Simultaneous suppresion of both sources of fluctuations.

<p>For all panels, shown in the main window is the variance as a function of the strength of regulation and shown in the inset is the mean as a function of the strength of regulation. Low implies noisier input <i>u</i> and higher extrinsic variability. When the dominant source of variability is input fluctuations, then a strong regulation strategy is preferable. However when the contributions of input fluctuations and chemical reaction stochasticity are comparable, then strategies with smaller gains are preferable. This is especially true for the decoupled implementations. <b>A</b>. Coupled FB implementation <b>B</b>. Coupled IFF implementation <b>C</b>. Decoupled FB implementation <b>D</b>. Decoupled IFF implementation. Values were obtained using 40,000 SSA simulations of models shown in Table B in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004958#pcbi.1004958.s001" target="_blank">S1 Text</a>.</p

Adaptation in feedback (FB) and incoherent feedforward (IFF) architectures.

<p><b>A.</b> Cartoon representation of FB and IFF architectures. Input <i>u</i> positively regulates species <i>X</i> and <i>Y</i> (black arrows). <i>X</i> inhibits the production of both <i>X</i> and <i>Y</i> in the FB architecture (red line, left panel) and mediates the degradation of <i>Y</i> (negative regulation) in the IFF architecture (red line, right panel). Both <i>X</i>, <i>Y</i> degrade at rates proportional to their respective populations (black arrows). <b>B.</b> Illustration of coupled and decoupled production. <i>X</i> and <i>Y</i> represent genes whose expression is under the control of the same promoter, but their production is considered coupled only if they are on the same transcript. <b>C</b>. Product inhibition functions considered in the FB implementations. <b>D</b>. The distribution of <i>y</i> at steady state for different values of regulation strength <i>α</i>. As the strength of regulation <i>α</i> increases, the distribution of y gets tighter around the nominal value . <i>u</i> is a Poisson distribution with mean . The distribution is the same distribution for both FB and IFF, with <i>g</i> = <i>g</i>_<i>R</i> used for FB.<b>E</b>. The distribution of <i>y</i> at steady state for different classes of inhibition functions, <i>g</i>_<i>R</i> and <i>g</i>_<i>H</i>. <b>F</b>. Circuit response to white noise. is shown by black stars (for FB) and red stars (for IFF). The bars show the corresponding variance of <i>y</i>(<i>t</i>) of 10000 sample runs at <i>t</i> = 20. The noise in <i>u</i> modeled by adding to the right-hand side of the ODE models, where is the standard Brownian motion.<b>G, H</b> Circuit response to input <i>u</i> given by a birth-death process (with birth rate <i>b</i>_<i>r</i> = 10 and death rate 1). Left panel shows sample trajectory in response to <i>u</i> for different gain implementations. Right panel shows the empirical probability distribution derived from each sample trajectory until final time <i>t</i> = 1000. Panel G shows the response of FB implementations using <i>g</i> = <i>g</i>_<i>H</i> and panel H that of IFF implementations. <b>I</b>. Log plot of the coefficient of variation (cv) of sample<i>y</i>-trajectories in response to <i>u</i> given by a birth-death process with birth rate <i>b</i>_<i>r</i> and death rate 0.1<i>b</i>_<i>r</i> (<i>t</i> ∈ [0, 1000]). Parameters used in D-I: , <i>k</i>₁ = <i>l</i>₁ = 5, <i>l</i>₂ = 1, <i>f</i>(<i>u</i>) = 5<i>u</i>. For , <i>n</i> = 1.1 (for <i>α</i> = 1), <i>n</i> = 11(for <i>α</i> = 10), <i>n</i> = 110 (for <i>α</i> = 100).</p

Feedback and incoherent feedforward models.

<p>Feedback and incoherent feedforward models.</p

Implementation Considerations, Not Topological Differences, Are the Main Determinants of Noise Suppression Properties in Feedback and Incoherent Feedforward Circuits

<div><p>Biological systems use a variety of mechanisms to deal with the uncertain nature of their external and internal environments. Two of the most common motifs employed for this purpose are the incoherent feedforward (IFF) and feedback (FB) topologies. Many theoretical and experimental studies suggest that these circuits play very different roles in providing robustness to uncertainty in the cellular environment. Here, we use a control theoretic approach to analyze two common FB and IFF architectures that make use of an intermediary species to achieve regulation. We show the equivalence of both circuits topologies in suppressing static cell-to-cell variations. While both circuits can suppress variations due to input noise, they are ineffective in suppressing inherent chemical reaction stochasticity. Indeed, these circuits realize comparable improvements limited to a modest 25% variance reduction in best case scenarios. Such limitations are attributed to the use of intermediary species in regulation, and as such, they persist even for circuit architectures that combine both IFF and FB features. Intriguingly, while the FB circuits are better suited in dealing with dynamic input variability, the most significant difference between the two topologies lies not in the structural features of the circuits, but in their practical implementation considerations.</p></div

Expected time to return from competence vs. .

<p>This figure shows the expected value it takes for a cell that started from competence to return to its vegetative state as varies. The results are obtained using FSP (<i>blue line</i>) and SSA (<i>red line</i>). SSA results were generated by averaging over runs. For each data point, the error indicated by the errorbar is no larger than with a certainty no smaller than . These results should be compared with the results obtained using FSP whose error has an upper bound of .</p

CIAT en la década de los ochenta: segunda aproximación del plan a largo plazo: para discusión con dirigentes de instituciones agrícolas nacionales en un seminario especial en el CIAT, 7-9 abril, 1981

<p>This figure shows the probability of entering in competence when is varied. The three plots show simulations from the full model using SSA (<i>black</i>), the reduced model using FSP (<i>blue</i>) and the reduced model using SSA (<i>red</i>). SSA results were generated by averaging over runs. For each data point, the error indicated by the errorbar is no larger than with a certainty no smaller than . This is to be compared to an upper bound of when using FSP.</p

Seven SSA runs.

<p>This figure shows seven different SSA simulation runs of the competence regulatory genetic circuit. Each run is shown by a different color. Long trajectories correspond to high levels of ComK indicating that the cell has entered in a state of competence. This figure illustrates the stochastic nature of competence, by showing that starting from the same initial conditions, only two out out of seven cells enter in competence.</p

Projection of infinite lattice into a finite subspace.

<p>The probability density vector evolves on an infinite integer lattice as shown by the arrows. A boundary region of interest is chosen (shown as a <i>box</i> in the figure). In this region all the reactions are maintained. Outside the region all the states are aggregated into one absorbing state, and the reactions leaving the region are maintained, while return from the outside to the inside of the region is prohibited by deletion of the reactions. We chose the maximum value of , so that we detect the probability of leaving this boundary region within the reactions run time we are interested in.</p

Expected time to return from competence vs. .

<p>This figure shows the expected value it takes for a cell that started from competence to return to its vegetative state as varies. The results are obtained using FSP (<i>blue line</i>) and SSA (<i>red line</i>). SSA results were generated by averaging over runs. For each data point, the error indicated by the errorbar is no larger than with a certainty no smaller than . These results should be compared with the results obtained using FSP whose error has an upper bound of .</p

Single SSA run for hours.

<p>This figure shows a single SSA run. The high level of ComK (shown in <i>blue</i>), as well as the negative correlations between ComK and ComS (shown in <i>red</i>) is a characteristic of competence.</p