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

Complementary Session: Introduction to the Design of Biological circuits

Foreword

Design of a synthetic biological pathway (whether it is computer-assisted or not) is in general a very complicated affair. Typically, a list of specifications (and tolerances) has been drawn for the synthetic pathway. Based on pre-existing designs (found in nature or not) and their inspiration, a biological designer will then propose a pathway (topology+genes) that may meet this specifications. Computer simulations are very valuable tools to check and if need be modify the design of a synthetic pathway. They are not however, without their problems and it is crucial that synthetic biologists are aware of the practical limitations of computer modelling.

The first set of limitations concern the verification phase. you must, by now, be aware of the complexity of biological pathways and how fast unpredictable behaviours may emerge. In the case of the repressilator, 3 genes are enough to generate a pathway with 'interesting' properties. More generally:

• the behaviour depends on the (potentially many) parameters of the system
  • 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)

Browsing the space of admissible parameters to check whether a proposed design meet some initial specifications therefore becomes - very quickly- a very difficult computational problem.

The second set of limitations is far worse unfortunately . Even if there is a subspace of parameters for which the synthetic pathway seems to meet your initial specifications, your model and simulations may mislead you. It may indeed be too simple or not have any predictive power. Possible reasons include:

• 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.

Now, all is not lost! Designing simple pathways with predictable properties/functions is indeed 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. But please 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...).

The Ramp Approximation

Ideal Induction As you must know by now, the relation between trancription rate and inductor concentration is modelled with a sigmoidal Hill function. to simplify things it is assumed it is ideal that is of infinite sharpness. Induction therefore depends only on the switch value Km.

A Basic Timer

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]

• With the ramp model, estimate how long it will take