Team:Grenoble-EMSE-LSU/Project/Modelling/Building

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<p>The construction of our model was not as linear as it is described, quite a few modelswere built, tried then abandonned. The aim was to find an explanation as simple as possible the results of the experiments carried out, equations describing the behaviour of our bacterial cells with as few parameters as possible. Thus our equations consider the <a href=#MatTime>maturation of fluorescent proteins</a> and the <a href=#AccDam>ability of the bacteria to repair themselves</a>.</p>  
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<p>The construction of our model was not a linear process: quite a few models were built, tried then abandoned. The aim was to find an explanation as simple as possible of the results of our experiments. This meant designing equations describing the behaviour of our bacterial cells with as few parameters as possible. Thus our equations consider the <a href=#MatTime>maturation of fluorescent proteins</a> and the <a href=#AccDam>ability of the bacteria to repair themselves</a>.</p>  
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<h2>Initial Model</h2>
<h2>Initial Model</h2>
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<h3>The equation</h3>
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<h3 id="equation1">The equation</h3>
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<p>    Our system is made of bacterial cells and ‘KillerRed’ proteins. Bacteria divide and produce KillerRed proteins, and KillerRed proteins respond to light: they fluoresce, degrade (photobleaching) and produce Radical Oxygen Species or ROS (phototoxicity). These reactions are exhibited by all fluorescent proteins, but <a href="/Team:Grenoble-EMSE-LSU/Project/Biology">the 3D structure of KillerRed</a> makes its degradation quicker and its high concentration allows ROS to reach proteins, DNA and membrane within the bacteria and damage its vital functions.</p>
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<p>    Our system is made of bacterial cells and ‘KillerRed’ proteins. Bacteria divide and produce KillerRed proteins, and KillerRed proteins respond to light: they fluoresce, degrade (photobleaching) and produce Radical Oxygen Species or ROS (phototoxicity). These reactions are exhibited by all fluorescent proteins, but <a href="/Team:Grenoble-EMSE-LSU/Project/Biology">the 3D structure of KillerRed</a> is responsible for a ROS production 1000-fold greater than that of other fluorescent proteins.</p>
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<p>$\bullet$ $I$ the amount of incident (white) light.</p>
<p>$\bullet$ $I$ the amount of incident (white) light.</p>
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<p>The evolution of C and K is linked to I by the set of equations : </p>
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<p>The evolution of C and K is linked to I by the set of equations :<br></p>
<center style="font-size:150%;">
<center style="font-size:150%;">
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<p>$\diamond$ $rC$ describes bacterial growth.</p>
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<p><br>$\diamond$ $rC$ describes bacterial growth.</p>
<p>$\diamond$ $kIK=kI\frac{K}{C}C$ the amount of bacteria killed by KillerRed and light.</p>
<p>$\diamond$ $kIK=kI\frac{K}{C}C$ the amount of bacteria killed by KillerRed and light.</p>
<p>$\diamond$ $aC$ the production of KillerRed.</p>
<p>$\diamond$ $aC$ the production of KillerRed.</p>
<p>$\diamond$ $bIK$ the amount of KillerRed photobleached.</p>
<p>$\diamond$ $bIK$ the amount of KillerRed photobleached.</p>
<p>$\diamond$ $kIK\frac{K}{C}$ the amount of KillerRed in the bacteria killed in the final time step.</p>
<p>$\diamond$ $kIK\frac{K}{C}$ the amount of KillerRed in the bacteria killed in the final time step.</p>
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<p>$r$, $a$, $k$ and $b$ are constants that we will determinate <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Parameters">later</a>.</p>
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<p>$r$, $a$, $k$ and $b$ are constants that are described <a href=#param1>a bit lower</a>.</p>
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<p>    Unfortunately, $C$ and $K$ are not measurable variables. The only quantities we can quickly and easily measure are the optical density (OD) associated the amount of dead AND living bacteria, and the global fluoresence associated with the amount of KillerRed in the dead AND living bacteria. In order to compare our model with experimental results, we need two additional variables : </p>
<p>    Unfortunately, $C$ and $K$ are not measurable variables. The only quantities we can quickly and easily measure are the optical density (OD) associated the amount of dead AND living bacteria, and the global fluoresence associated with the amount of KillerRed in the dead AND living bacteria. In order to compare our model with experimental results, we need two additional variables : </p>
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<p>$\bullet$ $D$ the amount of <b>dead</b> bacteria per milliter of cell suspension.</p>
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<p>$\bullet$ $D$ the amount of <b>dead</b> bacteria per milliliter of cell suspension. We consider that dead bacteria have the same Optical Density as living ones, because ROS damage does not lyse the cell. </p>
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<p>$\bullet$ $K_D$ the amount of KillerRed inside the <b>dead</b> bacteria per milliter of cell suspension.</p>
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<p>$\bullet$ $K_D$ the amount of KillerRed inside the <b>dead</b> bacteria per milliliter of cell suspension.</p>
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<p> $D$ and $K_D$ are necessary for the model, because they appear in the measurement od $OD_{600}$ and fluorescence, but dead bacteria don't grow anymore, and a KillerRed protein that is in a dead bacteria has no more effect on the $OD_{600}$ evolution. </p>  
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<p> $D$ and $K_D$ are necessary for the model, because they appear in the measurement of $OD_{600}$ and fluorescence, but dead bacteria don't grow anymore, and a KillerRed protein that is in a dead bacteria has no more effect on the $OD_{600}$ evolution. </p>  
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<p> The simplest possible units were used, corresponding to the measurable quantities : </p>
<p> The simplest possible units were used, corresponding to the measurable quantities : </p>
<p> $C$ and $D$ are in '$OD_{600}nm$' units.</p>
<p> $C$ and $D$ are in '$OD_{600}nm$' units.</p>
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<p> $K$ and $K_D$ are in units of fluorescence (UF). Bacterial auto-fluorescence is considered as negligible compared to KillerRed fluorescence. </p>
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<p> $K$ and $K_D$ are in Relative Fluorescent Unit (RFU). Bacterial auto-fluorescence is considered negligible compared to KillerRed fluorescence. </p>
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<p id="param1"> 4 parameters appear in those equations:</p>
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<p> $r$: the rate of growth of bacteria in $min^{-1}$</p>
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<p> $a$: the production of KillerRed per bacteria in $RFU.OD^{-1}.min^{-1}$</p>
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<p> $b$: the efficiency of photobleaching in $RFU.UL^{-1}.min^{-1}$</p>
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<p> $k$: the toxicity of KillerRed in $OD.RFU^{-1}.UL^{-1}.min^{-1}$</p>
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<br id="linGrowth">
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<p > The evolution of the absorbance ($C+D$) and the global fluorescence ($K+K_D$) can also be written:</p>
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<center style="font-size:150%;">
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$
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\left\{
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  \begin{array}{l l}
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    \frac{d(C+D)}{dt}=rC \\
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    \frac{d(K+K_D)}{dt}=aC-bI(K+K_D)\\
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  \end{array}
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\right.
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$
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</center>
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<p> The derivative of absorbance is proportional to the amount of living bacteria, therefore, a linear growth of the absorbance is characteristic of a constant population of bacteria, and this will stay true even with the more complete models. The evolution of fluorescence is simply the combination of the production and the photobleaching terms.</p> 
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<h3>Analytical Solution</h3>
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<h3 id="AnaSol">Analytical Solution</h3>
<p> This simple model can be partially solved, for $C(t)$ or $I(t)$ for example constant  : </p>
<p> This simple model can be partially solved, for $C(t)$ or $I(t)$ for example constant  : </p>
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<p> Thus, in the specific case that $I_0=\frac{r^2}{ak-rb}$, we have $\lim_{t\to\infty}\frac{dC}{dt}(t)=0$.</p>
<p> Thus, in the specific case that $I_0=\frac{r^2}{ak-rb}$, we have $\lim_{t\to\infty}\frac{dC}{dt}(t)=0$.</p>
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<p> The resolution of this equation have shown the possibility to stabilize the system.</p>   
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<p> The resolution of this equation shows the possibility of stabilizing the system.</p>   
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<p>The resolution of the latter equation shows that it is possible to stabilize the system by means of a suitable (constant) light intensity.</p>
<p>The resolution of the latter equation shows that it is possible to stabilize the system by means of a suitable (constant) light intensity.</p>
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<h3>Comparison with experiments</h3>
<h3>Comparison with experiments</h3>
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<p> This first model is very interesting to understand which parameters govern the evolution of the living cell population and to show that conditions exist to stabilize it. But unfortunately this set of equation is insufficient account for the results of the experiments. </p>
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<p> This first model is very interesting for understanding which parameters govern the evolution of the living cell population and to show that conditions exist to stabilize it. But unfortunately this set of equations is insufficient to account for the results of the experiments.<br><br></p>
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<p align="center"><img src="https://static.igem.org/mediawiki/2013/7/76/Build_Exp1_OD.png"></p>
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<p id="legend">Figure 1.<br>Evolution of absorbance in OD.<br><br></p>
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<p align="center"><img src="https://static.igem.org/mediawiki/2013/6/61/Build_Exp1_fluo.png"></p>
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<p id="legend">Figure 2.<br>Evolution of fluorescence in UF.<br><br></p>
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<p> Whereas we observe a lag between the onset of light and the decrease of fluorescence, the first model predicts an immediate decrease. This discrepancy requires the introduction of other phenomena to be introduced to explain the lag between the stimulus (the light) and the reaction (the decrease of fluorescence and the OD stabilization). Of course this explanation should be borne supported by biological facts. </p>
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<p> Whereas we observe a lag between the onset of light and the decrease of fluorescence, the first model predicts an immediate decrease. This discrepancy requires the introduction of other phenomena to explain the lag between the stimulus (the light) and the reaction (the decrease of fluorescence and the OD stabilization). Of course this explanation should be supported by biological facts. </p>
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<p>$\bullet$ $K_m$ the amount of mature KillerRed inside the living bacteria per milliter of cell suspension.</p>
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<p>$\bullet$ $K_m$ the amount of mature KillerRed inside the living bacteria per milliliter of cell suspension.</p>
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<p>$\bullet$ $K_i$ the amount of immature KillerRed inside the living bacteria per milliter of cell suspension.</p>
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<p>$\bullet$ $K_i$ the amount of immature KillerRed inside the living bacteria per milliliter of cell suspension. As an immature fluorescent protein does not have a chromophore, it does not degrade with light, and so is not affected by photobleaching.</p>
<center style="font-size:150%;">
<center style="font-size:150%;">
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<p>$\diamond$ $mK_i$ is the term expressing the maturation of KillerRed </p>
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<p>$\diamond$ $mK_i$ is the term expressing the maturation of KillerRed. We have a new parameter:</p>
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<p> $m$: the maturation rate of KillerRed in $min^{-1}$</p>
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<p> Similarly, immature Killer Red is also found in dead cells and its evolution is described by the following set of equations : </p>
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<p> Similarly, immature KillerRed is also found in dead cells and its evolution is described by the following set of equations : </p>
<center style="font-size:150%;">
<center style="font-size:150%;">
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<h3> Comparison with experiments</h3>
<h3> Comparison with experiments</h3>
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<p> The curves drawn from the model gives the right trend, observed in the experiments: the lag of the reaction, the peak of fluorescence short after light is switched on and then the swift decrease of fluorescence in the long term are qualitatively described. </p>
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<p> The curves drawn from the model give the right trend, observed in the experiments: the lag of the reaction, the peak of fluorescence short after light is switched on and then the swift decrease of fluorescence in the long term are qualitatively described.<br><br></p>
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<p align="center"><img src="https://static.igem.org/mediawiki/2013/4/41/Build_Exp2_OD.png"></p>
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<p id="legend">Figure 3.<br>Evolution of absorbance in OD.<br><br></p>
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<p align="center"><img src="https://static.igem.org/mediawiki/2013/1/11/Build_Exp2_fluo.png"></p>
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<p id="legend">Figure 4.<br>Evolution of fluorescence in UF.<br><br></p>
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<p>Nonetheless, it is impossible to get a good fit between the prediction of the model and the experiment. The maturation step alone does not explain why the production of KillerRed is so high two hours after the beginning of the illumination and the decrease of fluorescence is so rapid four hours after the illumination.</p>
<p>Nonetheless, it is impossible to get a good fit between the prediction of the model and the experiment. The maturation step alone does not explain why the production of KillerRed is so high two hours after the beginning of the illumination and the decrease of fluorescence is so rapid four hours after the illumination.</p>
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<p>[1]<a ref=http://www.chem.ufl.edu/~fanucci/courses/BiochemistryJournalClub/Spring2007/ChromophoreFormationinGFP_biochemistry_1997.pdf">REID Brian G., FLYNN Gregiry C. Chromophore Formation in Green Fluorescent Protein. Biochemistry, 1997, 36, p 6786-6791</a>.</p>
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<h3> Third Model</h3>
<h3> Third Model</h3>
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<p> Until now, our equations describe the phototoxic effect of KillerRed as instantaneous: in the presence of light Killer Red produces ROS, which immediately kills a certain proportion of the bacteria but has no lasting effect. It is well-known, however, that cells can repair damages due to ROS, up to a certain level. We can thus consider that bacteria are unable to instantly repair all the damages caused by ROS, and with damages mounting up, they are more and more fragile and close to death. Considering this accumulation allows us to shift the effect of illumination, and so to have a more accurate model.</p>
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<p> Until now, our equations describe the phototoxic effect of KillerRed as instantaneous: in the presence of light Killer Red produces ROS, which immediately kills a certain proportion of the bacteria but has no lasting effect. It is well-known, however, that cells can repair damages due to ROS, up to a certain level. We can thus consider that bacteria are unable to instantly repair all the damages caused by ROS, and with damages accumulating, they are more and more fragile and close to death.In other words, the effect of a certain amount of KillerRed at a certain time $u$, $K(u)$, illuminated by a light intensity $I(u)$, will affect cell growth at time $t$ later than $u$, weighted by a factor $e^{-p(t-u)}$ that vanishes as $t$ increases. The effect of this ROS production at time $u$ will thus exponentially decrease with time. The term $– kI.K$  (photokilling) was thus replaced by the integral:</p>
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<center style="font-size:150%">
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$-\int_{u=0}^t k'I(u)K(u)e^{-p(t-u)}du$
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</center>
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<p> And the equation of bacterial growth:</p>
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<center style="font-size:150%;">
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$
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\frac{dC}{dt}(t)=rC(t)-\int_{u=0}^t k'I(u)K(u)e^{-p(t-u)}du
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$
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</center>
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<p> $p$: the ability of the cell to repair in one minute, in $min^{-1}$.</p>
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<p> There is no analytical solution for this equation. But it is simple to treat it in its discrete-time form.</p>
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<p> Written in its discrete-time form, the evolution of $C$ was described by </p>
<p> Written in its discrete-time form, the evolution of $C$ was described by </p>
<center style="font-size:150%;">
<center style="font-size:150%;">
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$\tau$ the value of a step of time:
$
$
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C(t+1)-C(t)=rC(t)-kI(t)K(t)
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C(t+\tau)-C(t)=rC(t)-kI(t)K(t)
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$
</center>
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\left\{
\left\{
   \begin{array}{l l}
   \begin{array}{l l}
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     C(t+1)-C(t)=rC(t)-\mbox{tox}(t) \\
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     C(t+\tau)-C(t)=rC(t)-\mbox{tox}(t) \\
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     \mbox{tox}(t+1)=l.\mbox{tox}(t)+k'I(t)K(t)\\
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     \mbox{tox}(t+\tau)=l.\mbox{tox}(t)+k'I(t)K(t)\\
   \end{array}
   \end{array}
\right.
\right.
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with $l\in[0,1]$
with $l\in[0,1]$
</center>
</center>
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The variable 'tox' is representative of the amount of damages inflicted to bacteria, and so of their probability of dying. During a time step (for us, a minute), bacteria cures part of their injuries ($l<1$) and suffers new damages ($k'I(t)K(t)$). At first, few bacteria die : $\mbox{tox}(t)\cong k'I(t)K(t)$, then, 'tox' increases until it reach $\mbox{tox}(t)\cong\frac{k'}{1-l}I(t)K(t)$
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The variable 'tox' is representative of the amount of damages inflicted to bacteria, and so of their probability of dying. During a time step (for us, a minute), bacteria heal part of their injuries ($l<1$) and suffer new damages ($k'I(t)K(t)$). At first, few bacteria die : $\mbox{tox}(t)\cong k'I(t)K(t)$, then, 'tox' increases until it reach $\mbox{tox}(t)\cong\frac{k'}{1-l}I(t)K(t)$. $k'$ is the same in the continuous-time and the discrete-time forms, it will from now be named simply $k$. $p$ and $l$ are directly related : $p=-\frac{\ln(l)}{\tau}$.
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<p> This new equation can also be written in continuous-time form:</p>
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<p>We have our last variable:</p>
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<center style="font-size:150%;">
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<p> $l$: the rate of healing of the bacteria by step of time. unit less</p>
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$
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\frac{dC}{dt}(t)=rC(t)-\int_{u=0}^t k'I(u)K(u)e^{\ln(l)(t-u)}du
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$
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</center>
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<p>Considering this accumulation allows us to shift the effect of illumination, and so to have a more accurate model.</p>
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<p> But there is no analytical solution for this equation. We will therefore rely on numerical solutions in what follows.</p>
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<h3> Comparison with experiments</h3>
<h3> Comparison with experiments</h3>
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<p> With this model, we can now properly describe our data : </p>
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<p> With this model, we can now properly describe our data:<br><br></p>
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<p align="center"><img src="https://static.igem.org/mediawiki/2013/e/e6/Build_Exp3_OD.png"></p>
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<p id="legend">Figure 5.<br>Evolution of absorbance in OD.<br><br></p>
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<p align="center"><img src="https://static.igem.org/mediawiki/2013/c/c9/Build_Exp3_fluo.png"></p>
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<p id="legend">Figure 6.<br>Evolution of fluorescence in UF.<br><br></p>
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<p> As this fit seems to decribe well the kinetics observed, we will use this model to predict all our systems. But we still have parameters to adjust to find the best fit possible.</p>
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<h2>Reference</h2>
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<p>[1]<a ref=http://www.chem.ufl.edu/~fanucci/courses/BiochemistryJournalClub/Spring2007/ChromophoreFormationinGFP_biochemistry_1997.pdf">REID Brian G., FLYNN Gregiry C. Chromophore Formation in Green Fluorescent Protein. Biochemistry, 1997, 36, p 6786-6791</a>.</p>
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<li id="next"><a href="/Team:Grenoble-EMSE-LSU/Project/Modelling/Parameters">Next Page</a></li>
<li id="next"><a href="/Team:Grenoble-EMSE-LSU/Project/Modelling/Parameters">Next Page</a></li>

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