Team:NYMU-Taipei/Model
Modeling is an extremely important part to our project, because it helps us accurately check and predict the results of the experiments, which are worked in the wet lab. In our project, there are two essential types of microalgae that play very important roles, Synechococcus PCC7942 and Chlorella vulgaris. The following will show our success in modeling.
Synechococcus PCC7942
The modeling from figure 1 to figure 5 belongs to the experiments of Synechococcus PCC7942 pigments.
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 * in / [ki * exp(i*m) + in]
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| R | CO2 productive rate | mol/g*min | 0.000046 |
| i | irradiance | uE/m2 | - |
| n | irradiance exponential constant | - | 1.19 |
| ki | productive coefficient | uE/(m2)*s | 174 |
| m | constant | (m2)*s /uE | 0.0022 |
fig.1-1 Influence of irradiance on photosynthesis rate
R = A1 exp(-E1rT) - A2 exp(-E2/rT)
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| R | CO2 productive rate | - | - |
| A1 | preexponential factor at i=400 | - | 1147.7 |
| A2 | preexponential factor at i=200 | - | 3.818*108 |
| E1 | activation energy at i=400 | mol/J | 42700 |
| E2 | activation energy at i=200 | mol/J | 77100 |
| T | temperature | K | - |
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 = 0.01x - 3.5, 400<=x<=500
y = 1.5, 501<=x<=600
y = 3 - 0.0025*x, 601<=x<= 800
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| x | wavelength | nm | - |
| y | irradiance | W/m2 | - |
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)
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| U | specific growth rate | day-1 | - |
| Umax | maximum specific growth rate | day-1 | - |
| Kss | substrate parameter | - | 1 |
| A | constant | day-1 | 1.0114*1010 |
| E | activation energy | cal / mol | 6842 |
| R | gas constant | cal / K*mol | 8.314 |
fig.3 Microalgae productivity in different temperature
When our Synechococcus 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 = A1 exp(-B1/pH) - A2 exp(-B2/pH)
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| 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 * en / [ke * exp(e*m) + en]
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| Rmax | maximum rate | mol / g*min | 0.000046 |
| e | absorbed energy | w/m2 | - |
| n | energy exponential constant | - | 1.252 |
| ke | productive coefficient | uE / (m2)*s | 157.88 |
| m | constant | (m2)*s / uE | 0.0035 |
fig.5 The relation between photosynthetic rate and total yield
Chlorella vulgaris
The modeling from figure 6 to figure 12 belongs to the experiments of Chlorella vulgaris nitrogen starvation.
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 + B exp[-C(t-M)]
= μ (specific growth rate)
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| X | biomass concentration | g/l | - |
| t | time | hr | - |
| A | the asymptotic of ln Xt/X0 as t decrese indefinitely | - | 1.252 |
| B | the asymptotic of ln Xt/X0 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 = αdX/dt + βX;
dN/dt = -V*X;
V = [(qM-Q)/(qM-q)] * [(Vm*N)/(N+Vh)]
Q = (X0*Q0 + N0 - N) / X
| Symbol | Definition | Symbol | Definition |
|---|---|---|---|
| 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 | P | lipid |
| Vh | Half-saturation coefficient | Vm | Maximum uptake rate of nitrogen |
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};
| Symbol | Definition | Symbol | Definition |
|---|---|---|---|
| n1 | biomass at frist state | n2: | biomass at secind state |
| x | biomass concentration(g/l) | t | time(hr) |
| Symbol | Definition | Value |
|---|---|---|
| A | the asymptotic of ln Xt/X0 as t decrese indefinitely | -39.9532 |
| B | the asymptotic of ln Xt/X0 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
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| 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
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| p | lipid concentrtion | - | - |
| K1 | growth correlation coefficient | g2/g2 | 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-x2/(1+b*x*z)] / Rx + x / Yx
z2 = [cz-z2/(1+g*z*x)] / Rz + z / Yz
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| 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
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| l | lipid proportion in cell | - | - |
| k | constant | g/100g | 1.13372 |
| b | yield coefficient | - | 1.57172 |
| ns | initial nitrogen concentration | - | - |
| a | correlation coefficient | - | 2.276 |
| a | regression constant | - | 0.51653 |
| e | a perturbation | g/100g | -55.2776 |
fig.12 Nitrogen-lipid plot
NrtA is an endocrine secretion protein and this characteristic is a bound to reach our goal because it do not has 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 changement 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
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| - | - | ||
| - | |||
| - | - | ||
| - | |||
| - | |||
fig.13 NrtA exocrine secretion
