Introduction

Acetic acid is considered harmful to E. coli. High concentrations of acetate ions are thought to inhibit growth. Therefore, the more acetate ions are synthesized, the more E. coli will suffer from acetate concentrations. I set up modeling to see if there is a stable amount of acetate ion synthesis.

method

We considered the amount of IPTG added as changing the amount of expression. The amount of protein per cell when IPTG was added could be predicted as follows.
\[\frac{dc}{dt}=k(i, µ)-µc\]
"c" is the protein concentration, "μ" is the growth rate, and "k" is the protein synthesis rate, which depends on the IPTG concentration i, and in some cases also on the growth rate, hence the notation k(i, μ)[2]. The term k(i, μ) is used because it depends on the IPTG concentration i and in some cases also depends on the growth rate [3].
We predict that the amount of acetate emitted by the transporter per unit time is proportional to the amount of PoxB expressed. We assume that the amount of acetate ions produced by this is proportional to the amount of protein c, which we denote as ac. That is, a is the amount of acetate ion produced per protein produced.

It is possible that the acetate ions that are synthesized are not successfully eliminated and accumulate in the body, leading to death or not. I would like to reconsider this in the modeling.

The synthesized acetate ions are not properly eliminated and may or may not accumulate in the body and cause death. I would like to rethink this in modeling. I would like to discuss a modified logistic model. In the logistic equation, Acetate has an exponential effect on the growth of E. coli. Acetate constrains the growth of E. coli [4].

The model predicted from this is as follows. Acetate not only inhibits growth, but in some cases may even kill it. Therefore, the growth rate can be negative. The following equation was created using p as the effect of Acetate on E. coli.

\[\frac{d}{dt}N=r\left(1-\frac{N}{K}\right)N-pac=r(1-\frac{N}{K})N-pa\left(\frac{kt}{µ}\right)-pa\left(c_0-\frac{k}{µ}\right)e^{-µt}\]

This equation will be considered in simplified form. Let Q be the rate limited by the synthesis of acetate.

\[\frac{dN}{dt}=r\left(1-\frac{N}{K}\right)N-Q・・・(*)\]
Furthermore, the acetate required per cell can be considered proportional to this Q. Therefore, by doing mQN (where m is a constant), it is expected that the overall acetate yield can be determined by calculating the area of this integral. The overall acetate yield is expected to be obtained by calculating the area of this integral.

results

The results of Equation (＊) are as follows. The right axis represents time and the vertical axis represents the predicted increase in the number of E. coli. m is a parameter that links the amount of acetate discharged from the body to the amount of acetate that accumulates per unit of time in the counterpart.

Q=0.001 Q=0.18 Q=0.5

Total mQN was obtained as m=100.

Q=0.001 Q=0.18 Q=0.5

Discusssion

When the wading ability of Q is high, E. coli is unable to grow. In particular, at Q=0.5, the number of E. coli decreases exponentially after leaving a peak value larger than the maximum value at Q=0.18. The number of E. coli is stable at Q=0.18. Furthermore, it should be noted that when Q (Q=0.18) is given, which allows the T. megacephaly to remain stable from the initial value, it is possible to collect high acetate in a stable manner. This suggests that the maximum effective rate of acetate collection may be achieved by selecting a promoter with a moderate expression frequency, since the strength of the RBS and promoter can determine the protein mass fraction and its optimal synthesis rate, both of which are dependent on cell proliferation [5]. We hope to discuss the appropriate concentration of IPTG by using a number of combinations of these factors. Furthermore, although we are only focusing on acetate synthesis this time, we are considering a co-residence pathway with methanogenic bacteria in the future. In this way, acetate is quickly consumed and converted to methane. In this case, it is known that a higher expression promoter and RBS will produce more acetate. This will require a new discussion.

[1]Stéphane Pinhal(2019)Journal of Bacteriology "Acetate Metabolism and the Inhibition of Bacterial Growth by Acetate"
https://journals.asm.org/doi/10.1128/JB.00147-19
[2]・[3]Niclas Nordholt(2017) nature scientific report “Effects of growth rate and promoter activity on single-cell protein expression“
https://www.nature.com/articles/s41598-017-05871-3
[4]Stéphane Pinhal(2019) Journal of Bacteriology” Acetate Metabolism and the Inhibition of Bacterial Growth by Acetate”
https://journals.asm.org/doi/10.1128/JB.00147-19
[5]Fernando N. Santos-Navarro(2021) ACS SyntheticBiology” RBS and Promoter Strengths Determine the Cell-Growth-Dependent Protein Mass Fractions and Their Optimal Synthesis Rates”
https://pubs.acs.org/doi/10.1021/acssynbio.1c00131