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Team:ULB-Brussels/Project/Mathematical
From 2009.igem.org
Contents |
Introduction
Synthetic biology is a recent development in biology which aims at producing useful material via biological agents. In this context, a biological system can be seen as a complex network composed of different functional parts ( [14]). Mathematical tools allow one to make prediction on the dynamical behaviour of a given bio-system. We aim at studying ft the system which has been built in the second section of this report. We already know from our experiment that our system is able to produce glue in the presence of IPTG inductor. We now address the following question: what is the influence of the different parameters on the global dynamics (degradation rate, production rate, level of initial quantity for the C2P22 repressor,...) with and without IPTG. In order to achieve this task, we will consider three different models: the first one shows the basal property we are looking for: a system able to produce glue. The two next models can be seen as two different steps of improvement. The purpose of these models is to improve the experimental control of the glue production. Parameters values result from the literature and previous iGEM team wiki’s. We know these remain qualitative and that lab work should be carried out to specify them more precisely.
- Hill function for activator
EQUATION1
- Hill function for repressor
EQUATION2
In these expressions, [x] is the concentration of activated genes, p is called the Hill coefficient, k is the activation coefficient and α is the maximum expression level of the promotor.
For each block of the system, we can obtain a dynamical equation by considering its interactions with the other blocks of the system. For each block we can build an equation of the following form:
EQUATION3
Where, [x]-i is the concentration of the gene i, R(Ha([x]);Hp([x])) is the regulating function which is a combination of Hill functions. The second term of the right side is the destruction term, where γi is the maximum destruction rate of the gene i.
We must bear in mind, however that the robustness of a given operational regime with respect to external perturbations strongly depends on the value of the Hill coefficients. [16] In particular, the robustness is expected to increase with the value of the hill coefficient. The cooperativity behaviour is also a function of the Hill coefficient. For these reasons we will consider a situation for which p = 2.
A first simplified model
In this first step, we present a simplified model of our system. We make the following assumptions: we consider that the LuxR+HSL complex is formed quickly at the beginning of the dynamics. This assumption, allows us to modelize the quorum sensing system by considering the complex LuxR+HSL only. The simplified schema is shown on figure 14(b): the effect of the LuxI and LuxR is represented by the autoregulation arrow on the box of the complex LuxR+HSL. We also neglect the effect of the block parE in this first approach. For the system shown in figure 14(b) we can obtain the following equations (writing here [L] for the LuxR+HSL concentration:
FIGURE 14
- Equation for the c1 repressor block (designed by c1 in the equation):
EQUATION4
In this equation, the parameter Β has the following explicit form:
EQUATION5
In our case, [LacI] can be considered as a constant, then we have Β = Β([IPTG]). In this first model we consider only the situation with IPTG inside the system: Β ≠ 0.
- Equation for the c2 repressor block (designed by c2 in the equation):
EQUATION 6
- Equation for LuxR+HSL block (designed by L in the equation):
EQUATION 7
- Equation for the glue production (designed by Gl in the equation, or by hfsGH in the text)
EQUATION 8
We choose a set of values for the different parameters:
EQUATION 9
Stationary state
The first step of our analysis is the study of the stationary point. In order to do that we consider the following algebraic system:
SYSTEM EQUATION
By solving this system of equations for the previous set of parameters we can find different sets of stationary states. So we have (considering only the states with real values):
SYSTEM EQUATION
Linear stability analysis
EQUATION 10
ω is the growth rate of the perturbation
SYSTEM EQUATION
is the vector representing the perturbation of each concentration. Linearizing the equations, we can rewrite them in the following form [17]:
EQUATION 11
where
EQUATION
and
EQUATION 12
is the Jacobian matrix. Taking into account the form of the perturbation (26), we can write the equation (11) as follows:
EQUATION 13
and the eigenvalues ω satisfy the characteristic equation:
EQUATION 14
Given the roots ω of the above equation, we can easily deduce the steady state. From (26) it is easy to see that a given state is unstable if any of the !i has a positive real part. In our case we obtain the following expression for det |Lij - ωδij |:
EQUATION 15
Equation (14) is a polynomial of 4th order and its first root is of the form:
EQUATION 16
The others are the roots of a polynomial expression of the third order. We could obtain these roots analytically but their expressions are too complicated to be useful. So, we solve numerically the equation for the previous sets of parameters and we put into the equation (15) the corresponding values for the different sets of stationary states. We have:
- set one = {ω1,2 = -0.100036 ± 0.0002iω3 = -0.0994, ω4 = -0:01}
- set two = {ω1 = -0.1, ω2 = -0.1ω3 = -0.1ω -4 = -0.01}
- set three = {ω1,2 = -0.1014 ± 0.0240i,; &omega 3 = -0.01&omega 4 = 0.1023}
