# Why Modelling?:

# Model Overview:

## Model Application

## What´s inside the box?

# Extended Model

# Applying the model: Synechocystis promoters

Why?

To express bioluminescence under circadian oscillation it was essential to identify the ideal promoters that could yield highest substrate concentrations at the exact time our biolamp is on i.e. by night.

Based on literature (ref) we found that suitable promoters to start our model were sigE (sll1689), Transaldolase (slr1793) and Cytochrome aa3 (sll1898).

Therefore, in the following section we explain how to model the oscillation in terms of product generation under three circadian promoters: sigE (sll1689), Transaldolase (slr1793) and Cytochrome aa3 (sll1898).

Methodology

Product generation by promoter

From known sigE (sll1689) production data we could build up a model using MatLab software. (Please check our script!).

Basically, what we have after data adjustment is a sine function consistent with the oscillatory nature of the circadian rhythm.

Generation = C*sen(2π(t-(peak-6))/12)+1

The C parameter was optimized using Excel Solver’s algorithm of least squares.

For Transaldolase (slr1793) and Cytochrome aa3 (sll1898) a microarray data was adjusted (using the same algorithm of least squares) to obtain:

Promoter Peak hour
PsigE 8
Pta 14
Paa3 9

So, the protein generation models for each of the promoters are:

sigE production=0,8028*sen(π(t-2)/12)+1

Pta production=0,6246*sen(π(t-8)/12)+1

Pcaa3 production=0,9493*sen(π(t-3)/12)+1

Solving ODE’s

Assumptions: reaction only depends on promoter activity.
Protein half life expression of 8 hours.

LuxAB concentrations

We start with the fundamental balance law:

Entries-exits+generation-consumption= accumulation

As there is no entries nor exits (closed system) the expression becomes

Generation-consumption= accumulation

For LuxAB we know

Generation= k_1*sen((π*(t-8))/12)+1 (obtained above)

Consumption=ln(2)/t_m *X(t)

Equation to solve

dX/dt=k_1*sen((π*(t-8))/12)+1-ln(2)/t_m *X(t)

Using Matlab to solve this ODE

For LuxCDEG

Generation=k2*sen((π*(t-2))/24)+1

Consumption=ln(2)/t_m *Y(t)

Equation to solve

dY/dt=k_1*sen((π*(t-8))/12)+1-ln(2)/t_m *Y(t)

MatLab result

Finally we have the protein concentrations at different times

The files for the following models are:

Digitize2.m
UC-chileDiffs.m

The following are the models we obtained by MatLab

And this is the best model because it yields highest substrate concentration at the desired time (12 hr)