Sequestillator Model 1: A simple model

Parameters:

• At= total amount of NifA
• tmax= maximal transcription rate
• tl=translation rate
• r = conversion rate from La to Lb
• Kd= dissociation constant of NifL and Nifa (= kr/kf, where kf is the forward binding rate of NifL and NifA and kr is the rate of decomposition of the NifL/NifA complex)
• any d= degradation rate of that species

Professor Daniel Forger came up with this model for the Sequestillator. This simple system assumes the total amount of NifA in the system is constant, and considers three variable: NifL mRNA (mL), NifL (La), and a simple covalent modification of NifL (Lb). The quadratic mRNA production function comes from making rapid equilibrium assumptions (see box to the right of the equations. A= free unbound NifA, L = unbound NifL). From analyzing this small model, we were able to see that in order for oscillations to exist, there needed to be a tight binding between NifL and NifA (i.e., Kd is very small) and one-to-one titration of NifL and NifA (the mRNA production function = tmax*A/At, where A is free NifA in the system. A graph of the function vs. Lb is a oblique line).
We used the Indexilator to make some "relative" Ninfa index searches. Look at table 1 below:

Table 1

We see that if we increase our search range for the dissociation constant to 0 to .1, then we get a substantially smaller Ninfa index, illustrating the importance of having a tight binding between NifA and NifL.
Some "sequential" searches (i.e., instead of randomizing, we picked incremental values for each parameters: i.e., one parameter would be varied from 1 to 10 in increments of 1, and another from 1 to 5 in increments of .5. Given below is a three-dimensional cloud picture, of -log(Kd)vs At vs tmax.:

Picture 1

-log(Kd) was tested at values 1 to 10 in increments of 1 (for some reason, the -log(Kd)=10 values got cut off). Tmax was ranged from 0 to 5 in .2 increments, and r was ranged from 0 to 1 in .1 increments. Each dot in the plot corresponds to a set of parameters that gave oscillations. Notice that for -log(Kd)= 2 or 1, we see no oscillations at all - this yet another testament to necessity to have a small Kd in order for oscillations to occur.

There was another aspect of our model that we can also verify: the importance of a one-to-one NifA/NifL titration:

Table 2

We see in trial 2, by increasing the range of At to 0 to 15 (and hence increasing the chance of having larger values of At), we have a reduction in the Ninfa index, as the system is unbalanced (too much At around). We return to similar Trial 1 results when we also vary tmax between 0 to 15 in Trial 3.

FURTHER WORK THAT WILL BE PRESENTED DURING iGEM POSTER TALKS

• More simulations; search over larger regions of parameter space.
• A more continuous cloud picture - the one given here has too much empty space....