Team:Valencia UPV/Modeling

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Modeling

Do you think it is possible to mathematically describe a cell? Would you like to know the possibilities that Printeria modeling offers you?

One of the fundamental bases of the Printeria project has undoubtedly been mathematical modeling. Thanks to the development and application of new mathematical models, it is possible to quantify the expression of proteins in cells, and therefore characterize through different experiments the parts designed by Printeria. From the Printeria modeling team, we intend to reach different goals:

Design simple mathematical models based on differential equations that describe the biochemical processes of a cell. With them, we can simulate the different genetic circuits that Printeria allows us to build.
Develop a Simulation Tool that allows the user to visualize a prediction of the results of their experiment before running it in Printeria.
Optimize model parameters to match simulation results to experimental data obtained from Printeria constructions.
Characterize the parts of our Part Collection from the optimization results and provide the user with all the information about the Printeria kit.

Although in the development of the project we have dealt with all these aspects, all of them have a single purpose: to understand and describe in a mathematical way the biological processes that take place inside the cell.

Models & Experiments

Characterisation procedures

Simulation Tool

Models & Experiments

The development of new and simple mathematical models has been one of the essential bases of the Printeria project. The models offer us a multitude of applications: to describe the basic biochemical reactions of the cell, to characterize the parts of our Printeria kit by means of different experiments or to elaborate new tools that facilitate to our user the learning of Synthetic Biology.

At the beginning of the design of any mathematical model, we have started from a cell schemein which all the cellular biochemical reactions we wish to model are represented. From the reactions, a set of equations is deduced. Each of these equations describes the temporal variation of the biochemical species in the cell (DNA, RNA, proteins...), and depends on a set of parameters with a physical meaning.

Because they are differential and generally non-linear equations, mathematical models are simulated using software tools such as MATLAB. The results obtained from the simulation reflect the evolution of the concentration of biochemical species over time. These theoretical results can be contrasted with the experimental data obtained in the laboratory and and thereby adjust the parameters so that the model adapts to the data in an optimal way. This process, called multiobjective parameter optimization, is the process that allows us to validate our models and see if they respond to what happens inside the cells.

In the development of the set of mathematical models that describe the genetic circuits printed by Printeria, we have prioritized simplicity over complexity of the model. A more complex model may be more precise, but it requires a large number of parameters whose value is often unknown. Therefore, our models are composed of few equations and parameters, and they are all deterministic models, that is, their parameters take a concrete, non-probabilistic value.

In Printeria Modeling team, we have grouped the models designed in two fields: constitutive expression models and inducible expression models. We have also established a multi-objective optimization protocol in order to contrast theoretical and experimental results. With this, we can describe practically all the cells modified with Printeria.

Want to find out more about how we did it?

Constituve expression models

Constitutive expression models are those in which the expression of the cell's messenger RNA (mRNA) is not regulated. The gene does not depend on any inducing biochemical species, so it will produce RNA molecules continuously, and therefore the translation of the protein will also be constant.

Model design

A simple scheme that summarizes the cellular transcription and translation processes in a constitutive expression model is shown in the following Figure. We only consider as biochemical species involved in reactions to mRNA and our protein of interest (PoI).

From the cell schema, we can deduce the following biochemical reactions.

Parameter	Description	Units	Value
C_R	Constitutive transcription rate	molecules.min^-1	---
p	Translation rate	min^-1	---
d_m	mRNA degradation rate	min^-1	---
d_p	PoI degradation rate	min^-1	---
μ	Dilutio rate	min^-1	---
K_max	Maximum growth capacity	cells	---

With the reactions and approximations already established, we apply the Mass Action Law (MAL) to deduce the differential equations. The MAL establishes that the variation of the species resulting from a reaction is proportional to the product of the reactants. In our case, to the set of differential equations deduced from the reactions we add a cell growth equation. Biochemical species and our model of equations obtained are shown below.

Variable	Biochemical species	Units
n₁	mRNA	Molecules
n₂	PoI	Molecules
n₃	Number of cells	Cells

By obtaining the equations, and seeking the simplicity of the model, a series of quasi-static approximations have been made. We have established that mRNA is generated with a constant C_R constitutive transcription rate. Another consideration has been to assume that the copy number of plasmids in cell is constant. We have also proposed that other species such as ribosomes, which also participate in transcription, are constant over time.

After developing the model and raising the reactions and equations, two files have been programmed in MATLAB: the mc_simple.m function, which describes the system of differential equations, and the script model_const_simple.m in which the parameters are defined and the model is solved by the ode45 MATLAB function. The result of the simulation of the model is shown in the following Figure.

At this point anyone could ask us: Which are the advantages of a constituent model like this? Why is a model like this really useful to us? Faced with these questions, the Printeria Modeling team has looked for the answers in the great characteristics that the model offers us.

A simple, easy-to-understand model that clearly explains the processes of cellular transcription and translation.
It is valid for any Printeria construction that has a constitutive expression promoter. The variation between different constructions occurs in the parameters values, not in the model equations!
It has few parameters, with a clear physical meaning, and easily optimizable.
As a compact model, simulations and optimizations are performed at high speed.

Optimization

Inducible expression models

Multi-objective parameter optimization

To what extent is our model valid? What values should the parameters take? To what extent do the theoretical results resemble the experimental ones? Is our model able to explain the behaviour of cells in reality?

Characterisation procedures

This year, Valencia UPV iGEM team has designed an extensive Part Collection in purpose of allowing the user to design multiple genetic constructions and experiments. One of our main objectives has been to show the user clear and structured information about the pieces that make up the Printeria kit. For this reason, we have considered the characterization of the parts as a priority when developing the project. In this way, we have elaborated some procedures which have allowed us to systematically obtain and structure information from the parts.

Procedure for obtaining protein spectra
Comparison between sfGFP and GFP relative fluorescence intensity

Procedure for obtaining protein spectra

Obtaining excitation and emission spectra is a fundamental aspect in the process of characterization of a fluorescent protein. Each protein has a characteristic spectrum, which indicates the energy in which the molecule is excited or emits at a certain wavelength.

The characterization of the reporter proteins by excitation and emission spectra are of great importance in experimentation. When contrasting experimental information with the theoretical results of mathematical models, we experiment with numerous reporter proteins, such as fluorescence proteins or chromoproteins. However, the fluorescence data obtained must be corrected applying diferent operations in order to obtain representative fluorescence data:

The subtraction of the blank fluorescence:
F_{subs} = F - F_{blank}

The quotient of the fluorescence with the gain:
F_{gain} = \frac{F_{subs}}{G}

The quotient of the fluorescence with excitation and emission efficiency:
F_{corrected} = \frac{F_{gain}}{\frac{exc(\%)em(\%)}{100}}

Until now, the corrections applied to the experiments performed with these reporter proteins were the subtraction of the fluorescence of the blank, and the division by the gain factor of the measuring equipment. However, with the protein spectra we can also normalize the fluorescence data to values that would have been obtained with maximum excitation and emission.

Owing to this reason, a protocol has been established in the lab by Lab and Modeling team to obtain the spectrum of any reporter protein.

Materials:

Measuring equipment: Biotek Cytation3
96 well plate
MATLAB 2018a software

Procedure:

We look for the theoretical spectra of the protein to be measured or similar molecules in order to determine the wavelength at which the protein is excited or emitted at maximum energy, i.e. where the theoretical spectral peaks occurs.

We define the protocol of our equipment to get the absorbance and fluorescence dataset. In our protocol, the most important parameters to be established are summarized in the following Table.

Parameters	Description
Number of samples	From 3 to 6 samples of reporter protein and 3 samples of blank
Temperature	37 ºC
Shake	Double Orbital. 01:00 (MM:SS)
Absorbance. Optical Density (OD)	Wavelenght at 600 nm emission
Excitacion and emission scans	The scans occur between two wavelength limit values. The established range will depend on the theoretical spectrum of the protein.
Excitacion and emission wavelengths	These values will depend on the range, and therefore on the spectrum. Values far from the theoretical peak lead to more attenuated fluorescence curves, and values very close to the peak can lead to overlap and error in reading the data. Therefore, a compromise must be reached between curve resolution and reading overlap.
Gain (G)	Normally the gain value G = 60, although for proteins with lower fluorescence, it is recommended that G takes higher values.

The experiment is introduced, and the experimental absorbance data and the fluorescence curves of the samples are obtained with the blank correction applied.
The dataset is exported to an Excel file.
The MATLAB script spectrum.m for fluorescent proteins is executed:
1. Dataset is extracted from the Excel file. We discard readings that have suffered overlap, or that take negative values.
2. The fluorescence curves of all samples are averaged, and the result is normalised (from 0 to 100%).
3. Graphs of the normalized absorption and emission spectra are plotted. The X-axis represents the wavelength (nm), and the Y-axis represents the normalized fluorescence intensity (%).

In practical terms, the protocol has been applied to all of our reporter proteins: GFP, sfGFP, YFP and mRFP. All results can be found in our parts collection as well as in the iGEM catalog.

In the particular case of a reporter chromoprotein, such as amilCP, we do not measure fluorescence, but absorbance. In this case, in rder to obtain the corrected absorbance curve, we must subtract from the cell absorbance data with the reporter protein the absorbance of a medium with cells without chromoprotein. Once the data have been corrected, we normalize them between values of 0 and 100 and with this we elaborate the graph. The protocol used can be found in the MATLAB script amilCP_spectrum.m.

Finally, we have also obtained the spectra of the fluorescein molecule. These spectra have been used to correct the fluorescence data used in the Interlab Study to obtain the Relative Fluorescence Units (RFU) to Molecules of Equivalent Fluorochrome (MEFL) conversion factor. In addition, the comparison between GFP and sfGFP proteins RFU has been possible thanks to fluorescein spectra.

Parameters	Value
Number of samples	6 samples. 3 samples of blank
Temperature	37 ºC
Shake	Double Orbital. 01:00 (MM:SS)
Absorbance. Optical Density (OD)	Wavelenght at 600 nm emission
Emission range	[495 - 580] nm
Fixed excitation wavelength	480 nm
Excitation range	[430 - 520] nm
Fixed emission wavelength	545 nm
Gain (G)	60

Comparison between sfGFP and GFP relative fluorescence intensity

One of the main problems we encounter when processing the results of experiments in Synthetic Biology are the units of measurement of fluorescence data. Unlike absorbance, where there is a simple conversion between Optical Density (OD) and cell number, there is no direct relationship between the Relative Units of Fluorescence (RFU) and the number of protein molecules in the cell. Moreover, RFUs vary among reporter fluoroproteins: for example, an RFU of the GFP protein does not have to be equivalent to an RFU of the mRFP protein, sfGFP, etc.

Thanks to initiatives such as the Interlab Study, we have been able to go a step further and obtain a MEFL/RFU_GFP factor of equivalence between the RFU of the GFP protein and the Molecules of Equivalent Fluorochrome (MEFL). This relationship is an important breakthrough, as it can give us a more accurate estimation of the amount of GFP molecules in the cell.

However, another reporter protein widely used in the experiments is the sfGFP. This protein has a much faster folding than GFP, which translates into a higher fluorescence intensity per molecule. In order to obtain the MEFL/RFU_sfGFP factor from sfGFP, the Printeria Modeling and Lab teams have designed a comparative experiment between both proteins. The experiment consists of designing two identical transcriptional units (TU), changing only the CDS sequence so that each TU will produce GFP and sfGFP, respectively. It should be added that this experiment is based on two fundamental assumptions:

The number of GFP molecules produced in the cells is equivalent to the number of MEFL.
Given two TU with identical promoters, RBS and terminators, but with different CDS, under the same experimental conditions, the number of molecules produced by each TU is the same.

Taking these axioms into account, the materials and procedure followed to calculate the MEFL/RFU_sfGFP factor were as follows.

Materials:

Measuring equipment: Biotek Cytation3
96 well plate
MATLAB 2018a software

Procedure:

From the fluorescein spectrum, and the fluorescence data obtained from the Interlab Study experiment, blank, gain and efficiency corrections have been made in the data set.
The data are introduced in the Excel file of the Interlab Study. With them we obtain the factor MEFL/RFU_GFP. In our case, MEFL/RFU_GFP = 271.2 (GFP equivalent molecules/RFU)

We establish the protocol of the experiment and the parameters of our equipment. The experiment consists in the measurement of the absorbance and fluorescence of two TU of identical structure, whose CDS codifies for GFP and sfGFP proteins. The most relevant information of the experiment is detailed in the following Table:

Parameters	Description
Time	06:00:00 (HH:MM:SS). Measurement interval: 05:00 (MM:SS)
Number of samples	8 samples for each TU. Total samples: 16
Temperature	37 ºC
Shake	Double Orbital Continuous
Absorbance. Optical Density (OD)	Wavelenght at 600 nm emission
Excitacion wavelength	485 nm
Emission wavelength	528 nm
Gain	60

The experiment is introduced into the equipment and the absorbance and fluorescence curves are obtained, as well as the curve of Fluorescence F/Absorbance OD or FOD ratio with the applied blank correction.
The absorbance and fluorescence data are exported to an Excel file.
Running the MATLAB GFP_sfGFP_comparison.m script:
1. We extract the FOD data from the GFP and sfGFP from the Excel file.
2. We apply the gain and efficiency corrections of the spectrum to the FOD.
3. We plot the FOD curves and look for a stationary equilibrium time interval in the expression of the GFP and sfGFP proteins. In our experiment, we have decided to chose the interval [145,290] min
4. We obtain the average value of both regions.
5. We calculate the number of GFP MEFLs by multiplying the FOD data by the MEFL/RFUGFP factor.
6. If we assume that the number of molecules expressed by both TU is the same, the number of MEFL for the calculated GFP is the same as the number of MEFL for the sfGFP.
7. We calculate the MEFL/RFU_sfGFP factor by dividing the MEFL number by the average FOD value of the sfGFP in the stationary region. In our case, MEFL/RFU_sfGFP = 121.4.

Reporter protein	MEFL/RFU factor
GFP	273.7
sfGFP	121.4

With MEFL/RFU factors, FOD data obtained in any experiment in which GFP or sfGFP was used as reporter proteins can be transformed into equivalent fluorescein molecules by applying the following ratio: Molecules = (MEFL/RFU) · FOD.

The new data give us a more accurate estimation of the number of molecules in the cell. Consequently, by relating the experimental results with the theoretical mathematical models in the optimization process, the parameters of the model acquire values more consistent with their physical significance, working in equivalent molecules and not in arbitrary units.