Difference between revisions of "Team:Oxford/Modeling"

Arabinose-induced expression

We have decided to use an arabinose induced promoter for the expression of a number of our proteins. This promoter can be modelled as the following chemical system:

\[[Arab:AraC]\overset{K}{\rightarrow}mRNA\overset{\alpha}{\rightarrow}P\] \[mRNA\overset{\gamma_{1}}{\rightarrow}\phi\quad P\overset{\gamma_{2}}{\rightarrow}\phi\]

Our pBAD promoter is known as a double repressor. AraC binds to pBAD which represses transcription of mRNA. By introducing Arabinose into the system, AraC will bind to Arabinose and form the Arab:AraC compound. Transcription can then occur.

For this system we will assume that AraC is always in large concentration and that its binding to arabinose happens on a faster time scale to transcription. Therefore, we do not need to consider the individual concentrations of arabinose and AraC, instead we just need to include the concentration of the complex [Arab:AraC]. The rate \(K\) is not just a simple constant and is given as the hill function in the equations below.

Using Michaelis-Mentin kinematics, we arrive at the equations:

\[\dfrac{d[mRNA]}{dt}=K_{max}\dfrac{[Arab:AraC]^{n}}{K_{half}^{n}+[Arab:AraC]^{n}}-\gamma_{1}[mRNA]\] \[\dfrac{d\left[P\right]}{dt}=\alpha\left[mRNA\right]-\gamma_{2}\left[P\right]\]

Where we define the symbols as:

Symbol	Definition	Initial Value/Literature Value	Fitted
\([Arab:AraC]\)	The concentration of associated Arabinose and AraC	\(0\)	-
\([mRNA]\)	The concentration of mRNA	\(0\)	-
\([P]\)	The concentration of our product	\(0\)	-
\(\alpha\)	Translation rate	\(15ntd\: s^{-1}\)/length of sequence [6]	\(6.6ntd\: s^{-1}\)/length of sequence
\(\gamma_{1}\)	Degradation rate of mRNA	\(5.13\times10^{-4}s^{-1}\) [5]	\(1.1\times10^{-2}s^{-1}\)
\(\gamma_{2}\)	Degradation rate of product	\(5.13\times10^{-4}s^{-1}\) [5]	\(1.1\times10^{-2}s^{-1}\)
\(K_{max}\)	Maximal transcription rate	\(50ntd\: s^{-1}\)/length of sequence [6]	\(47ntd\: s^{-1}\)/length of sequence
\(K_{half}\)	Half-maximal transcription rate	\(160\mu M\) [8]	\(100\mu M\)
\(n\)	Hill coefficient	\(2.65\) [3]	\(2.73\)

This table contains literature values for the parameters, found from a number of sources. We measured GFP expression in E. coli to create data to fit parameters to. This is how our fitting function works, and our data is given below.

We can now calculate the limiting concentrations that our products will be expressed at. There are mutliple products being expressed, each of different sequence lenghts. Here is a table showing the relevant proteins and sequence lengths:

Product	Sequence Length (/bp)
pBAD HisB DNase DsbA	621
pBAD HisB DspB YebF
pBAD HisB DspB
pBAD HisB MccS	414
pBAD HisB DspB Fla
pBAD HisB Art-175 DsbA	987
pBAD HisB Art-175 YebF	1284
pBAD HisB Art-E	632
pBAD HisB Art-175 Fla	1095
pBAD HisB Art-175	936
pBAD HisB DNase	570

We now can run our model of the system by solving the set of equations using the MATLAB equation solver ode15s. Below is a plot of the concentration of product against time for each protein expressed with this inducer-promoter pair where the expression is induced by a step function:

To calculate if our project will work in real life, we can model the number of cells required to reach a sufficient enzyme secretion rate to stop a biofilm from forming. To do this, we need to measure the rate of secretion of enzymes from our cells.

This can be difficult to do when the protein itself is mixed in with the fluid our cells grow in. GFP is much easier to measure in the lab, so we induced its expression and measured the fluorescence. From this we can find the parameters given in our first table, and then apply the model to a longer - or shorter - gene and predict expression.

Using the graph below, we can fit our expression data to the model presented in the tutorial, where we have included an arbitrary scaling factor (instead of a calibration curve) to match GFP fluorescence to molecule concentrations.

By comparing the base-pair length of the GFP gene and of our enzymes, we can expect expression concentrations of the order of \(nM\). This results feeds into our modelling of a mass-exchange system which delivers the enzymes from our cells out to the fluid they're needed in. That model will give us the number of cells required.

Dispersin B

Dispersin B is one of the anti-biofilm agents we are using in our project and will be the focus of this delivery section. As such we will assume that conclusions reached apply to all of our enzymes.

A concentration of Dispersin B of 60μg/ml is required to destroy a biofilm that has already formed on a surface. This equates to a concentration of 1.50μM. This is higher than the steady-state gene expression concentration we can expect from our cells, meaning that our system cannot rely solely on diffusion to transport our enzymes to the biofilm. We will therefore model these diffusion systems assuming that our cells are expressing at a 2μM concentration and later we will look at optimising the gene expression to this level.

Beads