Location: Elowitz, Leibler, 2000 @ cb2913da4514 / elowitz_leibler_2000.cellml

Author:
Dougal Cowan <devnull@localhost>
Date:
2011-05-13 09:11:25+12:00
Desc:
added cmeta:id to appropriate variables in the cellml fixed the file:// references within and rebuilt the session file
Permanent Source URI:
https://teaching.physiomeproject.org/workspace/elowitz_leibler_2000/rawfile/cb2913da4514ce15e56e9c2b46cfb3096525e49b/elowitz_leibler_2000.cellml

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				<title>A synthetic oscillatory network of transcriptional regulators. </title>
				<author>
					<firstname>Jeelean</firstname>
					<surname>Lim</surname>
					<affiliation>
						<shortaffil>Auckland Bioengineering Institute, University of Auckland</shortaffil>
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			<section id="sec_status">
				<title>Model Status</title>
				<para>
            This CellML version of this model is not able to reproduce the results shown in Fig1c of the original publication, as the initial conditions for protein concentrations were not provided. The original published equations were scaled and modified with reference to the same model on the Biomodels database (<ulink url="http://biomodels.caltech.edu/BIOMD0000000012">BIOMD0000000012 - Elowitz2000_Repressilator</ulink>). Once the model comes to equilibrium (t &gt; 400 minutes,) its output is correct. 
</para>
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The units have been checked in this model and are consistent. 
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				<title>Model Structure</title>
				<para>
ABSTRACT:  Networks of interacting biomolecules carry out many essential functions in living cells, but the 'design principles' underlying the
functioning of such intracellular networks remain poorly understood, despite intensive efforts including quantitative analysis of relatively simple systems. Here we present a complementary approach to this problem: the design and construction of a synthetic network to implement a particular function.
				</para>
				<para>
We used three transcriptional repressor systems that are not part of any natural biological clock to build an oscillating network, termed
the repressilator, in Escherichia coli. The network periodically induces the synthesis of green fluorescent protein as a readout of
its state in individual cells. The resulting oscillations, with typical periods of hours, are slower than the cell-division cycle, so the
state of the oscillator has to be transmitted from generation to generation. This artificial clock displays noisy behaviour, possibly
because of stochastic fluctuations of its components. Such 'rational network design' may lead both to the engineering of
new cellular behaviours and to an improved understanding of naturally occurring networks.
				</para>
				<para>
The complete original paper reference is cited below:
				</para>
				<para>
A synthetic oscillatory network of transcriptional regulators, Michael B. Elowitz and Stanislas Leibler, 2000,<emphasis> Nature: International Weekly Journal of Science</emphasis>, 403, 335-338. <ulink url="http://www.ncbi.nlm.nih.gov/pubmed/10659856">PubMed ID: 10659856</ulink>
				</para>
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								<title>Figure 1a</title>
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					<caption>The repressilator network.</caption>
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			<rdf:value xml:lang="en">This is the CellML description of Elowitz and Leibler's mathematical model on the synthetic oscillatory network of transcriptional regulators</rdf:value>
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			<rdf:value xml:lang="en">There was an error in rescaling previously:
m in the equations in box 1 is not rescaled to the mRNA divided by the translation efficiency, M/eff,  but is just M*eff/K_m, with the efficiency taken in units of rescaled protein p = P/K_m.
The 2 forms of the differential equations are therefore:
dM_i/dt = K_m/(eff*tavg) * alpha * Km^n/(Km^n+P_j^n) + alpha0 - M_i/tavg
and
dPi/dt = beta*(Mi*eff/tavg - Pi/tavg)

Rescaled protein numbers were also used for the alphas:
beta = proteindecay/mRNAdecay = rnahalflife/proteinhalflife = 0.2
and
alpha0 = number of maximal rescaled proteins per cell in steady state under full repression:
with a0 = leaky promotor strength = 5*10^-4 mRNA per second
max. translation = eff*a0
protein decay = Pi/average_protein_lifetime
in steady state: max. translation = protein decay =&gt;
P_max = eff*a0*average_protein_lifetime(in seconds) = 20*5*10^(-4)*10/ln(2)*60 = 8.656
alpha0 = p_max = P_max/K_m =  0.216
and for the completely repressor free state:
a = fully induced promotor strength = 0.5 mRNAs per second
P_max= 20*0.5*10/ln(2)*60 = 8656.2
p_max =  216.4 = alpha + alpha0
alpha = 216.2

These corrections seem to give more sensible results. The protein numbers are still the same, but the mRNA numbers are only about 1/15th of the proteins.</rdf:value>
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			<rdf:_1 xml:lang="en">synthetic biology</rdf:_1>
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			<title xmlns="http://purl.org/dc/elements/1.1/" xml:lang="en">A Synthetic Oscillatory Network of Transciptional Regulators</title>
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			<rdf:value xml:lang="en">This CellML version of this model is not able to reproduce the results shown in Fig1c of the original publication, as the initial conditions for protein concentrations were not provided. The original published equations were scaled and modified with reference to the same model on the Biomodels database BIOMD0000000012 - Elowitz2000_Repressilator). Once the model comes to equilibrium (t &gt; 400 minutes,) its output is correct. </rdf:value>
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			<N xmlns="http://www.w3.org/2001/vcard-rdf/3.0#">
				<rdf:Description rdf:about="rdf:#3ee320fe-afa0-4755-84cc-dc3eecf46b85"/>
			</N>
		</rdf:Description>
	</rdf:RDF>   
	
   <units name="minute">
      <unit multiplier="60" units="second"/>
   </units>
   <units name="first_order_rate_constant">
      <unit exponent="-1" units="minute"/>
   </units>
   <units name="per_sec">
      <unit exponent="-1" units="second"/>
   </units>
   <units name="second_per_minute">
      <unit units="second"/>
	  <unit exponent="-1" units="minute"/>
   </units>
   
   <component name="environment">
      <variable cmeta:id="environment_time" name="time" public_interface="out" units="minute"/>
   </component>
   
   <component name="parameters">
      <variable initial_value="2" name="n" public_interface="out" units="dimensionless"/>
      <variable name="alpha_0" public_interface="out" units="dimensionless"/>
      <variable name="alpha" public_interface="out" units="dimensionless"/>
      <variable name="beta" public_interface="out" units="dimensionless"/>
      <variable initial_value="40" name="K_m" public_interface="out" units="dimensionless"/>
      <variable initial_value="20" name="efficiency" public_interface="out" units="dimensionless"/>
      <variable initial_value="2" name="mRNA_halflife" public_interface="out" units="minute"/>
      <variable name="t_ave" public_interface="out" units="minute"/>
      <variable name="kd_prot" public_interface="out" units="first_order_rate_constant"/>
      <variable name="kd_mRNA" public_interface="out" units="first_order_rate_constant"/>
      <variable initial_value="10" name="prot_halflife" public_interface="out" units="minute"/>
      <variable initial_value="5e-4" name="tps_repr" public_interface="out" units="per_sec"/>
      <variable initial_value="0.5" name="tps_active" public_interface="out" units="per_sec"/>
      <variable name="k_tl" public_interface="out" units="first_order_rate_constant"/>
      <variable name="a_tr" public_interface="out" units="first_order_rate_constant"/>
      <variable name="a0_tr" public_interface="out" units="first_order_rate_constant"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>t_ave</ci>
            <apply>
               <divide/>
               <ci>mRNA_halflife</ci>
               <apply>
                  <ln/>
                  <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>kd_prot</ci>
            <apply>
               <divide/>
               <apply>
                  <ln/>
                  <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn>
               </apply>
               <ci>prot_halflife</ci>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>kd_mRNA</ci>
            <apply>
               <divide/>
               <apply>
                  <ln/>
                  <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn>
               </apply>
               <ci>mRNA_halflife</ci>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>k_tl</ci>
            <apply>
               <times/>
               <ci>efficiency</ci>
               <ci>kd_mRNA</ci>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>a_tr</ci>
            <apply>
               <times/>
               <apply>
                  <minus/>
                  <ci>tps_active</ci>
                  <ci>tps_repr</ci>
               </apply>
               <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="second_per_minute">60</cn>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>a0_tr</ci>
            <apply>
               <times/>
               <ci>tps_repr</ci>
               <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="second_per_minute">60</cn>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>alpha</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <ci>a_tr</ci>
                  <ci>efficiency</ci>
               </apply>
               <apply>
                  <times/>
                  <ci>kd_prot</ci>
                  <ci>K_m</ci>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>alpha_0</ci>
            <apply>
               <divide/>
               <apply>
                  <times/>
                  <ci>a0_tr</ci>
                  <ci>efficiency</ci>
               </apply>
               <apply>
                  <times/>
                  <ci>kd_prot</ci>
                  <ci>K_m</ci>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>beta</ci>
            <apply>
               <divide/>
               <ci>kd_prot</ci>
               <ci>kd_mRNA</ci>
            </apply>
         </apply>
      </math>
   </component>
   <component name="M_lacl">
      <variable initial_value="0" name="M_lacl" public_interface="out" units="dimensionless"/>
      <variable name="a_tr" public_interface="in" units="first_order_rate_constant"/>
      <variable name="P_cl" public_interface="in" units="dimensionless"/>
      <variable name="n" public_interface="in" units="dimensionless"/>
      <variable name="a0_tr" public_interface="in" units="first_order_rate_constant"/>
      <variable name="K_m" public_interface="in" units="dimensionless"/>
      <variable name="kd_mRNA" public_interface="in" units="first_order_rate_constant"/>
      <variable name="time" public_interface="in" units="minute"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>M_lacl</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <plus/>
                  <ci>a0_tr</ci>
                  <apply>
                     <divide/>
                     <apply>
                        <times/>
                        <ci>a_tr</ci>
                        <apply>
                           <power/>
                           <ci>K_m</ci>
                           <ci>n</ci>
                        </apply>
                     </apply>
                     <apply>
                        <plus/>
                        <apply>
                           <power/>
                           <ci>K_m</ci>
                           <ci>n</ci>
                        </apply>
                        <apply>
                           <power/>
                           <ci>P_cl</ci>
                           <ci>n</ci>
                        </apply>
                     </apply>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>kd_mRNA</ci>
                  <ci>M_lacl</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="M_tetR">
      <variable initial_value="20" name="M_tetR" public_interface="out" units="dimensionless"/>
      <variable name="a_tr" public_interface="in" units="first_order_rate_constant"/>
      <variable name="P_lacl" public_interface="in" units="dimensionless"/>
      <variable name="n" public_interface="in" units="dimensionless"/>
      <variable name="a0_tr" public_interface="in" units="first_order_rate_constant"/>
      <variable name="K_m" public_interface="in" units="dimensionless"/>
      <variable name="kd_mRNA" public_interface="in" units="first_order_rate_constant"/>
      <variable name="time" public_interface="in" units="minute"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>M_tetR</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <plus/>
                  <apply>
                     <divide/>
                     <apply>
                        <times/>
                        <ci>a_tr</ci>
                        <apply>
                           <power/>
                           <ci>K_m</ci>
                           <ci>n</ci>
                        </apply>
                     </apply>
                     <apply>
                        <plus/>
                        <apply>
                           <power/>
                           <ci>K_m</ci>
                           <ci>n</ci>
                        </apply>
                        <apply>
                           <power/>
                           <ci>P_lacl</ci>
                           <ci>n</ci>
                        </apply>
                     </apply>
                  </apply>
                  <ci>a0_tr</ci>
               </apply>
               <apply>
                  <times/>
                  <ci>kd_mRNA</ci>
                  <ci>M_tetR</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="M_cl">
      <variable initial_value="0" name="M_cl" public_interface="out" units="dimensionless"/>
      <variable name="a_tr" public_interface="in" units="first_order_rate_constant"/>
      <variable name="P_tetR" public_interface="in" units="dimensionless"/>
      <variable name="n" public_interface="in" units="dimensionless"/>
      <variable name="a0_tr" public_interface="in" units="first_order_rate_constant"/>
      <variable name="K_m" public_interface="in" units="dimensionless"/>
      <variable name="kd_mRNA" public_interface="in" units="first_order_rate_constant"/>
      <variable name="time" public_interface="in" units="minute"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>M_cl</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <plus/>
                  <apply>
                     <divide/>
                     <apply>
                        <times/>
                        <ci>a_tr</ci>
                        <apply>
                           <power/>
                           <ci>K_m</ci>
                           <ci>n</ci>
                        </apply>
                     </apply>
                     <apply>
                        <plus/>
                        <apply>
                           <power/>
                           <ci>K_m</ci>
                           <ci>n</ci>
                        </apply>
                        <apply>
                           <power/>
                           <ci>P_tetR</ci>
                           <ci>n</ci>
                        </apply>
                     </apply>
                  </apply>
                  <ci>a0_tr</ci>
               </apply>
               <apply>
                  <times/>
                  <ci>kd_mRNA</ci>
                  <ci>M_cl</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="P_lacl">
      <variable cmeta:id="P_lacl_P_lacl" initial_value="0" name="P_lacl" public_interface="out" units="dimensionless"/>
      <variable name="k_tl" public_interface="in" units="first_order_rate_constant"/>
      <variable name="M_lacl" public_interface="in" units="dimensionless"/>
      <variable name="kd_prot" public_interface="in" units="first_order_rate_constant"/>
      <variable name="time" public_interface="in" units="minute"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>P_lacl</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>k_tl</ci>
                  <ci>M_lacl</ci>
               </apply>
               <apply>
                  <times/>
                  <ci>kd_prot</ci>
                  <ci>P_lacl</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="P_tetR">
      <variable cmeta:id="P_tetR_P_tetR" initial_value="0" name="P_tetR" public_interface="out" units="dimensionless"/>
      <variable name="k_tl" public_interface="in" units="first_order_rate_constant"/>
      <variable name="M_tetR" public_interface="in" units="dimensionless"/>
      <variable name="kd_prot" public_interface="in" units="first_order_rate_constant"/>
      <variable name="time" public_interface="in" units="minute"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>P_tetR</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>k_tl</ci>
                  <ci>M_tetR</ci>
               </apply>
               <apply>
                  <times/>
                  <ci>kd_prot</ci>
                  <ci>P_tetR</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component name="P_cl">
      <variable cmeta:id="P_cl_P_cl" initial_value="0" name="P_cl" public_interface="out" units="dimensionless"/>
      <variable name="k_tl" public_interface="in" units="first_order_rate_constant"/>
      <variable name="M_cl" public_interface="in" units="dimensionless"/>
      <variable name="kd_prot" public_interface="in" units="first_order_rate_constant"/>
      <variable name="time" public_interface="in" units="minute"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>P_cl</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>k_tl</ci>
                  <ci>M_cl</ci>
               </apply>
               <apply>
                  <times/>
                  <ci>kd_prot</ci>
                  <ci>P_cl</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <connection>
      <map_components component_1="M_lacl" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection>
      <map_components component_1="M_tetR" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection>
      <map_components component_1="M_cl" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection>
      <map_components component_1="P_lacl" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection>
      <map_components component_1="P_tetR" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection>
      <map_components component_1="P_cl" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection>
      <map_components component_1="M_lacl" component_2="parameters"/>
      <map_variables variable_1="a_tr" variable_2="a_tr"/>
      <map_variables variable_1="n" variable_2="n"/>
      <map_variables variable_1="a0_tr" variable_2="a0_tr"/>
      <map_variables variable_1="K_m" variable_2="K_m"/>
      <map_variables variable_1="kd_mRNA" variable_2="kd_mRNA"/>
   </connection>
   <connection>
      <map_components component_1="M_lacl" component_2="P_cl"/>
      <map_variables variable_1="P_cl" variable_2="P_cl"/>
   </connection>
   <connection>
      <map_components component_1="M_tetR" component_2="parameters"/>
      <map_variables variable_1="a_tr" variable_2="a_tr"/>
      <map_variables variable_1="n" variable_2="n"/>
      <map_variables variable_1="a0_tr" variable_2="a0_tr"/>
      <map_variables variable_1="K_m" variable_2="K_m"/>
      <map_variables variable_1="kd_mRNA" variable_2="kd_mRNA"/>
   </connection>
   <connection>
      <map_components component_1="M_tetR" component_2="P_lacl"/>
      <map_variables variable_1="P_lacl" variable_2="P_lacl"/>
   </connection>
   <connection>
      <map_components component_1="M_cl" component_2="parameters"/>
      <map_variables variable_1="a_tr" variable_2="a_tr"/>
      <map_variables variable_1="n" variable_2="n"/>
      <map_variables variable_1="a0_tr" variable_2="a0_tr"/>
      <map_variables variable_1="K_m" variable_2="K_m"/>
      <map_variables variable_1="kd_mRNA" variable_2="kd_mRNA"/>
   </connection>
   <connection>
      <map_components component_1="M_cl" component_2="P_tetR"/>
      <map_variables variable_1="P_tetR" variable_2="P_tetR"/>
   </connection>
   <connection>
      <map_components component_1="P_lacl" component_2="parameters"/>
      <map_variables variable_1="k_tl" variable_2="k_tl"/>
      <map_variables variable_1="kd_prot" variable_2="kd_prot"/>
   </connection>
   <connection>
      <map_components component_1="P_lacl" component_2="M_lacl"/>
      <map_variables variable_1="M_lacl" variable_2="M_lacl"/>
   </connection>
   <connection>
      <map_components component_1="P_tetR" component_2="parameters"/>
      <map_variables variable_1="k_tl" variable_2="k_tl"/>
      <map_variables variable_1="kd_prot" variable_2="kd_prot"/>
   </connection>
   <connection>
      <map_components component_1="P_tetR" component_2="M_tetR"/>
      <map_variables variable_1="M_tetR" variable_2="M_tetR"/>
   </connection>
   <connection>
      <map_components component_1="P_cl" component_2="parameters"/>
      <map_variables variable_1="k_tl" variable_2="k_tl"/>
      <map_variables variable_1="kd_prot" variable_2="kd_prot"/>
   </connection>
   <connection>
      <map_components component_1="P_cl" component_2="M_cl"/>
      <map_variables variable_1="M_cl" variable_2="M_cl"/>
   </connection>
</model>