15 research outputs found

### 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><sub><i>d</i></sub> 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><sub>*</sub> ≤ 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><sub>*</sub> or 〈<i>r</i>〉/<i>x</i><sub>*</sub>, model predictions agree with experimental data point.</p

### 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><sub>*</sub>. (C, D) Deterministic dynamics illustrate the effects of regulation. Parameters are <i>kt</i><sub>*</sub> = 20 and <i>K</i> = 15 in C; <i>μt</i><sub>*</sub> = 2.75, <i>K</i> = 2.6, and <i>N</i> = 15 in B and D; and <i>x</i><sub>*</sub> = 15 and <i>H</i> = 1 throughout. <i>t</i><sub>0</sub> is defined by in C and in D.</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><sub><i>d</i></sub> 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><sub>*</sub> = 〈<i>r</i>〉/<i>x</i><sub>*</sub> = 10, and <i>H</i> = 3, with <i>kt</i><sub>*</sub>, <i>μt</i><sub>*</sub>, 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><sub><i>d</i></sub> set to experimental values. In all cases, <i>α</i> is set to ensure that mean threshold crossing time equals <i>t</i><sub>*</sub>.</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><sub>*</sub> = 15 and <i>H</i> = 3 in both.</p

### Dependence of the population size on the inoculation density.

<p>Colony cultures inoculated with different cell densities grow to different population sizes. Circles are experiment data measured at 74 hr post inoculation, and error bars are s. e. m. The best-fit 3-d bacterial growth model reproduces these data within experimental error bars and computational confidence interval, without additional fitting.</p

### <i>E. coli</i> population dynamics.

<p>Experimental data, averaged over all experiments (symbols, error bars are s. e. m.), are compared with the fits of the mathematical model we developed (solid lines). For clarity, uncertainty of the numerical predictions is omitted and is shown instead in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005679#pcbi.1005679.g004" target="_blank">Fig 4</a>. Liquid cultures switch abruptly from the exponential growth to the saturation, and then decay slowly. In contrast, 3-d colonies gradually slow down before saturating (see Inset) at a population size larger than that in the liquid, and then decay. Note that the curves start with ∼50 CFU/ml, which corresponds to over ∼150 colonies started by individual cells in the 3-d colony setup.</p

### Mathematical model predictions.

<p>(A) Population growth in liquid culture and in 3-d colonies. The growth parameters are chosen as best fit values for our experimental data (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005679#pcbi.1005679.t001" target="_blank">Table 1</a>), except for <i>D</i>, which we vary to illustrate different growth regimes. The diffusion-limited regime in the limit of small <i>D</i> is consistent with the prediction <i>N</i> ∝ <i>t</i><sup>3/2</sup>. The time scales <i>τ</i><sub><i>i</i></sub> are illustrated for <i>D</i> = 1.4 × 10<sup>5</sup> <i>μ</i>m<sup>2</sup>/hr. (B) Profile of the nutrient concentration in space at different times using the same parameters as above and <i>D</i> = 5.5 × 10<sup>5</sup> <i>μ</i>m<sup>2</sup>/h, as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005679#pcbi.1005679.t001" target="_blank">Table 1</a>. The edge of the colony is illustrated by stars on each curve. The inset shows that the concentration decreases exponentially at the colony edge in the diffusion-limited growth regime. The penetration depth is about 3 <i>μ</i>m.</p

### 3-d colony growth.

<p>(A) Photograph of a representative <i>E. coli</i> colony inside 3-d agar at 22 hrs post inoculation. (B) A growing colony at 22 hrs as simulated using our mathematical model. Heatmap shows the spherically symmetric nutrient concentration, and the meshgrid sphere represents the colony. At this time, the nutrient at the center of the colony is fully consumed. Since the growth rate depends on local nutrient concentration, the cells at the center of the colony are not growing anymore.</p

### Cells in liquid and in colonies have different sizes.

<p>(Left) The fraction of non-filamentous cells (< 5 <i>μ</i>m in length) in liquid cultures and in colonies. Color convention is as in the previous figures. Error bars represent the square-root counting statistics. (Center) The mean and the median cell sizes in liquid and colony cultures. Error bars of the means are s. e. m. Error bars of the medians are the bootstrapped 95% confidence intervals. (Right) The ratio of cell sizes in liquid to those in colonies. The solid line shows the ratio of the means, and the dashed line is the ratio of the medians. Error bars are propagated from the error bars of the means and the medians. Cells in old colonies are 1.6 to 3.4 times shorter than in old liquid cultures.</p