Rendering of the source text

<?xml version="1.0"?>
<model xmlns="http://www.cellml.org/cellml/1.0#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" cmeta:id="alexander_2010" name="alexander_2010">

<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
  <articleinfo>
  <title>Self-tolerance and Autoimmunity in a Regulatory T Cell Model</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 represents system 1 (equations 1a-1d) in the original publication.  The model runs in both COR and OpenCell to replicate the published results (figure 2) with R0=0.8.  The units have been checked and they are consistent.
          </para>
  </section>
  <sect1 id="sec_structure">
<title>Model Structure</title>

<para>
ABSTRACT: The class of immunosuppressive lymphocytes known as regulatory T cells (Tregs) has been identified as a key component in preventing autoimmune diseases. Although Tregs have been incorporated previously in mathematical models of autoimmunity, we take a novel approach which emphasizes the importance of professional antigen presenting cells (pAPCs). We examine three possible mechanisms of Treg action (each in isolation) through ordinary differential equation (ODE) models. The immune response against a particular autoantigen is suppressed both by Tregs specific for that antigen and by Tregs of arbitrary specificities, through their action on either maturing or already mature pAPCs or on autoreactive effector T cells. In this deterministic approach, we find that qualitative long-term behaviour is predicted by the basic reproductive ratio R(0) for each system. When R(0) is less tHAN 1, only the trivial equilibrium exists and is stable; when R(0) is greater than 1, this equilibrium loses its stability and a stable non-trivial equilibrium appears. We interpret the absence of self-damaging populations at the trivial equilibrium to imply a state of self-tolerance, and their presence at the non-trivial equilibrium to imply a state of chronic autoimmunity. Irrespective of mechanism, our model predicts that Tregs specific for the autoantigen in question play no role in the system's qualitative long-term behaviour, but have quantitative effects that could potentially reduce an autoimmune response to sub-clinical levels. Our results also suggest an important role for Tregs of arbitrary specificities in modulating the qualitative outcome. A stochastic treatment of the same model demonstrates that the probability of developing a chronic autoimmune response increases with the initial exposure to self antigen or autoreactive effector T cells. The three different mechanisms we consider, while leading to a number of similar predictions, also exhibit key differences in both transient dynamics (ODE approach) and the probability of chronic autoimmunity (stochastic approach).
</para>

<informalfigure float="0" id="fig_reaction_diagram">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>model diagram</title>
    </objectinfo>
    <imagedata fileref="alexander_2010.png"/>
  </imageobject>
</mediaobject>
<caption>Flow chart illustrating interactions among populations and flow in/out of compartments in System 1. Populations to be modelled explicitly are in black boxes; those that are considered only implicitly or as intermediaries are in grey boxes. Movement in or out of a compartment is indicated by a black arrow; activating influences are indicated with green arrows; suppressive influences are indicated with red arrows; and other interactions or effects are indicated with blue arrows.</caption>
</informalfigure>

<para>
The original paper reference is cited below:
</para>

<para>
Self-tolerance and Autoimmunity in a Regulatory T Cell Model, Alexander HK and Wahl LM, 2010, <emphasis>Bulletin of Mathematical Biology</emphasis>.  <ulink url="http://www.ncbi.nlm.nih.gov/pubmed/20195912">PubMed ID: 20195912</ulink></para>

</sect1>
</article>
</documentation> 



  
  <units name="day">
    <unit multiplier="86400.0" units="second"/>
  </units>
  
  <units name="first_order_rate_constant">
    <unit exponent="-1" units="day"/>
  </units>
  
  
  <component name="environment">
    <variable cmeta:id="environment_time" name="time" public_interface="out" units="day"/>
  </component>
  
  <component cmeta:id="A" name="A">
      
    <variable cmeta:id="A_A" initial_value="1.0" name="A" public_interface="out" units="dimensionless"/>
    <variable initial_value="0.0025" name="v_tilday" public_interface="out" units="first_order_rate_constant"/>
    
    <variable initial_value="1e-4" name="f" units="dimensionless"/>
    <variable initial_value="3e-6" name="sigma1" units="first_order_rate_constant"/>
    <variable initial_value="0.25" name="b1" units="first_order_rate_constant"/>
	<variable initial_value="0.25" name="muA" units="first_order_rate_constant"/>
	
	<variable name="G" public_interface="in" units="dimensionless"/>
    <variable name="R" public_interface="in" units="dimensionless"/>
    <variable name="time" public_interface="in" units="day"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML"> 
      <apply>
        <eq/> 
        <apply>
          <diff/> 
          <bvar>
            <ci> time </ci>
          </bvar> 
          <ci> A </ci> 
        </apply>
        <apply>
          <minus/>
          <apply>
            <times/>
            <ci> f </ci>
            <ci> v_tilday </ci>
            <ci> G </ci>
          </apply>
          <apply>
            <plus/>
            <apply>
              <times/>
              <apply>
                <plus/>
                <apply>
                  <times/>
				  <ci> sigma1 </ci>
                  <ci> R </ci>
				</apply> 
			    <ci> b1 </ci>
		      </apply> 
			  <ci> A </ci>
		    </apply> 
            <apply>
              <times/>
              <ci> muA </ci> 
              <ci> A </ci>
            </apply> 
          </apply>
        </apply>
      </apply>    
    </math>
  </component>
  
  <component cmeta:id="R" name="R">
      
    <variable cmeta:id="R_R" initial_value="0.0" name="R" public_interface="out" units="dimensionless"/>
    
    <variable initial_value="0.016" name="pi1" units="first_order_rate_constant"/>
    <variable initial_value="200.0" name="beta" units="first_order_rate_constant"/>
    <variable initial_value="0.25" name="muR" units="first_order_rate_constant"/>
    
	<variable name="A" public_interface="in" units="dimensionless"/>
    <variable name="E" public_interface="in" units="dimensionless"/>
    <variable name="time" public_interface="in" units="day"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML"> 
      <apply>
        <eq/> 
        <apply>
          <diff/> 
          <bvar>
            <ci>time</ci>
          </bvar> 
          <ci>R</ci> 
        </apply>
        <apply>
          <minus/>
          <apply>
            <times/>
            <apply>
              <plus/>
              <apply>
                <times/>
                <ci> pi1 </ci> 
                <ci> E </ci>
              </apply> 
			  <ci> beta </ci>
            </apply>
			<ci> A </ci>
          </apply>
          <apply>
            <times/>
            <ci> muR </ci>
            <ci> R </ci>
          </apply> 
        </apply>
      </apply>    
    </math>
  </component>
  
  <component cmeta:id="E" name="E">
      
    <variable cmeta:id="E_E" initial_value="0.0" name="E" public_interface="out" units="dimensionless"/>
    
    <variable initial_value="1000.0" name="lambdaE" units="first_order_rate_constant"/>
    <variable initial_value="0.25" name="muE" units="first_order_rate_constant"/>
    
	<variable name="A" public_interface="in" units="dimensionless"/>
    <variable name="time" public_interface="in" units="day"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML"> 
      <apply>
        <eq/> 
        <apply>
          <diff/> 
          <bvar>
            <ci>time</ci>
          </bvar> 
          <ci>E</ci> 
        </apply>
        <apply>
          <minus/>
          <apply>
            <times/>
            <ci> lambdaE </ci> 
            <ci> A </ci>
          </apply>
          <apply>
            <times/>
            <ci> muE </ci>
            <ci> E </ci>
          </apply> 
        </apply>
      </apply>    
    </math>
  </component>
  
  <component cmeta:id="G" name="G">
      
    <variable cmeta:id="G_G" initial_value="1e8" name="G" public_interface="out" units="dimensionless"/>
    
    <variable initial_value="2000.0" name="gamma" units="first_order_rate_constant"/>
    <variable initial_value="5.0" name="muG" units="first_order_rate_constant"/>
    
	<variable name="v_tilday" public_interface="in" units="first_order_rate_constant"/>
	<variable name="E" public_interface="in" units="dimensionless"/>
    <variable name="time" public_interface="in" units="day"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML"> 
      <apply>
        <eq/> 
        <apply>
          <diff/> 
          <bvar>
            <ci>time</ci>
          </bvar> 
          <ci>G</ci> 
        </apply>
        <apply>
          <minus/>
          <apply>
            <times/>
            <ci> gamma </ci> 
            <ci> E </ci>
          </apply>
          <apply>
            <plus/>		  
			<apply>
              <times/>
              <ci> v_tilday </ci>
              <ci> G </ci>
            </apply> 
			<apply>
              <times/>
              <ci> muG </ci>
              <ci> G </ci>
            </apply> 
		  </apply>
        </apply>
      </apply>    
    </math>
  </component>

  
  <connection>
    <map_components component_1="A" component_2="environment"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
  
  <connection>
    <map_components component_1="R" component_2="environment"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
  
  <connection>
    <map_components component_1="E" component_2="environment"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
  
  <connection>
    <map_components component_1="G" component_2="environment"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
 
  <connection>
    <map_components component_1="A" component_2="R"/>
    <map_variables variable_1="R" variable_2="R"/>
    <map_variables variable_1="A" variable_2="A"/>
  </connection>
  
  <connection>
    <map_components component_1="A" component_2="G"/>
    <map_variables variable_1="G" variable_2="G"/>
    <map_variables variable_1="v_tilday" variable_2="v_tilday"/>
  </connection>
  
  <connection>
    <map_components component_1="R" component_2="E"/>
    <map_variables variable_1="E" variable_2="E"/>
  </connection>
  
  <connection>
    <map_components component_1="E" component_2="A"/>
    <map_variables variable_1="A" variable_2="A"/>
  </connection>
  
  <connection>
    <map_components component_1="G" component_2="E"/>
    <map_variables variable_1="E" variable_2="E"/>
  </connection>
  
  
  
  
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"><rdf:Description rdf:nodeID="n1"><W3CDTF xmlns="http://purl.org/dc/terms/" xml:lang="en">2010-03-02</W3CDTF></rdf:Description><rdf:Description rdf:nodeID="n2"><title xmlns="http://purl.org/dc/elements/1.1/" xml:lang="en">Bulletin of Mathematical Biology</title></rdf:Description><rdf:Description rdf:nodeID="n3"><JournalArticle xmlns="http://www.cellml.org/bqs/1.0#"><rdf:Description rdf:nodeID="n4"/></JournalArticle><Pubmed_id xmlns="http://www.cellml.org/bqs/1.0#" xml:lang="en">20195912</Pubmed_id></rdf:Description><rdf:Description rdf:nodeID="n5"><Given xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">Catherine</Given><Family xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">Lloyd</Family><Other xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">May</Other></rdf:Description><rdf:Description rdf:nodeID="n6"><rdf:type><rdf:Description rdf:about="http://www.w3.org/1999/02/22-rdf-syntax-ns#Bag"/></rdf:type><rdf:_2 xml:lang="en">autoimmune</rdf:_2><rdf:_1 xml:lang="en">immunology</rdf:_1><rdf:_3 xml:lang="en">regulatory T cell</rdf:_3></rdf:Description><rdf:Description rdf:nodeID="n7"><subject_type xmlns="http://www.cellml.org/bqs/1.0#" xml:lang="en">keyword</subject_type><rdf:value><rdf:Description rdf:nodeID="n6"/></rdf:value></rdf:Description><rdf:Description rdf:about="#A"><alternative xmlns="http://purl.org/dc/terms/" xml:lang="en">mature antigen presenting cells</alternative><alternative xmlns="http://purl.org/dc/terms/" xml:lang="en">dendritic cells</alternative><alternative xmlns="http://purl.org/dc/terms/" xml:lang="en">mature pAPCs</alternative><title xmlns="http://purl.org/dc/elements/1.1/" xml:lang="en">A</title></rdf:Description><rdf:Description rdf:about="#alexander_2010"><reference xmlns="http://www.cellml.org/bqs/1.0#"><rdf:Description rdf:nodeID="n3"/></reference><reference xmlns="http://www.cellml.org/bqs/1.0#"><rdf:Description rdf:nodeID="n8"/></reference><simulation xmlns="http://www.cellml.org/metadata/simulation/1.0#"><rdf:Description rdf:nodeID="n9"/></simulation></rdf:Description><rdf:Description rdf:nodeID="n10"><N xmlns="http://www.w3.org/2001/vcard-rdf/3.0#"><rdf:Description rdf:nodeID="n11"/></N></rdf:Description><rdf:Description rdf:nodeID="n8"><subject xmlns="http://purl.org/dc/elements/1.1/"><rdf:Description rdf:nodeID="n7"/></subject></rdf:Description><rdf:Description rdf:about="#E"><alternative xmlns="http://purl.org/dc/terms/" xml:lang="en">effector T cells</alternative><title xmlns="http://purl.org/dc/elements/1.1/" xml:lang="en">E</title></rdf:Description><rdf:Description rdf:nodeID="n12"><Person xmlns="http://www.cellml.org/bqs/1.0#"><rdf:Description rdf:nodeID="n13"/></Person></rdf:Description><rdf:Description rdf:nodeID="n14"><Orgunit xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">Auckland Bioengineering Institute</Orgunit><Orgname xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">The University of Auckland</Orgname></rdf:Description><rdf:Description rdf:nodeID="n15"><Person xmlns="http://www.cellml.org/bqs/1.0#"><rdf:Description rdf:nodeID="n10"/></Person></rdf:Description><rdf:Description rdf:nodeID="n16"><N xmlns="http://www.w3.org/2001/vcard-rdf/3.0#"><rdf:Description rdf:nodeID="n5"/></N><ORG xmlns="http://www.w3.org/2001/vcard-rdf/3.0#"><rdf:Description rdf:nodeID="n14"/></ORG><EMAIL xmlns="http://www.w3.org/2001/vcard-rdf/3.0#"><rdf:Description rdf:nodeID="n17"/></EMAIL></rdf:Description><rdf:Description rdf:nodeID="n18"><rdf:type><rdf:Description rdf:about="http://www.w3.org/1999/02/22-rdf-syntax-ns#Seq"/></rdf:type><rdf:_2><rdf:Description rdf:nodeID="n12"/></rdf:_2><rdf:_1><rdf:Description rdf:nodeID="n15"/></rdf:_1></rdf:Description><rdf:Description rdf:nodeID="n4"><title xmlns="http://purl.org/dc/elements/1.1/" xml:lang="en">
            Self-tolerance and Autoimmunity in a Regulatory T Cell Model
          </title><creator xmlns="http://purl.org/dc/elements/1.1/"><rdf:Description rdf:nodeID="n18"/></creator><issued xmlns="http://purl.org/dc/terms/"><rdf:Description rdf:nodeID="n1"/></issued><Journal xmlns="http://www.cellml.org/bqs/1.0#"><rdf:Description rdf:nodeID="n2"/></Journal></rdf:Description><rdf:Description rdf:nodeID="n13"><N xmlns="http://www.w3.org/2001/vcard-rdf/3.0#"><rdf:Description rdf:nodeID="n19"/></N></rdf:Description><rdf:Description rdf:nodeID="n20"><rdf:first><rdf:Description rdf:nodeID="n21"/></rdf:first><rdf:rest><rdf:Description rdf:about="http://www.w3.org/1999/02/22-rdf-syntax-ns#nil"/></rdf:rest></rdf:Description><rdf:Description rdf:about=""><title xmlns="http://purl.org/dc/elements/1.1/" xml:lang="en">
        Self-tolerance and Autoimmunity in a Regulatory T Cell Model: System 1
      </title><creator xmlns="http://purl.org/dc/elements/1.1/"><rdf:Description rdf:nodeID="n16"/></creator></rdf:Description><rdf:Description rdf:nodeID="n19"><Given xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">L</Given><Family xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">Wahl</Family><Other xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">M</Other></rdf:Description><rdf:Description rdf:nodeID="n17"><rdf:type><rdf:Description rdf:about="http://imc.org/vCard/3.0#internet"/></rdf:type><rdf:value xml:lang="en">c.lloyd@auckland.ac.nz</rdf:value></rdf:Description><rdf:Description rdf:nodeID="n9"><boundIntervals xmlns="http://www.cellml.org/metadata/simulation/1.0#"><rdf:Description rdf:nodeID="n20"/></boundIntervals></rdf:Description><rdf:Description rdf:about="#G"><alternative xmlns="http://purl.org/dc/terms/" xml:lang="en">antigen</alternative><title xmlns="http://purl.org/dc/elements/1.1/" xml:lang="en">G</title></rdf:Description><rdf:Description rdf:nodeID="n21"><pointDensity xmlns="http://www.cellml.org/metadata/simulation/1.0#nonstandard-" xml:lang="en">10000</pointDensity><endingValue xmlns="http://www.cellml.org/metadata/simulation/1.0#" xml:lang="en">10</endingValue><maximumStepSize xmlns="http://www.cellml.org/metadata/simulation/1.0#" xml:lang="en">0.25</maximumStepSize></rdf:Description><rdf:Description rdf:about="#R"><alternative xmlns="http://purl.org/dc/terms/" xml:lang="en">regulatory T cells</alternative><alternative xmlns="http://purl.org/dc/terms/" xml:lang="en">active regulatory T cells</alternative><title xmlns="http://purl.org/dc/elements/1.1/" xml:lang="en">R</title></rdf:Description><rdf:Description rdf:nodeID="n11"><Given xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">H</Given><Family xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">Alexander</Family><Other xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xml:lang="en">K</Other></rdf:Description></rdf:RDF></model>