Threshold crossing of a regulated molecular species.

<p>(A) A target species <i>X</i> is regulated by either an accumulating activator <i>A</i> or a degrading repressor <i>R</i>. (B) Temporal precision is quantified by the variance of the first-passage time, at which the stochastic molecule number <i>x</i> first crosses the threshold <i>x</i>_*. (C, D) Deterministic dynamics illustrate the effects of regulation. Parameters are <i>kt</i>_* = 20 and <i>K</i> = 15 in C; <i>μt</i>_* = 2.75, <i>K</i> = 2.6, and <i>N</i> = 15 in B and D; and <i>x</i>_* = 15 and <i>H</i> = 1 throughout. <i>t</i>₀ is defined by in C and in D.</p

Model predictions agree with neuroblast migration data.

<p>(A) Number of <i>mig-1</i> mRNA molecules per cell as a function of time <i>t</i>, obtained by single-molecule fluorescent in situ hybridization, from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.ref005" target="_blank">5</a>]. Magenta shows approximate range of times when cell migration terminates. Black lines show mean (dashed) and standard deviation <i>σ</i>_<i>d</i> of cell division times (black points). (B) Timing variance vs. linearity of <i>x</i>(<i>t</i>), both for experimental data in A (blue circle) and our model (curves, Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.e049" target="_blank">16</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.e050" target="_blank">17</a>). Data analyzed using ranges of threshold 10 ≤ <i>x</i>_* ≤ 25 and bin size 3 ≤ Δ<i>x</i> ≤ 12; error bars show standard deviations of these results. We see that for sufficiently large cost 〈<i>a</i>〉/<i>x</i>_* or 〈<i>r</i>〉/<i>x</i>_*, model predictions agree with experimental data point.</p

Results are robust to additional complexities including cell division.

<p>(A, B) Green solid curves show slices from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.g002" target="_blank">Fig 2</a> with <i>K</i> = 10 while black dashed line shows unregulated limit . We see that regulation can reduce timing variance even with bursts in activator production of mean size <i>b</i> (A, cyan and magenta dashed), initial Poisson noise in repressor number (B, green dashed), or steady state <i>k</i>/<i>μ</i> in regulator dynamics (blue) unless it approaches regulation threshold <i>K</i> (red). (C) Mean dynamics of activator model (solid) and repressor model (dashed) in which cell division occurs at time on average. Abrupt reductions in molecule numbers are smoothed by noise in <i>t</i>_<i>d</i> and by binomial partitioning of molecules. (D) Timing variance approaches that with no division (dashed) within experimental division region (gray). In A and B, parameters are as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.g002" target="_blank">Fig 2</a>. In C and D, parameters are <i>x</i>* = 15, 〈<i>a</i>〉/<i>x</i>_* = 〈<i>r</i>〉/<i>x</i>_* = 10, and <i>H</i> = 3, with <i>kt</i>_*, <i>μt</i>_*, and <i>K</i> set to optimal values (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.g002" target="_blank">Fig 2</a>) and and <i>σ</i>_<i>d</i> set to experimental values. In all cases, <i>α</i> is set to ensure that mean threshold crossing time equals <i>t</i>_*.</p

Optimal regulatory strategies.

<p>Timing variance as a function of the regulatory parameters reveals (A) a trajectory along which the variance decreases in the case of the activator and (B) a global minimum in the case of the repressor. White dashed line in A and white dot in B show the analytic approximations in Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.e029" target="_blank">9</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.e031" target="_blank">11</a>, respectively. Parameters are <i>N</i> = 15 in B, and <i>x</i>_* = 15 and <i>H</i> = 3 in both.</p