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<li><a class="reference internal" href="#">Glycolysis Model and Metabolic Control Analysis</a><ul>
<li><a class="reference internal" href="#A-Simple-Model-of-the-Glycolysis-of-Human-Erythrocytes">A Simple Model of the Glycolysis of Human Erythrocytes</a></li>
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<p>This page was generated from <a class="reference external" href="https://github.com/kaizu/ecell4_docs/blob/latest/en/examples/example05.ipynb">examples/example05.ipynb</a>.</p>
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<div class="section" id="Glycolysis-Model-and-Metabolic-Control-Analysis">
<h1>Glycolysis Model and Metabolic Control Analysis<a class="headerlink" href="#Glycolysis-Model-and-Metabolic-Control-Analysis" title="Permalink to this headline">¶</a></h1>
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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="A-Simple-Model-of-the-Glycolysis-of-Human-Erythrocytes">
<h2>A Simple Model of the Glycolysis of Human Erythrocytes<a class="headerlink" href="#A-Simple-Model-of-the-Glycolysis-of-Human-Erythrocytes" title="Permalink to this headline">¶</a></h2>
<p>This is a model for the glycolysis of human erythrocytes which takes into account ATP-synthesis and -consumption. This model is based on the model introduced in the following publications:</p>
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<span></span><span class="n">citation</span><span class="p">(</span><span class="mi">125616</span><span class="p">)</span>
</pre></div>
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<p>Heinrich R, Rapoport TA, Mathematical analysis of multienzyme systems. II. Steady state and transient control. <em>Bio Systems</em>, <strong>1</strong>(7), 130-6, 1975. <a class="reference external" href="https://doi.org/10.1016/0303-2647(75)90050-7">10.1016/0303-2647(75)90050-7</a>. PubMed PMID: <a class="reference external" href="https://www.ncbi.nlm.nih.gov/pubmed/125616">125616</a>.</p>
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<span></span><span class="n">citation</span><span class="p">(</span><span class="mi">168932</span><span class="p">)</span>
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<p>Rapoport TA, Heinrich R, Mathematical analysis of multienzyme systems. I. Modelling of the glycolysis of human erythrocytes. <em>Bio Systems</em>, <strong>1</strong>(7), 120-9, 1975. <a class="reference external" href="https://doi.org/10.1016/0303-2647(75)90049-0">10.1016/0303-2647(75)90049-0</a>. PubMed PMID: <a class="reference external" href="https://www.ncbi.nlm.nih.gov/pubmed/168932">168932</a>.</p>
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<p>The model consists of seven reactions and is at the steady state.</p>
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<span></span><span class="k">with</span> <span class="n">reaction_rules</span><span class="p">():</span>
<span class="mi">2</span> <span class="o">*</span> <span class="n">ATP</span> <span class="o">></span> <span class="mi">2</span> <span class="o">*</span> <span class="n">A13P2G</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">ADP</span> <span class="o">|</span> <span class="p">(</span><span class="mf">3.2</span> <span class="o">*</span> <span class="n">ATP</span> <span class="o">/</span> <span class="p">(</span><span class="mf">1.0</span> <span class="o">+</span> <span class="p">(</span><span class="n">ATP</span> <span class="o">/</span> <span class="mf">1.0</span><span class="p">)</span> <span class="o">**</span> <span class="mf">4.0</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="n">A23P2G</span> <span class="o">></span> <span class="n">PEP</span> <span class="o">|</span> <span class="mf">0.15</span>
<span class="n">A13P2G</span> <span class="o">+</span> <span class="n">ADP</span> <span class="o">></span> <span class="n">PEP</span> <span class="o">+</span> <span class="n">ATP</span> <span class="o">|</span> <span class="mf">1.57e+4</span>
<span class="n">PEP</span> <span class="o">+</span> <span class="n">ADP</span> <span class="o">></span> <span class="n">ATP</span> <span class="o">|</span> <span class="mi">559</span>
<span class="c1"># AMP + ATP > 2 * ADP | (1.0 * (AMP * ATP - 2.0 * ADP * ADP))</span>
<span class="n">AMP</span> <span class="o">+</span> <span class="n">ATP</span> <span class="o">></span> <span class="mi">2</span> <span class="o">*</span> <span class="n">ADP</span> <span class="o">|</span> <span class="mf">1.0</span> <span class="o">*</span> <span class="n">AMP</span> <span class="o">*</span> <span class="n">ATP</span>
<span class="mi">2</span> <span class="o">*</span> <span class="n">ADP</span> <span class="o">></span> <span class="n">AMP</span> <span class="o">+</span> <span class="n">ATP</span> <span class="o">|</span> <span class="mf">2.0</span> <span class="o">*</span> <span class="n">ADP</span> <span class="o">*</span> <span class="n">ADP</span>
<span class="n">ATP</span> <span class="o">></span> <span class="n">ADP</span> <span class="o">|</span> <span class="mf">1.46</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">get_model</span><span class="p">()</span>
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<span></span><span class="n">show</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
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2 * ATP > 2 * A13P2G + 2 * ADP | ((3.2 * ATP) / (1.0 + pow((ATP / 1.0), 4.0)))
A13P2G > A23P2G | 1500.0
A23P2G > PEP | 0.15
A13P2G + ADP > PEP + ATP | 15700.0
PEP + ADP > ATP | 559.0
AMP + ATP > 2 * ADP | (1.0 * AMP * ATP)
2 * ADP > AMP + ATP | (2.0 * ADP * ADP)
ATP > ADP | 1.46
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<span></span><span class="n">y0</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"A13P2G"</span><span class="p">:</span> <span class="mf">0.0005082</span><span class="p">,</span> <span class="s2">"A23P2G"</span><span class="p">:</span> <span class="mf">5.0834</span><span class="p">,</span> <span class="s2">"PEP"</span><span class="p">:</span> <span class="mf">0.020502</span><span class="p">,</span>
<span class="s2">"AMP"</span><span class="p">:</span> <span class="mf">0.080139</span><span class="p">,</span> <span class="s2">"ADP"</span><span class="p">:</span> <span class="mf">0.2190</span><span class="p">,</span> <span class="s2">"ATP"</span><span class="p">:</span> <span class="mf">1.196867</span><span class="p">}</span>
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<span></span><span class="n">ret</span> <span class="o">=</span> <span class="n">run_simulation</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="n">m</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">y0</span><span class="p">)</span>
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<span></span><span class="n">ret</span>
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<div class="section" id="Metabolic-Control-Analysis">
<h2>Metabolic Control Analysis<a class="headerlink" href="#Metabolic-Control-Analysis" title="Permalink to this headline">¶</a></h2>
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<span></span><span class="kn">import</span> <span class="nn">numpy</span>
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<span></span><span class="n">w</span> <span class="o">=</span> <span class="n">ret</span><span class="o">.</span><span class="n">world</span>
<span class="n">sim</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">w</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
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<p>First of all, <code class="docutils literal notranslate"><span class="pre">get_stoichiometry</span></code> gives a stoichiometry matrix from the given species and reactions as follows:</p>
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<span></span><span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">get_stoichiometry</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">list_species</span><span class="p">(),</span> <span class="n">m</span><span class="o">.</span><span class="n">reaction_rules</span><span class="p">()))</span>
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array([[ 2., -1., 0., -1., 0., 0., 0., 0.],
[ 0., 1., -1., 0., 0., 0., 0., 0.],
[ 2., 0., 0., -1., -1., 2., -2., 1.],
[ 0., 0., 0., 0., 0., -1., 1., 0.],
[-2., 0., 0., 1., 1., -1., 1., -1.],
[ 0., 0., 1., 1., -1., 0., 0., 0.]])
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<p>The <code class="docutils literal notranslate"><span class="pre">evaluate</span></code> method of <code class="docutils literal notranslate"><span class="pre">ode.ODEWorld</span></code> returns current fluxes of the given reactions.</p>
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<span></span><span class="n">numpy</span><span class="o">.</span><span class="n">array</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">m</span><span class="o">.</span><span class="n">reaction_rules</span><span class="p">()))</span>
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array([1.25489042, 0.76235178, 0.76235178, 1.74742906, 2.50978084,
0.0959183 , 0.0959183 , 1.74742906])
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<p><code class="docutils literal notranslate"><span class="pre">ode.ODESimulator</span></code> has methods for the fundamental properties related to metabolic control analysis.</p>
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<span></span><span class="n">x</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">sim</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="n">x</span>
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array([5.08234520e-04, 5.08234520e+00, 2.05016233e-02, 8.01410043e-02,
2.18995778e-01, 1.19686922e+00])
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<span></span><span class="n">dxdt</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">sim</span><span class="o">.</span><span class="n">derivatives</span><span class="p">())</span>
<span class="n">dxdt</span>
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array([ 0.00000000e+00, -7.32716110e-11, -1.27009514e-13, 1.07631959e-12,
1.09356968e-12, -2.16981988e-12])
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<span></span><span class="n">J</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">sim</span><span class="o">.</span><span class="n">jacobian</span><span class="p">())</span>
<span class="n">J</span>
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array([[-4.93823371e+03, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, -7.97928197e+00, -3.54260035e+00],
[ 1.50000000e+03, -1.50000000e-01, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00],
[ 3.43823371e+03, 1.50000000e-01, -1.22418640e+02,
0.00000000e+00, -3.48112548e+00, 0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
-1.19686922e+00, 8.75985548e-01, -8.01410043e-02],
[-3.43823371e+03, 0.00000000e+00, -1.22418640e+02,
2.39373844e+00, -2.11916605e+01, -1.92231834e+00],
[ 3.43823371e+03, 0.00000000e+00, 1.22418640e+02,
-1.19686922e+00, 2.03156750e+01, 2.00245935e+00]])
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<span></span><span class="n">E</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">sim</span><span class="o">.</span><span class="n">elasticity</span><span class="p">())</span>
<span class="n">E</span>
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array([[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, -1.77130018e+00],
[ 1.50000000e+03, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00],
[ 0.00000000e+00, 1.50000000e-01, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00],
[ 3.43823371e+03, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 7.97928197e+00, 0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 1.22418640e+02,
0.00000000e+00, 1.14604074e+01, 0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
1.19686922e+00, 0.00000000e+00, 8.01410043e-02],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 8.75985548e-01, 0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 1.46000000e+00]])
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<p><code class="docutils literal notranslate"><span class="pre">ode.ODESimulator</span></code> also provides a stoichiometry matrix and fluxes.</p>
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<span></span><span class="n">S</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">sim</span><span class="o">.</span><span class="n">stoichiometry</span><span class="p">())</span>
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array([[ 2., -1., 0., -1., 0., 0., 0., 0.],
[ 0., 1., -1., 0., 0., 0., 0., 0.],
[ 0., 0., 1., 1., -1., 0., 0., 0.],
[ 0., 0., 0., 0., 0., -1., 1., 0.],
[ 2., 0., 0., -1., -1., 2., -2., 1.],
[-2., 0., 0., 1., 1., -1., 1., -1.]])
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<span></span><span class="n">v</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">sim</span><span class="o">.</span><span class="n">fluxes</span><span class="p">())</span>
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array([1.25489042, 0.76235178, 0.76235178, 1.74742906, 2.50978084,
0.0959183 , 0.0959183 , 1.74742906])
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<p>These properties satisfy some relations at the steady state.</p>
<p><span class="math notranslate nohighlight">\(\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{x} = \mathbf{S} \mathbf{v}\)</span></p>
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<span></span><span class="n">numpy</span><span class="o">.</span><span class="n">isclose</span><span class="p">(</span><span class="n">dxdt</span><span class="p">,</span> <span class="n">S</span> <span class="o">@</span> <span class="n">v</span><span class="p">)</span>
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array([ True, True, True, True, True, True])
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<p><span class="math notranslate nohighlight">\(\mathbf{J} = \frac{\mathrm{d}^2}{\mathrm{d}t^2}\mathbf{x} = \mathbf{S}\left(\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{v}\right) = \mathbf{S} \mathbf{E}\)</span></p>
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<span></span><span class="n">numpy</span><span class="o">.</span><span class="n">isclose</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">S</span> <span class="o">@</span> <span class="n">E</span><span class="p">)</span>
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array([[ True, True, True, True, True, True],
[ True, True, True, True, True, True],
[ True, True, True, True, True, True],
[ True, True, True, True, True, True],
[ True, True, True, True, True, True],
[ True, True, True, True, True, True]])
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<p>Next, the <code class="docutils literal notranslate"><span class="pre">ecell4.mca</span></code> submodule provides useful functions for metabolic network and control analyses.</p>
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<span></span><span class="kn">from</span> <span class="nn">ecell4.mca</span> <span class="kn">import</span> <span class="o">*</span>
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<p><code class="docutils literal notranslate"><span class="pre">generate_full_rank_matrix</span></code> gives square matrix to be full rank. In this model, 5 out of 8 reactions are independent.</p>
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<span></span><span class="p">(</span><span class="n">link_matrix</span><span class="p">,</span> <span class="n">kernel_matrix</span><span class="p">,</span> <span class="n">independent_list</span><span class="p">)</span> <span class="o">=</span> <span class="n">generate_full_rank_matrix</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">link_matrix</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">kernel_matrix</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">independent_list</span><span class="p">)</span>
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[[ 1. 0. 0. 0. 0.]
[ 0. 1. 0. 0. 0.]
[ 0. 0. 1. 0. 0.]
[ 0. 0. 0. 1. 0.]
[ 0. 0. 0. 0. 1.]
[ 0. 0. 0. -1. -1.]]
[[-0. -0. 0.5]
[-0. -1. 1. ]
[-0. -1. 1. ]
[-0. 1. 0. ]
[ 0. 0. 1. ]
[ 1. 0. 0. ]
[ 1. 0. 0. ]
[ 0. 1. 0. ]]
[0 1 2 3 4]
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<span></span><span class="n">reduced_matrix</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">independent_list</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">reduced_matrix</span>
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array([[ 2., -1., 0., -1., 0., 0., 0., 0.],
[ 0., 1., -1., 0., 0., 0., 0., 0.],
[ 0., 0., 1., 1., -1., 0., 0., 0.],
[ 0., 0., 0., 0., 0., -1., 1., 0.],
[ 2., 0., 0., -1., -1., 2., -2., 1.]])
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<p>The original stoichiometry matrix can be reproduced from these reduced matrices:</p>
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<span></span><span class="n">S</span> <span class="o">==</span> <span class="n">link_matrix</span> <span class="o">@</span> <span class="n">reduced_matrix</span>
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array([[ True, True, True, True, True, True, True, True],
[ True, True, True, True, True, True, True, True],
[ True, True, True, True, True, True, True, True],
[ True, True, True, True, True, True, True, True],
[ True, True, True, True, True, True, True, True],
[ True, True, True, True, True, True, True, True]])
</pre></div></div>
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<span></span><span class="mi">0</span> <span class="o">==</span> <span class="n">S</span> <span class="o">@</span> <span class="n">kernel_matrix</span>
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array([[ True, True, True],
[ True, True, True],
[ True, True, True],
[ True, True, True],
[ True, True, True],
[ True, True, True]])
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<p>Finally, <code class="docutils literal notranslate"><span class="pre">ecell4.mca</span></code> provides two functions, <code class="docutils literal notranslate"><span class="pre">unscaled_control_coefficients</span></code> and <code class="docutils literal notranslate"><span class="pre">scaled_control_coefficients</span></code>, which calculate concentration and flux control coefficients from stoichiometry and elasticity matrices.</p>
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<span></span><span class="n">ccc</span><span class="p">,</span> <span class="n">fcc</span> <span class="o">=</span> <span class="n">unscaled_control_coefficients</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">E</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">ccc</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">fcc</span><span class="p">)</span>
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[[ 4.45754267e-04 -2.22877134e-04 0.00000000e+00 -1.93612289e-04
0.00000000e+00 -7.45294400e-04 7.45294400e-04 -2.92648441e-05]
[ 4.45754267e+00 4.43789533e+00 -6.66666667e+00 -1.93612289e+00
0.00000000e+00 -7.45294400e+00 7.45294400e+00 -2.92648441e-01]
[ 2.20583561e-02 -2.86048736e-03 0.00000000e+00 1.24794630e-03
-8.16869067e-03 -4.03592231e-02 4.03592231e-02 -4.10843366e-03]
[-1.22757774e-01 6.13788871e-02 0.00000000e+00 -2.67777988e-02
0.00000000e+00 -5.85505375e-01 5.85505375e-01 8.81566859e-02]
[-1.43377537e-01 7.16887683e-02 0.00000000e+00 -3.12756961e-02
0.00000000e+00 3.62033276e-01 -3.62033276e-01 1.02964464e-01]
[ 2.66135311e-01 -1.33067655e-01 0.00000000e+00 5.80534949e-02
0.00000000e+00 2.23472099e-01 -2.23472099e-01 -1.91121150e-01]]
[[ 5.28594477e-01 2.35702761e-01 0.00000000e+00 -1.02830166e-01
0.00000000e+00 -3.95836168e-01 3.95836168e-01 3.38532927e-01]
[ 6.68631401e-01 6.65684300e-01 0.00000000e+00 -2.90418434e-01
0.00000000e+00 -1.11794160e+00 1.11794160e+00 -4.38972662e-02]
[ 6.68631401e-01 6.65684300e-01 1.11022302e-16 -2.90418434e-01
0.00000000e+00 -1.11794160e+00 1.11794160e+00 -4.38972662e-02]
[ 3.88557554e-01 -1.94278777e-01 0.00000000e+00 8.47581026e-02
0.00000000e+00 3.26269264e-01 -3.26269264e-01 7.20963121e-01]
[ 1.05718895e+00 4.71405523e-01 0.00000000e+00 -2.05660332e-01
0.00000000e+00 -7.91672336e-01 7.91672336e-01 6.77065854e-01]
[-1.25596650e-01 6.27983250e-02 0.00000000e+00 -2.73970578e-02
0.00000000e+00 3.17135918e-01 6.82864082e-01 9.01953828e-02]
[-1.25596650e-01 6.27983250e-02 0.00000000e+00 -2.73970578e-02
0.00000000e+00 3.17135918e-01 6.82864082e-01 9.01953828e-02]
[ 3.88557554e-01 -1.94278777e-01 0.00000000e+00 8.47581026e-02
0.00000000e+00 3.26269264e-01 -3.26269264e-01 7.20963121e-01]]
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<p>Unscaled control coefficients satisfies connectivity theorem:</p>
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<span></span><span class="n">numpy</span><span class="o">.</span><span class="n">isclose</span><span class="p">(</span><span class="n">fcc</span> <span class="o">@</span> <span class="p">(</span><span class="n">E</span> <span class="o">@</span> <span class="n">link_matrix</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
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array([[ True, True, True, True, True],
[ True, True, True, True, True],
[ True, True, True, True, True],
[ True, True, True, True, True],
[ True, True, True, True, True],
[ True, True, True, True, True],
[ True, True, True, True, True],
[ True, True, True, True, True]])
</pre></div></div>
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<p>and</p>
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<span></span><span class="n">numpy</span><span class="o">.</span><span class="n">isclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">take</span><span class="p">(</span><span class="n">ccc</span><span class="p">,</span> <span class="n">independent_list</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">@</span> <span class="p">(</span><span class="n">E</span> <span class="o">@</span> <span class="n">link_matrix</span><span class="p">),</span> <span class="o">-</span><span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="n">link_matrix</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
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array([[ True, True, True, True, True],
[ True, True, True, True, True],
[ True, True, True, True, True],
[ True, True, True, True, True],
[ True, True, True, True, True]])
</pre></div></div>
</div>
<p>On the other hand, scaled control coefficients</p>
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<span></span><span class="n">ccc</span><span class="p">,</span> <span class="n">fcc</span> <span class="o">=</span> <span class="n">scaled_control_coefficients</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">ccc</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">fcc</span><span class="p">)</span>
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[[ 1.10061937 -0.3343157 0. -0.6656843 0. -0.14065824
0.14065824 -0.10061937]
[ 1.10061937 0.6656843 -1. -0.6656843 0. -0.14065824
0.14065824 -0.10061937]
[ 1.35017697 -0.10636707 0. 0.10636707 -1. -0.18882349
0.18882349 -0.35017697]
[-1.92220644 0.58387469 0. -0.58387469 0. -0.70077336
0.70077336 1.92220644]
[-0.82158249 0.2495576 0. -0.2495576 0. 0.15856752
-0.15856752 0.82158249]
[ 0.27903688 -0.0847581 0. 0.0847581 0. 0.01790928
-0.01790928 -0.27903688]]
[[ 5.28594477e-01 1.43190526e-01 0.00000000e+00 -1.43190526e-01
0.00000000e+00 -3.02559747e-02 3.02559747e-02 4.71405523e-01]
[ 1.10061937e+00 6.65684300e-01 0.00000000e+00 -6.65684300e-01
0.00000000e+00 -1.40658239e-01 1.40658239e-01 -1.00619373e-01]
[ 1.10061937e+00 6.65684300e-01 1.11022302e-16 -6.65684300e-01
0.00000000e+00 -1.40658239e-01 1.40658239e-01 -1.00619373e-01]
[ 2.79036879e-01 -8.47581026e-02 0.00000000e+00 8.47581026e-02
0.00000000e+00 1.79092784e-02 -1.79092784e-02 7.20963121e-01]
[ 5.28594477e-01 1.43190526e-01 0.00000000e+00 -1.43190526e-01
0.00000000e+00 -3.02559747e-02 3.02559747e-02 4.71405523e-01]
[-1.64316956e+00 4.99116585e-01 0.00000000e+00 -4.99116585e-01
0.00000000e+00 3.17135918e-01 6.82864082e-01 1.64316956e+00]
[-1.64316956e+00 4.99116585e-01 0.00000000e+00 -4.99116585e-01
0.00000000e+00 3.17135918e-01 6.82864082e-01 1.64316956e+00]
[ 2.79036879e-01 -8.47581026e-02 0.00000000e+00 8.47581026e-02
0.00000000e+00 1.79092784e-02 -1.79092784e-02 7.20963121e-01]]
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<p>satisfies summation theorem:</p>
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<span></span><span class="n">numpy</span><span class="o">.</span><span class="n">isclose</span><span class="p">(</span><span class="n">ccc</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
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array([ True, True, True, True, True, True])
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<p>and</p>
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<span></span><span class="n">numpy</span><span class="o">.</span><span class="n">isclose</span><span class="p">(</span><span class="n">fcc</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
</pre></div>
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array([ True, True, True, True, True, True, True, True])
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