The timing of adding engineering E.coli or purified protein to Chlorella vulgaris culture is critical for our project. Through the initial, final biomass concentration data, the instantaneous rate of a reference time and other lab environment datas, we simulate the change in biomass concentration throughout the culture cycle. The status information in the culture medium at each point is then obtained through the other calculus to obtain the best timing point and the 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.1-1 Growth curve of Chlorella vulgaris

fig.1-2 Growth rate of Chlorella vulgaris

Simulating common system of oil accumulation and nitrogen source consumption, not only get the reference of state before the improvement as well as the stage information, but also as a basic equation after some parameters or organisms join 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.2 Oil accumulation and nirogen source consumption at normal situation

To find the optimal amount of nitrogen removal, we model biomass decrease in different nitrogen concentration environments, and then we can find the best productivity.

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.3 Biomass in different nitrogen concentration

Put normal and modified nitrogen source system together,see their demonstration, like speed and occasion.by constructing this model,we can find out the declining rate of each state,then adjust experiment.

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.4 Nitrogen source in nitrogen starvation

We predict total lipid increase under nitrogen starvation. The model provide theoretical information of top yield. This graph show that if we use symbiotic microbe isolating nitrogen source 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.5 Oil accumulation in nitrogen starvation

According to our reference experiment data, we find that e.coli can build a relationship with chlorella like symbiosis. So we build a model and use 3 kinds of situations’ value to simulate their change when they are co-cultured. According to it,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.6-1 Population of co-cultured Chlorella and modified E.coli

fig.6-2 Population of co-cultured Chlorella and modified E.coli

fig.6-3 Population of co-cultured Chlorella and modified E.coli

This chart demonstrate the connection between initial nitrogen concentration and final lipid proportion in algae cell.tell us the approximately trend of it.

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.7 Nitrogen-lipid plot

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

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

R: co2 productive rate
Rmax: maximum rate mol/g*min //0.000046
i: irradiance uE/m^2
n: irradiance exponential constant//1.19
ki: productive coefficient uE/(m^2)*s //174
m: constant (m^2)*s /uE//0.0022

fig.8-1 Influence of irradiance on photosynthesis rate

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

R: co2 productive rate
A1:preexponential factor at i=400 //1147.7
A2:preexponential factor at i=200 //3.818*10^8
E1:activation energy at i=400 mol/J //42700
E2:activation energy at i=200 mol/J //77100
T:temperature K

fig.8-2 Influence of temperature on photosynthesis rate

The simplified graph can be used to calculate how much energy be absorbed by each pigments approximately, also knowing 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.9 Simulation of energy absorption of each pigment

After we get the influence degree on temperature,with this model ,we can predict the microalgae productivity at each temperature with other condition keeping stable.It is one of those which use to ensure the experiment in 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.10 Microalgae productivity in different temperature

During microalgae grow at each phase,the equilibrium of pH value is different. This model can be used to collocate with our device,also for 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.11 Microalgae productivity in different pH

The model tells us that theoretically there is no faster photosynthetic rate only if more energy be absorbed, so after work with other model, 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.12 The relation between photosynthetic rate and total yield