Parameters Estimation

Getting a "complexation function" from convenient datas

Therefore, we have written a little module which can estimate the parameters of a complexation function, even with some noise and few data available.

The method we have employed is essentially based on a least-square optimization.

Getting convenient datas

Thus, we need experimental datas. To quantify the strength of a transcription factor on a promoter, we will use measurements of GFP fluorescence, and compare to the strength of the constitutive promoter http://partsregistry.org/Measurement/SPU/Learn J23101, as it was proposed by the iGEM competition. The datas we are looking for must appear as a table of values, giving several "expression rates" with their corresponding "transcription factor concentrations".

First hypothesis

For this aim, we made several hypothesis, which we will verify as good as it is possible for us :

(1) We do not take into acount the 'traduction' phase (see however considerations on RBS), so we directly correlate the transcription of a gene with the concentration of its protein.

(2) We assume that, whatever is the gene behind the promoter, its expression depends only of the transcription factor of the promoter, and not, for instance, of the weight of this gene. That's why comparing promoter strength is relevent only if the genes behind have similar length.

(3) We consider that the activity of a promoter is well described as a a transduction rate ( b , we will later note β = b*p_tot, where p_tot denotes the total number of promoters p present in the cell), multiplied by the amount of efficient promoters (p_eff), depending on the quantity of its transcription factor (TF), and on the coefficients of their complexation reaction ( K et n ), typically leading to a Hill function. Thus, we suppose that the protein concentration (Prot) follows this equation :

where γ is a constant, due to degradation and of dilution of the protein, along time and cell divisions.

(4) Endly, knowing γ will give us the data we are looking for. In a first approach, we assume that, as long as the barcteria are in their phase of exponential growth, the degradation is far smaller than the dilution, and can be omitted. But we will probably discuss that later (see considerations on RBS). First, we wanted to keep a permanent exponential phase of growth with the system of Ron Weiss (see our project), but we finally (to simplify our already huge construction!) decided to put the cells-culture in a chemostat (see Bio-Chemical General Assumptions).

(5) Unless we find further documents dealing with the relation between the intensity of fluorescence and the concentration of GFP, we will directly use the measure in fluorescence, that we will treat as a protein concentration, more or less arbitrary normalised.
Actually, we will use the linear relation between GFP mut3B concentration (nM) and fluorescence (au), given in the registry [6]. The conversion factor is 79,429.

How to control the concentration of the transcription factor ?

Using {aTc, TetR, pTet}

Now, we must use as a variable of reference an element that could be introduced in the bacteria, well-controlled, and from which all the concentrations of our transcription factor will depend. We propose a construction in which our transcription factor is put after the promoter pTet, which is under the repression of TetR. Since aTc is a small diffusive molecule that binds to TetR and inhibits this way the repression of pTet, we can use it as an "inducer". To do so, we must place in the bacterium the gene tetR after a constitutive promoter (like J23101). According to previous hypothesis, this will provide at steady-state a "constant concentration" of TetR (we note [TetR]_tot, and it is supposed to be the TOTAL concentration of TetR, under every form) in the bacterium. If we consider the binding reaction this way (where aTc><TetR denotes the complex)

with a dissociation constant K_aTc, we find at the steady-state

where [aTc]_i denotes the concentration of aTc we introduced in the medium, that will stay constant in all the bacteria along time, assuming that its degradation is near 0, and that the diffusion is quick. That determines the amount of TetR unbound :

According to the hypothesis (3), the activity of pTet would verify (keeping the same notations) :

leading to this steady-state :

In the last equation, we will have 'access' (see hypothesis (5)) to [prot]_eq and possibly to γ_prot, and we are looking for β_tet, K_tet and n_tet, thanks to our program (see getting Hill function with convenient datas). But we need to know [TetR_free], or [aTc><TetR]. We control [aTc]_i and we could estimate [TetR_tot], that depends of the constitutive promoter (J23101) we will put before, which we will caracterise (see the function we are looking for).

Using {AraC, Arabinose, pBad}

An other well-known promoter called pBad, is induced by the complex AraC><arab, where AraC appears to be a protein constitutively produced by an operon attached to pBad, and arabinose is a sugar, that we can add at will in the medium, and diffuses through cells. By considering this reaction of complexation :

and treat it exactly as for the pTet expression.

RBS Issue

The promoter is not the only one factor which control the expression of the protein... in particular, the traduction phenomenon is almost as important as the transduction. As we decided not to take into acount the traduction, that means that we do not want to deal with mRNA and ribosomes ; nevertheless, as we aimed in the "hill approach" to simulate as precisely as possible the involved concentrations, we must integrate in the traduction simulation the most important influences of the transduction'.

That is why, rigorously, we must describe an induction (or repression) by the association "promoter + specific RBS".

Still, as it is proposed by iGEM, we will use the GFPgenerator (E0240) in association with its RBS (E0032), to caracterise the expression of the gene behind promoters. However, the RBS before the genes which codes for the transcriptions factors we want to induce, are the natural RBS (specific respectively to tetR, flhDC, fliA, etc... ). Therefore, we must pay attention on what we are measuring.

The idea is that, by considering that the traduction rate depends near linearly of the Ribosome Binding Site (as we guess ; it gives the affinity between the mRNA and the ribosome !), we introduce a constant factor between the expected value of a protein and its real value.

Moreover, we observe that at the steady state, the degradation rate γ has got a linear effect on the concentration of the protein at equilibrium, too.

For instance, if we consider a protein prot coded by its corresponding gene, put behind one of our "caracterised promoters", with its natural RBS, we will have between our construction (expected value, given by GFP) and the real expression of prot the following relation

Because we want to stay as near as possible to the "natural construction of the flagella" (see our project), we will keep the natural RBS of every genes we want to put in our final construction (particularly flhDC and fliA).

Also we must remember that the we are not necessarly dealing with real quantities, but with abstract quantities that are coherents together, in our construction. Finally, in our research of caraterising the expression of the genes in a so simplified way, those methods allow us to skip evaluations of the relative strength of the RBS, and of the degradation rates of our proteins.

To conclude, we nevertheless must notice that the effect of the coefficient γ in our model has got an other effect : it changes the dynamics of the system, in the time in which the steady states are reached. As our system intend to produce oscillations, the transcient states are important, and here is the biggest weakness of this model.

What are we looking for ?

The different functions we would like to determine are the followings. They are linked to a basic description of the "theoretical protocol" that will allow us to get the expected datas. We decide to let the original promoters in the bacteria, so that the strength that we are measuring is "the strength for an additional promoter in the cell", keeping those which already exist : this makes sense for our construction, and probably for most of the constructions of synthetic biology wih E.coli.

Because of the undetermined strength of the RBS and degradation rate of the different proteins, the parameters we will have acces to are biased. When a parameter is noted K_ref and is referenced later in K_ref/{coef}, that means that we will calculate the latter, and that this value will be the one instead of the former in the equations we will introduce in the system of ODE.

• [expr.(J23101)] = ƒ0

• [expr.(pTet)] = ƒ1([TetR],[aTc])

• [expr.(pBad)] = ƒ2([Arabinose])

According to the hypothesis (1) and (2), we assume this will directly give us [Protein] = ƒ0, [Protein] = ƒ1([aTc]) or [Protein] = ƒ2([Arabinose]) for a given Protein coded by a gene put behind the J23101, pBad or pTet promoter.

• [expr.(pFlhDC)] = ƒ3([OmpR*],[FliA])

• [expr.(pFlhDC)] = ƒ3bis([EnvZ*],[FliA])

• [expr.(pFlhDC)] = ƒ4([FlhDC],[FliA])

• [expr.(pFliL)] = ƒ5([FlhDC],[FliA])

• [expr.(pFlgA)] = ƒ6([FlhDC],[FliA])

• [expr.(pFlgB)] = ƒ7([FlhDC],[FliA])

• [expr.(pFlhB)] = ƒ8([FlhDC],[FliA])

• [expr.(pLas)] = ƒ9([LasR><HSL])

• [expr.(pLux)] = ƒ9([LuxR><HSL])

Noise estimation

Firstly, we could compute the standard deviation for each set of points at a given inducer concentration, and to normalize it according to the mean value of the set itself.

The next stage is to get the mean of those normalized sd values at every inducer concentration.

Since the β parameter of the Hill function has a linear influence, it is possible to translate this error directly on the parameter; for example: β_err = random('norm',β,err) where β is the estimated value of the parameter and err is the mean of the normalized standard deviations of the experimental values.

For each cell in the model, we could use such noised values for Vmax parameter, in order to reproduce randomness estimated in the wet lab.