Quorum Production by the Brush E.coli

Mathematical modeling is essential in qualitatively describing de novo genetic circuits that frequently arise in synthetic biology. We can use such models for two objectives: (1) predicting the behavior of combinations of BioBrick parts designed for the synthetic circuit that performs some task, and (2) choosing the appropriate promoter and ribosome binding site (RBS) with suitable strengths for the circuit. Also, it will serve as a reference for others who use the BioBrick in the future. In summary, the model and the computer simulation are our beginning point for making testable predictions about the behavior of our system. We construct a computational model describing the genetic network encompassing relevant signal transduction pathways in order to help build E.coli that can draw pictures!

1. Modeling E.coli Type I. (Brush E.coli)

1.1. Modeling Approach

There are several known Quorum sensing (QS) networks. All known QS networks operate as an “on-off” gene expression switch by controlling the level of a certain transcription factor whose expression is suppressed in the “off” state and is strongly induced in the “on” state.[2] Usually, the intracellular network that is controlled by the quorum sensing remains in the “off” state until the quorum reaches a certain concentration. After quorum reaches the threshold concentration, the genetic circuit changes its state into “on” state and activates the expression of the relevant genes. In this model, we hypothesized that the typical E.coli cell volume is ~7.0×10-16L and cells are freely permeable to quorums. We used a standard chemical kinetic approach based on the mass-action rate law. The kinetic parameters used in our model are based on the published data. The rate constants were taken from several papers on the mathematical modeling of quorum sensing pathways.

1.2. Model

Before moving on to our model system, let us review how to model a general case of protein production from a single gene.

1.2.A. Protein Production of a Single Gene Modeling

The actual protein production from a single gene is composed of complex processes. However, in this model, protein production is simplified into two processes: transcription and translation.

Fig 1 | The schematic diagram of protein production modeling

The model can be represented by these chemical reactions with corresponding ordinary differential equations (ODEs):

where p_m is the mRNA production rate constant, and p_p is the protein production rate constant. Although the choice of parameters depends on many factors such as the gene of interest and the internal and external environment of gene expression, the commonly accepted estimation of parameters is sufficient for our gene expression model. Therefore, we choose the average transcription/translation rate constants which are taken from published data. [1]

The degradation rate of the mRNA and protein can be calculated from

where t_1/2is the degradation half-life in minutes. From this equation, we can calculate k_md(mRNA degradation rate), k_pd(protein degradation rate). The values are in the constants table. [3. Constants Table of this page] Based on these facts and the law of mass action, we can write these equations:

The above models an average protein production from a single gene. However, in synthetic biology, we can control the transcription rate and the translation rate by appropriately changing the promoter and RBS parts. Therefore, we can represent the rate constants differently:

where χ_p is the strength ratio of the promoter relative to the average rate of transcription. Also, χ_R is the strength ratio of RBS relative to the average rate of translation. It is determined by the promoter (or RBS) BioBrick part that we use.

We will refer to the aforementioned protein production rate of a single gene model to build our own models. If specific values for the production and degradation rates exist for a certain protein, we will use those values. Otherwise, we will use a general, averaged value like above model. Let’s jump right into the details of the model.

1.2.B. Modelling of Brush(Type I) E.coli

The above schematic diagram represents the genetic circuit implmented in the Brush E.coli after it has been genetically recombined by the Cre-Lox system[HYPERLINK TO THE WIKIPEDIA]. The Cre-Lox recombination system is commonly used to knockout a specific gene conditionally, in which a gene is only knocked out at a specific time. By using this Cre-Lox recombination system, one of the four quorum sensing systems is chosen and installed in Brush E.coli (In this model, the Lux based quorum sensing system is installed). In our model, we will only discuss the system embedded with the Lux system and assume that the model is applicable to the other three installations because they operate with an almost identical mechanism and have similar reaction constants. Now, we briefly explain each component of the model in the following paragraphs.

1. DNA1 translates LacI constituitively. As a derivative of the lac operon, LacI binds to the gene lacO to suppress the expression of the genes downstream relative to lacO. Although four LacI must bind to lacO for this to occur, we assume one is sufficient so that we can simplify the analysis.

2. DNA2 produces LuxI, the component of the Lux based quorum sensing. LacI suppresses expression of DNA2 by binding to the LacO which is upstream to the luxI gene. As a result, DNA2 is not translated until we add IPTG.

3. When we add IPTG, it binds with LacI. IPTG acts as a competitive inhibitor to the binding of LacI to lacO; IPTG prevents LacI from binding to DNA2, clearing way for the RNA Polymerase to begin transcription. That is, IPTG induces the translation of luxI in DNA2; it is an inducing signal that turns the switch on. [2] In this model, we assumed that IPTG is applied in the form of a step function.

In order to describe the behaviour of the circuit with a deterministic model, we used the mass action rate laws in part A to represent changes of each quantity with respect to time in a set of ordinary differential equations. By solving these equations, we can obtain the number of each variable at a specific time. (Note, we used the Michaelis-Menten equation to account for the affinity difference of LacI for both for LacO and IPTG. Detailed explanation will follow.)

Lists of Modeling