Difference between revisions of "Team:FAFU-CHINA/Model/Growth Model"

 
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= <b style="text-align:center;text-indent:21pt"><span style="font-family:'Times New Roman';font-size:20.0000pt">Growth Model of </span></b><b style="text-align:center;text-indent:21pt"><i><span style="font-family:'Times New Roman';font-size:20.0000pt">Chlamydomonas reinhardtii</span></i></b><b style="text-align:center;text-indent:21pt"><span style="font-family:'Times New Roman';font-size:20.0000pt">&nbsp;– Supported by BNU-CHINA</span></b><br/> =
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== <span style="font-family:'Times New Roman'">1.Introduction</span> ==
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<p style="text-indent:20.9pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">It’s essential for us to get accurate growth condition of </span><i><span style="font-family:'Times New Roman';font-size:12pt">Chlamydomonas reinhardtii</span></i><span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;in the natural environment to keep the concentration of toxin at a lethal level. But in fact, it is almost impossible to test concentration anywhere due to the lack of equipment and skills. Therefore, building the growth model can help determine the amount of </span><i><span style="font-family:'Times New Roman';font-size:12pt">Chlamydomonas reinhardtii</span></i><span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;they should use and when they need to add more. To build an accurate growth model, BNU-China team members who have much experience in the mathematics helped us to achieve it.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:20.9pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">Contacting with data provided by wet laboratory, we can draw the diagram of variation trend of algae population. Then we can get the key point where rate of algae population increment meets the maximal value so that the results can guide to culture of algae in their wet part. To control quantity of aquatic larva of mosquito by applying expression of specific protein in algae. There is an impressive impact of establishing mathematic modeling in population of algae.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:20.9000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">They helped us to establish a mathematic model to illustrate the whole temporal change of algae population. In general, it’s an original differential equations based on light intensity, mineral nutrient, organism and carbon dioxide, which are four main parameters in that. As for the temporal changing rate of population of algae growing in ideal conditions, there has been a lot of methods to solve this question. They referred to Huisman model and combined with practice factors. Then we got our deducted model. This model has a few parameters and it’s easy to get the solution.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:20.9000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">We provided the data of wet laboratory for us, and they run the model to get result. Finally, these results can help us to complete experiment. <font face="宋体">　</font></span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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== <span style="font-family:'Times New Roman'">2.Hypothesis of Model</span> ==
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">The fundamental way that the algae grow is through photosynthesis, in which the inorganic carbon in the water (carbon dioxide) can be transformed into the organic carbon (carbohydrate). However, the photosynthesis of the algae, which is fixed in the water, is influenced and limited by lots of factor.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">Not only in the laboratory but also the factory, Culturing algae in the stirred-well mixed culturing vessel with fixed volume is a common way. Under the circumstances, we believe that those important parameters, which is related to the growth of the algae, are all isotropy. In another word, by only considering the result which is dependent on time, we can meet the requirements of the experiments, or even the production.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">Thus, after weighing up the actual conditions, based on the combination of the existing model of the first order ordinary differential equations, we set up some second order ordinary differential equations to simulate the growth velocity - the algae’s dry weight of the time derivative. Furthermore, the results, which the model has simulated the growth condition of the algae in limited light and nutritive substance, quite tally with the actual situation. In some limiting cases, such as limited light with unlimited nutritive substance or unlimited light with limited nutritive substance, these equations can be simplified into some common first order ordinary differential equations.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p align="center" style="text-indent:24.0000pt;text-align:center">[[File:T--FAFU-CHINA--gmm1.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="center" style="text-indent:24.0000pt;text-align:center"><i><span style="font-family:'Times New Roman';font-size:12pt">Figure1: The logical relationships among four parameters</span></i><i><span style="font-family:'Times New Roman';font-size:12pt"></span></i></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">Firstly, according to the flow figure below, we explain the logical relationship of the four important parameters of the model. </span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">The growing speed of the alga in the incubator, A, which means the change of dry weight of the alga in unit interval is equal to the increase minus the decrease of the organic substance in the cells. The decrease is mainly based on two ways. One is the natural death and the other is the artificial separation of the useful mature alga.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">The carbon dioxide in the water area is converted to saccharides by the photosynthesis of alga and then stored in the cells. This also shows the conclusion that the growth of the population density of the alga will accelerate the speed of the decrease of carbon dioxide. Hence the content of carbon is equal to the inflow minus the part which are converted by photosynthesis.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">Similarly, the content of mineral substance in the water area M is also a factor which has its influence on photosynthesis. It will decrease faster when the amount of the alga is increasing as well.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">In a certain space, the sum of the carbohydrate, S (exclusive that in the cell), can be considered as the result which the fixed sum of the photosynthesis subtracts the total sum of the increment of the dry weight and the decrement of the dry weight in the culturing medium. (death, artificial extraction).</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p align="center" style="text-indent:24.0000pt;text-align:center">[[File:T--FAFU-CHINA--gmm2.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="center" style="text-align:center"><i><span style="font-family:'Times New Roman';font-size:12pt">Figure2: The quantitative relationships among four parameters</span></i><i><span style="font-family:'Times New Roman';font-size:12pt"></span></i></p>
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== <span style="font-family:'Times New Roman'">3.Mathematic Formulation</span> ==
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">According to the expression of photosynthesis as following:</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p align="center" style="text-indent:24.0000pt;text-align:center">[[File:T--FAFU-CHINA--gmm3.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="center" style="text-indent:24.0000pt;text-align:center"><i><span style="font-family:'Times New Roman';font-size:12pt">Figure3: The reaction of photosynthesis</span></i><i><span style="font-family:'Times New Roman';font-size:12pt"></span></i></p>
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<p><span style="font-family:'Times New Roman';font-size:12.0000pt">We know that the coefficient of the carbon dioxide which is consumed by carbohydrates is 44/33g [CO2] g[CH2O]</span><sup><span style="font-family:'Times New Roman';font-size:12.0000pt;vertical-align:super">-1</span></sup><span style="font-family:'Times New Roman';font-size:12.0000pt">. If the increment of the carbohydrates’ dry weight is influenced only by the content of the organic matters and mineral substance, and the proportion of the two is 1:9 approximately, the coefficient of the consumed mineral substance, k</span><sub><span style="font-family:'Times New Roman';font-size:12.0000pt;vertical-align:sub">2</span></sub><span style="font-family:'Times New Roman';font-size:12.0000pt">, is:</span><span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:24pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="center" style="text-align:center">[[File:T--FAFU-CHINA--gmm4.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="center" style="text-align:center"><i><span style="font-family:'Times New Roman';font-size:12pt">Figure4: The equation of k</span></i><i><sub><span style="font-family:'Times New Roman';font-size:12pt;vertical-align:sub">2</span></sub></i><i><span style="font-family:'Times New Roman';font-size:12pt"></span></i></p>
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<p><span style="font-family:'Times New Roman';font-size:12.0000pt">The coefficient of the consumed organic matters, k3, is:</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="center" style="text-align:center">[[File:T--FAFU-CHINA--gmm5.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="center" style="text-align:center"><i><span style="font-family:'Times New Roman';font-size:12pt">Figure5: The equation of k</span></i><i><sub><span style="font-family:'Times New Roman';font-size:12pt;vertical-align:sub">3</span></sub></i><i><span style="font-family:'Times New Roman';font-size:12pt"></span></i></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">From the above, what can be seen is that the main idea of these ordinary differential equations is element conservation. They work out the growth velocity indirectly by analyzing the transform of the substances in the fixed space.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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== <span style="font-family:'Times New Roman'">4.Result</span> ==
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<p align="center" style="text-indent:24.0000pt;text-align:center">[[File:T--FAFU-CHINA--gmm6.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="center" style="text-indent:24.0000pt;text-align:center"><i><span style="font-family:'Times New Roman';font-size:12pt">Figure 6. The model result of Algae concentration </span></i><i><span style="font-family:'Times New Roman';font-size:12pt"></span></i></p>
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<center>
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== <span style="font-family:'Times New Roman'">Reference</span> ==
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</center>
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<p style="margin-left:20.9000pt;text-indent:-20.9000pt"><span style="font-family:'Times New Roman';font-size:12pt">[1] Jayaraman S K, Rhinehart R R. Modeling and Optimization of Algae Growth[J]. Industrial &amp; Engineering Chemistry Research, 2010, 54(33).</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p align="center" style="text-align:center"><b><span style="font-family:'Times New Roman';font-size:12pt">Acknowledgement</span></b><b><span style="font-family:'Times New Roman';font-size:12pt"></span></b></p>
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<p align="center" style="text-align:center"><span style="font-family:'Times New Roman';font-size:12.0000pt">Zhiyao Chen and BNU-CHINA team.</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="right" style="text-align:right"><span style="font-family:'Times New Roman';font-size:12.0000pt">Written by Zhiyao Chen (BNU-CHINA team member) and Junhao Lu</span><span style="font-family:'Times New Roman';font-size:12.0000pt"></span></p>
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<p style="text-indent:24pt"><span style="font-family:'Times New Roman';font-size:12.0000pt">&nbsp;</span></p>
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<p align="center" style="text-align:center"><span style="font-family:'Times New Roman';font-size:14pt">&nbsp;</span></p>
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<p align="center" style="text-align:center"><span style="font-family:'Times New Roman';font-size:14pt">Demo Model</span><span style="font-family:'Times New Roman';font-size:14pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">Abstract </span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21pt"><span style="font-family:'Times New Roman';font-size:12pt">Demo model can be divided into two main parts, the first part is the establishment of algae growth model, the second part is to describe the changes in the number of larvae. In the first part, we model and the transformation of material based on the idea of the establishment of changes in the number of differential equations of algae on time, then consider the distribution of the number of algae in space, to obtain more accurate results. The results showed that the algal growth curve was similar to that of the S type curve, which was consistent with our expectations.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">In the second part, we establish the relationship between the number of larvae change on time using predator-prey model, and extract the parameters required by the model from the real experimental data, to predict the change of the number of larvae. This model can predict algae in different experiments used to kill mosquito larvae.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">Basic assumption </span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">1.The distribution of algae is spatially isotropic.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">2.The existence of water and algae can make the transmission of light intensity attenuation</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">3.The death rate of the algae and the concentration of the algae were fixed proportion.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">4.the external incident light intensity changes in 24h cycle</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p align="center" style="text-align:center"><span style="font-family:'Times New Roman';font-size:12pt">Algae growth model</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">Model One</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">First, we based on the logical relationship between the four important parameters in the flow chart and built our model. This part of the basic theory by BNU-CHINA students as we provide, we added a richer explanation for their model.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p align="center" style="text-indent:24.0000pt;text-align:center">[[File:T--FAFU-CHINA--gmm1.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">We define A as the concentration of algae, so the A derivative to time is the growth rate of algae. The content of mineral substance in the water area is M. S, C represent the carbohydrate and the carbon dioxide in the water respectively. Suppose that the pool is spatially isotropic, and let the input of mineral substance and co2 is </span>https://static.igem.org/mediawiki/2016/8/85/T--FAFU-CHINA--gmm8.png<span style="font-family:'Times New Roman';font-size:12pt"><font face="宋体">，</font></span>https://static.igem.org/mediawiki/2016/1/1f/T--FAFU-CHINA--gmm9.png<span style="font-family:'Times New Roman';font-size:12pt"><font face="宋体">。</font></span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">The Natural algae mortality is </span>https://static.igem.org/mediawiki/2016/f/f3/T--FAFU-CHINA--gmm10.png<span style="font-family:'Times New Roman';font-size:12pt">, which will decrease the amount of A, and will also take away the sugar stored in algae. The sugar will be converted from a part of carbon dioxide, and we put it as </span>https://static.igem.org/mediawiki/2016/e/e2/T--FAFU-CHINA--gmm11.png<span style="font-family:'Times New Roman';font-size:12pt">, the cost of C is </span>https://static.igem.org/mediawiki/2016/d/d5/T--FAFU-CHINA--gmm12.png<span style="font-family:'Times New Roman';font-size:12pt">.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">From the photosynthetic reaction, we can easily obtain the coefficient K1 </span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt">[[File:T--FAFU-CHINA--gmm3.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">so &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>https://static.igem.org/mediawiki/2016/6/61/T--FAFU-CHINA--gmm14.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21pt"><span style="font-family:'Times New Roman';font-size:12pt">New algae are produced inside the existing algae at a rate </span>https://static.igem.org/mediawiki/2016/2/23/T--FAFU-CHINA--gmm15.png<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;from nutrients and sugar, where </span>https://static.igem.org/mediawiki/2016/a/ab/T--FAFU-CHINA--gmm16.png<span style="font-family:'Times New Roman';font-size:12pt">is the rate constant and f m (M) denotes the concentration of nutrients inside the cells. This depletes nutrients and sugar by a rate of </span>https://static.igem.org/mediawiki/2016/1/1e/T--FAFU-CHINA--gmm17.png<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;and </span>https://static.igem.org/mediawiki/2016/7/7d/T--FAFU-CHINA--gmm18.png<span style="font-family:'Times New Roman';font-size:12pt">, respectively. The ratio is about 1: 9</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24pt"><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:24pt"><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:156.0000pt">[[File:T--FAFU-CHINA--gmm4.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:132.0000pt">[[File:T--FAFU-CHINA--gmm5.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">So far, we can list a set of differential equations to describe the changes in the amount of algae, mineral substance, sugar and carbon dioxide</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>https://static.igem.org/mediawiki/2016/f/fd/T--FAFU-CHINA--gmm21.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:48.0000pt">https://static.igem.org/mediawiki/2016/5/54/T--FAFU-CHINA--gmm22.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:48.0000pt">https://static.igem.org/mediawiki/2016/6/6e/T--FAFU-CHINA--gmm23.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:48.0000pt">https://static.igem.org/mediawiki/2016/7/72/T--FAFU-CHINA--gmm24.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="margin-left:27.0000pt;text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">The model also obeys the following conservation law</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:48.0000pt">https://static.igem.org/mediawiki/2016/3/34/T--FAFU-CHINA--gmm25.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">There are many coefficients in the above equation that have not been explained, then we will derive the specific expression of each coefficient and its value.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt">https://static.igem.org/mediawiki/2016/a/ab/T--FAFU-CHINA--gmm16.png<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;is supposed to be a parameter that relate with the current concentration of algae. We borrowed from the Logistic model as the following form:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:168.0000pt">https://static.igem.org/mediawiki/2016/a/ac/T--FAFU-CHINA--gmm27.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:36.0000pt">https://static.igem.org/mediawiki/2016/f/f9/T--FAFU-CHINA--gmm28.png<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;represent the max algae that the pool can support, k4 is a proportional coefficient</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The form of &nbsp;</span>https://static.igem.org/mediawiki/2016/1/16/T--FAFU-CHINA--gmm29.png<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;is similar</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>https://static.igem.org/mediawiki/2016/8/82/T--FAFU-CHINA--gmm30.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:36.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">As for </span>https://static.igem.org/mediawiki/2016/0/0c/T--FAFU-CHINA--gmm31.png<span style="font-family:'Times New Roman';font-size:12pt">, we think It should be a function of temperature , light , A and C. Let us suppose that </span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>https://static.igem.org/mediawiki/2016/f/ff/T--FAFU-CHINA--gmm32.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The Influence of Light Intensity on this coefficient are borrowed from Huisman model:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:84.0000pt">https://static.igem.org/mediawiki/2016/f/fa/T--FAFU-CHINA--gmm33.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This function considered the influence that comes from A, the depth of the pool d. H is half saturation coefficient,</span>https://static.igem.org/mediawiki/2016/9/98/T--FAFU-CHINA--gmm34.png<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;and </span>https://static.igem.org/mediawiki/2016/0/06/T--FAFU-CHINA--gmm35.png<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;is the light absorption constants of algae and background. We use a simple fractional expression to describe The effect of temperature </span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:96.0000pt">https://static.igem.org/mediawiki/2016/d/dd/T--FAFU-CHINA--gmm36.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">Similarly, we also deal with the effects of C on the parameters in the same form:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:96.0000pt">https://static.igem.org/mediawiki/2016/f/f1/T--FAFU-CHINA--gmm37.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Now we update the terms and rewrite the equations as follows:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:120.0000pt">https://static.igem.org/mediawiki/2016/7/7c/T--FAFU-CHINA--gmm38.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:120.0000pt">https://static.igem.org/mediawiki/2016/a/a4/T--FAFU-CHINA--gmm39.pnghttps://static.igem.org/mediawiki/2016/c/cc/T--FAFU-CHINA--gmm40.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>https://static.igem.org/mediawiki/2016/d/d9/T--FAFU-CHINA--gmm41.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p>https://static.igem.org/mediawiki/2016/a/a5/T--FAFU-CHINA--gmm42.png<span style="font-family:'Times New Roman';font-size:12pt"> </span><span style="font-family:'Times New Roman';font-size:12pt">When solving the equations, the value of each parameter is taken from the real environment of the experiment. Some of them is inconvenient to measure, and we found them from the Internet and the literature t. The two terms that represent the external input we treat as follows:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:60.0000pt">https://static.igem.org/mediawiki/2016/3/37/T--FAFU-CHINA--gmm43.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">In order to maintain the concentration of carbon dioxide in the water to maintain near the ideal value, we set the input amount</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>https://static.igem.org/mediawiki/2016/2/2d/T--FAFU-CHINA--gmm44.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">And the growth curve of algae could be solved by MATLAB.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt">[[File:T--FAFU-CHINA--gmm45.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">As we can see from the image, it is a curve that is similar to the S type, which is in line with our expectations.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">Model 2</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">We only consider the relationship between time and algae’ concentration in model 1. In this part, we add the factor of space, and combine it with certain time.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">The biomass of algae in the water body is related to the growth rate (which Is largely decided by the rate of photosynthesis) and the process of mixing and death. In this part, we mainly focus on the influence of availability of light to the photosynthesis rate. The death rate includes both the rate of being token in as well as the natural death rate of the algae. Advection is assumed to be absent which corresponds to the still water body. In the horizontal plane, we consider no variation and hence, the growth rate is independent of x and y coordinates. The depth in the water body is denoted by z.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt">https://static.igem.org/mediawiki/2016/e/e1/T--FAFU-CHINA--gmm46.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt">https://static.igem.org/mediawiki/2016/6/6c/T--FAFU-CHINA--gmm47.png<span style="font-family:'Times New Roman';font-size:12pt;position:relative;top:6pt">is a constant related to diffusion efficiency,</span>https://static.igem.org/mediawiki/2016/3/3d/T--FAFU-CHINA--gmm48.png<span style="font-family:'Times New Roman';font-size:12pt"><font face="宋体">、</font></span>https://static.igem.org/mediawiki/2016/2/23/T--FAFU-CHINA--gmm49.png<span style="font-family:'Times New Roman';font-size:12pt"><font face="宋体">、</font></span>https://static.igem.org/mediawiki/2016/3/39/T--FAFU-CHINA--gmm50.png<span style="font-family:'Times New Roman';font-size:12pt"><font face="宋体">、</font></span>https://static.igem.org/mediawiki/2016/7/7f/T--FAFU-CHINA--gmm51.png<span style="font-family:'Times New Roman';font-size:12pt;position:relative;top:6pt">&nbsp;are correlated variables related to growth rate: light intensity, nutrition (phosphorus and nitrogen) and the concentration of carbon dioxide. </span>https://static.igem.org/mediawiki/2016/9/99/T--FAFU-CHINA--gmm52.png<span style="font-family:'Times New Roman';font-size:12pt;position:relative;top:6pt">means this equation has considered the decreased rate because of being token and natural death.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">For light intensity, we adopt dependency relationship in the form of Monod:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt">https://static.igem.org/mediawiki/2016/a/ab/T--FAFU-CHINA--gmm53.png<span style="font-family:'Times New Roman';font-size:12pt">lm is the efficiency for algae to absorb light. HL is semi-saturation concentration. This equation could guarantee that during less light irradiation, growth rate is near linear, and when light is intensified, growth rate is limited by the border of μ0. The light absorbed by algae is not consistent. Light intensity is affected by two aspects: the existence of algae (upper layer) and the quantity of water (upper layer). The following equation describes the light-tight phenomenon caused by algae, with the decrease of light due to the increase of depth</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt">https://static.igem.org/mediawiki/2016/f/f5/T--FAFU-CHINA--gmm54.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt">https://static.igem.org/mediawiki/2016/5/58/T--FAFU-CHINA--gmm55.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">I0(t) represents light intensity from the outside related to time (for example the cycle of day and night). k is the constant represents the decrease of water towards light. rs is correlated constant of light decrease with the existence of algae.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">Through the operator after discretization, the partial differential equation PDE with boundary conditions can be changed into ordinary differential equation ODE:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:24.0000pt">https://static.igem.org/mediawiki/2016/9/9b/T--FAFU-CHINA--gmm56.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">After approximation:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt">https://static.igem.org/mediawiki/2016/b/b4/T--FAFU-CHINA--gmm57.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt">https://static.igem.org/mediawiki/2016/3/34/T--FAFU-CHINA--gmm58.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">Use the accumulation of small quantity to replace integral operation.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">According to the parameters given in the essay, we draw a sketch after several adjustments and completion:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:24.0000pt">[[File:T--FAFU-CHINA--gmm59.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p style="text-indent:24.0000pt">[[File:T--FAFU-CHINA--gmm60.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">In this equation, when </span>https://static.igem.org/mediawiki/2016/c/cc/T--FAFU-CHINA--gmm61.png<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;<font face="宋体">、</font></span>https://static.igem.org/mediawiki/2016/9/9a/T--FAFU-CHINA--gmm62.png<span style="font-family:'Times New Roman';font-size:12pt;position:relative;top:5pt">&nbsp;</span><span style="font-family:'Times New Roman';font-size:12pt">we do discretization and solving, and the time period is 3 days; the quantity of mineral substance is constant with nitrogen of 3.64 · 10</span><span style="font-family:'Times New Roman';font-size:12pt">−</span><span style="font-family:'Times New Roman';font-size:12pt">10 [mol/(l · s)] and phosphorus of 2.78 · 10</span><span style="font-family:'Times New Roman';font-size:12pt">−</span><span style="font-family:'Times New Roman';font-size:12pt">10 [mol/(l · s)], without extra adding of carbon dioxide. Seeing from the figure we can find the periodicity of algae density, which is caused by the periodically changed light intensity</span>[[File:T--FAFU-CHINA--gmm63.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt">. Similar to our expectation, the concentration of algae is lower in the deep. Because of the parameter, the growth cannot be guaranteed at the beginning of simulation due to the lack of light. After a day, at night, algae density falls below the initial value, and then decrease gradually. Under certain depth, algae can no longer continue growth, and will be token or die naturally.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21pt"><span style="font-family:'Times New Roman';font-size:12pt">With different nutrition, light intensity and other boundary conditions, we will get different response system.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">Model of wriggler quantity</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">We have analyzed the relationship among algae concentration, space and time. In order to predict the effect of algae to kill pests, we construct a new model to describe this. Our goal is to establish a system to predict the killing effect under different circumstances.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">In this part, we use “predators and prey” to describe algae and wriggler. We use prey model to describe their mutual effect. In this model, W represents the quantity of wriggler, which is predators. Because they may die due to their own reasons (we suppose </span>https://static.igem.org/mediawiki/2016/6/63/T--FAFU-CHINA--gmm64.png<span style="font-family:'Times New Roman';font-size:12pt">as proportionality factor). Therefore, we should add an item on the right side of the equation,</span>https://static.igem.org/mediawiki/2016/c/c5/T--FAFU-CHINA--gmm65.png<span style="font-family:'Times New Roman';font-size:12pt">.</span>https://static.igem.org/mediawiki/2016/f/f5/T--FAFU-CHINA--gmm66.png<span style="font-family:'Times New Roman';font-size:12pt">represents the change of algae due to time. For simplification, we assume the denser the algae concentration, the better effect towards kill wrigglers. We use to</span>https://static.igem.org/mediawiki/2016/0/08/T--FAFU-CHINA--gmm67.png<span style="font-family:'Times New Roman';font-size:12pt">represent proportionality factor of inhibiting ability, which is decided by the principle to kill wrigglers. So, we could get:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p align="center" style="text-indent:21.0000pt;text-align:center">https://static.igem.org/mediawiki/2016/f/fe/T--FAFU-CHINA--gmm68.png<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21pt"><span style="font-family:'Times New Roman';font-size:12pt">From the experiment data, </span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24pt"><span style="font-family:'Times New Roman';font-size:12pt">when t=5, wrigglers begin to die</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24pt"><span style="font-family:'Times New Roman';font-size:12pt">when t=24, livability is 0.04885, mortality is 95.12%</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24pt"><span style="font-family:'Times New Roman';font-size:12pt">when t=48, livability </span>https://static.igem.org/mediawiki/2016/f/fa/T--FAFU-CHINA--gmm69.png]<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:21.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">We think when t=5, toxicity begins to appear, so we choose t=5 as our time starting point. Following is drawn by MATLAB. Apart from this, we can get the equation’s analytic solution using MATLAB, and the parameters are decided by data. Through this way, we can have a better prediction that under different circumstances, the quantity of wriggler will change in which way.</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">The change curve of wriggler quantity is:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p align="center" style="text-indent:21.0000pt;text-align:center">[[File:T--FAFU-CHINA--gmm70.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">The equation’s analytic solution is:</span><span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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<p align="center" style="text-align:center">[[File:T--FAFU-CHINA--gmm71.png | 825px | thumb | center ]]<span style="font-family:'Times New Roman';font-size:12pt"></span></p>
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<p style="text-indent:24.0000pt"><span style="font-family:'Times New Roman';font-size:12pt">&nbsp;</span></p>
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Latest revision as of 06:42, 8 November 2016


Team FAFU-CHINA T--FAFU-CHINA--FAFU_modeling3.png