Team:UANL Mty-Mexico/Modeling

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Math Model

Introduction

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.

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 expect the temperature to be the main factor involved in the regulation of the reporter gene. In an ideal situation, where temperature is changed over time and where cells are left to reach the maximum fluorescence at each temperature, we expect to first have an off state (basal fluorescence) at low temperatures followed by an increase in the relative fluorescence until an optimum is reached at a certain temperature. Then, we expect to observe a decline until an off state is reached again. Time will have almost the same effect as it does in any other giving biological phenomena involving gene expression.

The relative fluorescence units (RFUs) were calculated in relation to 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 17hr of growth at 37°C in LB medium.

All fluorescence measurements were normalised to the OD600 nm of their corresponding culture.

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 Fsample is the OD600nm-normalised fluorescence emitted by a sample, while Fstandard is the OD600-normalised fluorescence measurement for the corresponding standard culture (again, BBa_E1010 for RFP and BBa_E0040 for GFP).

In our experiments, we had the same unchanging global conditions (temperature, initial conditions, and medium used to grow cells).

For each measurement at a given temperature, the system was left growing until a point in which the OD normalised relative fluorescence reached its saturation point.

After a few exploratory tests at different incubation times, it was found that the optimal incubation time to reach signal saturation was around 15 hours to 20 hours. Taken this into account, all the measurements used for the model were done at 17 hours of incubation time. Thus, we can say these fluorescence measurements were Steady State measurements

Our measurements were made in LB medium at 25, 30, 37 and 42°C, for each one of our constructions that are part of the whole circuit.

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.

Another way to do this parameter estimation, is performing a Function Fit for the analytic solution of the differential equations, which is shown in the Appendix, at the end of this page.

Due to time restrictions, we didn't get time series data for the relative fluorescence at a fixed temperature. This data would have been useful for estimating the parameter /(/delta/) of our dynamic model.

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 growing for 8 hours at a given temperature, a culture would have reached the steady state OD600-normalised 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 \(F_{Rst}\).

Our dynamic model is reduced in complexity when \(F_{st}\) is known -which is the case when we let grow the cells until they reach a stable fluorescence measurement-. 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 behaviour 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 normalise OD600 data and growing their culture in the same conditions).

However, as explained above, we weren't able to take time series measurements of cultures growing at fixed temperatures. We simulated the behaviour of our system using arbitrary values for the \(\delta\) parameter spanning three orders of magnitude and the parameters obtained for the \(f(T)\) function.

We are also aware 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}

Nevertheless, this is a consideration that we'll leave for further analysis and for now, we'll assume that \(\delta)\ does not change significantly in the chosen temperature range (25-42°C).

In mathematics, a Gaussian Function is a function of the form:

\begin{equation} \ f(x) = a e^{\frac{-(x-b)^{2}}{2c^{2}}} \end{equation}

for some real constants a>0, b>0, c>0, and Euler's number).

For which MATLAB uses the following custom equation when fitting data to a Gaussian Function:

\begin{equation} \ f(x)= a e^{\frac{-(x-b)^{2}}{c^{2}}} \end{equation}

Where 2c is considered as c for simplicity the normalisation of x, b is the mean (average) and c the standard deviation.

For us, we are using this function as a model for the dependency of the Relative Fluorescence of each construction. Its equation would be the following:

\begin{equation} \ F_{Rst}(T)= a e^{\frac{-(T-b)^{2}}{c^{2}}} \end{equation}

The parameter a is related to the amplitude of the function. This is important for the model, because higher amplitudes will provide a better signal, and will not be lost among noise background.

The parameter b is related to the horizontal shift. A better interpretation of it is the value of x where the function has its higher value. For us this is the optimal temperature of the RNAT.

The parameter c is related to the width of the bell. This is important for the application of this model for RNAT, because smaller width will mean a more clear and defined signal.

Some important biological phenomena can be modelling as a Gaussian function, including macroscopic (e.g., the length of a specific bird beak) and also molecular phenomena (e.g., enzyme activity).

According to the model, the majority of the observations will show a range of values that culminates in a mean value being generated. The central peak of the bell curve will show the modal average, and the highest frequency of results will be shown in this space.

It is a basic and fairly accurate model of how different characteristics, or traits, are carried through a sample of organisms, and how certain conditions provide an optimal function or higher survival rate among a biological system.

We are choosing the Gaussian function, because of the relevance that its parameters have for our model. Defining these parameters will provide us a way to quantify the strength (a), optimal value (b), and definition or clearness (c) of our RNAT activity.

However, we still lack enough experimental information to develop a theory that explains why the behaviour of a RNAT can be fitted to a Gaussian function and to determine whether if this is an universal behaviour of all RNATs or if it is restricted to the RNATs we are working with.

The results for the Gaussian Function Fitting are shown in the following figures (all the following figures were done within Matlab, as well as the fitting of the functions):

Figure 1 (M1).
Figure 2 (M2).
Figure 3 (M11).
Figure 4(M12 *).
Figure 5(all)
Simulations for clones M1 and M2 at different \(\delta\) values.

Both M1 and M2 have a similar behaviour. They show an optimal Relative Fluorescence around the same temperatures (37°C). M2 seems to have a better fit to the data than M1, while M1 (around 0.5 RFU) has greater amplitude than M2 (around 0.4 RFU).

The results for M11 and M12 have a fit that doesn’t follow a biological interpretation, because they seem to have (according to a Gaussian Function Fit) an optimal temperature near to values higher than the viable temperature for the microorganism. This is because they show a slower (or shifted) increase of Relative Fluorescence in relation to the others.

In the following section, we show the result for the parameters.

Clone a amin/max b bmin/max c cmin/max
M1 0.5042 0.3594/0.6490 35.2700 32.9600/37.5800 8.0180 5.6810/10.3500
M2 0.3630 0.3309/0.3950 36.5300 35.5300/37.5200 8.5250 7.5210/9.5300
M11 0.6283 -5.4200/6.6770 63.1300 -141.3000/267.6000 24.0900 -63.4000/111.6000
M12plus 0.6646 -57.64/58.9700 57.7300 -789.0000/904.4000 13.1800 -291.1000/317.4000
M12 0.1914 -inf/+inf 41.49 -inf/+inf 1.225 -inf/+inf
RFP 1.0000 0.8088/1.1910 38.2300 34.5700/41.8900 10.3700 6.5230/14.2100

Parameter table

For the parameter a, which represent the strength of the signal, M1 is the better one among the clones, excluding the result of M11 and M12, because they show their respective peaks far away from the accepted range of temperatures approximately around 20°C and 50°C.

For the parameter b, which represent the optimal temperature for the RNAT (the ON state), M1 and M2 behave more likely to what we expected. The same happens for the control RFP. All these are around 37°C. For M11 and M12 (for its both cases), optimal temperatures are away from what we expected, and what would work for between the viable temperatures.

For the parameter c, which represents the width (clearness or definition) of the signal, the smallest (best) clone was M1. M12 is not considered because it has a very poor amplitude and a shifted optimal temperature.

Clone SSE R2 AdjustedR2 RMSE
M1 0.0066 0.9637 0.9396 0.0471
M2 0.0005 0.9953 0.9922 0.0134
M11 0.0055 0.9041 0.9041 0.0430
M12plus 0.0001 0.9956 0.9867 0.0083
M12 0.0092 0.7161 0.5268 0.0553
RFP 0.0266 0.9710 0.9516 0.0941

Fitting Confidence

The last table shows the confidence parameters (SSE, R2, Adjusted R2 and RMSE) that Matlab provides for each fit. For the first and last column, smaller values mean less error or deviation of the model to the data. For the other two parameters, the closer the values get to 1, it means it has a better fit.

M12 has a particular behaviour compared to the other clones, which is of great interest but need further and deeper analysis. For the previous analysis, M12 represent a restricted range of our data set for which we are more confident of the results.

For more insights around this special case, refer to Results, in the section Wetlab.

  1. ShahP ,Gilchrist MA(2010)Is Thermosensing Property of RNA Thermometers Unique? PLoS ONE,5(7):e11308.doi:10.1371/journal.pone.0011308.
  2. H. A. Von Fircks, T. Verwijst,(1993)Plant Viability as a Function of Temperature Stress (The Richards Function Applied to Data from Freezing Tests of Growing Shoots) Plant Physio ,103(1):125–130.
  3. Hoops S, et al. (2010)COPASI–a COmplex Pathway Simulator Bioinformatics ,22,3067-3074,2006,http://dx.doi.org/10.1093/bioinformatics/btl485
  4. COPASI Documentation 2.Steady State calculation(2013,May 16).Retrieved from http://www.copasi.org/tiki-integrator.php?repID=9&file=ch07s02.html
  5. Ting Chen, et al. Modelling gene expression with differential equations (1999) Pacic Symposium of Biocomputing
  6. Gaussian function(Consulted on 2013,September 27)Retrieved from http://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4282164/Gaussian%20function.pdf

Here we are going to show how we solved the differential equation 5.


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

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}

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