Team:Paris/Modeling/FromMolReactToNLOde

From 2008.igem.org

Revision as of 03:04, 27 October 2008 by Hugo (Talk | contribs)

From Molecular Reactions to Non-Linear ODEs


Return to "Workflow on an Example"

Contents

Idea and Assumptions

We must here propose a Mathematical Modeling of the elementary molecular reactions. The idea of the Characterization Approach is that this modelization must both accounts for every small steps of the system and allow the experimental characterizations.

Therefore, the following equations do not describe properly what really happens in the cells. For exemple, we know that the transcription factor FlhD-FlhC is actually an hexamere FlhD4C2. But, as we will surely not get access to the dissociation constant of the hexamerisation, we just treat it as an abstract inducer protein "FlhDC", with an order (n) in its complexation caracterization probably around 2*4 = 8 (but perhaps completly different ! ; the estimation of the error by the parameter finder program would tell us if it stays consistent).

Mathematical Interpretation of the Molecular Reactions

First, we consider the complexation phenomenon. We show, under the quasi steady-state hypothesis how it leads to Non-Linear interactions like "Hill functions".

↓ precisions ↑


Complexations


Quasi Steady-State Hypothesis

The problem is the following : for given values of molecules A and B at initial state, what is the final state if A and B can bind together ? With the answer at this question, we would be able to treat separately each step of our system as an "independant module", taking as input values of Ai and Bi and giving as output values of Aeq, Beq and Complexeq. By doing so, we are doing the hypothesis of quasi steady-state because it implies that the complexations reach immediatly their equilibrium. The kinetical behaviour of the final model is nevertheless taking into account, because the reactions of productions are considered slower, and the characterization would give us their speed, regards to the ratio with the (known) value of the degradation γ

Complexation Reaction

The first hypothesis is that a complexation reaction is fully determined by the following :

Reacthill.jpg

and that the rates k+ et k- stay constant under all conditions. Then, the second hypothesis is that these equations are (kinetically speaking) elementary :

Kinhill.jpg
and at steady-state :
Sshill.jpg

Thus, the solution is (at least if we take the smallest real root of the equation, it is useless to demonstrate the unicity, or even the existence, of such a solution) :

Eqnss.jpg

Equilibrium of a Complex

Now, if we imagine a given amount of Ai and Bi, that are calculated as their equilibrium without taking acount of their complexation (but, for instance, of other interactions, productions and disappearance), and that the produced complex C disappears along time with a degradation rate γ, we get :

GammaC.jpg

so that the equilibrium gives : EqGammaC.jpg with Keff.jpg

We will obviously be looking for the parameter Keff, which avoid us to evaluate the unavailable k+

Hill Functions

The previous system of complexation applied in particular to the association of the Promoters (P) and its Transcription Factor (TF).

Because the promoters on a "low copy plasmid" exists in the cell in ten exemplaries, in contrary to a protein, which, as long as it is produced (even weakly) exists in thousands of exemplaries, we assume can the quantities of TF and P are different by several orders of magnitude. Then, with the previous notations, if A, B and C stands respectively for TF, P and the complex TF><P, we will get

Approxhill.jpg

that we can easily solve with :

Reshill.jpg

Depending to the order n (also called cooperativity, because it correponds to the possibilities of the transcription factor to binds in a group on the promoter), this function is a sigmoïd, known as the Hill function. The parameter K , called activation constant, is often replaced in the previous expression by the following notation

ChangeK.jpg

It simplifies the manipulations of the expression ; we can notice that K represents now the amount of TF needed to bind half of the total P in the cell.

Hilldef.jpg


< Back - to "From Molecular Reactions to Non-Linear ODEs"|

Then, we apply these results to the Molecular Reactions of our system. That gives all the theoretical calculus of the complexations, and finally, with the previous assumptions, we get the equations of the full system of ODEs.