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ecell4
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<li><a class="reference internal" href="#">6. How to Solve ODEs with Rate Law Functions</a><ul>
<li><a class="reference internal" href="#6.1.-ReactionRuleDescriptor">6.1. ReactionRuleDescriptor</a></li>
<li><a class="reference internal" href="#6.2.-ReactionRuleDescriptorPyFunc">6.2. ReactionRuleDescriptorPyFunc</a></li>
<li><a class="reference internal" href="#6.3.-NetworkModel">6.3. NetworkModel</a></li>
<li><a class="reference internal" href="#6.4.-References-in-a-Rate-Law">6.4. References in a Rate Law</a></li>
<li><a class="reference internal" href="#6.5.-More-about-ODEs">6.5. More about ODEs</a></li>
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<div class="section" id="6.-How-to-Solve-ODEs-with-Rate-Law-Functions">
<h1>6. How to Solve ODEs with Rate Law Functions<a class="headerlink" href="#6.-How-to-Solve-ODEs-with-Rate-Law-Functions" title="Permalink to this headline">¶</a></h1>
<p>In general, <code class="docutils literal notranslate"><span class="pre">ReactionRule</span></code> describes a mass action kinetics with no more than two reactants. In case of a reaction with a complecated rate law, <code class="docutils literal notranslate"><span class="pre">ReactionRule</span></code> could be extensible with <code class="docutils literal notranslate"><span class="pre">ReactionRuleDescriptor</span></code>. Here, we explan the use of <code class="docutils literal notranslate"><span class="pre">ReactionRuleDescriptor</span></code> especially for <code class="docutils literal notranslate"><span class="pre">ode</span></code>.</p>
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<span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">from</span> <span class="nn">ecell4.prelude</span> <span class="kn">import</span> <span class="o">*</span>
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<div class="section" id="6.1.-ReactionRuleDescriptor">
<h2>6.1. ReactionRuleDescriptor<a class="headerlink" href="#6.1.-ReactionRuleDescriptor" title="Permalink to this headline">¶</a></h2>
<p><code class="docutils literal notranslate"><span class="pre">ReactionRule</span></code> defines reactants, products, and a kinetic rate.</p>
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<span></span><span class="n">rr1</span> <span class="o">=</span> <span class="n">ReactionRule</span><span class="p">()</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">add_reactant</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"A"</span><span class="p">))</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">add_reactant</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"B"</span><span class="p">))</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">add_product</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"C"</span><span class="p">))</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">set_k</span><span class="p">(</span><span class="mf">1.0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">reactants</span><span class="p">()))</span> <span class="c1"># => 2</span>
<span class="nb">print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">products</span><span class="p">()))</span> <span class="c1"># => 1</span>
<span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">k</span><span class="p">())</span> <span class="c1"># => 1.0</span>
<span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span> <span class="c1"># => A+B>C|1</span>
<span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">has_descriptor</span><span class="p">())</span> <span class="c1"># => False</span>
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2
1
1.0
A+B>C|1
False
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<p>In addition to that, <code class="docutils literal notranslate"><span class="pre">ReactionRule</span></code> could be extensible with <code class="docutils literal notranslate"><span class="pre">ReactionRuleDescriptor</span></code>.</p>
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<span></span><span class="n">desc1</span> <span class="o">=</span> <span class="n">ReactionRuleDescriptorMassAction</span><span class="p">(</span><span class="mf">1.0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">desc1</span><span class="o">.</span><span class="n">k</span><span class="p">())</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">set_descriptor</span><span class="p">(</span><span class="n">desc1</span><span class="p">)</span>
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1.0
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<p><code class="docutils literal notranslate"><span class="pre">ReactionRuleDescriptor</span></code> is accessible from <code class="docutils literal notranslate"><span class="pre">ReactionRule</span></code>.</p>
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<span></span><span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">has_descriptor</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">get_descriptor</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">get_descriptor</span><span class="p">()</span><span class="o">.</span><span class="n">k</span><span class="p">())</span>
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True
<ecell4_base.core.ReactionRuleDescriptorMassAction object at 0x147e2c4765e0>
1.0
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<p><code class="docutils literal notranslate"><span class="pre">ReactionRuleDescriptor</span></code> can store stoichiometric coefficients for each reactants:</p>
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<span></span><span class="n">desc1</span><span class="o">.</span><span class="n">set_reactant_coefficient</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="c1"># Set a coefficient of the first reactant</span>
<span class="n">desc1</span><span class="o">.</span><span class="n">set_reactant_coefficient</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c1"># Set a coefficient of the second reactant</span>
<span class="n">desc1</span><span class="o">.</span><span class="n">set_product_coefficient</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c1"># Set a coefficient of the first product</span>
<span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
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2*A+3*B>4*C|1
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<p>You can get the list of coefficients in the following way:</p>
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<span></span><span class="nb">print</span><span class="p">(</span><span class="n">desc1</span><span class="o">.</span><span class="n">reactant_coefficients</span><span class="p">())</span> <span class="c1"># => [2.0, 3.0]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">desc1</span><span class="o">.</span><span class="n">product_coefficients</span><span class="p">())</span> <span class="c1"># => [4.0]</span>
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[2.0, 3.0]
[4.0]
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<p>Please be careful that <code class="docutils literal notranslate"><span class="pre">ReactionRuleDescriptor</span></code> works properly only with <code class="docutils literal notranslate"><span class="pre">ode</span></code>.</p>
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<div class="section" id="6.2.-ReactionRuleDescriptorPyFunc">
<h2>6.2. ReactionRuleDescriptorPyFunc<a class="headerlink" href="#6.2.-ReactionRuleDescriptorPyFunc" title="Permalink to this headline">¶</a></h2>
<p><code class="docutils literal notranslate"><span class="pre">ReactionRuleDescriptor</span></code> provides a function to calculate a derivative (flux or velocity) based on the given values of <code class="docutils literal notranslate"><span class="pre">Species</span></code>. In this section, we will explain the way to define your own kinetic law.</p>
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<span></span><span class="n">rr1</span> <span class="o">=</span> <span class="n">ReactionRule</span><span class="p">()</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">add_reactant</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"A"</span><span class="p">))</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">add_reactant</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"B"</span><span class="p">))</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">add_product</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"C"</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
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A+B>C|0
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<p>First, define a rate law function as a Python function. The function must accept six arguments and return a floating number. The first and second lists contain a value for each reactants and products respectively. The third and fourth represent volume and time. The coefficients of reactants and products are given in the last two arguments.</p>
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<span></span><span class="k">def</span> <span class="nf">ratelaw</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">rc</span><span class="p">,</span> <span class="n">pc</span><span class="p">):</span>
<span class="k">return</span> <span class="mf">1.0</span> <span class="o">*</span> <span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mf">2.0</span> <span class="o">*</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
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<p><code class="docutils literal notranslate"><span class="pre">ReactionRuleDescriptorPyFunc</span></code> accepts the function.</p>
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<span></span><span class="n">desc1</span> <span class="o">=</span> <span class="n">ReactionRuleDescriptorPyfunc</span><span class="p">(</span><span class="n">ratelaw</span><span class="p">,</span> <span class="s1">'test'</span><span class="p">)</span>
<span class="n">desc1</span><span class="o">.</span><span class="n">set_reactant_coefficients</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">desc1</span><span class="o">.</span><span class="n">set_product_coefficients</span><span class="p">([</span><span class="mi">1</span><span class="p">])</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">set_descriptor</span><span class="p">(</span><span class="n">desc1</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">desc1</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
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test
1*A+1*B>1*C|0
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<p>A <code class="docutils literal notranslate"><span class="pre">lambda</span></code> function is available too.</p>
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<span></span><span class="n">desc2</span> <span class="o">=</span> <span class="n">ReactionRuleDescriptorPyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">r</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">rc</span><span class="p">,</span> <span class="n">pc</span><span class="p">:</span> <span class="mf">1.0</span> <span class="o">*</span> <span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mf">2.0</span> <span class="o">*</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s1">'test'</span><span class="p">)</span>
<span class="n">desc2</span><span class="o">.</span><span class="n">set_reactant_coefficients</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">desc2</span><span class="o">.</span><span class="n">set_product_coefficients</span><span class="p">([</span><span class="mi">1</span><span class="p">])</span>
<span class="n">rr1</span><span class="o">.</span><span class="n">set_descriptor</span><span class="p">(</span><span class="n">desc2</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">desc1</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="n">rr1</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
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test
1*A+1*B>1*C|0
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<p>To test if the function works properly, evaluate the value with <code class="docutils literal notranslate"><span class="pre">ode.World</span></code>.</p>
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<span></span><span class="n">w</span> <span class="o">=</span> <span class="n">ode</span><span class="o">.</span><span class="n">World</span><span class="p">()</span>
<span class="n">w</span><span class="o">.</span><span class="n">set_value</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"A"</span><span class="p">),</span> <span class="mi">10</span><span class="p">)</span>
<span class="n">w</span><span class="o">.</span><span class="n">set_value</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"B"</span><span class="p">),</span> <span class="mi">20</span><span class="p">)</span>
<span class="n">w</span><span class="o">.</span><span class="n">set_value</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"C"</span><span class="p">),</span> <span class="mi">30</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">w</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">rr1</span><span class="p">))</span> <span class="c1"># => 140 = 1 * 10 * 20 - 2 * 30</span>
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140.0
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<div class="section" id="6.3.-NetworkModel">
<h2>6.3. NetworkModel<a class="headerlink" href="#6.3.-NetworkModel" title="Permalink to this headline">¶</a></h2>
<p><code class="docutils literal notranslate"><span class="pre">NetworkModel</span></code> accepts <code class="docutils literal notranslate"><span class="pre">ReactionRule</span></code>s with and without <code class="docutils literal notranslate"><span class="pre">ReactionRuleDescriptor</span></code>.</p>
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<span></span><span class="n">m1</span> <span class="o">=</span> <span class="n">NetworkModel</span><span class="p">()</span>
<span class="n">rr1</span> <span class="o">=</span> <span class="n">create_unbinding_reaction_rule</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"C"</span><span class="p">),</span> <span class="n">Species</span><span class="p">(</span><span class="s2">"A"</span><span class="p">),</span> <span class="n">Species</span><span class="p">(</span><span class="s2">"B"</span><span class="p">),</span> <span class="mf">3.0</span><span class="p">)</span>
<span class="n">m1</span><span class="o">.</span><span class="n">add_reaction_rule</span><span class="p">(</span><span class="n">rr1</span><span class="p">)</span>
<span class="n">rr2</span> <span class="o">=</span> <span class="n">create_binding_reaction_rule</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"A"</span><span class="p">),</span> <span class="n">Species</span><span class="p">(</span><span class="s2">"B"</span><span class="p">),</span> <span class="n">Species</span><span class="p">(</span><span class="s2">"C"</span><span class="p">),</span> <span class="mf">0.0</span><span class="p">)</span>
<span class="n">desc1</span> <span class="o">=</span> <span class="n">ReactionRuleDescriptorPyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">r</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">rc</span><span class="p">,</span> <span class="n">pc</span><span class="p">:</span> <span class="mf">0.1</span> <span class="o">*</span> <span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s2">"test"</span><span class="p">)</span>
<span class="n">desc1</span><span class="o">.</span><span class="n">set_reactant_coefficients</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">desc1</span><span class="o">.</span><span class="n">set_product_coefficients</span><span class="p">([</span><span class="mi">1</span><span class="p">])</span>
<span class="n">rr2</span><span class="o">.</span><span class="n">set_descriptor</span><span class="p">(</span><span class="n">desc1</span><span class="p">)</span>
<span class="n">m1</span><span class="o">.</span><span class="n">add_reaction_rule</span><span class="p">(</span><span class="n">rr2</span><span class="p">)</span>
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<p>You can access to the list of <code class="docutils literal notranslate"><span class="pre">ReactionRule</span></code>s in <code class="docutils literal notranslate"><span class="pre">NetworkModel</span></code> via its member <code class="docutils literal notranslate"><span class="pre">reaction_rules()</span></code>.</p>
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<span></span><span class="nb">print</span><span class="p">([</span><span class="n">rr</span><span class="o">.</span><span class="n">as_string</span><span class="p">()</span> <span class="k">for</span> <span class="n">rr</span> <span class="ow">in</span> <span class="n">m1</span><span class="o">.</span><span class="n">reaction_rules</span><span class="p">()])</span>
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['C>A+B|3', '1*A+1*B>1*C|0']
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<p>Finally, you can run simulations in the same way with other solvers as follows:</p>
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<span></span><span class="n">run_simulation</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="n">m1</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="p">{</span><span class="s1">'A'</span><span class="p">:</span> <span class="mi">60</span><span class="p">,</span> <span class="s1">'B'</span><span class="p">:</span> <span class="mi">60</span><span class="p">})</span>
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<p>Modeling with Python decorators is also available by specifying a function instead of a rate (floating number). When a floating number is set, it is assumed to be a kinetic rate of a mass action reaction, but not a constant velocity.</p>
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<span></span><span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">reduce</span>
<span class="kn">from</span> <span class="nn">operator</span> <span class="kn">import</span> <span class="n">mul</span>
<span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">A</span> <span class="o">+</span> <span class="n">B</span> <span class="o">==</span> <span class="n">C</span> <span class="o">|</span> <span class="p">(</span><span class="k">lambda</span> <span class="n">r</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">:</span> <span class="mf">0.1</span> <span class="o">*</span> <span class="n">reduce</span><span class="p">(</span><span class="n">mul</span><span class="p">,</span> <span class="n">r</span><span class="p">),</span> <span class="mf">3.0</span><span class="p">)</span>
<span class="n">m1</span> <span class="o">=</span> <span class="n">get_model</span><span class="p">()</span>
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<p>For the simplicity, you can directory defining the equation with <code class="docutils literal notranslate"><span class="pre">Species</span></code> names as follows:</p>
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<span></span><span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">A</span> <span class="o">+</span> <span class="n">B</span> <span class="o">==</span> <span class="n">C</span> <span class="o">|</span> <span class="p">(</span><span class="mf">0.1</span> <span class="o">*</span> <span class="n">A</span> <span class="o">*</span> <span class="n">B</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">)</span>
<span class="n">m1</span> <span class="o">=</span> <span class="n">get_model</span><span class="p">()</span>
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<p>When you call a <code class="docutils literal notranslate"><span class="pre">Species</span></code> (in the rate law) which is not listed as a reactant or product, it is automatically added to the list as an enzyme.</p>
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<span></span><span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">S</span> <span class="o">></span> <span class="n">P</span> <span class="o">|</span> <span class="mf">1.0</span> <span class="o">*</span> <span class="n">E</span> <span class="o">*</span> <span class="n">S</span> <span class="o">/</span> <span class="p">(</span><span class="mf">30.0</span> <span class="o">+</span> <span class="n">S</span><span class="p">)</span>
<span class="n">m1</span> <span class="o">=</span> <span class="n">get_model</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">m1</span><span class="o">.</span><span class="n">reaction_rules</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="n">m1</span><span class="o">.</span><span class="n">reaction_rules</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_descriptor</span><span class="p">()</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
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1*S+1*E>1*P+1*E|0
((1.0 * E * S) / (30.0 + S))
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<p>where <code class="docutils literal notranslate"><span class="pre">E</span></code> in the equation is appended to both reacant and product lists.</p>
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<span></span><span class="n">run_simulation</span><span class="p">(</span><span class="mf">10.0</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="n">m1</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="p">{</span><span class="s1">'S'</span><span class="p">:</span> <span class="mi">60</span><span class="p">,</span> <span class="s1">'E'</span><span class="p">:</span> <span class="mi">30</span><span class="p">})</span>
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<p>Please be careful about typo in <code class="docutils literal notranslate"><span class="pre">Species</span></code>’ name. When you make a typo, it is unintentionally recognized as a new enzyme:</p>
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<span></span><span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">A13P2G</span> <span class="o">></span> <span class="n">A23P2G</span> <span class="o">|</span> <span class="mi">1500</span> <span class="o">*</span> <span class="n">A13B2G</span> <span class="c1"># typo: A13P2G -> A13B2G</span>
<span class="n">m1</span> <span class="o">=</span> <span class="n">get_model</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">m1</span><span class="o">.</span><span class="n">reaction_rules</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
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1*A13P2G+1*A13B2G>1*A23P2G+1*A13B2G|0
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<p>When you want to disable the automatic declaration of enzymes, inactivate <code class="docutils literal notranslate"><span class="pre">util.decorator.ENABLE_IMPLICIT_DECLARATION</span></code>. If its value is <code class="docutils literal notranslate"><span class="pre">False</span></code>, the above case will raise an error:</p>
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<span></span><span class="kn">import</span> <span class="nn">ecell4.util.decorator</span>
<span class="n">ecell4</span><span class="o">.</span><span class="n">util</span><span class="o">.</span><span class="n">decorator</span><span class="o">.</span><span class="n">ENABLE_IMPLICIT_DECLARATION</span> <span class="o">=</span> <span class="kc">False</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">A13P2G</span> <span class="o">></span> <span class="n">A23P2G</span> <span class="o">|</span> <span class="mi">1500</span> <span class="o">*</span> <span class="n">A13B2G</span>
<span class="k">except</span> <span class="ne">RuntimeError</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
<span class="n">ecell4</span><span class="o">.</span><span class="n">util</span><span class="o">.</span><span class="n">decorator</span><span class="o">.</span><span class="n">ENABLE_IMPLICIT_DECLARATION</span> <span class="o">=</span> <span class="kc">True</span>
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RuntimeError('[A13B2G] is unknown [(1500 * {0})].')
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<p>Although E-Cell4 is specialized for a simulation of biochemical reaction network, by using a synthetic reaction rule, ordinary differential equations can be translated intuitively. For example, the Lotka-Volterra equations:</p>
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\[\begin{split}\frac{dx}{dt} = Ax - Bxy\\\frac{dy}{dt} = -Cx + Dxy\end{split}\]</div>
<p>where <span class="math notranslate nohighlight">\(A=1.5, B=1, C=3, D=1, x(0)=10, y(0)=5\)</span>, are solved as follows:</p>
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<span></span><span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span>
<span class="o">~</span><span class="n">x</span> <span class="o">></span> <span class="n">x</span> <span class="o">|</span> <span class="n">A</span> <span class="o">*</span> <span class="n">x</span> <span class="o">-</span> <span class="n">B</span> <span class="o">*</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span>
<span class="o">~</span><span class="n">y</span> <span class="o">></span> <span class="n">y</span> <span class="o">|</span> <span class="o">-</span><span class="n">C</span> <span class="o">*</span> <span class="n">y</span> <span class="o">+</span> <span class="n">D</span> <span class="o">*</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span>
<span class="n">run_simulation</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="n">get_model</span><span class="p">(),</span> <span class="n">y0</span><span class="o">=</span><span class="p">{</span><span class="s1">'x'</span><span class="p">:</span> <span class="mi">10</span><span class="p">,</span> <span class="s1">'y'</span><span class="p">:</span> <span class="mi">5</span><span class="p">})</span>
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<div class="section" id="6.4.-References-in-a-Rate-Law">
<h2>6.4. References in a Rate Law<a class="headerlink" href="#6.4.-References-in-a-Rate-Law" title="Permalink to this headline">¶</a></h2>
<p>Here, we exlain the details in the rate law definition.</p>
<p>First, when you use simpler definitions of a rate law with <code class="docutils literal notranslate"><span class="pre">Species</span></code>, only a limited number of mathematical functions (e.g. <code class="docutils literal notranslate"><span class="pre">exp</span></code>, <code class="docutils literal notranslate"><span class="pre">log</span></code>, <code class="docutils literal notranslate"><span class="pre">sin</span></code>, <code class="docutils literal notranslate"><span class="pre">cos</span></code>, <code class="docutils literal notranslate"><span class="pre">tan</span></code>, <code class="docutils literal notranslate"><span class="pre">asin</span></code>, <code class="docutils literal notranslate"><span class="pre">acos</span></code>, <code class="docutils literal notranslate"><span class="pre">atan</span></code>, and <code class="docutils literal notranslate"><span class="pre">pi</span></code>) are available there even if you declare the function outside the block.</p>
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<span></span><span class="k">try</span><span class="p">:</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">erf</span>
<span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">S</span> <span class="o">></span> <span class="n">P</span> <span class="o">|</span> <span class="n">erf</span><span class="p">(</span><span class="n">S</span> <span class="o">/</span> <span class="mf">30.0</span><span class="p">)</span>
<span class="k">except</span> <span class="ne">TypeError</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
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TypeError('must be real number, not DivExp')
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<p>This error happens because <code class="docutils literal notranslate"><span class="pre">erf</span></code> is tried to be evaluated agaist <code class="docutils literal notranslate"><span class="pre">S</span> <span class="pre">/</span> <span class="pre">30.0</span></code>, which is not a floating number. In contrast, the following case is acceptable:</p>
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<span></span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">erf</span>
<span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">S</span> <span class="o">></span> <span class="n">P</span> <span class="o">|</span> <span class="n">erf</span><span class="p">(</span><span class="mf">2.0</span><span class="p">)</span> <span class="o">*</span> <span class="n">S</span>
<span class="n">m1</span> <span class="o">=</span> <span class="n">get_model</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">m1</span><span class="o">.</span><span class="n">reaction_rules</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_descriptor</span><span class="p">()</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
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(0.9953222650189527 * S)
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<p>where only the result of <code class="docutils literal notranslate"><span class="pre">erf(2.0)</span></code>, <code class="docutils literal notranslate"><span class="pre">0.995322265019</span></code>, is passed to the rate law. Thus, the rate law above has no reference to the <code class="docutils literal notranslate"><span class="pre">erf</span></code> function. Similarly, a value of variables declared outside is acceptable, but not as a reference.</p>
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<span></span><span class="n">kcat</span><span class="p">,</span> <span class="n">Km</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">30.0</span>
<span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">S</span> <span class="o">></span> <span class="n">P</span> <span class="o">|</span> <span class="n">kcat</span> <span class="o">*</span> <span class="n">E</span> <span class="o">*</span> <span class="n">S</span> <span class="o">/</span> <span class="p">(</span><span class="n">Km</span> <span class="o">+</span> <span class="n">S</span><span class="p">)</span>
<span class="n">m1</span> <span class="o">=</span> <span class="n">get_model</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">m1</span><span class="o">.</span><span class="n">reaction_rules</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_descriptor</span><span class="p">()</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
<span class="n">kcat</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="c1"># This doesn't affect the model</span>
<span class="nb">print</span><span class="p">(</span><span class="n">m1</span><span class="o">.</span><span class="n">reaction_rules</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_descriptor</span><span class="p">()</span><span class="o">.</span><span class="n">as_string</span><span class="p">())</span>
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((1.0 * E * S) / (30.0 + S))
((1.0 * E * S) / (30.0 + S))
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<p>Even if you change the value of a variable, it does <strong>not</strong> affect the rate law.</p>
<p>On the other hand, when you use your own function to define a rate law, it can hold a reference to variables outside.</p>
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<span></span><span class="n">k1</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">S</span> <span class="o">></span> <span class="n">P</span> <span class="o">|</span> <span class="p">(</span><span class="k">lambda</span> <span class="n">r</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">:</span> <span class="n">k1</span> <span class="o">*</span> <span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># referring k1</span>
<span class="n">m1</span> <span class="o">=</span> <span class="n">get_model</span><span class="p">()</span>
<span class="n">ret1</span> <span class="o">=</span> <span class="n">run_simulation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="n">m1</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="p">{</span><span class="s2">"S"</span><span class="p">:</span> <span class="mi">60</span><span class="p">})</span>
<span class="n">k1</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="c1"># This could change the result</span>
<span class="n">ret2</span> <span class="o">=</span> <span class="n">run_simulation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="n">m1</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="p">{</span><span class="s2">"S"</span><span class="p">:</span> <span class="mi">60</span><span class="p">})</span>
<span class="n">plotting</span><span class="o">.</span><span class="n">plot_number_observer</span><span class="p">(</span><span class="n">ret1</span><span class="p">,</span> <span class="s1">'-'</span><span class="p">,</span> <span class="n">ret2</span><span class="p">,</span> <span class="s1">'--'</span><span class="p">)</span>
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<p>However, in this case, it is better to make a new model for each set of parameters.</p>
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<span></span><span class="k">def</span> <span class="nf">create_model</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
<span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">S</span> <span class="o">></span> <span class="n">P</span> <span class="o">|</span> <span class="n">k</span>
<span class="k">return</span> <span class="n">get_model</span><span class="p">()</span>
<span class="n">ret1</span> <span class="o">=</span> <span class="n">run_simulation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="n">create_model</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="mf">1.0</span><span class="p">),</span> <span class="n">y0</span><span class="o">=</span><span class="p">{</span><span class="s2">"S"</span><span class="p">:</span> <span class="mi">60</span><span class="p">})</span>
<span class="n">ret2</span> <span class="o">=</span> <span class="n">run_simulation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="n">create_model</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="mf">2.0</span><span class="p">),</span> <span class="n">y0</span><span class="o">=</span><span class="p">{</span><span class="s2">"S"</span><span class="p">:</span> <span class="mi">60</span><span class="p">})</span>
<span class="n">plotting</span><span class="o">.</span><span class="n">plot_number_observer</span><span class="p">(</span><span class="n">ret1</span><span class="p">,</span> <span class="s1">'-'</span><span class="p">,</span> <span class="n">ret2</span><span class="p">,</span> <span class="s1">'--'</span><span class="p">)</span>
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<div class="section" id="6.5.-More-about-ODEs">
<h2>6.5. More about ODEs<a class="headerlink" href="#6.5.-More-about-ODEs" title="Permalink to this headline">¶</a></h2>
<p>In <code class="docutils literal notranslate"><span class="pre">ode.World</span></code>, a value for each <code class="docutils literal notranslate"><span class="pre">Species</span></code> is a floating number. However, for the compatibility, the common member <code class="docutils literal notranslate"><span class="pre">num_molecules</span></code> and <code class="docutils literal notranslate"><span class="pre">add_molecules</span></code> regard the value as an integer.</p>
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<span></span><span class="n">w</span> <span class="o">=</span> <span class="n">ode</span><span class="o">.</span><span class="n">World</span><span class="p">()</span>
<span class="n">w</span><span class="o">.</span><span class="n">add_molecules</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"A"</span><span class="p">),</span> <span class="mf">2.5</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">w</span><span class="o">.</span><span class="n">num_molecules</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"A"</span><span class="p">)))</span>
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2
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<p>To set/get a real number, use <code class="docutils literal notranslate"><span class="pre">set_value</span></code> and <code class="docutils literal notranslate"><span class="pre">get_value</span></code>:</p>
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<span></span><span class="n">w</span><span class="o">.</span><span class="n">set_value</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"B"</span><span class="p">),</span> <span class="mf">2.5</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">w</span><span class="o">.</span><span class="n">get_value</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"A"</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">w</span><span class="o">.</span><span class="n">get_value</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"B"</span><span class="p">)))</span>
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2.5
2.5
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<p>As a default, <code class="docutils literal notranslate"><span class="pre">ode.Simulator</span></code> employs the Rosenblock method, called <code class="docutils literal notranslate"><span class="pre">ROSENBROCK4_CONTROLLER</span></code>, to solve ODEs. In addition to that, two solvers, <code class="docutils literal notranslate"><span class="pre">EULER</span></code> and <code class="docutils literal notranslate"><span class="pre">RUNGE_KUTTA_CASH_KARP54</span></code>, are available. <code class="docutils literal notranslate"><span class="pre">ROSENBROCK4_CONTROLLER</span></code> and <code class="docutils literal notranslate"><span class="pre">RUNGE_KUTTA_CASH_KARP54</span></code> adaptively change the step size during time evolution due to error controll, but <code class="docutils literal notranslate"><span class="pre">EULER</span></code> does not.</p>
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<span></span><span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="n">A</span> <span class="o">></span> <span class="o">~</span><span class="n">A</span> <span class="o">|</span> <span class="mf">1.0</span>
<span class="n">m1</span> <span class="o">=</span> <span class="n">get_model</span><span class="p">()</span>
<span class="n">w1</span> <span class="o">=</span> <span class="n">ode</span><span class="o">.</span><span class="n">World</span><span class="p">()</span>
<span class="n">w1</span><span class="o">.</span><span class="n">set_value</span><span class="p">(</span><span class="n">Species</span><span class="p">(</span><span class="s2">"A"</span><span class="p">),</span> <span class="mf">1.0</span><span class="p">)</span>
<span class="n">sim1</span> <span class="o">=</span> <span class="n">ode</span><span class="o">.</span><span class="n">Simulator</span><span class="p">(</span><span class="n">w1</span><span class="p">,</span> <span class="n">m1</span><span class="p">,</span> <span class="n">ode</span><span class="o">.</span><span class="n">EULER</span><span class="p">)</span>
<span class="n">sim1</span><span class="o">.</span><span class="n">set_dt</span><span class="p">(</span><span class="mf">0.01</span><span class="p">)</span> <span class="c1"># This is only effective for EULER</span>
<span class="n">obs1</span> <span class="o">=</span> <span class="n">FixedIntervalNumberObserver</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
<span class="n">sim1</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mf">3.0</span><span class="p">,</span> <span class="n">obs1</span><span class="p">)</span>
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<p><code class="docutils literal notranslate"><span class="pre">ode.Factory</span></code> also accepts a solver type and a default step interval.</p>
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<span></span><span class="n">run_simulation</span><span class="p">(</span><span class="mf">3.0</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="n">m1</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="p">{</span><span class="s2">"A"</span><span class="p">:</span> <span class="mf">1.0</span><span class="p">},</span> <span class="n">solver</span><span class="o">=</span><span class="p">(</span><span class="s1">'ode'</span><span class="p">,</span> <span class="n">ode</span><span class="o">.</span><span class="n">EULER</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">))</span>
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<p>See also the examples listed below:</p>
<ul class="simple">
<li><p><a class="reference external" href="../examples/example5.html">Glycolysis of Human Erythrocytes</a></p></li>
<li><p><a class="reference external" href="../examples/example2.html">Drosophila Circadian Clock</a></p></li>
<li><p><a class="reference external" href="../examples/example1.html">Attractors</a></p></li>
</ul>
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