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.

R = R_max * iⁿ / [k_i * exp(i*m) + iⁿ]

Fig.1-1 Influence of irradiance on photosynthesis rate

R = A₁ exp(-E₁rT) - A₂ exp(-E₂/rT)

Fig.1-2 Influence of temperature on photosynthesis rate

　　The simplified graph from sunshine distribution is used to approximately calculate how much energy is absorbed by each pigment and quantifies the photon adsorption amount after conversion.

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

y = 1.5, 501<=x<=600

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

Fig.2 Simulation of energy absorption of each pigment

　　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

U_max = A*exp(-E/RT)

Fig.3 Microalgae productivity in different temperatures

　　When our Synechococcus PCC7942 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)

Fig.4 Microalgae productivity in different pH

　　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]

Fig.5 The relation between photosynthetic rate and total absorption

　　The timing of adding engineered E.coli or purified protein to Chlorella vulgaris 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

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

= μ (specific growth rate)

Fig.6-1 Growth curve of Chlorella vulgaris

Fig.6-2 Growth rate of Chlorella vulgaris

　　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;

dN/dt = -V*X;

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

Q = (X₀*Q₀ + N₀ - N) / X

Fig.7 Oil accumulation and nirogen source consumption at normal situation(lipid:green;nitrogen:blue)

　　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;

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

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)

　　We put normal and modified nitrogen source systems together to see their demonstration like speed and occasion. By constructing this model, we can find out the declining rate of each state and then adjust our experiments.

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

Fig.9 Nitrogen source in nitrogen starvation(normal:blue;starvation:red)

　　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

Fig.10 Oil accumulation in nitrogen starvation

　　According to our reference of experimental data, we find that E.coli can build a relationship, which is like symbiosis, with Chlorella vulgaris. Therefore, we build a model and use three kinds of values from different situations to simulate their change when they are co-cultured. According to this, we get the proper experimental proportion of them at each need.

x₂ = [ax-x²/(1+b*x*z)] / R_x + x / Y_x

z₂ = [cz-z²/(1+g*z*x)] / R_z + z / Y_z

Fig.11-1 Population of co-cultured Chlorella and modified E.coli(initial concentration 0.1g/l) (chlorella vulgaris:green;e.coli:orange)

Fig.11-2 Population of co-cultured Chlorella and modified E.coli(initial concentration 0.012g/l)(chlorella vulgaris:green;e.coli:orange)

Fig.11-3 Population of co-cultured Chlorella and modified E.coli(initial concentration 0.3g/l)

　　This chart demonstrates the connection between initial nitrogen concentration and final lipid proportion in algae cell, and it tell us the approximate trend.

l = k[ln(b(n_s+a))^-1] - e

Fig.12 Nitrogen-lipid plot

　　NrtA is an endocrine secretion protein and this characteristic is a bound to reach our goal because it does not have enough efficiency to make microalgae to produce a significant amount of biofuel. We have tried to turn NrtA into exocrine secretion protein but unfortunately, we didn’t make it in time. If we have successfully transform it into a exocrine secretion protein, and with the help of the connected constitutive promoter, we might have a better result than before theoretically. And this model provide the predictive quantity of change and productivity of new method.

dr₁/dt = [1/(1+p₁/a₁)]c1p1 - [1/(1+p₂/b₁)]v₁r₁

dr₂/dt = [1/(1+p₂/a₂)]c₂p₁ - [1/(1+p₂/b₂)]v₂r₂

dr₃/dt = [1/(1+p₃/a₃)]c₃p₁ - [1/(1+p₂/b₃)]v₃r₃

dp₁/dt = [1/(1+p₁/d₁)]l₁r₁ - u₁p₁

dp₂/dt = [1/(1+p₂/d₂)]l₂r₂ - u₂p₂

dp₃/dt = [1/(1+p₃/d₃)]l₃r₃ - u₃p₃

Fig.13 NrtA exocrine secretion (normal quantity in cell:green;exocrine quantity in cell:yellow;normal productive speed:purple;exocrine productive speed:pink)