|
|
| Line 1: |
Line 1: |
| - | {{template projet}}
| |
| | | | |
| - |
| |
| - | ==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}
| |
"
