Team:ULB-Brussels/Project/Mathematical

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

Revision as of 01:20, 22 October 2009