Team:UANL Mty-Mexico/Modeling

We took as reference for the RFUs the amount of fluorescence emitted by an E. coli K12 culture transformed with a constitutively expressed part BBa_E1010 (for RFP expression) or BBa_E0040 (for GFP expression) per unit of Optical Density at 600 nm light (OD600) after 8hr of growth at 37°C in LB medium.

In this way the amount of fluorescence emitted by our culture was calculated as follows:

\begin{equation} \large F_{R} = \frac{F_{sample}}{F_{standard}} \end{equation}

where F_sample is the OD₆₀₀-normalized fluorescence emited by a sample, while F_standard is the OD₆₀₀-normalized fluorescence measurement for the corresponding standard culture (again, BBa_E1010 for RFP and BBa_E0040 for GFP).

Steady State: Referrers to the condition or conditions of a physical system or process that does not endure a change over time, or in which any given change is continually balanced by another, such as the stable condition of a system in equilibrium.

Parameter estimation: is the process or method used when trying to calculate mathematical model parameters based on experimental data sets through numerical methods. These data sets can be the result of steady states of independent events or time course experiments. Although there are numerical methods to do this, more complex algorithms with great accuracy are already incorporated in most of the new mathematical software, such as COPASI and MATLAB.

More deeply, Estimation Theory is a branch of Statistics and Signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component (deviation or unknown tendency). The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements, trying to adjust these parameters to fit the model to the experimental data.

The best numerical method for the estimation of parameters through COPASI resulted to be the Evolutionary Programming.

Let's consider the change of the relative fluorescence of a sample with respect to time. This change can be described by the following differential equation:

\begin{equation} \frac{dF_{R}}{dt} = \alpha - \delta F_{R} \end{equation}

Here, \(\alpha\) is the production rate (in RFUs per unit time or \(RFU min^{-1}\)) and \(\delta\) is the degradation rate (in reciprocal unit time units or \(min^{-1}\)).

This is the familiar production and degradation model whose steady state can be expressed as follows:

\begin{equation} \ F_{Rst} = \frac{\alpha}{\delta} \end{equation}

Now, we assume that after growting for 8 hours at a given temperature, a culture would have reached the steady state OD₆₀₀-normalized fluorescence expression. Taking this assumption into account, we can plot the value of \(F_{Rst}\) after growth at different temperatures and find a function that we will call \(f(T)\). Note that we use capital \(T\) to distinguish temperature from time, for which we use lower case \(t\).

If we merge this assumption to equation 3, we have:

\begin{equation} \ F_{Rst} = f(T) \end{equation}

In Shah and Gilchrist, (2010), it was found that the probability of openness of a ribosome binding site (RBS) of an mRNA with respect to temperature, fits well into a logistic equation. However, the authors did not find significant differences in the behaviour of known RNATs and non-RNAT elements and admit that RBS openness cannot be assumed to be directly correlated to translational activity. Therefore, their RBS-melting probability equation would not be recommendable to be used directly in gene expression models for RNATs.

However, because other factors may be involved in RNAT-mediated gene regulation, such as the effect of ribosome binding and other structural features, we can still assume there is an unknown function that takes into account these effects. We initially tried with a logistic model, but our data seems to be more in concordance with a Gaussian distribution.

The most accessible measure of fluorescence emission that we can get to find a relation with temperature, is the steady state fluorescence emission of our system at a given temperature, which we already called \(F_{Rst}\).

The form that \(f(T)\) from equation 4 we expect to find is a Gaussian function:

\begin{equation} \ f(T) = a\large{(}1 - e^{b-cT}\large{)}^{-1/d} \end{equation}

After merging equation 3 and 4 and rearranging elements, we get:

\begin{equation} \frac{dF_{R}}{dt} = \delta\large{(}a\large{(}1 - e^{b-cT}\large{)}^{-1/d} - F_{R}\large{)} \end{equation}

Equation 6 is a dynamic model which we now proceed to describe and develop.

This equation is simplified when \(F_{st}\) is known. It can be used to construct a model in COPASI [Hoops, S., et al., (2006)] and experimental data can be fitted to the equation and empiric estimations for \(\delta\) can be obtained. This estimations can be used to compare the behavior of different RNATs, whenever they are tested under the same standard conditions that we have described through out the text (i.e., using the same standards to normalize OD₆₀₀ data and growing their culture in the same conditions).

However, we still need to take into account that as temperature increases, the overall degradation rate of proteins also does. In consequence, we expect to see a rise in the degradation rate in our dynamic model as temperature increases. If we plot the value of \(\delta\) obtained at different temperatures, we will end with a function \(\delta(T)\) that shows an unknown form:

\begin{equation} \frac{dF_{R}}{dt} = \delta(T) \large{(}f(T) - F_{R}\large{)} \end{equation}

Note that after fitting the temperature series of \(F_{Rst}\) to equation 4, the \(f(T)\) term in equation 5 will have a specific value. In this way, equation 5 can be simplified to:

\begin{equation} \ \frac{dF_{Rst}}{dt} = \delta(T_{i})\large{(}F_{Rsti} - F_{R}\large{)} \end{equation}

where \(T_{i}\) is the fixed temperature at which the \(F_{R}\) times series where taken and \(F_{Rsti}\) is the steady state value of \(F_{R}\) at \(T_{i}\).