View source for Optimization examples

Here we give some examples of the [[BioUML]] usage for solving the problem of parameter estimation applied to the models of biochemical pathways.
For details about creation your oun optimization document in BioUML, see the chapter [[Optimization document]]. All information about the optimization methods implemented in BioUML is done in the chapter [[Optimization problem]].

==Testing the convergence rate of the optimization methods==

<ul>
<li>'''Optimization document''': ''data'' > ''Examples'' > ''Optimization'' > ''Data'' > ''Documents'' > ''test_case_1A''</li>
<li>'''Model''': ''data'' > ''Examples'' > ''Optimization'' > ''Data'' > ''Diagrams'' > ''diagram_1A''</li>
<li>'''Experimental data''': ''data'' > ''Examples'' > ''Optimization'' > ''Data'' > ''Experiments'' > ''exp_data_1''</li>
</ul>

To analyze a convergence rate of the optimization methods implemented in [[BioUML]] [1], we considered a reaction chain extracted from the model by Neumann et al. [2] and representing activation of caspase-8 triggered by the receptor
CD95 (APO-1/Fas).

<table border="0" cellspacing="0" cellpadding="4">
  <tr>
    <td>[[File:optimization_examples_model_1.png|thumb|The test model of caspase-8 activation]]</td>
    <td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
    <td>
      <table  border="1" align="center" cellspacing="0" cellpadding="4">
        <tr>
          <td>'''ID'''</td>
          <td>'''Reactions'''</td>
          <td>'''Reaction rates'''</td>
          <td>'''Initial values'''</td>
        </tr>
        <tr>
          <td>r1</td>
          <td>CD95L + FADD:CD95R → DISC</td>
          <td>''k''<sub>1</sub> ⋅ [CD95L] ⋅ [CD95R:FADD]</td>
          <td>[CD95L]<sub>0</sub> = 113.220, [CD95R:FADD]<sub>0</sub> = 91.266</td>
        </tr>
        <tr>
          <td>r2</td>
          <td>DISC + pro8 → DISC:pro8</td>
          <td>''k''<sub>2</sub> ⋅ [DISC] ⋅ [pro8]</td>
          <td>[pro8]<sub>0</sub> = 64.477, [DISC]<sub>0</sub> = 0.0</td>
        </tr>
        <tr>
          <td>r3</td>
          <td>DISC:pro8 + pro8 → 2 · p43/p41</td>
          <td>''k''<sub>3</sub> ⋅ [DISC:pro8] ⋅ [pro8]</td>
          <td>[pro8]<sub>0</sub> = 64.477, [DISC:pro8]<sub>0</sub> = 0.0</td>
        </tr>
        <tr>
          <td>r4</td>
          <td>2 · p43/p41 → casp8</td>
          <td>''k''<sub>4</sub> ⋅ [p43/p41]<sup>2</sup></td>
          <td>[p43/p41]<sub>0</sub> = 0.0</td>
        </tr>
        <tr>
          <td>r5</td>
          <td>casp8 →</td>
          <td>''k''<sub>5</sub> ⋅ [casp8]</td>
          <td>[casp8]<sub>0</sub> = 0.0</td>
        </tr>
      </table>
    </td>
  </tr>
</table>

We performed estimation of parameters using the search space defined as:

[[File:optimization_examples_formula_1.png]]

where upper bounds were chosen based on the order of magnitude of parameter values proposed in [2].

Estimation was based on the experimental data obtained by Neumann ''et al''. [2] for procaspase-8 and its cleaved products
p43/p41 and caspase-8.

<table  border="1" cellspacing="0" cellpadding="4">
  <tr>
    <td>'''Time (min<sup>-1</sup>)'''</td>
    <td>'''p43/p41 (nM)'''</td>
    <td>'''pro-8 (nM)'''</td>
    <td>'''casp-8 (nM)'''</td>
  </tr>
  <tr>
    <td>0.0</td>
    <td>0.058</td>
    <td>59.963</td>
    <td>0.000</td>
  </tr>
  <tr>
    <td>10.0</td>
    <td>0.268</td>
    <td>57.565</td>
    <td>0.041</td>
  </tr>
  <tr>
    <td>20.0</td>
    <td>4.760</td>
    <td>58.590</td>
    <td>0.316</td>
  </tr>
  <tr>
    <td>30.0</td>
    <td>8.252</td>
    <td>59.422</td>
    <td>1.397</td>
  </tr>
  <tr>
    <td>45.0</td>
    <td>16.144</td>
    <td>48.190</td>
    <td>3.520</td>
  </tr>
  <tr>
    <td>60.0</td>
    <td>17.021</td>
    <td>38.950</td>
    <td>3.947</td>
  </tr>
  <tr>
    <td>90.0</td>
    <td>15.269</td>
    <td>23.502</td>
    <td>4.871</td>
  </tr>
  <tr>
    <td>120.0</td>
    <td>12.530</td>
    <td>13.127</td>
    <td>4.878</td>
  </tr>
  <tr>
    <td>150.0</td>
    <td>10.335</td>
    <td>10.703</td>
    <td>4.228</td>
  </tr>
</table>

We  reviewed solutions obtained by all optimization methods for 100 runs. Each run was based on the generation of 10<sup>7</sup> different  guesses.
The best result was obtained by the particle swarm optimization (PSO) and the cellular genetic algorithm (MOCell). Methods SRES, MOCell and PSO found similar solutions. Methods ASA and glbSolve
found other values for parameters ''k''<sub>1</sub> and ''k''<sub>2</sub> showing lower efficiency.

<table border="0" cellspacing="0" cellpadding="4">
  <tr>
    <td>[[File:optimization_examples_figure_2.png|thumb|The objective function mean values dynamics for 100 runs. The best value obtained by PSO is marked by the red line.]]</td>
    <td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
    <td>
'''The best guesses obtained by optimization methods for 100 runs'''
      <table  border="1" cellspacing="0" cellpadding="4">
        <tr>
          <td>'''Parameters'''</td>
          <td>'''SRES'''</td>
          <td>'''MOCell'''</td>
          <td>'''PSO'''</td>
          <td>'''ASA'''</td>
          <td>'''glbSolve'''</td>
        </tr>
        <tr>
          <td>''k''<sub>1</sub></td>
          <td>0.0004691</td>
          <td>0.0004611</td>
          <td>0.0004277</td>
          <td>0.0001028</td>
          <td>0.0020576</td>
        </tr>
        <tr>
          <td>''k''<sub>2</sub></td>
          <td>0.0002059</td>
          <td>0.0002046</td>
          <td>0.0002155</td>
          <td>0.0007875</td>
          <td>0.0001228</td>
        </tr>
        <tr>
          <td>''k''<sub>3</sub></td>
          <td>0.0009999</td>
          <td>0.0010000</td>
          <td>0.0009984</td>
          <td>0.0009930</td>
          <td>0.0009527</td>
        </tr>
        <tr>
          <td>''k''<sub>4</sub></td>
          <td>0.0007915</td>
          <td>0.0008225</td>
          <td>0.0008419</td>
          <td>0.0008117</td>
          <td>0.0007790</td>
        </tr>
        <tr>
          <td>''k''<sub>5</sub></td>
          <td>0.0325900</td>
         <td>0.0336720</td>
          <td>0.0334167</td>
          <td>0.0334118</td>
          <td>0.0313443</td>
        </tr>
      </table>
    </td>
    <td>&nbsp;&nbsp;&nbsp;&nbsp;</td>
    <td>
'''Values of the objective function for 100 runs'''
      <table  border="1" cellspacing="0" cellpadding="4">
        <tr>
          <td>'''Methods'''</td>
          <td>'''The best value'''</td>
          <td>'''The mean value'''</td>
          <td>'''The worst value'''</td>
        </tr>
        <tr>
          <td>'''PSO'''</td>
          <td>11.787</td>
          <td>13.164</td>
          <td>14.703</td>
        </tr>
        <tr>
          <td>'''MOCell'''</td>
          <td>12.082</td>
          <td>13.484</td>
          <td>14.771</td>
        </tr>
        <tr>
          <td>'''SRES'''</td>
          <td>12.466</td>
          <td>14.987</td>
          <td>18.283</td>
        </tr>
        <tr>
          <td>'''ASA'''</td>
          <td>13.728</td>
          <td>15.794</td>
          <td>16.610</td>
        </tr>
        <tr>
          <td>'''glbSolve'''</td>
          <td>16.614</td>
          <td>16.614</td>
          <td>16.614</td>
        </tr>
      </table>
    </td>
  </tr>
</table>

==References==
# Kutumova E., Ryabova A., Valeev T., Kolpakov F. BioUML plug-in for nonlinear parameter estimation using multiple experimental data. ''Virtual biology''. 2013. 1:47-58. 
# Neumann L., Pforr C., Beaudouin J., Pappa A., Fricker N., Krammer P.H., Lavrik I.N., Eils R. Dynamics within the CD95 death-inducing signaling complex decide life and death of cells. ''Molecular Systems Biology''. 2010. 6:352.