Team:NYMU-Taipei/Model
This year, our modeling focuses on predicting the effect of our modified microbes on productivity. It is an extremely important part to our project because it helps us accurately check and predict information from our experiments that are tested 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 descriptions will show our success in modeling.
Synechococcus PCC7942
The modeling from Figure 1 to Figure 5 belongs to the experiments of Synechococcus PCC7942 pigments for better photosynthetic efficiencies. We need to check if another microalgae contains an exogenous pigment that can successfully reach new photosynthesis rate and further increase the proportion of biomass. We already have models about the influence of energy adsorption, but pigments will certainly affect other factors. Therefore, we construct several models that each represents an important factor in the growth and cell composition. Thus, we can determine the best culturing collocation by combining these models.
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 = 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 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
| 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. Modeling is used 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 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)
| 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 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 = 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 absorption
Chlorella vulgaris
The modeling from Figure 6 to Figure 13 belongs to the experiments of Chlorella vulgaris for nitrogen starvation. To precisely calculate the timing of starting co-culturing and to ensure there are enough high-affinity E. coli in the bioreactor, we built several models that include the original and new system. They demonstrated the significant improvement of productivity after successfully deprived the microalgae from nitrogen. For instance, one of them provides a variety of information about population when two organisms in the pool start building some relationship.
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(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 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 = [(qM-Q)/(qM-q)] * [(Vm*N)/(N+Vh)]
Q = (X0*Q0 + N0 - N) / X
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| P | lipid | g/L | - |
| N | nitrogen | g/L | - |
| X | biomass | g/L | - |
| α | the instantaneous yield coefficient of product formation due to cell growth | g/g | 0.1973 |
| β | the specific formation rate of product | day-1 | 0.00037 |
| q | Minimum N quota | g/g | 0.0178 |
| qM | Maximum N quota | g/g | 0.0935 |
| Q | N quota | g/g | - |
| Vm | Maximum uptake rate of nitrogen | g/g*day | 0.596 |
| Vh | Half-saturation coefficient | g/m3 | 0.0000103 |
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(ns+a))-1]]-e};
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| n1 | biomass at frist state | g/l | - |
| n2 | biomass at secind state | g/l | - |
| x | biomass concentration | g/l | - |
| t | time | hr | - |
| Symbol | Definition | Unit | 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(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 = 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(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
| 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 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)] / Rx + x / Yx
z2 = [cz-z2/(1+g*z*x)] / Rz + z / Yz
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| Z | e.coil | - | - |
| X | chlorella vulgaris | - | - |
| 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(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(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 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
| Symbol | Definition | Unit | Value |
|---|---|---|---|
| Type 1 | protein initialize others | - | - |
| Type 2 | protein stabilize others | - | - |
| Type 3 | functional protein | - | - |
| t | time | min | - |
| r | mRNA | nMolar | - |
| p | protein | nMolar | - |
| c | relative transcription rate | mRNA/(protein·min) | 0.03/0.03/0.12 |
| l | relative translation rate | protein/(mRNA·min) | 2/2/2 |
| v | relative degradation rates of mRNA | min-1 | 0.03/0.03/0.023 |
| u | relative degradation rates of proteins | min-1 | 0.15/0.015/0.009 |
| an, bn, and dn | the effectiveness factors of the respective feedback loops for type n | - | a1:60 b1:120 d1:120 a2:140 b2:140 d2:150 a3:200 b3:140 d3:310 |
Fig.13 NrtA exocrine secretion (normal quantity in cell:green;exocrine quantity in cell:yellow;normal productive speed:purple;exocrine productive speed:pink)
