Team:UANL Mty-Mexico/Modeling

Mathematical models that represent the dynamic behavior of biological systems are a quite prolific field of work and are pillar for Systems Biology. A number of deterministic and stochastic formalisms have been developed at different abstraction levels that range from the molecular to the population levels.

In principle, a model that is simple but that is good enough to describe and make predictions, with a degree of certainty, about the phenomenon under scrutiny, would be desirable.

Deterministic mathematical models that describe the behavior of genetic circuits and the interactions of the proteins they encode are usually built upon mass action kinetics theory.

Aside from the common objection that they are not suitable to describe systems that show a low number of particles, we believe that a deterministc model at a molecular level of these kind of systems and the degree of certainty with which they can be used for inter-system comparison or usage, do not outweigh the costs of the experimental determination of parameters.

Here we propose a model for the description and comparison of the behavior of the effect of RNA thermometers or RNATs on the expression of a reporter protein. The model is tested with relative fluorescence units data, which are not hard to obtain, and the model and its parameters should allow for inter-system comparisons, i.e., to compare the temperature-dependent gene regulation features of different RNATs; an extension that works with protein concentration units is also proposed, along with a potential application in metabolic engineering, and waits for experimental validation.

We present a model for the relation between time, temperature and the change in fluorescence (measured in Relative Fluorescent Units or RFUs) of an E. coli culture that harbors a genetic construction where a fluorescent protein is under control of a RNAT.

We took as reference for the RFUs the amount of fluorescence emited 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 600nm light (OD₆₀₀) after 8hr of growth at 37°C in LB medium.

In this way the amount of fluorescence emited 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).

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 8hr 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 and that can also be fitted to a logistic model, where the parameters englobe the effect of both, temperature and those factors not considered in the Shah and Gilchrist, (2010) study.

This function would be related to the probability of finding the RBS open at a given temperature and now that we're considering the effects of the remaining factors for the RNAT function, we can safely assume that RBS openness is related to translation initiation and therefore, to fluorescence emission in our system.

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 logistic function, so we are going to use the generalized Richards logistic function:

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

which was also used to fit temperature-dependent data in Von Fircks and Verwijst, (1993). Experimental data can be fitted to this equation using Matlab.

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 \(T_{i}\).

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Here we are going to show how we solved the differential equation #.

\( \frac{dF_{R}}{dt} = \delta (F_{Rst} - F_{R}) \)

First we choose to solve it through separation of variables. Thus we move every term with \(F_{R}\) to the left side of the equation, and everything else in the right side:

\( \frac{dF_{R}}{{(}F_{R} - F_{Rst}\large{)}} = - \delta {dt} \)

Then we proceed to integrate but sides of the equation:

\( \int \frac{dF_{R}}{{(}F_{R} - F_{Rst}\large{)}} = - \int \delta {dt} \)

The left integral has the solution of a natural logarithm, while the right side, as \delta is constant for constant temperature, has a simple solution:

\( \ln \left |(F_{Rst} - F_{R})\right | = - \delta {t} + C \)

Then we apply the exponential function in both sides of the equation:

\( F_{R} - F_{Rst} = e^{- \delta {t} + C} \)

Rearranging the equation, through laws of exponents, we can change the right side like this:

\( F_{R} - F_{Rst} = e^{C} \ e^{- \delta {t}} \)

The exponential of \(C\) can be treated as another constant, that we will call \(K\):

\( F_{R} - F_{Rst} = K \ e^{- \delta {t}} \)

There by:

\begin{equation} \ F_{R}= F_{Rst} + K \ e^{- \delta {t}} \end{equation}