# ODE models of test constructs

The genetic circuitry involved in each test construct can be individually modelled.

These individual models can be compared with experimental results at each stage, and thus the individual models can be refined.

Individual models can be combined to generate a model of the complete circuit.

This model can be compared to the experimental results to evaluate how well it represents the behaviour of the system.

[1]

## Characterisation of Constitutive promoter and RBS

A simple synthesis-degradation model is assumed for the modelling of the expression of a protein under the control of a constitutive promoter, with the same model assumed for all four promoter-RBS constructs. The synthesis-degradation model assumes a steady state level of mRNA. $\frac{d[A]}{dt} = k_{1} - d_{1}[A]$ Where [A] represents the concentration of protein (in this case GFP), k1 represents the rate of sythesis of the protein and d1 represents the degradation rate. We can easily simulate this synthesis-degradation model using matlab:
ODE
Simulation File

We can also solve this ODE analytically.
At $\frac{d[A]}{dt}=0$, $[A]=\frac{k_1}{d_1}$ and you can see this relationship in the parameter scan graphs. From the wetlab experiments it is likely that we will obtain steady-state data for each of the four promoter-RBS constructs. If we assume the same rate of degradation of GFP in each case, we can have some measure of the relative rate of transcription through each promoter which will help us with the selection of the most appropriate promoter to use for Phase 2. In order to obtain an absolute measure of transcription (as opposed to a relative measure of transcriptional strength) we require constitutive expression in terms of molecules per cell (as opposed to fluorescene in arbitrary units).
Note from the parameter scan graphs:

• In the case where k1 = 0, no GFP is sythesised.
• In the case where d1 = 0, the concentration of protein does not reach a steady state.

## Characterisation of inducible promoter and RBS

Characterising the inducible promoter is a bit less straightforward. There are four inducible promoter + RBS combinations.
1. pHyper-spank-gsiB,
2. pHyper-spank-SpoVG,
3. pxyl-gsiB,
4. pxyl-SpoVG

For each of the above four inducible promoter + RBS combinations, we want to experimentally determine the maximum protein expression level. (In these test constructs the reporter protein is GFP). By varying the concentration of the inducer we want to characterise the transfer function between the inducer concentration and the steady-state GFP expression level. In addition, we want to characterise the time taken for induction, and how this is affected by concentrations of inducer.

The image on the right shows the genetic construct for the inducible promoter. A transcriptional regulator is constitutively expressed: In the case of the promoter pHyper-spank, the transcriptional regulator is LacI, and in the case of the promoter pxyl, the transcriptional regulator is XylR. Constitutive expression levels of these transcriptional repressors can be taken from the above characterisation using GFP as a reporter.

Factors affecting the steady state protein level downstream of the inducible promoter:

• Basal ("leaky") expression through the inducible promoter in the absence of inducer.
• Expression level of repressor.
• Concentration of inducer.
• Functional relationship between concentration of inducer, concentration of repressor and rate of transcription.

We describe repressor-inducer interactions by using LacI and IPTG in an example:

LacI reversibly binds IPTG.

$[LacI] + [IPTG] \longleftrightarrow [IPTG-LacI]$

With a kon of k2 and a koff of k3

LacI is constitutively expressed and enters into an equilibrium with the repressor, IPTG:

$\frac{d[LacI]}{dt}=k_1-d_1[LacI]-k_2[IPTG][LacI]+k_3[IPTG-LacI]$

The concentration of free IPTG is dependent on the concentration of free LacI:

$\frac{d[IPTG]}{dt}=-k_2[IPTG][LacI]+k_3[IPTG-LacI]$

And the concentration of [IPTG-LacI] must also be accounted for in the ODE model.

$\frac{d[IPTG-LacI]}{dt}=k_2[LacI][IPTG]-k_3[IPTG-LacI]$

The difficulty is that the interaction between the inducer, the repressor protein and the operator DNA is not simple. It depends on the relative concentrations and the state of the repressor protein (free or bound to the operator) [2].

### First Model

In the first model, the rate of synthesis GFP is affected by [LacI]. LacI binds cooperatively to the promoter and this relationship if described using a Hill function. Cooperative binding is a suitable model [3] and the promoter pHyper-spank has two LacI binding sites.

$\frac{d[GFP]}{dt}=\frac{k_4k_{m}^n}{k_m^n+[LacI]^n}-d_2[GFP]$

The trouble with this model is that [LacI] evolves to a steady state regardless of [IPTG]t = 0. As [GFP]steadystate depends soley on [LacI], the paramters of the Hill function and the degradation rate of GFP and is not dependent on [IPTG]t = 0, and [LacI]steadystate is the same for all [IPTG]t = 0, changing [IPTG]t = 0 affects only the dynamic and not the steady-state behaviour of [GFP].

Here are the simulations in matlab:

And here are four simulations at different [IPTG]t = 0:

Shown below:

By inspecting the simulations at different [IPTG]t = 0 you can see that the maximum [GFP] reached before [GFP] attains steady-state is affected by [IPTG]t = 0. In order to characterise the dynamic behaviour of the system these values are also shown. If the wetlab team is able to measure these maximum values this will aid us in model selection.

### Second Model

In this model, the interactions between IPTG and LacI are as in the first model, but the rate of expression of GFP is related to the concentration of free IPTG and not the concentration of LacI.

We use the "activator" form of the Hill function for this model:

$\frac{d[GFP]}{dt}=\frac{k_4[IPTG]^n}{k_m^n+[IPTG]^n}-d_2[GFP]$

## Characterisation of light-inducible promoter and RBS

Having established the transfer function between inducer concentration and GFP concentration (which will be useful for part characterisation and establishing the most appropriate RBS even if these inducible promoters won't be used in our final constructs) we need to establish:

1. Whether our model describing the constitutive expression of GFP can be applied to the constitutive expression of YtvA? If we can establish the concentration of GFP in molecules per cell this will be a better measure than arbitrary units/fluorescence but even so, YtvA will likely have a different degradation rate (can we characterise this)?
2. The relationship between YtvA (over)expression and expression through the sigma-B sensitive promoters. What is a good model for this relationship?

i.e. can we assume
$\frac{d[ytvA]}{dt}=k_1-d_1[ytvA]$
making use of the same values of k1 and d1 that were found from the characterisation of the constitutive promoters?
Would it be more relevant to use
$\frac{d[ytvA]}{dt}=k_1-d_2[ytvA]$
i.e. same expression level, different degradation rate, and how to characterise this degradation rate?
We could try the model
$\frac{d[ytvA]}{dt}=f(\frac{d[GFP]}{dt})$
Where f represents some function to be determined.
Once we have characterised this relationship, we need to characterise the relationship between ytvA expression, light intensity/duration of light exposure and protein expression downstream of the σB promoter.

## Characterisation of EpsE

Characterisation of EpsE has two considerations:

• Succeessful expression
• Desired functionality

It would be desirable to verify expression of EpsE by using anti-EpsE antibody and an immunohistochemical technique.

However we can functionally characterise EpsE by measuring motility as a function of inducer concentration.

## Characterisation of Biomaterials

Ideally we wish to quantitatively characterise Biomaterial expression and secretion to enable us to select the optimal signal peptide - biomaterial combination. We can use our data about GFP expression to estimate expression through the light-inducible system but secretion is more tricky. Hmm.

## Light induced expression of biomaterials

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