We want to use pigments to enhance the photosynthesis rate. Different pigments adsorb different wavelength of sunlight, and bring about different irradiance, temperature, and photosynthesis rate. These two models show the influence of irradiance and temperature on photosynthesis rate.

R=Rmax.i^n/(ki*exp(i.m)+i^n)

fig.1-1 Influence of irradiance on photosynthesis rate

R=A1exp(-E1/rT)-A2exp(-E2/rT)

fig.1-2 Influence of temperature on photosynthesis rate

　　The simplified graph is used to calculate how much energy is absorbed by each pigment approximately, and also help us know the photon adsorption amount after conversion.

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

y=1.5, 501<=x<=600

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

x:
y:

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. It is the modeling to ensure that our experiments are under control.

U = Umax*Kss

Umax = A*exp(-E/RT)

U: specific growth rate day^-1
Umax: maximum specific growth rate day^-1
Kss: substrate parameter //1
A: constant day^-1 //1.0114*10^10
E: activation energy cal/mol//6842
R: gas constant cal/K*mol //8.314

fig.3 Microalgae productivity in different temperature

　　When our Synechosistis PCC7942 grows at each phase, the equilibrium of pH value is different. This model can be used to collocate with our device, and also accomplishing the purpose of enhance productivity.

R=A1exp(-B1/ph)-A2exp(-B2/pH)

R: Co2 productive rate
A1: preexponential factor at i=400 //8.625*10^-5
A2: preexponential factor at i=200 //1.83885*10^-2
B1: activation energy at i=400 mol/J //6.45
B2: activation energy at i=200 mol/J //69.2

fig.4 Microalgae productivity in different pH

　　The model tells us that theoretically there is no faster photosynthetic rate, only if more energy is absorbed. After working with other modeling, we can establish the relation between photosynthetic rate and total yield for the purpose of best balance.

R=Rmax.e^n/(ke*exp(e.m)+e^n)

Rmax: maximum rate mol/g*min //0.000046
e: absorbed energy w/m^2
n: energy exponential constant//1.252
ke: productive coefficient uE/(m^2)*s //157.88
m: constant (m^2)*s /uE//0.0035

fig.5 The relation between photosynthetic rate and total yield

　　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, which is based on reference time and other lab environment data, would be gained. 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(Xt/X0)/t=A+Bexp(-C(t-M))=μ(specific growth rate)

X: biomass concentration(g/l)
t: time(hr)
A: the asymptotic of ln Xt/Xo as t decrese indefinitely
B: the asymptotic of ln Xt/Xo as t increase indefinitely
C: the relative growth rate at time M

fig.6-1 Growth curve of Chlorella vulgaris

fig.6-2 Growth rate of Chlorella vulgaris

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

dP/dt=*dX/dt+*X;

dN/dt=-V*X;

V=((qM-Q)/(qM-q))*((Vm*N)/(N+Vh));

Q=(X0*Q0+N0-N)/X;

P: lipid
N: nitrogen
X: biomass

α: the instantaneous yield coefficient of product formation due to cell growth
β: the specific formation rate of product

q: Minimum N quota
qM: Maximum N quota
Q: N quota
Vm: Maximum uptake rate of nitrogen
Vh: Half-saturation coefficient

fig.7 Oil accumulation and nirogen source consumption at normal situation

　　To find out the best quantity of nitrogen removal, we model several situations of decreasing the biomass in different environment with different concentration of nitrogen, and then we can find the best productivity by comparison.

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

x2=x1+(n2-n1)*((k(ln(b(ns+a))^-1))-e);

n1: biomass at frist state
n2: biomass at secind state
x: biomass concentration(g/l)
t: time(hr)

A: the asymptotic of ln Xt/Xo as t decrese indefinitely //-39.9532
B: the asymptotic of ln Xt/Xo as t increase indefinitely //-0.0222
C: the relative growth rate at time M hr //45.6931

k: constant //8.15229
b:yield coefficient//1207.569
ns:initial nitrogen concentration
a:regression constant//0.01
e:a perturbation//0.50678

fig.8 Biomass in different nitrogen concentration

　　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 experiments.

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

n: nitrogen concentration
Yxn: nitrate coefficient g/g 0.21016
m: maintenance parameter hr^-1 0.0014393
x: biomass concentration

fig.9 Nitrogen source in nitrogen starvation

　　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

p: lipid concentrtion
K1: growth correlation coefficient g^2/g^2 //122.40085
K2: non-growth correlation coefficient g^-1 //0.28736
e: a perturbation g/l*hr //-0.078

fig.10 Oil accumulation in nitrogen starvation

　　According to our reference of experiment 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 situation 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-x^2/(1+b*x*z))/Rx+x/Yx

z2=(cz-z^2/(1+g*z*x))/Rz+z/Yz

X: chlorella vugaris
Z: e.coil
Rx: symbiosis coefficient g/hr //1.0000023
Rz: symbiosis coefficient g/hr //1.178
Yx: correlation coefficient//12.576
Yz: correlation coefficient//2.276
a: population constant //0.80467
c: population constant//0.61198
b: relative parameter //0.00027
g: relative parameter //0.0013

fig.11-1 Population of co-cultured Chlorella and modified E.coli

fig.11-2 Population of co-cultured Chlorella and modified E.coli

fig.11-3 Population of co-cultured Chlorella and modified E.coli

　　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(ns+a))^-1)-e

l: lipid proportion in cell
k: constant g/100g //1.13372
b: yield coefficient//1.57172
ns: initial nitrogen concentration
a: regression constant//0.51653
e: a perturbation g/100g//-55.2776

fig.12 Nitrogen-lipid plot