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As functional amyloid fibers biosynthesis is still not totally understood, there are not many models other than descriptive sketches that represent the curli formation. From these observations we decided to gather the information we could and build models from them as incomplete as they may be, in order to provide future teams working on engineered CsgA with a basis to start from.
We therefore were able to build up two models:

1. the CurLy'On Simulator, a computed simulation of CsgA secretion and polymerisation that, provided with the right parameters, could make for a good alternative to a mathematical model for a protein kinetics study;
2. the implementation of the only two mathematical models we could find in the litterature that seemed relevant (with biological justification) in describing in vitro CsgA polymerisation in the C language in a fashion that can be given to a numerical solver, as these models require a heavy calculation power.

The CurLy'On Simulator is based on the principles of Tim Hutton's artificial chemistry. In this way of modeling, every particle of the environment, be it a protein, an inorganic molecule or simply an atom, is represented as a spherical particle, characterized by a radius, a type (that we will represent by a letter) that can never change and a state (represented by a number) that may change when encountering other particles. These particles abide by a set of basic "rules" provided by the user. These rules specify if two particles that meet may bond (or unbond if they are already tied together) according to both their type and state.

For instance let's say that we have an environment containing only particles of the 'a' type in state 0, and the set of rules where '+' signifies that the particles are not bound together when they met, while '.' means that the two particles are bound together. Then what may happen is something like this : super schéma de la mort tuante
Then, by creating lots of rules like that, involving many particles with different type and state, it is possible to schematically reproduce various biological phenomena, which is what we did for CsgA polymerisation into a curli fiber.

vidéo du CurLy'On Simulator Colours meaning :

• yellow is for the cell membrane;
• red is for the CsgG pores;
• green is for the nickel ions;
• lightblue is for soluble CsgA;
• pink is for polymerised CsgA ;
• blue is for CsgA with the His1-tag ;
• purple is for CsgA with the His2-tag.

Since for this model most of the work was coding in C++ language, we won't explain the whole process behind the program as it wouldn't bring any enlightment about the model here. The program also has a few additionnal features :

• before launching it, you can specify the initial composition of the environment : what particles are present, where, and which ones are linked together;
• you can add a flow of particles from above (modeling the arrival of nickel ions for instance), or from below (for the production rate of your protein for instance) ;
• it can be paused ;

However, the work on this program is still in progress, as there are many more features we would like to add for it to be a good starting point for the work of any future team that would want to use our work.lien vers la to-do list du bas

We also found a publication lien publi by John S. Schreck and Jian-Min Yuan where two mathematical models for in vitro soluble CsgA polymerisation were treated. Seeing how such models were scarce, we wanted to reproduce their results and so that future teams working on this kind of issue may use our work and integrate it in a more complex differential equations system involving gene expression and protein secretion for instance.

The two studied models will be referred to as the Smoluchowski model and the Knowles model. Though the expression may differ, both models' main idea is to follow the evolution throughout time of the concentrations c_r of the fibers of length r (containing r polymerised CsgA), where r goes from one (soluble CsgA) to a maximum length fixed by the user since a numerical resolution cannot go to the infinity.
Moreover, after a few mails with the authors, we were told that in order to get the same results as them, we should use a solver with a precision on par with the Runge-Kutta-Fehlberg fourth-fifth order runge-kutta method, as well as consider the fibers to be able to at least grow up to 30 000 in length. This means that for both models we had to solve over thirty thousands differential equations at a time, which is as you can guess, extremely ressources-consuming for any computer.

The Smoluchowski model is quite heavy as it takes into consideration every possible way to form a fiber of length r, by combination of two smaller fibers or by the breaking of a bigger one. First, the expression of the mass flux from aggregate concentrations c_r(t) and c_s(t) going to c_{r+s}(t) can be written as :

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