Team:Bologna/Modeling

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Model-based analysis of the genetic Flip-Flop

The genetic Flip-Flop


Figure 1: Scheme of the genetic Flip-Flop


The molecular circuit in Figure 1 works as a Flip-Flop (SR Latch), which can switch between two different states according to the external stimuli (UVc and IPTG). LacI-ON represents the stable state in which LacI gene is active and LacI protein represses the TetR gene expression, with a positive feedback. Therefore, the LacI-ON state coincides with the TetR-OFF condition. On the contrary, the TetR-ON represents the state with the TetR gene active and the LacI gene silenced (LacI-OFF). Owing to the coexistence of two stable states (bistability), this circuit is capable of serving as a binary of memory. We denominated it Flip-Flop since it works as a SR Latch: LacI state is the Q.jpg output and TetR state is the Qneg.jpg output. Uvc is the set signal and IPTG is the reset signal. Indeed, IPTG stimulation inhibits LacI repressor, thus can cause the transition from the LacI-ON state to the TetR-ON, whereas UVc radiation inactiving LexA repressor through the SOS response (Friedberg et al., 1995) can cause the opposite transition from LacI-ON to TetR-ON.




Mathematical model


Model equations


The Flip-Flop circuit in Figure 1 can be modeled by the following equations:

Equazioni.jpg


Symbol definition is listed in Table 1.

Tabella simboli.jpg


A common motif in repressor proteins is the presence of a dimeric nucleotide-binding site with dimeric structure. In accordance to this general structure the cooperativity coefficients Mu.jpg were assumed equal to 2. The maximum velocity of repressor synthesis Alfa.jpg accounts for the strength of the unregulated promoter and RBS. The value of the affinity constant for the binding of repressor to the promoter strictly depends on the sequence of operator site (OS block).

Adimensional equations


The equations (1.1) and (1.2) can be written dimensionless:


Sistemaequazioni.jpg

Where:

Accozz.jpg


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Equibrium conditions


In the absence of stimuli, the adimensional concentrations of LacI (I.jpg) and TetR (R.jpg) at equilibrium are related by the equations:

Equa3new.jpg


To obtain these relations the UVc-dependent term in equation (1.4) was ignored (Appr.jpg ). This is justified by the high binding constant of LexA for its operator, and the consequent negligible contribution to the LacI synthesis. Equations (1.6) and (1.7) can have one or three solutions that represent the equilibrium conditions of the circuit. The solutions, i.e. the equilibrium conditions, can be graphically identified as the intersections between R.jpg and I.jpg nullclines (see Figure 2). The case of multiple equilibrium conditions (bistability case) is shown in Figure 2 panel b). TetR-ON and LacI-ON are stable equilibriums separated by the unstable one (saddle point). Due to the bistability the circuit can operate as a binary memory. The existence of a bistability condition depends on the value of Kr.jpg and Ki.jpg parameters. If Kr.jpg decrease (see Figure 2 pannel (a)) a saddle-node bifurcation can occur, TetR-ON equilibrium vanishes and remain only the stable equilibrium LacI-ON. The contrary occurs when Ki.jpg is decreased (Figure 2 pannel (c)). Thus bistability is guaranteed only for a limited range of Ki.jpg and Kr.jpg values.

Equicondition.jpg


Bifurcation analysis


Assuming that LacI-ON exists the corresponding equilibrium value of I.jpg is higher than 1 (see Figure 2), then it can be assumed that Iquadro.jpg and the equation (1.7) simplies to:

Equa1 8b.jpg


Substituting this expression in equation (1.6) one obtain:

Equa1 9b.jpg

Then:

Equa1 10b.jpg


To be real the solutions of equation (1.10) it is necessary that Krdis.jpg (see Figure 3). Under this condition the existence of the LacI-ON state is assured. When Kreq.jpg the system undergoes a saddle-node bifurcation (LacI-ON and saddle point go in collision) and the two equilibrium points vanish.

SaddleNodebifurcation.jpg


An analogous result can be obtained for the existence of the TetR-ON state. Thus, a sufficient condition for bistability is:

Equa1 11 12.jpg


Figure 4 shows the log-log plot of (1.11) and (1.12)

Rangeofbistability.jpg


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Procedure for Ki.jpg-index identification


The procedure will be described for LacI, analogous procedure can be applied to the TetR case. The value of Ki.jpg -index can be identified comparing the experimental responses of the open loop and closed loop circuits:

  • Open loop circuit
    Molecularcircuit1.jpg
  • Closed loop circuit
    Molecularcircuit2.jpg


The LacI concentration in the open loop circuit is given by:
Equa1 13.jpg

Thus the equilibrium condition is:
Equa1 14.jpg
The time derivative of LacI concentration in the close loop circuit follows:
Equa1 15.jpg
Which gives the equilibrium condition:
Equa1 16.jpg
That can be rewritten
Equa1 17.jpg
The affinity coefficient can consequently be derived from this expression
Equa1 18.jpg
Inserting the (1.14) and (1.18) in the Ki.jpg-index definition one obtain:
Equa1 19b.jpg
We assume that GFP is proportion to Imai.jpg, then
Equa1 20.jpg
We introduce the ratio H.jpg between the fluorescence in open loop and in closed loop:
Equa1 21.jpg
After measuring the ratio H.jpg it is possible to calculate Ki.jpg by the curve in Figure 6 and then it is possible to establish by Figure 4 the Kr.jpg range that guarantees bistability. In the presence of an experimentally characterized library of regulated promoter, the procedure can be adopted to design genetic Flip-Flop with desired behaviors.

Indexforlaci.jpg

Numerical simulation

Sim.jpg
Iptg.jpg

Bibliography

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