# Imperial College/Courses/2010/Synthetic Biology/Computer Modelling Practicals/Practical 3

Practical 3: A Mystery Circuit

Objectives:

This is the third and last practical of your introduction into computer modelling for synthetic biology. The first practical was designed to give you an insight into:

• the notion of limiting step
• the law of mass action and its application
• the dymamics of possibly the most important class of reactions in biochemistry: enzymatic reactions

The second practical introduced you to the standard models for genetic expression:

• constitutive gene expression
• inducible gene expression

After these two practicals, you could be forgiven for thinking that genetic circuits always behave in a nice, predictable manner. As this final tutorial will show you, new behaviours emerge when genes interact with each other in a non-linear fashion. And these new behaviours do not require many genes to appear...

A Mystery Circuit

If you run the new version of Cell Designer, Download this File. If you run an older version of Cell Designer, Download that File.

Model CellDesigner Instructions

It can be shown that after some normalisation the ODE system can be written as:

\begin{alignat}{1} \frac{d[mRNA]_{i}}{dt} & = \frac{a}{1+{[Protein]_{j}}^n} - [mRNA]_{i} \\ \frac{d[Protein]_{i}}{dt} & = b[mRNA]_{i} - b[Protein]_{i} \\ \ i=1,2,3; \\ \ j=3,1,2; \\ \end{alignat}

The following questions constiture the last part of your coursework (Section E):

• Question 1 : Before studying the general properties of the mystery circuit, let us study a simplified version of it. The ODE system indeed contains a lot of symmetries that can be exploited and yield surprising properties. Let us consider a particular choice of initial conditions: the initial conditions of the mRNA terms are all equal and the initial conditions of the protein terms are also all equal.
• Q1.1 For Bioengineers only: It can be shown that for such a choice of initial conditions leads, the mRNA concentrations all remain equal (but vary with time) and so do the protein concentrations.Can you explain briefly why this is the case? Note: you do not have to submit a full mathematical proof.
• Q1.2: We now call X the mRNA concentration and Y the protein concentration. What system of ODE does X and Y satisfy? Show that this system corresponds to an auto-regulated sytem.
• Q1.3: What is the natural choice for the initial conditions of the system (justify)?
• Q1.4: Simulate the new simplified system for a=10, b=1000 and n=2 for these initial conditions and comment.
• Q1.5: Now simulate the new simplified system for a=200, b=5 and n=2 for these intial conditions and comment.
• Q1.6: For either of these choice of parameters, let the initial conditions of the simplified system vary and comment on your results.
• Q1.7: The simplified ODE system and corresponding auto-regulated behaviour is intersing from a theoretical point of view but has no experimental/practical relevance. Can you explain why?
• Question 2 : Now we return to the case a=10, b=1000 and n=2. The simplified system studied in the previous question is of course not representative of the overall behaviour of the mystery circuit.
• Q2.1: Let us assume that we can purify one of the proteins so that its initial condition is 1 and the other initial conditions are 0. Run the simulation. What happens?
• Q2.2: Describe the properties of the system (the simulation you have run is representative)
• Q2.3: Can you give a qualitative explanation for the behaviour of the system?
• Question 3 : Now we return to the case a=200, b=5 and n=2 and we seek to investigate its behaviour
• Q3.1: Again we assume that we can purify one of the proteins so that its initial condition is 1. Run the simulation again. What happens?
• Q3.2: Describe the properties of the system
• Q3.3: Can you give a qualitative explanation for the behaviour of the system?
• Question 4: The interesting property takes some time to emerge. From an experimental point of view, this is a problem ( the synthetic plasmid is liable to be rejected, the growth medium may run out of nutrients etc...). In iGEM 2007 the Imperial College team investigated playing on the initial conditions of their simple synthetic system so that their system had better proporties. We can do the same here.
• Let us assume that we can purify all the proteins so that you can set up their initial conditions. You want to determine the initial conditions that will make the property emerge fastest. How would you do this?
• Please note: You are not asked to run all the relevant simulations, just explain how you can solve such a problem.
• The question has at least two distinct components which must be adressed:
• A practical strategy to browse through the space of possible parameters
• A practical set of criteria to determine whether the property has emerged
• A few simulations or drawings are always welcome as they usually make explanations simpler...
• Question 5: The circuit is called the repressilator.Can you explain why?
• Hint: This circuit appeared was in an article published in 2000 by M.Elowitz. Track it if you need help!