DIMERIZATION OF TRANSCRIPTION FACTORS IN GENE EXPRESSION
MODELS: THE CONTROL OF HMP SYNTHESIS BY NSRR AS A CASE STUDY
MOHAMMAD HAMIDI
Master of Science in Physical Chemistry, Azad University, 2014
A thesis submitted
in partial fulfilment of the requirements for the degree of
MASTER OF SCIENCE
in
BIOCHEMISTRY
Department of Chemistry and Biochemistry
University of Lethbridge
LETHBRIDGE, ALBERTA, CANADA
© Mohammad Hamidi, 2022
DIMERIZATION OF TRANSCRIPTION FACTORS IN GENE EXPRESSION
MODELS: THE CONTROL OF HMP SYNTHESIS BY NSRR AS A CASE STUDY
MOHAMMAD HAMIDI
Date of Defence: December 20, 2022
Dr. M. Roussel Professor Ph.D.
Thesis Supervisor
Dr. S. Wetmore Professor Ph.D.
Thesis Examination Committee Member
Dr. T. Patel Associate Professor Ph.D.
Thesis Examination Committee Member
Dr. M. Gerken Professor Ph.D.
Chair, Thesis Examination Committee
Dedication
This thesis is dedicated to my wife, who is super supportive throughout my whole life.
iii
Abstract
Many transcription factors act as dimers or higher oligomers. The effects of dimerization of
transcription factors in gene expression models are not well studied. However, explicit con-
sideration of dimers and, especially, of their regulation, may affect the dynamical behavior
of a model. In the present work, I study how the effects of dimerization of transcriptional
factors impact gene expression and compare the results between models for the control of
transcription assuming monomeric or dimeric transcription factors. NsrR is a nitric oxide
(NO) sensor and a key regulator of NO metabolism in a number of bacterial species. NsrR
uses an iron-sulfur cluster to sense NO. NsrR binds to gene promoters as a dimer. In Strep-
tomyces, NsrR acts a repressor to regulate only two independent copies of the hmp gene
and the nsrR gene. Nitrosylation of an iron-sulfur cluster in each NsrR monomer affects
the ability of NsrR to bind to the promoters of these genes. Hmp catalyzes the detoxifying
reaction of NO to the less toxic nitrate ion. In this study, I have built a base model in which
NsrR acts as a (theoretical) monomer, and then I have generated two models where NsrR is
a dimer that differ in the assumed effects of uneven nitrosylation of the two monomers in a
dimer. Most of the parameters were estimated from the literature. The model equations for
the three models were generated by using the rule-based modelling software BioNetGen.
The monomer and dimer models behave essentially identically either at the default param-
eters estimated from the literature, or with changes in most of the parameters. However,
by changing binding parameters for NsrR-promoter complexes or the rate of recycling of
nitrosylated NsrR to the intact holo-protein, the monomer and dimer models can be made
to behave differently.
iv
Acknowledgments
First and foremost, I am deeply grateful to my supervisor, Marc Roussel, whose guidance
and encouragement has made this thesis possible. He has been an inspiration and molded
my perspective on biochemistry. He truly embodied the role of an advisor as he pushed me
to develop my own thoughts and abilities in order to make me a better, more well rounded
researcher. I am grateful for all the knowledge and experience I have gained under Marc’s
supervision.
I would like to thank my thesis committee: Stacey and Trushar for giving me all their
valuable comments.
I would also like to thank Professor James R. Faeder of the University of Pittsburgh for
answering my questions about BioNetGen.
v
Contents
Dedication iii
Abstract iv
Acknowledgments v
List of Tables viii
List of Figures ix
1 Introduction 1
1.1 Biological Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Biological roles of nitric oxide . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Antibiotics and resistance . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Eliminating the toxicity of NO . . . . . . . . . . . . . . . . . . . . 3
1.1.4 Regulation of NO-responsive genes . . . . . . . . . . . . . . . . . 3
1.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Justifying a study based on Streptomyces . . . . . . . . . . . . . . . . . . . 6
1.5 Rule-based modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Plan of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Methods 10
3 Dimerization of transcription factors in gene expression models: the control of
Hmp synthesis by NsrR as a case study 12
Abstract 13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Description of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 Hmp kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.2 Monomeric model of NsrR dynamics (MM model) . . . . . . . . . 20
3.2.3 Dimer model 1 (DM1) . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.4 Dimer model 2 (DM2) . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.5 Parameter estimates . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Methods and software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
vi
CONTENTS
3.4.1 Dynamical behavior of the models at the default parameters . . . . 33
3.4.2 Conditions making the models behave differently . . . . . . . . . . 41
3.5 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Conclusion 51
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Bibliography 55
A BioNetGen code 63
A.1 Monomer (MM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A.2 Dimer model 1 (DM1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.3 Dimer model 2 (DM2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
B Differential equations 74
B.1 Monomer (MM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B.2 Dimer model 1 (DM1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
B.3 Dimer model 2 (DM2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
vii
List of Tables
3.1 Default model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
viii
List of Figures
1.1 Structures of NADPH and NADH. . . . . . . . . . . . . . . . . . . . . . . 4
3.1 Bifurcation diagram for model MM at the default parameters values. . . . . 34
3.2 Two-parameter phase diagram showing a cusp bifurcation. . . . . . . . . . 35
3.3 Bifurcation diagrams for models DM1 and DM2. . . . . . . . . . . . . . . 36
3.4 NsrR family dimer concentrations vs [NO] at the default parameters for DM1 38
3.5 NsrR family dimer concentrations vs [NO] for model DM1 when kAB =
kBC = kCD = kDE = kEF = 1µM−1s−1, other parameters being at their de-
fault values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Bifurcation diagram for model MM with all parameters as in Table 3.1
except [O2] = 16µM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.7 Bifurcation diagrams for the three models at kr = 0.001 s−1. . . . . . . . . 43
3.8 Limit cycles for model DM2 at kr = 0.001 s−1. . . . . . . . . . . . . . . . 44
3.9 Active promoter concentrations vs [NO] . . . . . . . . . . . . . . . . . . . 45
3.10 Bifurcation diagram for models MM, DM1 and DM2 when k−B = 1×
10−1s−1 and k −3 −1−C = 1×10 s . . . . . . . . . . . . . . . . . . . . . . . . 47
ix
Chapter 1
Introduction
1.1 Biological Background
1.1.1 Biological roles of nitric oxide
Nitric oxide (NO) is an important molecule in biological systems as it is a signalling
molecule in dilute solution and can be a toxic molecule when it is concentrated. For ex-
ample, NO can act as a retrograde neurotransmitter in synapses [1], may decrease muscle
soreness [2], can cause lower blood pressure [3] and may help manage type 2 diabetes [4].
Additionally, the mammalian immune system uses NO to try to kill bacteria [5], and bacte-
ria have defences against it [6]. Also, the immune system makes NO as a general response
to bacteria because NO is a signal used by the immune system in order to tell certain com-
ponents to turn on [7]. Thus, studying NO and its biological roles can be helpful for modern
medicine.
Enzymes play an important role in production and removal of NO in microorganisms.
Nitric oxide is synthesized by an enzyme called NOS (NO synthase) [8] that catalyzes
the conversion of L-arginine to L-citrulline [9]. On the other hand, NORs (nitric oxide
reductases) which are bacterial enzymes, catalyze the reaction of NO to N2O (nitrous oxide)
and H2O [10]. In general, four families of reductases catalyze the conversion of NO −3
(nitrate) to N2 (nitrogen) as follows [11, 12]: NO − −
N−→ar NO − −N−→ir3 2 NO−
N−→or −N−2−O→rN2O N2
where Nar is the nitrate reductase enzyme, Nir is the nitrite reductase enzyme, Nor is the
nitric oxide reductase enzyme and N2Or is the nitrous oxide reductase enzyme.
1
1.1. BIOLOGICAL BACKGROUND
1.1.2 Antibiotics and resistance
The most effective treatment for bacterial infection is applying an antibiotic when the
immune system cannot kill them. How can antibiotics help our immune system to kill the
bacteria? There are some possibilities here [13]. Most traditional antibiotics attack the
bacterial ribosome which stops translation. Some antibiotics interfere with the production
of the cell wall. One of the big problems of developing antibiotics right now is that most
of them have the same parts of the cell as targets. So, it is fairly easy for bacterial cells to
evolve resistance to new antibiotics working in the same way.
Some bacteria show resistance to antibiotics and are called antibiotic-resistant bacte-
ria. For instance, Escherichia coli, Klebsiella pneumoniae, Staphylococcus aureus, and
Streptococcus pneumoniae are bacteria that commonly develop antibiotic resistance [14].
Studying antibiotic-resistant bacteria can give us new approaches to the problem of antibi-
otic resistance by discovering how bacteria can handle antibiotic effects. One possibility is
that the bacteria use efflux transporters to pump the antibiotic out of the cell. Bacteria can
also evolve to change an antibiotic’s target by mutating the target or using post-translational
modifications in order to prevent binding of an antibiotic to its target [15].
According to the WHO’s 2018 report entitled Global Antimicrobial Surveillance Sys-
tem [14], 500,000 patients were placed under bacterial infection surveillance in 22 different
countries in a sampling process. The statistical review revealed that the percentage of bac-
teria showing defences against antibiotics ranged from 0 to 82 percent by country (resistant
to at least one antibiotic), 0 to 51 percent resistant against penicillin and 8 to 65 percent of
E. coli resistant against ciprofloxacin. Thus, studying a bacterial defence system such as the
defence against NO could give us new approaches to treat patients with antibiotic-resistant
infections, which are sorely needed.
2
1.1. BIOLOGICAL BACKGROUND
1.1.3 Eliminating the toxicity of NO
Bacterial cells need a pathway to eliminate NO as NO damages the proteins and DNA
of bacteria if they cannot eliminate it [16]. There are variations between types of bacteria,
but typically an NO sensor based on an iron-sulfur cluster controls the expression of genes
necessary to detoxify NO. For example, in Streptomyces, when NO is penetrating into a
bacterial cell, it reacts with the iron-sulfur cluster of a transcription factor called NsrR. It
takes several NO to completely react with the cluster, but as each NO reacts, it loosens the
binding between NsrR and the promoters to which the former binds. This eventually allows
the hmp gene to be expressed.
Hmp is the key enzyme that eliminates NO in most bacteria [16]. This detoxifying
reaction is catalyzed by the Hmp enzyme [17]:
Hmp
2NO+2O − + +2 +NAD(P)H−−→ 2NO3 +NAD(P) +H
In this reaction, NAD(P)H binds to the enzyme first, and then O2 and NO respectively [18].
If NO binds before O2, what will happen? NO binding before O2 prevents O2 binding to
Hmp. Thus, Hmp cannot catalyze this reaction until the inhibitory NO dissociates.
The only difference between NADPH and NADH is that NADPH has one additional
phosphate group (Fig. 1.1). Hmp can use both of them interchangeably.
1.1.4 Regulation of NO-responsive genes
One of the most important parts of any gene is its promoter which plays an integral role
for the control of protein expression [19]. Transcription will be continued or stopped based
on the type of regulatory proteins bound to the promoters of a gene [20]. Regulators of
transcription can be either repressors or activators. If a repressor is bound to a promoter,
transcription is stopped while an activator helps RNA polymerase bind to promoters that
the polymerase cannot bind to on its own.
3
1.1. BIOLOGICAL BACKGROUND
Figure 1.1: Structures of NADPH and NADH.
(a) NADPH structure. PubChem for CID 5884, Dihydronicotinamide-adenine dinucleotide
phosphate beta-NADPH. https://pubchem.ncbi.nlm.nih.gov/compound/5884, in the
public domain. (b) NADH structure. National Center for Biotechnology Information
(2022). PubChem Compound Summary for CID 439153, 1,4-Dihydronicotinamide ade-
nine dinucleotide. Retrieved October 29, 2022 from https://pubchem.ncbi.nlm.nih.
gov/compound/1_4-Dihydronicotinamide-adenine-dinucleotide, in the public do-
main
Transcription factors often act as dimers or higher oligomers [21]. Some transcription
factors are able to generate a number of distinct dimers with different binding partners. A
protein can have different functions depending on its binding partner. A homodimer is a
dimer which is formed from identical monomers and heterodimers are dimers consisting of
different monomers. Homodimers are the most prevalent type of dimers across all species
[22]. The transcription factor NsrR is a good example of a homodimer.
NsrR is a repressor of gene expression which is a member of the Rrf2 transcriptional
regulator family [23]. The Rrf2 family belongs to the winged helix-turn-helix superfamily
of prokaryotic transcriptional regulators [24]. A majority of its members act primarily as
repressors [25]. Some of the other members of this superfamily are RsrR (Redox-sensitive
response regulator) and IscR (Iron-sulfur cluster regulator) [25]. In most bacteria, there is
at least one gene from the Rrf2 family. In this family, the N-terminus acts for binding to the
promoter and the C-terminal is involved in signalling [26].
NsrR is widespread in bacteria. It was previously known to act as a repressor for redox
reactions in Nitrosomonas spp, and has more recently been discovered to act as a regulator
of nitric oxide metabolism in a number of bacterial species including gamma- and beta-
proteobacteria, as well as gram-positive Bacillus and Streptomyces, based on the analysis
4
1.3. PREVIOUS WORK
of Arkin’s group [27].
NsrR requires a [4Fe–4S] cluster to act as a repressor. NO can react with the iron-sulfur
cluster of NsrR. Since interfering with NsrR functions could sensitize bacterial cells to NO
and make them vulnerable to attack by the mammalian immune system, it might be a new
antibiotic target because of this role.
1.2 Statement of the problem
NsrR acts as a dimer in its biological functions. As mentioned, the effects of dimer-
ization of transcription factors in gene expression models are not well studied. The reason
is that generating and writing down a large number of differential equations for a dimeric
system is a complicated and error-prone process. To address this issue, we used rule-based
modeling to create models, as computers are good at repetitive tasks.
In a previous study, Roussel ignored the effects of dimerization of FNR in the Hmp
control system in E. coli [18]. In E. coli, hmp is regulated by both FNR and NsrR, and FNR
senses both nitric oxide and oxygen. Thus, pursuing the Hmp control system in E. coli
might not be a good idea because of its complexity. On the other hand, in Streptomyces,
hmp is regulated only by NsrR which only responds to nitric oxide.
Since NsrR is a representative of a group of iron-sulfur proteins that respond to NO [28,
29] and, more importantly, the kinetics of the reaction of NO with NsrR has been studied
for Streptomyces [23], my main focus is on considering the effects of NsrR dimerization in
models for the control of Hmp synthesis by NsrR in Streptomyces with the use of rule-based
modeling.
1.3 Previous work
Roussel [18] has developed a delayed mass-action model for the control system for Hmp
by another iron-sulfur protein called FNR in E. coli. He assembled a detailed biochemical
model for the control of the synthesis of Hmp when FNR acts as a repressor of hmp tran-
5
1.4. JUSTIFYING A STUDY BASED ON STREPTOMYCES
scription. Also, he estimated parameters for the reactions, showed bistability in this model,
and carried out the steady-state analysis for this model to show how the system dynami-
cally behaves. In E. coli, hmp expression is controlled by both FNR and NsrR; however, he
didn’t consider regulation by NsrR. To my knowledge, there are no other detailed models
of an hmp control system in any bacteria.
Robinson and Brynildsen described models for nitric oxide metabolism in E. coli [30,
31]. They demonstrated that Hmp is not just an enzyme that detoxifies NO, but that in fact
it provides the main pathway for detoxifying NO. This is really important because it means
that I can really focus on Hmp without worrying about other pathways that may consume
NO. Their models are comprehensive ones for NO metabolism including empirical expres-
sions for the control of Hmp expression. Although their models answer important questions
about nitric oxide metabolism, they didn’t take the molecular details of this system into ac-
count and they didn’t model the control system for the synthesis of Hmp at a mechanistic
level. Then, this limits the possibility of using these models to think about possible new
antibacterial strategies.
Also, Crack and his colleagues [23] demonstrated that in Streptomyces, the NsrR iron-
sulfur cluster reacts with NO. They mentioned that hmpA1 and hmpA2 are two hmp paralogs
in Streptomyces. The ratios of NO to NsrR required to abolish binding to the three relevant
promoters differ among hmpA1, hmpA2 and nsrR. They experimentally showed that reac-
tion with 4, 2 and 8 equivalents of NO respectively, will completely abolish the binding of
NsrR to those three promoters. Also, they studied the kinetics of the reaction of NO with
the iron-sulfur cluster in NsrR.
1.4 Justifying a study based on Streptomyces
As mentioned before, NO is a toxic molecule at higher concentration. When bacterial
cells are exposed to NO, they can remove it and avoid its toxic effects by making Hmp.
Currently, NO is not attacking bacteria effectively because many bacteria have an hmp
6
1.5. RULE-BASED MODELING
gene.
Since a number of bacteria live in the human digestive system, and probably most of
the bacteria that are of medical concern have at least one hmp gene, Hmp and its control
system are interesting targets for new antibacterial agents [30, 31].
In E. coli, NsrR regulates different genes such as hmp, hcp, hcr, ytfE and ygbA [27].
Similarly, in Klebsiella pneumoniae, NsrR regulates different genes including hcp, hcr,
hmp, tehB and dnrN [27]. But in Streptomyces, NsrR regulates only three genes: two
independent copies of the hmp gene, each with its own promoter, and also nsrR itself [23].
On one hand, Streptomyces is a non-pathogenic bacteria [32]. On the other hand, a small
number of genes are controlled by NsrR in Streptomyces. Additionally, the kinetics of the
reaction of NO with the NsrR iron-sulfur cluster has been studied for the Streptomyces
protein. Hmp is only controlled by NsrR in Burkholderia pseudomallei [27] which is a
medically relevant bacteria. Since the Hmp control system in Burkholderia pseudomallei
is of a similar simplicity to the control system in Streptomyces, and since the latter is a
well-studied Hmp control system involving only repression by NsrR, Streptomyces’ Hmp
control system can be a good starting point for studying the function of NsrR in detail.
1.5 Rule-based modeling
Rule-based modeling generally uses a formal pattern-matching language to describe the
rules by which molecules of certain classes interact [33]. A rule-based model is a chem-
ical model including a number of rules from which differential equations are translated.
By having differential equations, we can integrate them, compute steady states, study the
bifurcation diagram, etc.
BioNetGen is software for rule-based modeling that uses a pattern-matching language
called BNGL. Based on the observations of Crack et al [23], there are five kinetically distin-
guishable reactions of NsrR containing iron-sulfur clusters with NO, so we have six states
for an NsrR monomer and then, the total number of states for an NsrR dimer is 62=36 states.
7
1.5. RULE-BASED MODELING
So, BioNetGen as a rule-based modeling software can generate the differential equations in
the dimeric system.
1.5.1 Modeling
As mentioned previously, NsrR binds DNA as a dimer in bacterial cells. A similar
monomer-based model has been studied by Roussel [18] focusing on the effects of delays
in gene expression. Roussel ignored the dimeric nature of FNR, so the dynamical effects of
dimerization of this transcription factor were not studied. Therefore, studying these effects
for NsrR would be dynamically interesting.
In this work, I built several models for the detoxification of nitric oxide in Streptomyces
based on the control of the expression of the Hmp enzyme by NO, mediated by NsrR. For
the baseline model, the dimeric nature of NsrR has been ignored. Afterwards, the model
was exported from BioNetGen in order to study its dynamics in XPPAUT [34]. Meantime, a
majority of the parameters were estimated. Then, I carried out simulation studies using both
XPPAUT and BioNetGen. Finally, we carried out the numerical bifurcation analysis for the
model. After that, we could develop new models by considering the effects of dimerization
of NsrR and could compare the results between the models in which NsrR acts as monomer
and the models where NsrR is a dimer. The baseline model we have been developing is
based on Roussel’s model [18]. Roussel created a model for the synthesis of Hmp in E. coli
under the control of FNR. In three models, we considered the effects of binding of NsrR to
two different promoters, hmpA1 and hmpA2. For the nsrR promoter, binding of the NsrR
complex needs 8 NO’s to be entirely abolished [23], so the degree of nitrosylation is eight
when binding of NsrR to nsrR promoters has been lost. Since nsrR expression is much less
sensitive to NO than expression of hmp, we assumed NsrR is constant in order to focus on
the effects of dimerization of NsrR on expression of the hmpA1 and hmpA2 genes. Then,
two models based on NsrR dimers were developed and studied.
The dynamics of models including dimeric NsrR were studied and we will see that
8
1.6. PLAN OF THESIS
the dynamics of these systems can be different from those of simpler systems considering
only NsrR monomers. These studies will also reveal under which conditions we can use a
reduced model not considering dimerization.
1.6 Plan of thesis
Methods and software applied in this thesis are described in chapter 2. Hmp kinetics is
studied in chapter 3. Additionally, in chapter 3, a base model is built where the dimerization
of the transcription factor NsrR has been ignored and afterwards, two other models are
created where NsrR acts as a dimer. The reason for creating two different dimer models is
that we do not know how differences in the nitrosylation states of two monomers in a dimer
affect binding of NsrR to the promoters. Moreover, rule-based modeling is used to generate
the differential equations for the dimer models. Parameter estimates from experimental
data are obtained and as a result, a set of baseline parameters used throughout the study
is reported in table 3.1. The effects of dimerization of NsrR in the Hmp gene expression
models is explored. Then, the three models are compared in order to find the conditions that
make the three models behave differently as well as the situations where the three models
behave identically. Chapter 4 contains a summary as well as showing that our results are a
platform for new studies in the future.
9
Chapter 2
Methods
In nonlinear dynamics, a bifurcation is a qualitative change in the behavior of the solutions
of a system of equations as a parameter changes. The parameter-dependent behavior is
displayed in a bifurcation diagram. For instance, it shows stable and unstable steady states
and limit cycles [35].
There is software available to compute and plot bifurcation diagrams. In this study,
BioNetGen [33], Copasi [36], XPPAUT [34] and AUTO were used.
BioNetGen [33] was used for generating the differential equations.
Copasi [36] is software for simulating biochemical systems and analyzing their dynam-
ical behavior. It was used as an intermediate for generating ODE files (XPPAUT input files)
from SBML (Copasi input files). The systems biology markup language, SBML, is a file
format for description of biological systems [37, 38].
XPPAUT [34] was used for two purposes: we can numerically solve differential equa-
tions with it (i.e. simulate our system), and we can use the built-in version of AUTO to
carry out bifurcation analysis.
AUTO can calculate branches of steady states (stable or unstable), periodic solutions
(stable or unstable) and bifurcation points of different kinds such as saddle-node, Hopf
and so on. In a saddle-node bifurcation, an unstable saddle point collides with a stable
equilibrium, and the two annihilate. Additionally, by varying a parameter, a stable focus
may be converted into an unstable focus creating a limit cycle. This kind of bifurcation is
called a Hopf bifurcation [35].
10
2. METHODS
BioNetGen integrates differential equations forward in time, so it provides only stable
solutions. Since BioNetGen is able to show only stable steady state branches, which gives
us an incomplete picture of dynamics, we exported the models from BioNetGen’s .bngl
format to SBML files and imported the SBML files to Copasi. Then, we exported the
.ode files to XPPAUT to enable the computation of both stable and unstable steady state
branches, limit cycles and bifurcation points.
11
Chapter 3
Dimerization of transcription factors in
gene expression models: the control of
Hmp synthesis by NsrR as a case study
CRediT authorship contribution statement
Mohammad Hamidi: Investigation, Methodology, Visualization, Writing — original
draft. Marc Roussel: Conceptualization, Investigation, Funding acquisition, Supervision,
Validation, Writing — review & editing.
12
Abstract
Many transcription factors are active as dimers or higher oligomers. Moreover, the activity
of transcription factors is often controlled by chemical modification of or by ligand binding
to the individual monomers. In these cases, a multiplicity of states of the oligomeric tran-
scription factor results. How important is the high-dimensional state space of the oligomer
to the behavior of a gene control system? In this paper, we study a model for the control
of the transcription of the hmp genes by NsrR in Streptomyces, a dimeric protein whose
active form carries one [4Fe–4S] cluster per monomer. In Streptomyces, NsrR represses
the transcription of two independent copies of the hmp gene whose product, Hmp, is an
enzyme that catalyzes the conversion of toxic nitric oxide (NO) to nitrate. Reaction of the
iron-sulfur cluster of NsrR with NO eventually alleviates the repression of the hmp genes,
but little is known about the effect of unequal nitrosylation of the two monomers form-
ing the NsrR dimer. To study the importance of the dimeric nature of NsrR on its control
characteristics, we built a model based on a fictitious monomeric NsrR repressor and two
models in which we account for the dimeric nature NsrR but making different assumptions
about the kinetic characteristics of unevenly nitrosylated monomers within a dimer. For
our default parameters, which were largely based on experimental data available from the
literature, we found little difference between the three models provided repair of damaged
iron-sulfur clusters was fast. However, significant differences were found when repair was
slow. Leaving aside experimental parameter estimates, differences could also be observed
when the first nitrosylation step tightened binding of the repressor to the hmp promoters
rather than loosening it.
Keywords: Transcription factor; Dimerization; Nitric oxide; Hmp synthesis; NsrR; Rule-
13
3.1. INTRODUCTION
based modeling; BioNetGen
3.1 Introduction
Gene expression is almost always controlled at least in part by the binding of regulatory
proteins, known as transcription factors, to the gene’s promoter [19]. Transcription factors
can either be activators or repressors, and some transcription factors can assume different
roles on different promoters. The active form of a transcription factor is sometimes a dimer
or higher oligomer. Some transcription factors are able to form a number of distinct dimers
with different binding partners, each dimer having different biological properties [21].
From a modeling perspective, dimeric transcription factors pose some challenges due
to combinatorial complexity. Although every situation is different, it is occasionally tempt-
ing to ignore dimerization and to treat the transcription factor as a monomer. What are the
effects of ignoring the dimeric nature of a transcription factor in a model of gene expres-
sion? This is the question broached in this paper via a case study of a particular sensing and
control system, namely the nitric oxide (NO) sensing system mediated by the dimeric iron-
sulfur protein NsrR, which controls the expression of the NO detoxifying enzyme Hmp in
a number of bacterial species [27].
NsrR is a member of the Rrf2 transcriptional regulator family, generally acting as a
repressor in a wide variety of bacterial species [39]. NsrR was originally discovered as a
repressor controlling the expression of nitrite reductase in Nitrosomonas europaea [40]—
hence its name, which stands for nitrite-sensitive repressor—and was later found to act as
a global regulator of nitric oxide metabolism in a number of bacterial species including
gamma- and beta-proteobacteria, and the more evolutionarily distant Bacillus and Strep-
tomyces [27]. The holo form of NsrR contains a [4Fe–4S] cluster [39]. This iron-sulfur
cluster reacts with NO. After several equivalents of NO have reacted, NsrR becomes un-
able to bind DNA, and the genes under its control can therefore be transcribed [23].
One of the genes frequently controlled by NsrR in this manner is hmp. Hmp is an NO-
14
3.1. INTRODUCTION
detoxifying enzyme that catalyzes the conversion of NO to nitrate [17]. Because it allows
bacteria to survive NO generated by the immune system, Hmp is a virulence factor [31, 41].
Thus, Hmp and its NsrR-mediated control system could be interesting targets for the design
of new antibiotics.
To better understand this system, we built a mathematical model of Hmp and its control
system. We specifically chose to model this control system in the soil bacterium Strepto-
myces because of its relative simplicity: in Streptomyces, NsrR regulates only three genes;
two independent copies of the hmp gene, and also nsrR itself [23]. As we were building
this model, we realized that the dimeric nature of the NsrR repressor posed a problem: NO
will necessarily react at each of the two iron-sulfur clusters contained in a dimer. What is
the effect of considering the effect of the pair of iron-sulfur clusters on the behavior of a
model for the control of the expression of Hmp by dimeric NsrR? A previous model for
the similar FNR-mediated control system in Escherichia coli had swept this issue under the
carpet [18], treating the hmp gene as if it was controlled by a monomeric FNR repressor.
We therefore decided to focus on the role of dimers in this control system. We set aside
the regulation of nsrR by NsrR for now, on the basis that the latter regulatory interaction
presumably exists to maintain homeostasis in the NsrR concentration.
The family of hmp control systems presents an interesting advantage from the perspec-
tive of these studies: the E. coli FNR-mediated control system displays bistability [18],
i.e. a pair of saddle-node bifurcations, when NO is supplied to the cells at constant rate.
Because the two control systems are in many ways similar, a Streptomyces model for the
expression of Hmp under the control of NsrR would be expected to also display bistabil-
ity, which is exactly what we found, as outlined below. Bifurcation points can be sensitive
to the details of a model, so it seemed to us that a study of this system would illuminate
the conditions under which dimerization of transcription factors might be quantitatively or
qualitatively important.
In order to study this question, we devised three variations of a model for the control
15
3.2. DESCRIPTION OF THE MODELS
of Hmp expression by NsrR. All three models share a submodel for the detoxification of
NO by Hmp described in section 3.2.1. The remainder of section 3.2 describes three treat-
ments of NsrR and of its interaction with the two hmp promoters in Streptomyces: a simple
monomer model (section 3.2.2), and two slightly different models for the interaction of
the dimer with the promoters (sections 3.2.3 and 3.2.4). A full set of parameter estimates
is provided in section 3.2.5. Because of their complexity, the differential equations for
the dimer models were generated using the rule-based modeling software BioNetGen [33].
Section 3.4 presents the results of this study. Specifically, in section 3.4.1, we study the
three models with our best estimates of the parameters. Under these conditions, the three
models behave essentially identically. Then in section 3.4.2, we deliberately study parame-
ter sets for which differences are observed between the monomer and dimer models in order
to illuminate the conditions that will make it necessary to consider dimerization in models
with similar features to ours. Finally, in section 3.5, we offer some concluding thoughts.
3.2 Description of the models
3.2.1 Hmp kinetics
We begin with a description of Hmp kinetics, which is common to all the models studied
in this contribution. The core of this submodel is based on one previously described for the
synthesis of Hmp in E. coli under the control of FNR [18].
We consider a cell bathed in an external pool of NO, whose concentration is assumed
constant. The following model reaction represents the permeation of NO through the cell
membrane:
NO −−
km
[ext] ↽⇀− NO (3.1)
km
where NO without any subscripts indicates nitric oxide inside the cell, while NO[ext] repre-
sents the constant, external pool of NO.
In Streptomyces, NO can be produced internally through a series of conversions from
16
3.2. DESCRIPTION OF THE MODELS
nitrate [42]:
NO−3 → NO
−
2 → NO. (3.2)
Thus, we should also consider endogenous generation of NO. For simplicity, we assume
this occurs at constant rate:
k
0/ −→p NO. (3.3)
This is equivalent to assuming that the internal pools of nitrate and nitrite are roughly con-
stant.
Nitric oxide can be converted back to nitrate by the action of Hmp [17]. The reaction
catalyzed by Hmp is
Hmp
2NO+2O2 +NAD(P)H−−→ 2NO−+NAD(P)++H+3 . (3.4)
The catalyzed reaction proceeds by an ordered mechanism where, for catalysis to occur,
NAD(P)H binds in the active site of the enzyme first, then O2, and then NO [43]. If NO
binds to the enzyme before oxygen, the result is a non-productive complex which is not
able to carry out catalysis. Before writing down our model, we consider the cellular con-
centrations of NADH and NADPH.
According to Kumar and Bruheim [44], metabolite concentrations ranges for NADH
and NADPH are 0.004–0.356 nmol/mg DCW (dry cell weight) and 0.016–0.203 nmol/mg DCW
respectively. The Michaelis constants of Hmp for NADH and NADPH are 3.2µM and
180µM respectively [43]. Owing to the discrepancy in units, for comparing the cellular
concentrations of NADH and NADPH with their Michaelis constants, a relationship be-
tween dry cell weight and volume for Streptomyces is needed.
Streptomyces grows by developing hyphae which are subdivided into compartments that
contain many copies of its single linear chromosome [45, 46, 47]. For simplicity, we assume
that the volume occupied per chromosome is roughly equivalent to the volume of a sphere
of the same diameter as a hypha. We treat this volume as that of a ‘cell’ in the calculations
17
3.2. DESCRIPTION OF THE MODELS
below. Streptomyces have typical diameters in the range of 0.5–1 µm [48]. The median
radius is therefore approximately 0.4µm, from which we calculate a volume of 0.27µm3.
Loferer-Krössbacher et al determined a general relationship between dry cell weight
and volume for bacteria, namely DCW = 435V 0.86, where DCW is in fg, and V is in µm3
[49]. The dry weight of a typical cell is then, from Loferer-Krössbacher’s formula, 140 fg,
from which we calculate a dry weight to volume ratio of 523fg/µm3 or 0.523g/cm3. Thus,
the range of cellular concentrations of NADH and NADPH are estimated to be 2–186 µM
and 8–106 µM. The smallest values for NADH are only observed when the cells are grown
in phosphate-limited batch culture for a lengthy period. Otherwise, NADH is found in
Streptomyces cells in concentrations above 46µM.
Since Hmp can use NADH and NADPH interchangeably and by comparing the Michaelis
constants with the estimated range of cellular concentrations of NADH and NADPH, we
would conclude that the enzyme will be saturated with one of these two substrates at all
times provided they are not starved of phosphate. Assuming that phosphate levels are suf-
ficient to sustain NADH levels much greater than the Michaelis constant of Hmp for this
cofactor, and given that the rate constant for NADH binding to Hmp is substantial [43],
the saturating concentration of NADH implies that an NAD(P)H molecule should bind to
the enzyme roughly as fast as the active site is cleared. We therefore do not include the
NAD(P)H binding step in the model below.
The key reactions of the catalytic mechanism are then
k
Hmp O −−1+ 2 ↽⇀− Hmp ·O2, (3.5a)
k−1
k
Hmp ·O NO−−22 + ↽⇀− Hmp ·O2 ·NO−
k−−→3 Hmp+NO−
k 3
, (3.5b)
−2
k
Hmp+NO− 4↽−⇀− Hmp ·NO. (3.5c)
k−4
Compare reactions (3.5a) and (3.5c), and note that the latter reaction represents an ex-
ample of competitive substrate inhibition by NO. Substrate inhibition often generates bista-
18
3.2. DESCRIPTION OF THE MODELS
bility [50, 51, 52, 53].
Because we want to focus on the effects of dimerization of NsrR, for simplicity, we
ignore transcriptional and translational delays, as well as clearance delays [54], although
delays can have significant effects on the behavior of a model. An ordinary differential
equation model with NsrR monomers will act as the control to which dimer models will be
compared.
Since Streptomyces has two hmp genes, hmpA1 and hmpA2, we need transcription re-
actions for both. In principle, each transcription process could have different rate constants
since their promoters are different [23]. However, since there are no measurements avail-
able of the rates of transcription at these two promoters, we assume that the rate constants
are identical. Thus we write
k
ProhmpA1 −→d ProhmpA1 +RBShmpA1, (3.6a)
ProhmpA2 −k→d ProhmpA2 +RBShmpA2, (3.6b)
where RBShmpA1 and RBShmpA2 represent the ribosome binding sites of the respective
mRNAs. We focus on the RBS rather than on the full mRNA transcript because, in prokary-
otic cells, translation can be initiated as soon as the RBS becomes available. We treat tran-
scription as a pseudo-first-order process assuming that bacterial cells have a constant level
of free RNA polymerase.
In theory, the two hmp genes generate two distinct mRNAs, which could also have
distinct transcription kinetics. However, for lack of data on the transcription rates of these
mRNAs, we assume in fact that the two mRNAs are identical, and henceforth simply write
RBShmp. Again, assuming a constant concentration of free ribosomes, mRNA translation
can be modeled by the following reaction:
RBShmp −→ke RBShmp +Hmp. (3.7)
19
3.2. DESCRIPTION OF THE MODELS
We also need to consider the decay of Hmp and of its mRNA. For the mRNA:
RBShmp −k→5 0/ . (3.8)
In the case of Hmp, given that the function of this enzyme requires it to be in intimate
contact with highly reactive substances, we would expect it to have some rate of breakdown.
Indeed, in Salmonella, Hmp has a half-life of 66 minutes [55]. We assume that when Hmp
breaks down, any substrates to which it was bound are released. Of course, if one of these
substrates has reacted with the protein, that may not be correct. However, the behavior
of the model is insensitive to this detail. Moreover, lacking data to the contrary, we also
assume that binding substrates does not affect the rate of degradation.
k
Hmp−→6 0/ , (3.9a)
Hmp · −kO →62 O2 +0/ , (3.9b)
Hmp ·O2 · −
k
NO →6 O2 +NO+0/ , (3.9c)
Hmp · kNO−→6 NO+0/ . (3.9d)
3.2.2 Monomeric model of NsrR dynamics (MM model)
Both the FNR [56] and NsrR [39] holo-protein monomers contain an iron-sulfur cluster,
specifically a [4Fe—4S] cluster. This iron-sulfur cluster is essential for binding of these
transcription factors to specific sequences in the promoter regions of the genes they regulate.
These proteins sense nitric oxide due to the reactions of NO with their iron-sulfur clusters.
The accumulation of these reactions eventually results in the loss of ability of these proteins
to bind their respective promoter sequences.
In a previous model of the control of Hmp synthesis by FNR in E. coli, the dimeric
nature of the FNR transcription factor was neglected [18]. Two reasons were given for
treating FNR as a monomer: First, reaction of the iron-sulfur clusters in the two monomers
20
3.2. DESCRIPTION OF THE MODELS
appears to be independent [57], i.e. there is no detectable cooperativity or anti-cooperativity.
Secondly, the formation of dimers under cellular conditions is rapid and nearly quantitative
[56]. An additional kinetic factor in favor of a monomer model not discussed in this earlier
contribution is that there is a reasonable time-scale separation between the nitrosylation
steps, with early nitrosylation steps being substantially faster than later ones [57]. NsrR
similarly forms dimers that are likely the dominant species in vivo [58], and the nitrosylation
of its iron-sulfur cluster displays significant time-scale separation [23]. It may be hoped
that a monomer model, which will be the baseline model in this study, can capture the key
dynamics of this system, perhaps even quantitatively.
NsrR almost universally acts as a repressor [39]. In Streptomyces, NsrR controls just
three genes, the two copies of the hmp gene and nsrR itself. The latter is almost certainly a
homeostatic mechanism, which we set aside in this study in order to focus on the main feed-
back loop between iron-sulfur cluster damage by NO, expression of Hmp, and removal of
NO by Hmp. The two hmp promoters do not respond identically to nitrosylation of NsrR’s
iron-sulfur cluster. Crack and coworkers found that, for the hmpA2 promoter, binding was
abolished after reaction of two molecules of NO per cluster, while four molecules of NO
were required to abolish binding at the hmpA1 promoter [23]. Interestingly, NsrR continues
to bind to the nsrR promoter until eight equivalents of NO have reacted with the repressor’s
iron-sulfur cluster. This lesser sensitivity to NO at the nsrR promoter presumably curbs the
synthesis of a repressor that would inhibit the synthesis of Hmp while the cell is coping
with elevated levels of NO. This regulatory arrangement further supports treating the total
cellular concentration of NsrR as a constant in this model.
The iron-sulfur cluster of NsrR reacts with up to eight NO molecules [23]. Only five
of the nitrosylation steps are kinetically distinguishable, but each of these steps displays
first-order kinetics with respect to NO. The latter piece of data suggests that there are in
fact eight distinct nitrosylation steps. Our model includes only the five kinetically distin-
guishable steps. We have experimented with variations in which we insert the “missing”
21
3.2. DESCRIPTION OF THE MODELS
NO molecules into the stoichiometries of some of the reactions, but this had a completely
negligible effect on our results. Moreover, there was no easy way to implement a similar
“fix” in the dimer models described in sections 3.2.3 and 3.2.4. In the interests of maximiz-
ing the consistency between models, we therefore left out these rapidly-added nitric oxide
molecules. The only remaining puzzle was then the assignment of the remaining states to
the two- and four-NO reaction products, which we return to shortly. The modeled reactions
are thus
k
NsrR+NO−−A→B NsrRB, (3.10a)
k
NsrRB +NO−−B→C NsrRC, (3.10b)
−kNsrR +NO −CDC → NsrRD, (3.10c)
NsrR kD +NO−−D→E NsrRE, (3.10d)
NsrR +NO−k−EE →F NsrRF. (3.10e)
In these equations, NsrR is the holo-protein with an intact iron-sulfur cluster, while NsrB,C,D,E,F
represent nitrosylation products of NsrR.
Because of the kinetically invisible steps, we have no way of knowing which of the
NsrR nitrosylation products explicitly represented corresponds to which nitrosylation state
of NsrR. Time-scale separation between these steps is least distinct between steps (3.10c)
and (3.10d), and between steps (3.10b) and (3.10c) [23]. It is tempting to speculate that this
made the identification of other kinetic steps with similar kinetics impossible, i.e. that the
kinetically invisible steps are, for lack of a better term “hiding” in a part of the mechanism
where time-scale separation is less clear. Step (3.10a) would then represent a single nitrosy-
lation step. NsrRC could then be the 2-NO intermediate and NsrRD the 4-NO intermediate.
We did test other possibilities, but again, they made no difference to the modeling results
presented below.
Binding of NsrR to the hmpA1 and hmpA2 promoters is modeled by the following reac-
22
3.2. DESCRIPTION OF THE MODELS
tions:
k
ProhmpA1 +NsrR− A hmpA1↽−⇀− Pro ·NsrR, (3.11a)
k−A
k
ProhmpA1 +NsrR −−B⇀− ProhmpA1B ↽ ·NsrRB, (3.11b)
k−B
k
ProhmpA1 +NsrR −−C⇀− ProhmpA1C ↽ ·NsrRC, (3.11c)
k−C
ProhmpA2 −−
k
NsrR A+ ↽⇀− ProhmpA2 ·NsrR, (3.11d)
k−A
k
ProhmpA2 +NsrR −−B⇀− ProhmpA2B ↽ ·NsrRB. (3.11e)
k−B
In accordance with the foregoing discussion, these reactions allow binding of NsrR to the
hmpA1 promoter until four equivalents of NO have reacted with NsrR’s iron-sulfur cluster
(the NsrRD state), and to the hmpA2 promoter until two equivalents of NO have been added
(the NsrRC state).
NO can react with NsrR even when it is complexed with promoter [23]. We assume
similar kinetics as for reactions of NO with free NsrR.
ProhmpA1 · kNsrR+NO−−A→B ProhmpA1 ·NsrRB, (3.12a)
ProhmpA1 · kNsrR +NO−−B→C ProhmpA1B ·NsrRC, (3.12b)
k
ProhmpA1 ·NsrR +NO−−C→D ProhmpA1C +NsrRD, (3.12c)
ProhmpA2 · kNsrR+NO−−A→B ProhmpA2 ·NsrRB, (3.12d)
ProhmpA2 · kNsrRB +NO−−B→C ProhmpA2 +NsrRC. (3.12e)
Note the dissociation of NsrR from the promoters in reactions (3.12c) and (3.12e).
Overton et al showed that many bacteria have enzymes that restore the iron-sulfur clus-
ters in NsrR, among other iron-sulfur proteins [59]. Also, they showed that Streptomyces
has such an enzyme. We therefore include a recycling reaction for NsrR species with dam-
aged iron-sulfur clusters. Since little is known about the kinetics of these recycling pro-
23
3.2. DESCRIPTION OF THE MODELS
cesses, we treat these reactions as simple first-order processes with a common rate constant:
NsrR krγ −→ NsrR, (3.13)
with γ ∈ {B,C,D,E,F}. Including a recycling pathway will allow us to treat the total con-
centration of NsrR as a constant.
The monomer model (MM) consists of reactions (3.1), (3.3) and (3.5)–(3.13). Under
the assumptions already presented, this model admits three conservation relations:
[NsrR]0 = [NsrR]+ [NsrRB]+ [NsrRC]+ [NsrRD]
+ [NsrRE]+ [NsrRF]+ [ProhmpA1 ·NsrR]
(3.14a)
+[ProhmpA1 ·NsrR ]+ [ProhmpA1B ·NsrRC]
+ [ProhmpA2 ·NsrR]+ [ProhmpA2 ·NsrRB],
[ProhmpA1] = [ProhmpA10 ]+ [ProhmpA1 ·NsrR]
+ [ProhmpA1 ·NsrRB] (3.14b)
+[ProhmpA1 ·NsrRC],
[ProhmpA2]0 = [ProhmpA2]+ [ProhmpA2 ·NsrR]
(3.14c)
+[ProhmpA2 ·NsrRB].
These conservation relations can be used to eliminate, respectively, [NsrR], [ProhmpA1] and
[ProhmpA2].
3.2.3 Dimer model 1 (DM1)
As mentioned previously, NsrR exists as a dimer in vivo. Since the ability of NsrR
to bind to promoters depends on the nitrosylation of NsrR, we need to consider the nitro-
sylation states of both monomers within a dimer. This leads to a combinatorial problem:
with six possible nitrosylation states (NsrR, NsrRB, . . . NsrRF), there are 36 possible dimer
24
3.2. DESCRIPTION OF THE MODELS
states. Trying to write down reactions or differential equations for all of these states as
well as the possible complexes of the various NsrR dimers with the two promoters con-
sidered in this study would be extremely error-prone. To solve this problem, we turn to
rule-based modeling [60]. Rule-based modeling generally uses a formal pattern-matching
language to describe the rules by which molecules of certain classes interact. In this work,
we used BioNetGen and its rule-based language BNGL [33]. In this section, we build dimer
counterparts to the reactions described in section 3.2.2.
In FNR, nitrosylation reactions at the two iron-sulfur clusters proceed independently
[57]. We assume the same to be true for NsrR. Then the reactions of NsrR dimers with NO
can be represented as follows:
NsrR ·NsrRγ +NO−
k−A→B NsrRB ·NsrRγ, (3.15a)
k
NsrRB ·NsrR BCγ +NO−−→ NsrRC ·NsrRγ, (3.15b)
NsrR ·NsrR +NO−k−CDC γ → NsrRD ·NsrRγ, (3.15c)
NsrRD ·NsrR +NO−
k−DEγ → NsrRE ·NsrRγ, (3.15d)
NsrR ·NsrR NO−k+ −EE γ →F NsrRF ·NsrRγ. (3.15e)
where NsrRγ can be any possible state of an NsrR monomer including NsrR, NsrRB, NsrRC,
NsrRD, NsrRE and NsrRF.
Modeling the binding of NsrR dimers to the promoters requires a decision about the
rule obeyed by the dimers. Available experimental data tells us at what nitrosylation states
binding is lost to different promoters, but it does not tell us how mixed states behave. We
implemented promoter binding rules in the simplest possible way in BioNetGen as follows:
ProhmpA1
kA
+NsrR ·NsrR −−⇀− ProhmpA1γ ↽ ·(NsrR ·NsrRγ), (3.16a)
k−A
ProhmpA1
k
+NsrRB ·NsrR −−
B⇀− ProhmpA1γ ↽ ·(NsrRB ·NsrRγ), (3.16b)
k−B
25
3.2. DESCRIPTION OF THE MODELS
ProhmpA1
k
+NsrRC ·NsrR −−
C⇀− ProhmpA1γ ↽ ·(NsrRC ·NsrRγ), (3.16c)
k−C
ProhmpA2
k
+NsrR ·NsrR − A hmpA2γ ↽−⇀− Pro ·(NsrR ·NsrRγ), (3.16d)
k−A
k
ProhmpA2 +NsrRB ·NsrR
B
γ ↽−−⇀− ProhmpA2 ·(NsrRB ·NsrRγ). (3.16e)
k−B
where, again, NsrRγ represents an arbitrary monomer state.
Because of the pattern-matching syntax of BNGL, NsrRB ·NsrRC, for example, will
match the rules corresponding to both of the reactions (3.16b) and (3.16c), as well as re-
action (3.16e). This has several important implications. First, binding of NsrRB ·NsrRC to
the hmpA1 promoter will have an effective dissociation constant
( )
K ProhmpA1
k + k
d · (NsrR
−B −C
B ·NsrRC) = .kB + kC
This can be seen by writing down the mass-action terms corresponding to the two match-
ing binding processes, and then considering the implied equilibrium relationship. In other
words, the dissociation constant for NsrRB ·NsrRC at the hmpA1 promoter will be a com-
promise between the monomeric dissociation constants. In fact, it is easy to show that
k ( )−B
< K ProhmpA1d ·
k
(NsrRB ·NsrR −CC) < .kB kC
This is perhaps not an unreasonable assumption given our lack of knowledge of the ef-
fects of mixed nitrosylation states on the binding. However, kinetically, the binding and
dissociation reactions will be faster than in the monomer model because they will proceed
with effective rate constants that are sums of microscopic rate constants for the two match-
ing reactions. This is unrealistic, although irrelevant in this study since we focus on the
steady-state structure of the model.
A slightly more serious consequence is that any dimer NsrRB ·NsrRγ with γ ≥ B will
bind to the hmpA2 promoter with the same affinity. While this is not impossible, a plausible
26
3.2. DESCRIPTION OF THE MODELS
guess would be that higher nitrosylation states γ weaken the ProhmpA2 · (NsrB ·NsrRγ) com-
plex. Still, lacking data on the binding of dimers in mixed nitrosylation states, the simple
BNGL rule used in this study is as good a hypothesis as any.
As in the monomer model, we consider reaction of NO with NsrR complexed to DNA:
ProhmpA1 · k(NsrR ·NsrRγ)+NO−−A→B ProhmpA1 ·(NsrRB ·NsrRγ), (3.17a)
k
ProhmpA1 ·(NsrRB ·NsrRγ)+NO−−B→C ProhmpA1 ·(NsrRC ·NsrRγ), (3.17b)
k
ProhmpA1 ·(NsrRC ·NsrRγ)+NO−−C→D ProhmpA1 +NsrRD ·NsrRγ, (3.17c)
ProhmpA2 ·(NsrR ·NsrRγ)+NO−
k−A→B ProhmpA2 ·(NsrRB ·NsrRγ), (3.17d)
ProhmpA2 ·(NsrRB ·NsrRγ)+NO−
k−B→C ProhmpA1 +NsrRC ·NsrRγ.. (3.17e)
Again, the pattern-matching BNGL language has some interesting consequences. For ex-
ample, reaction (3.17c) causes the immediate dissociation from the hmpA1 promoter of an
NsrRD ·NsrRγ dimer formed on that promoter. However, if γ ≤ C, then this dissociative
process will be counteracted to some extent by binding via reactions (3.16a) to (3.16c).
Finally, the NsrR recycling pathway is described as follows, assuming that the iron-
sulfur cluster repair enzyme operates on one monomer at a time:
NsrRB ·NsrR −→
kr
γ NsrR ·NsrRγ, (3.18a)
NsrRC ·NsrRγ −→
kr NsrR ·NsrRγ, (3.18b)
NsrRD ·NsrR −→
kr
γ NsrR ·NsrRγ, (3.18c)
NsrRE ·NsrR
kr
γ −→ NsrR ·NsrRγ, (3.18d)
NsrRF ·NsrRγ −→
kr NsrR ·NsrRγ. (3.18e)
with γ≥ B.
Model DM1 includes reactions (3.1), (3.3), (3.5)–(3.9) and (3.15)–(3.18). The final
model consists of 168 reactions linking 57 species (not counting the sink species NO−3 ).
27
3.2. DESCRIPTION OF THE MODELS
Like model MM, DM1 is subject to conservation relations for the total concentrations of
NsrR and of the two promoters. These conservation relations are automatically detected by
BioNetGen and used to eliminate three differential equations.
3.2.4 Dimer model 2 (DM2)
As discussed above, DM1 has some odd and, arguably, unphysical characteristics in
terms of allowing binding of complexes in which one NsrR monomer is in a high nitrosyla-
tion state. Because of the limited programmability of the BNGL pattern-matching language,
one alternative is to explicitly list cases. This is tedious and loses some of the advantages
of rule-based modeling, but it does allow us to implement a different logic for binding of
NsrR dimers to the promoters.
One of many possible alternatives is to have the binding strength of an NsrR dimer to
the promoters be determined by the ‘most damaged’ cluster. Fortunately in this particular
instance, only a limited number of dimers are able to bind to the promoters under this rule.
The relevant reactions are
ProhmpA1
k
+NsrR ·NsrR−−A⇀− ProhmpA1↽ ·(NsrR ·NsrR), (3.19a)
k−A
ProhmpA1
k
+NsrRB ·NsrR −
B hmpA1
B ↽−⇀− Pro ·(NsrRB ·NsrRB), (3.19b)
k−B
k
ProhmpA1 +NsrR ·NsrR−−B⇀− ProhmpA1B ↽ ·(NsrRB ·NsrR), (3.19c)
k−B
ProhmpA1
k
+NsrRC ·NsrR −−
C⇀− ProhmpA1C ↽ ·(NsrRC ·NsrRC), (3.19d)
k−C
k
ProhmpA1 +NsrRC ·NsrR
C
B −↽−⇀− ProhmpA1 ·(NsrRC ·NsrRB), (3.19e)
k−C
ProhmpA1
k
+NsrR ·NsrR−−CC ↽⇀− ProhmpA1 ·(NsrRC ·NsrR), (3.19f)
k−C
ProhmpA2
k
+NsrR ·NsrR− A hmpA2↽−⇀− Pro ·(NsrR ·NsrR), (3.19g)
k−A
ProhmpA2
k
+NsrR ·NsrR − B hmpA2B B ↽−⇀− Pro ·(NsrRB ·NsrRB), (3.19h)
k−B
ProhmpA2
k
+NsrR BB ·NsrR−↽−⇀− ProhmpA2 ·(NsrRB ·NsrR). (3.19i)
k−B
28
3.2. DESCRIPTION OF THE MODELS
Model DM2 consists of reactions (3.1), (3.3), (3.5)–(3.9), (3.15) and (3.17)–(3.19) rep-
resenting 109 distinct reactions linking 40 chemical species (NO−3 excepted).
3.2.5 Parameter estimates
We now turn to the problem of estimating the parameters of the model.
To calculate [ProhmpA1] or [ProhmpA20 ]0, we consider that the compartments from the
subdivision of Streptomyces during growth will typically contain many copies of its single
linear chromosome per cell. We assume that the volume occupied per chromosome is
roughly equivalent to the volume of a sphere with the same diameter as a hypha in the
calculation below, as in section 3.2.1. Assuming once again a typical radius of 0.4µm, a
straightforward calculation then gives a promoter concentration of 6×10−9 mol/L. Some
simulations not reported here show that the results are not very sensitive to this parameter,
presumably because of the control of the Hmp concentration by NO.
We note in passing that the small population of any given gene in a cell means that the
model will only describe the behavior averaged over a large population [61]. Given the
objective of this study, this should not be a significant limitation.
To our knowledge, there have been no measurements of the concentration of NsrR in
vivo. We assume, arbitrarily, that the concentration of NsrR in Streptomyces is similar to
the concentration of FNR in E. coli [18].
The rate constant km represents permeation of NO through the cell membrane. We as-
sume that the cell is surrounded by an aqueous medium. This may not represent the situa-
tion for a Streptomyces cell in its natural environment, soil, but it would describe laboratory
experiments in which, for example, Streptomyces is grown in a chemostat. (Streptomyces
grows well in chemostat culture, although some precautions must be taken to avoid the
formation of pellets and biofilms [62].)
The rate of change of the concentration of NO inside the cell due to permeation is [63]
AP
vperm = ([NO]V ext
− [NO]) , (3.20)
29
3.2. DESCRIPTION OF THE MODELS
where P is the permeability coefficient of the cell for NO, A is the surface area of the cell
and V is its volume. By comparing this equation with the permeation process (3.1), we see
that
km = AP/V. (3.21)
It is difficult to find permeability coefficients for NO through biological membranes or
artificial analogs thereof. The octanol-water partition coefficient of NO suggests poor per-
meation of membranes given NO’s small size and low polarity [64]. On this basis, one
could reasonably guess that the permeability coefficient of NO might be somewhere be-
tween those of water and carbon dioxide, i.e. in the range 3.4× 10−3–3.5× 10−1 cms−1
[65]. As a convenience, because our immediate objective is to study the behavior of models
and not to draw conclusions about the detoxification properties of the system, we adopt a
value towards the bottom of this range of 10−2 cms−1. Values of P in this range place the
system in a regime where bistability is observed over a substantial range of kp and [NO]ext
values, as we will see later. Again using a typical cell radius of 0.4µm, km then works out
to 750s−1.
External NO concentrations encountered by bacteria in various environments can occa-
sionally be as high as 10−6 M [66], setting an upper bound for values of [NO]ext. Similarly,
maximum intracellular NO concentrations of 10−7 M will provide a basis for interpreting
some of the results as well as constraining possible values of kp.
For Streptomyces in the wild, the maximum oxygen pressure the cells are likely to
experience is sea-level air, with pO2 = 0.21atm. The Henry’s law constant of oxygen in
water at 25◦C is 1.3× 10−8 molL−1Pa−1 [67]. Assuming a laboratory experiment at a
controlled temperature of 25◦C, the maximum possible intracellular oxygen concentration
is therefore approximately 277µM. Streptomyces’ aerobic metabolism will decrease the
intracellular concentration of O2, and in its natural habitat, there is a further decrease in the
availability of oxygen with depth in the soil [68]. Accordingly, we use a somewhat lower
O2 concentration of 200µM for most of our calculations.
30
3.3. METHODS AND SOFTWARE
We assume that the half-life of Hmp sets the rate constant k6. This will be roughly
correct provided cells grow slowly, which they might if they are subjected to prolonged
nitrosylation stress. The half-life of Hmp in Salmonella is 66 minutes [55]. Lacking data
specific to Streptomyces, we calculate k6 using this value.
Isabella et al have measured dissociation constants for NsrR with several promoters,
obtaining values in the low-nM range [69]. If we assume Kd ≈ 35nM (the NsrR-nsrR disso-
ciation constant measured by Isabella) and a moderate binding constant k 7 −1 −1A = 10 M s ,
then from K(A)d = k−A/kA, we get k−A = 0.35s. A progressive loosening of the NsrR-
promoter complexes as NO reacts with NsrR’s iron-sulfur cluster would lead to increasing
values of Kd . However, the results reported by Crack et al do not suggest a large drop in
binding affinity prior to each promoter passing the threshold where it completely loses its
ability to bind NsrR. We therefore assume a modest drop in affinity prior to the sharp cutoff
indicated by [23]. Table 3.1 summarizes the baseline set of parameters used in this study.
For lack of appropriate data for NsrR in Streptomyces, parameters relating to transcription
and translation rates were reused from the study of Roussel on the FNR control system in
E. coli [18].
3.3 Methods and software
As previously mentioned, we built the dimer models in BioNetGen 2.5.1 [33]. These
models were converted to SBML, imported into Copasi 4.37 [36], and then exported to
XPPAUT input files. While BioNetGen has a primitive “bifurcate” command that finds
attractors based on forward time integration, the version of AUTO built into XPPAUT 8.0
is able to generate true bifurcation diagrams based on standard continuation methods [34].
The XPPAUT input files generated by this process require minimal editing, namely the
removal of the differential equation for [NO−3 ] which accumulates without bound since we
have not provided a sink for this species. As there is no dynamical role for this species,
removing it does not change the dynamics of the model.
31
3.4. RESULTS
Table 3.1: Default model parameters. Notes: [c] value calculated in the text; [a] assumed
value; [s] value from a different species used for lack of Streptomyces data.
Parameter Value Notes Parameter Value Notes
Hmp dynamics NsrR dynamics
k1 3.5 µM−1s−1 [43][s] kAB 4.52 µM−1s−1 [23]
k−1 0.44 s−1 [43][s] kBC 0.113 µM−1s−1 [23]
k 850 µM−1s−12 [43][s] kCD 1.34×10−2 µM−1s−1 [23]
k−2 2×10−4 s−1 [43][s] kDE 7.48×10−3 µM−1s−1 [23]
k−3 200 s−1 [43][s] k 0.43×10−3 µM−1s−1EF [23]
k4 26 µM−1s−1 [43][s] k 10 µM−1A s−1 [a]
k 2×10−4 s−1−4 [43][s] kB 10 µM−1s−1 [a]
k 0.014 s−1 [70][s] k 10 µM−1s−15 C [a]
k6 1.8×10−4 s−1 [55][s] k 0.35 s−1−A [c]
[ProhmpA1]0 6×10−3 µM [c] k−B 0.4 s−1 [c]
[ProhmpA2] 6×10−30 µM [c] k−C 0.45 s−1 [c]
[NsrR]0 2.5 µM [a] k 0.1 s−1r [a]
[O2] 200 µM [a] NO transport
k 0.71 s−1d [18][s] km 750 s−1 [c]
ke 0.83 s−1 [18][s]
Because of its relative simplicity, we originally created an XPPAUT input file directly
for the monomer model. To verify our BioNetGen modeling, we also generated a BNGL
file describing the monomer model, and verified that identical results were obtained after
translating it into an XPPAUT input file as were obtained with our hand-written input file.
BioNetGen has a useful $ syntax to fix the concentrations of desired variables. This
allowed us to study the effect of intracellular [NO] on the concentrations of the various
species (Figs. 3.4 and 3.9). In these calculations, we scan [NO] (via its initial concentration,
which appears as a parameter in the BNGL script) and BioNetGen computes (by forward
integration) the steady-state concentrations of dimers at a set of fixed NO concentrations.
3.4 Results
In this paper, we investigate the dynamical behavior of MM, DM1 and DM2, which
differ mainly in their treatment of the rules by which NsrR binds to the hmp promoters.
32
3.4. RESULTS
Bifurcation diagrams are sensitive probes of dynamics, and since models of this type fre-
quently have saddle-node bifurcations [18], we carried out a set of bifurcation studies for
these models, reported below. Since kp represents endogenous production of NO, it acts
as a natural control parameter reflecting changing conditions within a cell. Alternatively,
[NO]ext could be varied.
3.4.1 Dynamical behavior of the models at the default parameters
We started by computing a bifurcation diagram of the monomer model (MM) at the
default parameters. This bifurcation diagram is shown in Fig. 3.1(a). We found a range of
kp values where we observe bistability, i.e. two stable states, delimited by a pair of saddle-
node bifurcations (the turning points where a branch of stable steady states meets a branch
of unstable steady states) [18, 71].
We note in passing that varying either kp or [NO]ext is equivalent due to the appear-
ance of these two parameters in the combination ptotal = kp + km[NO]ext in the equation for
d[NO]/dt. Thus, witness the straight-line boundaries of the bistable region shown in the
phase diagram of Fig. 3.1(b), which are lines of constant ptotal. While we focus on kp as a
control parameter in our bifurcation diagrams, similar results would therefore be obtained
using [NO]ext as the control parameter, give or take a transformation of the scale of the
parameter axis. In other words, either internal NO production (kp) or external nitric oxide
can drive this system into or out of the bistable region.
There is substantial uncertainty in the permeability rate constant, km. We therefore
generated a phase diagram in the (kp,km) parameter plane, shown in Fig. 3.2. This shows a
relatively broad range of km values at which bistability could be observed. However, based
on the discussion of section 3.2.5, km might be as high as 2.6×104 s−1. Thus, at the higher
end of the plausible range of values for this parameter, there is a single steady state.
We followed up this brief study of the bifurcation structure of model MM with a similar
study of the bifurcations of models DM1 and DM2. Figure 3.3 shows that the two dimer
33
3.4. RESULTS
Figure 3.1: Bifurcation diagram for model MM at the default parameters values. (a)
Bifurcation diagram, showing a bistable region between two saddle-node bifurcations at
[NO]ext = 0. In this and all subsequent bifurcation diagrams, solid red lines represent stable
steady states and dashed black lines represent unstable steady states. (b) Phase diagram
showing the bistable region in the (kp, [NO]ext) parameter plane. Outside of the bistable
region, the system has one stable steady state. The boundaries of the bistable region have
common slope k−1m .
34
3.4. RESULTS
Figure 3.2: Two-parameter phase diagram showing a cusp bifurcation. All parameters are
at the default values for the MM model.
models behave essentially identically to the MM model. Thus, given our best estimates of
the parameters, the simpler monomer model is a perfectly adequate predictive model for
the behavior of this entire class of models. But why would that be?
Thinking of the possible dimer states as analogous to elements of a matrix, we define
diagonal dimer states, which are the complexes including the same monomer states, i.e.
NsrRX ·NsrRX complexes. Pursuing the matrix analogy, we define sub-diagonal states as
those that lie just below the diagonal states and super-diagonal states as those just above the
diagonal. To put it another say, the sub- and super-diagonal states consist of monomers
that differ from the diagonal states by a single nitrosylation event, e.g. states such as
NsrRC ·NsrRB and NsrRB ·NsrRC. Sub- and super-diagonal complexes that differ by a
permutation of the states such as the latter two represent the same chemical species since
the two NsrR dimers are indistinguishable. We thus refer to sub- and super-diagonal states
collectively as near-diagonal states. We also define the central band to consist of states that
are either diagonal or near-diagonal.
Our dimer models were built on the assumption, supported by experimental evidence
35
3.4. RESULTS
Figure 3.3: Bifurcation diagrams for models DM1 and DM2. (a) Bifurcation diagram for
model DM1. (b) Bifurcation diagram for model DM2. All parameters are as shown in table
3.1.
36
3.4. RESULTS
[57], that nitrosylation reactions at the two iron-sulfur clusters in a dimer are indepen-
dent events. Examining the values of kAB to kEF in Table 3.1, we see that there is a rapid
decrease in the nitrosylation rate constants with increasing nitrosylation. Thus, despite
the assumption of independence, the consequent separation of time scales implies that we
should mostly see an orderly ‘walk’ of individual dimers from the diagonal to the next near-
diagonal state, then to the next diagonal state, with states farther from the diagonal rarely
visited.
Figure 3.4(a) shows the concentrations of a ‘family’ of NsrR dimers in model DM1,
in this case the NsrRC ·NsrRX family, as we vary the intracellular NO concentration as
described in the Methods section. As expected, the highest dimer concentrations are seen
in the diagonal and near-diagonal states, NsrRC ·NsrRB, NsrRC ·NsrRC and NsrRC ·NsrRD.
NsrRC ·NsrRE, which is not in the central band, grows to be almost as large as NsrRC ·NsrRD
because of the relatively small ratio of kCD to kDE (less than 2) than for any other adja-
cent pair of nitrosylation rate constants in Table 3.1, the next smallest ratio being greater
than 8. Thus, for families that include NsrRX ·NsrRD among the near-diagonal states, the
NsrRX ·NsrRE complex may reach relatively high concentrations even when it is not in the
central band, but otherwise the diagonal and near-diagonal complexes dominate the dynam-
ics. Moreover, in DM1, NsrRX ·NsrRE dimers with X > C do not bind any promoter, and
in DM2, no NsrRX ·NsrRE dimer is able to bind a promoter, so the concentrations of these
dimers are less significant to the overall dynamics of the model than would be the case for
dimers that are directly involved in the control of hmp expression.
Other NsrR dimer families behave similarly to the NsrRC family, with the caveat noted
above about the D and E states.
Figure 3.4(b) shows the concentrations of the diagonal dimers as a function of the NO
concentration. Because of the separation of time scales implied by the nitrosylation rate
constants, at most concentrations, a single dimer form will dominate and there are never
high concentrations of three diagonal dimers simultaneously. Thus, there is generally a
37
3.4. RESULTS
Figure 3.4: NsrR family dimer concentrations vs [NO] at the default parameters for DM1.
(a) NsrRC family dimer concentrations vs [NO] at the default parameters. The concentration
of NsrRC ·NsrRF reaches a maximum of 0.064µM near [NO] = 102 µM (not shown). (b)
Concentrations of diagonal dimers as a function of [NO] plotted on a logarithmic scale.
38
3.4. RESULTS
simple ordered transition from one state to the next of the dimers. NsrRD ·NsrRD is an
exception due to the small ratio of kCD to kDE noted above. This allows the formation
of NsrRE ·NsrRE via the pathway NsrRC ·NsrRD → NsrRC ·NsrRE → NsrRD ·NsrRE →
NsrRE ·NsrRE, bypassing NsrRD ·NsrRD.
DM2 behaves identically both with respect to the dominance of the central band states
and to the orderly succession of dimer states as [NO] increases. The orderly walk of dimer
states up the central band explains the sharp results obtained experimentally with respect
to the number of NO molecules required to lose binding at the hmpA1, hmpA2 and nsrR
promoters in Streptomyces [23]. Had all of kAB to kEF been similar in size, for instance,
we would not obtain the clean separation between dimer states as [NO] increased. Instead,
at any given NO concentration, a complex mixture of NsrR states would result (Fig. 3.5),
which would weaken the straightforward stoichiometric relationship between administered
NO and the population of bound promoters observed in the experiments. Interestingly
however, the bifurcation diagrams of the three models (not shown) continue to agree nearly
perfectly despite the disordered nitrosylation sequence in the dimer models. In fact, we
have tried changing the nitrosylation rate constants kAB to kEF in various ways, including
reversing their order and randomly shuffling their values, and the bifurcation diagrams of
the three models always agree closely holding the other rate constants at their default val-
ues. The selection of a monomer model vs a dimer model is therefore indifferent to the
nitrosylation kinetics.
Streptomyces are soil-dwelling bacteria, and the oxygen concentration decreases with
depth in the soil. The dependence of [O2] on soil depth depends on a number of factors,
but for reference we consider the measurements of Sheppard et al, which were obtained in
a grassland environment [68]. At an 11 cm depth, these measurements give an O2 concen-
tration of roughly 16µM. Figure 3.6 shows a bifurcation diagram obtained at this concen-
tration for model MM. There is a shift of the bistable region to lower concentrations, but
otherwise the behavior of the model is unaffected.
39
3.4. RESULTS
Figure 3.5: NsrR family dimer concentrations vs [NO] for model DM1 when kAB = kBC =
kCD = k −1 −1DE = kEF = 1µM s , other parameters being at their default values. (a) NsrRC
family dimer concentrations vs [NO]. (b) Concentrations of diagonal dimers as a function
of [NO].
40
3.4. RESULTS
Figure 3.6: Bifurcation diagram for model MM with all parameters as in Table 3.1 except
[O2] = 16µM.
3.4.2 Conditions making the models behave differently
In the previous section, we found that the three models agree in almost perfect quanti-
tative detail for our best-estimate, default parameters, as well as for a variety of different
arrangements of the nitrosylation rate constants, and for different oxygen concentrations.
But is this necessarily the case? In this section, we explore this question in order to obtain
insights that will hopefully extend to other models.
Since the differences between the models are all related to the treatment of the NsrR
dimers, and since changing the nitrosylation rate constants (holding other parameters con-
stant) maintains the equivalence between the models, the only rate constants that are directly
implicated in the dimer dynamics that remain to be investigated are the recycling rate con-
stant kr and the rate constants associated with binding of NsrR (monomers or dimers) to the
hmp promoters.
The steady states do not depend strongly on kr for values of this rate constant similar to
or larger than the default value of 0.1s−1 we have assumed. However, if recycling is slow,
the three models make distinctly different predictions, as seen in Fig. 3.7. In the case of
41
3.4. RESULTS
models MM and DM1 [panels (a) and (b)], the differences are only quantitative, with the bi-
furcation points being shifted to slightly lower values of kp in DM1 relative to MM. Unlike
the situation with the default parameters, the MM model displays a supercritical Andronov-
Hopf bifurcation [18, 71] on the upper branch of steady states, the limit-cycle (oscillatory)
regime being observed over an extremely narrow range of parameters and ending in a ho-
moclinic bifurcation [71]. An Andronov-Hopf bifurcation also occurs in the DM1 model,
but much closer to the saddle-node point. The type of Andronov-Hopf bifurcation and
the ultimate fate of the limit cycle created are difficult to ascertain due to numerical diffi-
culties in the vicinity of the Andronov-Hopf bifurcation. Nevertheless, it is clear that the
two bifurcation diagrams make similar predictions and that, if we did not already know that
NsrR binds to DNA as a dimer, an experimental bifurcation diagram varying kp (or [NO]ext)
would not be sufficient to distinguish between these models.
DM2 on the other hand behaves very differently than the other two models at kr =
0.001s−1 [Fig. 3.7(c)]. The classic S-shaped steady-state curve leading to regions of bista-
bility has been lost. Instead, the model displays large-amplitude oscillations over a wide
range of parameters. Again, the AUTO continuation computations are difficult near the
Andronov-Hopf bifurcations, so the maxima and minima of the limit cycles were calculated
using the ‘Poincaré map’ feature of XPPAUT. The emergence of large-amplitude oscilla-
tions very near the Andronov-Hopf points, as well as the shapes of the limit cycles near
these points [Fig. 3.8(a) and (d)] are suggestive of a canard explosion [72]. Moreover, the
complex shapes of the limit cycles at some values of kp are suggestive of mixed-mode os-
cillations [Fig. 3.8(c) and (d)]. We leave a deeper investigation of these dynamics for the
future.
Our immediate concern is the reason for the significant difference between DM2 and
the other two models at smaller values of kr. As [NO] increases, the hmpA2 gene is the first
to be transcribed (Fig. 3.9), as expected given that fewer equivalents of NO are required to
abolish binding of NsrR at this gene’s promoter [23]. Lowering the value of kr generally
42
3.4. RESULTS
Figure 3.7: Bifurcation diagrams for the three models at kr = 0.001 s−1. All other parame-
ters are as shown in table 3.1. Green dots show Andronov-Hopf bifurcation points and blue
dots depict maxima and minima of periodic trajectories. (a) MM. (b) DM1. (c) DM2.
43
3.4. RESULTS
Figure 3.8: Limit cycles for model DM2 at kr = 0.001 s−1. (a) kp = 85µMs−1. (b) kp =
105. (c) kp = 115. (d) kp = 165. All other parameters are at default values as shown in
table 3.1.
makes the control system more sensitive to [NO] in two ways: First, the dose-response
curves shift towards the left, i.e. a smaller concentration of NO is required to activate the
transcription of each hmp gene. Second, the dose-response curves become steeper as mea-
sured by, e.g., the slope of a graph of [Prohmp] vs [NO] at half-maximal activation. (Note
that this increase in steepness is not visually evident from Fig. 3.9 due to the logarithmic
scale of the abscissa.) The increased sensitivity to [NO] at smaller values of kr arises be-
cause slower recycling tends to result in the accumulation of higher nitrosylation states of
NsrR, which are incompetent to bind the hmp promoters (data not shown). While all three
models have the same qualitative behavior with respect to decreasing kr, the effect is much
stronger in DM2. The slopes at half-max of the dose-response curves increase by factors of
less than 20 when kr decreases from 0.1s−1 to 0.001s−1 in models MM and DM1, but by
more than 50 in DM2. In other words, the response to NO becomes much more switch-like
in DM2 with falling kr than in the other two models. The increased sensitivity to [NO], by
both measures, means that threshold-crossing events that switch the genes on and off are
44
3.4. RESULTS
Figure 3.9: Active promoter concentrations vs [NO] at k −1r = 0.1s [panels (a) to (c)] and
kr = 0.001s−1 [panels (d) to (f)] for the three models. (a), (d) MM; (b), (e) DM1; and (c),
(f) DM2. All other parameters are at the default values of table 3.1.
more likely to occur with relatively small changes in [NO]. Delays in the response due to the
biochemical reactions intervening between a change in [NO] and a change in the transcrip-
tional states of the two promoters then result in overcompensation, leading to oscillations.
Altering the rate constants for binding of the NsrR monomers or dimers at the hmp
promoters—k±A, k±B, k±C—could reasonably be expected to differentiate the models since
the main differences between them are in these binding steps. In order to answer the larger
question about conditions that might lead to monomer and dimer models giving different
45
3.5. DISCUSSION AND CONCLUSION
results, we tried different values of these rate constants, setting aside the experimental data
[69] on which we based our original parameter estimates. We explored a number of dif-
ferent combinations of rate constants, most of which resulted in bifurcation diagrams that
were negligibly different between the three models. Available experimental data suggest
a progressive weakening of the binding as additional equivalents of NO react with NsrR’s
iron-sulfur cluster [23, 73], i.e. an increasing dissociation constant Kd , and we generated
the rate constants presented in Table 3.1 on this assumption. It was only when we tried to
reverse this trend completely, i.e. to assume that the dissociation constants decrease with
increasing nitrosylation, that we started to see differences between the models. Figure 3.10
shows one example with k−A k−B k−C, keeping the other parameters at their default
values. Under these conditions, models MM and DM2 behave essentially identically, but
DM1 displays bistability over a much wider range of kp values. Similar results can be ob-
tained with k−A = k−B = k−C and kA kB kC. It is also not strictly necessary to change
all of these rate constants in order for DM1 to behave differently from MM and DM2. For
example, keeping all other parameters at their default values, reducing k−B to 0.01s−1 re-
sults in MM and DM2 having a very narrow range of kp values over which bistability is
observed (kp ≈ 9–12µMs−1), whereas bistability is observed over a much wider range for
DM1 (kp ∈ (13.6,35.9)µMs−1). The key factor seems to be to have a large decrease in the
dissociation constant from binding step A to binding step B.
3.5 Discussion and conclusion
In previous work, Roussel had argued that an adequate model for the control of Hmp
synthesis by an iron-sulfur protein could be obtained by treating the dimeric transcription
factor as a monomer [18]. In this paper, we tested this proposition by comparing models
for the control of Hmp synthesis in Streptomyces based on monomeric and dimeric NsrR
repressors. Using, insofar as it was possible, parameters obtained from the literature, we
found that at our default set of parameters, the monomer model and both dimer models give
46
3.5. DISCUSSION AND CONCLUSION
Figure 3.10: Bifurcation diagram for models MM, DM1 and DM2 when k = 1×10−1s−1−B
and k = 1×10−3s−1−C . (a) Bifurcation diagram for MM. (b) Bifurcation diagram for DM1.
(c) Bifurcation diagram for DM2. All parameters are as shown in table 3.1 except k−B and
k−C.
47
3.5. DISCUSSION AND CONCLUSION
essentially identical results provided recycling of the nitrosylated NsrR proteins to their
undamaged state is rapid. Slow recycling can produce very significant differences in the
behavior of the models, not only quantitatively, but qualitatively (Fig. 3.7).
Since we had not anticipated the importance of the repair pathway, we modeled recovery
of holo-NsrR as a simple first-order reaction. This assumption and its implications for the
adequacy of a monomer-based transcription factor model clearly needs deeper examination.
The uncertainty surrounding the actors in this pathway [74, 75] makes detailed modeling
awkward. Nevertheless, the attempt to build a more refined model of recycling would be
worthwhile given its importance in determining the behavior of this class of models.
Recycling of a transcription factor, as in the models presented here, is a conservative
process. (See, e.g., Eq. (3.14a).) A production-destruction system in which transcription
factors that have reached a certain modification state are degraded and replaced by freshly
synthesized proteins lacks a conservation relation and may behave quite differently. How-
ever, it is possible that homeostatic control of the protein level would have a somewhat
similar effect to a conservation relation so that some of the observations made in this study
could find counterparts in a production-destruction system. It would therefore be of con-
siderable interest to study the effects of dimerization or oligomerization of transcription
factors in production-destruction systems.
In addition to recycling, we found that assumptions made about the effect of differential
nitrosylation between monomers in a dimer on the binding of the transcription factor to a
promoter can have important effects on model behavior. DM1 and DM2 can be thought of
as two extremes along a continuum of possibilities. In DM1, it is sufficient for one monomer
to be in a sufficiently low nitrosylation state for an NsrR dimer to bind to a promoter. In
DM2, the highest nitrosylation state in a pair determines the binding properties. Of course,
there are many possibilities between these two extremes, and we await experimental data
that would allow us to determine how unequal nitrosylation affects binding.
Homodimeric transcription factors that are activated either by covalent modification or
48
3.5. DISCUSSION AND CONCLUSION
by binding a ligand are not at all unusual, so some of the lessons learned in this study may
be applicable to other situations. For example, Smad2 dimers, which are key players in
the transduction of TGF-β signals, only form when the individual units are phosphorylated.
Dephosphorylation results in dissociation of the dimer into its components, and recycling
of the Smad2 monomers to the cytosol [76]. Basic helix-loop-helix proteins also frequently
form homodimers whose activity is controlled by phosphorylation [77]. In these models,
we would expect that both the rate of recycling of the transcription factors and assumptions
about the effect of modifying just one monomer would be key parameters in determining
model behavior.
In every version of the model and for a wide range of rate constants, we found that
bistability was observed. This was useful in this study because it allowed us to visualize
the effects of various assumptions about NsrR and its binding properties to the hmp pro-
moters on the dynamics of the overall control system. Experimentally, bistability would
manifest itself as a hysteresis loop, i.e. different states of the cell observed depending on
whether the control parameter was being swept in an increasing or decreasing direction.
The simplest way to observe this effect, if indeed it occurs in cells, would be to use the
external NO concentration as a control parameter [Fig. 3.1(b)]. However, there is an upper
limit in the membrane permeability to NO above which bistability would not be observed
[Fig 3.2]. It is therefore an open question whether bistability could be observed in real
cells. The Streptomyces system is particularly appealing for experimental study of this and
many other questions about feedback between NO, the state of NsrR and the Hmp con-
centration because of its relative simplicity and of the small number of directly interacting
components.
We did not take into account the expression of NsrR, which is itself under the control
of the NsrR transcription factor, nor did we consider the kinetics of synthesis of NsrR
monomers, which includes loading the apo-enzyme with an iron-sulfur cluster, and of
the assembly of the monomers into dimers. Instead, we assumed that NsrR dimers (or
49
3.5. DISCUSSION AND CONCLUSION
monomers in model MM) are present at constant concentration in the cell. In some cases,
the expression of the transcription factor, its post-translational modification and its oligomer-
ization kinetics will be critical to the dynamics of the control system. In such cases, it would
not be possible to ignore the oligomers, and rule-based modeling would become a critical
tool for the study of such systems. The ongoing development of rule-based systems with
the ability to describe a larger range of relationships will therefore greatly facilitate research
involving the construction and study of models of gene expression.
We created DM2 by explicitly enumerating the states that are able to bind the promoter
and manually assigning their rate constants. We were forced to do this because BNGL lacks
a comparison operator that would allow us to express rules in terms of “the less nitrosylated
monomer”, or other equivalent formulations. Were we to add the nsrR gene to our model,
this would become a real chore given that NsrR binds to its promoter until 8 NO molecules
have reacted per monomer [23]. At that point, it would likely make sense for us to write
a program to instantiate any given set of rules rather than use BioNetGen for the purpose.
This observation highlights one important limitation of the BNGL language, which we hope
future releases will address.
50
Chapter 4
Conclusion
4.1 Summary
Holo-NsrR, containing an iron-sulfur cluster, generally acts as a repressor in a number
of genes associated with nitrogen oxide metabolism in bacteria and binds to promoters as
a dimer. Since Roussel ignored the effects of dimerization of FNR [18], studying these
effects for NsrR is dynamically interesting. We therefore built a base model where NsrR
acts as a monomer and two dimer models, DM1 and DM2, to study the effect of dimer-
ization of NsrR on Hmp gene expression. Afterwards, we compared the dynamics of the
dimer models with the monomer model. Based on the experimental data, binding of NsrR
to hmpA1 and hmpA2 is entirely abolished at 4 and 2 NO’s per cluster, respectively. Ac-
cording to these data, the kinetically distinguishable nitrosylated states of NsrR are NsrRB,
NsrRC, NsrRD, NsrRE and NsrRF. We showed that the behaviors of the monomer and
dimer models are similar at the default values of the parameters shown in table 3.1. We
also made many changes in the parameters, and generally the monomer and dimer mod-
els behaved similarly. Thus, for a wide range of parameters, binding only depends on the
properties of the monomers and not on the dimer states. Then, we found that by changing
the binding parameters of NsrR to the promoter, DM1 behaves differently than MM and
DM2. Additionally, we showed that small values of the recycling rate of NsrR results in
different behavior of DM2. In summary, changing the binding parameters or altering the
recycling rate of NsrR to the intact holo-protein makes a difference to the results, however
the recycling rate is more salient.
51
4.2. FUTURE DIRECTIONS
Depending on the depth where Streptomyces lives in the soil, the oxygen concentration
is different [68]. We showed that if Streptomyces lives near the soil surface (higher O2
concentration), the bistable region will be bigger and there is a shift of the bistable region
to higher amount of NO and conversely, if Streptomyces lives in the deeper layers of the
soil (lower O2 concentration), the bistable region gets smaller and there is a shift of the
bistable region to lower concentrations of NO. Therefore, the O2 concentration affects the
dynamical behavior of the system.
4.2 Future directions
The aim of this thesis was to study the effects of dimerization of transcription factors in
gene expression models. Through a quantitative analysis of mathematical models, several
results were obtained that can help direct future research in the field. This section provides
some key points for further research such as additional work on the Hmp control system for
pathogenic bacteria, the effects of oligomerization of transcription factors in gene expres-
sion, and considering the delays in gene transcription and translation.
We did not take into account the expression of NsrR, which is itself under the control of
the dimeric NsrR transcription factor, nor we did not consider the kinetics of association of
NsrR monomers into dimers. Instead, we assumed that NsrR dimers are present at constant
concentration in the cell. In some cases, the expression of the transcription factor and its
oligomerization kinetics will be critical to the dynamics of the control system. In such
cases, it would not be possible to ignore the oligomers, and rule-based modeling would
become a critical tool for the study of such systems. The ongoing development of rule-based
systems with the ability to describe a larger range of relationships will therefore greatly
facilitate research involving the construction and study of models of gene expression.
A consistent theme throughout the thesis work was studying the control system of Hmp.
As mentioned in the first chapter, nsrR is much less sensitive to NO, so we assumed NsrR
is constant. Allowing NsrR to be variable could be explored in future studies.
52
4.2. FUTURE DIRECTIONS
In the models presented here, the NsrR recycling pathway is described as a simple
first-order process. We found that the recycling pathway is a salient factor in determining
whether or not the models behave differently. Thus, studying he enzymic nature of the re-
pair pathway and modeling the repair process of damaged iron-sulfur clusters by an enzyme
should be of high priority for future studies.
In chapter three, the dynamical behavior of three different models were investigated.
The DM2 model behaved differently at lower kr. In DM2, the complex shapes of the limit
cycles in Fig.3.7(c) at some values of kp are suggestive of mixed-mode oscillations. Future
studies should probe the dynamical behavior around the bifurcation points in this model in
detail.
Furthermore, we ignored the transcriptional and translational delays, as well as clear-
ance delays in our study, although delays can have significant effects on the behavior of
a model [78]. Thus, new studies could examine the use of delayed mass-action kinetics
for modeling of the transcription of genes [79]. The existing computational methods and
software we applied do not handle delays, so new tools need to be developed to combine
rule-based modeling with delays for studying complex transcription factor systems with
delays.
Building on this theoretical study of the control system for Hmp in Streptomyces, the
Hmp control system in organisms of biomedical interest could be studied. Streptomyces
is rarely pathogenic [32]. Although the Hmp control system in Streptomyces differs from
those in pathogenic bacteria [27], future study would be expanded to pathogenic bacteria
such as Burkholderia pseudomallei by modifying the Hmp control system in our models. In
B. pseudomallei, NsrR regulates different genes including hmp, nnrS2, norB and nnrS1 [27]
but, as in Streptomyces, hmp is only controlled by NsrR. In other words, studying a simple
control system can be a stepping stone to studying more complex systems. So, my work on
Streptomyces will be directly relevant to those bacteria which are of medical concern and
my studies are a platform for thinking about the development of new antibiotics as Hmp is
53
4.2. FUTURE DIRECTIONS
a virulence factor [80].
Additionally, rule-based modeling can be applied for other transcription factors that act
as dimers or higher oligomers. For example, the EhPgp5 protein confers multi-drug re-
sistance in Entamoeba histolytica by altering ion conductance [81], and EhHSTF7 controls
the EhPgp5 gene. Studying EhHSTF7, which acts as an oligomer in Entamoeba [82], could
be a starting point for future studies.
54
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62
Appendix A
BioNetGen code
A.1 Monomer (MM)
begin model
begin parameters
kAB 4.52 # /micromol.s
kBC 0.113 # /micromol.s
kCD 1.34e-2 # /micromol.s
kDE 7.48e-3 # /micromol.s
kEF 0.43e-3 #/micromol.s
kr 0.009
NsrR_init 2.5 # micromolar
NO_init 10 # micromolar
NOext_init 0.1
O2_init 200
ke1 0.03
kp 0
k1 8
k_1 1.4
k2 2.5e+3
k_2 6.2e-4
k_3 6.2e+2
k4 80
k_4 6.2e-4
k5 0.014
k6 2.4e-4
ProhmpA1_init 6.22e-3
ProhmpA2_init 6.22e-3
kd 0.71
ke 0.83
kA 10
k_A 0.35
kB 10
k_B 0.4
kC 10
k_C 0.45
63
A.1. MONOMER (MM)
end parameters
begin molecule types
NsrR(a)
NsrRB(a)
NsrRC(a)
NsrRD(a)
NsrRE(a)
NsrRF(a)
NO(a)
NOext()
Hmp(n,o)
O2(a)
ProhmpA2(b)
ProhmpA1(b)
RBS()
NO3(a)
end molecule types
begin species
NsrR(a) NsrR_init
NO(a) NO_init
ProhmpA2(b) ProhmpA2_init
ProhmpA1(b) ProhmpA1_init
$O2(a) O2_init
$NOext() NOext_init
end species
begin observables
#Species ProhmpA2 ProhmpA2(b)
#Species ProhmpA1 ProhmpA1(b)
Species No NO(a)
#Species Hmp_NO Hmp(n!+)
#Species RBS RBS()
end observables
begin reaction rules
# Hmp/NO dynamics
# Permeation and intracellular production
NOext() <-> NO(a) ke1,ke1
0 -> NO(a) kp
# Enzyme kinetics
64
A.1. MONOMER (MM)
Hmp(n,o) + O2(a) <-> Hmp(n,o!1).O2(a!1) k1,k_1
Hmp(n,o!1).O2(a!1) + NO(a) <-> Hmp(n!0,o!1).O2(a!1).NO(a!0) k2,k_2
Hmp(n!0,o!1).O2(a!1).NO(a!0) -> Hmp(n,o) + NO3(a) k_3
Hmp(n,o) + NO(a) <-> Hmp(n!1,o).NO(a!1) k4,k_4
# Production/decay of mRNA (ribosome binding sites) and of Hmp
ProhmpA2(b) -> ProhmpA2(b) + RBS() kd
ProhmpA1(b) -> ProhmpA1(b) + RBS() kd
RBS() -> RBS() + Hmp(n,o) ke
RBS() -> 0 k5
# Decay of Hmp and its complexes
Hmp(n,o) -> 0 k6
Hmp(n,o!1).O2(a!1) -> O2(a) + 0 k6
Hmp(n!0,o!1).O2(a!1).NO(a!0) -> O2(a) + NO(a) k6
Hmp(n!1,o).NO(a!1) -> NO(a) k6
# Reactions of NsrR with NO
NsrR(a) + NO(a) -> NsrRB(a) kAB
NsrRB(a) + NO(a) -> NsrRC(a) kBC
NsrRC(a) + NO(a) -> NsrRD(a) kCD
NsrRD(a) + NO(a) -> NsrRE(a) kDE
NsrRE(a) + NO(a) -> NsrRF(a) kEF
# Binding equilibria
ProhmpA1(b) + NsrR(a) <-> ProhmpA1(b!1).NsrR(a!1) kA,k_A
ProhmpA1(b) + NsrRB(a) <-> ProhmpA1(b!1).NsrRB(a!1) kB,k_B
ProhmpA1(b) + NsrRC(a) <-> ProhmpA1(b!1).NsrRC(a!1) kC,k_C
ProhmpA2(b) + NsrR(a) <-> ProhmpA2(b!1).NsrR(a!1) kA,k_A
ProhmpA2(b) + NsrRB(a) <-> ProhmpA2(b!1).NsrRB(a!1) kB,k_B
# Reactions of NsrR in complex with promoters with NO
ProhmpA1(b!1).NsrR(a!1) + NO(a) -> ProhmpA1(b!1).NsrRB(a!1) kAB
ProhmpA1(b!1).NsrRB(a!1) + NO(a) -> ProhmpA1(b!1).NsrRC(a!1) kBC
ProhmpA1(b!1).NsrRC(a!1) + NO(a) -> ProhmpA1(b) + NsrRD(a) kCD
ProhmpA2(b!1).NsrR(a!1) + NO(a) -> ProhmpA2(b!1).NsrRB(a!1) kAB
ProhmpA2(b!1).NsrRB(a!1) + NO(a) -> ProhmpA2(b) + NsrRC(a) kBC
# NsrR recycling
NsrRB(a) -> NsrR(a) kr
NsrRC(a) -> NsrR(a) kr
NsrRD(a) -> NsrR(a) kr
NsrRE(a) -> NsrR(a) kr
NsrRF(a) -> NsrR(a) kr
end reaction rules
65
A.2. DIMER MODEL 1 (DM1)
end model
generate_network({overwrite=>1})
#writeSBML()
#parameter_scan({parameter=>"NO_init",par_min=>0,par_max=>100,n_scan_pts=>100,\
#method=>"ode",t_end=>10})
#Varying kp
bifurcate({parameter=>"kp",par_min=>0,par_max=>100,n_scan_pts=>200,\
method=>"ode",t_end=>1e7})
#writeLatex()
A.2 Dimer model 1 (DM1)
begin model
begin parameters
kAB 4.52 # /micromol.s
kBC 0.113 # /micromol.s
kCD 1.34e-2 # /micromol.s
kDE 7.48e-3 # /micromol.s
kEF 0.43e-3 #/micromol.s
kr 0.001
NsrR_init 2.5
logNO -4
NO_init 10ˆlogNO
NOext_init 0
O2_init 200
km 750
kp 0
k1 3.5
k_1 0.44
k2 850
k_2 2e-4
k_3 200
k4 26
k_4 2e-4
k5 0.014
k6 1.8e-4
ProhmpA1_init 6e-3
ProhmpA2_init 6e-3
kd 0.71
ke 0.83
kA 10
66
A.2. DIMER MODEL 1 (DM1)
k_A 0.35
kB 10
k_B 0.4
kC 10
k_C 0.45
end parameters
begin molecule types
NsrR(b,dim,s˜0˜1˜2˜3˜4˜5)
NO(a)
NOext()
Hmp(n,o)
O2(a)
ProhmpA2(b,b)
ProhmpA1(b,b)
RBS()
NO3(a)
end molecule types
begin species
NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜0) NsrR_init
$NO(a) NO_init
ProhmpA2(b,b) ProhmpA2_init
ProhmpA1(b,b) ProhmpA1_init
$O2(a) O2_init
$NOext() NOext_init
end species
begin observables
#Species Hmp_NO Hmp(b,c!1).NO(a!1)
#Species Hmp_O2_NO Hmp(b!0,c!1).O2(a!0).NO(a!1)
#Species Hmp_O2 Hmp(b,c!1).O2(a!1)
Species NO NO(a)
Species ProhmpA2 ProhmpA2(b)
Species ProhmpA1 ProhmpA1(b)
#Species Hmp Hmp(c,c)
#Species RBS RBS()
#Species NsrRC_NsrR NsrR(s˜2).NsrR(s˜0)
#Species NsrRC_NsrRB NsrR(s˜2).NsrR(s˜1)
#Species NsrRC_NsrRC NsrR(s˜2).NsrR(s˜2)
#Species NsrRC_NsrRD NsrR(s˜2).NsrR(s˜3)
#Species NsrRC_NsrRE NsrR(s˜2).NsrR(s˜4)
67
A.2. DIMER MODEL 1 (DM1)
#Species NsrRC_NsrRF NsrR(s˜2).NsrR(s˜5)
#Species NsrR_NsrR NsrR(b,s˜0).NsrR(b,s˜0)
#Species NsrRB_NsrRB NsrR(b,s˜1).NsrR(b,s˜1)
#Species NsrRC_NsrRC NsrR(s˜2).NsrR(s˜2)
#Species NsrRD_NsrRD NsrR(s˜3).NsrR(s˜3)
#Species NsrRE_NsrRE NsrR(s˜4).NsrR(s˜4)
#Species NsrRF_NsrRF NsrR(s˜5).NsrR(s˜5)
end observables
begin reaction rules
# Hmp/NO dynamics
# Permeation and intracellular production
NOext() <-> NO(a) km,km
0 -> NO(a) kp
# Enzyme kinetics
Hmp(n,o) + O2(a) <-> Hmp(n,o!1).O2(a!1) k1,k_1
Hmp(n,o!1).O2(a!1) + NO(a) <-> Hmp(n!0,o!1).O2(a!1).NO(a!0) k2,k_2
Hmp(n!0,o!1).O2(a!1).NO(a!0) -> Hmp(n,o) + NO3(a) k_3
Hmp(n,o) + NO(a) <-> Hmp(n!1,o).NO(a!1) k4,k_4
# Production/decay of mRNA (ribosome binding sites) and of Hmp
ProhmpA2(b,b) -> ProhmpA2(b,b) + RBS() kd
ProhmpA1(b,b) -> ProhmpA1(b,b) + RBS() kd
RBS() -> RBS() + Hmp(n,o) ke
RBS() -> 0 k5
# Decay of Hmp and its complexes
Hmp(n,o) -> 0 k6
Hmp(n,o!1).O2(a!1) -> O2(a) + 0 k6
Hmp(n!0,o!1).O2(a!1).NO(a!0) -> O2(a) + NO(a) k6
Hmp(n!1,o).NO(a!1) -> NO(a) k6
# Reactions of NsrR with NO
{MatchOnce}NsrR(b,s˜0).NsrR(b,s) + NO(a) -> NsrR(b,s˜1).NsrR(b,s) kAB
{MatchOnce}NsrR(b,s˜1).NsrR(b,s) + NO(a) -> NsrR(b,s˜2).NsrR(b,s) kBC
{MatchOnce}NsrR(b,s˜2).NsrR(b,s) + NO(a) -> NsrR(b,s˜3).NsrR(b,s) kCD
{MatchOnce}NsrR(b,s˜3).NsrR(b,s) + NO(a) -> NsrR(b,s˜4).NsrR(b,s) kDE
{MatchOnce}NsrR(b,s˜4).NsrR(b,s) + NO(a) -> NsrR(b,s˜5).NsrR(b,s) kEF
# Binding equilibria using as simple a binding rule as possible
{MatchOnce}ProhmpA1(b,b) + {MatchOnce}NsrR(b,s˜0).NsrR(b,s) <-> \
{MatchOnce}ProhmpA1(b!1,b!2).NsrR(b!1,s˜0).NsrR(b!2,s) kA,k_A
{MatchOnce}ProhmpA1(b,b) + {MatchOnce}NsrR(b,s˜1).NsrR(b,s) <-> \
68
A.3. DIMER MODEL 2 (DM2)
{MatchOnce}ProhmpA1(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) kB,k_B
{MatchOnce}ProhmpA1(b,b) + {MatchOnce}NsrR(b,s˜2).NsrR(b,s) <-> \
{MatchOnce}ProhmpA1(b!1,b!2).NsrR(b!1,s˜2).NsrR(b!2,s) kC,k_C
{MatchOnce}ProhmpA2(b,b) + {MatchOnce}NsrR(b,s˜0).NsrR(b,s) <-> \
{MatchOnce}ProhmpA2(b!1,b!2).NsrR(b!1,s˜0).NsrR(b!2,s) kA,k_A
{MatchOnce}ProhmpA2(b,b) + {MatchOnce}NsrR(b,s˜1).NsrR(b,s) <-> \
{MatchOnce}ProhmpA2(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) kB,k_B
# Reactions of NsrR in complex with promoters with NO
{MatchOnce}ProhmpA1(b!1,b!2).NsrR(b!1,s˜0).NsrR(b!2,s) + NO(a) -> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) kAB
{MatchOnce}ProhmpA1(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) + NO(a) -> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜2).NsrR(b!2,s) kBC
{MatchOnce}ProhmpA1(b!1,b!2).NsrR(b!1,s˜2).NsrR(b!2,s) + NO(a) -> \
ProhmpA1(b,b) + NsrR(b,s˜3).NsrR(b,s) kCD
{MatchOnce}ProhmpA2(b!1,b!2).NsrR(b!1,s˜0).NsrR(b!2,s) + NO(a) -> \
ProhmpA2(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) kAB
{MatchOnce}ProhmpA2(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) + NO(a) -> \
ProhmpA2(b,b) + NsrR(b,s˜2).NsrR(b,s) kBC
# NsrR recycling
NsrR(b,s˜0).NsrR(b,s) -> NsrR(b,s˜0).NsrR(b,s˜0) kr
NsrR(b,s˜1).NsrR(b,s) -> NsrR(b,s˜1).NsrR(b,s˜0) kr
NsrR(b,s˜2).NsrR(b,s) -> NsrR(b,s˜2).NsrR(b,s˜0) kr
NsrR(b,s˜3).NsrR(b,s) -> NsrR(b,s˜3).NsrR(b,s˜0) kr
NsrR(b,s˜4).NsrR(b,s) -> NsrR(b,s˜4).NsrR(b,s˜0) kr
NsrR(b,s˜5).NsrR(b,s) -> NsrR(b,s˜5).NsrR(b,s˜0) kr
end reaction rules
end model
generate_network({overwrite=>1})
bifurcate({parameter=>"kp",par_min=>0,par_max=>150,n_scan_pts=>200,\
method=>"ode",t_end=>1e7})
#parameter_scan({parameter=>"km",par_min=>0,par_max=>1,n_scan_pts=>500,\
#method=>"ode",t_end=>100})
#writeSBML()
#writeLatex()
A.3 Dimer model 2 (DM2)
begin model
69
A.3. DIMER MODEL 2 (DM2)
begin parameters
kAB 4.52 # /micromol.s
kBC 0.113 # /micromol.s
kCD 1.34e-2 # /micromol.s
kDE 7.48e-3 # /micromol.s
kEF 0.43e-3 #/micromol.s
kr 0.001
NsrR_init 2.5
NO_init 0
NOext_init 0
O2_init 200
km 750
kp 0
k1 3.5
k_1 0.44
k2 850
k_2 2e-4
k_3 200
k4 26
k_4 2e-4
k5 0.014
k6 1.8e-4
ProhmpA1_init 6e-3
ProhmpA2_init 6e-3
kd 0.71
ke 0.83
kA 10
k_A 0.35
kB 10
k_B 0.4
kC 10
k_C 0.45
end parameters
begin molecule types
NsrR(b,dim,s˜0˜1˜2˜3˜4˜5)
NO(a)
NOext()
Hmp(n,o)
O2(a)
ProhmpA2(b,b)
ProhmpA1(b,b)
RBS()
NO3(a)
end molecule types
70
A.3. DIMER MODEL 2 (DM2)
begin species
NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜0) NsrR_init
ProhmpA2(b,b) ProhmpA2_init
ProhmpA1(b,b) ProhmpA1_init
# Note: O2 and NOext are fixed.
$O2(a) O2_init
$NOext() NOext_init
$NO(a) NO_init
end species
begin observables
#Species Hmp_NO Hmp(b,c!1).NO(a!1)
#Species Hmp_O2_NO Hmp(b!0,c!1).O2(a!0).NO(a!1)
#Species Hmp_O2 Hmp(b,c!1).O2(a!1)
Species ProhmpA2 ProhmpA2(b)
Species ProhmpA1 ProhmpA1(b)
#Species No NO(a)
#Species Hmp Hmp(c,c)
#Species RBS RBS()
#Species NsrRC_NsrR NsrR(s˜2).NsrR(s˜0)
#Species NsrRC_NsrRB NsrR(s˜2).NsrR(s˜1)
#Species NsrRC_NsrRC NsrR(s˜2).NsrR(s˜2)
#Species NsrRC_NsrRD NsrR(s˜2).NsrR(s˜3)
#Species NsrRC_NsrRE NsrR(s˜2).NsrR(s˜4)
#Species NsrRC_NsrRF NsrR(s˜2).NsrR(s˜5)
end observables
begin reaction rules
# Hmp/NO dynamics
# Permeation and intracellular production
NOext() <-> NO(a) km,km
0 -> NO(a) kp
# Enzyme kinetics
Hmp(n,o) + O2(a) <-> Hmp(n,o!1).O2(a!1) k1,k_1
Hmp(n,o!1).O2(a!1) + NO(a) <-> Hmp(n!0,o!1).O2(a!1).NO(a!0) k2,k_2
Hmp(n!0,o!1).O2(a!1).NO(a!0) -> Hmp(n,o) + NO3(a) k_3
Hmp(n,o) + NO(a) <-> Hmp(n!1,o).NO(a!1) k4,k_4
# Production/decay of mRNA (ribosome binding sites) and of Hmp
ProhmpA2(b,b) -> ProhmpA2(b,b) + RBS() kd
71
A.3. DIMER MODEL 2 (DM2)
ProhmpA1(b,b) -> ProhmpA1(b,b) + RBS() kd
RBS() -> RBS() + Hmp(n,o) ke
RBS() -> 0 k5
# Decay of Hmp and its complexes
Hmp(n,o) -> 0 k6
Hmp(n,o!1).O2(a!1) -> O2(a) + 0 k6
Hmp(n!0,o!1).O2(a!1).NO(a!0) -> O2(a) + NO(a) k6
Hmp(n!1,o).NO(a!1) -> NO(a) k6
# Reactions of NsrR with NO
{MatchOnce}NsrR(b,s˜0).NsrR(b,s) + NO(a) -> NsrR(b,s˜1).NsrR(b,s) kAB
{MatchOnce}NsrR(b,s˜1).NsrR(b,s) + NO(a) -> NsrR(b,s˜2).NsrR(b,s) kBC
{MatchOnce}NsrR(b,s˜2).NsrR(b,s) + NO(a) -> NsrR(b,s˜3).NsrR(b,s) kCD
{MatchOnce}NsrR(b,s˜3).NsrR(b,s) + NO(a) -> NsrR(b,s˜4).NsrR(b,s) kDE
{MatchOnce}NsrR(b,s˜4).NsrR(b,s) + NO(a) -> NsrR(b,s˜5).NsrR(b,s) kEF
# Binding equilibria using the rule that binding strength is determined by
# the weakest binding part of a dimer
{MatchOnce}ProhmpA1(b,b) + NsrR(b,s˜0).NsrR(b,s˜0) <-> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜0).NsrR(b!2,s˜0) kA,k_A
{MatchOnce}ProhmpA1(b,b) + NsrR(b,s˜1).NsrR(b,s˜1) <-> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s˜1) kB,k_B
{MatchOnce}ProhmpA1(b,b) + NsrR(b,s˜1).NsrR(b,s˜0) <-> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s˜0) kB,k_B
{MatchOnce}ProhmpA1(b,b) + NsrR(b,s˜2).NsrR(b,s˜2) <-> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜2).NsrR(b!2,s˜2) kC,k_C
{MatchOnce}ProhmpA1(b,b) + NsrR(b,s˜2).NsrR(b,s˜1) <-> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜2).NsrR(b!2,s˜1) kC,k_C
{MatchOnce}ProhmpA1(b,b) + NsrR(b,s˜2).NsrR(b,s˜0) <-> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜2).NsrR(b!2,s˜0) kC,k_C
{MatchOnce}ProhmpA2(b,b) + NsrR(b,s˜0).NsrR(b,s˜0) <-> \
ProhmpA2(b!1,b!2).NsrR(b!1,s˜0).NsrR(b!2,s˜0) kA,k_A
{MatchOnce}ProhmpA2(b,b) + NsrR(b,s˜1).NsrR(b,s˜1) <-> \
ProhmpA2(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s˜1) kB,k_B
{MatchOnce}ProhmpA2(b,b) + NsrR(b,s˜1).NsrR(b,s˜0) <-> \
ProhmpA2(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s˜0) kB,k_B
# Reactions of NsrR in complex with promoters with NO
{MatchOnce}ProhmpA1(b!1,b!2).NsrR(b!1,s˜0).NsrR(b!2,s) + NO(a) -> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) kAB
{MatchOnce}ProhmpA1(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) + NO(a) -> \
ProhmpA1(b!1,b!2).NsrR(b!1,s˜2).NsrR(b!2,s) kBC
{MatchOnce}ProhmpA1(b!1,b!2).NsrR(b!1,s˜2).NsrR(b!2,s) + NO(a) -> \
72
A.3. DIMER MODEL 2 (DM2)
ProhmpA1(b,b) + NsrR(b,s˜3).NsrR(b,s) kCD
{MatchOnce}ProhmpA2(b!1,b!2).NsrR(b!1,s˜0).NsrR(b!2,s) + NO(a) -> \
ProhmpA2(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) kAB
{MatchOnce}ProhmpA2(b!1,b!2).NsrR(b!1,s˜1).NsrR(b!2,s) + NO(a) -> \
ProhmpA2(b,b) + NsrR(b,s˜2).NsrR(b,s) kBC
# NsrR recycling
NsrR(b,s˜0).NsrR(b,s) -> NsrR(b,s˜0).NsrR(b,s˜0) kr
NsrR(b,s˜1).NsrR(b,s) -> NsrR(b,s˜1).NsrR(b,s˜0) kr
NsrR(b,s˜2).NsrR(b,s) -> NsrR(b,s˜2).NsrR(b,s˜0) kr
NsrR(b,s˜3).NsrR(b,s) -> NsrR(b,s˜3).NsrR(b,s˜0) kr
NsrR(b,s˜4).NsrR(b,s) -> NsrR(b,s˜4).NsrR(b,s˜0) kr
NsrR(b,s˜5).NsrR(b,s) -> NsrR(b,s˜5).NsrR(b,s˜0) kr
end reaction rules
end model
generate_network({overwrite=>1})
bifurcate({parameter=>"kp",par_min=>0,par_max=>150,n_scan_pts=>200,\
method=>"ode",t_end=>1e7})
#parameter_scan({parameter=>"km",par_min=>0,par_max=>1,n_scan_pts=>500,\
#method=>"ode",t_end=>100})
#writeSBML()
#writeLatex()
73
Appendix B
Differential equations
B.1 Monomer (MM)
MM has 22 species and 42 reactions.
ẋ1 = −p26x4x1− p26x3x1− p1x1x2 + p27x8
+p27x9 + p6x10 + p6x14 + p6x18 + p6x20
+p6x22
ẋ2 = +p12x6− p12x2− p1x1x2− p1x8x2
−p1x9x2− p2x10x2−2p18x11x2− p2x12x2− p2x13x2
−p3x14x2− p15x15x2 + p19x16 + p21x16− p3x17x2
−p4x18x2 + p16x19 + p21x19− p5x20x2
ẋ3 = −p26x3x1 + p27x9− p28x3x10 + p29x13
+p2x13x2
ẋ4 = −p26x4x1 + p27x8− p28x4x10 + p29x12
−p30x4x14 + p31x17 + p3x17x2
ẋ5 = −2p13x11x5 + p14x15 + p21x15 + p21x19
ẋ6 = −p12x6 + p12x2
ẋ7 = +p24x3 + p24x4− p20x7
ẋ8 = +p26x4x1− p27x8− p1x8x2
ẋ9 = +p26x3x1− p27x9− p1x9x2
x1˙0 = +p1x1x2− p28x4x10− p28x3x10− p2x10x2
−p6x10 + p29x12 + p29x13
x1˙1 = +p25x7−2p13x11x5−2p18x11x2− p21x11
+p14x15 + p19x16 + p17x19
x1˙2 = +p28x4x10 + p1x8x2− p29x12− p2x12x2
x1˙3 = +p28x3x10 + p1x9x2− p29x13− p2x13x2
x1˙4 = +p2x10x2− p30x4x14 + p2x13x2− p3x14x2
−p6x14 + p31x17
x1˙5 = +2p13x11x5− p14x15− p15x15x2− p21x15
74
B.1. MONOMER (MM)
+p16x19
x1˙6 = +2p18x11x2− p19x16− p21x16
x1˙7 = +p30x4x14 + p2x12x2− p31x17− p3x17x2
x1˙8 = +p3x14x2 + p3x17x2− p4x18x2− p6x18
x1˙9 = +p15x15x2− p16x19− p17x19− p21x19
x2˙0 = +p4x18x2− p5x20x2− p6x20
x2˙1 = +p17x19
x2˙2 = +p5x20x2− p6x22
BioNetGen generates a .net file to show the meanings of the parameters and variables
in each model. The relevant sections of the .net file for MM are:
# Created by BioNetGen 2.5.1
begin parameters
1 kAB 4.52 # Constant
2 kBC 0.113 # Constant
3 kCD 1.34e-2 # Constant
4 kDE 7.48e-3 # Constant
5 kEF 0.43e-3 # Constant
6 kr 0.001 # Constant
7 NsrR_init 2.5 # Constant
8 logNO -0.4 # Constant
9 NO_init 10.0ˆlogNO # ConstantExpression
10 NOext_init 0 # Constant
11 O2_init 200 # Constant
12 km 750 # Constant
13 k1 3.5 # Constant
14 k_1 0.44 # Constant
15 k2 850 # Constant
16 k_2 2e-4 # Constant
17 k_3 200 # Constant
18 k4 26 # Constant
19 k_4 2e-4 # Constant
20 k5 0.014 # Constant
21 k6 1.8e-4 # Constant
22 ProhmpA1_init 6e-3 # Constant
23 ProhmpA2_init 6e-3 # Constant
24 kd 0.71 # Constant
25 ke 0.83 # Constant
26 kA 10 # Constant
27 k_A 0.35 # Constant
28 kB 10 # Constant
29 k_B 0.4 # Constant
75
B.2. DIMER MODEL 1 (DM1)
30 kC 10 # Constant
31 k_C 0.45 # Constant
end parameters
begin species
1 NsrR(a) NsrR_init
2 $NO(a) NO_init
3 ProhmpA2(b) ProhmpA2_init
4 ProhmpA1(b) ProhmpA1_init
5 $O2(a) O2_init
6 $NOext() NOext_init
7 RBS() 0
8 NsrR(a!1).ProhmpA1(b!1) 0
9 NsrR(a!1).ProhmpA2(b!1) 0
10 NsrRB(a) 0
11 Hmp(c,c) 0
12 NsrRB(a!1).ProhmpA1(b!1) 0
13 NsrRB(a!1).ProhmpA2(b!1) 0
14 NsrRC(a) 0
15 Hmp(c!1,c).O2(a!1) 0
16 Hmp(c!1,c).NO(a!1) 0
17 NsrRC(a!1).ProhmpA1(b!1) 0
18 NsrRD(a) 0
19 Hmp(c!1,c!2).NO(a!2).O2(a!1) 0
20 NsrRE(a) 0
21 NO3(a) 0
22 NsrRF(a) 0
end species
B.2 Dimer model 1 (DM1)
DM1 has 58 species and 168 reactions.
ẋ1 = −p1x1x2− p26x4x1− p26x3x1 + p27x9
+p27x10 + p6x8 + p6x13 + p6x19 + p6x27
+p6x35
ẋ2 = +p11x6− p11x2 + p12− p1x1x2
−p1x8x2− p2x8x2− p1x9x2− p1x10x2− p18x11x2
−p1x13x2− p2x12x2− p3x13x2− p1x14x2− p2x14x2
−p1x15x2− p2x15x2− p15x16x2 + p19x17 + p21x17
−p1x19x2− p2x18x2− p3x18x2− p4x19x2− p1x20x2
−p2x21x2− p3x20x2− p1x22x2− p2x23x2 + p16x24
+p21x24− p1x27x2− p2x25x2− p3x26x2− p4x25x2
76
B.2. DIMER MODEL 1 (DM1)
−p5x27x2− p1x28x2− p2x29x2− p3x29x2− p1x30x2
−p2x31x2− p1x35x2− p2x33x2− p3x34x2− p4x34x2
−p5x33x2− p1x36x2− p2x37x2− p3x38x2− p1x39x2
−p2x40x2− p2x41x2− p3x42x2− p4x43x2− p5x42x2
−p1x44x2− p2x45x2− p3x46x2− p1x47x2− p2x48x2
−p3x49x2− p4x50x2− p5x50x2− p2x51x2− p3x52x2
−p2x53x2− p4x54x2− p5x55x2− p3x56x2− p5x57x2
ẋ3 = −p26x3x1− p26x3x8 + p27x10− p28x3x8
−p26x3x13 + p27x15− p28x3x12 + p29x15 + p2x15x2
−p26x3x19 + p27x22− p28x3x18 + p29x23 + p2x23x2
−p26x3x27 + p27x30− p28x3x25 + p29x31 + p2x31x2
−p26x3x35 + p27x39− p28x3x33 + p29x40 + p2x40x2
+p27x47− p28x3x41 + p29x48 + p2x48x2 + p29x53
+p2x53x2
ẋ4 = −p26x4x1− p26x4x8 + p27x9− p28x4x8
−p26x4x13 + p27x14− p28x4x12 + p29x14− p30x4x13
−p26x4x19 + p27x20− p28x4x18 + p29x21− p30x4x18
+p31x20 + p3x20x2− p26x4x27 + p27x28− p28x4x25
+p29x29− p30x4x26 + p31x29 + p3x29x2− p26x4x35
+p27x36− p28x4x33 + p29x37− p30x4x34 + p31x38
+p3x38x2 + p27x44− p28x4x41 + p29x45− p30x4x42
+p31x46 + p3x46x2 + p29x51− p30x4x49 + p31x52
+p3x52x2 + p31x56 + p3x56x2
ẋ5 = −p13x11x5 + p14x16 + p21x16 + p21x24
ẋ6 = −p11x6 + p11x2
ẋ7 = +p24x3 + p24x4− p20x7
ẋ8 = +p1x1x2− p1x8x2− p2x8x2− p26x4x8
−p28x4x8− p26x3x8− p28x3x8− p6x8 + p27x14
+p29x14 + p27x15 + p29x15 +2p6x12 + p6x18
+p6x25 + p6x33 + p6x41
ẋ9 = +p26x4x1− p27x9− p1x9x2
x1˙0 = +p26x3x1− p27x10− p1x10x2
x1˙1 = +p25x7− p13x11x5− p18x11x2− p21x11
+p14x16 + p19x17 + p17x24
x1˙2 = +p1x8x2− p2x12x2− p28x4x12− p28x3x12
−2p6x12 + p29x21 + p29x23
x1˙3 = +p2x8x2− p1x13x2− p3x13x2− p26x4x13
77
B.2. DIMER MODEL 1 (DM1)
−p30x4x13− p26x3x13 + p2x15x2− p6x13 + p27x20
+p31x20 + p27x22 + p6x18 +2p6x26 + p6x34
+p6x42 + p6x49
x1˙4 = +p26x4x8 + p28x4x8 + p1x9x2− p27x14
−p29x14− p1x14x2− p2x14x2
x1˙5 = +p26x3x8 + p28x3x8 + p1x10x2− p27x15
−p29x15− p1x15x2− p2x15x2
x1˙6 = +p13x11x5− p14x16− p15x16x2− p21x16
+p16x24
x1˙7 = +p18x11x2− p19x17− p21x17
x1˙8 = +p1x13x2 + p2x12x2− p2x18x2− p3x18x2
−p28x4x18− p30x4x18− p28x3x18 + p2x23x2− p6x18
−p6x18 + p29x29 + p31x29 + p29x31
x1˙9 = +p3x13x2− p1x19x2− p4x19x2− p26x4x19
−p26x3x19 + p3x20x2− p6x19 + p27x28 + p27x30
+p6x25 + p6x34 +2p6x43 + p6x50 + p6x54
x2˙0 = +p26x4x13 + p30x4x13 + p2x14x2− p27x20
−p31x20− p1x20x2− p3x20x2
x2˙1 = +p28x4x12 + p1x14x2− p29x21− p2x21x2
x2˙2 = +p26x3x13− p27x22− p1x22x2
x2˙3 = +p28x3x12 + p1x15x2− p29x23− p2x23x2
x2˙4 = +p15x16x2− p16x24− p17x24− p21x24
x2˙5 = +p1x19x2 + p3x18x2− p2x25x2− p4x25x2
−p28x4x25− p28x3x25 + p3x29x2− p6x25− p6x25
+p29x37 + p29x40
x2˙6 = +p2x18x2− p3x26x2− p30x4x26 + p2x31x2
−2p6x26 + p31x38
x2˙7 = +p4x19x2− p1x27x2− p5x27x2− p26x4x27
−p26x3x27− p6x27 + p27x36 + p27x39 + p6x33
+p6x42 + p6x50 +2p6x55 + p6x57
x2˙8 = +p26x4x19− p27x28− p1x28x2
x2˙9 = +p28x4x18 + p30x4x18 + p1x20x2 + p2x21x2
−p29x29− p31x29− p2x29x2− p3x29x2
x3˙0 = +p26x3x19− p27x30− p1x30x2
x3˙1 = +p28x3x18 + p1x22x2− p29x31− p2x31x2
x3˙2 = +p17x24
x3˙3 = +p1x27x2 + p4x25x2− p2x33x2− p5x33x2
78
B.2. DIMER MODEL 1 (DM1)
−p28x4x33− p28x3x33− p6x33− p6x33 + p29x45
+p29x48
x3˙4 = +p2x25x2 + p3x26x2− p3x34x2− p4x34x2
−p30x4x34 + p3x38x2 + p2x40x2− p6x34− p6x34
+p31x46
x3˙5 = +p5x27x2− p1x35x2− p26x4x35− p26x3x35
−p6x35 + p27x44 + p27x47 + p6x41 + p6x49
+p6x54 + p6x57 +2p6x58
x3˙6 = +p26x4x27− p27x36− p1x36x2
x3˙7 = +p28x4x25 + p1x28x2− p29x37− p2x37x2
x3˙8 = +p30x4x26 + p2x29x2− p31x38− p3x38x2
x3˙9 = +p26x3x27− p27x39− p1x39x2
x4˙0 = +p28x3x25 + p1x30x2− p29x40− p2x40x2
x4˙1 = +p1x35x2 + p5x33x2− p2x41x2− p28x4x41
−p28x3x41− p6x41− p6x41 + p29x51 + p29x53
x4˙2 = +p2x33x2 + p4x34x2− p3x42x2− p5x42x2
−p30x4x42 + p2x48x2− p6x42− p6x42 + p31x52
x4˙3 = +p3x34x2− p4x43x2 + p3x46x2−2p6x43
x4˙4 = +p26x4x35− p27x44− p1x44x2
x4˙5 = +p28x4x33 + p1x36x2− p29x45− p2x45x2
x4˙6 = +p30x4x34 + p2x37x2− p31x46− p3x46x2
x4˙7 = +p26x3x35− p27x47− p1x47x2
x4˙8 = +p28x3x33 + p1x39x2− p29x48− p2x48x2
x4˙9 = +p2x41x2 + p5x42x2− p3x49x2− p30x4x49
+p2x53x2− p6x49− p6x49 + p31x56
x5˙0 = +p3x42x2 + p4x43x2− p4x50x2− p5x50x2
+p3x52x2− p6x50− p6x50
x5˙1 = +p28x4x41 + p1x44x2− p29x51− p2x51x2
x5˙2 = +p30x4x42 + p2x45x2− p31x52− p3x52x2
x5˙3 = +p28x3x41 + p1x47x2− p29x53− p2x53x2
x5˙4 = +p3x49x2 + p5x50x2− p4x54x2 + p3x56x2
−p6x54− p6x54
x5˙5 = +p4x50x2− p5x55x2−2p6x55
x5˙6 = +p30x4x49 + p2x51x2− p31x56− p3x56x2
x5˙7 = +p4x54x2 + p5x55x2− p5x57x2− p6x57
−p6x57
x5˙8 = +p5x57x2−2p6x58
79
B.2. DIMER MODEL 1 (DM1)
The relevant sections of the .net file for DM1 are:
# Created by BioNetGen 2.5.1
begin parameters
1 kAB 4.52 # Constant
2 kBC 0.113 # Constant
3 kCD 1.34e-2 # Constant
4 kDE 7.48e-3 # Constant
5 kEF 0.43e-3 # Constant
6 kr 0.1 # Constant
7 NsrR_init 2.5 # Constant
8 NO_init 0 # Constant
9 NOext_init 0 # Constant
10 O2_init 200 # Constant
11 km 750 # Constant
12 kp 0 # Constant
13 k1 3.5 # Constant
14 k_1 0.44 # Constant
15 k2 850 # Constant
16 k_2 2e-4 # Constant
17 k_3 200 # Constant
18 k4 26 # Constant
19 k_4 2e-4 # Constant
20 k5 0.014 # Constant
21 k6 1.8e-4 # Constant
22 ProhmpA1_init 6e-3 # Constant
23 ProhmpA2_init 6e-3 # Constant
24 kd 0.71 # Constant
25 ke 0.83 # Constant
26 kA 10 # Constant
27 k_A 0.35 # Constant
28 kB 10 # Constant
29 k_B 0.4 # Constant
30 kC 10 # Constant
31 k_C 0.45 # Constant
end parameters
begin species
1 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜0) NsrR_init
2 NO(a) NO_init
3 ProhmpA2(b,b) ProhmpA2_init
4 ProhmpA1(b,b) ProhmpA1_init
5 $O2(a) O2_init
6 $NOext() NOext_init
80
B.2. DIMER MODEL 1 (DM1)
7 RBS() 0
8 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜1) 0
9 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜0).ProhmpA1(b!1,b!3) 0
10 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜0).ProhmpA2(b!1,b!3) 0
11 Hmp(n,o) 0
12 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜1) 0
13 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜2) 0
14 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜1).ProhmpA1(b!1,b!3) 0
15 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜1).ProhmpA2(b!1,b!3) 0
16 Hmp(n,o!1).O2(a!1) 0
17 Hmp(n!1,o).NO(a!1) 0
18 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜2) 0
19 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜3) 0
20 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜2).ProhmpA1(b!1,b!3) 0
21 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜1).ProhmpA1(b!1,b!3) 0
22 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜2).ProhmpA2(b!1,b!3) 0
23 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜1).ProhmpA2(b!1,b!3) 0
24 Hmp(n!1,o!2).NO(a!1).O2(a!2) 0
25 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜3) 0
26 NsrR(b,dim!1,s˜2).NsrR(b,dim!1,s˜2) 0
27 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜4) 0
28 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜3).ProhmpA1(b!1,b!3) 0
29 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜2).ProhmpA1(b!1,b!3) 0
30 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜3).ProhmpA2(b!1,b!3) 0
31 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜2).ProhmpA2(b!1,b!3) 0
32 NO3(a) 0
33 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜4) 0
34 NsrR(b,dim!1,s˜2).NsrR(b,dim!1,s˜3) 0
35 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜5) 0
36 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜4).ProhmpA1(b!1,b!3) 0
37 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜3).ProhmpA1(b!1,b!3) 0
38 NsrR(b!1,dim!2,s˜2).NsrR(b!3,dim!2,s˜2).ProhmpA1(b!1,b!3) 0
39 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜4).ProhmpA2(b!1,b!3) 0
40 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜3).ProhmpA2(b!1,b!3) 0
41 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜5) 0
42 NsrR(b,dim!1,s˜2).NsrR(b,dim!1,s˜4) 0
43 NsrR(b,dim!1,s˜3).NsrR(b,dim!1,s˜3) 0
44 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜5).ProhmpA1(b!1,b!3) 0
45 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜4).ProhmpA1(b!1,b!3) 0
46 NsrR(b!1,dim!2,s˜2).NsrR(b!3,dim!2,s˜3).ProhmpA1(b!1,b!3) 0
47 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜5).ProhmpA2(b!1,b!3) 0
48 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜4).ProhmpA2(b!1,b!3) 0
49 NsrR(b,dim!1,s˜2).NsrR(b,dim!1,s˜5) 0
50 NsrR(b,dim!1,s˜3).NsrR(b,dim!1,s˜4) 0
51 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜5).ProhmpA1(b!1,b!3) 0
81
B.3. DIMER MODEL 2 (DM2)
52 NsrR(b!1,dim!2,s˜2).NsrR(b!3,dim!2,s˜4).ProhmpA1(b!1,b!3) 0
53 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜5).ProhmpA2(b!1,b!3) 0
54 NsrR(b,dim!1,s˜3).NsrR(b,dim!1,s˜5) 0
55 NsrR(b,dim!1,s˜4).NsrR(b,dim!1,s˜4) 0
56 NsrR(b!1,dim!2,s˜2).NsrR(b!3,dim!2,s˜5).ProhmpA1(b!1,b!3) 0
57 NsrR(b,dim!1,s˜4).NsrR(b,dim!1,s˜5) 0
58 NsrR(b,dim!1,s˜5).NsrR(b,dim!1,s˜5) 0
end species
B.3 Dimer model 2 (DM2)
DM2 has 41 species and 109 reactions.
ẋ1 = −p1x1x2− p26x4x1− p26x3x1 + p27x9
+p27x10 + p6x8 + p6x13 + p6x19 + p6x26
+p6x31
ẋ2 = +p11x6− p11x2 + p12− p1x1x2
−p1x8x2− p2x8x2− p1x9x2− p1x10x2− p18x11x2
−p1x13x2− p2x12x2− p3x13x2− p1x14x2− p2x14x2
−p1x15x2− p2x15x2− p15x16x2 + p19x17 + p21x17
−p1x19x2− p2x18x2− p3x18x2− p4x19x2− p1x21x2
−p2x20x2− p3x21x2− p2x22x2 + p16x23 + p21x23
−p1x26x2− p2x24x2− p3x25x2− p4x24x2− p5x26x2
−p2x27x2− p3x27x2− p1x31x2− p2x29x2− p3x30x2
−p4x30x2− p5x29x2− p3x32x2− p2x33x2− p3x34x2
−p4x35x2− p5x34x2− p3x36x2− p4x37x2− p5x37x2
−p4x38x2− p5x39x2− p5x40x2
ẋ3 = −p26x3x1 + p27x10− p28x3x8− p28x3x12
+p29x15 + p2x15x2 + p29x22 + p2x22x2
ẋ4 = −p26x4x1 + p27x9− p28x4x8− p28x4x12
+p29x14− p30x4x13 + p29x20− p30x4x18 + p31x21
+p3x21x2− p30x4x25 + p31x27 + p3x27x2 + p31x32
+p3x32x2
ẋ5 = −p13x11x5 + p14x16 + p21x16 + p21x23
ẋ6 = −p11x6 + p11x2
ẋ7 = +p24x3 + p24x4− p20x7
ẋ8 = +p1x1x2− p1x8x2− p2x8x2− p28x4x8
−p28x3x8− p6x8 + p29x14 + p29x15 +2p6x12
+p6x18 + p6x24 + p6x29 + p6x33
82
B.3. DIMER MODEL 2 (DM2)
ẋ9 = +p26x4x1− p27x9− p1x9x2
x1˙0 = +p26x3x1− p27x10− p1x10x2
x1˙1 = +p25x7− p13x11x5− p18x11x2− p21x11
+p14x16 + p19x17 + p17x23
x1˙2 = +p1x8x2− p2x12x2− p28x4x12− p28x3x12
−2p6x12 + p29x20 + p29x22
x1˙3 = +p2x8x2− p1x13x2− p3x13x2− p30x4x13
+p2x15x2− p6x13 + p31x21 + p6x18 +2p6x25
+p6x30 + p6x34 + p6x36
x1˙4 = +p28x4x8 + p1x9x2− p29x14− p1x14x2
−p2x14x2
x1˙5 = +p28x3x8 + p1x10x2− p29x15− p1x15x2
−p2x15x2
x1˙6 = +p13x11x5− p14x16− p15x16x2− p21x16
+p16x23
x1˙7 = +p18x11x2− p19x17− p21x17
x1˙8 = +p1x13x2 + p2x12x2− p2x18x2− p3x18x2
−p30x4x18 + p2x22x2− p6x18− p6x18 + p31x27
x1˙9 = +p3x13x2− p1x19x2− p4x19x2 + p3x21x2
−p6x19 + p6x24 + p6x30 +2p6x35 + p6x37
+p6x38
x2˙0 = +p28x4x12 + p1x14x2− p29x20− p2x20x2
x2˙1 = +p30x4x13 + p2x14x2− p31x21− p1x21x2
−p3x21x2
x2˙2 = +p28x3x12 + p1x15x2− p29x22− p2x22x2
x2˙3 = +p15x16x2− p16x23− p17x23− p21x23
x2˙4 = +p1x19x2 + p3x18x2− p2x24x2− p4x24x2
+p3x27x2− p6x24− p6x24
x2˙5 = +p2x18x2− p3x25x2− p30x4x25−2p6x25
+p31x32
x2˙6 = +p4x19x2− p1x26x2− p5x26x2− p6x26
+p6x29 + p6x34 + p6x37 +2p6x39 + p6x40
x2˙7 = +p30x4x18 + p1x21x2 + p2x20x2− p31x27
−p2x27x2− p3x27x2
x2˙8 = +p17x23
x2˙9 = +p1x26x2 + p4x24x2− p2x29x2− p5x29x2
−p6x29− p6x29
83
B.3. DIMER MODEL 2 (DM2)
x3˙0 = +p2x24x2 + p3x25x2− p3x30x2− p4x30x2
+p3x32x2− p6x30− p6x30
x3˙1 = +p5x26x2− p1x31x2− p6x31 + p6x33
+p6x36 + p6x38 + p6x40 +2p6x41
x3˙2 = +p30x4x25 + p2x27x2− p31x32− p3x32x2
x3˙3 = +p1x31x2 + p5x29x2− p2x33x2− p6x33
−p6x33
x3˙4 = +p2x29x2 + p4x30x2− p3x34x2− p5x34x2
−p6x34− p6x34
x3˙5 = +p3x30x2− p4x35x2−2p6x35
x3˙6 = +p2x33x2 + p5x34x2− p3x36x2− p6x36
−p6x36
x3˙7 = +p3x34x2 + p4x35x2− p4x37x2− p5x37x2
−p6x37− p6x37
x3˙8 = +p3x36x2 + p5x37x2− p4x38x2− p6x38
−p6x38
x3˙9 = +p4x37x2− p5x39x2−2p6x39
x4˙0 = +p4x38x2 + p5x39x2− p5x40x2− p6x40
−p6x40
x4˙1 = +p5x40x2−2p6x41
The relevant sections of the .net file for DM2 are:
# Created by BioNetGen 2.5.1
begin parameters
1 kAB 4.52 # Constant
2 kBC 0.113 # Constant
3 kCD 1.34e-2 # Constant
4 kDE 7.48e-3 # Constant
5 kEF 0.43e-3 # Constant
6 kr 0.1 # Constant
7 NsrR_init 2.5 # Constant
8 NO_init 0 # Constant
9 NOext_init 0 # Constant
10 O2_init 200 # Constant
11 km 750 # Constant
12 kp 0 # Constant
13 k1 3.5 # Constant
14 k_1 0.44 # Constant
15 k2 850 # Constant
84
B.3. DIMER MODEL 2 (DM2)
16 k_2 2e-4 # Constant
17 k_3 200 # Constant
18 k4 26 # Constant
19 k_4 2e-4 # Constant
20 k5 0.014 # Constant
21 k6 1.8e-4 # Constant
22 ProhmpA1_init 6e-3 # Constant
23 ProhmpA2_init 6e-3 # Constant
24 kd 0.71 # Constant
25 ke 0.83 # Constant
26 kA 10 # Constant
27 k_A 0.35 # Constant
28 kB 10 # Constant
29 k_B 0.4 # Constant
30 kC 10 # Constant
31 k_C 0.45 # Constant
end parameters
begin species
1 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜0) NsrR_init
2 NO(a) NO_init
3 ProhmpA2(b,b) ProhmpA2_init
4 ProhmpA1(b,b) ProhmpA1_init
5 $O2(a) O2_init
6 $NOext() NOext_init
7 RBS() 0
8 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜1) 0
9 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜0).ProhmpA1(b!1,b!3) 0
10 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜0).ProhmpA2(b!1,b!3) 0
11 Hmp(n,o) 0
12 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜1) 0
13 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜2) 0
14 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜1).ProhmpA1(b!3,b!1) 0
15 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜1).ProhmpA2(b!3,b!1) 0
16 Hmp(n,o!1).O2(a!1) 0
17 Hmp(n!1,o).NO(a!1) 0
18 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜2) 0
19 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜3) 0
20 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜1).ProhmpA1(b!1,b!3) 0
21 NsrR(b!1,dim!2,s˜0).NsrR(b!3,dim!2,s˜2).ProhmpA1(b!3,b!1) 0
22 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜1).ProhmpA2(b!1,b!3) 0
23 Hmp(n!1,o!2).NO(a!1).O2(a!2) 0
24 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜3) 0
25 NsrR(b,dim!1,s˜2).NsrR(b,dim!1,s˜2) 0
26 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜4) 0
27 NsrR(b!1,dim!2,s˜1).NsrR(b!3,dim!2,s˜2).ProhmpA1(b!3,b!1) 0
85
B.3. DIMER MODEL 2 (DM2)
28 NO3(a) 0
29 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜4) 0
30 NsrR(b,dim!1,s˜2).NsrR(b,dim!1,s˜3) 0
31 NsrR(b,dim!1,s˜0).NsrR(b,dim!1,s˜5) 0
32 NsrR(b!1,dim!2,s˜2).NsrR(b!3,dim!2,s˜2).ProhmpA1(b!1,b!3) 0
33 NsrR(b,dim!1,s˜1).NsrR(b,dim!1,s˜5) 0
34 NsrR(b,dim!1,s˜2).NsrR(b,dim!1,s˜4) 0
35 NsrR(b,dim!1,s˜3).NsrR(b,dim!1,s˜3) 0
36 NsrR(b,dim!1,s˜2).NsrR(b,dim!1,s˜5) 0
37 NsrR(b,dim!1,s˜3).NsrR(b,dim!1,s˜4) 0
38 NsrR(b,dim!1,s˜3).NsrR(b,dim!1,s˜5) 0
39 NsrR(b,dim!1,s˜4).NsrR(b,dim!1,s˜4) 0
40 NsrR(b,dim!1,s˜4).NsrR(b,dim!1,s˜5) 0
41 NsrR(b,dim!1,s˜5).NsrR(b,dim!1,s˜5) 0
end species
86