Regularly log-periodic functions and some applications

We prove a Tauberian theorem for the Laplace--Stieltjes transform and Karamata-type theorems in the framework of regularly log-periodic functions. As an application we determine the exact tail behavior of fixed points of certain type smoothing transforms

Cadherin-mediated cell segregation.

<p>Four combinations of L-cell lines are shown: <i>a)</i> L-cells+L-cells(E-cad), <i>b)</i> L-cells+L-cells, <i>c)</i> L-cells(N-cad)+L-cells(E-cad) and <i>d)</i> L-cells(N-cad)+L-cells, with the cell type named in the order green+red and the expression of each cadherin type (N and E) indicated in parentheses.</p

Cadherin mediated cell segregation.

<p>Frame <i>a</i> shows the digitised cell positions at the end of the experiment with the corresponding RDF plotted in part <i>b</i>, coloured as in previous RDF plots (See legend to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043226#pone-0043226-g002" target="_blank">Figure 2</a>). The black line on the RDF plots is the expected random value and the light blue bar marks the region over which the peaks were summed to calculate the peak-ratio segregation score (S).</p

Clustering with increasing cross-link strength.

<p><i>a</i>. The effect of increasing stickiness on clustering was plotted as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043226#pone-0043226-g001" target="_blank">Figure 1</a> for simulations of different lengths from 10 K to 500 K, which appear as a series of traces with the longer runs in bolder lines. These results differ from those in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043226#pone-0043226-g001" target="_blank">Figure 1</a> by having the strength of links between cells increase with their linkage time, controlled by the parameter sieze = 100 (the results for unhook = 0 are not plotted in this graph). <i>b</i>. Using simulated cells at the same density as the cadherin cells and with a value of sieze = 600, 25 simulations were run for 4000 cycles over a range of unhook values (red lines) and compared to the results with no time dependent adhesion (blue lines), corresponding to a very large value of sieze. The dashed lines mark the mean of the top 20% of runs at the optimal value of unhook for the plots of corresponding colour.</p

RDF plot for long Eph/ephrin segregation taken from the images obtained from two experiments in which the cells were allowed to segregate for two days.

<p>Plot <i>a</i> was calculated from the image and data shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043226#pone-0043226-g012" target="_blank">Figure 12</a>. The configuration of cells giving rise to plot <i>b</i> had separated into two green clusters giving rise to the more extreme values (see legend to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043226#pone-0043226-g005" target="_blank">Figure 5</a> for an explanation of the plots and note the change in Y-axis scale in plot <i>b</i>). In both plots it can be seen that the separation of the red cells has increased, shifting the red peak in the plots away from the measured region (light-blue bar). This makes the value of the segregation score less reliable.</p

Cell segregation with simulation length.

<p>The degree of cell segregation as measured by the like∶mixed type ratio score (S, Y-axis) is plotted against ‘virtual time’ measured by the square-root of the number of time-steps in thousands (X-axis. e.g., 30 = 30×30 thousand = 900,000 steps). The filled circles are data from full simulations starting with a random mix of cell types and the open circles started from artificially segregated cells. The latter are plotted at the predicted time when the best-fit line to the full simulation data (solid line) cuts the mean value of the segregated data (horizontal dashed line).</p

Eph/ephrin cell segregation after two days.

<p>Part <i>a</i> shows an image of the cells at the end of the experiment and part <i>b</i> shows the corresponding automatically digitised data derived from the image.</p

Score variation with peak width and displacement.

<p>Values of the peak-ratio score are plotted for displacements to the start and end points of the segment over which the peaks are summed (corresponding to variations of the light-blue bar in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0043226#pone-0043226-g008" target="_blank">Figure 8d</a>). Data are plotted for symmetry generated variations for each of the ten final frames for the Eph/ephrin data (red) and the cadherin data (blue). The three dimensional plot is viewed down diagonal lines of data that have the same peak width (increasing to the left). The green crosses mark the observed data at the “default” peak size of 10 microns used throughout. Variation in this region spans roughly 2 units of segregation score (5–7) which is 6+/−1 or 17% variation.</p

The effect of “stickiness” on segregation was measured by plotting the ratio of like-contacts and mixed contacts (S in <b>Equ. 2</b> on the Y-axis) against the value of the parameter unhook which controls the degree of cell cross-linking (X-axis, as negative log-value. i.e. 3 = 1/1000).

<p>The segregation score is plotted for simulations of different lengths from 1000 to 500 K steps, which appear as a series of traces with the longer runs always more segregated (higher). For the two longest runs (100 K and 500 K steps) the bold trace is the average over three runs shown by dashed lines. The results for unhook = 0 (i.e. no release) are plotted as squares at an arbitrary point to the right (marked by the infinity symbol) and connected to each trace by a dotted line for clarity. The large dot marks the maximum degree of segregation obtained with unhook (plotted at 4.3).</p

MSD plot for cell populations.

<p>The Mean square displacement, MSD (Y-axis, square microns), is plotted against time (X-axis, seconds) for the two cell lines: red, in which the EphB2 and ephrinB1 were expressed and blue for cells with differential expression of E and N cadherins. The fine lines are linear fits to the data with slopes: red = 0.16 and blue = 0.04 (the latter is such a close fit that it is almost invisible).</p