Criticality mirrored by the unimodal-bimodal transition through the flattened unimodality.

<p>Criticality of the whole expression at 10–15 min of MCF-7 cell stimulated by HRG exhibits three distinct response domains going from higher to lower <i>nrmsf</i> (left to right in the figure): (left) dynamic domain (<i>nrmsf</i>> 0.16; unimodal profile: <i>N</i> = 3269 mRNAs), (middle) transit domain (0.08 <<i>nrmsf</i>< 0.16 flattened unimodal profile: <i>N</i> = 9707 mRNAs), (right) static domain (<i>nrmsf</i> < 0.21; bimodal profile: <i>N</i> = 9059 mRNAs). First row shows the corresponding putative energy profiles (<i>x</i>-axis: states; <i>y</i>-axis: energy, here specified in abstract terms referring to a physical system undergoing a transition) from a single-well to double-well profiles through flattened single-well profile (blue: 10 min; 15 min: red). These energy profiles should correspond to free energy in terms of the symmetry argument of Landau. Second row shows frequency distributions of mRNA expression from unimodal to bimodal distribution through a flattened unimodal distribution (<i>b</i>: Sarle's bimodality coefficient; <i>x</i>: natural log of expression, <i>ln</i>(<i>ε</i>(t)) and y: natural log of frequency; blue polygonal line: 10 min; red histogram: 15 min); Third row reports the density profile in the regulatory space (<i>x</i>: natural log of expression, <i>ln</i>(<i>ε</i>(10min)) at 10 min vs. <i>y</i>: log of the change in expression at 10–15 min, <i>ln</i>(<i>ε</i>(15min)/<i>ε</i>(10min))) showing clear unimodal to bimodal transition (color bars: probability density). Match of peaks of the histograms and density profiles confirms the statistical reliability of the unimodal-bimodal transition of frequency distribution. The temporal invariant flatness of energy profile suggests the existence of the critical point (CP) (<i>ln</i>(<i>ε</i>(CP)), black solid circle), which is the point where up- and down-regulation balance, i.e., the point where the change in expression between different time points is around zero.