Modeling

Growth curve of Chlorella vulgaris

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

Oil accumulation & Nirogen source consumption

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

Biomass in different nitrogen concentration

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

Nitrogen concentration in nitrogen starvation

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

Oil accumulation 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

Population of co-cultured Chlorella and modified E.coli

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.5-1 Population of co-cultured Chlorella and modified E.coli

fig.5-2 Population of co-cultured Chlorella and modified E.coli

fig.5-3 Population of co-cultured Chlorella and modified E.coli

Nitrogen-lipid plot

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.6 Nitrogen-lipid plot

123

descibe

fig.5