Variability of the ML estimators, represented by log₁₀ <i>E</i>_c (left), log₁₀ <i>E</i>_nc (center), and the correlation coefficient <i>ρ</i> between and (right) as functions of the ratio of unbinding rates.

<p>Here we use <i>r</i>_nc = 1, <i>k</i>_c = <i>k</i>_nc = 1, <i>c</i>_c + <i>c</i>_nc = 1. The legend and the colors represent different ratios of concentrations of the cognate and the non-cognate ligands . We plot averages over 30,000 randomly generated binding/unbinding sequences for each combination of the rates. Each sequence itself consists of <i>n</i> = 30,000 binding events, simulated using the Gillespie algorithm. Standard errors are too small to show.</p

Histogram of phenotype values of uniformly random sequences for the inferred epistatic model.

<p>Random sequences have very low inferred phenotype values because of the specificity of binding sites. The peak of the distribution indicates what phenotype values evolve under neutral conditions. The the wild-type value, (green line), is much higher than the neutral value indicating selective pressure.</p

Generalizing the fitted function by replacing the output values with a non-linear function improves the least squares fit.

<p>Constrained non-linear optimization found the optimal for the linear model with . The non-linearity is due to the first few bins being dominated by background fluorescence and not gene expression.</p

The interaction coefficients for are clustered around the subunits of the system: CRP, RNAP, and their constituent binding sites (defined by white rectangles in figure 3a).

<p>The total amount of interaction (sum of the magnitude of coefficients) is shown in the first column. The interactions are categorized into three exclusive types of epistasis: synergistic, , , and share the same sign (and are non-zero), antagonistic, and share the same sign, but has opposite sign, and sign epistasis, , and are of opposite sign and is non zero.</p

Comparison of errors of the approximate and the ML solutions.

<p>We plot (left), (center) and the covariance of the approximate estimates (right) as functions of on- and off-rates. Simulations are performed in the same way as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005490#pcbi.1005490.g002" target="_blank">Fig 2</a>. Legends and color scheme are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005490#pcbi.1005490.g002" target="_blank">Fig 2</a>.</p

The LASSO solution of the quadratic model was computed for 100 values of .

<p>Blue is the value, and red is the 10-fold cross-validated . The green curve is the variance of for randomly generated sequences. The variance is too large even for values of that are larger than the optimal value predicted by the maximum of the curve. We choose the model with (dashed line) for further analysis. This model has non-zero coefficients, most of which are epistatic.</p

The model.

<p>(a). Two ligands, cognate and non-cognate having concentrations <i>c</i>_c and <i>c</i>_nc, bind to a receptor R with binding rates <i>k</i>_c and <i>k</i>_nc, respectively. The cognate unbinding rate is defined as lower than the non-cognate one (<i>r</i>_c < <i>r</i>_nc). (b) Time series of receptor occupancy is used to determine both on-rates.</p

Sensitivity of the epistatic coefficients to the choice of the regularization parameter .

<p>As in Fig. 3, we show the matrices of the sums of the absolute values of the pair interaction coefficients for each pair of sites . a) Coefficients for the model with maximum (). b) Coefficients for the full model: . Notice the same general structure of the coefficients for varying , including in Fig. 3. This indicates stability under changes of the parameter.</p

(blue) coefficients for the non-epistatic model with no-glucose (normal levels of cAMP) (red) with glucose (no cAMP).

<p>CRP is activated by cAMP and does not bind without it.</p

Stem plot of the linear coefficients.

<p>Three circles on each stem represent the changes in phenotype for each of the three possible mutations per site. CRP and RNAP are known to each bind at two sites (magenta and cyan areas). Red circles correspond to the mutations needed to get the consensus sequences.</p