Difference between revisions of "Team:Valencia UPV/Model"

Models & Experiments

The development of new and simple mathematical models has been one of the essential bases of the Printeria project. The ODE models offer a multitude of applications: describing the basic biochemical reactions of the cell, to characterizing the parts of our Printeria kit by means of different experiments, and elaborating new tools that facilitate the user learn about Synthetic Biology.

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

Want to find out more about how we did it?

Modeling process

At the beginning of the design of any mathematical model, we have started from a cell scheme in which all the biochemical reactions happen. From the reactions, a set of equations is inferred. Each of these equations describe the temporal evolution of the main biochemical species in the cell (DNA, RNA, proteins and transcription factors), and depend on a set of parameters with a physical meaning.

Modeling process schema.

Because they are ordinary differential and generally non-linear equations (ODE), 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 thereby adjust the parameters. The ODE model fits 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.

The development of the set of mathematical models of the genetic circuits printed by Printeria, we have prioritized simplicity over complexity. A more complex model may be more precise, but it requires a large number of parameters whose value is oftenly unknown. Therefore, our models are reduced order but rigorous deterministic models , where their parameters take a concrete, non-probabilistic value.

Constitutive expression models

Constitutive expression models are those in which protein expression is not regulated. The gene that encodes a protein does not depend on any transcription factor, so it will continuously transcribe messenger RNA molecules, and then translation of the protein will be unregulated.

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).

Constitutive expression model. Cell schema.

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

Constitutive expression model. Biochemical reactions.

Parameter	Description	Units	Value
CR	Constitutive transcription rate is the transcription rate KR times the mean plasmid copy number in cell cn. In our case, we are using the pMBI replication origin, so cn ∼ 500, and CR = KR·cn	molecules.min-1	326 [1, 7]
p	mRNA translation rate	min-1	2.38 [2]
dm	mRNA degradation rate	min-1	0.247 [7]
dp	PoI degradation rate	min-1	0.156 [5]
μ	Dilution rate	min-1	0.017 [5]
Kmax	Maximum growth capacity	cells	1.6·108 [5]

Constitutive expression model. Model parameters obtained from literature.

With the reactions and approximations already established, we apply the Law of Mass Action kinetics[1] (LMA) to deduce the differential equations. The LMA establishes that the variation of the species resulting from a reaction is proportional to the product of the reactants. In our case, we also take into account an ODE for the cell growth. The resulting ODE model and the main biochemical species are shown below.

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

Constitutive expression model. Biochemical species.

Constitutive expression model. Model equations.

After this, a series of quasi-steady state approximations [2] have been made. We have established that mRNA is generated at a constant C_R constitutive transcription rate. In addition, the copy number of plasmids c_n is kept as a constant. We consider that other species such as RNAP polymerases and ribosomes are kept constant as well.

The computational simulation of this ODE model uses two MATLAB files: the mc_simple.m function, which describes the system of differential equations, and the script model_const_simple.m where the parameters are defined and the model is solved by the ode45 MATLAB function. The simulation results are shown in the following Figure.

Constitutive expression model. Model predictions using parameter values from literature.

In the the temporal evolution of the mRNA and the protein PoI we can distinguish two phases: the transient phase and the stationary phase. If we perform different simulations varying some parameters of the model, we will observe how the equilibrium point of the stationary phase varies. MATLAB script mc_sim_analysis.m repeatedly simulates the model by changing the C_R and p parameters. The following Figure shows how an increase of both parameteres implies a greater expression of the protein PoI.

Constitutive expression model analysis. The graph represents de PoI stationary value as function of C_R and p ratios.

At this point anyone could ask us: Which are the advantages of a constitutive expression 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.

Experiments & Optimization

The simple model of constitutive expression is a model that characterizes a large number of Printeria Transcriptional Units (TU), specifically, all those constructions whose promoter is constitutive. We know that all these TU follow the same model, but each one has different parameters, and therefore different experimental values of fluorescence and absorbance. Faced with this situation, the Printeria Lab and Modeling teams have designed some experiments, in which we have applied the multi-objective parameter optimization, and thus check whether the designed constitutive expression model responds to experimental reality.

Experiments changing RBS

We have designed two experiments following the same experimental protocol to assemble different Printeria TUs with the same promoter, CD (sfGFP reporter protein) and transcriptional terminator, but with different RBSs. Once a TU is made, we estimated the model parameters and particularly the RBS of every TU using the multiobjective optimization protocol.

Printeria RBS:

• Strong expression: BBa_K2656009.

• Medium expression: BBa_K2656011.

• weak expression: BBa_K2656010.

• Very weak expression: BBa_K2656008, BBa_K2656012.

Experiment parameters	Description
Time	06:00:00 (HH:MM:SS). Measurement interval: 05:00 (MM:SS)
Number of samples	8 samples for each TU colony
Number of medium samples	8 samples of M9 medium
Temperature	37 ºC
Shake	Double Orbital. Continuously
Absorbance. Optical Density (OD) measure	Wavelenght at 600 nm emission
Excitacion wavelength	Wavelenght at 485 nm
Emission wavelength	Wavelenght at 528 nm
Gain (G)	60

Biotek Cytation 3 experimental parameters.

Optimization specifications	Description
Parameters	Constitutive transcription rate CR: fixed Translation rate p: to optimize PoI degradation rate dp: to optimize mRNA degradation rate dm: fixed Dilution rate μ: to optimize Maximum growth capacity Kmax: experimental value
Objetives to optimize	For each TU, we set 2 objetives: FOD (Fluorescence per cell) and OD (Absorbance). In each experiment we measured 3 TU, so we are optimizing 6 objectives per experiment
Parameter ranges	Translation rate p: [0.001 - 6] min-1 PoI degradation rate dp: [0.0058 - 0.0087] min-1 Dilution rate μ: [0.0058 - 0.035] min-1
MATLAB files	spMODEparam.m: it defines the parameters to be optimized, their value ranges, number of objectives, the cost function, the identification and validation experimental data and other spMODE algorithm parameters. When the script is executed, a spMODEDat structure variable is defined. This structure contains all the standardized optimization information that the spMODE algorithm will need for its execution. Experiment 1: spMODEparam_exp1sfGFP.m Experiment 2: spMODEparam_exp2sfGFP.m spMODE.m: it contains the Multi-objective Differential Evolution Algorithm with Spherical Pruning, which optimizes parameters for our experimental results (to execute spMODE.m file we need also SphPruning.m file). CostFunction_RBS_RMS_2n.m: it simulates the model with different vector parameters, and calculates the Root Mean Square Error with the identification dataset. Then, it returns the error for each objective and parameter vector. levelDiagram.m: plots Pareto front and Pareto set that gives us back the algorithm. execute_RBS_2n.m: this script launches the optimization by executing spMODEparam.m, spMODEm algorithm and levelDiagram.m. Then, allows the user to enter the best parameters, and plots validation experimental data and simulation results in the same graph.

Optimization specifications for the experiment.

Experiment 1 changing RBS. Experimental and optimized simulated model data of strong, medium and weak protein expression RBS

Experiment 2 changing RBS. Experimental and optimized simulated model data of strong and very weak protein expression RBS

Experiment 2 changing RBS. Experimental and optimized simulated model data of very weak protein expression RBS

The results above were obtained after an optimization and decision-making process, where one set of the model parameters were selected to simulate the ODE model of every TU. These simulations were compared with the experimental data showing good agreement. The optimized set of model parameters are summarized in the following table.

Optimized parameters	Values
Translation rate p	Experiment 1: For strong RBS BBa_K2656009: p = 0.2059 min-1 For medium RBS BBa_K2656011: p = 0.082 min-1 For weak RBS BBa_K2656010: p = 0.01 min-1 Experiment 2: For strong RBS BBa_K2656009: p = 0.9256 min-1 For very weak RBS BBa_K2656008: p = 0.04089 min-1 For very weak RBS BBa_K2656012: p = 0.02889 min-1
PoI degradation rate dp	Experiment 1: dp = 0.0058 min-1 Experiment 2: dp = 0.00818 min-1
Dilution rate μ	Experiment 1: For strong RBS BBa_K2656009: μ = 0.01492 min-1 For medium RBS BBa_K2656011: μ = 0.01631 min-1 For weak RBS BBa_K2656010: μ = 0.01641 min-1 Experiment 2: For strong RBS BBa_K2656009: μ = 0.01197 min-1 For very weak RBS BBa_K2656008: μ = 0.01288 min-1 For very weak RBS BBa_K2656012: μ = 0.0118 min-1

List of optimized parameters for RBS experiment.

In addition, the Printeria Modeling team has also calculated the relative strength between the different RBS, taking BBa_K2656009 strong RBS as a reference. The relative strength is the quotient between the values of the protein at the equilibrium point and the reference. The characterization of the RBS parts by their relative strength is shown below.

Original BioBrick RBS part	Printeria RBS part	Relative strength	p parameter ratio (pRBS/pref)
BBa_B0030 (Reference part)	BBa_K2656009 (Reference part)	1	1
BBa_B0034	BBa_K2656011	0.371	0.398
BBa_B0032	BBa_K2656010	0.045	0.048
BBa_J61100	BBa_K2656008	0.042	0.044
BBa_J61101	BBa_K2656012	0.031	0.031

Relative strength between different RBS parts.

Experiments changing promoters

We have designed two experiments following the same experimental protocol to assemble different Printeria TUs with the same RBS, CD (GFP reporter protein) and transcriptional terminator, but with different promoters. Then, following the optimization protocol we have obtained the model parameters and particularly the promoter values.

Printeria promoters:

• Strong promoters: BBa_K2656005

• Medium promoters: BBa_K2656007

• Weak promoters: BBa_K2656004

Experiment parameters	Description
Time	06:00:00 (HH:MM:SS). Measurement interval: 05:00 (MM:SS)
Number of samples	8 samples for each TU colony
Number of medium samples	8 samples of M9 medium
Temperature	37 ºC
Shake	Double Orbital. Continuously
Absorbance. Optical Density (OD) measure	Wavelenght at 600 nm emission
Excitacion wavelength	Wavelenght at 485 nm
Emission wavelength	Wavelenght at 528 nm
Gain (G)	60

Biotek Cytation 3 experimental parameters.

Optimization specifications	Description
Parameters	Constitutive transcription rate CR: to optimize Translation rate p: fixed PoI degradation rate dp: to optimize mRNA degradation rate dm: fixed Dilution rate μ: to optimize Maximum growth capacity Kmax: experimental value
Objetives to optimize	For each TU, we set 2 objetives: FOD (Fluorescence) and OD (Absorbance). In each experiment we have measured 3 TU, so we are optimizing 6 objectives per experiment
Parameter ranges	Constitutive transcription rate CR: 500 x [0.001 - 5] min-1 PoI degradation rate dp: [0.0058 - 0.0087] min-1 Dilution rate μ: [0.0058 - 0.035] min-1
MATLAB files	spMODEparam.m: it defines the parameters to be optimized, their value ranges, number of objectives, the cost function, the identification and validation experimental data and other spMODE algorithm parameters. When the script is executed, a spMODEDat structure variable is defined. This structure contains all the standardized optimization information that the spMODE algorithm will need for its execution. In this experiment, we have used the spMODEparam_exp6GFP.m MATLAB script. spMODE.m: it contains the Multi-objective Differential Evolution Algorithm with Spherical Pruning, which optimizes parameters for our experimental results (to execute spMODE.m file we need also SphPruning.m file). CostFunction_Prom_RMS_2n.m: it simulates the model with different vector parameters, and calculates the Root Mean Square Error with the identification dataset. Then, it returns the error for each objective and parameter vector. levelDiagram.m: plots Pareto front and Pareto set that gives us back the algorithm. execute_Prom_2n.m: this script launches the optimization by executing spMODEparam.m, spMODEm algorithm and levelDiagram.m. Then, allows the user to enter the best parameters, and plots validation experimental data and simulation results in the same graph.

Optimization specifications for the experiment.

Experiment 1 changing promoters. Experimental and optimized simulated model data of strong, medium and weak promoters.

The results above were obtained after an optimization and decision-making process, where one set of the model parameters were selected to simulate the ODE model of every TU. These simulations were compared with the experimental data showing good agreement.

Optimized parameters	Values
Constitutive transcription rate CR	For strong promoter BBa_K2656005: CR = 539.5 min-1 For medium promoter BBa_K2656007: CR = 486.25 min-1 For weak promoter BBa_K2656004: CR = 171.15 min-1
PoI degradation rate dp	dp = 0.008492 min-1
Dilution rate μ	For strong promoter BBa_K2656005: μ = 0.0072 min-1 For medium promoter BBa_K2656007: μ = 0.0068 min-1 For weak promoter BBa_K2656004: μ = 0.011 min-1

List of optimized parameters for promoters experiment.

In addition, the Printeria Modeling team has also calculated the relative strength between the different promoters, taking BBa_K2656005 strong promoter as a reference. The characterization of the promoter parts by their relative strength is shown below.

Original BioBrick RBS part	Printeria RBS part	Relative strength	p parameter ratio (pProm/pref)
BBa_J23102 (Reference part)	BBa_K2656005 (Reference part)	1	1
BBa_J23101	BBa_K2656007	0.9367	0.9013
BBa_J23106	BBa_K2656004	0.2603	0.3172

Relative strength between different promoter parts.

Experiment to compare reporter proteins with and without LVA tag

In the following experiment we want to compare the degradation rate of different protein reporters in two Printeria TUs with identical promoters, RBS and transcriptional terminators. An LVA degradation tag has been added to one of the protein sequences. This tag causes an increase of protease activity, so it means a faster degradation rate of the reporter protein. A priori, the degradation rate in the TU with the LVA tag will be higher.

With these experiments, the team aims:

• To analyze the effect of the protein degradation rate variation on our constitutive expression model, and determine if the model fits the experimental results of the reporter proteins with LVA degradation tag.

• To characterize the YFP reporter protein with LVA tag as a new iGEM part and as Improve project .

The experiments and optimization process performed for each reporter protein are described below. We have followed the experimental protocol and multi-objective optimization protocol.

YFP with and without LVA tag

Experiment parameters	Description
Time	06:00:00 (HH:MM:SS). Measurement interval: 05:00 (MM:SS)
Number of samples	8 samples for each TU colony
Number of medium samples	8 samples of M9 medium
Temperature	37 ºC
Shake	Double Orbital. Continuously
Absorbance. Optical Density (OD) measure	Wavelenght at 600 nm emission
Excitacion wavelength	Wavelenght at 500 nm
Emission wavelength	Wavelenght at 540 nm
Gain (G)	60

Biotek Cytation 3 experimental parameters.

Optimization specifications	Description
Parameters	Constitutive transcription rate CR: fixed Translation rate p: to optimize PoI degradation rate dp: to optimize mRNA degradation rate dm: fixed Dilution rate μ: to optimize Maximum growth capacity Kmax: experimental value
Objetives to optimize	For each TU, we set 2 objetives: FOD (Fluorescence) and OD (Absorbance). In this experiment we have measured 2 TU, so we are optimizing 4 objectives per experiment
Parameter ranges	Constitutive transcription rate CR: [0.01 - 2] min-1 PoI degradation rate dp: [0.0058 - 0.018] min-1 Dilution rate μ: [0.0058 - 0.035] min-1
MATLAB files	spMODEparam.m: it defines the parameters to be optimized, their value ranges, number of objectives, the cost function, the identification and validation experimental data and other spMODE algorithm parameters. When the script is executed, a spMODEDat structure variable is defined. This structure contains all the standardized optimization information that the spMODE algorithm will need for its execution. In this experiment, we have used spMODEparam_exp1YFP.m spMODE.m: it contains the Multi-objective Differential Evolution Algorithm with Spherical Pruning, which optimizes parameters for our experimental results (to execute spMODE.m file we need also SphPruning.m file). CostFunction_improve_RMS_2n.m: it simulates the model with different vector parameters, and calculates the Root Mean Square Error with the identification dataset. Then, it returns the error for each objective and parameter vector. levelDiagram.m: plots Pareto front and Pareto set that gives us back the algorithm. execute_improve_2n.m: this script launches the optimization by executing spMODEparam.m, spMODEm algorithm and levelDiagram.m. Then, allows the user to enter the best parameters, and plots validation experimental data and simulation results in the same graph.

Optimization specifications for the experiment.

After the multiobjective optimization process, we selected one of the solution for the model parameters and then we compared the model simulations with the experimental data. For both TUs, the simulation results match with the experimental ones.

Experiment YFP vs. YFP tag. Experimental data and optimized model of the YFP and YFP tag proteins

Optimized parameters	Values
Translation rate p	For YFP with LVA tag BBa_K2656020: p = 0.4622 min-1 For YFP without LVA tag BBa_K2656021: p = 1.053 min-1
PoI degradation rate dp	For YFP with LVA tag BBa_K2656020: dp = 0.01492 min-1 For YFP without LVA tag BBa_K2656021: dp = 0.0080 min-1
Dilution rate μ	For YFP with LVA tag BBa_K2656020: μ = 0.01166 min-1 For YFP without LVA tag BBa_K2656021: μ = 0.01307 min-1

List of optimized parameters.

At this point, it is easy to see that the degradation rate is significantly higher in the reporter protein with the degradation LVA tag than the YFP protein. Computing the ratio of the degradation rates for both YFP proteins, we observe that the value of the protein with the LVA tag is about twice the value of the protein without LVA tag.

Inducible expression models

Inducible expression models are those where protein expression depends on one or more inducing molecules. They are called activators or repressors. The concentration of these biochemical species determines if the transcription is activated or repressed, respectively.

Our Printeria device gives us the possibility to create some inducible genetic constructs:

• P_BAD/araC inducible model

• LuxR-LuxI inducible model

P_BAD/araC inducible model

The inducible promoter P_BAD/araC [5] is a promoter whose expression depends on two inducing complexes: the L-arabinose (activator) and the araC (repressor). The dimeric protein araC binds DNA to compound a DNA loop and thus prevents the binding between DNA and RNA polymerase. However, when two arabinose molecules bind the araC dimer, the DNA loop is broken and the binding between the RNA polymerase and the promoter region is possible.

We have proposed an ODE model describing the P_BAD/araC mechanism. This theoretical model has been included in the Simulation Tool, so Printeria is able to predict the behavior of the P_BAD/araC inducible promoter circuits.

Model design

We propose the following scheme that describe the biochemical reactions of the P_BAD/araC system. We have assumed a very high concentration of araC inside the cell because the gene encoding for the araC protein is found in the cell genome of some strains (e.g. in DH5α strains).

P_BAD/araC inducible model. Cell schema.

From the cell schema we infer the following biochemical reactions:

P_BAD/araC inducible model. Biochemical reations.

Parameter	Description	Units	Value
D	Diffusion coefficient	min-1	2 [2]
Vc	Cell volume(1.1x10-9 μL)/External volume(1x10-3 μL)	adimensional	1.1x10-6 [2]
ku	Dissociation rate of (PBAD.araC.arabi) complex	min-1	10 [10]
kd	Dissociation constant of (PBAD.araC.arabi) complex	molec2	100 [2, 10 ]
kb	Association rate of (PBAD.araC.arabi) complex. ku/kd	molec-2min-1	0.1 (Estimated)
K	Transcription rate is the product of transcription rate per plasmid Kt and the mean number of plasmids in cell cn . In our case, we are using the pMBI replication origin, so cn ∼ 500, and K = Kt·cn.	min-1	326 [1, 7]
α	(PBAD.araC) basal activity constant	adimensional	0.01 (Estimated)
p	mRNA translation rate	min-1	2.38 [2]
darab	L-arabinose degradation rate. We assume a very slow degradation.	min-1	∼10-6
dm	mRNA degradation rate	min-1	0.247 [7]
dp	PoI degradation rate	min-1	0.0063 [5]
μ	Dilution rate	min-1	0.017 [5]
Kmax	Maximum growth capacity	cells	1.6·108 [5]

P_BAD/araC inducible model. Model parameters.

We apply the Law of Mass Action kynetics (LMA) to obtain the equivalent ODE model. Considerin the same previous assumptions for the constitutive models, and also the fact the the plasmid copy number c_n is kept as a constant, we can obtain a reduced oreder model. Thus, we get a model of six differential equations and one algebraic equation as below.

Variable	Biochemical species	Units
n1	(PBAD.araC.arab) complex	Molecules
n2	Extracellular L-arabinose	Molecules
n3	Intracellular L-arabinose	Molecules
n4	mRNA	Molecules
n5	PoI	Molecules
n6	Number of cells	Cells

P_BAD/araC inducible model. Biochemical species.

P_BAD/araC inducible model. Model equations.

Now we have two MATLAB files: the mi_araC_simplest.m function, which describes the ODE model, and the script model_inducible_simplest.m where the parameters are defined. The model was simulated by using the ode23t MATLAB function. The simulation results are shown below.

P_BAD/araC inducible model. Model predictions using parameter values from literature.

Once the P_BAD/araC model is developed and coded, we simulate the model by varying the parameters through our MATLAB script analysis_mi_araC_simplest.m. In this case, we have varied the K transcription and p translation rates and we have represented in a 3D graph the values of the PoI in the steady state.

P_BAD/araC inducible model analysis. The graph represents de PoI stationary value as function of p and K ratios.

As we can see in the 3D graph, the PoI expression growths exponentially when translation rate p and transcription rate K increase.

But we can also vary the initial concentration of L-arabinose in P_BAD/araC model. The set of simulations varying p and arab₀ parameters has been programmed in the analysis_mi_araC_simplest_arab.m MATLAB script. We have represented our results of the PoI in the steady state in a 3D graph.

P_BAD/araC inducible model analysis. The graph represents de PoI stationary value as function of p ratio and L-arabinose concentration (arab₀).

LuxR-LuxI inducible model

The LuxR-LuxI genetic circuit [2] is a more complex genetic circuit than the models presented so far. It couples two functional subsystems: i) quorum sensing interconnecting cell population with the external inducer AHL, and ii) a positive feedback loop regulating expression of the protein of interest.

The first subsystem implements a cell-to-cell communication mechanism via quorum sensing. It is based on the exchange of the small signaling autoinducer molecule N-acyl-L-homoserine lactone (AHL) to induce population cell consensus. AHL molecules passively diffuses across the cellular membrane from inside the cell to the external environment, and viceversa, following Fick’s Law.

The second subsystem introduces an intracellular feedback loop to control expression of the protein of interest. First, LuxR protein is expressed by luxR under the constitutive promoter Pc. Then, LuxR and AHL bind forming the heterodimer (LuxR.AHL), which subsequently dimerizes as the heterotetramer transcription factor (LuxR.AHL)₂. The dimer represses protein expression when it is attached to the promoter Plux.

Model design

Once we know the biochemical functioning of the LuxR-LuxI circuit, we propose the following cellular schema, based on the Boada et al.(2017) model [2].

LuxR-LuxI inducible model. Cell schema.

Based on the cell schema, we deduct the following biochemical reactions. We assume that our reporter protein will be GFP.

LuxR-LuxI inducible model. Biochemical reactions.

Parameter	Description	Units	Value
CR	Constitutive transcription rate is the transcription rate KR times the mean plasmid copy number in cell cn. In our case, we are using the pACYC replication origin, so cn ∼ 10, and CR = KR·cn	molecules.min-1	7.9
keG	gGFP transcription rate	molecules.min-1	17.5
α	PluxR promoter basal activity	adimensional	0.1 (Estimated)
pR	mR translation rate	min-1	10
pG	mGFP translation rate	min-1	3.09
kA	Synthesis rate of AHL by LuxI	min-1	0.04
k-1	Dissociation rate of (R·A)	min-1	10
k-2	Dissociation rate of (R·A)2	min-1	1
kd1	Dissociation constant of (R·A)	molecules	100
kd2	Dissociation constant of (R·A)2	molecules	20
kdlux	Dissociation constant of (R·A)2 to the PluxR promoter	molecules	10
dG	GFP degradation rate	min-1	0.027
dR	R degradation rate	min-1	0.2
dA	A degradation rate	min-1	0.057
dAe	A degradation rate in culture medium	min-1	0.057
dRA	(R·A) degradation rate	min-1	0.156
dRA2	(R·A)2 degradation rate	min-1	0.017
dmR	mR degradation rate	min-1	0.247
D	Diffusion coefficient of AHL through the cell membrane	min-1	2
Vc	Cell volume(1.1x10-9 μL)/External volume(1x10-3 μL)	adimensional	1.1x10-6
N	Number of cells	cells	240

LuxR-LuxI inducible model. Model parameters.

Applying the Law of Mass Action kynetics (LMA) and some quasi-static approximations and a model reduction, we can obtain the following differential and algebraic equations of the LuxR-LuxI inducible model.

Variable	Biochemical species	Units
n1	GFP protein	Molecules
n2	LuxR protein	Molecules
n3	Dimer of (R.A)2	Molecules
n4	AHL intracellular inducer	Molecules
n5	AHLext intracellular inducer	Molecules
n6	Monomer (R.A)	Molecules

LuxR-LuxI inducible model. Biochemical species.

LuxR-LuxI inducible model. Model equations.

As we can see, we have obtained five differential equations and one algebraic equation. It should be noted that this model does not describe cell growth, but considers a constant number of cells N. This is because during LuxR-LuxI experiments a flow is established in the culture that allows to maintain a constant number of cells.

Once the model has been developed and the model equations have been obtained, we have programmed a MATLAB function called mi_lux.m containing the equations and a MATLAB script called model_ind_lux_analysis.m that performs model simulation and its analysis (to solve the model we have used ode15s MATLAB function).

LuxR-LuxI inducible model. Model predictions using parameter values from literature.

LuxR-LuxI inducible model analysis. The graph shows the relation between AHL concentration and protein expression.

As we can see, the results of the analysis show a clear dependence between the expression of the protein and the inducing AHL molecule concentration. When comparing the experimental data measured and normalized for different concentrations of AHL with the simulated results with our literature parameters, we can see that both take the form of a sigmoid function, or also known as Hill function.

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?

One of the most important phases in the field of Printeria Modeling has been the validation of theoretical models with experimental data. In the Printeria team we wanted to confirm that the models designed are consistent with our experiments. However, we know that the parameters will not take a fixed value, but vary depending on the genetic construction with which we experiment. With this idea in mind, the need to apply an optimization process arises.

The process of multiobjective parameter optimization [3] consists in minimize two or more design objectives that have a trade-off between them as function of some decision variables. In our case, these objectives are estimation errors between experimental data and model predictions, and the decission variables are some of our model parameters. From a dataset, and applying a mathematical algorithm, we can find the optimal solutions, also known as Pareto-optimal. These optimal solutions are such that there are no more solutions that do not improve one objective without worsening the rest.

As this is a multi-objective optimization, there is usually no single optimal solution, but rather a set of them which form the Pareto front. For each of these solutions, we find a set of decision variables associated: the Pareto set [6]. In our case, the Pareto front solutions represent the RMS error between experimental data and simulations, and the Pareto set is the set of parameters that give rise to these solutions.

From the Printeria Modeling team we have established a protocol that allows us to obtain the optimal solutions of the model parameters for any experiment:

1. We define the objectives to be optimized. These will be the values of Fluorescence F/Unit of Absorbance OD or FOD, and the cellular growth (OD) of a Transcriptional Unit (TU). Therefore, if we have different n TU in an experiment, we will optimize 2n objectives.

2. We choose the parameters to optimize and establish the range of values they can take.

3. We select the identification and validation data. The identification data are those used in the optimization process. The validation data are used to compare the experiment with the simulated theoretical model and the optimized parameters.

4. We define the cost function, which specifies the error function to be minimized, simulates the model with different parameter values, and calculates the error between the simulation data and the experimental identification data.

5. The mathematical optimization algorithm is executed. In our case we use the Multi-objective Differential Evolution Algorithm with Spherical Pruning. This algorithm uses the cost function to test different parameter values and thus search for optimal parameter values.

6. The Pareto set and Pareto front diagrams and the different solutions of our optimized parameters for the different objectives are obtained.

7. Decision Making is carried out by the designer. The optimal parameters are chosen for our 2n objectives.

8. The model is simulated with the optimized parameters and compared with the validation data.

Following this protocol, and particularizing it for each experiment, we have been able to determine the validity of our models and obtain the best parameters for our experiments.

Simulation Tool

One of the main challenges of the Printeria project has been to give the user a chance to really understand what our device is doing when it prints genetic circuits on bacteria. Facing this challenge, we have also sought to provide as much information as possible about the Printeria product. And what better way than to give our user a quantitative, mathematical description of what happens in our cells?

In our project we believe that mathematical modeling can be a very powerful tool for transmitting knowledge, so the Printeria Software and Modeling teams have brought together the most outstanding mathematical models in a single Simulation Tool.

Our Simulation Tool has been conceived as a mathematical model simulation program that has been perfectly integrated into our Printeria Controller software. When our user wants to assemble a genetic construct and selects the parts, the Simulation Tool offers the possibility to experiment in silico and predict cell behavior before printing the bacteria. All this, just by pressing the Show model results button... and instantly!

But how does the Simulation Tool work?

The Printeria Modeling and Software teams have jointly designed a program developed in a single Python script (have a look at our simulate.py script). The program combines the functions of the main mathematical models of Printeria with the information of the Printeria kit stored in the Printeria database. In addition, as it is a single Python file, it is stored and run on the Raspberry Pi 3 server.

Now let's see what happens if the user has chosen the Printeria parts and presses the Simulate button...

Simulation Tool working scheme.

1. The Printeria client connects to the Printeria MongoDB database where we store all the information of our Printeria kit.

2. The four parts of the genetic construction (promoter, RBS, CDS and transcriptional terminator) are extracted from the database thanks to their identifiers.

3. The type of model (constitutive or inducible expression) is identified from the type of promoter selected.

4. The model parameters of each construction part are extracted.

5. The initial conditions of the simulation and its duration are established.

6. The model is simulated using Python's odeint function

7. A temporary CSV file is generated in which the we store the results of the simulation in a certain format.

8. From the CSV file, the simulation results are represented graphically on the Printeria site. The user can view or download their results.

And what are the advantages of an application like our Simulation Tool?

• User-friendly. It is very simple to operate for the user. Just select your Printeria parts, press a button... And that's it!

• Integrated. All our software is included in the Raspberry Pi 3 server. You don't need to download the program.

• Instant experimental data. Our Simulation Tool runs simulations at high speed. In a few seconds you'll have your results.

• Customizable parameters. You can modify your parameters and adapt them to your experiment, to obtain more precise results.

• All-in-one. Our Simulation Tool includes several mathematical models, constitutive and inducible, in a single code and solves them particularizing for each Printeria genetic construction.

Would you like to try the Simulation Tool? Go to our Printeria Controller site and discover all the possibilities offered by our software.

What are you waiting for to start discovering it?

References

1. Picó, J., Vignoni, A., Picó-Marco, E., & Boada, Y. (2015). Modelado de sistemas bioquímicos: De la ley de acción de masas a la aproximación lineal del ruido. Revista Iberoamericana de Automática e Informática Industrial RIAI, 12(3), 241-252.

2. Y. Boada, A. Vignoni, and J. Picó. (2017). Engineered control of genetic variability reveals interplay among quorum sensing, feedback regulation, and biochemical noise. ACS Synthetic Biology, 6(10):1903–1912, 2017a.

3. Segel, L. A., & Slemrod, M. (1989). The quasi-steady-state assumption: a case study in perturbation. SIAM review, 31(3), 446-477.

4. Boada, Y., Vignoni, A., Reynoso-Meza, G., & Picó, J. (2016). Parameter identification in synthetic biological circuits using multi-objective optimization. Ifac-Papersonline, 49(26), 77-82.

5. R. Milo and R. Phillips. Cell Biology by the Numbers. First edition, 2015. ISBN9780815345374.

6. Boada, Y., Vignoni, A., & Picó, J. (2017). Reduction of population variability in protein expression: A control engineering approach. Actas de las XXXVIII Jornadas de Automática.

7. U. Alon. An Introduction to Systems Biology. Desing Principles of Biological Circuits. Champan and Hall/CRC, Edition, 2007.

8. Schleif, R. (2000). Regulation of the L-arabinose operon of Escherichia coli. Trends in Genetics, 16(12), 559-565.

9. Boada, Y. (2018). A systems engineering approach to model, tune and test synthetic gene circuits. PhD. Thesis, Universitat Politècnica de València.

10. N. E. Buchler, U. Gerland, and T. Hwa. Nonlinear protein degradation and the function of genetic circuits. Proceedings of the National Academy of Sciences of the United States of America, 102(27):9559–9564, 2005.