MODELLING

Introduction

Our model predicts how the absorption spectrum of a modified E. coli changes over time in response to a given incident light spectrum. It is used to estimate key properties of our system such as stability, development time of image, the balance of colours and has three steps to its operation.

The model is split into three sections. The first part calculates the reaction of each sensor species to a given incident light spectrum. The second is the heart of the model and it simulates the biochemistry of our designed pathways according to how the sensors respond to the light. The third and final part takes the data from the simulation and plots a surface showing how the absorption spectrum of a cell changes with time.

Currently the model is based on results from literature and theoretical conjecture as experimental results are not available. The motivation for the model was to numerically characterize our bio-bricks for future use and to help us create the first colour coliroid. With experimental results the model could readily be updated to fulfill its original mandate.

The Team

Angus Laurenson

Callum Vincent

Henri Laakso

Modelling Software

The model uses two programs; [http://www.mathworks.co.uk/products/matlab/ MATLAB®] and KaSiM. Together they provide the necessary tools to create an accurate model of our system.

KaSiM and Kappa

KaSiM is a [http://en.wikipedia.org/wiki/Stochastic stochastic] simulator that executes files written in [http://www.kappalanguage.org/ Kappa]. Kappa is a rule based modelling language for protein interaction networks. To download KaSiM or the manual please visit the KaSiM page.

MATLAB®

[http://www.mathworks.co.uk/products/matlab/ MATLAB®] is a high-level language and interactive environment for numerical computation, visualization, and programming. It can be used to analyze data, develop algorithms, and create models and applications.

The model

The model describes the absorption spectrum of the cell as a function of time and a given time invariant incident light spectrum. This is achieved by calculating the effect of the known light input on the individual light sensitive proteins. This information is then fed into a stochastic simulation of the biological pathways that computes how the population of sensors and pigments vary over time. The total absorption spectrum over time is given by the sum of the product of every protein's population and the corresponding absorption spectrum of the protein. In this section each of these steps in considered in the following subsections.

It will be helpful to read the theory page and the modelling software section of this page before reading further as an understanding of the biological pathways and software is important for understanding the

Sensor activation rate calculation

The rule for each sensor is that it is either on or off. In the on state it catalyses the phosphorylation of the primary intermediate and in the off state it does not. Activation by light either turns a sensor on or off depending on the sensor species; the red and blue light sensors are switched off when activated while the green light sensor is switched on when activated. The activation rate is the forward rate of the rule that governs the state of a sensor. Its counterpart, the deactivation rate is the backward rate of the same rule.

The activation rate is calculated as the integral over all space of the product of the sensor's frequency response with the frequency spectrum of the incident light.

This images shows the red light sensor (Cph8) being activated by incident light with a wavelength spectrum in the form of a sine wave. The red line shows the absorbtion spectrum of Cph8, the black line is the incident light spectrum and the filled in blue area is the area under the curve of the product between the two. This gives us the activation rate for the sensor which in this case is a very high 913.3643. Such a high rate will cause the sensors to be activated permenantly

This has two important implications. Firstly it means that the sensor will not be activated by light that has no overlap between its spectrum and the frequency response of the sensor. Secondly there will be a saturation point at high intensity light where increasing the intensity will not appreciably affect the system as the activation rate will dominate the deactivation rate. Causing the sensors to be effectively permanently activated. These characteristics are faithful to the light sensitive proteins we are using and that is why we are using this method.

Biological pathway simulation

This subsection will explain the kappa model. How it works, the justification for its use,

Absorption spectrum

Each protein species has an associated absorption spectrum. This multiplied by the population of the protein at a point in time gives the contribution of that protein to the total absorption spectrum of the cell at that moment. The sum of the contributions from each protein plus the absorption spectrum of the background cell gives the total absorption spectrum of the cell at that moment. This calculation repeated for every time step gives the total absorption of the cell over time.

The absorption spectrum over time is plotted as a surface of intensity as a function of time and frequency.

The absorption spectrum at a point in time tells us what colour and tone the cell will be at that moment. Over time it shows us how the mix of colours change as the image develops, how long it takes to reach an equilibrium, the moment of greatest contrast and many other important features.

It will also reveal how the expression of a gene is controlled by each light sensitive pathway. Something which has implications for future projects.

Results

This section will list in detail the results of our model

Preliminary results suggest that the mix of colours in an individual e.coli is very unpredicatable. But the average for a cohort of cells is more predicatable.

This images shows the populations of the cyan, magenta and yellow pigments over time. It is an alpha run of our stochastic model (KaSiM/Kappa).

This image shows the combined absorbtion spectra of the cyan, magenta and yellow pigments over time calculated from the data in the graph alhpa1 above. It is a beta run of our absorbtion spectrum plot (MATLAB).

This images show the combined absorbtion spectra of the cyan, magenta and yellow pigments over time calculated from the data in the graph alhpa1 above. This is the same surface as in the previous plot but viewed from another direction (90 degrees anticlockwise from the previous image)

Analysis

This section will contain what the data means

Conclusions

This section will contain our conclusions about the model

Future

This section will contain our opinions about the future of the model

Bibliography