Distributed delay can either increase of decrease pulse duration in an incoherent feedforward loop.

<p>(A) <i>Top</i>: The longer pathway consists of the sum of two shorter pathways: . (A) <i>Bottom</i>: The expected value of signaling time as a function of the relative standard deviation of the delay time. (B) <i>Top</i>: The shorter pathway is simply the signaling of the first gene to the third. (B) <i>Bottom</i>: Expected signaling time, . (C) <i>Top</i>: The output pulse is determined by the amount of time gene is actively transcribing. This time is simply the difference of the longer path duration () and the shorter path duration (). (C) <i>Bottom</i>: Depending on the thresholds , , and , the expected pulse duration can either increase or decrease as a function of the delay variability. In each of the three plots, the data on the vertical axis are presented relative to the mean pulse duration at . Here, the colored lines correspond to (blue), (green), and (brown), while , . In addition, the protein degradation rates are each , all delays are gamma distributed with mean .</p

The effects of distributed delay on transcriptional signaling.

<p>(A) For the simplified symmetric distribution where the delay takes values and with equal probability, the mean signaling time decreases with increasing variability in delay time, Eq. (8). Shown are the signaling times (normalized by the time at ), versus CV of the delay time for signaling threshold values from (red), through (green) to in steps of 1. Here and . When (brown) increasing randomness in delay time has little effect on the mean. (B) Same as panel (A) but with the probability distribution, , for different values of . (C) The transition from the small regime to the large regime occurs when . Here we fix and between the different curves vary from (magenta) to (orange) in steps of 1. Dashed lines show the asymptotic approximations, Eqs. (9) and (10), which meet at the black line. Panels (D) and (E) are equivalent to panels (A) and (B), with following a gamma distribution, , and . (F) The coefficient of variation of the signaling time, , as a function of .</p

Network schematics for the coherent and incoherent feedforward loops.

<p>Each pathway in the networks has an associated signaling threshold () and mean delay time (). The random time between the initiation of transcription of gene to the full formation of a total of proteins is denoted , which is an implicit function of .</p

Signaling time depends on the number of initiation events.

<p> can <i>increase or decrease</i> as a function of depending on the value of . Here . (A) vs. CV of for varying from (red) to (green) to (blue) using the Bernoulli delay distribution in Example with . Note the transition that occurs at . (B) Equivalent to (A), but plotting CV of the signaling time instead of conditional expectation. (C) and (D) Contour plots corresponding to (A) and (B), respectively. Notice that for fixed , signaling time CV can change non-monotonically with . For instance, at , signaling time CV starts low (red), increases to (green) and then decreases thereafter. Plots were obtained through stochastic simulation with trials.</p

Distributed delay in the delayed negative feedback oscillator.

<p>Shown are the analytically predicted (solid lines) and numerically obtained (symbols with standard deviation error bars) mean peak heights of the negative feedback oscillator with Hill coefficients of (orange), (red), and (i.e. step function, black). The top inset shows the shape of the Hill function for the three values of , with colors matching those in the main figure. The lower inset shows one realization of the oscillator at parameter values corresponding to the large black circle on the orange () curve of the main figure. The average and the standard deviation of the peak heights were calculated from stochastic simulations of oscillations. Here , , , and .</p

The effects of distributed delay on transcriptional signaling.

<p>(A) PDFs for the signaling time using the delay distribution from Example with . The PDFs in red correspond to signal threshold value , green to and brown to . Here and . (B) A 2D view of panel (A) with . Solid lines show analytical results which are nearly indistinguishable from those obtained through stochastic simulation (black lines). Note that the discontinuity in the green curve is due to the discrete nature of the Bernoulli delay distribution. The CDF, , has jump discontinuities that, in light of Eq. (18), produce jump discontinuities in the signaling time PDF. The discontinuity is apparent in both the theoretical prediction (green line) and the stochastic simulations (black line). Panels (C) and (D) are equivalent to panels (A) and (B) with following a gamma distribution. The PDFs were discretized over 200 bins using trials.</p

Experimentally tracking single cells during glucose depletion assays.

<p>A) Cells were trapped within a microfluidic flow chamber and the environmental concentration of glucose was depleted as a function of time, while holding the galactose concentration constant. As glucose levels dropped, individual cells heterogeneously activated Gal1p production, and the resulting fluorescence trajectories were recorded. B) Glucose concentration as a function of time (red line). Here, the depletion time is 4 hrs. Also shown is the experimentally measured fluorescence trajectory of an individual cell to the 4 hr depletion time (green line). This cell first initiates Gal1p production and then accumulates protein (below). C) Images of yeast cells in the microfluidic device at successive events from a 4 hr glucose-depletion assay. [glu] = 2%, t = 0 (hr) indicates the beginning of glucose depletion; [glu] = 0% indicates total depletion of glucose; labels FI = 0 and 200 (AU) correspond to the times at which FI reaches these values; end of run, the end of glucose depletion assay.</p

Results of the mathematical model.

<p>A) Simulated standard deviation of the accumulation time as a function of the depletion time for the full model (red dots) and a model lacking the energy scaling (blue dashed line). Note that the energy scaling recreates the non-monotonicity observed in the experiments. B) Box plots of the distributions of the different types of cell cycle lengths for the 3 hr depletion scheme. The red box plots are the experimental data, whereas the blue box plots are obtained from a simulation of Eq. (5). The distribution of the energy availability in the diauxic phase (<i>ϵ</i>) is obtained by fitting the model to the diauxic cell cycle data.</p

An energy model for the glucose/galactose switch.

<p>A) A schematic of the regulation in the heuristic stochastic model. Glucose inhibits expression of the GAL genes, but increases the cellular energy needed for protein production. Gal4p up-regulates transcription of <i>gal2</i>, whose gene product, Gal2p, imports galactose. The ability of the cell to metabolize galactose is therefore dependent on the availability of Gal2p. In addition, intercellular galactose increases cellular energy and up-regulates <i>gal2</i>. B) We used piecewise linear functions for <i>E</i>_<i>glu</i>([<i>glu</i>]) and <i>E</i>_<i>gal</i>([<i>Gal</i>2<i>p</i>]). The <i>E</i>_<i>glu</i> term is assumed to only depend on the environmental glucose concentration. The <i>E</i>_<i>gal</i> term does not depend on environmental galactose because it is held constant; what changes is the amount of galactose that is utilized by the cell, a good proxy for which is the concentration of Gal2p. The maximum (steady-state) energy levels </p><p></p><p></p><p></p><p><mi>E</mi></p><p><mi>g</mi><mi>l</mi><mi>u</mi></p><p><mi>m</mi><mi>a</mi><mi>x</mi></p><p></p><p></p><p></p><p></p> and <p></p><p></p><p></p><p><mi>E</mi></p><p><mi>g</mi><mi>a</mi><mi>l</mi></p><p><mi>m</mi><mi>a</mi><mi>x</mi></p><p></p><p></p><p></p><p></p> are inferred from the cell-cycle lengths in the two conditions. The threshold <i>th</i>_<i>glu</i> is the glucose threshold at which the cell is assumed to obtain maximal energy. The equivalent threshold for galactose is <i>th</i>_<i>gal</i>; this is the threshold at which the cell has sufficient Gal2p to maximally utilize environmental galactose.<p></p

Experimentally measured single-cell responses to different glucose-depletion times.

<p>Ten glucose-depletion times were tested. Red lines depict changes in glucose concentration from 2% to 0% (w/v). Gray curves depict individual fluorescence trajectories of Gal1-YFP. Green curves show mean FI. For reference, the fluorescence of cells grown in just galactose was approximately 8200 ± 2000 AU (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004399#pcbi.1004399.s001" target="_blank">S1 Dataset</a>, sheet 14).</p