1. The Interplay between Feedback and Buffering in Cellular Homeostasis
Edward J. Hancock, Jordan Ang, Antonis Papachristodoulou, Guy-Bart Stan 
Cell Systems 
Volume 5, Issue 5, Pages e23 (November 2017)
DOI: /j.cels
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2. Cell Systems 2017 5, 498-508.e23DOI: (10.1016/j.cels.2017.09.013)
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3. Figure 1 The Basic Buffering-Feedback Topology in this Study
(A) Network diagram for the generic minimal model in Equation 1. y represents the regulated species, x the buffering species, and dy the disturbance. x is the buffer reservoir that compensates for changes to y via Le Chatelier-driven y↔x conversion. Feedback, represented by the dashed line, senses the y amount and actuates the y production rate, py, to compensate for changes.
(B) Minimal model for ATP regulation. ATP is the regulated species, pCr is the buffering species. ATP is supplied via the glycolysis pathway and consumed via downstream ATP demand. Perturbations to the supply and demand of ATP represent disturbances to the model.
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4. Figure 2
Feedback and Buffering Each Work More Effectively at Different Timescales
Comparison of the effects of oscillatory and impulse production disturbances on the regulated species for various buffering and feedback strengths (see the Appendix for simulation parameters). Contour plots of the sensitivity functions ϕω (Equation 2) or ϕimp (Equation 3) are shown in the top row. Smaller values of ϕω or ϕimp imply better regulation. Corresponding temporal plots are shown on the bottom row, where trajectories correlate to similarly colored dots in the contour plots. Here, y¯ is the set point of the regulated species, and smaller deviations from y¯ imply better regulation. Note that in the oscillatory cases, temporal plots show the response some time after the onset of the disturbance, once oscillations have settled to a steady state.
(A–F) Low-frequency (A and D), medium-frequency (B and E), and high-frequency (C and F) oscillatory disturbances. For the temporal plots, the disturbance amplitude is 50% of the nominal production rate py. Feedback is the dominant regulatory mechanism at lower frequencies, while buffering dominates at higher frequencies. Note that as their frequency increases, disturbances become increasingly rejected independent of feedback and buffering (i.e., for all regulation cases). This is due to the damping effect of the y removal term (−γyy in the model).
(G and H) Impulse disturbance. For the temporal plot, the disturbance occurs at t = 0 with a magnitude of 50% of the set point y¯. In comparison with feedback, buffering initially works much quicker in moving y back toward its set point but suffers from a longer-lived residual deviation.
(I) Comparing impulse disturbance rejection between rapid and slower buffering. Rejection is improved for faster buffering, especially in the short term.
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5. Figure 3 Buffering Improves Feedback Stability
(A and B) Responses to a step disturbance at t = 0 for the minimal model with feedback delays under various buffering-feedback combinations. The disturbance amplitude is 50% of the nominal production rate py. In (A) (weak buffering), increasing the feedback gain causes oscillations, while in (B) (strong buffering), increasing the feedback gain instead improves steady-state regulation beyond what is possible when buffering is absent.
(C) The feedback gain and buffering equilibrium ratio for the six cases in plots (A and B). The shaded area corresponds to the unstable region, where the stability boundary is determined by Equation 4.
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6. Figure 4
Feedback and Buffering Have Opposite Effects on Molecular Noise
(A) Contour plot of the F term in the noise sensitivity function ϕnoise (Equation 5) versus buffering and feedback strengths, where a smaller F value implies lower noise in the regulated species.
(B and C) Stochastic simulations illustrating the molecular noise levels in the regulated species when different regulation methods are used (computed using the Gillespie algorithm; see STAR Methods A.10). (B) Temporal plots during a small window of time after the system has become stationary. (C) The probability mass function (PMF) of each simulation (computed over an extended window of time). The theoretical probability mass functions (from chemical Langevin equations) is indicated by the smooth black curve with the corresponding theoretical ϕnoise value in the upper right corner of each plot. These simulations demonstrate that rapid buffering introduces high-frequency noise, and therefore effective noise reduction is better achieved through feedback devoid of buffering.
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7. Figure 5 Alternative Topologies and Glycolysis
(A–D) Slower buffering filters out high-frequency reservoir disturbances. (A) Disturbances located at the reservoir, dx. (B and C) A slower buffer filters out high-frequency reservoir disturbances, dx, from the pool of regulated species, y. In these plots, the disturbance amplitude is 50% of the nominal production rate py. (D) A slower buffer causes an impulse reservoir disturbance to be released more slowly into the regulated pool. In this plot, the disturbance occurs at t = 0 with a magnitude of 50% of the set point y¯.
(E) Slow buffering with reservoir feedback acts as integral feedback. Production p(x) with feedback from the reservoir species. Slow buffering provides integral action because any changes to x are slow to be reversed and therefore accumulate.
(F) Glycolytic feedback and creatine phosphate buffering of ATP. Network diagram for a minimal model describing ATP production by glycolysis and ATP buffering by pCr. The primary feedback gain, HPFK, in this model is represented by the (negative) slope of ATP concentration to PFK activity. Here, m represents intermediate metabolites in glycolysis.
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8. Figure 6
Glycolytic Feedback versus Creatine Phosphate Buffering of ATP in Different Species
Using the ratios of normalized −ΔPFK activity/Δ[ATP] (see Equation 16 in the Appendix) and [pCr]/[ATP] as proxies for feedback gain and buffering ratio, respectively (see STAR Methods B), we find empirical cross-species support for a correlation between glycolytic feedback via PFK and ATP buffering by pCr. The shaded area in the figure corresponds to the unstable region, where the stability boundary is roughly and qualitatively estimated for illustrative purposes. This correlation, together with the theoretical results of Equation 10, and previously observed experimental cases of glycolytic oscillations induced by the removal of pCr, suggest that buffering enables higher-gain glycolytic feedback. Red data point, measurements and estimates of in vitro systems; blue data points, in vivo measurements and estimates.
Cell Systems 2017 5, e23DOI: ( /j.cels )
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