<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : cui_2008.xml
CREATED : 30th April 2008
LAST MODIFIED : 30th April 2008
AUTHOR : Catherine Lloyd
Bioengineering Institute
The University of Auckland
MODEL STATUS : This model conforms to the CellML 1.1 Specification.
DESCRIPTION : This file contains a CellML description of Cui and Kaandorp's 2008 mathematical model of the transcriptional response of zinc homeostasis system in E. coli.
CHANGES:
--><model xmlns="http://www.cellml.org/cellml/1.0#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bqs="http://www.cellml.org/bqs/1.0#" xmlns:cellml="http://www.cellml.org/cellml/1.0#" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" cmeta:id="cui_2008" name="cui_2008">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Transcriptional response of zinc homeostasis system in E. coli</title>
<author>
<firstname>Catherine</firstname>
<surname>Lloyd</surname>
<affiliation>
<shortaffil>Auckland Bioengineering Institute, The University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>
This CellML model runs in both PCEnv and COR to recreate the published results. The units have been checked and they are consistent. In this particular version of the model the zinc-buffering effects of TPEN are considered.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
ABSTRACT: BACKGROUND: The zinc homeostasis system in Escherichia coli is one of the most intensively studied prokaryotic zinc homeostasis systems. Its underlying regulatory machine consists of repression on zinc influx through ZnuABC by Zur (Zn2+ uptake regulator) and activation on zinc efflux via ZntA by ZntR (a zinc-responsive regulator). Although these transcriptional regulations seem to be well characterized, and there is an abundance of detailed in vitro experimental data available, as yet there is no mathematical model to help interpret these data. To our knowledge, the work described here is the first attempt to use a mathematical model to simulate these regulatory relations and to help explain the in vitro experimental data. RESULTS: We develop a unified mathematical model consisting of 14 reactions to simulate the in vitro transcriptional response of the zinc homeostasis system in E. coli. Firstly, we simulate the in vitro Zur-DNA interaction by using two of these reactions, which are expressed as 4 ordinary differential equations (ODEs). By imposing the conservation restraints and solving the relevant steady state equations, we find that the simulated sigmoidal curve matches the corresponding experimental data. Secondly, by numerically solving the ODEs for simulating the Zur and ZntR run-off transcription experiments, and depicting the simulated concentrations of zntA and znuC transcripts as a function of free zinc concentration, we find that the simulated curves fit the corresponding in vitro experimental data. Moreover, we also perform simulations, after taking into consideration the competitive effects of ZntR with the zinc buffer, and depict the simulated concentration of zntA transcripts as a function of the total ZntR concentration, both in the presence and absence of Zn(II). The obtained simulation results are in general agreement with the corresponding experimental data. CONCLUSION: Simulation results show that our model can quantitatively reproduce the results of several of the in vitro experiments conducted by Outten CE and her colleagues. Our model provides a detailed insight into the dynamics of the regulatory system and also provides a general framework for simulating in vitro metal-binding and transcription experiments and interpreting the relevant experimental data.
</para>
<para>
In 1999-2001, Outten <emphasis>et al.</emphasis> published a series of papers presenting data from several <emphasis>in vitro</emphasis> transcription and metal-binding competition experiments in <emphasis>E. coli.</emphasis> These studies provide a detailed data set on which a mathematical model can be based. In the study described here, Cui and Kaandorp have developed such a mathematical model, and they use it to simulate the <emphasis>in vitro</emphasis> transcriptional response of zinc homeostasis system in <emphasis>E. coli.</emphasis> <ulink url="http://www.cellerator.org/">Cellerator</ulink>, an open source software, was used to automatically generate the equations, and the model was subsequently translated into CellML to facilitate future model exchange, reuse and implementation.
</para>
<informalfigure float="0" id="fig_reaction_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>model diagram</title>
</objectinfo>
<imagedata fileref="cui_2008.png"/>
</imageobject>
</mediaobject>
<caption>A schematic diagram of the reactions described in the model of zinc homeostasis system in Escherichia coli. Extracellular zinc enters the cytoplasm through ZnuABC and ZupT, where its presence can cause Zur to bind to the znu operator and repress the transcription of the znuACB gene cluster. Excess intracellular zinc ions are exported by ZntA and ZitB, and cytosolic zinc can bind with protein ZntR to form a strong transcriptional activator of the zntA gene. Cytoplasmic zinc trafficking may involve chaperone-like proteins. Abbreviations: Zur* : active Zur; ZntR* : active ZntR; C? : (a possible zinc chaperone protein whose existence is still under debate).</caption>
</informalfigure>
<para>
The original paper reference is cited below:
</para>
<para>
Jiangjun Cui, Jaap A. Kaandorp and Catherine M. Lloyd 2008, <emphasis>BMC Systems Biology</emphasis>, 2:89. <ulink url="http://www.ncbi.nlm.nih.gov/pubmed/18950480">PubMed ID: 18950480</ulink>
</para>
</sect1>
</article>
</documentation>
<units name="minute">
<unit units="second" multiplier="60.0"/>
</units>
<units name="molar">
<unit units="mole"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="per_molar">
<unit units="molar" exponent="-1"/>
</units>
<units name="nanomolar">
<unit units="mole" prefix="nano"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="first_order_rate_constant">
<unit units="second" exponent="-1"/>
</units>
<units name="second_order_rate_constant">
<unit units="nanomolar" exponent="-1"/>
<unit units="second" exponent="-1"/>
</units>
<units name="third_order_rate_constant">
<unit units="nanomolar" exponent="-2"/>
<unit units="second" exponent="-1"/>
</units>
<component name="environment">
<variable units="second" public_interface="out" name="time"/>
</component>
<!-- The 7 ODEs describing the Zur transcription assay -->
<component name="Py" cmeta:id="Py">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Py">
<dc:title>Py</dc:title>
<dcterms:alternative>Zur dimer containing one zinc ion per monomer</dcterms:alternative>
<dcterms:alternative>Zn2Zur</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Py" initial_value="25.0"/>
<variable units="nanomolar" public_interface="in" name="Zn"/>
<variable units="nanomolar" public_interface="in" name="Py1"/>
<variable units="third_order_rate_constant" public_interface="in" name="r3"/>
<variable units="first_order_rate_constant" public_interface="in" name="r4"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
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<apply>
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<bvar>
<ci>time</ci>
</bvar>
<ci> Py </ci>
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<apply>
<minus/>
<apply>
<times/>
<ci> r4 </ci>
<ci> Py1 </ci>
</apply>
<apply>
<times/>
<ci> r3 </ci>
<apply>
<power/>
<ci> Zn </ci>
<cn cellml:units="dimensionless"> 2 </cn>
</apply>
<ci> Py </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Py1" cmeta:id="Py1">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Py1">
<dc:title>Py1</dc:title>
<dcterms:alternative>active Zur</dcterms:alternative>
<dcterms:alternative>Zur dimer containing two zinc ions per monomer</dcterms:alternative>
<dcterms:alternative>Zn4Zur</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Py1" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Zn"/>
<variable units="nanomolar" public_interface="in" name="Py"/>
<variable units="nanomolar" public_interface="in" name="Dw"/>
<variable units="nanomolar" public_interface="in" name="Qw2"/>
<variable units="third_order_rate_constant" public_interface="in" name="r3"/>
<variable units="first_order_rate_constant" public_interface="in" name="r4"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_1"/>
<variable units="second_order_rate_constant" public_interface="in" name="k1a"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Py1 </ci>
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<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> r3 </ci>
<apply>
<power/>
<ci> Zn </ci>
<cn cellml:units="dimensionless"> 2 </cn>
</apply>
<ci> Py </ci>
</apply>
<apply>
<times/>
<ci> k_1 </ci>
<ci> Qw2 </ci>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> r4 </ci>
<ci> Py1 </ci>
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<apply>
<times/>
<ci> k1a </ci>
<ci> Dw </ci>
<ci> Py1 </ci>
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</math>
</component>
<component name="Dw" cmeta:id="Dw">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Dw">
<dc:title>Dw</dc:title>
<dcterms:alternative>DNA of ZnuC</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Dw" initial_value="4.0"/>
<variable units="nanomolar" public_interface="in" name="Qw2"/>
<variable units="nanomolar" public_interface="in" name="Py1"/>
<variable units="nanomolar" public_interface="in" name="Qw1"/>
<variable units="nanomolar" public_interface="in" name="Rw"/>
<variable units="second_order_rate_constant" public_interface="in" name="k1a"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_1"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Dw </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> k_1 </ci>
<ci> Qw2 </ci>
</apply>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qw1 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qw1 </ci>
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</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k1a </ci>
<ci> Dw </ci>
<ci> Py1 </ci>
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<apply>
<times/>
<ci> k2 </ci>
<ci> Dw </ci>
<ci> Rw </ci>
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</math>
</component>
<component name="Rw" cmeta:id="Rw">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Rw">
<dc:title>Rw</dc:title>
<dcterms:alternative>RNA polymerase for znuC transcription</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Rw" initial_value="50.0"/>
<variable units="nanomolar" public_interface="in" name="Dw"/>
<variable units="nanomolar" public_interface="in" name="Qw1"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Rw </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qw1 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qw1 </ci>
</apply>
</apply>
<apply>
<times/>
<ci> k2 </ci>
<ci> Dw </ci>
<ci> Rw </ci>
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</apply>
</apply>
</math>
</component>
<component name="Qw1" cmeta:id="Qw1">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Qw1">
<dc:title>Qw1</dc:title>
<dcterms:alternative>transcription initiation complex of ZnuC</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Qw1" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Dw"/>
<variable units="nanomolar" public_interface="in" name="Rw"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Qw1 </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> k2 </ci>
<ci> Dw </ci>
<ci> Rw </ci>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qw1 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qw1 </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Qw2" cmeta:id="Qw2">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Qw2">
<dc:title>Qw2</dc:title>
<dcterms:alternative>Zn4Zur-DNA complex</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Qw2" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Dw"/>
<variable units="nanomolar" public_interface="in" name="Py1"/>
<variable units="second_order_rate_constant" public_interface="in" name="k1a"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_1"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Qw2 </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> k1a </ci>
<ci> Dw </ci>
<ci> Py1 </ci>
</apply>
<apply>
<times/>
<ci> k_1 </ci>
<ci> Qw2 </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Mw" cmeta:id="Mw">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Mw">
<dc:title>Mw</dc:title>
<dcterms:alternative>mRNA of ZnuC</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Mw" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Qw1"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Mw </ci>
</apply>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qw1 </ci>
</apply>
</apply>
</math>
</component>
<!-- The 10 ODEs to describe the ZntR transcription assay -->
<component name="Px" cmeta:id="Px">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Px">
<dc:title>Px</dc:title>
<dcterms:alternative>apo-ZntR</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Px" initial_value="50.0"/>
<variable units="nanomolar" public_interface="in" name="Zn"/>
<variable units="nanomolar" public_interface="in" name="Px1"/>
<variable units="nanomolar" public_interface="in" name="Dz"/>
<variable units="nanomolar" public_interface="in" name="Qz4"/>
<variable units="second_order_rate_constant" public_interface="in" name="r1"/>
<variable units="first_order_rate_constant" public_interface="in" name="r2"/>
<variable units="second_order_rate_constant" public_interface="in" name="k1b"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_1"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Px </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> r2 </ci>
<ci> Px1 </ci>
</apply>
<apply>
<times/>
<ci> k_1 </ci>
<ci> Qz4 </ci>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> r1 </ci>
<ci> Zn </ci>
<ci> Px </ci>
</apply>
<apply>
<times/>
<ci> k1b </ci>
<ci> Dz </ci>
<ci> Px </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Px1" cmeta:id="Px1">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Px1">
<dc:title>Px1</dc:title>
<dcterms:alternative>active ZntR</dcterms:alternative>
<dcterms:alternative>ZnZntR</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Px1" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Zn"/>
<variable units="nanomolar" public_interface="in" name="Px"/>
<variable units="nanomolar" public_interface="in" name="Dz"/>
<variable units="nanomolar" public_interface="in" name="Qz2"/>
<variable units="second_order_rate_constant" public_interface="in" name="r1"/>
<variable units="first_order_rate_constant" public_interface="in" name="r2"/>
<variable units="second_order_rate_constant" public_interface="in" name="k1"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_1"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Px1 </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> r1 </ci>
<ci> Zn </ci>
<ci> Px </ci>
</apply>
<apply>
<times/>
<ci> k_1 </ci>
<ci> Qz2 </ci>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> r2 </ci>
<ci> Px1 </ci>
</apply>
<apply>
<times/>
<ci> k1 </ci>
<ci> Dz </ci>
<ci> Px1 </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Tp" cmeta:id="Tp">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Tp">
<dc:title>Tp</dc:title>
<dcterms:alternative>TPEN</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Tp" initial_value="10000.0"/>
<variable units="nanomolar" public_interface="in" name="Zn"/>
<variable units="nanomolar" public_interface="in" name="Tp1"/>
<variable units="second_order_rate_constant" public_interface="in" name="r5"/>
<variable units="first_order_rate_constant" public_interface="in" name="r6"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Tp </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> r6 </ci>
<ci> Tp1 </ci>
</apply>
<apply>
<times/>
<ci> r5 </ci>
<ci> Zn </ci>
<ci> Tp </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Tp1" cmeta:id="Tp1">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Tp1">
<dc:title>Tp1</dc:title>
<dcterms:alternative>zinc-bound TPEN</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Tp1" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Zn"/>
<variable units="nanomolar" public_interface="in" name="Tp"/>
<variable units="second_order_rate_constant" public_interface="in" name="r5"/>
<variable units="first_order_rate_constant" public_interface="in" name="r6"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Tp1 </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> r5 </ci>
<ci> Zn </ci>
<ci> Tp </ci>
</apply>
<apply>
<times/>
<ci> r6 </ci>
<ci> Tp1 </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Zn" cmeta:id="Zn">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Zn">
<dc:title>Zn</dc:title>
<dcterms:alternative>zinc</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Zn" initial_value="10000.0"/>
<variable units="nanomolar" public_interface="in" name="Tp"/>
<variable units="nanomolar" public_interface="in" name="Tp1"/>
<variable units="nanomolar" public_interface="in" name="Px"/>
<variable units="nanomolar" public_interface="in" name="Px1"/>
<variable units="second_order_rate_constant" public_interface="in" name="r1"/>
<variable units="first_order_rate_constant" public_interface="in" name="r2"/>
<variable units="second_order_rate_constant" public_interface="in" name="r5"/>
<variable units="first_order_rate_constant" public_interface="in" name="r6"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Zn </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> r2 </ci>
<ci> Px1 </ci>
</apply>
<apply>
<times/>
<ci> r6 </ci>
<ci> Tp1 </ci>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> r1 </ci>
<ci> Zn </ci>
<ci> Px </ci>
</apply>
<apply>
<times/>
<ci> r5 </ci>
<ci> Zn </ci>
<ci> Tp </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Dz" cmeta:id="Dz">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Dz">
<dc:title>Dz</dc:title>
<dcterms:alternative>DNA of ZntA</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Dz" initial_value="2.0"/>
<variable units="nanomolar" public_interface="in" name="Qz2"/>
<variable units="nanomolar" public_interface="in" name="Px"/>
<variable units="nanomolar" public_interface="in" name="Qz1"/>
<variable units="nanomolar" public_interface="in" name="Px1"/>
<variable units="nanomolar" public_interface="in" name="Qz4"/>
<variable units="nanomolar" public_interface="in" name="Rz"/>
<variable units="second_order_rate_constant" public_interface="in" name="k1"/>
<variable units="second_order_rate_constant" public_interface="in" name="k1b"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_1"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2a"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Dz </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> k_1 </ci>
<ci> Qz2 </ci>
</apply>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz1 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qz1 </ci>
</apply>
<apply>
<times/>
<ci> k_1 </ci>
<ci> Qz4 </ci>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k1b </ci>
<ci> Dz </ci>
<ci> Px </ci>
</apply>
<apply>
<times/>
<ci> k1 </ci>
<ci> Dz </ci>
<ci> Px1 </ci>
</apply>
<apply>
<times/>
<ci> k2a </ci>
<ci> Dz </ci>
<ci> Rz </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Rz" cmeta:id="Rz">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Rz">
<dc:title>Rz</dc:title>
<dcterms:alternative>RNA polymerase for zntA transcription</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Rz" initial_value="100.0"/>
<variable units="nanomolar" public_interface="in" name="Qz1"/>
<variable units="nanomolar" public_interface="in" name="Dz"/>
<variable units="nanomolar" public_interface="in" name="Qz3"/>
<variable units="nanomolar" public_interface="in" name="Qz4"/>
<variable units="nanomolar" public_interface="in" name="Qz5"/>
<variable units="nanomolar" public_interface="in" name="Qz2"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2b"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2c"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2a"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Rz </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz1 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qz1 </ci>
</apply>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz3 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qz3 </ci>
</apply>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz5 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qz5 </ci>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k2a </ci>
<ci> Dz </ci>
<ci> Rz </ci>
</apply>
<apply>
<times/>
<ci> k2b </ci>
<ci> Qz4 </ci>
<ci> Rz </ci>
</apply>
<apply>
<times/>
<ci> k2c </ci>
<ci> Qz2 </ci>
<ci> Rz </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Qz1" cmeta:id="Qz1">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Qz1">
<dc:title>Qz1</dc:title>
<dcterms:alternative>transcription initiation complex formed by Dz and Rz</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Qz1" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Dz"/>
<variable units="nanomolar" public_interface="in" name="Rz"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2a"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Qz1 </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> k2a </ci>
<ci> Dz </ci>
<ci> Rz </ci>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz1 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qz1 </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Qz2" cmeta:id="Qz2">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Qz2">
<dc:title>Qz2</dc:title>
<dcterms:alternative>ZnZutA-DNA complex</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Qz2" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Dz"/>
<variable units="nanomolar" public_interface="in" name="Rz"/>
<variable units="nanomolar" public_interface="in" name="Qz3"/>
<variable units="nanomolar" public_interface="in" name="Px1"/>
<variable units="second_order_rate_constant" public_interface="in" name="k1"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_1"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2c"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Qz2 </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> k1 </ci>
<ci> Dz </ci>
<ci> Px1 </ci>
</apply>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz3 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qz3 </ci>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k_1 </ci>
<ci> Qz2 </ci>
</apply>
<apply>
<times/>
<ci> k2c </ci>
<ci> Qz2 </ci>
<ci> Rz </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Qz3" cmeta:id="Qz3">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Qz3">
<dc:title>Qz3</dc:title>
<dcterms:alternative>transcription initiation complex formed by Qz2 and Rz</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Qz3" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Qz2"/>
<variable units="nanomolar" public_interface="in" name="Rz"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2c"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Qz3 </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> k2c </ci>
<ci> Qz2 </ci>
<ci> Rz </ci>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz3 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qz3 </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Qz4" cmeta:id="Qz4">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Qz4">
<dc:title>Qz4</dc:title>
<dcterms:alternative>apo-ZntA-DNA complex</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Qz4" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Dz"/>
<variable units="nanomolar" public_interface="in" name="Rz"/>
<variable units="nanomolar" public_interface="in" name="Qz5"/>
<variable units="nanomolar" public_interface="in" name="Px"/>
<variable units="second_order_rate_constant" public_interface="in" name="k1b"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_1"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2b"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Qz4 </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> k1b </ci>
<ci> Dz </ci>
<ci> Px </ci>
</apply>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz5 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qz5 </ci>
</apply>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k_1 </ci>
<ci> Qz4 </ci>
</apply>
<apply>
<times/>
<ci> k2b </ci>
<ci> Qz4 </ci>
<ci> Rz </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Qz5" cmeta:id="Qz5">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Qz5">
<dc:title>Qz5</dc:title>
<dcterms:alternative>transcription initiation complex formed by Qz4 and Rz</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Qz5" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Qz4"/>
<variable units="nanomolar" public_interface="in" name="Rz"/>
<variable units="second_order_rate_constant" public_interface="in" name="k2b"/>
<variable units="first_order_rate_constant" public_interface="in" name="k_2"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Qz5 </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> k2b </ci>
<ci> Qz4 </ci>
<ci> Rz </ci>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz5 </ci>
</apply>
<apply>
<times/>
<ci> k_2 </ci>
<ci> Qz5 </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Mz" cmeta:id="Mz">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#Mz">
<dc:title>Mz</dc:title>
<dcterms:alternative>mRNA of ZnuC</dcterms:alternative>
</rdf:Description>
</rdf:RDF>
<variable units="nanomolar" public_interface="out" name="Mz" initial_value="0.0"/>
<variable units="nanomolar" public_interface="in" name="Qz1"/>
<variable units="nanomolar" public_interface="in" name="Qz3"/>
<variable units="nanomolar" public_interface="in" name="Qz5"/>
<variable units="first_order_rate_constant" public_interface="in" name="k3"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci> Mz </ci>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz1 </ci>
</apply>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz3 </ci>
</apply>
<apply>
<times/>
<ci> k3 </ci>
<ci> Qz5 </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="model_parameters">
<variable units="second_order_rate_constant" public_interface="out" name="r1" initial_value="2.73E2"/>
<variable units="first_order_rate_constant" public_interface="out" name="r2" initial_value="3.437E-4"/>
<variable units="third_order_rate_constant" public_interface="out" name="r3" initial_value="4.41E10"/>
<variable units="first_order_rate_constant" public_interface="out" name="r4" initial_value="9E-3"/>
<variable units="second_order_rate_constant" public_interface="out" name="r5" initial_value="3E4"/>
<variable units="first_order_rate_constant" public_interface="out" name="r6" initial_value="1.506E-2"/>
<variable units="second_order_rate_constant" public_interface="out" name="k1" initial_value="0.025"/>
<variable units="second_order_rate_constant" public_interface="out" name="k1a" initial_value="1.0"/>
<variable units="second_order_rate_constant" public_interface="out" name="k1b" initial_value="1.253E-2"/>
<variable units="first_order_rate_constant" public_interface="out" name="k_1" initial_value="0.9"/>
<variable units="second_order_rate_constant" public_interface="out" name="k2" initial_value="0.02"/>
<variable units="second_order_rate_constant" public_interface="out" name="k2a" initial_value="0.00005"/>
<variable units="second_order_rate_constant" public_interface="out" name="k2b" initial_value="0.0002"/>
<variable units="second_order_rate_constant" public_interface="out" name="k2c" initial_value="0.0037"/>
<variable units="first_order_rate_constant" public_interface="out" name="k_2" initial_value="0.3"/>
<variable units="first_order_rate_constant" public_interface="out" name="k3"/>
<variable units="second" name="td0" initial_value="1800.0"/>
<variable units="second" name="td" initial_value="2700"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci> k3 </ci>
<piecewise>
<piece>
<cn cellml:units="first_order_rate_constant"> 0.0 </cn>
<apply>
<and/>
<apply>
<geq/>
<ci> time </ci>
<cn cellml:units="second"> 0.0 </cn>
</apply>
<apply>
<lt/>
<ci> time </ci>
<ci> td0 </ci>
</apply>
</apply>
</piece>
<piece>
<cn cellml:units="first_order_rate_constant"> 0.011 </cn>
<apply>
<and/>
<apply>
<geq/>
<ci> time </ci>
<ci> td0 </ci>
</apply>
<apply>
<lt/>
<ci> time </ci>
<ci> td </ci>
</apply>
</apply>
</piece>
</piecewise>
</apply>
</math>
</component>
<connection>
<map_components component_2="environment" component_1="Py"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Py1"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Dw"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Rw"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Qw1"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Qw2"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Mw"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Px"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Px1"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Tp"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Tp1"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Zn"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Dz"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Rz"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Qz1"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Qz2"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Qz3"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Qz4"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Qz5"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Mz"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="model_parameters"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="Py" component_1="Py1"/>
<map_variables variable_2="Py" variable_1="Py"/>
<map_variables variable_2="Py1" variable_1="Py1"/>
</connection>
<connection>
<map_components component_2="Py" component_1="Zn"/>
<map_variables variable_2="Zn" variable_1="Zn"/>
</connection>
<connection>
<map_components component_2="Py" component_1="model_parameters"/>
<map_variables variable_2="r3" variable_1="r3"/>
<map_variables variable_2="r4" variable_1="r4"/>
</connection>
<connection>
<map_components component_2="Py1" component_1="Dw"/>
<map_variables variable_2="Dw" variable_1="Dw"/>
<map_variables variable_2="Py1" variable_1="Py1"/>
</connection>
<connection>
<map_components component_2="Py1" component_1="Qw2"/>
<map_variables variable_2="Qw2" variable_1="Qw2"/>
<map_variables variable_2="Py1" variable_1="Py1"/>
</connection>
<connection>
<map_components component_2="Py1" component_1="Zn"/>
<map_variables variable_2="Zn" variable_1="Zn"/>
</connection>
<connection>
<map_components component_2="Py1" component_1="model_parameters"/>
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