# Team:Amsterdam/project/diary/modelling

## August 2nd - Fusion protein activity

There are no $k_{cat}$ values available for the methyltransferase that we use, M.ScaI.

## July 31st - Stochastic model of the Lac operon

We are using a glucose-insensitive lac operon to control the expression of our Zinc-Finger-methyltransferase fusion protein (FP), by replacing the three naturally controlled genes (LacY, LacZ and LacA) with our fusion protein. The signal to be registered in our proof of principle, lactose, controls the expression this operon. The lac operon has been extensively studied in the past fifty years and is known to display some 'leaky' expression of the three genes. This means that some operon expression will occur independent of the presence of the natural inducer lactose. How should we control for this leaky expression? Will the small amounts of leakily expressed fusion protein cause the majority of bits in our Cellular Logbook cell to be methylated? This would mean we need to adapt our genetic circuits, as we want methylation only to occur in reaction to the signal to be measured. Or is the dreaded leaky expression too low and/or are the diffusion time and the methylation rate of the FP too low to form a serious threat?

To answer these questions, we thought it would be valuable to model the behaviour of the lac operon stochastically using Gillespie's Stochastic Simulation Algorithm. Only this way, the activity of the low copies of FP can be accurately accounted for. This will also allow us to account for natural variation occurring between individuals in the population of cells and to estimate probability distribution functions of characteristics of interest. Performing and analyzing stochastic simulations are more challenging than setting up systems of differential equations. We are currently bundling some tips on how to do this effectively. They have been gathered 'the hard way' in an trial-and-error like fashion, hopefully they can save you some trouble should you decide to do stochastic simulations as well. Expect them on this page shortly.

As a starting point, the stochastic model for the lac operon by Stamatakis & Mantzaris (2007) has been used along with parameter values they have gathered. We did a test for the basic leaky expression of the single cell-system by simulating using the following initial values: 5 for the operon number and 0 for the external IPTG (artifical lac operon activator). In a first simulation, with only three seconds of simulation time, we see uninduced increase of operon controlled mRNA followed by increase of the corresponding protein. [This is the transcription/translation of LacY in the model and we will probably have to adjust the rates at which the contributing reactions occur to our FP.] In a longer 200 second simulation of the system we see a less clear pattern. What causes the sudden outbreaks of protein amounts? The lac operon network topology contains no positive feedback loops, outside of signal induced permease and betagalactosidase transcription increase which are irrelevant here, since the inducer is set to 0. Analytically, the FP amounts should reach some kind of a steady state, determined by the mRNA production, translation & degradation rates and the protein degradation rate. This simulation indicates that wild deviations from the steady state/mean level of protein are possible however.

Now that we can gather a basic idea of the amounts of fusion protein in the cell, the next step would be to model the activity of the FP. Should we find that the leaky expression of the FP leads to too much activity, we still have various means of increasing the protein degradation rate. Alternatively we could lower the operon's leaky expression by overexpression of LacI. More on this to come!