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Curly'on - IGEM 2014 INSA-LYON

CURLI FORMATION

As functional amyloid fibers biosynthesis is still not totally understood, there aren't 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:

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;
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.

CurLy'On Simulator

Principle

The CurLy'On Simulator is based on the principles of Tim Hutton's artificial chemistry. In this way of modeling, every particle in 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. Their movements are brownian, and their interactions 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

\left\{ \begin{array}{lcl} a0+a0 &\rightarrow &a1.a0 \\ a1.a1 &\rightarrow &a1+a2\\ \end{array} \right.

where '+' signifies that the particles are not bound together, 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 types and states, it is possible to schematically reproduce various biological phenomena, which is what we did for CsgA polymerisation into a curli fiber.

The simulator

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

Mathematical model

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.

Smoluchowski model

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 :

W_{r+s}(t) = k_+c_r(t)c_s(t) - k_-c_{r+s}(t)

From there it is easy to deduct the expression of every way to combine two smaller fibers into one of length r, as well as the rate of formation of fibers longer than r from a fiber of r length. Since the case of soluble CsgA and of dimeric fibrils are particular since the term of smaller fibers combination cannot apply, we have for r>2:

\frac{dc_r(t)}{dt} = \frac{1}{2}\sum_{s=1}^{r_1}W_{s,r-s}(t) - \sum_{s=1}^{+\infty}W_{r,s}

As for r=2, the combination of two soluble CsgA gets its own aggregation constant as the polymerisation can obviously be a bit different than when an already polymerised CsgA is involved. Finally, the soluble CsgA concentration corresponds to the total variation that cannot be explained by the interaction of two existing fibers. Hence for the Smoluchowski model we get the system :

\left\{ \begin{array}{lcl} \frac{dc_r(t)}{dt} = \frac{1}{2}\sum_{s=1}^{r-1}W_{s,r-s}(t) - \sum_{s=1}^{+\infty}W_{r,s} &for & r>=3 \\ \frac{dc_2(t)}{dt} = k_nc_1(t)^2 - \sum_{s=1}^{+\infty}W_{2,s}\\ \frac{dc_1(t)}{dt} = - \sum_{j=2}^{+\infty}j\frac{dc_r(t)}{dt} \end{array} \right.

This model lead to a stabilised polymerised mass of CsgA as well as average length of the fibers, so we would rather advise to use this one over the Knowles model, though it's way heavier.
However, since both models were able to fit quite well to experimental data in the publication, we thought it may still be interesting to develop Knowles model as well.

Knowles model

The Knowles model is simpler than the Smoluchowski model as it only considers the addition of one soluble CsgA at a time for the fiber growth, and doesn't try to trace the sizes of the pieces from a fiber break up. Thus we have the system for this model :

\left\{ \begin{array}{ll} \frac{dc_r(t)}{dt} &= 2k_+c_1(t)[c_{r-1}(t)-c_r(t)] - k_-[(r-1)c_r(t) +\sum_{i=r+1}^{+\infty}c_i(t)]+ k_nc_1(t)^2*\delta_{r,2} \\ &for\ r>1\ and\ \delta_{r,2}=1\ only\ if\ r=2,\ and\ 0\ otherwise \\ \frac{dc_1(t)}{dt} &= - \sum_{j=2}^{+\infty}j\frac{dc_r(t)}{dt} \end{array} \right.

Please notice that here the 'k' have a different than for the smoluchowski model: they stand for the monomer addition rate, and for the rate constant of any type of breaking up of an aggregate into two pieces, regardless of the sizes of the fragments.

Though simpler, this model somehow leads to an equilibrium where the fibers are mostly dimeric, which isn't what can be observed on cells surface. That is the reason why we think the Smoluchowski model may be more relevant for the study of curli synthesis.

Team:INSA-Lyon/CurliSynthesis