Team:Toulouse/Modelling

From 2014.igem.org

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Modeling is a tool used to simplify and study systems. We can try to predict behavior with bibliographic informations or informations obtained from experiment.</br>
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Modeling is a tool used to simplify and study systems. It helps us to predict behavior thanks to bibliographic or experimental informations.</br>
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Our project focuses on the development of our engineered bacterium in tree. The bacterial growth in trees seems to be unknown, so we must infer <i>Bacillus subtilis</i> behavior.</p>
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The following modelisation focuses on the development of our engineered bacterium (called SubtiTree) in tree. The bacterial growth in trees seems to be unknown, thus we must infer <i>Bacillus subtilis</i> behavior.</p>
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<i>Bacillus subtilis</i> is a tree endophyte strain. A study showed that <i>Bacillus subtilis</i> could develop and fully colonize a tree, reaching a concentration of 10⁵ cells per gram of fresh plant. We need to know in which conditions the growth of <i>B. subtilis</i> is optimum in a tree and if the weather can stop its development during winter. So we decided to work on the <i>Bacillus subtilis</i> growth in function of the temperature during the year. Modeling bacterial growth in a tree section generates some difficulties, we need to know the distance between two tree extremities (treetops and root) or the speed sap flow which can vary with temperatures during the day and seasons, cause of the type of sap (phloem, xylem). Furthermore a tree is not an homogeneous system, its roots, trunk and branch do not contain same amount of sap and wood. The average speed of the plane tree sap is 2.4m/h, which means that in a day the sap will flow from one end to the other of a tree 30m. Tree is reduced to a bioreactor.
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<i>Bacillus subtilis</i> is a tree endophyte strain. A study showed that <i>Bacillus subtilis</i> could develop and fully colonize a tree, reaching a concentration of 10⁵ cells per gram of fresh plant. We need to know in which conditions the growth of <i>B. subtilis</i> is optimum in a tree and if the weather can stop its development during winter. Therefore we decided to work on the <i>Bacillus subtilis</i> growth in function of the temperature during the year.  
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<br>Modeling bacterial growth in a tree section generates some difficulties. We need to know the distance between two tree extremities (treetops and root) or the speed sap flow. However the speed sap flow can vary with temperature during the day and seasons cause of the type of sap (phloem, xylem). Furthermore a tree is not an homogeneous system: its roots, trunk and branches do not contain the same amount of sap and wood. <br>The average speed of the plane tree sap is 2.4m/h, which means that in a day the sap of a 30m tree will flow from one extremity to the other. Tree is reduced to a bioreactor.
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According to the publication of <b>Xianling Ji</b>, after 6 months of <i>Bacillus subtilis</i> growth in a tree, bacteria cells reach a concentration of 10⁵ cells per gram of fresh plant. We assume that 10⁵ cells / g is the maximum concentration.
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According to the publication of <b>Xianling Ji</b> (See References), after six months of <i>Bacillus subtilis</i> growth in a tree, bacteria cells reach a concentration of 10⁵ cells per gram of fresh plant. We assume that 10⁵ cells/g is the maximum concentration.
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Only temperature impact on bacterial growth.
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Only temperature impacts on bacterial growth.
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An assessment of the <i>Bacillus subtilis</i> growth in a similar sap, the birch sap was performed in laboratory conditions with optimum growth medium for <i>Bacillus subtilis</i>. Thus, a growth rate μ opt. From this value we can extrapolate a growth curve as a function of temperature.
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An assessment of the <i>Bacillus subtilis</i> growth in a similar sap was performed in laboratory conditions with optimum growth medium for <i>Bacillus subtilis</i>. The composition sap used was the one from birch sap.<br>
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For this we used to <b>cardinal temperature model</b>: </p>
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In these conditions, the growth rate μ is optimal. From this value we can extrapolate a growth curve as a function of temperature. We used to <b>cardinal temperature model</b>: </p>
   
   
<center style="margin-bottom:50px;"><img style="" src="http://2014.igem.org/wiki/images/8/85/Formules_Rosso.png" alt="cardinal temperature model"></center>
<center style="margin-bottom:50px;"><img style="" src="http://2014.igem.org/wiki/images/8/85/Formules_Rosso.png" alt="cardinal temperature model"></center>
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T: Temperature.</br>
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T: Temperature</br>
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µ_opt: optimal growth rate.</br>
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µ<sub>opt</sub>: Optimal growth rate</br>
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µ: growth rate at T.</br>
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µ: growth rate at temperature T</br>
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T_max = max temperature supported by bacteria.</br>
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T<sub>max</sub> = Maximum temperature supported by bacteria</br>
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T_min = min temperature supported by bacteria.</br>
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T<sub>min</sub> = Minimum temperature supported by bacteria</br>
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T_opt = optimum temperature for the growth.</br></br>
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T<sub>opt</sub> = Optimum temperature for the growth</br></br>
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     Necessary parameters for this function are minimun temperature T_min and maximum temperature T_max, optimal temperature for the growth T_opt and optimal growth rate µ_opt.</br>
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     Necessary parameters for this function are minimun temperature T<sub>min</sub> and maximum temperature T<sub>max</sub>, optimal temperature for the growth T<sub>opt</sub> and optimal growth rate µ<sub>opt</sub>.</br>
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     T_min: 10°C</br>
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     T<sub>min</sub>: 10°C</br>
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     T_max: 52°C</br>
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     T<sub>max</sub>: 52°C</br>
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     T_opt: 37°C</br>
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     T<sub>opt</sub>: 37°C</br>
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     µ_opt: 8.5968 cfu/d</br></br>
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     µ<sub>opt</sub>: 8.5968 cfu/d</br></br>
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The optimal growth rate (µ_opt) is obtained experimentally with a similar birch sap environment.</br>
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The optimal growth rate (µ<sub>opt</sub>) is obtained experimentally with a similar birch sap environment.</br>
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The growth rate is negative below 10°C (growth test performed at 10°C and 4°C under similar conditions for the measurement of μ_opt), survival rate after 24h was 0.3 % at 10°C and null at 4°C.
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The growth rate is negative below 10°C (according to growth tests performed at 10°C and 4°C under similar conditions for the measurement of μ<sub>opt</sub>), survival rate after 24h was 0.3 % at 10°C and null at 4°C.<br>
Conditions apply:</p>
Conditions apply:</p>
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<center style="margin-top: -52px;"><img style="" src="http://2014.igem.org/wiki/images/b/b1/Plot_growth_rate.png" alt="Figure1"></center>
<center style="margin-top: -52px;"><img style="" src="http://2014.igem.org/wiki/images/b/b1/Plot_growth_rate.png" alt="Figure1"></center>
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<p class="legend">Fig 1: bacterial growth (µ) as a function of temperature</p>
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<p class="legend">Figure 1: bacterial growth (µ) as a function of temperature</p>
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<p class="texte"> A logistic model developed by <b>Hiroshi Fujikawa</b> is used to study bacterial growth.</p>
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<p class="texte"> A logistic model developed by <b>Hiroshi Fujikawa</b> (See References) is used to study bacterial growth.</p>
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<p class="legend">General logistics formulas</p>
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<p class="legend">General logistics formulas:</p>
<center style="margin:-44px 0 65px;"><img style="" src="http://2014.igem.org/wiki/images/c/c3/Form_general_fonction.png" alt="General logistics formulas"></center>
<center style="margin:-44px 0 65px;"><img style="" src="http://2014.igem.org/wiki/images/c/c3/Form_general_fonction.png" alt="General logistics formulas"></center>
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In our cases, µ depends on the temperature. N corresponds to the bacterial population, Nmin and Nmax are two asymptotes. Parameter "m" is a curvature parameter; larger is m, smaller is the curvature of the deceleration phase with the model. Parameter n is a parameter related to the period lag; larger is n, shorter is the period of lag. Nmin is slightly lower than N0, when N is small, close to Nmin, as the initial state (N is equal to N0), Nmin / N is almost equal to 1 so the term (1-(Nmin/N)) is less than 1, growth is very slow. If N decrease until reach Nmin the term (1-(Nmin/N)) is equal to 0 thus there can not be any growth. Similarly when N is equal to Nmax the term (1-(N/Nmax)) is equal to 0 and the growth is blocked.</br>
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In our case, the growth rate µ depends on the temperature.  
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<br>N corresponds to the bacterial population, N<sub>min</sub> and N<sub>max</sub> are two asymptotes.  
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<br>The parameter m is a curvature parameter. Larger m is, smaller is the curvature of the deceleration phase with the model.  
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<br>The parameter n is a parameter related to the period lag. larger n is, shorter is the period of lag.  
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<br>N<sub>min</sub> is slightly lower than N<sub>0</sub>, when N is small, close to N<sub>min</sub>, as the initial state (N is equal to N<sub>0</sub>), N<sub>min</sub> / N is almost equal to 1 so the term (1-(N<sub>min</sub>/N)) is less than 1, growth is very slow. If N decrease until reach N<sub>min</sub> the term (1-(N<sub>min</sub>/N)) is equal to 0 thus there can not be any growth. Similarly when N is equal to N<sub>max</sub> the term (1-(N/N<sub>max</sub>)) is equal to 0 and the growth is blocked.</br>
To overcome this we labor under two conditions , positive growth and negative growth, so two equations.This led to the writing of this model:</p>
To overcome this we labor under two conditions , positive growth and negative growth, so two equations.This led to the writing of this model:</p>

Revision as of 14:59, 16 October 2014