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

Timers are in theory very easy to design (in practice they are fiendishly hard to build, you will see why). They also are ideal toy-systems to learn the basics of design in biological engineering. The simplest design rests on the following 3 ideas

• 1) Switching on a gene is achieved when the concentration of the gene activator crosses the activation threshold
• 2) With an inducible gene, the rate of protein production depends on the concentration of inductor
• 3) consequently the time it takes for that protein to go over a defined threshold will depend on the concentration of inductor (if the gene is strong enough)

The simplest timer design therefore consists of two activated genes in cascade. We note A the activator of gene 1. Gene 1 synthesizes protein 1 (P_1) which activates Gene 2, which itself produces protein (P_2).

Model	The simplest timer
The synthetic pathway can be modelled as: [math]\displaystyle{ \begin{alignat}{1} \frac{d[P_1]}{dt} & = K_1\frac{{[A]}^n}{K_m1+{[A]}^n} - d_1[P_1] \\ \frac{d[P_2]}{dt} & = K_2*\frac{{[P_1]}^n}{K_m2+{[P_1]}^n} - d_2[P_2] \\ \end{alignat} }[/math]

• Preliminary analysis
  • sketch the expected behaviour of the system (for both protein 1 and 2).
  • Prove that the first gene has to be strong enough for the second gene to be switched on
  • With the ramp model, estimate how long it will take before the production of the second protein is switched on
• Now let's run simulations for the following parameters and comment the results
  • Activation parameters
    • For promoter 1, K_m=100
    • for promoter 2, K_m = 500
    • all the hill functions are infinitely sharp
    • initial concentration of inducer A_0=250;
  • both genes are non-leaky
  • Protein 1: production rate s_1=100 ; degradation rate = 0.1
  • Protein 2: production rate s_2=50 ; degradation rate = 0.5
• Unfortunately genes are leaky and hill exponents are not infinite...
  • Redo the simulations for n_1=1 and n_2=2;
  • Swap the hill exponents and do the simulations - comment your results
  • the first gene is now leaky witha leakiness coefficient of 10% -n_1=1 and n_2=2 - comment your results

A Basic Biosensor

Finally, we want to produce a protein for a limited amount of time. A constitutive gene is always on, so it is not a suitable solution.

• A possible solution is to activate a gene. The gene will then be switched off once concentration of activator to drop below the activation threshold. But this is not very practical (explain why)
• Fortunately a simple solution with two genes exists.

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]

• Preliminary analysis
  • sketch the expected behaviour of the system (for both protein 1 and 2, and molecule A).
  • Prove that the first gene has to be strong enough for the second gene to be switched off
  • With the ramp model, estimate how long it will take before the production of the second protein is switched on
• Now let's run simulations for the following parameters and comment the results
  • Activation parameters
    • For promoter 1, K_m=100
    • for promoter 2, K_m = 500
    • all the hill functions are infinitely sharp
    • initial concentration of inducer A_0=250;
  • both genes are non-leaky
  • Protein 1: production rate s_1=100 ; degradation rate = 0.1
  • Protein 2: production rate s_2=50 ; degradation rate = 0.5
• Unfortunately genes are leaky and hill exponents are not infinite...
  • Redo the simulations for n_1=1 and n_2=2;
  • Swap the hill exponents and do the simulations - comment your results
  • the first gene is now leaky witha leakiness coefficient of 10% -n_1=1 and n_2=2 - comment your results

A bump generator

Finally, we want to produce a protein for a limited amount of time. A constitutive gene is always on, so it is not a suitable solution.

• A possible solution is to activate a gene. The gene will then be switched off once concentration of activator to drop below the activation threshold. But this is not very practical (explain why)
• Fortunately a simple solution with two genes exists.

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]

• Preliminary analysis
  • sketch the expected behaviour of the system (for both protein 1 and 2, and molecule A).
  • Prove that the first gene has to be strong enough for the second gene to be switched off
  • With the ramp model, estimate how long it will take before the production of the second protein is switched on
• Now let's run simulations for the following parameters and comment the results
  • Activation parameters
    • For promoter 1, K_m=100
    • for promoter 2, K_m = 500
    • all the hill functions are infinitely sharp
    • initial concentration of inducer A_0=250;
  • both genes are non-leaky
  • Protein 1: production rate s_1=100 ; degradation rate = 0.1
  • Protein 2: production rate s_2=50 ; degradation rate = 0.5
• Unfortunately genes are leaky and hill exponents are not infinite...
  • Redo the simulations for n_1=1 and n_2=2;
  • Swap the hill exponents and do the simulations - comment your results
  • the first gene is now leaky witha leakiness coefficient of 10% -n_1=1 and n_2=2 - comment your results