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The basics

Synthetic biology relies heavily on genetic constructs designed to perform a specific task. In order to predict the behaviour of such systems, we need to be able to model it. Knowing in advance how well or how poorly a construct will work can help us during the genetic design phase or the preparation of our experiments. We need to describe the future reactions of our system by creating a mathematical model of it. The parameters of the model are then fitted to the experimental results.

The bible for synthetic biology modeling is Uri Alon's book An Introduction to Systems Biology: Design Principles of Biological Circuits [1]. Most of our models are based on his approach to biological circuits. Below is the explanation of the basic structure behind our simulations.

The gene geneX is responsible for the production of the corresponding protein X. The promoter pX controlling the expression geneX can be:

• A constitutive promoter, geneX is activated whatever the conditions
• Positively regulated, geneX is more active when an inducer is present
• Negatively regulated, geneX is less active when an repressor is present

Constitutive promoter

Positive autoregulation

Negative autoregulation

The inducer or repressor can be another protein or even the product itself. In the latter case, the gene is auto-regulated, wether positively or negatively. To begin, let's assume pX is a constitutive promoter. We will make another simplification by not taking into account the translation step and assuming that the gene "directly" produces protein X.

We will know take a look at the parameters involved in modeling this network.

• [X] is the concentration of protein X
• is the expression rate of protein X (molecule.s^-1). It mainly depends on the constitutive promoter.
• is the dilution rate, due to cell division (s^-1)
• is the degradation rate of protein X (s^-1)

The equation and solution modeling the behaviour of this system are the following:

Constitutive expression of protein X over time

Now, let's assume that pX is auto-regulated, either positively or negatively. We need to introduce a few new parameters.

• K is the dissociation constant, representing the binding of a inducer/repressor to the promoter
• n is the Hill coefficient of the function
• is the maximum production rate of our gene (molecule.s^-1)
• now represents the basal expression ("leaking") of the gene (molecule.s^-1)

The constants used in our examples below are arbitrary and chosen to amplify the behaviours expected. The equations and solutions are now:

Positive autoregulation equation and behaviour

Negative autoregulation equation and behaviour

You will note that the regulation is modeled as a Hill function. This type of function is used to represent most of the regulation that takes place in a genetic network.

Of course most genetic networks are more complex than simply auto-regulated nodes. The product of one gene can regulate the activation of another which in turn inhibits a third, etc. By coupling this kind of equations together, we achieved modeling most of our genetic networks pretty easily.