Validación, verificación y credibilidad (I)

The analysis of the previous section suggests that two of the main potential features of the looping motif are an increase in the level of repression and a robustness with respect to fluctuations in the concentration of repressor. The wild-type architecture of thelac operon is different, however, from the two-operator case considered there. Three operators are present in that case allowing for three different loops. In this section we dissect this case in order to determine which features of the simpler looping motif with only one possible loop still apply to the wild-type case.

Having three binding sites translates into three possible loops: O1-O2 (401 bp), O1-O3 (92 bp) and O2-O3 (492 bp). Each loop corresponds to a different interoperator distance and, therefore, to a different looping free energy. It is of importance to point out the presence of a binding site for the activator CRP between O3 and O1. CRP has been reported to interact with LacI or the LacI-mediated O3-O1 loop [44]. One possible explanation for this is the fact that CRP bends the DNA it binds to, potentially facilitating the formation of the DNA loop [44–46]. As a result of this interaction we might expect the looping free energies for looping to be different in the presence or absence of CRP. In order to determine the respective values of Floop we invoke the different lac operon deletions characterized by Oehler et al. [22]. These constructs are shown in figure 4.10(A), where it is shown that they measured the fold-change in gene expression for all possible deletions of the wild-type lac operon in the context of two different Lac repressor concentrations and in the presence of the CRP binding site.

Since we already have the binding energies corresponding to each operator we can use equation 4.23 in order to determine the looping free energies for the O1-O2 and O3-O1 loops on the basis of the two-operator measurements. However, this strategy does not allow us to determine the looping free energy of the O3-O2 loop from such constructs because this loop does not affect repression directly. To obtain its looping energy we turn to a model of repression in the wild-type lac operon. The full lac operon involves a significant proliferation of states beyond those shown in figure 4.7 and as a result, we go straight to the expression for the fold-change. In order to do so in a more compact fashion we define the weight corresponding to a single

repressor bound to any of the sites as

ri= 2R NN S

e−β∆εrdi_{, (4.25)}

where the subscriptilabels the operator that is bound. When two binding sites are occupied simultaneously we define rij = 4R(R−1) (NN S) 2 e −β(∆εrdi+∆εrdj) _(4.26)

and when all three are bound we use

rijk =

8R(R−1)(R−2) (NN S)3 e

−β(∆εrdi+∆εrdj+∆εrdk)_{. (4.27)}

The states with a loop are described by

rloop(ij)=

2R NN S

e−β(∆εrdi+∆εrdj+∆Floop,ij)_{. (4.28)}

Finally, we can also have a loop and the remaining site occupied by another Lac repressor molecule. We represent this by ri,loop(jk)= 4R(R−1) (NN S) 2 e −β(∆εrdi+∆εrdj+∆εrdk+∆Floop,jk)_{. (4.29)}

In all of these definitions the indices i, j and kcorrespond to the different operators. Using this notation the fold-change in gene expression for the wild-typelacoperon is

fold-change = [1 +r2+r3+r23+r23,loop]/ (4.30)

1 +r1+r2+r3+r12+r13+r23+r123+rloop(12)+ (4.31)

rloop(13)+rloop(23)+r1,loop(23)+r2,loop(13)+r3,loop(12)

.

Notice that the only unknown in this expression is the looping free energy between O3 and O2. We obtain it by fitting this formula to the data for the wild-typelacoperon from figure 4.10(A). In table 4.2 we show the various looping energies for the different operator combinations obtained so far. In section 4.4.1 we obtain looping energies for loops of the same lengths as thelacoperon loops in the absence of CRP. These looping energies are also shown in table 4.2. Notice that for those loops harboring the CRP site within the loop that the difference in looping energy is roughly 2kBT. This stabilization of the loops by CRP is in quantitative agreement with previous results bothin vitro[44] andin vivo [46].

Now that we have all the parameters of equation 4.30 we can predict the fold-change in gene expression for the lac operon and its mutants at any concentration of Lac repressor. These predictions are shown in figure 4.10(A). It is interesting to note that even though a construct bearing only O1 and O2 shows the plateau at wild-type concentrations of Lac repressor that was identified in the previous section, the complete wild-type construct does not maintain this feature. As a result, we conclude that in the natural lac operon

with all three binding sites the plateau in the repression curve does not exist. Wild-typeE. coliis not robust against fluctuations in repressor by the mechanism suggested when only the O1-O2 loop is considered. It has been proposed that the absence of CRP can recover this “robustness” in thelacoperon [39, 47, 48].

Interestingly, all three looping constructs shown in figure 4.10(A) have very similar levels of gene expres- sion at the wild-type concentration of 10 repressors per cell. This becomes more clear in figure 4.10(B) where we plot the probability of the different loops as a function of the number of Lac repressors. It can be seen that the probability of looping between O1 and O2 is approximately equal to that corresponding to O1 and O3 at the wild-type concentration. It is possible that the three-operator system is simply maximizing the amount of repression in the cell over the entire range of repressor concentrations. In figure 4.10(A) it can be seen that over the entire range of repressor concentrations the fold-change resulting from the presence of 3 binding sites is always larger than or equal to the fold-change with 2 binding sites. That is, the fold-change effect is enhanced by the presence of all three operators. Still, this then leaves the question open: if the looping probabilities and the repression levels attained by each loop are the same, what is the functional and evolutionary significance of the full three operator case?

From figure 4.10(B) it is also evident that the predominant loop can be selected by changing the concentra- tion of transcription factor. The fact that such a titration might select for a predominant DNA conformation might be of special interest for eukaryotic domain intercommunication proteins such as SpGCF1 [49].

Finally, a caveat of the model for the lac operon proposed here is that it does not account for “deacti- vation” of CRP when repressor is bound to O3. Oehler et al. [22] observed that there is residual repression in the absence of both O1 and O2 sites. Another possible caveat has to do with an experimental subtlety related to how the operators where deleted. For some of the operator deletions some of the base pairs of the operators were mutated. The choice of bases to mutate corresponded to the ones that had been deter- mined to be most relevant for binding [50]. Still, it was recently proposed that a residual binding energy corresponding to the deleted operators can cause a significant change in the predicted behavior [20, 48]. As a result, a more complex model accounting for both CRP inactivation and residual binding to the deleted operators might be necessary in order to analyze the results by Oehler et al. in more detail.

Table 4.2: Looping energies between the wild-type operators. The energies corresponding to loops in the wild-typelacoperon are shown in the presence and absence of CRP.

Loop Distance (bp) ∆Floop in the presence of CRPa (kBT) ∆Floop in the absence of CRPb(kBT)

O3−O1 92 6.3 7.9

O1−O2 401 8.4 10.3

O3−O2 493 11c ₁₁

a_{Obtained from Oehler et al. [22]} b_{Obtained from M¨uller et al. [25]}

c_{energy is obtained by using the two previous looping energies and solving for the wild-type case with the three operators}

10-1 100 101 102 103 104 10 10 10 10 10 100 [LacI₄] (nM) fold-change -1 -2 -3 -4 -5 401 bp 92 bp O1 O2 O3 CRP lacZ CRP O1 O2 O3 lacZ CRP O1 O2 O3 lacZ CRP O1 O2 O3 lacZ 10-1 100 101 102 103 104 10 10 10 10 100 [LacI₄] (nM) Probability of looping -1 -2 -3 -4 401 bp 92 bp O1 O2 O3 CRP lacZ O3 O1 O1 O2 O3 O2 (A) (B)

Figure 4.10: Contribution of the various DNA loops in the lac operon. (A) Experimental data measured by Oehler et al. [22] and theoretical fits corresponding to the wild-type lac operon and various simpler constructs derived from it. (B) Theoretical prediction of the probability of formation of the different loops in the lac operon as a function of the repressor concentration. The vertical dashed line in (A) and (B) corresponds to the wild-type concentration of Lac repressor.