Imperial College/Courses/2010/Synthetic Biology/Computer Modelling Practicals/Design

Complementary Session: Introduction to the Design of Biological circuits

Objectives:

This page includes extra material for the course of synthetic biology. The material presented in this session is not part of your coursework. It is however, useful (very useful) for the rest of the course, especially the mini-iGEM project.

Computer-Assisted Design of synthetic biological pathways is in general a very complicated affair. You must, by now, be aware of some of the reasons for this.

• the behaviour depends on the parameters of the system
  • there may be many
  • we may not know them with enough accuracy - sometimes not at all
  • a small change in a parameter may lead to a totally different behaviour (bifurcation)
• initial conditions are also liable to have an influence (the arguments regarding the parameters mostly apply to the initial conditions too)

As you must have seen with the case of the repressilator, 3 genes are enough to generate a pathway with 'interesting' properties.

The situation is unfortunately worse. Even if there is a combination of parameters that

• some basic properties of the cell have a significant impact on the effective dynamics of pathways. Take for instance the growth rate:
  • it appears in the dilution term of proteins (easy to incorporate into the model)
  • but is also affects in a highly nonlinear way the gene copy number
  • it affects the concentration of free and bound RNAp and therefore the level of transcription etc..
• some modules in your system may be very hard to model (if at all possible)
  • for instance transport of molecules through a membrane and diffusion phenomena can be modelled but it becomes complicated fast
  • in a model, errors pile up so much so that after a while the predictive power of your model is negligible.
• your synthetic pathway may 'cross-talk' with natural pathways; since we are not able to model the whole metabolism of the cell this crosstalk effect can not be assessed.

Fortunately, designing simple pathways with predictable properties/functions is possible, even without the extensive use of software. This session aims at introducing to you the basic tools and techniques of design (without which no computer-assisted design is possible. Remember: in practice it gets very complicated, very fast...

Preliminary Simplifications

A great deal of the design work takes place on a sheet of paper. It is therefore important to develop an intuition of the functioning of the various elements and how they combine. To make our task simpler, it is customary to make a few (usually easily justifiable) assumptions. The following assumptions are the most common ones:

Initial Conditions

• for a constititutive gene, both protein and mRNA are at steady state
• for an inducible gene, the same assumption holds but the steady state depends on the concentration of inducer
  • write the general expression of the steady state of protein and mRNA for a activated gene
  • write the general expression of the steady state of protein and mRNA for a repressed gene

Time scales

• Binding reactions occur very fast, so fast we can reliably assume they are instant
• mRNA reaches its steady state after a few minutes
• Proteins reach their steady states in hours

Simplified Gene Expression Model

Finally it is customary to approximate the gene expression profile by the simplified model of practical 2, where it is assumed that mRNA is at steady state. It is also custom to overlook the evolution of mRNA unless it is stricly needed as for instance with riboswitches... In practice it is assumed that the production rate of proteins is constant.

In the case of a constitutive promoter or an inducible promoter for which the inducer does not enter any other biochemical pathway (this includes degradation) the gene expression profile simplifies into a simple ramp profile.

• Let us deal with the case of a constitutive promoter first; How do the parameters of the ramp model refer to parameters of the standard constitutive gene expression model?
• Same question for an activated gene
• Same question for a repressed gene

Although very simple, the ramp model is very powerful and has been widely used in software such as rovergene that check whether a proposed-network topology may meet certain requirements (for instance oscillations, steady state of protein 1 between two specified values etc...).

Model	The Ramp Approximation
A ramp is very simple piecewise function It can be shown that after some normalisation the ODE system can be written as: [math]\displaystyle{ \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} }[/math]

Negative Auto-Regulation

Model	The basic delay
It can be shown that after some normalisation the ODE system can be written as: [math]\displaystyle{ \begin{alignat}{1} \frac{d[mRNA]}{dt} & = \frac{a}{1+{[Protein]}^n} - [mRNA] \\ \frac{d[Protein]}{dt} & = b[mRNA] - b[Protein] \\ \end{alignat} }[/math]

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