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<p>　　This year, our modeling focuses on predicting the effect of our modified microbes on productivity. It is an extremely important part to our project because it helps us accurately check and predict information from our experiments that are tested in the wet lab. In our project, there are two essential types of microalgae that play very important roles, <i>Synechococcus PCC7942</i> and <i>Chlorella vulgaris</i>. The following descriptions will show our success in modeling.

<!--PCC7942-->

<p>　　The modeling from Figure 1 to Figure 5 belongs to the experiments of <i>Synechococcus PCC7942</i> pigments for better photosynthetic efficiencies. We need to check if another microalgae contains an exogenous pigment that can successfully reach new photosynthesis rate and further increase the proportion of biomass. We already have models about the influence of energy adsorption, but pigments will certainly affect other factors. Therefore, we construct several models that each represents an important factor in the growth and cell composition. Thus, we can determine the best culturing collocation by combining these models.

<!-- Photosynthesis rate of algae -->

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Photosynthesis rate of algae

<p>　　We want to use pigments to enhance the photosynthesis rate. Different pigment absorbs different wavelength of sunlight and bring about different irradiance, body temperature, and photosynthesis rate. These two models show the influence of irradiance and temperature on photosynthesis rate.

<img src='https://static.igem.org/mediawiki/2017/8/89/T--NYMU-Taipei--model_i-rco2.png'  

alt='irradiance-photosynthesis rate plot'

<p style='font-size:20px'>Fig.1-1 Influence of irradiance on photosynthesis rate</p>

 

 

<img src='https://static.igem.org/mediawiki/2017/6/6a/T--NYMU-Taipei--model_t-rco2.png'  

alt='temperature-photosynthesis rate plot'

<p style='font-size:20px'>Fig.1-2 Influence of temperature on photosynthesis rate</p>

 

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<!-- Simulation of energy absorption of each pigment -->

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Simulation of energy absorption of each pigment

y = 0.01x - 3.5, 400<=x<=500

<br>y = 1.5, 501<=x<=600

<br>y = 3 - 0.0025*x, 601<=x<= 800

x:

<br>y:

</blockquote>-->

 

<img src='https://static.igem.org/mediawiki/2017/f/fb/T--NYMU-Taipei--model_energy-pigment.png'  

alt='Simulation of energy absorption of each pigment'

style='width:55%'>

<p style='font-size:20px'>Fig.2 Simulation of energy absorption of each pigment</p>

<p></p>

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<!-- Microalgae productivity in different temperatures -->

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Microalgae productivity in different temperatures

<p>　　After we get the influential degree on temperature, we can use our modeling to predict the productivity of microalgae at different temperature without other affecting factors. Modeling is used to ensure that our experiments are under control.

U = U_max * K_ss

<br>U_max = A*exp(-E/RT)

<img src='https://static.igem.org/mediawiki/2017/8/8c/T--NYMU-Taipei--model_t-spr.png'  

alt='Microalgae productivity in different temperatures'

<p style='font-size:20px'>Fig.3 Microalgae productivity in different temperatures</p>

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<!-- Microalgae productivity in different pH values-->

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Microalgae productivity in different pH values

<p>　　When our <i>Synechococcus PCC7942</i> grows at each phase, the equilibrium of the pH value is different. This model can be used to collocate with our device, and also accomplish the purpose of enhancing productivity.

R = A₁ exp(-B₁/pH) - A₂ exp(-B₂/pH)

                        <table>

                        <tr>

                            <th scope="col">Definition</th>

                            <th scope="col"&>Value</th>

                            <th>8.625*10^-5</th>

                        </tr>

                              <th>-</th>

                              <th>1.83885*10^-2</th>

                        </tr>

                              <th>B₁</th>

                              <th>activation energy at i=400</th>

<img src='https://static.igem.org/mediawiki/2017/3/36/T--NYMU-Taipei--model_ph-rco2.png'  

alt='Microalgae productivity in different pH'

<p style='font-size:20px'>Fig.4 Microalgae productivity in different pH</p>

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<!-- The relation between photosynthetic rate and total absorption -->

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The relation between photosynthetic rate and total absorption

<p>　　The model tells us that theoretically, there is no faster photosynthetic rate unless more energy is absorbed. After working with other models, we established the relationship between photosynthetic rate and total absorption for the purpose of best balance.

R = R_max * eⁿ / [k_e * exp(e*m) + eⁿ]

                      <table>

                        <tr>

                            <th scope="col">Definition</th>

                            <th scope="col"&>Value</th>

                            <th>mol / g*min </th>

                            <th>w/m²</th>

                            <th>-</th>

                              <th>-</th>

                              <th>1.252</th>

                        </tr>

                        <tr>

                            <th>k_e</th>

                              <th>uE / (m²)*s</th>

                              <th>157.88</th>

                              <th>0.0035</th>

                        </tr>

                      </table>

<img src='https://static.igem.org/mediawiki/2017/a/a2/T--NYMU-Taipei--model_e-rco2.png'  

alt='The relation between photosynthetic rate and total yield'

<p style='font-size:20px'>Fig.5 The relation between photosynthetic rate and total absorption</p>

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<!--chorella-->

<p>　　The modeling from Figure 6 to Figure 13 belongs to the experiments of <i>Chlorella vulgaris</i> for nitrogen starvation. To precisely calculate the timing of starting co-culturing and to ensure there are enough high-affinity E. coli in the bioreactor, we built several models that include the original and new system. They demonstrated the significant improvement of productivity after successfully deprived the microalgae from nitrogen. For instance, one of them provides a variety of information about population when two organisms in the pool start building some relationship.

<!--Growth curve of Chlorella vulgaris-->

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Growth curve of <i>Chlorella vulgaris</i>

<p>　　The timing of adding engineered <i>E.coli</i> or purified protein to <i>Chlorella vulgaris</i> culture is critical to our project. By analyzing the initial and final biomass concentration data the instantaneous rate would be gained. This instantaneous rate is based on reference time and other lab environment data. We have simulated the change in biomass concentration throughout the culture cycle. The intermittent information in the culture medium at each point is ultimately gained through combining other modeling results, which aims to determine the best timing and corresponding state.

ln(X_t/X₀) / t

<br>

<br>= A + B exp[-C(t-M)]

<br>= μ (specific growth rate)</h6>

                            <th>Symbol</th>

                            <th scope="col"&>Unit</th>

                            <th scope="col"&>Value</th>

                            <th>X</th>

                            <th>biomass concentration</th>

                            <th>g/l</th>

                            <th>-</th>

                        </tr>

                            <th>t</th>

                            <th>time</th>

                            <th>hr</th>

                            <th>-</th>

                        </tr>

                        <tr>

                            <th>A</th>

                              <th>the asymptotic of ln X_t/X₀ as t decrese indefinitely</th>

                              <th>1.252</th>

                        </tr>

                            <th>B</th>

                              <th>the asymptotic of ln X_t/X₀ as t increase indefinitely</th>

                              <th>-</th>

                              <th>-</th>

                        </tr>

                              <th>the relative growth rate at time</th>

                              <th>M</th>

                              <th>-</th>

                        </tr>

<img src='https://static.igem.org/mediawiki/2017/0/04/T--NYMU-Taipei--model_growth_curve.gif'  

alt='Growth curve of Chlorella vulgaris'

<p style='font-size:20px'>Fig.6-1 Growth curve of <i>Chlorella vulgaris</i></p>

<img src='https://static.igem.org/mediawiki/2017/e/e5/T--NYMU-Taipei--model_growth_rate.gif'  

alt='Growth rate of Chlorella vulgaris'

<p style='font-size:20px'>Fig.6-2 Growth rate of <i>Chlorella vulgaris</i></p>

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</div>

<!-- oil accumulation and nitrogen source consumption -->

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Oil accumulation & Nitrogen source consumption

<p>　　By simulating common systems of oil accumulation and nitrogen source consumption, we cannot only get the reference data before the improvement, but also make it a basic equation after joining some parameters or organisms into the system.

dP/dt = αdX/dt + βX;

<br>V = [(q_M-Q)/(q_M-q)] * [(V_m*N)/(N+V_h)]

<br>Q = (X₀*Q₀ + N₀ - N) / X

                      <table>

                        <tr>

                            <th scope="col">Definition</th>

                            <th>lipid</th>

                            <th>g/L</th>

                            <th>-</th>

                        <tr>  

                            <th>the instantaneous yield coefficient of product formation due to cell growth</th>

                            <th>g/g</th>

                            <th>day^-1</th>

                            <th>0.00037</th>

                        <tr>

                        <tr>

                            <th>q</th>

                            <th>Minimum N quota</th>

                            <th>0.0178</th>

<th>g/g</th>

                            <th>0.0935</th>

                            <th>-</th>

                            <th>0.596</th>

                        </tr>

                      </table>

<img src='https://static.igem.org/mediawiki/2017/c/ca/T--NYMU-Taipei--model_L%26N.gif'  

alt='Oil accumulation and nirogen source consumption at normal situation'

style='width:90%'>

<p style='font-size:20px'>Fig.7 Oil accumulation and nirogen source consumption at normal situation(lipid:green;nitrogen:blue)</p>

<p></p>

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</div>

<!-- Biomass in different nitrogen concentrations-->

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Biomass in different nitrogen concentrations

<p>　　To find out the best quantity of nitrogen removal, we modeled several situations of decreasing the biomass in different environments with different concentration of nitrogen. We then find the best productivity by comparing these two situations.  

n₂ = exp{[A + C*exp(-exp(-B(t-M)))] * (t₂-t₁)} * n₁;

<br>x₂ = x₁ + (n₂-n₁) * {[k[ln(b(n_s+a))^-1]]-e};

                      <table>

                        <tr>

                            <th scope="col">Definition</th>

                            <th>-</th>                       

                            <th>-</th>

 

                            <th scope="col">Definition</th>

                            <th scope="col"&>Value</th>

                        </tr>

                            <th>A</th>

                            <th>the asymptotic of ln X_t/X₀ as t decrese indefinitely</th>

                            <th>-39.9532</th>

                        </tr>

                            <th>B</th>

                            <th>the asymptotic of ln X_t/X₀ as t increase indefinitely </th>

                              <th>constant</th>

                              <th>-</th>

                              <th>8.15229</th>

                        </tr>

                              <th>yield coefficient</th>

                            <th>n_s</th>

                              <th>initial nitrogen concentration</th>

                              <th>regression constant</th>

                              <th>-</th>

                              <th>0.01</th>

                              <th>0.50678</th>

                        </tr>

                        </table>

<img src='https://static.igem.org/mediawiki/2017/e/e1/T--NYMU-Taipei--model_biomass.gif'  

alt='Biomass in different nitrogen concentration'

<p style='font-size:20px'>Fig.8 Biomass in different nitrogen concentration(concentration after nitrogen deletion,black:0.1;red:0.03;green:0.02;blue:0.01;yellow:0.005  g/l)</p>

<center>

<a href="#!" onclick="toggleHeight3(this, 1250);" style='font-size:20px;color:#2c498c'>

 

<!-- Nitrogen source in nitrogen starvation -->

dn/dt = Y_xn * dx/dt + m*x

                        <tr>

                            <th>Symbol</th>

                            <th scope="col">Definition</th>

                            <th scope="col"&>Unit</th>

                            <th scope="col"&>Value</th>

                        </tr>

                        <tr>

                            <th>n</th>

                            <th>nitrogen concentration</th>

                            <th>0.21016</th>

                        </tr>

                            <th>m</th>

                              <th>maintenance parameter</th>

                              <th>hr^-1</th>

                              <th>0.0014393</th>

                        </tr>

                            <th>x</th>

                              <th>biomass concentration</th>

                              <th>-</th>

                        </tr>

                      </table>

<img src='https://static.igem.org/mediawiki/2017/5/5b/T--NYMU-Taipei--model_ns_nitrogen.gif'  

alt='Nitrogen source in nitrogen starvation'

<p style='font-size:20px'>Fig.9 Nitrogen source in nitrogen starvation(normal:blue;starvation:red)</p>

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</center>

</div>

<!-- Oil accumulation in nitrogen starvation -->

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Oil accumulation in nitrogen starvation

<p>　　We predict that total lipid will increase under nitrogen starvation. The modeling provides the theoretical information of the maximum of productivity. This graph shows that if we use symbiotic microbe to make nitrogen source isolated from the system temporarily and successfully, the productivity will be enhanced.

dp/dt = k₁(dx/dt)² + k₂(dx/dt)(x) + e

                      <table>

                        <tr>

                            <th scope="col">Definition</th>

                            <th scope="col"&>Value</th>

                        </tr>

                            <th>p</th>

                        </tr>

                      </table>

<img src='https://static.igem.org/mediawiki/2017/e/e8/T--NYMU-Taipei--model_ns_oil.gif'  

alt='Oil accumulation in nitrogen starvation'

style='width:65%'>

<p style='font-size:20px'>Fig.10 Oil accumulation in nitrogen starvation</p>

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<!-- Population of co-cultured Chlorella and modified E.coli -->

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