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| − | <h3>Introduction</h3>
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| − | Based on a set of ordinary differential equations (ODE) describing the kinetics of the cells differentiation, we designed a model to find the best
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| − | differentiation rate.
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| − | First we developed a deterministic algorithm based on the ordinary differential equations solutions.
| + | Just ask me if you want my wiki. Work is done but I don't want to share it with all igem teams ! |
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| − | Then we find out that a stochastic algorithm could be an other solution to solve our problem.
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| − | For system involving large cell counts, the ordinary differential equations model give an accurate representation of the behavior. But with small cell
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| − | counts, the stochastic and discrete method has a significant influence on the observed behaviour.
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| − | <br />
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| − | These reasons led us to write both a deterministic programm based on the mass action law and a stochastic programm based on the Gillespie’s stochastic
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| − | simulation algorithm (SSA). With these two programms we obtain an accurate analysis of the vitamin production.
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| − | <h3>Mass action law model</h3>
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| − | <h4>Parameters</h4>
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| − | We conceived a simple model with the minimum number of parameters.
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| − | We find seven significant parameters for our model.
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| − | <br />
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| − | <ul>
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| − | <li>\(t\) : fermentation period.</li>
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| − | <li>\(k_{1}\) : rate constant of the mother cell differentiation.</li>
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| − | <li>\(k_{2}\) : rate constant of the mother cell doubling time.</li>
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| − | <li>\(k_{3}\) : rate constant of the differentiate cell doubling time.</li>
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| − | <li>\(k_{4}\) : rate constant of the differentiate cell vitamin production.</li>
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| − | <li>\([MC]_0\) : initial concentration of mother cells in the medium.</li>
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| − | <li>\([DC]_0\) : initial concentration of differentiate cells in the medium.</li>
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| − | </ul>
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| − | <h4>Kinetic equations</h4>
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| − | We wrote four simple kinetic equations.
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| − | <br />
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| − | \[
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| − | \begin{align}
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| − | MC \xrightarrow[]{k_{1}} DC
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| − | \\
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| − | MC \xrightarrow[]{k_{2}} 2.MC
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| − | \\
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| − | DC \xrightarrow[]{k_{3}} 2.DC
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| − | \\
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| − | DC \xrightarrow[]{k_{4}} DC + Vitamin
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| − | \end{align}
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| − | \]
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| − | <h4>Formal mathematical solution</h4>
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| − | <h5>Translation in ordinary differential equations</h5>
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| − | We used the law of mass action to write the ordinary differential equations based on kinetic equations \((1)\), \((2)\), \((3)\) and \((4)\).
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| − | <br />
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| − | We found the following equations.
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| − | <br />
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| − | \[
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| − | \begin{align}
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| − | \frac{d[MC]}{dt}(t) = (k_{2} - k_{1}).[MC](t)
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| − | \end{align}
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| − | \]
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| − | \[
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| − | \begin{align}
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| − | \frac{d[DC]}{dt}(t) = k_{1}.[MC](t) + k_{3}.[DC](t)
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| − | \end{align}
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| − | \]
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| − | \[
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| − | \begin{align}
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| − | \frac{d[Vitamin]}{dt}(t) = k_{4}.[DC](t) \Rightarrow [Vitamin](t) = k_{4}.\int_{0}^{t}{[DC](t').dt'}
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| − | \end{align}
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| − | \]
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| − | <h5>Mathematical resolution</h5>
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| − | Simple ordinary differential equations resolution methods are used to find the solutions of the previous equations \((5)\), \((6)\), and \((7)\).
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| − | We expressed the time evolution of the mother cell, differentiate cell and vitamin concentrations.
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| − | <br />
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| − | \[
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| − | \begin{align}
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| − | [MC](t) = [MC]_{0}.e^{(k_{2} - k_{1}).t}
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| − | \end{align}
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| − | \]
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| − | <br />
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| − | \[
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| − | \begin{align}
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| − | [DC](t) =
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| − | \begin{cases}
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| − | [DC]_{0}.e^{k_{3}.t} + [MC]_{0}.\frac{k_{1}}{k_{2}-k_{1}-k_{3}}.(e^{(k_{2} - k_{1}).t} - e^{k_{3}.t}) & \mbox{if } k_{1} \ne k_{3} - k_{2}
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| − | \\
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| − | \\
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| − | ([MC]_{0}.(k_{2} - k_{3}).t + [DC]_{0}).e^{k_{3}.t} & \mbox{if } k_{1} = k_{3} - k_{2}
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| − | \end{cases}
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| − | \end{align}
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| − | \]
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| − | <br />
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| − | \[
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| − | \begin{align}
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| − | [Vitamin](t) =
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| − | \begin{cases}
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| − | [DC]_{0}.\frac{k_{4}}{k_{3}}.(e^{k_{3}.t} - 1) + [MC]_{0}.\frac{k_{4}.k_{1}}{(k_{2}-k_{1}-k_{3}).(k_{2} - k_{1})}.(e^{(k_{2} - k_{1}).t} -
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| − | 1) + [MC]_{0}.\frac{k_{4}.k_{1}}{(k_{2}-k_{1}-k_{3}).k_{3}}.(1 - e^{k_{3}.t}) & \mbox{if } k_{1} \ne k_{3} - k_{2}
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| − | \\
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| − | \\
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| − | [DC]_{0}.\frac{k_{4}}{k_{3}}.(e^{k_{3}.t} - 1) + [MC]_{0}.\frac{k_{4}}{k_{3}}.(k_{3}.t - 1).(\frac{k_{2}}{k_{3}} - 1).(e^{k_{3}.t} - 1) +
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| − | [MC]_{0}.\frac{k_{4}}{k_{3}}.(\frac{k_{2}}{k_{3}} - 1) &
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| − | \mbox{if } k_{1} = k_{3} - k_{2}
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| − | \end{cases}
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| − | \end{align}
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| − | \]
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| − | <h4>Vitamin optimization</h4>
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| − | Our goal is to optimize the vitamin production. We can only change three parameters : \(k_{1}\), \([MC]_0\) and \([DC]_0\).
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| − | <br />
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| − | \(t\), \(k_{2}\), \(k_{3}\) and \(k_{4}\) are constants. \([MC]_0\) and \([DC]_0\) are not relevant parameters. It seems logical that the
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| − | more cells are in the medium, the more vitamin are producted.
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| − | <br />
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| − | However, it could be interesting to compare different \(\frac{[MC]_0}{[DC]_0}\) ratios but we prefer to focus on the rate constant \(k_{1}\).
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| − | <br />
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| − | We try to find the best \(k_{1}\). To this end we maximize the vitamin function numerically with MATLAB.
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| − | <br />
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| − | As an example, lets consider the following parameters.
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| − | <br />
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| − | <ul>
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| − | <li>\(t = 10\)</li>
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| − | <li>\(k_{2} = 0.66\)</li>
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| − | <li>\(k_{3} = 0.33\)</li>
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| − | <li>\(k_{4} = 1\)</li>
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| − | <li>\([MC]_0 = 5\)</li>
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| − | <li>\([DC]_0 = 0\)</li>
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| − | </ul>
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| − | We obtain the following graph with a simple MATLAB programm available <a>here</a>.
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| − | <br />
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| − | <img src="https://static.igem.org/mediawiki/2015/4/45/OptimizeK1.png" title="Vitamin optimization" alt="Vitamin optimization" style="align:center;">
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| − | As you can see, we are able to find the best \(k_{1}\) to optimize the vitamin production.
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| − | <br />
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| − | We noticed that two differents \(k_{1}\) exists to optimize the maximum concentration of differentiate cell \([DC]\) or the maximum concentration of vitamin
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| − | \([Vitamin]\).
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| − | <br />
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| − | In our case, we chose of course the \(k_{1}\) that optimize the vitamin production <i>ie</i> \(k_{1} = 0.207\).
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| − | <h4>Deterministic evolution of mother cell, differentiate cell and vitamin concentrations</h4>
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| − | We wrote a deterministic algorithm with MATLAB using the previous solutions \((8)\), \((9)\) and \((10)\). For those interested, the source code is available <a>here</a>.
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| − | <br />
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| − | In order to optimize the vitamin production, we use the same parameters as previously and set \(k_{1}\) with the previous resulting value <i>ie</i> \(k_{1} =
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| − | 0.207\).
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| − | <br />
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| − | We obtained the following graph.
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| − | <img src="https://static.igem.org/mediawiki/2015/f/f7/DeterministicEvolution.png" title="Deterministic evolution of the system" alt="Deterministic evolution of the
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| − | system" style="align:center;">
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| − | As you can see, the mother cell, differentiate cell and vitamin concentrations follow an exponential law of time.
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| − | <br />
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| − | This result seems relevant. The model does not take into account the cells death and the nutrients present in the medium.
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| − | <h3>Stochastic model</h3>
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| − | <h3>Conclusion</h3>
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