Curve fitting

Based on laboratory experiments was possible to plotting fluorescence standard curve as a function of GFP concentration (Green Fluorescent Protein). This adjustment was essential to facilitated the process of obtaining the GFP molecules number just using the fluorescence values obtained in lab.

The curve that better fitted the fluorescence standard curve points was the straight: (1) $$ F_{luo}=a[GFP]+b $$

Where the parameters a and b are constants, [GFP] is GFP concentrations given by nM and F_{luo} is fluorescence in u.a. (IMAGEM 1)

The graph 1 fitting constants results and their respective uncertainties is described in table 1 on appendix (link). The result was: (2) $$ F_{luo}=113[GFP]+94$$

For further analyses we will transform this equation for GFP molecules number, resulting in the following equation: (3) $$ F_{luo}=1.85 x 10^⁻13[GFP]+117 $$

Using the lab results (link) of uspA and J23101, we fitted the GFP molecules number in function of PEG (polyethylene glycol) concentration with the exponential function: $$ y=y_0+Ae^[R_0x] $$ and made the graph 2. (IMAGEM 2)

This curves have both the coefficient of determination value (/r^2/) = 0.97,it is the number that indicates how well data fit a statistical model , in other words, almost 97% of the experimental dots are described by this fit. Moreover, the constants uncertainties are relatively low for both adjustment as can be seen in the tables 2 and 3 on appendix (link).

The exponential fitting curve for uspA promoter: (4)$$ $$

And the exponential fitting curve for J23101 promoter: (5)$$ $$

However, we need to associate this values not with PEG concentrations, but with the osmotic pressure. So was used the equations described in the works of Braccini et al. (1996) to correlate osmotic pressure as a function of PEG concentration and temperature in the moment of the experiment. Where \psi_osm is osmotic pressure (bar), C represent PEG 6000 concentration (g/L) and T is environmental temperature given in degrees Celsius. (6)$$ $$

This way was possible to create a new graph that associate GFP and PEG values of graph 2 with their respective values in osmotic pressure at 37ºC calculated by equation (6). (IMAGEM 3)

Using the same exponential fitting applied in the PEG dates we reached coefficient of determination values of 0.92 for uspA and about 0.96 for J23101, according to the tables 4 and 5 on appendix(link). The new fitting equation for uspA promoter: (7)$$ $$

And the for J23101 promoter: (8)$$ $$

Simulations and Results

For reach one of our main goals, that is model Zn concentration in cellular environment as a function of time, we have to find one function that describes production of smtA and zur, proteins that associate with zinc, versus time. Thus we will consider that the smtA production equals of GFP produciton by uspA promoter, what means that if we modeled GFP production versus time, will be possible to use the same curve to describe smtA.

As we have not obtained specific experimental points to adjust an empiric curve, we will consider that the GFP molecules number in function of time is rule by a logistic function, because is the behavior that appears in majority of studies.

Logistic function: (9)$$ $$