Team:Trieste/project/modeling

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<h1 id="h1_lf" class="main_tit"><div>Modeling</div></h1>
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    <h1 id="h1_lf" class="main_tit"><div>Modeling</div></h1>
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<h1 id="h1_rt" class="main_tit"><div>More</div></h1>
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    <h1 id="h1_rt" class="main_tit"><div>More</div></h1>
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<h2>Assumptions</h2>
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                                <h2>Assumptions</h2>
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<p>
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                                <p>
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    <ul class="fancy">
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                                    <ul class="fancy">
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<li>No interaction with living tissue</li>
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                                        <li>No interaction with living tissue</li>
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    </ul>
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                                    </ul>
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</p>
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                                </p>
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<h2>Model description</h2>
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<h2>Model description</h2>
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<p>
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<p>
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    We chose to use a deterministic, empirical model because it gives simple,
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We chose to use a deterministic, empirical model because it gives simple,
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    clean solutions.
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clean solutions.
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</p>  
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</p>  
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<p>
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<p>
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    The model is composed of 6 differential equations, each one describing
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The model is composed of 6 differential equations, each one describing
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    the concentration of a substance inside or outside the bacteria.
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the concentration of a substance inside or outside the bacteria.
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    The variables are:
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</p>
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    <ul>
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<p>
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        <li>\(b\) - bacteria concentration</li>
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All the \(\delta_i\) parameters represent the variable's decay over time.
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        <li>\(p\) - CymR concentration</li>
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</p>
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        <li>\(c\) - p-cumate concentration</li>
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<p>
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        <li>\(l_i\) - internal toxin concentration</li>
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The first equation, describing the bacteria concentration in the media (bacteria growth) is based
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        <li>\(l_e\) - external toxin concentration</li>
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on the logistics function:
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        <li>\(a\) - antibody concentration</li>
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\begin{equation}
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    </ul>
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\dot{x} = x (1-x)
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    All the \(\delta_i\) parameters represent the variable's decay over time.
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\end{equation}
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</p>
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We added a term to model bacteria death due to toxin, so the complete equation
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<p>
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is:
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    The first equation, describing the bacteria concentration in the media (bacteria growth) is based
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\begin{equation}
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    on the logistics function:
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\dot{b} = k_1  b  (N - b) - k_2  b  (l_i + l_e)
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    \begin{equation}
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\end{equation}
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    \dot{x} = x (1-x)
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There are three parameters that control this model's behaviour.
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    \end{equation}
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The \(N\) parameter specifies the maximum concentration of bacteria.
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    We added a term to model bacteria death due to toxin, so the complete equation
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The \(k_1\) parameter controls the growth speed and the \(k_2\) parameter sets how
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    is:
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powerful the toxin is.
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    \begin{equation}
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</p>
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    \dot{b} = k_1  b  (N - b) - k_2  b  (l_i + l_e)
 +
    \end{equation}
 +
    There are three parameters that control this model's behaviour.
 +
    The \(N\) parameter specifies the maximum concentration of bacteria.
 +
    The \(k_1\) parameter controls the growth speed and the \(k_2\) parameter sets how
 +
    powerful the toxin is.
 +
</p>
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<p>
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<p>
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The next equation models the CymR concentration inside the bacteria.
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    The next equation models the CymR concentration inside the bacteria.
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\begin{equation}
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    \begin{equation}
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\dot{p} = k_3 - \delta_1 p - \alpha  p  c
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    \dot{p} = k_3 - \delta_1 p - \alpha  p  c
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\end{equation}
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    \end{equation}
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The \(k_3\) parameter determines the rate of synthesis of CymR and \(\alpha\) the rate of cumate ligation.
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    The \(k_3\) parameter determines the rate of synthesis of CymR and \(\alpha\) the rate of cumate ligation.
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</p>
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</p>
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<p>
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<p>
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The p-cumate concentration function is in fact a parameter, because it is
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    The p-cumate concentration function is in fact a parameter, because it is
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directly controllable by adding p-cumate into the bacteria solution.
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    directly controllable by adding p-cumate into the bacteria solution.
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\begin{equation}
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    \begin{equation}
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\dot{c} = k_4 - \delta_2 c
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    \dot{c} = k_4 - \delta_2 c
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\end{equation}
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    \end{equation}
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The \(k_4\) parameter models p-cumate increase (adding p-cumate at a steady rate).
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    The \(k_4\) parameter models p-cumate increase (adding p-cumate at a steady rate).
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</p>
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</p>
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<p>
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<p>
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We use two equations to better model the toxin concentration, taking into
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    We use two equations to better model the toxin concentration, taking
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account the concentration of toxin both inside and outside the bacteria. The equation
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    into account the concentration of toxin both inside and outside
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for the inside concentration is a Hill equation with a decay term. We used
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    the bacteria. The equation for the inside concentration is a Hill
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it because the concentration increases until the saturation is
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    equation for the inhibitory effect of CymR (\(p\)) on the biosynthesis
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reached, and from then on the bacteria stops producing the toxin. The point of
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    of the toxin, and the decay term. We used
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saturation is never reached, though, because the bacteria dies due to the
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    it because the concentration increases until the saturation is reached,
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toxin itself. The outside equation is the most complex one; the outside toxin
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    and from then on the bacteria stops producing the toxin. The outside
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concentration depends on bacteria death: when the bacteria dies, all the toxin
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    equation is the most complex
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it has produced is released outside.
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    one; the outside toxin concentration depends on bacteria death: when
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\begin{align}
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    the bacteria dies, all the toxin it has produced is released outside.
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\dot{l_i} &= \frac{A}{k_5 + p} -\delta_3  l_i \\  
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    \begin{align}
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\dot{l_e} &= v (l_i + l_e) b l_i - \delta_4 l_e
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    \dot{l_i} &= \frac{A}{k_5 + p} -\delta_3  l_i \\  
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\end{align}
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    \dot{l_e} &= v (l_i + l_e) b l_i - \delta_4 l_e
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The \(A\) parameter sets the maximum toxin concentration inside the bacteria, while
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    \end{align}
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the \(v\) parameter takes into account the toxin dilution in the bacteria
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    The \(A\) parameter sets the maximum toxin concentration inside the bacteria, while
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solution.
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    the \(v\) parameter takes into account the toxin dilution in the bacteria
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</p>
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    solution and the toxin strength.
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<p>
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</p>
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Last but not least, the antibody density, which we suppose always increasing in
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<p>
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time.
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    Last but not least, the antibody density, which we suppose always increasing in
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\begin{equation}
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    time.
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\dot{a} = k_6 - \delta_5  a
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    \begin{equation}
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\end{equation}
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    \dot{a} = k_6 - \delta_5  a
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</p>
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    \end{equation}
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                                <h2>Tools</h2>
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</p>
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                                <p>
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<h2>Tools</h2>
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                                    We decided to use Octave, the open source alternative to  
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<p>
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                                    Matlab as the framework for our model.
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    We decided to use Octave, the open source alternative to  
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                                </p>
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    Matlab as the framework for our model.
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                                <p>
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</p>
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                                  <img src="https://static.igem.org/mediawiki/2012/5/55/Trieste-Cumato.png" alt="Model with cumato" />
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<p>
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                                  <img src="https://static.igem.org/mediawiki/2012/9/9f/Trieste-No_cumato.png" alt="Model without cumato" />
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    <img src="https://static.igem.org/mediawiki/2012/5/55/Trieste-Cumato.png" alt="Model with cumato" />
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                                </p>
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    <img src="https://static.igem.org/mediawiki/2012/9/9f/Trieste-No_cumato.png" alt="Model without cumato" />
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<li><a href="https://2012.igem.org/Team:Trieste/project">Abstract</a></li>
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<li><a href="https://2012.igem.org/Team:Trieste/project">Abstract</a></li>
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                <li><a href="https://2012.igem.org/Team:Trieste/project/overview">Project Overview</a></li>                 
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<li><a href="https://2012.igem.org/Team:Trieste/project/overview">Project Overview</a></li>                 
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<li><a href="https://2012.igem.org/Team:Trieste/project/applications">Applications</a></li>         
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<li><a href="https://2012.igem.org/Team:Trieste/project/applications">Applications</a></li>         
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                                <li class="select" ><a href="https://2012.igem.org/Team:Trieste/project/modeling">Modeling</a></li>
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<li class="select"><a href="https://2012.igem.org/Team:Trieste/project/modeling">Modeling</a></li>
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<h2>Contact us</h2>
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<h2>Contact us</h2>
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<p>For other information, write to:</p>
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<p>For other information, write to:</p>
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Revision as of 08:50, 26 September 2012

Modeling

More

Assumptions

  • No interaction with living tissue

Model description

We chose to use a deterministic, empirical model because it gives simple, clean solutions.

The model is composed of 6 differential equations, each one describing the concentration of a substance inside or outside the bacteria. The variables are:

  • \(b\) - bacteria concentration
  • \(p\) - CymR concentration
  • \(c\) - p-cumate concentration
  • \(l_i\) - internal toxin concentration
  • \(l_e\) - external toxin concentration
  • \(a\) - antibody concentration
All the \(\delta_i\) parameters represent the variable's decay over time.

The first equation, describing the bacteria concentration in the media (bacteria growth) is based on the logistics function: \begin{equation} \dot{x} = x (1-x) \end{equation} We added a term to model bacteria death due to toxin, so the complete equation is: \begin{equation} \dot{b} = k_1 b (N - b) - k_2 b (l_i + l_e) \end{equation} There are three parameters that control this model's behaviour. The \(N\) parameter specifies the maximum concentration of bacteria. The \(k_1\) parameter controls the growth speed and the \(k_2\) parameter sets how powerful the toxin is.

The next equation models the CymR concentration inside the bacteria. \begin{equation} \dot{p} = k_3 - \delta_1 p - \alpha p c \end{equation} The \(k_3\) parameter determines the rate of synthesis of CymR and \(\alpha\) the rate of cumate ligation.

The p-cumate concentration function is in fact a parameter, because it is directly controllable by adding p-cumate into the bacteria solution. \begin{equation} \dot{c} = k_4 - \delta_2 c \end{equation} The \(k_4\) parameter models p-cumate increase (adding p-cumate at a steady rate).

We use two equations to better model the toxin concentration, taking into account the concentration of toxin both inside and outside the bacteria. The equation for the inside concentration is a Hill equation for the inhibitory effect of CymR (\(p\)) on the biosynthesis of the toxin, and the decay term. We used it because the concentration increases until the saturation is reached, and from then on the bacteria stops producing the toxin. The outside equation is the most complex one; the outside toxin concentration depends on bacteria death: when the bacteria dies, all the toxin it has produced is released outside. \begin{align} \dot{l_i} &= \frac{A}{k_5 + p} -\delta_3 l_i \\ \dot{l_e} &= v (l_i + l_e) b l_i - \delta_4 l_e \end{align} The \(A\) parameter sets the maximum toxin concentration inside the bacteria, while the \(v\) parameter takes into account the toxin dilution in the bacteria solution and the toxin strength.

Last but not least, the antibody density, which we suppose always increasing in time. \begin{equation} \dot{a} = k_6 - \delta_5 a \end{equation}

Tools

We decided to use Octave, the open source alternative to Matlab as the framework for our model.

Model with cumato Model without cumato

Team iGEM 2012

Contact us

For other information, write to:

igem2012@gmail.com
Università degli studi di Trieste ICGEB Illy Fondazione Cassa di Risparmio
iGEM 2012 iGEM 2012 iGEM 2012 iGEM 2012 iGEM 2012 iGEM 2012
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