Fixed Points

Fixed points are the points where and equation's rate of change, or slope, is zero. there are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value towards which the function converges, whereas an unstable equilibrium is a value away from which the function will diverge. A saddle-node is here the function both converges towards and diverges away from (Wikipedia).

Why are fixed points important?

fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analysed. The numerical method for directly calculating fixed points is to find the roots of the system of equations using built-in root finding function such as fzero in MATLAB or similar functions in Maple and C. The indirect method is to plot and find the intersections of these nullclines, which represent fixed points.

Results

We were able to find the equilibrium points of the system analytically (for Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analysing the probability that our system would exhibit istable behaviour.

Nullclines

In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).

Why are nullclines important?

the intersections of the cullclines gives the equilibrium points of the system of differential equations. From graphs of the sullclines, it is obvious whether or not a system will be bistable. If the nullclines only intersect in one place the sestem is not viable. If there are more than two interesctions, the middle equilibrium point is often an unstable saddle point.

How are the nullclines calculated?

Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.

Simulations involved

nullclines1.png	Figure 1. This figure is a plot of how much galactose (in green) and methionine (in blue) must be added to produce GFP and CFP. In this plot, both Hill coefficients are equal to two. The nullclines cross over at three equilibrium points, where the middle is a saddle-node fixed point. This is an 'ideal' bistability plot.
nullclines2.png	Figure 2. This figure again shows the required amounts of galactose and methionine to produce GFP and CFP respectively. In this plot, both Hill cefficients are equal to one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were equal to one.
Figure 3. This figure again shows the concentrations of galactose and methionine to produce GFP and CFP. In this graph, we experimented with setting the initial values of galactose and methionine and seeing how the intersection points are offset. The blue line represents the GFP = CFP line. The further from this line the equilibrium points are, the harder it will be to switch between stable states.	nullclines3.png

Results

We focused our efforts on plotting the mullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combination would most likely produce stable bistability. We cound that all combinations gave bistability, except when both Hill coefficents were equal to one. the optimal Hill coefficient combination occurred when both Hill coefficients were equal to two. We informed the biology team of this, and that is we wanted our system to successfully switch, we could not have both Hill coefficents equal to one.

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