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Signalling is nothing without control...
MODELLING
BASIC CONCEPT
(se trovo articoli e riferimenti posso cambiare la prima frase dicendo "It is well recognized that...")
Synthetic biology greatly relies on modelling as a tool for quantitatively analysing the behaviour of a system. Specifically, it allows to conduct dynamical and stationary phase analyses and parameter sensitivity estimations, based on a mathematical description of the biological system of interest. Therefore it is a valuable predictive tool of the desired behaviour, allowing to test different configurations of the system prior to its implementation. Moreover, it is an essential constituent of the experimental approach, as regards the design of the experiments for the characterization of the parts of the system and data elaboration.
According to this, modelling plays a central role in the development of our project, from the characterization of the individual components to the design of the final device.
Indeed, for every functional part designed, we have realized several variants differing in the entity of the input-output relation. For example, we have assembled various combinations of the promoters of our circuit (namely, pTet and pLux) with different RBSs, resulting in a particular POPSin-POPSout relation.
"mettere disegno?"
After the quantitative characterization of every part, modelling of the final assembled device allows us to predict the global input-output relation of the circuit. This in turn gives an insight into the best choice for the final device, based on criteria such as the achievement of the desired behaviour, the degree of sensitivity and biological tolerance.
In this section the mathematics of our project is provided: first, the system of equations is introduced, together with an explanation of the variables and parameters involved. Subsequently, experimental procedures for parameters estimation are presented.
SYSTEM OF EQUATIONS Equazioni
"\frac{d[HSL]}{dt}= N*Vmax_L_u_x_I*\frac{LuxI}{(km_L_u_x_I + LuxI)} - N*Vmax_A_i_i_A*\frac{[HSL]}{(km_A_i_i_A + [HSL])} - %gamma * [HSL]"
We have condensed in a unique equation transcription and translation processes. (riferimenti ad altri studi). 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), the term delta stands for the leakage activity of the promoter, who is supposed to be partially active, even in the absence of inducer. 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.
The second term in equation (1) and (2) is composed of two parts. The first one (gamma*LuxI/HSL) describes with a linear relation the degradation rate per cell of the protein. The second one (mu*(Nmax-N)/Nmax)*luxI/HSL) is a dilution term and is related to the cell replication process. To understand this, let's consider the simplest case of a single cell's division.
DISEGNO CELLULA CHE SI DIVIDE
When this happens, we can assume that all the content of the mother cell equally distributes in the two derived cells. Consequently, if we had for example ten molecules of LuxI per cell, after the cellular division they would become five molecules per cell.
Now consider equation (3). The processes described herein 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 eta=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. 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.
Similarly, HSL degradation rate is limited by HSL availability; even if HSL concentration varies with time, there is always a corresponding limit in AiiA concentration, which determines a saturation in the degradation rate.
The third term in equation three is similar to the corresponding ones present in the first two equations and describes protein degradation.
Equation (4) is the common equation describing logistic cell growth.
| Parameter | Unit of Measurement | Value |
| αpTet | [(mRFP/min)/cell] | - |
| δpTet | [-] | - |
| kpTet | [nM] | - |
| ηpTet | [-] | - |
| γpTet | [1/min] | - |
| αpLux | [(mRFP/min)/cell] | - |
| δpLux | [-] | - |
| kpLux | [nM] | - |
| ηpLux | [-] | - |
| γpLux | [1/min] | - |
| Vmax_LuxI | [nM/(min*cell)] | - |
| km_LuxI | [nM] | - |
| Vmax_AiiA | [nM/(min*cell)] | - |
| km_AiiA | [nM] | - |
| γHSL | [1/min] | - |
| Nmax | [cell] | - |
| μ | [1/min] | - |
Spiegazione parametri e test con cui misurarli
Equations (1) and (2)
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.
Relating to the equations (1) and (2), we assume acquainted the protein degradation rate, which equals gamma_i=0.0173 due to the presence of the LVA tag (see registry...).
Moreover the dilution term is exactly the specific growth rate, to be determined through an apposite experiment.
What remains is the term describing the synthesis rate.
The parameters involved were experimentally determined through ad hoc designed experiments. We exploited specifically assembled parts, formed by the sequence of
promoter-RBS-mRFP-TT FARE DISEGNO AL RIGUARDO
Indeed, this parts are also amenable to be applied as components for subsequent iGem projects.
promoter-RBS-mRFP-TT FARE DISEGNO AL RIGUARDO
Indeed, this parts are also amenable to be applied as components for subsequent iGem projects.
What we want to characterize is the promoter+RBS complex. 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 is exactly the same as the number given by any other protein that would be espressed 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. Clearly this is a strong hypotesis, however its level of approximation is considered to be adequate.
The construct realized is the best suited for our laboratory equipment instruments. Specifically, our TECAN spectrophotometer allows us to measure Scell(dmRFP/dt/OD) 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.
alfa_pTet and alfa_pLux 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.
delta_ptet and delta_plux, as previously explained, are responsible for basal protein expression (given by alfa_pTet*delta_pTet/pLux), liable for protein production (LuxI and AiiA respectively) even in the absence of autoinducer.
eta_ptet and eta_plux are thed Hill's cooperativity constants, and determine the ripidity of the switch like curve relating Scell with the concentration of inducer.
Katc and kplux are the semi-saturation constant, and in case of a unity value for eta_atc and eta_plux, indicate 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.
[dGFP/dt/cell]=[dGFP/dt/cell]*([-]+([.]-[-])/([-]+[nM]/[nM]))
Third equation
It can be recognized that the characterization of the processes implied in this equation required a greatest level application and formalization, since it involved us to design ad hoc experimental measures we hadn't previously engaged in. Both the experiments can be regarded as made of tho phases, which we can define the stimulation/inducing phase and the reading phase respectively. Each of them relies on a specific construct appropiately induced; the inducing phase construct is different for LuxI dependent HSL synthesis rate and AiiA dependent HSL degradation rate characterization, while the reading phase construct is the same for the two experiments; indeed, it relies on T9002, and allows to determine HSL concentration based on the Scell produced Further details are given in the following paragraphs.
LuxI dependent HSL synthesis rate
As we have previously seen, the first term in equation (3) describes HSL synthesis rate dependence on LuxI concentration. We have also yet explained why we took such a formalism. This is not the typical mathematical modelling approach used to describe pLux promoter activity, usually represented as a function of HSL-LuxR complex concentration. There is plenty of literature references in this context, while little is available in the case of LuxR abundance. Theoretically, this is a particular situation of the more general relating to the complex contemplation, but obviously it is not easy to derive from it.
In the stimulation phase of the experimental set up, we used the following construct:pTet-RBS-LuxI-TT
Theoretically, the biological processes related to this biobrick are those of LuxI synthesis and degradation and the LuxI driven HSL synthesis. In order to isolate the last one, namely HSL synthesis, we should make sure that it is possible to consider LuxI in stationary phase (and so at a constant concentration) and to leave aside LuxI degradation.
The condition on LuxI degradation subsists, since we operate on a shorter time scale (order of hours).
Moreover, the adopted experimental protocol allowed us to consider LuxI in stationary phase.
So we can hypotesize that the HSL produced only depends on the relation between LuxI and HSL, given a specific stationary concentration of LuxI, that in turn depends on aTc concentration.
In order to measure this HSL concentration, we used the T9002 device (reading phase).It is worth mentioning that we can predict LuxI "concentration" (in terms of (dmRFP/dt)/OD) based on the previously characterized pTet-RBS constructs. Moreover, varying the RBS, we can span a relatively large range of LuxI concentrations, providing us with more experimental points.
Hereafter the single passes of the experiment are schematically proposed, in order to better understand it:
1) Transform a MGZ1 E. coli strain with the pTet-RBS-LuxI-TT construct, and wait three hours for reaching the exponential phase growth.
2) Induce the culture with a proper amount of aTc.
3) 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
4) Retrieve the supernatants prepared and use them to induce the T9002 constructs contained in the TECAN spectrophotometer wells
5) Wait until sensing is completed and retrieve the results from TECAN.
AiiA dependent HSL degradation rate
AiiA dependent HSL degradation rate experiment is the same as the previous one as regards the passes involved, and simply differs in that there is no part producing HSL, so we have to inject it at a proper concentration. Then, the construct used in the stimulation phase, namely
pTet-RBS-AiiA-TT,
brings to the production of a known amount of AiiA (expressed in terms of dmRFP/dt/cell). Therefore, this time, when we take samples at different times from aTc induction (after having waited for AiiA to become in stationary phase) we will see AiiA dependent HSL degradation.
So the following are the passes involved in the experiment:
1) Transform a MGZ1 E. coli strain with the pTet-RBS-AiiA-TT construct, and wait three hours for reaching the exponential phase growth.
2) Induce the culture with a proper amount of aTc.
3) 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
4) Retrieve the supernatants prepared and use them to induce the T9002 constructs contained in the TECAN spectrophotometer wells
5) Wait until sensing is completed and retrieve the results from TECAN.
Equation (4)
The parameters Nmax and μ, can be calculated from the analysis of the OD600 produced by our MGZ1 culture. In particular, μ is derived as the slope of the log(OD600) growth curve. Nmax is determined with a proper procedure. After having reached saturation phase and having retrieved the corresponding OD600, we take a sample of the culture and make serial dilutions of it, then we plate the final diluted culture on a Petri and wait for the formation of colonies. The dilution serves to avoid the growth of too many and too close colonies in the Petri. Finally, we count the number of colonies, which correspond to Nmax.
CLOSED LOOP VS OPEN LOOP
Now that we have gone deep into the various aspects of the mathematical model of our closed loop, it's time to explain why it is advantageous with respect to the open loop.
In order to see that, we implemented and simulated in Matlab our closed loop circuit and the open loop one, consisting of the same construct without the feedback loop, that is the part pLux-RBS-AiiA-TT (E37-E40). The table below provides the values for the parameters of the model.
| Parameter | Unit of Measurement | Value |
| αpTet | [(mRFP/min)/cell] | - |
| δpTet | [-] | - |
| kpTet | [nM] | - |
| ηpTet | [-] | - |
| γpTet | [1/min] | - |
| αpLux | [(mRFP/min)/cell] | - |
| δpLux | [-] | - |
| kpLux | [nM] | - |
| ηpLux | [-] | - |
| γpLux | [1/min] | - |
| Vmax_LuxI | [nM/(min*cell)] | - |
| km_LuxI | [nM] | - |
| Vmax_AiiA | [nM/(min*cell)] | - |
| km_AiiA | [nM] | - |
| γHSL | [1/min] | - |
| Nmax | [cell] | - |
| μ | [1/min] | - |
The following graphs represent the corresponding results from the two models. As can be seen, both reach a stable equilibrium point, since that there isn't any positive feedback loop capable of bringing instability. However, starting from the same initial conditions (and with the same values for the parameters), the closed loop settles to a HSL steady state level far lower than the open loop, highlighting its capability to limit HSL concentration to a treshold level.
[[File:closed_loop_simul.jpg]]INSERISCI GRAFICI
On the same basis, it is interesting to observe what happens if we introduce a HSL impulse/stimulus, regarded as a noise. In the open loop model, it adds to the steady state bringing to a higher value of the equilibrium point. On the contrary, the feedback loop circuit is able to partially counteract it and to avoid great changes in the steady state value.
INSERISCI GRAFICI
On a biological level, the ability to control the concentration of a given molecule reveals fundamental in limiting the metabolic burden of the cell; moreover, in the particular case of HSL signalling molecules, this would give the possibility to regulate quorum sensing based population's behaviours.
