1
−2
−2
1
observed values
2
−2
−2
predicted values
−2
−2
0.6
0.4
predicted values
0
−1
min Flux
2
predicted values
1
1
predicted values
1.2
max ATP
2
predicted values
Predictive Fidelity
1.4
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
predicted values
1.6
predicted values
2
1.8
max BM
2
2
1
0
−1
−2
−2
−1
0
1
2
observed values
(d)
Figure 4.3: Predictive fidelities for objective functions with constraints (a,c) and corresponding individual flux
predictions (b,d) between predicted and experimental fluxes for all the objective functions evaluated without
additional constraint of E. coli batch culture conditions on the Genome-scale model.
Figure 4.4 shows the effect of the metabolic models type and size on the qualitative and quantitative predictions for the same dataset. It is clear that the predictive fidelity patterns varies depending on
the metabolic network and the dataset used. In general, for the chemostat cultures (Figure 4.4a, 4.4b,
4.4c, 4.4d), the Genome-scale model reported the best fidelities since their prediction errors seems
to be smallest. A possible reason is that for more detailed model reconstruction a better agreement
between predictive and experimental fluxes is obtained. When comparing the predictive performance
of FBA simulations on the Core and Schuetz model, there were similarities beside several discrepancies. These two reconstructions are on the same level of scale and share the common metabolic
structure, but there exist some differences on their specific pathway reactions. In fact, Schuetz et
al. [9] had constructed a highly interconnected stoichiometric network model based on known reactions of E. coli central carbon metabolism. This could be the reason for the superior of prediction
results on Schuetz model when compare to other with the same scale. However, in term of extension,
Genome-scale model iAF1260 is on a higher level than that of Schuetz model since the number of
reactions is about tenfold than the original Core reconstruction. The results on Figure 4.4, especially
4.4a and 4.4b, supported this comparison since the improvement made by changing from Core model
to Genome-scale model is even better than that when only replacing the former by Schuetz model.
The complete resuts for all datasets is presented in Section B of Appendix B.
34
4.2. The impact of objective functions by FBA in different metabolic models
Ishii 0.1h−1, C−limited
max BM
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
1
0
max BM/flux
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
min Flux
1
0
max ATP/flux
1
0
min ATPprod
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
Ishii 0.7h−1, C−limited
max ATP
0.5
Predictive Fidelity
0.5
Predictive Fidelity
max BM
1
min Rd
1
0
0.5
Predictive Fidelity
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
max ATPprod
1
0
0.5
Predictive Fidelity
max ATP
0.5
Predictive Fidelity
1
0
max BM/flux
0.5
Predictive Fidelity
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
1
1
0
0.5
Predictive Fidelity
1
0
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
min Flux
1
0
max ATP/flux
1
0
min ATPprod
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
min Rd
0.5
Predictive Fidelity
1
0
0
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
0
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
1
0
min ATPprod
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
1
1
0
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
1
0
0
0.5
Predictive Fidelity
(e)
0.5
Predictive Fidelity
min Flux
1
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
min Rd
1
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
1
1
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
max ATP
1
0
max BM/flux
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
max ATPprod
1
1
max ATPprod
max BM
min Rd
0.5
Predictive Fidelity
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
min Flux
1
0
max ATP/flux
1
0
min ATPprod
Genome−scale model
Core model
Schuetz model
0.5
Predictive Fidelity
0.5
Predictive Fidelity
Perrenoud 0.65h−1, batch
min Flux
max ATP/flux
min ATPprod
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
1
(d)
max ATP
max BM/flux
0.5
Predictive Fidelity
Genome−scale model
Core model
Schuetz model
Holm 0.67h−1, batch
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0
max ATP/flux
(c)
max BM
1
max ATP
max BM/flux
max ATPprod
1
1
min Rd
Genome−scale model
Core model
Schuetz model
max BM
Genome−scale model
Core model
Schuetz model
0.5
Predictive Fidelity
0.5
Predictive Fidelity
0.5
Predictive Fidelity
Emmerling 0.09h−1, C−limited
max ATP
max BM/flux
0
(b)
Emmerling 0.09h−1, N−limited
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
1
max ATPprod
(a)
max BM
0.5
Predictive Fidelity
min Flux
max ATP/flux
min ATPprod
Genome−scale model
Core model
Schuetz model
0.5
Predictive Fidelity
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
max ATPprod
Genome−scale model
Core model
Schuetz model
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
1
min Rd
1
(f)
Figure 4.4: Predictive fidelities of pFBA simulation for each objective function in several datasets with three
different metabolic models. Values above 1 are out of range.
To conclude, the difference reflects the fact that beside agreements on the most effective functions
for certain situations, using different metabolic models could significantly affect the conclusions for
certain cases. The factors related to the model reconstructions such as the biomass composition can
35
1
Chapter 4. Results and Discussion
be one possible reason for this difference.
4.3
Evaluation of pairwise constraints
In this section, the effect of adding cellular constraints to the genome-scale system are evaluated.
Here FBA was run for pairwise constraints to predict the fluxes. In order to understand how the
phenotype predictions vary across the different constraints, a particular case is selected, namely the
chemostat fermentation under the highest dilution rate of 0.7h−1 . This is a typical case where FBA
simulations are less accurate, since the cells were sub-optimally grown due to overflow metabolism
[39]. For this case the flux predictions behaved qualitatively well for the same optimal criteria and
is well-described by maximizing biomass yield (Figure 4.5a). It can be observed in Figure 4.5a that
all single or even pairwise constraints to this model did not significantly improve flux prediction in a
consistent manner for all tested cases. The same behaviour could be observed in the other objective
functions. In fact, the uselessness of pairwise cellular criteria also can be observed on other datasets
or model reconstructions (see Section B of Appendix B).
Ishii 0.7h−1, C−limited, Genome−scale model
Ishii 0.7h−1, C−limited, Genome−scale model
2
1.6
Predictive Fidelity
1.4
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.8
1.6
1.4
Predictive Fidelity
1.8
2
1.2
1
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
0
max
BM
max
ATP
min
Flux
(a)
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
(b)
Figure 4.5: Comparison between pFBA predictive fidelities using single (4.5a) and pairwise (4.5b) constraints
for the dataset from Ishii under the dilution rate of 0.7h−1 .
In general, no improvement in predictive fidelity for any of the single objective functions when
the combination of two constraints are used instead. On the other hand, when the complete set
of constraints “all constraints scenario” are used simultaneously, its contribution was not significant
compared to the “none” constrain scenario as reported previously in [9]. Identical conclusions were
obtained when repeating all these simulations with the other datasets (see Appendix B).
4.4
Evaluation of multi-objective functions
In this section, every possible combinations of the objective functions max BM, max ATP and min Flux
was examined instead of the use of only single objective in the previous simulations. These three objective functions had been reported as the best in a previous multi-objective research [50]. However,
in this previous work, metabolic flux analysis simulations were limited. Therefore, obtained studies
are needed to determine the optimality principles in different metabolic models. There are also good
36
4.4. Evaluation of multi-objective functions
biological meanings to evaluate the Pareto fronts of these three objective. In fact, growth rate, energy usage and enzymatic parsimony are among the most important but usually conflict aspects of
metabolism. It means that the improvement of one of them usually lead to the degradation of the other
two. Thus, using the Pareto optimization to understand the trade-off between these three meaningful
objective was required.
Table 4.1, 4.2 and 4.3 show the best predictive fidelities from experimental fluxes to the optimize
solutions of using single objective and to the Pareto fronts of combinations of two or three objective
functions for the Core model, Schuetz model and Genome-scale model, respectively.
Table 4.1: Predictive fidelities of pFBA simulation using single and multi-objective functions in the Core model.
max BM
max ATP
min Flux
max BM + max ATP
max BM + min Flux
max ATP + min Flux
max BM + max ATP +
min Flux
Emmerling
0.09
h−1 ,
N-limited
0.4085
0.3458
0.1491
0.1169
0.1490
0.1023
0.0856
Ishii
0.1 h−1 ,
C-limited
0.1373
0.3725
0.6321
0.1373
0.1367
0.3559
0.1367
Ishii
0.4 h−1 ,
C-limited
0.0327
0.1891
0.5951
0.0212
0.0327
0.1843
0.0212
Ishii
0.7 h−1 ,
C-limited
0.0688
0.2452
0.6612
0.0677
0.0677
0.1807
0.0649
Perrenoud
0.65
h−1 ,
batch
0.2307
0.6508
0.2408
0.2284
0.1820
0.1967
0.1490
Holm
0.67 h−1 ,
batch
0.3046
0.7334
0.3212
0.3046
0.0707
0.1399
0.0321
As observed in a previous work [50] no dual combination could describe all measured fluxes
adequately, and only the three efficiency objectives (max BM, maxATP and minFlux) achieved the
highest optimality for the datasets and models analyzed (Table 4.1, 4.2 and 4.3). Our results show
that the multi-objective approach improved the best prediction obtained with traditional FBA using a
single objective function. However, the improvement is not the same for different datasets. Overall,
our study showed that the multi-objective approach using two objective is better than single objective
but usually inferior than the combination of three objectives simultaneously.
Table 4.2: Predictive fidelities of pFBA simulation using single and multi-objective functions in the Schuetz model.
max BM
max ATP
min Flux
max BM + max ATP
max BM + min Flux
max ATP + min Flux
max BM + max ATP +
min Flux
Emmerling
0.09
h−1 ,
N-limited
0.3412
0.3369
0.2298
0.1544
0.2282
0.2291
0.1545
Ishii
0.1 h−1 ,
C-limited
0.2551
0.6662
0.4019
0.2551
0.2551
0.3146
0.2551
Ishii
0.4 h−1 ,
C-limited
0.1595
0.3166
0.3237
0.1517
0.1595
0.2514
0.1521
Ishii
0.7 h−1 ,
C-limited
0.1491
0.3072
0.2761
0.1491
0.1491
0.2499
0.1491
Perrenoud
0.65
h−1 ,
batch
0.3099
0.5433
0.3530
0.2952
0.3099
0.3481
0.2953
Holm
0.67 h−1 ,
batch
0.1841
0.6457
0.2131
0.1501
0.1681
0.2052
0.1505
Similar results are shown on Table 4.2 when Schuetz model reconstruction is applied. Moreover,
for the Genome-scale model, the multi-objective approach significantly improve the flux prediction for
37
Chapter 4. Results and Discussion
all datasets in a more consistent manner (Table 4.3).
Table 4.3: Predictive fidelities of pFBA simulation using single and multi-objective functions in the Genome-scale
model.
max BM
max ATP
min Flux
max BM + max ATP
max BM + min Flux
max ATP + min Flux
max BM + max ATP +
min Flux
Emmerling
0.09
h−1 ,
N-limited
0.3658
0.3563
0.2332
0.2314
0.2241
0.2904
0.2238
Ishii
0.1 h−1 ,
C-limited
0.1401
0.3741
0.1295
0.1401
0.1007
0.1288
0.0982
Ishii
0.4 h−1 ,
C-limited
0.0375
0.1882
0.1213
0.0304
0.0348
0.1207
0.0261
Ishii
0.7 h−1 ,
C-limited
0.0697
0.2472
0.1403
0.0693
0.0679
0.1267
0.0661
Perrenoud
0.65
h−1 ,
batch
0.2314
0.6782
0.4984
0.1247
0.2304
0.1334
0.1127
Holm
0.67 h−1 ,
batch
0.3302
0.7327
0.3519
0.3302
0.2962
0.1545
0.1047
Overall, the multi-objective function is very helpful in improving the prediction of FBA model. However, the pattern is not the same with different model reconstructions and datasets. The common
behaviour that could be observed is that the efficiency of objective combination depends on the performance of every single objective. It means, for example, if one objective among them is outperform
the remaining then the result of multi-objective FBA is close to that of the superior function alone
and the improvement seems not so significant. In some circumstances, all the objective functions
giving similar prediction errors and the combination of them would make quite strong promotion of
the fidelity. This phenomenon happens frequently and gives a reasonable meaning for the trade-off
between biological behaviours of living organisms. Sometimes, the cell needs to focus on one dominating objective but in certain conditions, several objectives are considered simultaneously in an equal
manner.
38
5
Conclusions and Future Work
Contents
5.1 Main conclusions and work summary . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Ongoing and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
40
39
Chapter 5. Conclusions and Future Work
5.1
Main conclusions and work summary
The work developed during this thesis explored the effect of different optimal principles in FBA for
three metabolic models using different conditions and comparison dataset than previous evaluations.
Although the fidelity patterns of FBA can differ considerably under different conditions, the classical
“maximization of biomass” was shown in general as one of the best objectives. Moreover, our results
show that the model size has a high impact on the predictive fidelities.
Despite the observed variations, several generalities emerged. In agreement with previous studies, the single objective of maximization of biomass yield achieves the best predictive accuracy. For
the batch growth condition the most consistent optimality criteria appears to be described by the
maximization of the biomass yield per flux or by the objective of maximization of ATP yield per flux
(see above definitions). Moreover, under N-limited continuous cultures, the criteria minimization of
the flux distribution or maximization of biomass yield was determined as the most significant. On
the other hand, the predictions obtained by flux balance analysis using additional combined standard
constraints are not better than those obtained using the single constraint or even none of them. For
the multi-objective optimization by Pareto-optimal flux distributions improve the best predictions obtained by each single objective function alone. The improvement, however, is not the same for all
models with different reconstructions or datasets. The combination in some cases provide significant
advances but there exist situations when it returns equal performance with the best single objective
function.
Although some optimal criteria gives reasonable predictions under certain conditions, there is no
universal criteria that performs well under all conditions. Therefore, systems biologists should perform
a careful evaluation and analysis of the objective functions case-by-case for each particular condition
and application.
Finally, Matlab code for investigation the effect of cellular objective function and constrains on
metabolic models has been also developed. This implementation of all objective functions and constraints can be easily adapted to test new metabolic systems and can be evaluated by comparing its
results with those reported here.
5.2
Ongoing and future work
There are some limitations in this project that should be improved in future work. Firstly, nonlinear
objective functions were not included in Pareto optimization due to the computational cost. By taking
these options into consideration, one could study more about the trade-off mechanisms of cell under
different conditions. Ongoing efforts are currently directed for investigating the challenge of multiobjective optimization formulations to reduce the computational cost of the simulations performed. It
also would be a big help if the trade-off behaviours of single objective in multi-objective modelling is
thoroughly elaborated. Also, new big data (fluxomics), should be added to make a comprehensive
knowledge about the metabolic flux distribution of E. coli more accurate. Another possible improvement is the application of more genome-scale reconstruction models of other strains/species, together
40
5.2. Ongoing and future work
with different experimental in vivo datasets. The aim is to have more information to investigate the
predictive power of the models and also the hidden factors involved in the optimizing mechanism
of other cells. Additionally, cells with genetic pertubation should be included in the simulations to
study the essential relationship between genotype and phenotype in metabolism. Finally, a consensus study is required to have a systematic framework for this case-by-case analysis. It should contain
a common standard formulation for all flux prediction methods, as well as reference dataset and the
way to evaluate the performance of each algorithm. It is hoped that the work developed in this thesis
contributes with methods to apply Systems Biology in biotechnology industries, and it becomes an
open door for new applications using the presented study.
41
Chapter 5. Conclusions and Future Work
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48
A
Experimental Datasets and Matches
between experimental and model
reactions
49
A. Experimental Datasets and Matches between experimental and model reactions
No
GLC + ATP → G6P
G6P → 6PG + NADPH
6PG → P5P + CO2 + NADPH
G6P → F6P
6PG → T3P + PYR
F6P + ATP → 2*T3P
2P5P → S7P + T3P
P5P + E4P → F6P + T3P
S7P + T3P → E4P + F6P
T3P → PGA + ATP + NADH
PGA → PEP
PEP → PYR + ATP
PYR → AcCoA + CO2 + NADH
OAA + AcCoA → ICT
ICT → OGA + CO2 +NADPH
OGA → FUM + CO2 + 1.5*ATP + 2*NADH
FUM → MAL
MAL → OAA + NADH
MAL → PYR + CO2 + NADH
OAA + ATP → PEP + CO2
PEP + CO2 → OAA
AcCoA → Acetate + ATP
NADPH → NADH
O2 + 2NADH → 2P/O*ATP
Biomass production
ICT + AcCoA → MAL + FUM + NADH
DHAP → PYR
ETH → ETHxt
Reaction
6.7 ± 0.7
0.10h−1 C-limited
100.0 ± 10.0
36.0 ± 3.6
36.0 ± 3.6
62.0 ± 6.2
0.0 ± 0.0
80.0 ± 8.0
11.0 ± 1.1
8.0 ± 0.8
11.0 ± 1.1
167.0 ± 16.7
157.0 ± 15.7
53.0 ± 5.3
136.0 ± 13.6
82.0 ± 8.2
56.0 ± 5.6
74.0 ± 7.4
74.0 ± 7.4
96.0 ± 9.6
4.0 ± 0.4
0.0 ± 0.0
0.0 ± 0.0
0.0 ± 0.0
7.9 ± 0.8
Ishii
0.40h−1 C-limited
100.0 ± 10.0
36.0 ± 3.6
36.0 ± 3.6
63.0 ± 6.3
0.0 ± 0.0
79.0 ± 7.9
11.0 ± 1.1
7.0 ± 0.7
11.0 ± 1.1
165.0 ± 16.5
153.0 ± 15.3
25.0 ± 2.5
101.0 ± 10.1
70.0 ± 7.0
70.0 ± 7.0
60.0 ± 6.0
60.0 ± 6.0
60.0 ± 6.0
0.0 ± 0.0
0.0 ± 0.0
25.0 ± 2.5
0.0 ± 0.0
5.2 ± 0.5
0.70h−1 C-limited
100.0 ± 10.0
42.0 ± 4.2
42.0 ± 4.2
56 ± 5.6
0.0 ± 0.0
80.0 ± 8.0
13.0 ± 1.3
11.0 ± 1.1
13.0 ± 1.3
170.0 ± 17.0
162.0 ± 16.2
43.0 ± 4.3
127.0 ± 12.7
89.0 ± 8.9
89.0 ± 8.9
83.0 ± 8.3
83.0 ± 8.3
83.0 ± 8.3
0.0 ± 0.0
0.0 ± 0.0
17.0 ± 1.7
14.0 ± 1.4
0.09h−1 N-limited
100.0 ± 10.0
67.0 ± 6.7
1.0 ± 0.1
96.0 ± 9.6
0.0 ± 1.0
94.0 ± 9.4
0.0 ± 2.0
−2.0 ± 0.2
0.0 ± 2.0
186.0 ± 18.6
180.0 ± 18.0
131.0 ± 13.1
147.0 ± 14.7
73.0 ± 7.3
73.0 ± 7.3
67.0 ± 6.7
67.0 ± 6.7
34.0 ± 3.4
33.0 ± 3.3
1.0 ± 0.1
47.0 ± 4.7
6.6 ± 0.7
15.0 ± 1.5
33.7 ± 3.4
4.0 ± 0.4
0.0 ± 1.0
0.0 ± 10.0
0.0 ± 1.0
Table A.1: Experimental flux data for growth of E. coli in different conditions from different sources (Ishii and Emmerling).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
6.4 ± 0.6
83.0 ± 8.3
Emmerling
0.09h−1 C-limited
100.0 ± 10.0
83.0 ± 8.3
44.0 ± 4.4
54.0 ± 5.4
0.0 ± 0.0
77.0 ± 7.7
17.0 ± 1.7
15.0 ± 1.5
17.0 ± 1.7
169.0 ± 16.9
159.0 ± 15.9
123.0 ± 12.3
108.0 ± 10.8
93.0 ± 9.3
93.0 ± 9.3
83.0 ± 8.3
83.0 ± 8.3
75.0 ± 7.5
7.0 ± 0.7
28.0 ± 2.8
56.0 ± 5.6
8.3 ± 0.8
−20.0 ± 2.0
0.40h−1 C-limited
100.0 ± 10.0
68.0 ± 6.8
21.0 ± 2.1
77.0 ± 7.7
0.0 ± 0.0
85.0 ± 8.5
6.0 ± 0.6
2.0 ± 0.2
6.0 ± 0.6
172.0 ± 17.2
159.0 ± 15.9
119.0 ± 11.9
99.0 ± 9.9
80.0 ± 8.0
80.0 ± 8.0
68.0 ± 6.8
68.0 ± 6.8
61.0 ± 6.1
7.0 ± 0.7
12.0 ± 1.2
45.0 ± 4.5
50
51
Reaction
GLC + ATP → G6P
G6P → 6PG + NADPH
6PG → P5P + CO2 + NADPH
G6P → F6P
6PG → T3P + PYR
F6P + ATP → 2*T3P
2P5P → S7P + T3P
P5P + E4P → F6P + T3P
S7P + T3P → E4P + F6P
T3P → PGA + ATP + NADH
PGA → PEP
PEP → PYR + ATP
PYR → AcCoA + CO2 + NADH
OAA + AcCoA → ICT
ICT → OGA + CO2 +NADPH
OGA → FUM + CO2 + 1.5*ATP + 2*NADH
FUM → MAL
MAL → OAA + NADH
MAL → PYR + CO2 + NADH
OAA + ATP → PEP + CO2
PEP + CO2 → OAA
AcCoA → Acetate + ATP
NADPH → NADH
O2 + 2NADH → 2P/O*ATP
Biomass production
ICT + AcCoA → MAL + FUM + NADH
DHAP → PYR
ETH → ETHxt
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Nanchen
0.09h−1 C-limited 0.40h−1 C-limited
100.0 ± 10.0
100.0 ± 5.0
30.0 ± 5.0
36.0 ± 3.0
23.0 ± 5.0
29.0 ± 3.0
69.0 ± 9.0
64.0 ± 4.0
7.0 ± 6.0
7.0 ± 4.0
82.0 ± 11.0
78.0 ± 5.0
8.0 ± 2.0
9.0 ± 1.0
5.0 ± 2.0
6.0 ± 1.0
8.0 ± 2.0
9.0 ± 1.0
176.0 ± 21.0
168.0 ± 9.0
168.0 ± 21.0
156.0 ± 9.0
201.0 ± 26.0
120.0 ± 9.0
190.0 ± 28.0
113.0 ± 9.0
121.0 ± 26.0
96.0 ± 9.0
65.0 ± 27.0
96.0 ± 9.0
56.0 ± 29.0
86.0 ± 9.0
113.0 ± 27.0
86.0 ± 9.0
169.0 ± 28.0
78.0 ± 8.0
0.0 ± 0.4
8.0 ± 2.0
70.0 ± 10.0
15.0 ± 0.2
31.0 ± 6.0
45.0 ± 4.0
0.0 ± 1.0
0.0 ± 0.7
29.0 ± 55.0
45.0 ± 18.0
372.0 ± 80.0
301.0 ± 29.0
5.0 ± 1.0
7.0 ± 1.0
56.0 ± 10.0
0.0 ± 1.0
0.0 ± 10.0
0.0 ± 10.0
0.0 ± 1.0
0.0 ± 1.0
7.1 ± 0.7
8.6 ± 0.9
Yang
0.1h−1 C-limited 0.55h−1 C-limited
100.0 ± 10.0
100.0 ± 10.0
34.0 ± 3.4
26.0 ± 2.6
34.0 ± 3.4
26.0 ± 2.6
65.0 ± 6.5
73.0 ± 7.3
0.0 ± 0.0
0.0 ± 0.0
83.0 ± 8.3
85.0 ± 8.5
11.0 ± 1.1
8.0 ± 0.8
8.0 ± 0.8
4.0 ± 0.4
11.0 ± 1.1
8.0 ± 0.8
173.0 ± 17.3
172.0 ± 17.2
163.0 ± 16.3
161.0 ± 16.1
130.0 ± 13.0
124.0 ± 12.4
107.0 ± 10.7
93.0 ± 9.3
87.0 ± 8.7
70.0 ± 7.0
87.0 ± 8.7
70.0 ± 7.0
78.0 ± 7.8
58.0 ± 5.8
78.0 ± 7.8
58.0 ± 5.8
75.0 ± 7.5
58.0 ± 5.8
3.0 ± 0.3
0.0 ± 0.0
67.0 ± 6.7
23.0 ± 2.3
94.0 ± 9.4
52.0 ± 5.2
Table A.2: (continue) from Nanchen, Yang, Perrenoud and Holm.
Perrenoud
0.65h−1 batch
100.0 ± 1.2
29.5 ± 1.3
25.7 ± 1.5
70.0 ± 1.5
3.8 ± 1.7
80.4 ± 1.8
7.0 ± 0.5
4.0 ± 0.5
7.0 ± 0.5
167.6 ± 2.8
153.1 ± 2.9
119.7 ± 2.9
102.6 ± 2.9
24.2 ± 3.1
24.2 ± 3.1
13.8 ± 3.1
13.8 ± 3.1
10.7 ± 2.4
3.1 ± 1.8
0.2 ± 0.3
27.1 ± 1.9
59.7 ± 2.4
−52.5 ± 9.3
142.0 ± 9.2
8.3 ± 0.1
17.0 ± 1.7
0.0 ± 0.0
22.0 ± 2.2
43.0 ± 4.3
60.0 ± 6.0
7.0 ± 0.7
159.0 ± 15.9
7.0 ± 0.7
27.0 ± 2.7
27.0 ± 2.7
18.0 ± 1.8
78.0 ± 7.8
14.0 ± 1.4
12.0 ± 1.2
14.0 ± 1.4
167.0 ± 16.7
153.0 ± 15.3
26.0 ± 2.6
Holm
0.67h−1 batch
100.0 ± 10
48.0 ± 4.8
48.0 ± 4.8
52.0 ± 5.2
A. Experimental Datasets and Matches between experimental and model reactions
No
GLC + ATP → G6P
G6P → 6PG + NADPH
6PG → P5P + CO2 + NADPH
G6P → F6P
6PG → T3P + PYR
F6P + ATP → 2*T3P
2P5P → S7P + T3P
P5P + E4P → F6P + T3P
S7P + T3P → E4P + F6P
T3P → PGA + ATP + NADH
PGA → PEP
PEP → PYR + ATP
PYR → AcCoA + CO2 + NADH
OAA + AcCoA → ICT
ICT → OGA + CO2 +NADPH
OGA → FUM + CO2 + 1.5*ATP + 2*NADH
FUM → MAL
MAL → OAA + NADH
MAL → PYR + CO2 + NADH
OAA + ATP → PEP + CO2
PEP + CO2 → OAA
AcCoA → Acetate + ATP
NADPH → NADH
O2 + 2NADH → 2P/O*ATP
Biomass production
ICT + AcCoA → MAL + FUM + NADH
DHAP → PYR
ETH → ETHxt
Experimental reaction
Core model
R GLCpts
R G6PDH2r
R GND
R PGI
R FBA, R FBP, R PFK, R TPI
R TKT1
R TKT2
R TALA
R GAPD, -R PGK
R ENO, -R PGM
R PYK
R PDH
R ACONTa, R ACONTb, R CS
R ICDHyr
R AKGDH, R SUCOAS
R FUM
R MDH
R ME1, R ME2
R PPCK
R PPC
R PTAr, -R ACKr
R NADTRHD
R O2t
R Biomass Ecoli core w/GAM
R ICL, R MALS
Model reaction
Schuetz model
glk, ptsG/H/I
zwf, pgl
gnd
pgi
edd, eda
pfkA, pfkB, fbaA, fbaB, tpiA
tktA, tktB
tktA r2, tktB r2
talA, talB
gapA, pgk
gpmA, gpmB, eno
pykF, pykA
aceEF
gltA, prpC, acnA, acnA r2, acnB, acnB r2
icd
sucAB, sucCD
fumA, fumB, fumC
mdh, mgo
maeA, maeB
pck
ppc
pta, ackA, ackB, tdcD, purT, acs, ac
pntAB, udhA
o2
biomass
aceA, aceB
Genome-scale model
GLCptspp
G6PDH2r
GND
PGI
EDD, EDA
FBA, FBP, PFK, TPI
TKT1
TKT2
TALA
GAPD
ENO, -PGM
PYK
PDH
ACONTa, ACONTb, CS
ICDHyr
AKGDH, SUCOAS
FUM
MDH
ME1, ME2
PPCK
PPC
PTAr, -ACKr
NADTRHD
O2tpp
Ec biomass iAF1260 core 59p81M
ICL, MALS
Table A.3: Mapping from experimental reactions to the corresponding ones in model reconstructions. The “-” symbol indicates opposite direction of the reaction. Note that beside
this common mapping, some dataset may provide information about additional specific reactions in the model reconstruction also.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
52
53
max ATPprod
min ATPprod
Objective
function
max BM
max BM/flux
max ATP
max ATP/flux
min Flux
min Redox
ATP producing reactions
Biomass objective function
ATP maintenance requirement
Intracellular reactions
All reactions that produce
NADH or NADPH
Involved reactions
pgk f, pykA, pykF, sucCD f, atp,
ackA f, ackB f, tdcD f, purT f
All
gapA f, aceEF, maeA, sucAB,
mdh f, udhA, fdhF, fdoGHI,
fdnGHI r2,
ldhA f,
adhE b,
mhpF b,
adhP b,
adhC b,
maeB, awf f, gnd, icd f, pntAB,
frdABCD f, sdhAB r, dld, sdhABCD b
All
GAPD f,
PHD,
ME2,
AKGDH, MDH f, NADTRHD,
LDH D f, ACALD f, ALCD2x f,
G6PDH2r r, GND ICDHyr f,
THD2, FRD7, SUCDi
SUCOAS f,
maint
ATPM
PGK f,
PYK,
ATPS4r f, ACKr b
biomass
Model reaction
Schuetz model
Biomass Ecoli core w GAM
Core model
PGK f,
PYK,
SUCOAS f,
ATPS4rpp f, ACKr b
All
GAPD f, PDH, ME2, AKGDH,
MDH f, NADTRHD, LDH D f,
ACALD f,
ALCD2x f,
ME2,
G6PDH2r f, GND, ICDHyr f,
THD2pp, FRD2, FRD3, SUCDi
ATPM
Genome-scale model
Ec biomass iAF1260
core 59p81M
Table A.4: Specific reaction fluxes used in each objective function. Reactions with ‘ f’ and ‘ b’ suffixes mean forward and backward irreversible reactions respectively which are
originated from a reversible reaction.
A. Experimental Datasets and Matches between experimental and model reactions
54
B
FBA simulation results
55
B. FBA simulation results
Simulation with single constraint
Figure B.1: Predictive fidelities for objective functions with single constraints for E. coli Core model.
(a)
(b)
−1
−1
Ishii 0.1h , C−limited, Core model
Ishii 0.4h , C−limited, Core model
2
1.6
Predictive Fidelity
1.4
1.8
1.6
1.4
Predictive Fidelity
1.8
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
1
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
max
BM
max
ATP
min
Flux
(c)
Ishii 0.7h−1, C−limited, Core model
Predictive Fidelity
1.4
1.6
1.4
1
0.8
1
0.8
0.6
0.4
0.4
0.2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
max
BM
max
ATP
min
Flux
(e)
Emmerling 0.09h−1, C−limited, Core model
Predictive Fidelity
1.4
1.6
1.4
1
0.8
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
56
min
Rd
Emmerling 0.4h−1, C−limited, Core model
1.8
1.2
0
max
ATP/flux
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
Predictive Fidelity
1.6
max
BM/flux
(f)
2
1.8
min
max
ATPprod ATPprod
Emmerling 0.09h−1, N−limited, Core model
1.8
1.2
0
min
Rd
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
Predictive Fidelity
1.6
max
ATP/flux
(d)
2
1.8
max
BM/flux
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
(g)
(h)
−1
−1
Nanchen 0.09h , C−limited, Core model
Nanchen 0.4h , C−limited, Core model
2
1.6
Predictive Fidelity
1.4
1.8
1.6
1.4
Predictive Fidelity
1.8
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
1
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
max
BM
max
ATP
min
Flux
max
BM/flux
(i)
Yang 0.1h , C−limited, Core model
Yang 0.55h , C−limited, Core model
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.8
1.6
1.4
Predictive Fidelity
Predictive Fidelity
1.4
1.2
1
0.8
1
0.8
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0
max
BM
max
ATP
min
Flux
(k)
Perrenoud 0.65h , batch, Core model
min
max
ATPprod ATPprod
Holm 0.67h , batch, Core model
1.8
1.6
1.4
1.2
1
0.8
1
0.8
0.6
0.4
0.4
0.2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0
min
Rd
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
Predictive Fidelity
Predictive Fidelity
1.4
max
ATP/flux
−1
2
1.6
max
BM/flux
(l)
−1
1.8
min
max
ATPprod ATPprod
−1
2
1.6
min
Rd
(j)
−1
1.8
max
ATP/flux
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
57
B. FBA simulation results
Figure B.2: Predictive fidelities for objective functions with single constraints for E. coli Schuetz model.
(a)
(b)
−1
−1
Ishii 0.1h , C−limited, Schuetz’s model
Ishii 0.4h , C−limited, Schuetz’s model
2
1.6
Predictive Fidelity
1.4
1.8
1.6
1.4
Predictive Fidelity
1.8
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
1
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
max
BM
max
ATP
min
Flux
max
BM/flux
(c)
Ishii 0.7h , C−limited, Schuetz’s model
Emmerling 0.09h , N−limited, Schuetz’s model
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.8
1.6
1.4
Predictive Fidelity
Predictive Fidelity
1.4
1.2
1
0.8
1
0.8
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0
max
BM
max
ATP
min
Flux
max
BM/flux
(e)
Emmerling 0.09h , C−limited, Schuetz’s model
Emmerling 0.4h , C−limited, Schuetz’s model
58
1.8
1.6
1.4
1.2
1
0.8
1
0.8
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0
min
max
ATPprod ATPprod
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
Predictive Fidelity
Predictive Fidelity
1.4
min
Rd
−1
2
1.6
max
ATP/flux
(f)
−1
1.8
min
max
ATPprod ATPprod
−1
2
1.6
min
Rd
(d)
−1
1.8
max
ATP/flux
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
(g)
(h)
−1
−1
Nanchen 0.09h , C−limited, Schuetz’s model
Nanchen 0.4h , C−limited, Schuetz’s model
2
1.6
Predictive Fidelity
1.4
1.8
1.6
1.4
Predictive Fidelity
1.8
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
1
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
max
BM
max
ATP
min
Flux
max
BM/flux
(i)
Yang 0.1h , C−limited, Schuetz’s model
Yang 0.55h , C−limited, Schuetz’s model
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.8
1.6
1.4
Predictive Fidelity
Predictive Fidelity
1.4
1.2
1
0.8
1
0.8
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0
max
BM
max
ATP
min
Flux
max
BM/flux
(k)
Perrenoud 0.65h , batch, Schuetz’s model
Holm 0.67h , batch, Schuetz’s model
1.8
1.6
1.4
1.2
1
0.8
1
0.8
0.6
0.4
0.4
0.2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0
min
max
ATPprod ATPprod
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
Predictive Fidelity
Predictive Fidelity
1.4
min
Rd
−1
2
1.6
max
ATP/flux
(l)
−1
1.8
min
max
ATPprod ATPprod
−1
2
1.6
min
Rd
(j)
−1
1.8
max
ATP/flux
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
59
B. FBA simulation results
Figure B.3: Predictive fidelities for objective functions with single constraints for E. coli iAF1260 Genome-scale
model.
(a)
(b)
Ishii 0.1h−1, C−limited, Genome−scale model
Ishii 0.4h−1, C−limited, Genome−scale model
2
1.6
Predictive Fidelity
1.4
1.8
1.6
1.4
Predictive Fidelity
1.8
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
1
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
max
BM
max
ATP
min
Flux
(c)
Ishii 0.7h−1, C−limited, Genome−scale model
Predictive Fidelity
1.4
1.6
1.4
1
0.8
1
0.8
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
max
BM
max
ATP
min
Flux
(e)
Emmerling 0.09h , C−limited, Genome−scale model
60
min
max
ATPprod ATPprod
Emmerling 0.4h , C−limited, Genome−scale model
1.8
1.6
1.4
1.2
1
0.8
1
0.8
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0
min
Rd
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
Predictive Fidelity
Predictive Fidelity
1.4
max
ATP/flux
−1
2
1.6
max
BM/flux
(f)
−1
1.8
min
max
ATPprod ATPprod
Emmerling 0.09h−1, N−limited, Genome−scale model
1.8
1.2
0
min
Rd
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
Predictive Fidelity
1.6
max
ATP/flux
(d)
2
1.8
max
BM/flux
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
(g)
(h)
−1
−1
Nanchen 0.09h , C−limited, Genome−scale model
Nanchen 0.4h , C−limited, Genome−scale model
2
1.6
Predictive Fidelity
1.4
1.8
1.6
1.4
Predictive Fidelity
1.8
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
1
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
max
BM
max
ATP
min
Flux
(i)
Yang 0.1h , C−limited, Genome−scale model
Yang 0.55h , C−limited, Genome−scale model
1.8
1.6
1.4
1.2
1
0.8
1
0.8
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
0
min
max
ATPprod ATPprod
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0
max
BM
max
ATP
min
Flux
(k)
Perrenoud 0.65h , batch, Genome−scale model
min
max
ATPprod ATPprod
Holm 0.67h , batch, Genome−scale model
1.8
1.6
1.4
1.2
1
0.8
1
0.8
0.6
0.4
0.4
0.2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
1.2
0.6
0
min
Rd
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
Predictive Fidelity
Predictive Fidelity
1.4
max
ATP/flux
−1
2
1.6
max
BM/flux
(l)
−1
1.8
min
max
ATPprod ATPprod
2
None
P/O=1
qO2max=15
Maintenance
Bounds
NADPH
All constraints
Predictive Fidelity
Predictive Fidelity
1.4
min
Rd
−1
2
1.6
max
ATP/flux
(j)
−1
1.8
max
BM/flux
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
max
ATPprod ATPprod
61
B. FBA simulation results
Simulation with pairwise constraint
Figure B.4: Predictive fidelities for objective functions with pairwise constraints for E. coli Core model.
(a)
(b)
−1
−1
Ishii 0.1h , C−limited, Core model
Ishii 0.4h , C−limited, Core model
2
1.6
Predictive Fidelity
1.4
1.2
1
1.8
1.6
1.4
Predictive Fidelity
1.8
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
max
BM
max
ATP
min
Flux
(c)
Ishii 0.7h , C−limited, Core model
1.2
1
1.8
1.6
1.4
1.2
1
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
max
BM
max
ATP
min
Flux
(e)
Emmerling 0.09h , C−limited, Core model
1.2
1
1.8
1.6
1.4
1.2
max
ATPprod
1
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
0.6
0.4
0.4
0.2
62
min
ATPprod
Emmerling 0.4h , C−limited, Core model
0.8
0
min
Rd
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
Predictive Fidelity
1.4
max
ATP/flux
−1
2
1.6
max
BM/flux
(f)
−1
1.8
max
ATPprod
Emmerling 0.09h , N−limited, Core model
0.8
0
min
ATPprod
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
Predictive Fidelity
1.4
min
Rd
−1
2
1.6
max
ATP/flux
(d)
−1
1.8
max
BM/flux
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
(g)
(h)
−1
−1
Nanchen 0.09h , C−limited, Core model
Nanchen 0.4h , C−limited, Core model
2
1.6
Predictive Fidelity
1.4
1.2
1
1.8
1.6
1.4
Predictive Fidelity
1.8
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
max
BM
max
ATP
min
Flux
max
BM/flux
(i)
Yang 0.1h , C−limited, Core model
1.2
1
1.8
1.6
1.4
1.2
1
0.6
0.4
0.4
0.2
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
max
BM
max
ATP
min
Flux
Perrenoud 0.65h−1, batch, Core model
1
min
Rd
max
ATPprod
Holm 0.67h−1, batch, Core model
1.8
1.6
1.4
0.8
1.2
1
0.6
0.4
0.4
0.2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
0
max
ATP/flux
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
Predictive Fidelity
1.2
max
BM/flux
(l)
2
1.4
min
ATPprod
0.2
max
BM
(k)
1.6
max
ATPprod
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
1.8
min
ATPprod
Yang 0.55h , C−limited, Core model
0.8
0
max
ATPprod
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
Predictive Fidelity
1.4
min
ATPprod
−1
2
1.6
min
Rd
(j)
−1
1.8
max
ATP/flux
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
63
B. FBA simulation results
Figure B.5: Predictive fidelities for objective functions with pairwise constraints for E. coli Schuetz model.
(a)
(b)
−1
−1
Ishii 0.1h , C−limited, Schuetz’s model
Ishii 0.4h , C−limited, Schuetz’s model
2
1.6
Predictive Fidelity
1.4
1.2
1
1.8
1.6
1.4
Predictive Fidelity
1.8
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
max
BM
max
ATP
min
Flux
(c)
Ishii 0.7h−1, C−limited, Schuetz’s model
Predictive Fidelity
1.4
1.2
1
1.8
1.6
1.4
1.2
1
0.6
0.4
0.4
0.2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
max
BM
max
ATP
min
Flux
Emmerling 0.09h−1, C−limited, Schuetz’s model
1
1.8
1.6
1.4
0.8
min
ATPprod
max
ATPprod
1
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
0.4
0.4
0.2
64
1.2
0.6
0
min
Rd
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
Predictive Fidelity
1.2
max
ATP/flux
Emmerling 0.4h−1, C−limited, Schuetz’s model
2
1.4
max
BM/flux
(f)
(e)
1.6
max
ATPprod
0.8
0.6
1.8
min
ATPprod
Emmerling 0.09h−1, N−limited, Schuetz’s model
0.8
0
min
Rd
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
1.6
max
ATP/flux
(d)
2
1.8
max
BM/flux
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
(g)
(h)
−1
−1
Nanchen 0.09h , C−limited, Schuetz’s model
Nanchen 0.4h , C−limited, Schuetz’s model
2
1.6
Predictive Fidelity
1.4
1.2
1
1.8
1.6
1.4
Predictive Fidelity
1.8
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
max
BM
max
ATP
min
Flux
max
BM/flux
(i)
Yang 0.1h , C−limited, Schuetz’s model
1.2
1
Yang 0.55h , C−limited, Schuetz’s model
1.8
1.6
1.4
0.8
1.2
1
0.6
0.4
0.4
0.2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
0
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
max
BM
max
ATP
min
Flux
(k)
Perrenoud 0.65h−1, batch, Schuetz’s model
Predictive Fidelity
1.4
1.2
1
min
Rd
min
ATPprod
max
ATPprod
min
ATPprod
max
ATPprod
Holm 0.67h−1, batch, Schuetz’s model
1.8
1.6
1.4
0.8
1.2
1
0.6
0.4
0.4
0.2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
0
max
ATP/flux
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
1.6
max
BM/flux
(l)
2
1.8
max
ATPprod
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
Predictive Fidelity
1.4
min
ATPprod
−1
2
1.6
min
Rd
(j)
−1
1.8
max
ATP/flux
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
65
B. FBA simulation results
Figure B.6: Predictive fidelities for objective functions with pairwise constraints for E. coli iAF1260 Genome-scale
model.
(a)
(b)
Ishii 0.1h−1, C−limited, Genome−scale model
Ishii 0.4h−1, C−limited, Genome−scale model
2
1.6
Predictive Fidelity
1.4
1.2
1
1.8
1.6
1.4
Predictive Fidelity
1.8
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
max
BM
max
ATP
min
Flux
max
BM/flux
(c)
Ishii 0.7h , C−limited, Genome−scale model
1.2
1
Emmerling 0.09h , N−limited, Genome−scale model
1.8
1.6
1.4
0.8
1.2
1
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
0
max
BM
max
ATP
min
Flux
max
BM/flux
(e)
Emmerling 0.09h , C−limited, Genome−scale model
1.2
1
66
max
ATPprod
Emmerling 0.4h , C−limited, Genome−scale model
1.8
1.6
1.4
0.8
1.2
1
0.6
0.4
0.4
0.2
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
0
min
ATPprod
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
Predictive Fidelity
1.4
min
Rd
−1
2
1.6
max
ATP/flux
(f)
−1
1.8
max
ATPprod
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
Predictive Fidelity
1.4
min
ATPprod
−1
2
1.6
min
Rd
(d)
−1
1.8
max
ATP/flux
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
(g)
(h)
−1
−1
Nanchen 0.09h , C−limited, Genome−scale model
Nanchen 0.4h , C−limited, Genome−scale model
2
1.6
Predictive Fidelity
1.4
1.2
1
1.8
1.6
1.4
Predictive Fidelity
1.8
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
1.2
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
max
BM
max
ATP
min
Flux
(i)
Yang 0.1h , C−limited, Genome−scale model
1.2
1
1.8
1.6
1.4
1.2
1
0.6
0.4
0.4
0.2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
0
max
ATPprod
max
BM
max
ATP
min
Flux
(k)
Perrenoud 0.65h−1, batch, Genome−scale model
Predictive Fidelity
1.4
1.2
1
min
Rd
min
ATPprod
max
ATPprod
Holm 0.67h−1, batch, Genome−scale model
1.8
1.6
1.4
0.8
1.2
1
0.6
0.4
0.4
0.2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
0.8
0.6
0
max
ATP/flux
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
1.6
max
BM/flux
(l)
2
1.8
max
ATPprod
Yang 0.55h , C−limited, Genome−scale model
0.8
0
min
ATPprod
2
P/O=1 & qO2max=15
P/O=1 & Maintenance
P/O=1 & Bounds
P/O=1 & NADPH
qO2max=15 & Maintenance
qO2max=15 & Bounds
qO2max=15 & NADPH
Maintenance & Bounds
Maintenance & NADPH
Bounds & NADPH
Predictive Fidelity
Predictive Fidelity
1.4
min
Rd
−1
2
1.6
max
ATP/flux
(j)
−1
1.8
max
BM/flux
0.2
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
0
max
BM
max
ATP
min
Flux
max
BM/flux
max
ATP/flux
min
Rd
min
ATPprod
max
ATPprod
67
B. FBA simulation results
Scatter plots
Figure B.7: Individual flux predictions between predicted and experimental fluxes for all the objective functions
evaluated without additional constraint of the Core model.
(a)
(b)
Ishii 0.1h−1, C−limited, Core model
−1
0
1
2
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
−1
1
2
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
−1
0
1
2
2
0
−1
−2
−2
−1
0
1
2
predicted values
1
0
−1
−1
0
1
1
0
−1
−2
−2
observed values
2
−2
−2
2
2
−1
0
1
2
0
−1
−2
−2
−1
max ATPprod
0
−1
−1
0
1
2
−1
0
1
2
1
0
−1
−1
0
0
1
0
−1
−2
−2
−1
0
1
2
1
2
1
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
observed values
predicted values
2
2
1
2
−1
0
1
2
0
−1
−2
−2
−1
0
−1
−1
0
1
2
−1
0
1
2
0
−1
0
−2
−2
−1
0
1
1
0
−1
−2
−2
−1
0
1
2
1
2
max BM
2
1
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
observed values
2
−1
−2
−2
−1
0
1
2
1
0
−1
−1
0
1
observed values
2
1
0
−1
−2
−2
observed values
2
−2
−2
0
predicted values
predicted values
2
2
1
2
−1
0
1
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
2
1
0
−1
−2
−2
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
observed values
predicted values
1
−1
0
−1
observed values
predicted values
min Flux
2
−2
−2
1
(f)
predicted values
predicted values
0
2
Emmerling 0.4h−1, C−limited, Core model
max ATP
−1
2
2
2
observed values
predicted values
max BM
1
1
max ATPprod
(e)
2
0
observed values
min ATPprod
1
−2
−2
1
observed values
2
observed values
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
1
max BM
2
Emmerling 0.09h−1, C−limited, Core model
predicted values
−1
observed values
observed values
predicted values
−2
−2
observed values
predicted values
min Flux
2
−2
−2
0
−1
observed values
predicted values
−1
1
(d)
predicted values
0
2
Emmerling 0.09h−1, N−limited, Core model
predicted values
predicted values
1
2
2
2
observed values
max ATP
predicted values
predicted values
predicted values
predicted values
max BM
1
min ATPprod
(c)
2
0
observed values
1
−2
−2
1
observed values
2
observed values
2
predicted values
−1
1
predicted values
0
2
predicted values
0
1
predicted values
predicted values
−1
2
Ishii 0.7h−1, C−limited, Core model
predicted values
1
min Rd
observed values
68
0
observed values
max ATP/flux
−1
−2
−2
−1
observed values
max BM/flux
0
−2
−2
−2
−2
observed values
min Rd
1
−2
−2
2
0
−1
observed values
2
−2
−2
1
1
max ATP/flux
−1
−2
−2
0
2
observed values
0
−2
−2
min Flux
2
predicted values
1
max BM/flux
1
−2
−2
−2
−2
max BM
2
observed values
2
−2
−2
0
−1
predicted values
−2
−2
1
predicted values
0
−1
min Flux
2
predicted values
predicted values
1
predicted values
max ATP
2
predicted values
predicted values
predicted values
predicted values
max BM
Ishii 0.4h−1, C−limited, Core model
−1
0
1
observed values
2
2
1
0
−1
−2
−2
−1
0
1
observed values
2
1
0
−1
−2
−2
−1
0
1
observed values
2
2
2
2
(g)
(h)
−1
−1
Nanchen 0.09h , C−limited, Core model
−1
0
1
2
−1
0
1
2
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
−1
max ATP/flux
min Rd
1
2
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
−1
0
1
2
2
0
−1
−2
−2
−1
0
1
2
predicted values
1
0
−1
−1
0
1
1
0
−1
−2
−2
observed values
2
−2
−2
2
2
−1
0
1
2
0
−1
−2
−2
−1
0
−1
−1
0
1
2
−1
0
1
2
−2
−2
−1
0
1
0
−1
−2
−2
−1
0
1
2
observed values
0
1
2
0
−1
−2
−2
−1
0
1
2
max ATP
2
predicted values
1
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
observed values
predicted values
2
2
1
2
−1
0
1
2
0
−1
−2
−2
−1
max ATPprod
0
−1
−1
0
1
2
−1
0
1
2
0
−1
0
−1
0
1
0
−1
−2
−2
−1
0
1
2
1
2
max BM
2
1
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
observed values
2
−1
−2
−2
−1
0
1
2
1
0
−1
−1
0
1
observed values
2
1
0
−1
−2
−2
observed values
2
−2
−2
0
predicted values
predicted values
2
2
1
2
−1
0
1
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
2
1
0
−1
−2
−2
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
observed values
predicted values
1
−1
−2
−2
observed values
predicted values
min Flux
2
−2
−2
0
−1
(l)
predicted values
−1
1
Holm 0.67h−1, batch, Core model
predicted values
predicted values
0
2
1
observed values
max ATP
1
2
2
2
(k)
max BM
1
min ATPprod
Perrenoud 0.65h−1, batch, Core model
2
0
observed values
1
−2
−2
1
observed values
2
observed values
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
1
max BM
2
predicted values
0
−1
observed values
predicted values
0
−1
observed values
predicted values
min Flux
1
−1
1
(j)
2
−2
−2
2
Yang 0.55h−1, C−limited, Core model
predicted values
predicted values
0
−1
2
2
2
observed values
max ATP
predicted values
predicted values
predicted values
predicted values
max BM
1
1
max ATPprod
(i)
2
0
observed values
min ATPprod
1
−2
−2
1
observed values
2
observed values
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
0
predicted values
predicted values
−1
2
Yang 0.1h−1, C−limited, Core model
predicted values
1
max BM/flux
observed values
predicted values
0
min Rd
−1
−2
−2
−1
observed values
max ATP/flux
0
−2
−2
−2
−2
max BM/flux
1
−2
−2
2
0
−1
observed values
2
−2
−2
1
1
observed values
−1
−2
−2
0
2
observed values
0
−2
−2
min Flux
2
predicted values
−2
−2
predicted values
−2
−2
0
−1
max ATP
1
observed values
1
−2
−2
0
−1
1
2
observed values
2
−2
−2
1
max BM
2
predicted values
0
−1
min Flux
2
predicted values
predicted values
1
−2
−2
Nanchen 0.4h , C−limited, Core model
max ATP
predicted values
predicted values
predicted values
predicted values
max BM
2
−1
0
1
observed values
2
2
2
1
0
−1
−2
−2
−1
0
1
observed values
2
1
0
−1
−2
−2
−1
0
1
2
2
observed values
69
2
B. FBA simulation results
Figure B.8: Individual flux predictions between predicted and experimental fluxes for all the objective functions
evaluated without additional constraint of the Schuetz model.
(a)
(b)
Ishii 0.1h−1, C−limited, Schuetz’s model
−1
0
1
2
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
−1
max ATP/flux
min Rd
1
2
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
−1
0
1
2
2
0
−1
−2
−2
−1
0
1
2
predicted values
1
0
−1
−1
0
1
1
0
−1
−2
−2
observed values
2
−2
−2
2
2
−1
0
1
2
0
−1
−2
−2
−1
0
−1
−1
0
1
2
−1
0
1
2
1
2
0
1
0
−1
−2
−2
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
observed values
predicted values
2
2
1
2
−1
0
1
2
0
−1
−2
−2
−1
max ATPprod
0
−1
−1
0
1
2
−1
0
1
2
0
−1
0
−2
−2
−1
0
1
1
0
−1
−2
−2
−1
0
1
2
1
2
max BM
2
1
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
observed values
2
−1
−2
−2
−1
0
1
2
1
0
−1
−1
0
1
observed values
2
1
0
−1
−2
−2
observed values
2
−2
−2
0
predicted values
predicted values
2
2
1
2
−1
0
1
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
2
1
0
−1
−2
−2
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
observed values
predicted values
1
−1
0
−1
observed values
predicted values
min Flux
2
−2
−2
1
(f)
predicted values
predicted values
0
2
Emmerling 0.4h−1, C−limited, Schuetz’s model
max ATP
−1
2
2
2
observed values
predicted values
max BM
1
1
min ATPprod
(e)
2
0
observed values
1
−2
−2
1
observed values
2
observed values
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
observed values
predicted values
1
Emmerling 0.09h−1, C−limited, Schuetz’s model
predicted values
−1
observed values
observed values
predicted values
−2
−2
1
max BM
2
predicted values
0
−1
0
0
−1
observed values
predicted values
min Flux
1
−1
1
(d)
2
−2
−2
2
Emmerling 0.09h−1, N−limited, Schuetz’s model
predicted values
predicted values
0
−1
2
2
2
observed values
max ATP
predicted values
predicted values
predicted values
predicted values
max BM
1
1
max ATPprod
(c)
2
0
observed values
min ATPprod
1
−2
−2
1
observed values
2
observed values
2
predicted values
−1
1
predicted values
0
2
predicted values
0
1
predicted values
predicted values
−1
2
Ishii 0.7h−1, C−limited, Schuetz’s model
predicted values
1
max BM/flux
observed values
70
0
min Rd
−1
−2
−2
−1
observed values
max ATP/flux
0
−2
−2
−2
−2
max BM/flux
1
−2
−2
2
0
−1
observed values
2
−2
−2
1
1
observed values
−1
−2
−2
0
2
observed values
0
−2
−2
min Flux
2
predicted values
1
observed values
1
−2
−2
−2
−2
max BM
2
observed values
2
−2
−2
0
−1
predicted values
−2
−2
1
predicted values
0
−1
min Flux
2
predicted values
predicted values
1
predicted values
max ATP
2
predicted values
predicted values
predicted values
predicted values
max BM
Ishii 0.4h−1, C−limited, Schuetz’s model
−1
0
1
observed values
2
2
1
0
−1
−2
−2
−1
0
1
observed values
2
1
0
−1
−2
−2
−1
0
1
observed values
2
2
2
2
(g)
(h)
−1
−1
Nanchen 0.09h , C−limited, Schuetz’s model
−1
0
1
2
−1
0
1
2
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
−1
max ATP/flux
min Rd
1
2
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
−1
0
1
2
2
0
−1
−2
−2
−1
0
1
2
predicted values
1
0
−1
−1
0
1
1
0
−1
−2
−2
observed values
2
−2
−2
2
2
−1
0
1
2
0
−1
−2
−2
−1
0
−1
−1
0
1
2
−1
0
1
2
−2
−2
−1
0
1
0
−1
−2
−2
−1
0
1
2
observed values
0
1
2
0
−1
−2
−2
−1
0
1
2
max ATP
2
predicted values
1
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
observed values
predicted values
2
2
1
2
−1
0
1
2
0
−1
−2
−2
−1
max ATPprod
0
−1
−1
0
1
2
−1
0
1
2
0
−1
0
−1
0
1
0
−1
−2
−2
−1
0
1
2
1
2
max BM
2
1
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
observed values
2
−1
−2
−2
−1
0
1
2
1
0
−1
−1
0
1
observed values
2
1
0
−1
−2
−2
observed values
2
−2
−2
0
predicted values
predicted values
2
2
1
2
−1
0
1
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
2
1
0
−1
−2
−2
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
observed values
predicted values
1
−1
−2
−2
observed values
predicted values
min Flux
2
−2
−2
0
−1
Holm 0.67h−1, batch, Schuetz’s model
predicted values
−1
1
(l)
predicted values
predicted values
0
2
1
observed values
max ATP
1
2
2
2
(k)
max BM
1
min ATPprod
Perrenoud 0.65h−1, batch, Schuetz’s model
2
0
observed values
1
−2
−2
1
observed values
2
observed values
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
1
max BM
2
predicted values
0
−1
observed values
predicted values
0
−1
observed values
predicted values
min Flux
1
−1
1
(j)
2
−2
−2
2
Yang 0.55h−1, C−limited, Schuetz’s model
predicted values
predicted values
0
−1
2
2
2
observed values
max ATP
predicted values
predicted values
predicted values
predicted values
max BM
1
1
max ATPprod
(i)
2
0
observed values
min ATPprod
1
−2
−2
1
observed values
2
observed values
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
0
predicted values
predicted values
−1
2
Yang 0.1h−1, C−limited, Schuetz’s model
predicted values
1
max BM/flux
observed values
predicted values
0
min Rd
−1
−2
−2
−1
observed values
max ATP/flux
0
−2
−2
−2
−2
max BM/flux
1
−2
−2
2
0
−1
observed values
2
−2
−2
1
1
observed values
−1
−2
−2
0
2
observed values
0
−2
−2
min Flux
2
predicted values
−2
−2
predicted values
−2
−2
0
−1
max ATP
1
observed values
1
−2
−2
0
−1
1
2
observed values
2
−2
−2
1
max BM
2
predicted values
0
−1
min Flux
2
predicted values
predicted values
1
−2
−2
Nanchen 0.4h , C−limited, Schuetz’s model
max ATP
predicted values
predicted values
predicted values
predicted values
max BM
2
−1
0
1
observed values
2
2
2
1
0
−1
−2
−2
−1
0
1
observed values
2
1
0
−1
−2
−2
−1
0
1
2
2
observed values
71
2
B. FBA simulation results
Figure B.9: Individual flux predictions between predicted and experimental fluxes for all the objective functions
evaluated without additional constraint of the Genome-scale model.
(a)
(b)
Ishii 0.1h−1, C−limited, Genome−scale model
−1
0
1
2
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
−1
max ATP/flux
min Rd
1
2
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
−1
0
1
2
2
0
−1
−2
−2
−1
0
1
2
predicted values
1
0
−1
−1
0
1
1
0
−1
−2
−2
observed values
2
−2
−2
2
2
−1
0
1
2
0
−1
−2
−2
−1
0
−1
−1
0
1
2
−1
0
1
2
1
2
0
1
0
−2
−2
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
observed values
predicted values
2
2
1
2
−1
0
1
2
0
−1
−2
−2
−1
max ATPprod
0
−1
−1
0
1
2
−1
0
1
2
0
−1
0
−2
−2
−1
0
1
1
0
−1
−2
−2
−1
0
1
2
1
2
max BM
2
1
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
observed values
2
−1
−2
−2
−1
0
1
2
1
0
−1
−1
0
1
observed values
2
1
0
−1
−2
−2
observed values
2
−2
−2
0
predicted values
predicted values
2
2
1
2
−1
0
1
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
2
1
0
−1
−2
−2
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
observed values
predicted values
1
−1
0
−1
observed values
predicted values
min Flux
2
−2
−2
1
(f)
predicted values
predicted values
0
2
Emmerling 0.4h−1, C−limited, Genome−scale model
max ATP
−1
2
2
2
observed values
predicted values
max BM
1
1
min ATPprod
(e)
2
0
observed values
1
−2
−2
1
observed values
2
observed values
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
−1
predicted values
1
Emmerling 0.09h−1, C−limited, Genome−scale model
predicted values
−1
observed values
observed values
observed values
predicted values
−2
−2
observed values
predicted values
0
−1
0
0
−1
1
max BM
2
predicted values
1
−1
1
Emmerling 0.09h−1, N−limited, Genome−scale model
min Flux
2
−2
−2
2
(d)
predicted values
predicted values
0
−1
2
2
2
observed values
max ATP
predicted values
predicted values
predicted values
predicted values
max BM
1
1
max ATPprod
(c)
2
0
observed values
min ATPprod
1
−2
−2
1
observed values
2
observed values
2
predicted values
−1
1
predicted values
0
2
predicted values
0
1
predicted values
predicted values
−1
2
Ishii 0.7h−1, C−limited, Genome−scale model
predicted values
1
max BM/flux
observed values
72
0
min Rd
−1
−2
−2
−1
observed values
max ATP/flux
0
−2
−2
−2
−2
max BM/flux
1
−2
−2
2
0
−1
observed values
2
−2
−2
1
1
observed values
−1
−2
−2
0
2
observed values
0
−2
−2
min Flux
2
predicted values
1
observed values
1
−2
−2
−2
−2
max BM
2
observed values
2
−2
−2
0
−1
predicted values
−2
−2
1
predicted values
0
−1
min Flux
2
predicted values
predicted values
1
predicted values
max ATP
2
predicted values
predicted values
predicted values
predicted values
max BM
Ishii 0.4h−1, C−limited, Genome−scale model
−1
0
1
observed values
2
2
1
0
−1
−2
−2
−1
0
1
observed values
2
1
0
−1
−2
−2
−1
0
1
observed values
2
2
2
2
(g)
(h)
−1
−1
Nanchen 0.09h , C−limited, Genome−scale model
−1
0
1
2
−1
0
1
2
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
−1
max ATP/flux
min Rd
1
2
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
−1
0
1
2
2
0
−1
−2
−2
−1
0
1
2
predicted values
1
0
−1
−1
0
1
1
0
−1
−2
−2
observed values
2
−2
−2
2
2
−1
0
1
2
0
−1
−2
−2
−1
0
−1
−1
0
1
2
−1
0
1
2
−2
−2
−1
0
1
0
−1
−2
−2
−1
0
1
2
0
1
2
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
predicted values
1
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
2
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
2
1
0
−1
−2
−2
observed values
predicted values
2
2
1
2
−1
0
1
2
0
−1
−2
−2
−1
max ATPprod
0
−1
−1
0
1
2
−1
0
1
2
0
−1
0
−1
0
1
0
−1
−2
−2
−1
0
1
2
1
2
max BM
2
1
0
−1
−2
−2
−1
0
1
2
max ATP
2
1
0
−1
−2
−2
−1
0
1
2
min Flux
2
1
0
−1
−2
−2
−1
0
1
2
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
observed values
observed values
observed values
observed values
max BM/flux
max ATP/flux
min Rd
max BM/flux
max ATP/flux
min Rd
−1
0
1
2
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
1
0
−1
−1
0
1
observed values
2
−1
−2
−2
−1
0
1
2
1
0
−1
−1
0
1
observed values
2
1
0
−1
−2
−2
observed values
2
−2
−2
0
predicted values
predicted values
2
2
1
2
−1
0
1
2
1
0
−1
−2
−2
−1
0
1
observed values
observed values
min ATPprod
max ATPprod
2
1
0
−1
−2
−2
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
observed values
predicted values
1
−1
−2
−2
observed values
predicted values
min Flux
2
−2
−2
0
−1
Holm 0.67h−1, batch, Genome−scale model
predicted values
−1
1
(l)
predicted values
predicted values
0
2
1
observed values
max ATP
1
2
2
2
(k)
max BM
1
min ATPprod
Perrenoud 0.65h−1, batch, Genome−scale model
2
0
observed values
1
−2
−2
1
observed values
2
observed values
2
predicted values
0
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
predicted values
2
2
observed values
observed values
predicted values
0
−1
observed values
predicted values
0
−1
1
max BM
2
predicted values
1
−1
1
Yang 0.55h−1, C−limited, Genome−scale model
min Flux
2
−2
−2
2
(j)
predicted values
predicted values
0
−1
2
2
2
observed values
max ATP
predicted values
predicted values
predicted values
predicted values
max BM
1
1
max ATPprod
(i)
2
0
observed values
min ATPprod
1
−2
−2
1
observed values
2
observed values
2
predicted values
−1
1
predicted values
0
2
predicted values
1
predicted values
0
predicted values
predicted values
−1
2
Yang 0.1h−1, C−limited, Genome−scale model
predicted values
1
max BM/flux
observed values
predicted values
0
min Rd
−1
−2
−2
−1
observed values
max ATP/flux
0
−2
−2
−2
−2
max BM/flux
1
−2
−2
2
0
−1
observed values
2
−2
−2
1
1
observed values
−1
−2
−2
0
2
observed values
0
−2
−2
min Flux
2
predicted values
−2
−2
predicted values
−2
−2
0
−1
max ATP
1
observed values
1
−2
−2
0
−1
max BM
1
2
observed values
2
−2
−2
1
2
predicted values
0
−1
min Flux
2
predicted values
predicted values
1
−2
−2
Nanchen 0.4h , C−limited, Genome−scale model
max ATP
predicted values
predicted values
predicted values
predicted values
max BM
2
−1
0
1
observed values
2
2
2
1
0
−1
−2
−2
−1
0
1
observed values
2
1
0
−1
−2
−2
−1
0
1
2
2
observed values
73
2
B. FBA simulation results
Comparing three model reconstructions
Figure B.10: Predictive fidelities of pFBA simulation for each objective function in six remaining datasets with
three different metabolic models. Values above 1 are out of range.
(a)
(b)
Ishii 0.4h−1, C−limited
Nanchen 0.4h−1, C−limited
max BM
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
max ATP
1
0
max BM/flux
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
1
0
1
min Rd
1
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
1
0.5
Predictive Fidelity
max ATP
1
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
min Flux
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
1
0
max ATP/flux
1
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
min Rd
1
0
0.5
Predictive Fidelity
1
max ATPprod
0.5
Predictive Fidelity
1
0
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
1
0.5
Predictive Fidelity
0.5
Predictive Fidelity
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
0
0
1
0.5
Predictive Fidelity
0
0.5
Predictive Fidelity
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
min Rd
1
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
min Rd
1
0
0.5
Predictive Fidelity
1
max ATPprod
Genome−scale model
Core model
Schuetz model
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
max ATP
1
0
max BM/flux
1
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
min Flux
1
0
max ATP/flux
1
0
min ATPprod
1
0
max ATP/flux
max BM
max ATPprod
1
0.5
Predictive Fidelity
−1
min Flux
Genome−scale model
Core model
Schuetz model
0.5
Predictive Fidelity
0
(f)
max ATP/flux
1
1
Yang 0.55h , C−limited
1
1
min Flux
−1
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
(e)
max ATP
1
0
max ATP
min ATPprod
Genome−scale model
Core model
Schuetz model
1
Genome−scale model
Core model
Schuetz model
max BM/flux
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
max ATPprod
max BM
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0
min Rd
(d)
min ATPprod
74
1
max ATP/flux
min ATPprod
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
Emmerling 0.4h−1, C−limited
max BM/flux
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0
min Flux
(c)
max BM
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
1
max BM/flux
Yang 0.1h , C−limited
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
max ATP
Nanchen 0.09h−1, C−limited
min ATPprod
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
0.5
Predictive Fidelity
Genome−scale model
Core model
Schuetz model
max BM/flux
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0
max BM
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
max ATPprod
max BM
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
1
max ATP/flux
min ATPprod
All constraints
NADPH
Bounds
Maintenance
qO2max=15
P/O=1
None
0
0.5
Predictive Fidelity
min Flux
0.5
Predictive Fidelity
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
max ATPprod
Genome−scale model
Core model
Schuetz model
0.5
Predictive Fidelity
1
0
0.5
Predictive Fidelity
1
min Rd
1
1
C
Published abstract and article
75
C. Published abstract and article
• Poster: Rafael S. Costa, Son Nguyen and Susana Vinga.“Comparison of cellular objectives
in flux balance constraint-based model”. The 13th European Conference on Computational
Biology (ECCB 2014), 9-12 September 2014, Strasbourg, France (poster C14).
• Article: Rafael S. Costa*, Son Nguyen*, Andras Hartman, Susana Vinga.“Exploring the cellular objective in flux balance constraint-based models”. Lecture Notes in Computer Science,
LNBI 8859, pp. 211-224. Springer International Publishing Switzerland (2014) (to appear ).
76

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