http://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&feed=atom&action=historyTeam:Paris Saclay/Modeling/bacterial Growth - Revision history2024-03-29T11:05:04ZRevision history for this page on the wikiMediaWiki 1.16.5http://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=381029&oldid=prevSC: /* Bacterial Population Growth */2014-10-18T01:16:43Z<p><span class="autocomment">Bacterial Population Growth</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Each time, we will expose both a deterministic model and a stochastic one.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Each time, we will expose both a deterministic model and a stochastic one.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The major advantage of the deterministic model is <del class="diffchange diffchange-inline">his </del>simplicity. But this simplicity has a cost : while every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables, the deterministic models perform the same way for a given set of initial conditions which is not really desirable. Conversely, in a stochastic model, randomness is present, and the variable states are not described by unique values, but rather by probability distributions.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The major advantage of the deterministic model is <ins class="diffchange diffchange-inline">its </ins>simplicity. But this simplicity has a cost: while every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables, the deterministic models perform the same way for a given set of initial conditions which is not really desirable. Conversely, in a stochastic model, randomness is present, and the variable states are not described by unique values, but rather by probability distributions.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Menu==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Menu==</div></td></tr>
</table>SChttp://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=369519&oldid=prevNdiaye: /* Bacterial Population Growth */2014-10-17T23:39:50Z<p><span class="autocomment">Bacterial Population Growth</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=Bacterial Population Growth=</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=Bacterial Population Growth=</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">The oxygen diffusion model confirms the intuition we could have: while oxygen can't easily move into the medium, the growth on the surface is sufficiently more important than the growth inside the medium for neglecting the latter. We will tackle here the process of bacterial growth in different aspects.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We are here considering a bacterial population uniformly spread on a surface in the Euclidean space, in crowd-free conditions and with unlimited food resources.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We are here considering a bacterial population uniformly spread on a surface in the Euclidean space, in crowd-free conditions and with unlimited food resources.</div></td></tr>
</table>Ndiayehttp://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=368223&oldid=prevJulietteC: /* Coefficient of Variation $\star\star$ */2014-10-17T23:29:06Z<p><span class="autocomment">Coefficient of Variation $\star\star$</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ V(t) = N_0\frac{\lambda+\mu}{\lambda-\mu}e^{(\lambda-\mu)t}(e^{(\lambda-\mu)t}-1) \]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ V(t) = N_0\frac{\lambda+\mu}{\lambda-\mu}e^{(\lambda-\mu)t}(e^{(\lambda-\mu)t}-1) \]</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Unlike $m(t)$ <del class="diffchange diffchange-inline">and </del>$V(t)$ depend not only on the difference between the birth and the death rates, but also on their absolute magnitudes. This is what we should expect, because predictions about the future size of a population will be less precise if birth and death occur in rapid succession than if they occur only occasionally.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Unlike $m(t)$<ins class="diffchange diffchange-inline">, </ins>$V(t)$ depend not only on the difference between the birth and the death rates, but also on their absolute magnitudes. This is what we should expect, because predictions about the future size of a population will be less precise if birth and death occur in rapid succession than if they occur only occasionally.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We define the coefficient of variation $\displaystyle CV(t)~:=~\frac{\sqrt{V(t)}}{m(t)}$ which qualify the variation of the system and we will study the effect of the relative values of $\lambda$ and $\mu$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We define the coefficient of variation $\displaystyle CV(t)~:=~\frac{\sqrt{V(t)}}{m(t)}$ which qualify the variation of the system and we will study the effect of the relative values of $\lambda$ and $\mu$.</div></td></tr>
</table>JulietteChttp://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=359508&oldid=prevPierre R: /* General conclusion : What about lemon? */2014-10-17T22:03:40Z<p><span class="autocomment">General conclusion : What about lemon?</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==General conclusion : What about lemon?==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==General conclusion : What about lemon?==</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>To come back to our lemon porject, let's see what is <del class="diffchange diffchange-inline">appended </del>when we consider now "''our''" own bacteria and not only a general bacterium species !</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>To come back to our lemon porject, let's see what is <ins class="diffchange diffchange-inline">appening </ins>when we consider now "''our''" own bacteria and not only a general bacterium species !</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In practise, we use the genetically modified ''E.coli''. As the modification that we carried out do not touch to the vital metabolism of bacteria we may assume that the original rates of birth and <del class="diffchange diffchange-inline">of </del>death for ''E.coli'' have not been changed.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In practise, we use the genetically modified ''E.coli''. As the modification that we carried out do not touch to the vital metabolism of bacteria we may assume that the original rates of birth and death for ''E.coli'' have not been changed.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>So, according to researches of Eric Stewart [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#References [Ste]], we have</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>So, according to researches of Eric Stewart [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#References [Ste]], we have</div></td></tr>
</table>Pierre Rhttp://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=329915&oldid=prevCtn: /* General conclusion : What about lemon? */2014-10-17T15:58:58Z<p><span class="autocomment">General conclusion : What about lemon?</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If we want to be really sure, we just have to choose $N_0$ as</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If we want to be really sure, we just have to choose $N_0$ as</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ N_0\geqslant\frac{3}{log_{10}\left(\frac{\lambda}{\mu}\right)} = \frac{3}{log_{10}\left(\frac{3,6 \times 10^{-2}}{4,6 \times 10^{-4}}\right)} = \frac{3}{log_{10}(78)} = \frac{3}{1,9} = 1,6 \]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ N_0\geqslant\frac{3}{log_{10}\left(\frac{\lambda}{\mu}\right)} = \frac{3}{log_{10}\left(\frac{3,6 \times 10^{-2}}{4,6 \times 10^{-4}}\right)} = \frac{3}{log_{10}(78)} = \frac{3}{1,9} = 1,6 \]</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>what, with our model, ensures us that if we <del class="diffchange diffchange-inline">put down </del>at least two bacteria on our lemon it will be wholly covered.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>what, with our model, ensures us that if we <ins class="diffchange diffchange-inline">deposit </ins>at least two bacteria on our lemon it will be wholly covered.</div></td></tr>
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</table>Ctnhttp://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=329843&oldid=prevCtn: /* General conclusion : What about lemon ? */2014-10-17T15:57:43Z<p><span class="autocomment">General conclusion : What about lemon ?</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[File:Paris_Saclay_Simple_Birth_Death_Probability_of_Extinction.png|660px|center]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[File:Paris_Saclay_Simple_Birth_Death_Probability_of_Extinction.png|660px|center]]</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>==General conclusion : What about lemon ?==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==General conclusion : What about lemon?==</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>To come back to our lemon porject, let's see what is appended when we consider now "''our''" own bacteria and not only a general bacterium species !</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>To come back to our lemon porject, let's see what is appended when we consider now "''our''" own bacteria and not only a general bacterium species !</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If we want to be really sure, we just have to choose $N_0$ as</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If we want to be really sure, we just have to choose $N_0$ as</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ N_0\geqslant\frac{3}{log_{10}\left(\frac{\lambda}{\mu}\right)} = \frac{3}{log_{10}\left(\frac{3,6 \times 10^{-2}}{4,6 \times 10^{-4}}\right)} = \frac{3}{log_{10}(78)} = \frac{3}{1,9} = 1,6 \]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ N_0\geqslant\frac{3}{log_{10}\left(\frac{\lambda}{\mu}\right)} = \frac{3}{log_{10}\left(\frac{3,6 \times 10^{-2}}{4,6 \times 10^{-4}}\right)} = \frac{3}{log_{10}(78)} = \frac{3}{1,9} = 1,6 \]</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">those</del>, with our model, ensures that if we put at least two bacteria on our lemon it will be wholly covered.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">what</ins>, with our model, ensures <ins class="diffchange diffchange-inline">us </ins>that if we put <ins class="diffchange diffchange-inline">down </ins>at least two bacteria on our lemon it will be wholly covered.</div></td></tr>
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</table>Ctnhttp://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=329598&oldid=prevCtn: /* General conclusion : What about lemon ? */2014-10-17T15:53:53Z<p><span class="autocomment">General conclusion : What about lemon ?</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>which is really near to $0$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>which is really near to $0$.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In practice, there is almost no chance that the population extinct and that there is <del class="diffchange diffchange-inline">place </del>bacteria will develop indefinitely.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In practice, there is almost no chance that the population <ins class="diffchange diffchange-inline">will be </ins>extinct and that there is <ins class="diffchange diffchange-inline">places where </ins>bacteria will develop indefinitely.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Actually, our model can be considerably improved but we feared it was not <del class="diffchange diffchange-inline">on </del>the level of the iGEM competition.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Actually, our model can be considerably improved but we feared it was not <ins class="diffchange diffchange-inline">of </ins>the level of the iGEM competition.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==References==</div></td></tr>
</table>Ctnhttp://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=329296&oldid=prevCtn: /* General conclusion : What about lemon ? */2014-10-17T15:48:24Z<p><span class="autocomment">General conclusion : What about lemon ?</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Moreover, we know, through the part [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#Probability_of_extinction_.24.5Cstar.24 probability of extinction], that the probability of extinction is, if we denote $N_0$ the initial population size,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Moreover, we know, through the part [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#Probability_of_extinction_.24.5Cstar.24 probability of extinction], that the probability of extinction is, if we denote $N_0$ the initial population size,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ p_0(\infty) \sim \left(\frac{\mu}{\lambda}\right)^{N_0} = \left(\frac{4,6 \times 10^{-4}}{\lambda = 3,6 \times 10^{-2}}\right)^{N_0} \simeq (1.3 \times 10^{-2})^{N_0} \simeq 10^{-2~N_0} \]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ p_0(\infty) \sim \left(\frac{\mu}{\lambda}\right)^{N_0} = \left(\frac{4,6 \times 10^{-4}}{\lambda = 3,6 \times 10^{-2}}\right)^{N_0} \simeq (1.3 \times 10^{-2})^{N_0} \simeq 10^{-2~N_0} \]</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>which is really near $0$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>which is really near <ins class="diffchange diffchange-inline">to </ins>$0$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In practice, there is almost no chance that the population extinct and that there is place bacteria will develop indefinitely.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In practice, there is almost no chance that the population extinct and that there is place bacteria will develop indefinitely.</div></td></tr>
</table>Ctnhttp://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=329201&oldid=prevCtn: /* General conclusion : What about lemon ? */2014-10-17T15:46:50Z<p><span class="autocomment">General conclusion : What about lemon ?</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==General conclusion : What about lemon ?==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==General conclusion : What about lemon ?==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>To come back to our lemon, let see what <del class="diffchange diffchange-inline">append </del>when we <del class="diffchange diffchange-inline">now </del>consider "''our''" bacteria and not only a general bacterium species !</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>To come back to our lemon <ins class="diffchange diffchange-inline">porject</ins>, let<ins class="diffchange diffchange-inline">'s </ins>see what <ins class="diffchange diffchange-inline">is appended </ins>when we consider <ins class="diffchange diffchange-inline">now </ins>"''our''" <ins class="diffchange diffchange-inline">own </ins>bacteria and not only a general bacterium species !</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In practise, we use genetically modified ''E.coli''. As the modification that we carried out do not touch to the vital metabolism of bacteria we may assume that the original ''E.coli''<del class="diffchange diffchange-inline">'s birth and death rate </del>have not been changed.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In practise, we use <ins class="diffchange diffchange-inline">the </ins>genetically modified ''E.coli''. As the modification that we carried out do not touch to the vital metabolism of bacteria we may assume that the original <ins class="diffchange diffchange-inline">rates of birth and of death for </ins>''E.coli'' have not been changed.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>So, according to <del class="diffchange diffchange-inline">the research </del>of Eric Stewart [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#References [Ste]], we have</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>So, according to <ins class="diffchange diffchange-inline">researches </ins>of Eric Stewart [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#References [Ste]], we have</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ \lambda = 3,6 \times 10^{-2} ~ min^{-1} \]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ \lambda = 3,6 \times 10^{-2} ~ min^{-1} \]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>and</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>and</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>So we are in the case $\lambda>\mu$ <del class="diffchange diffchange-inline">and the </del>partition [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#Coefficient_of_Variation_.24.5Cstar.5Cstar.24 coefficient of variation] ensure us that the system is asymptotically stable.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>So we are in the case $\lambda>\mu$<ins class="diffchange diffchange-inline">. The </ins>partition [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#Coefficient_of_Variation_.24.5Cstar.5Cstar.24 coefficient of variation] ensure us that the system is asymptotically stable.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Moreover, we know , through the part [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#Probability_of_extinction_.24.5Cstar.24 probability of extinction], that the probability of extinction is, if we denote $N_0$ the initial population size,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Moreover, we know, through the part [https://2014.igem.org/Team:Paris_Saclay/Modeling/bacterial_Growth#Probability_of_extinction_.24.5Cstar.24 probability of extinction], that the probability of extinction is, if we denote $N_0$ the initial population size,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ p_0(\infty) \sim \left(\frac{\mu}{\lambda}\right)^{N_0} = \left(\frac{4,6 \times 10^{-4}}{\lambda = 3,6 \times 10^{-2}}\right)^{N_0} \simeq (1.3 \times 10^{-2})^{N_0} \simeq 10^{-2~N_0} \]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ p_0(\infty) \sim \left(\frac{\mu}{\lambda}\right)^{N_0} = \left(\frac{4,6 \times 10^{-4}}{\lambda = 3,6 \times 10^{-2}}\right)^{N_0} \simeq (1.3 \times 10^{-2})^{N_0} \simeq 10^{-2~N_0} \]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>which is really near $0$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>which is really near $0$.</div></td></tr>
</table>Ctnhttp://2014.igem.org/wiki/index.php?title=Team:Paris_Saclay/Modeling/bacterial_Growth&diff=326790&oldid=prevCtn: /* Probability of extinction $\star$ */2014-10-17T15:12:01Z<p><span class="autocomment">Probability of extinction $\star$</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>As $t$ increases,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>As $t$ increases,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ p_0(t)~\underset{\begin{array}{c}{\scriptstyle \lambda\rightarrow\mu} \\ {\scriptstyle t\rightarrow\infty}\end{array}}\sim~\left(\frac{\mu t}{\mu t}\right)^{N_0}~\underset{\begin{array}{c}{\scriptstyle \lambda\rightarrow\mu} \\ {\scriptstyle t\rightarrow\infty}\end{array}}\longrightarrow~1 \]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\[ p_0(t)~\underset{\begin{array}{c}{\scriptstyle \lambda\rightarrow\mu} \\ {\scriptstyle t\rightarrow\infty}\end{array}}\sim~\left(\frac{\mu t}{\mu t}\right)^{N_0}~\underset{\begin{array}{c}{\scriptstyle \lambda\rightarrow\mu} \\ {\scriptstyle t\rightarrow\infty}\end{array}}\longrightarrow~1 \]</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>and so ultimate extinction is certain even <del class="diffchange diffchange-inline">tough </del>the birth and the death rate are equal.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>and so ultimate extinction is certain even <ins class="diffchange diffchange-inline">if </ins>the birth <ins class="diffchange diffchange-inline">rate </ins>and the death rate are equal.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[File:Paris_Saclay_Simple_Birth_Death_Probability_of_Extinction.png|660px|center]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[File:Paris_Saclay_Simple_Birth_Death_Probability_of_Extinction.png|660px|center]]</div></td></tr>
</table>Ctn