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Mathematical modelling page
Mathematical modelling plays nowadays a central role in Synthetic Biology, due to its ability to serve as a crucial link between the concept and realization of a biological circuit: what we propose in this page is a modelling approach to our project, which has proved extremely useful and very helpful before and after the "wet lab".
Thus, immediatly at the beginning, when there was little knowledge, a mathematical model based on a system of differential equations was derived and performed using a set of reasonable parameters, so as to validate the feasibility of the project. Once it became clear, starting from the characterization of the single subparts created in the wet lab, some of the parameters of the mathematical model were estimated (the others are known from literature) and they have been implemented in the same model, in order to predict the final behaviour of the whole engineered closed-loop circuit.
Therefore here, after a brief overview about the advantages that modelling engineered circuits can bring, we deeply analyze the system of equation formulas, underlining the role and the function of the parameters involved.
Experimental procedures for parameters estimation are discussed and, finally, a different type of circuit is presented and simulations performed, using ODE's with MATLAB and explaining the difference between a closed-loop model and an open one.
Thus, immediatly at the beginning, when there was little knowledge, a mathematical model based on a system of differential equations was derived and performed using a set of reasonable parameters, so as to validate the feasibility of the project. Once it became clear, starting from the characterization of the single subparts created in the wet lab, some of the parameters of the mathematical model were estimated (the others are known from literature) and they have been implemented in the same model, in order to predict the final behaviour of the whole engineered closed-loop circuit.
Therefore here, after a brief overview about the advantages that modelling engineered circuits can bring, we deeply analyze the system of equation formulas, underlining the role and the function of the parameters involved.
Experimental procedures for parameters estimation are discussed and, finally, a different type of circuit is presented and simulations performed, using ODE's with MATLAB and explaining the difference between a closed-loop model and an open one.
The importance of the mathematical model
The purposes of writing mathematical models for gene networks can be:
- Prediction: in the initial steps of the project, a good a-priori identification in silico allows to suppose the kinetics of the enzymes (aiiA, Luxi) and HSL involved in our gene network, basicly to understand if the complex circuit's structure and functioning could be achievable and to investigate the value's range of parameters for which the behavior is that expected.
- Parameter identification: using the lsqnonlin function of MATLAB it was possible to get all the parameters involved in the model, in order to perform realistic simulations not only of the single subparts created, but also of the whole final circuit, according to the a-posteriori identification.
- Modularity: studing and characterizing simple BioBrick Parts can allow to reuse this knowledge in other studies, facing with the same basic modules.
Equations for gene networks
Equations (1) and (2)
We have condensed in a unique equation transcription and translation processes. Equations (1) and (2) have identical structure, differing only in the parameters involved.
The first term describes, through Hill's equation formalism, the synthesis rate of the protein of interest (either LuxI or AiiA) depending on the concentration of the inducible protein (anhydrotetracicline -aTc- or HSL respectively). As can be seen in the parameters table (see below),α refers to the maximum activation of the promoter, δ stands for its leakage activity (this means that the promoter is quite induced even if there is no input). In particular, in equation (1), the quite total inhibition of pTet promoter is due to the constitutive production of TetR by our MGZ1 strain, while, in equation (2), Plux is almost repressed in the absence of the complex given by LuxR and HSL.
In the first term of equation (2) we have described the inducer as being represented only by HSL. This formalism stems from the fact that our final device offers a constitutive production of LuxR (due to the upstream constitutive promoter pLac), so that, assuming it abundant in the cytoplasm, we can derive the semplification of attributing pLux promoter induction only by HSL: this is the reason why we didn' t consider LuxR in the equations system as well as LuxI and AiiA.
Furthermore, in both equations (1) and (2) k and η stands respectively for the other parameter of the Hill relationship. The second term in equation (1) and (2) is composed of two parts. The first one (γ*LuxI/AiiA) describes with a linear relation the degradation rate per cell of the protein. The second one (μ*(Nmax-N)/Nmax)*LuxI/AiiA) takes into account the dilution term and is related to the cell replication process.
In the first term of equation (2) we have described the inducer as being represented only by HSL. This formalism stems from the fact that our final device offers a constitutive production of LuxR (due to the upstream constitutive promoter pLac), so that, assuming it abundant in the cytoplasm, we can derive the semplification of attributing pLux promoter induction only by HSL: this is the reason why we didn' t consider LuxR in the equations system as well as LuxI and AiiA.
Furthermore, in both equations (1) and (2) k and η stands respectively for the other parameter of the Hill relationship. The second term in equation (1) and (2) is composed of two parts. The first one (γ*LuxI/AiiA) describes with a linear relation the degradation rate per cell of the protein. The second one (μ*(Nmax-N)/Nmax)*LuxI/AiiA) takes into account the dilution term and is related to the cell replication process.
Equation (3)
The processes described here are not those of transcription and translation, but in principle are enzymatic reactions either related to the production or the degradation of HSL. Based on the experiments performed, we derived Hill's equation in the case of η=1. They cannot be exactly defined Michaelis Menten's equations since that in our formalism, LuxI and AiiA aren't described as enzymes (since they appear also in the denominator). We simply derived empirical formulas relating either LuxI or AiiA to HSL, and treated them with the typical Michaelis Menten formalism since they presented the corresponding sigmoidal shape/switching like behaviour. Regarding to this, we believe that the saturation phenomenon observed either in HSL production rate due to LuxI, or HSL degradation rate due to AiiA, underlies limiting elements in cell metabolism.
In the cell HSL binds to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the promoter Plux.
Intuitively, LuxI activity as an enzyme encounters an intrinsic limit in HSL synthesis depending on the finite and hypothetically fixed substrate concentration (namely SAM and hexanoyl-ACP, see ref.); this means that at a certain LuxI concentration, all the substrate forms activation complexes with LuxI, so that there is no more substrate available for the other LuxI produced. HSL degradation rate is limited by its availability; even if the concentration varies with time, there is always a corresponding limit in AiiA concentration, which determines a saturation in the degradation rate.
Moreover, both the formulas relating either LuxI or AiiA to HSL are multiplied by the number of cells N, due to the property of the lactone to diffuse free inside/outside bacteria. The third term in equation (3) is similar to the corresponding ones present in the first two equations and describes the intrinsic protein degradation.
Intuitively, LuxI activity as an enzyme encounters an intrinsic limit in HSL synthesis depending on the finite and hypothetically fixed substrate concentration (namely SAM and hexanoyl-ACP, see ref.); this means that at a certain LuxI concentration, all the substrate forms activation complexes with LuxI, so that there is no more substrate available for the other LuxI produced. HSL degradation rate is limited by its availability; even if the concentration varies with time, there is always a corresponding limit in AiiA concentration, which determines a saturation in the degradation rate.
Moreover, both the formulas relating either LuxI or AiiA to HSL are multiplied by the number of cells N, due to the property of the lactone to diffuse free inside/outside bacteria. The third term in equation (3) is similar to the corresponding ones present in the first two equations and describes the intrinsic protein degradation.
Equation (4)
Equation (4) is the common equation describing logistic cell growth, depending on the rate μ and the maximum number NMAX of cells per well reachable.
Table of parameters
| Parameter | Description | Unit of Measurement | Value |
| αPTet | maximum transcription rate of Ptet | [(AUr/min)/cell] | - |
| δPTet | leakage factor of promoter Ptet basic activity | [-] | - |
| ηPTet | Hill coefficient of Ptet | [-] | - |
| kPTet | dissociation costant of Ptet ? | [nM] | - |
| αPLux | maximum transcription rate of Plux | [(AUr/min)/cell] | - |
| δPLux | leakage factor of promoter Plux basic activity | [-] | - |
| ηPLux | Hill coefficient of Plux | [-] | - |
| kPLux | dissociation costant of Plux ? | [nM] | - |
| γPLux | LuxI costant degradation | [1/min] | - |
| γAiiA | AiiA costant degradation | [1/min] | - |
| γHSL | HSL costant degradation | [1/min] | - |
| Vmax_LuxI | maximum transcription rate of LuxI | [nM/(min*cell)] | - |
| km_LuxI | dissociation costant ? | [AUr/cell] | - |
| kCAT | ?? | [1/(min*cell)] | - |
| km_AiiA | dissociation costant ? | [AUr/cell] | - |
| NMAX | maximum number of bacteria per well | [cell] | - |
| μ | rate of bacteria groth | [1/min] | - |
According to the table above, the unit of the state variables are:
| State variable | Unit of Measurement |
| d[LuxI]⁄dt | [AUr⁄(min*cell)] |
| d[AiiA]⁄dt | [AUr⁄(min*cell)] |
| d[HSL]⁄dt | [nM⁄(min)] |
| d[N]⁄dt | [cell⁄(min)] |
Parameter estimation
In this section we examine the parameters of the model and justify the units of measure, relating them to the experiments performed for the characterization of the parts. We want to underline again our concept of modelling: beginning to caractherize simplier parts, we get their parameters and we try to predict the behaviour of the final engineerd closed-loop.
Promoter (Ptet & Plux)
These are the first subparts tested.
Firstly, in the figure above "RBSx" stands for, respectively,
RBS30,
RBS31,
RBS32,
RBS34. So we get 4 biobricks for each promoter, in order to investigate what happens in different conditions of RBS's efficiency. In this phase of the project we aim to increase our knowledge about promoter Ptet and Plux but it must be said that, here, it' s quite impossible to focus separately on the only activity of the promoter and RBSx; for this reason, when we "characterize promoters", we mean promoter and RBS together.
We realize this by introducing the mRFP fuorescent protein (followed by a double terminator), and we make the assumption that the number of fluorescent protein produced, due to the concentration of induction (aTc, HSL for Ptet, Plux respectively) is exactly the same as the number given by any other protein that would be expressed instead of the mRFP. In other words, in our hypotesis, if we would substitute the mRFP coding region with a region coding for another protein, we would obtain the same synthesis rate: this is the reason why the strength of the complex promoter-RBSx is expressed in Arbitrary Units [AUr]. Clearly this is a strong hypotesis, however its level of approximation is considered to be adequate.
Keeping this idea in mind, let's talk about the steps to estimate parameters.
As shown in the box below, we consider a wide (more or less, depending on the type of test) range of induction and we monitor, during the time, absorbance (line1, line2) and fluorescence (line3); the two vertical segments for each figure highlight the exponential phase of bacteria' s groth. We are able to make these measurement due to the Tecan Infinite F200, spectrophotometer that allows to know the Scell (explained few lines below) as a function of inducer concentration, thereby providing the desired input-output relation (inducer concentration versus promoter+RBS activity), which was modelled as a Hill curve.
After that, we can calculate the Scell as:
In the end, plotting Scell VS induction, we obtain the activation Hill curve of the promoter considered.
As shown in the box above, α as already mentioned, represent the protein maximum synthesis rate, which is reached, in accordance with Hill's formalism, when the inducer concentration tends to infinite, and, more practically, for sufficently high concentrations of inducer, meanwhile the product α*δ stands for the leakage activity (induction=0ng/µL), liable for protein production (LuxI and AiiA respectively) even in the absence of autoinducer. The paramenter η is the Hill's cooperativity constant and it affects the rapidity and ripidity of the switch like curve relating Scell with the concentration of inducer.
Lastly, k stands for the semi-saturation constant and, in case of a unity value for η, it indicates the concentration of substrate at which half the synthesis rate is achieved.
The unities of the various parameters can be easily derived considering the hill equation and the unity of its left handed side (for more details see the Table of parameters above).
Keeping this idea in mind, let's talk about the steps to estimate parameters.
As shown in the box below, we consider a wide (more or less, depending on the type of test) range of induction and we monitor, during the time, absorbance (line1, line2) and fluorescence (line3); the two vertical segments for each figure highlight the exponential phase of bacteria' s groth. We are able to make these measurement due to the Tecan Infinite F200, spectrophotometer that allows to know the Scell (explained few lines below) as a function of inducer concentration, thereby providing the desired input-output relation (inducer concentration versus promoter+RBS activity), which was modelled as a Hill curve.
After that, we can calculate the Scell as:
AiiA
These experiments aims to learn approximately the degradation rate of HSL due to the expression of AiiA. In these case, we are able to quantify exactly the concentration of HSL, using the well-characterized part BBa_T9002 in the previous iGEM.
This biobrick receives in input HSL concentration, and returns in output the intensity of fluorescence.
So, our idea is to control the degradation of HSL reading the fluorescence of T9002 due to a certain concentration of HSL; monitoring it in precise samples of time since aTc induction (after having waited enough for AiiA to become in stationary phase), we can estimate the degradation rate, also compared with other constructs, which would't degradate it. According to this, it' s necessary to know very well the reationship input-output of the biosensor: a curve of "calibration" of T9002 is obtain for each test performed, even if, in theory, it should be always the same.
Summarizing in few points, the following are the passes involved in the experiment:
- Transform a MGZ1 E. coli strain with the pTet-RBS-AiiA-TT construct, and wait three hours for reaching the exponential phase growth.
- Induce the culture with a proper amount of aTc.
- Take samples of the supernatant at different times (i.e. 0 h,1 h,4 h) and store them in the freezer at -20°C
- Retrieve the supernatants prepared and use them to induce the T9002 construct contained in the TECAN spectrophotometer wells
- Wait until sensing is completed and retrieve the results from TECAN.
