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.

Lists of Modeling