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• Model
• GR Proton Pumping Model
• Growth and Consumption Model
• Glucose Optimization Model
• Intelligence Production Model

Model

Purpose

E. Hybrid is designed to be served as a phototrophic platform available for multiple applications. For instance, they can make target proteins and valuable molecules. The overall process can be first simulated, fitted with experiment data points, validated with the prediction, and finally optimized considering growing factors. The general stepwise construction of the model is shown below.

The structure of GR-GFP was first simulated using Pymol. It consists of a seven helix transmembrane protein, a flexible linker, and a green fluorescence protein, as figure below shows.

3D structure simulation of GR linker GFP.

GR Proton Pumping Model

The construction of GR efficiency model stems from the proton pumping activity of GR-GFP, based on the photocurrent experiment we designed. As a result, we first analyzed the photocurrent data from the experiment (See EXPERIMENT RESULT ) by taking several factors into consideration.

We first described the chip (the insulation gap between the electrode is 0.2 [mm] length, 6 [mm] width, and 0.0262 [mm] height) and assumed the electric conductivity in the dark meets the pure theoretical water of 0.055 [μS/cm]. Therefore, we get the following formula below:

\[Current = \frac {Voltage\cdot Conductivity \cdot Volumn}{Length^{2}}\]

According to the formula above, the steady basis current was 0.4323 [nA]. The current we observed from the experiment is higher owing to the conductivity of low-salt unbuffered saline) and bacteria.

The induced current during illumination in the test could be deemed proportional to the number of protons pumped n, following the equation

\[I=nAvQ\]

, where I is the electric current. n is the number of charged particles per unit volume. A is the cross-sectional area of the conductor v is the drift velocity, and Q is the charge on each particle.

We proposed the following postulations to describe the proton pumping activity of GR.

1. The cytoplasm of Lemo21 pET32a (GR harmonized) is pH=7.5 ( [H⁺ 3.16×10 ^-8 [M]^[1]), the outside region of E. coli and periplasm is unbuffered saline, pH=7.
2. Proton motive force is only affected by two factors--one is the gloeobacter rhodopsin, and the other is the diffusion through the membrane caused by the proton gradient. Other factors can be eliminated with the vector control.
3. In illumination, the proton would be pumped to the periplasm by GR. It could diffuse extracellularly and change the number of charged particles, for the bacterial outer membrane is freely permeable to ions and small molecules, thus inducing photocurrent, which is proportional to the concentration of protons.

\[I\propto [proton]+\frac{K_{w}}{[proton]}\]

K_w is ion product constant.

We should calculate the volume ratio because of the volume differences of cytoplasm, periplasm, and extracellular space, considering that the changing concentration of protons would be different when the protons are pumped to another area.

The volume ratio of cytoplasm, periplasm, and extracellular space is 4:1:5952.

The data show in the last of the GR proton pumping model.

When protons are translocated, the proton gradient would also influence the proton pumping rate ^[2]. The concentration difference of protons would also affect the diffusion rate. Owing to abundant ions in the cytoplasm, the ions in the cytoplasm gives little changes of pH, which is similar to the function of buffer.

We assumed that the proton pump of GR is similar to the proton pump of respiration^[2], and the diffusion rate would be affected by pH value^[2]. Thus, GR pumping rate and diffusion rate can be represented as following:

\[Pump \propto \frac {V_{Max}}{1+(\Delta pH)^{2}}\]

\[Diffusion \propto V_{const}\cdot (-\Delta pH)\]

By inputting data, we can train our model and know the most important parameter V _max.

V_max=5*10^-4[M/sec]

V_const(between periplasm and cytoplasm)=5*10^-5[M/sec]

V_const (between periplasm and outside)=5*10^-7[M/sec]

The figure is the same as expected. At first, the light-induced proton pump protein-GR pumps the protons to the periplasm and permeate outside. When we turn off the light, the GR stops pumping protons. Some protons flow back to the cytoplasm gradually, and the rate would be very slow if the proton motion force is weak.

Figure 1: The current of a liquid including Lemo21 pET32a (GR harmonized) and Lemo21 pET32a. The illumination duration was 120 seconds, and the subsequent dark duration was 120 seconds.

Last but not least, the V_max is 5*10^-4 M/sec (0.16 [extracellular, ΔH+ × 10⁻⁷/min OD]), whereas the value of the proton pumping rate of GR by Pil Kim et al was 0.38.

After analyzing the function of GR, we need to simulate the result of Lemo21 pET32a (GR harmonized) on the growth to discuss how the function of GR affects the growth.

The data for calculating the volume ratio:

The density of E. coli is 1.589 (O.D.600)

Assume (O.D.600) = 0.3 gram dry weight /Liter ^[3]

Assuming a cytosolic volume of 1.77 × 10^-3 L/gDW ^[4] and the periplasmic volume represents 20% of the cell volume ^[5].

Growth and Consumption Model

Introduction

The growth and consumption model simulates and predicts the results of our experiments. We are mainly focusing on the limitation of glucose, different cloned genes and cultivated environment. Furthermore, we can get some important parameters that show the differences between Lemo21 pET32a (GR harmonized) and Lemo21 pET32a.

In order to prove the hypothesis, "cloning rhodopsin into E. coli would optimize the growth curve.” We first construct logical ODEs (Ordinary Differential Equations) to describe the growth curves of wild type E. coli and input the parameters about E. Hybrid from the review article (Claassens, N. J., et al., Potential of proton-pumping rhodopsins: engineering photosystems into microorganisms.", 2003)^[6] and do the same method to simulate the glucose consumption. Furthermore, we combine the two parts by some assumptions and mathematical methods.^[7] To improve our model, we fit in our experimental data and validate the prediction results of growth and consumption.

Besides, we also perform the experiment that Lemo21 pET32a (GR harmonized) and Lemo21 pET32a cultivated with sodium azide to confirm the functionality of GR. In our hypothesis, the growth of Lemo21 pET32a would be slower due to the inhibition of the electron transport chain. For the experiment, we construct a model on the basis of the hypothesis and successfully proved by our experiment.

Growth and Consumption

Growth Simulation

Assumption:

1. The nutrition of growth is sufficient to maintain a steady nutrition uptake rate.
2. The cultivation environment is finite, and there is a stationary phase for the growth of E. coli.
3. The bacteria mutation does not affect the growth curve.

Considering the common bacterial exponential growth, we set a parameter to stand for the base death rate.

$$\begin{equation} \frac{d[B]}{dt}=\mu_{Max}\cdot [B]=(g-D)\cdot [B] \end{equation}$$

In equation 1, B is the E. coli’s density [OD600] and t represents the time [hour]. We assume that μ_Max stands for the growth rate with sufficient nutrition [1/hour].

The base death rate is D, denoting the death rate of E. coli devoid of nutrition. ^[8] It is supposed that the E. coli would die and stop growing when the nutrition is running out, and g is the growth rate without considering base death rate. ^[8]

\[If\: g = 0\] then $$\begin{equation} \frac{d[B]}{dt}=-D\cdot B \end{equation}$$

The base death rate D is a constant for E. coli; nonetheless, the growth rate with enough nutrition g would change according to the cultivating environment. Therefore, we took the logistic differential equation to replace the unsteady growth rate with sufficient nutrition μ_Max.

logistic differential equation:

\[\frac{d[B]}{dt}=g_{const.d}\cdot (1-\frac{[B]}{B_{Max.d}})\cdot B \]

In the ODEs, g_const.d is the growth rate constant consider base death [1/hour]. And the maximum E. coli density considering base death [OD600] is B_Max.d

The logistic differential equation assumed the dynamic equilibrium of bacteria in the end; nevertheless, describing the decline of E. coli as the nutrition is depleted is necessary for our model. Moreover, the decline of E. coli when the nutrition is depleted has been considered in the base death rate D. Thus, the domain of logistic differential equations should exclude the base death rate.

\[g_{const}\cdot (1-\frac{[B]}{B_{Max}})\cdot [B]=g\cdot [B] \]

$$\begin{equation} \frac{d[B]}{dt}=\mu_{Max}\cdot [B]=(g_{const.d}\cdot (1-\frac{[B]}{B_{Max}})-D)\cdot [B] \end{equation}$$

Last but not least, we get growth rate constant [1/hour] g_const. The maximum density of E. coli is B_max.

In order to visualize our derivation ODEs, we simulate the growth curve of wild type E. coli. First, we adopted the parameters from the review article^[6]. These parameters were calculated under standard anaerobic conditions and didn’t consider the environmental limit. Thus, we need to assume the maximum density of E. coli B_max and to transform the growth rate to growth rate constant g_const with enough nutrition μ_Max by assuming the environmental limit is very low at first.

Figure 2: The simulation of wild type E. coli for 10 hours. The base death rate is D , which is 0.0175 [1/hour]^[8]. Because the μ and q aren’t considered the environment limit, the part of B_Max could be ignored. The growth rate constant g_const would become the growth rate with enough nutrition μ_Max^[6] + D base death rate D = 0.9975. We also assume that E. coli is 0.02 gram dry weight on a 1L container.

Then, we input the growth rate with enough nutrition μ_Max of E. coli with rhodopsin into our model to simulate the growth curve.

Figure 3: The simulation results of E. coli and wild type E. coli with rhodopsin for 10 hours. The parameter is the same as fig.2 The growth rate constant g_const is 1.0175.

We can see that the growth curve of E. coli with rhodopsin would grow slightly faster and it would be affected by the B_Max. Above all, we could expect that the E. coli with rhodopsin might grow faster than wild type E. coli; nevertheless, we still need to fit our model to predict and solve the unknown parameters.

Glucose Simulation

E. coli consumes nutrients from the environment mostly to satisfy the following two requirements: Physiological maintenance^[9] and mitosis. Physiological maintenance can be referred to many biological reactions that require energy, such as protein synthesis or chemical catalysis. As for mitosis, the energy is also required for nucleic acids synthesis or cytokinesis.

To clarify the energy consumption change of the two requirements, we assume that the energy is only supplied from glucose.

Assumption:

1. Glucose is a necessary substance for E. coli to proliferate and survive.
2. Glucose is a limited resource, and the concentration only decreases due to the consumption of E. coli.
3. The maintenance and growth energy are supplied by glucose, which is uptake by E. coli.^[7][10]

$$\begin{equation} \mu=Y\cdot (q-m) \end{equation}$$

$$\begin{equation} \frac{d[GLC]}{dt}=q\cdot [B] \end{equation}$$

The μ is growth rate [1/hour] and Y is biomass yield coefficient on glucose [OD600/mM], but we call it energy efficiency for short in the model. Otherwise, the maintenance is set as m [mM].

According to the research “A low-complexity metabolic network model for the respiratory and fermentative metabolism of Escherichia coli.”^[7], we can simulate that the glucose uptake rate [mM/OD600/hour] q is a function of the extracellular glucose concentration [mM] GLC, and the equation can be represented as:

$$\begin{equation} q=\frac{q_{Max}\cdot [GLC]}{k_{GLC}+[GLC]} \end{equation}$$

q_Max stand for the maximum of glucose uptake rate [mM/hour] and k_GLC represent the half-saturation Monod constant of glucose uptake [mM]

By substituting glucose uptake rate q in equation 4 and equation 5 with the maximum of glucose uptake rate q_max in equation 6, we get equation 7 and equation 8 as below, which are the growth rates of E. coli in the limited glucose environment and the glucose change rate as a function of extracellular glucose concentration, respectively:

$$\begin{equation} \mu=(\frac{q_{Max}\cdot [GLC]}{k_{GLC}+[GLC]}-m)\cdot Y \end{equation}$$

$$\begin{equation} \frac{d[GLC]}{dt}=(\frac{q_{Max}\cdot [GLC]}{k_{GLC}+[GLC]})\cdot [B] \end{equation}$$

In order to visualize our derivation ODEs, we simulate the glucose consumption curve of wild type E. coli by extracting the glucose uptake rate q from the review article ^[6]. The parameter is set on standard anaerobic conditions without considering the nutrition limit. Therefore, the glucose uptake rate q in the review article is equal to the maximum of glucose uptake rate q_Max in our model.

Figure 4: The simulation result of wild type E. coli for 10 hours. The density of wild type E .coli is the simulation results on fig.2 (the Maximum =1.6). The maximum glucose uptake rateq_Max is 10 [mmol/gram dry weight/hour]^[6]. The half-saturation Monod constant of glucose uptake k_GLC is 0.003 [mM] ^[11]. The glucose to cultivate is 22 mM.

Then, we input the maximum glucose uptake rate q of E. coli with rhodopsin into our model and simulate the growth curve.

Figure 5: The simulation result of wild type E. coli and E .coli with rhodopsin for 10 hours. The density of wild type E .coli is the simulation results on fig. 3(the Maximum =1.6). The maximum glucose uptake rateq_Max is 10 [mmol/gram dry weight/hour]^[6].

By observing the figure, the E .coli with rhodopsin would consume more glucose even if the maximum glucose uptake rate q_Max. However, the difference is not significant, and we cannot determine that the GR wouldn’t affect the glucose uptake.

Growth with Limited Glucose

Based on limited glucose experimental conditions, couldn't describe our experimental design. Therefore, combining the glucose and growth part is necessary to simulate our experiment results.

On the basis of the assumption: "The nutrition to growth is enough to maintain steady nutrition uptake rate.", the glucose uptake rate q would reach its maximum q_Max in equation 6.

The steady glucose uptake rate means that glucose uptake rate q wouldn’t change sensitively owing to the changing of extracellular glucose concentration.

if \[[GLC]\gg k_{GLC} \] then $$\begin{equation} q\approx q_{Max} \end{equation}$$

Moreover, the growth rate μ in equation 4 could be replaced by the growth rate with enough nutrition μ_Max in equation 3

if \[[GLC]\gg k_{GLC}\] then $$\begin{equation} \mu\approx \mu_{Max}\implies \mu_{Max}=Y\cdot (q_{Max}-m) \end{equation}$$

Substituting the growth rate with enough nutrition μ_Max in equation 3 by equation 10 could describe the change rate of E. coli with parameters about glucose. Deriving the equation could determine the maximum glucose uptake rate q_Max.

\[\frac{d[B]}{dt}=\mu_{Max}\cdot [B]=(g_{const}\cdot (1-\frac{[B]}{B_{Max}})-D)\cdot [B]=Y\cdot (g_{Max}-m)\cdot[B] \]

$$\begin{equation} q_{Max}=\frac{g_{const}\cdot (1-\frac{[B]}{B_{Max}})-D}{Y}+m \end{equation}$$

However, the assumption of " The nutrition of growth is enough to maintain a steady nutrition uptake rate." doesn't match the experiment condition. Therefore, the growth rate with enough nutrition μ_Max couldn’t match the experimental condition as well and should be replaced by the growth rate μ

\[\frac{d[B]}{dt}=\mu\cdot [B] \]

Substitute the growth rate μ by equation7

\[\frac{d[B]}{dt}=\mu\cdot [B]=(\frac{Y\cdot q_{Max}\cdot [GLC]}{k_{GLC}+[GLC]}-m)\cdot [B] \]

Substitute the glucose uptake rate q_Max by equation 11

$$\begin{equation} \frac{d[B]}{dt}=((g_{const}\cdot (1-\frac{[B]}{B_{Max}})-D+m\cdot Y)\cdot \frac{[GLC]}{k_{GLC}+[GLC]}-m)\cdot [B] \end{equation}$$

From the derivation above, the equation of E. coli change rate also needs to follow equation 2. Simply speaking, if the nutrition for growth (glucose in our model) is 0, the growth rate μ that would be affected by glucose should be a negative base death rate -d.

if \[[GLC]=0 \] then \[\frac{d[B]}{dt}=-D\cdot [B]=(-m\cdot Y)\cdot B\implies m=\frac{D}{Y} \]

In the end, we could replace the maintenance m in by base death rate d and energy efficiency Y. The equation of the change rate of E. coli could be determined.

\[By\: m=\frac{D}{Y}\]then \[\frac{d[B]}{dt}=(g_{const}\cdot (1-\frac{[B]}{B_{Max}})\cdot (\frac{[GLC]}{k_{GLC}+[GLC]})-D)\cdot [B]=\mu \cdot B \]

Taking the maximum of glucose uptake rate q_Max in equation 11 to equation 8, the change rate of glucose could be expressed simply.

\[\frac{d[GLC]}{dt}=\frac{g_{const}\cdot (1-\frac{[B]}{B_{Max}})\cdot (\frac{[GLC]}{k_{GLC}+[GLC]})}{Y}\cdot B=q\cdot B \]

Summary:

By the previous derivations, we obtain the following two differential equations:

\[\frac{d[B]}{dt}=(g_{const}\cdot (1-\frac{[B]}{B_{Max}})\cdot (\frac{[GLC]}{k_{GLC}+[GLC]})-D)\cdot [B]=\mu \cdot B \]

\[\frac{d[GLC]}{dt}=\frac{g_{const}\cdot (1-\frac{[B]}{B_{Max}})\cdot (\frac{[GLC]}{k_{GLC}+[GLC]})}{Y}\cdot B=q\cdot B \]

, which satisfies the assumptions below:

1. The cultivation volume is finite and restricts the growth of E. coli.
2. The mutation will not significantly influence the growth curve.
3. Glucose is a necessary substance for the survival and growth of E. coli.
4. The glucose is limited and the concentration only decreases due to the consumption of E. coli.
5. The physiological maintenance and growth energy are supplied by glucose only.

After combining the growth and glucose concentration, we can use the growth and consumption model to simulate the overall growth curve.

Figure 6: The figure shows the overall growth curve prediction of wild type E. coli and Lemo21 pET32a (GR harmonized) for 48 hours. The growth curve is simulated by the growth and consumption model. We could calculate energy efficiency Y = 0.09975 and 0.10175 [gram dry weight/mmol] for wild type E. coli and Lemo21 pET32a (GR harmonized). Assume the maximum density of E. coli B_Max 1.6~2.0 [gram dry weight].

We find that higher energy efficiency resulting from rhodopsin could extend the stationary phase under a similar maximum density of E. coli B_Max. If the maximum density of E. coli is much higher than wild type E. coli, the E. coli with rhodopsin might decline earlier. Because the more E. coli would consume more glucose resources even if each E. coli with rhodopsin needs less glucose. Consequently, even if Lemo21 pET32a (GR harmonized) is an efficient fundamental strain, it may decline earlier in the glucose limited environment owing to the higher density.

Otherwise, the glucose uptake rate is assumed in the review article; however, the glucose uptake rate q might change due to the growth rate environment limit in reality. We assume that the glucose uptake rate would change from 9 to 12 [mmol/ gram dry weight] compared with wild type E. coli under standard aerobic condition.

Figure 7: The figure shows the growth curve simulation of wild type E. coli and Lemo21 pET32a (GR harmonized) for 8 hours. The growth curve is simulated by the growth and consumption model. The growth of Lemo21 pET32a (GR harmonized) is faster than wild type E. coli and the higher glucose uptake rate q would make E. coli grow faster.

In short, our Lemo21 pET32a (GR harmonized) may have differences in the characters and parameters. Faster growth and lower need for glucose will reflect on the higher growth rate constant g_const and higher energy efficiency Y ; nonetheless, the time of Lemo21 pET32a (GR harmonized) decline is undetermined. Glucose uptake is also unclear. Therefore, we would take the experiment results to fit our model by the experiment data and revise the parameters.

Curve Fitting

After simulating the overall growth curve of wild type E. coli and Lemo21 pET32a (GR harmonized), we fit our growth and consumption model by the experimental data to obtain the parameters.

Figure 8: The fitting result for wild type E. coli and Lemo21 pET32a (GR harmonized) for 11 hours. The g_const of Lemo21 pET32a (GR harmonized) are 0.44 and 0.36 [1/hour] for wild type E. coli. Otherwise, the maximum density of E. coli B_Max are 1.14 and 1.24 for wild type E. coli and Lemo21 pET32a (GR harmonized). The energy efficiency parameters Y are 0.0382 and 0.039 for wild type E. coli and Lemo21 pET32a (GR harmonized), respectively.

Furthermore, we predict the growth curve with limited glucose to see how the GR effect the E. coli

The following figures are the growth curve under the 22.2 mM Glucose and glucose consumption curve. It is significant that Lemo21 pET32a (GR harmonized) grows faster due to the growth rate constant of Lemo21 pET32a (GR harmonized), increasing about 22.2%.

In the case without glucose, the maximum density of E. coli B_Max of Lemo21 pET32a (GR harmonized) is 8% higher than Lemo21 pET32a. Nevertheless, the glucose depleted before Lemo21 pET32a (GR harmonized) was close to 1.24 O.D.600 which is the maximum density of E. coli B_Max. Otherwise, the energy efficiency Y increases about 2.1% which is similar to the results “the energy efficiency Y” in the review article.

Figure 9, 10: The prediction result of Lemo21 pET32a and Lemo21 pET32a (GR harmonized) for 14 hours.

Figure 11: The difference of change rate of Lemo21 pET32a (GR harmonized) and wild type E. coli. The change rate is the instantaneous change rate of the density of E. coli.

It is significantly different than Lemo21 pET32a (GR harmonized) would grow faster than wild type; however, it would also decline earlier owing to the glucose.

After predicting the growth curve, we can validate with testing data to ensure credibility.

Table1:The table is the error rate of growth curve prediction under 22mM Glucose.

We not only validate the model by growth curve under 22 mM glucose but also validate the glucose consumption.

Table2: The table is the error rate of glucose consumption prediction.

Sodium Azide Treatment

After the analysis of the growth of Lemo21 pET32a (GR harmonized) and wild type E. coli, we consider the growth of E. coli under the sodium azide. Due to the function of GR like providing another way of energy input, Lemo21 pET32a (GR harmonized) might be less affected by sodium azide.

To know how the sodium azide affects the growth curve of Lemo21 pET32a and Lemo21 pET32a (GR harmonized). We fit the data into our growth and consumption model and calculate some important parameters to analyze the effect of sodium azide.

Figure 12: The fitting results of Lemo21 pET32a and Lemo21 pET32a (GR harmonized) growth curves with sodium azide treatment. The g_const is 0.398 and 0.228 for Lemo21 pET32a and Lemo21 pET32a (GR harmonized), respectively.

It is intuitive that Lemo21 pET32a (GR harmonized) grow faster than wild type E. coli, and gets into steady-state earlier than Lemo21 pET32a. This phenomenon proves the functionality of GR, which takes light energy as the supplement of the growth.

The overall growth curves are predicted in this part. We could analyze the growth curves to optimize the growth of Lemo21 pET32a (GR harmonized) by the following section.

Glucose Optimization Model

Introduction

After understanding the overall growth curve of Lemo21 pET32a (GR harmonized) and wild type E. coli by the growth and consumption model, we need to analyze the character of E. coli that is changed by GR, predict how glucose affects growth, and how to set glucose concentration to reach the best.

We use the glucose optimization model to get important time points by analyzing the growth curve of different initial glucose concentrations. The time points include the fastest-changing, steady, and decline. After prediction, the optimization of different goals would be presented clearly.

Growth Optimization

First, the glucose optimization model predicts the fastest-changing time point, which defines “The time E. coli attains the maximum change rate of E. coli.” After predicting the time point, we can maximize the E. coli amount more easily. The growth rate would gradually decrease after the fastest-changing time point. Thus, we can take some action to maintain fast growth before the growth is inhibited.

e.g., Refresh or change the cultivated culture before the time to maintain faster-changing.

Figure 13, 14: The maximum change rate and time point of E. coli under M9 with different glucose cultivated conditions.

We can see that Lemo21 pET32a (GR harmonized) would attain the maximum change rate time point earlier because the density of Lemo21 pET32a (GR harmonized) would increase faster and be limited by the cultivated environment. The other result is that the maximum change rate of Lemo21 pET32a (GR harmonized) is higher on any glucose concentration. Consequently, Lemo21 pET32a (GR harmonized) could attain the more powerful fastest-changing time earlier. Therefore, we can say, “Lemo21 pET32a (GR harmonized) is very efficient”.

Otherwise, the prediction shows the optimal fast growth condition. Take Lemo21 pET32a (GR harmonized). For example, we can cultivate Lemo21 pET32a (GR harmonized) with light and continuously express GR harmonized with 12mM glucose. Lemo21 pET32a (GR harmonized) could use less glucose to attain the same fastest-changing time point. Then, refresh it to the start unit after cultivating 9 hours. By returning the step, the total amount of E. coli could be maximized with the least resource.

Decline Prediction

Second, we predict the time of E. coli enters the decline fast. The E. coli will decline fast when exceeding the death time point. In the glucose optimization model, the death time point is defined as “The change rate of E. coli would be less than -0.002 on the time point.” This simulation could help the users when E. coli couldn’t work efficiently. e.g., If we want to express a protein continuously, we should change the medium before the death time point.

Figure 15: The death time point of E. coli under M9 with different glucose concentrations.

We can see that Lemo21 pET32a (GR harmonized) would decline in the early time compared with wild type E. coli. The glucose consumption in the log phase is higher than wild type E. coli due to the faster-growth and higher density of E. coli.

The prediction result also can be applied to the bioreactor we developed for the purpose of understanding the timing to control the cultivated conditions.

Steady Prediction

Third, Lemo21 pET32a (GR harmonized) could be a stain for photosynthesis and bioengineering. The stationary phase and maximum of E. coli density directly affect production efficiency. Simulating the effect of glucose on the stationary phase of Lemo21 pET32a (GR harmonized) could help us evaluate how to adjust the cultivated condition in our device.

For example, we can set 37 ℃ during the log phase of Lemo21 pET32a (GR harmonized) and cool down the temperature when attaining the predicted steady time point. The method can make the growth of Lemo21 pET32a (GR harmonized) could be extended.

Figure 16: The steady time point of E. coli under different glucose concentrations.

The figure shows that Lemo21 pET32a (GR harmonized) would enter the stationary phase earlier than wild type E. coli. Moreover, the longest time of entering the steady-state is about 23 hours.

Figure 17: The steady density of E. coli under different glucose concentrations.

The figure shows that Lemo21 pET32a (GR harmonized) would have a larger steady density of E. coli, which makes greater capacity on any concentration of glucose. Furthermore, the capacity would stop increasing after exceeding 25 mM glucose.

To test the prediction above, we should test the growth curve under different glucose conditions.

We can validate the model with testing data to ensure credibility by calculating the error rate of the growth curve under different glucose concentrations because each prediction of the glucose optimization model is from analyzing the growth curve under different glucose concentrations.

Table 3: The table is the error rate of growth curve prediction under 222.2 mM Glucose.

Table 4: The table is the error rate of growth curve prediction under 2.22 mM Glucose.

After optimizing by setting glucose and validating the model, we need to consider the E. Hybrid in our device environment.

Intelligence Production Model

Introduction

The intelligence production model is to analyze the environmental elements that may affect the growth and production of E. Hybrid. Support the device to optimize the production of E. Hybrid by setting cultivated conditions and determining the refreshed strategy.

Temperature effect

To activate GR, the exposure of sunlight is necessary for Lemo21 pET32a (GR harmonized), which must lead to the change of temperature. Discussing the effect on the fastest-changing time point, steady time point, and decline time point could help us overview the change of growth.

To activate GR, the exposure of sunlight is necessary for Lemo21 pET32a (GR harmonized), which must lead to the change of temperature. Discussing the effect on the fastest-changing time point, steady time point, and decline time point could help us overview the change of growth.

The parameters and formula of temperature effect on the growth rate come from the 2018 NCTU_Formosa.

For temperature, we began with the Ratkowsky equation, which describes effect of temperature on general bacterial growth rate, modeled as

\[Logistic\: Function:\: f(t)=\frac{C}{1+A\cdot e^{-B\cdot t}}\]

Then we combine the growth rate under certain condition with Logistic function, which we have used in the growth and consumption model.

\[f(R_{temp},t)=\frac{C}{1+A\cdot e^{-R_{temp}\cdot t}}\]

\[T_{min}=284K;\: T_{max}=326K\]

\[R_{temp}(T)=a\cdot[(T-T_{min})\cdot(1-e^{(b\cdot(T-T_{max}))})]^2\]

\[a=1.915\cdot 10^{-5};\: b=0.105\]

Figure 18: The effect on the fastest-changing time point, steady time point, and decline time point by temperature.

It is obvious that the stationary phase would approach 0 under low ( <30℃) and high (>45℃) temperature. Simply speaking, if we want to take action on the stationary phase of Lemo21 pET32a (GR harmonized), keeping the temperature at 30℃~45℃ is important.

All the time points would delay at much higher and lower temperatures and the steady time point is unstable at 30℃~45℃. The result means the control of temperature is still an essential task for our device.

Optimize Production

To simulate the production of Lemo21 pET32a (GR harmonized), we assume a constant protein expression rate and no degradation to only the problem. Then, the protein expression is shown as below:

\[\frac{d[Protein]}{dt}=\alpha\cdot [B] \]

α is the protein expression rate of E. coli. It is significant that the protein concentration is proportional to the area under the growth curve. To optimize the production potential of the photosynthesis of Lemo21 pET32a (GR harmonized), we calculate the area under the growth curve of Lemo21 pET32a (GR harmonized).

And, we take the area under the growth curve as production potential.

Figure 19: The production potential of Lemo21 pET32a (GR harmonized) with respect to temperatures.

For the production of Lemo21 pET32a (GR harmonized), we divide it into two types. One is the expression in the stationary phase; the other is the overall growth expression, excluding the death phase. Keeping Lemo21 pET32a (GR harmonized) maintaining 35℃~43℃ could ensure the production potential, which only considers the stationary phase of about 15 (OD600*hr). Nevertheless, less total output in the temperature condition if we think the other is the overall growth expression, excluding the death phase.

Figure 20: The yield of Lemo21 pET32a (GR harmonized) under different temperatures.

Refresh Strategy

Furthermore, calculating the yield of the overall growth expression, excluding the death phase. We can see that the optimized temperature returns to 37℃. Simply speaking, the much higher or lower temperature would maximize production.

Figure 21: The simulation result of production potential and density of Lemo21 pET32a (GR harmonized) for 400 hours. It would refresh once a day, and the refresh ratio is 33%, 50%, and 66% under 37℃.

The refresh ratio is equal to the volume of the old culture medium divided by the volume of the new culture medium. The production potential is higher for two to one refresh ratio in three days. If the refresh ratio is one to two, it can keep more steady in 400 hours. Above all, the model can make the device more intelligent.

Figure 22: The production potential and density of Lemo21 pET32a (GR harmonized) for 400 hours. It would refresh once a day, and the refresh ratio is 33%, 50%, and 66% under 25℃.

The figure is similar to Fig.21. However, we find that the difference of production potential and density of Lemo21 pET32a (GR harmonized) are not significant. Therefore, if we want to have higher production at low temperature, we can use a low refresh ratio to decrease the consumption of the medium.

These predictions are based on the experimental data and parameters from the 2018 NCTU_Formosa. To make the model apply to our device, we need to adjust the parameter, and initial condition.

Apply on Device

First, we simulate the growth on the 30°C and the initial condition is 0.44 O.D.600 Lemo21 pET32a (GR harmonized) and LB broth (we assume the nutrition is enough)

Figure 23: The Simulation of Lemo21 pET32a (GR harmonized) in device with enough nutrition (the assumption of LB broth) and refresh at 12 hours.

To make our model apply to the device, we fit the data to train the model.

We choose the data during (3~5 hours) because the bacterial fluid will not be evenly distributed in the culture at first.

Figure 24: The Fitting of Lemo21 pET32a (GR harmonized) before 12 hours (the time to refresh). The maximum density of E. coli B_max is 0.66. the growth rate constant is 0.44.

After fitting, we can predict the growth curve after 12 hours (the time to refresh). We still need to validate the prediction by taking testing data.

Figure 25: The prediction of Lemo21 pET32a (GR harmonized).

To test the credibility of the model, we still need to validate the prediction by taking testing data. We want to confirm the influence of refresh, so we take the testing data after 12 hours.

Figure 26: The validation of Lemo21 pET32a (GR harmonized) after 12 hours (the time to refresh). The root-mean-square error (RMSE) is 0.0345.

After validating the growth of E. coli on the device, we can make the device connect to the experiment by model and make the production more intelligent.

Reference

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