1. Computational and Systems Biology
  2. Physics of Living Systems
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Biophysical clocks face a trade-off between internal and external noise resistance

  1. Weerapat Pittayakanchit
  2. Zhiyue Lu
  3. Justin Chew
  4. Michael J Rust
  5. Arvind Murugan  Is a corresponding author
  1. University of Chicago, United States
Research Article
Cite this article as: eLife 2018;7:e37624 doi: 10.7554/eLife.37624
10 figures and 1 additional file

Figures

Free running clocks and damped ‘hourglass’ clocks are equally good time-keepers in noiseless conditions but internal and external fluctuations reveal significant differences.

(a) Free running circadian clocks, such as the KaiABC protein clock in S. elongatus, show rhythms in both oscillating and constant light (top) or dark (bottom) conditions. (b) In contrast, damped circadian clocks, such as that in P. marinus which lacks Kai A, show rhythms only in changing light conditions and decay to a fixed state in constant conditions. (c) When subject to external noise (i.e., weather-related amplitude fluctuations in light), simulations of the free running clock show low population variance while the damped clock shows high variance. In contrast, Gillespie simulations with high internal noise due to low copy number of Kai molecule reveals that damped clocks are much more robust than free running clocks. (d) A systematic study of clock precision (i.e., mutual information between clock state and time) at fixed external noise but decreasing Kai protein copy number N reveals that free running clocks are preferred at low internal noise but damped clocks are preferable at sufficiently high internal noise.

https://doi.org/10.7554/eLife.37624.003
A diverse range of biochemical oscillators show the trade-off between resistance to external and internal noise.

For each oscillator, the regime (green) of largest free running amplitude relative to the driving strength is most robust to external fluctuations but is most fragile to internal noise. In contrast, damped oscillations (red) are robust to internal noise and thus preferable at sufficiently high internal noise. Regimes (purple) of intermediate free running amplitude are preferred at intermediate internal noise levels. (a–g) Diverse biochemical oscillators from the literature were simulated with increasing internal noise ϵint=1/N while driven by a periodic square wave light signal with fixed strength external noise, using the external coupling and parameters specified in the original publications (Leloup et al., 1999; Schmal et al., 2014; Locke et al., 2005; Leloup and Goldbeter, 2003; Goldbeter, 1991; Goodwin, 1965; Gonze and Abou-Jaoudé, 2013; Kondepudi and Prigogine, 2014; Elowitz and Leibler, 2000; Buşe et al., 2009). Clock precision is defined as mutual information between outputs and time. The original publications identified a Hopf bifurcation parameter in these models, with free running oscillations on one side and damped oscillations on the other. Green and purple data correspond to parameter regimes with large and smaller amplitude free running oscillations relative to driving amplitude while the red data corresponds to a damped oscillator. Details in Appendix 2.

https://doi.org/10.7554/eLife.37624.004
Experiments and models of biological clocks show that external driving can be viewed as a switch between distinct day-time and night-time dynamics.

(a) Experiments on the Kai system at distinct ATP levels corresponding to day and night conditions reveal limit cycles shifted relative to each other in a phosphorylation space for Kai (reproduced from [Leypunskiy et al., 2017[). Similar behavior (Winfree, 2001) is seen in models of diverse biochemical oscillators studied in Figure 2. (b) We build a minimal model of such driven clocks as a limit cycle of radius R whose center is shifted by a distance L between day and night. In cycling conditions (see signal in (d)), an entrained clock’s state executes a trajectory that encompasses both limit cycles as shown (bottom). (c) For damped clocks (Saunders, 2002), phenomenology suggests that the day and night limit cycle dynamics are replaced by a point attractor whose position changes between day and night. The relaxation time between the day and night attractors is comparable to 12 hours, giving rise to the trajectory shown in cycling conditions. (d) The plot shows cycling conditions of light intensities that couple to (b) and (c).

https://doi.org/10.7554/eLife.37624.005
External weather-related light fluctuations are filtered out by limit cycle attractors but not by point attractors.

(a) Light intensity levels fluctuate on a range of time scales due to weather (power spectrum reproduced from Gu et al., 2001). (b) A population of limit cycle clocks of identical fixed geometry, subject to different realizations of weather conditions, show non-overlapping distributions (purple blobs) at different times of the day. Point attractor clocks form larger and more overlapping distributions. (c) A single representative dark pulse of 2.4 duration causes only a ΔΦ30 min phase lag in limit cycles since the trajectory’s deviation (purple) is fundamentally bounded by the circular attractor. In contrast, ΔΦ4 hr for the point attractor since the trajectory is in free-fall towards the blue night-time attractor. (d) The geometrically computed ΔΦ2 phase shift for a dark pulse of any fixed duration and time of occurrence (see Appendix 5) drops rapidly as (R/L)-2 for large-R/L limit cycles; this theoretical prediction agrees well with the population variance gain over a day in simulations. (e) Consequently, weakly driven limit cycles (i.e., high R/L) can tell time with high precision.

https://doi.org/10.7554/eLife.37624.006
Internal fluctuations severely affect continuous attractors but not point attractors.

(a) We model fluctuations due to finite copy number N as Langevin noise with mean zero and standard deviation ϵint, resulting in a diffusion constant ϵint21/N for the clock state. (b) The flat direction of limit cycles cannot contain diffusion, leading to large increases ϵint2Tday in population variance of clock state during each day (and night). In contrast, point attractor dynamics have constant curvature at all times, leading to a constant population variance over time. (c) The variance drops σ2σ2/s2 at dawn and dusk for limit cycles during the off-attractor dynamics between the day and night cycles. As with external noise, the variance drop is predicted by the slope dP(θ)/dθ of the circle map between the cycles. This dawn/dusk drop goes to zero for large R/L limit cycles but variance still increases during the day and night. (d) The variance for point attractors is Dτrelax, a constant determined by the curvature τrelax-1 of the harmonic potential. (e) Thus, with only internal noise present, the precision of limit cycle clocks increases with increasing separation L/R, asymptotically approaching the performance of point attractors.

https://doi.org/10.7554/eLife.37624.007
Large-R/L limit cycle attractors, which correspond to large amplitude free running clocks, outperform all other oscillators in projecting out external noise but are least robust to internal noise.

(a) Point attractors and smaller R/L limit cycles (red and purple curves) show low precision (i.e., low mutual information) but do not degrade as much as large-R/L limit cycles with increasing internal noise ϵint. Thus this simple dynamical systems model of clocks reproduces and explains the trade-off seen in the complex biochemical clocks shown in Figures 1 and 2. (b,c) Speed-precision trade-off. (b) With external noise alone, the most precise clocks (i.e., large R/L limit cycles) average over longer signal history and are thus the slowest to entrain, that is, slow to transform a population with uniform phase distribution to the steady state distribution. (c) However, with internal noise alone, there is no trade-off between speed and precision; faster entraining clocks (i.e., point attractors) are more accurate since slow clocks are exposed to more internal noise.

https://doi.org/10.7554/eLife.37624.008
Appendix 1—figure 1
Explicit biochemical KaiABC model simulated using the Gillespie algorithm.

(a) The experimentally well-characterized clock in S. elongatus consists of a negative feedback-enabled self-sustained oscillator. KaiBC complexes sequester KaiA, preventing runaway KaiC molecules from going through the cycle independently. (b) The genome of P. marinus lacks kaiA. We assume a minimal model consistent with known facts (Rust et al., 2007) about this clock; KaiC phosphorylation proceeds without KaiA and hence different KaiC hexamers can proceed independently through the cycle. (c) We combine both clocks in one model with an interpolating parameter γ that selects between an S. elongatus-like KaiA-dependent pathway and an P. marinus-like KaiA-independent pathway. All reactions shown are assumed to be first order mass-action kinetics. We simulate such a system at different overall copy numbers N using the Gillespie algorithm. (d) We find limit cycles for γ>0.9. The resulting limit cycles for γ=1,0.95 violate the simplifying assumptions used in our dynamical systems (e.g., non-circular cycles of different size); and yet our results are qualitatively validated by this model (Figure 1d from the main text).

https://doi.org/10.7554/eLife.37624.011
Appendix 4—figure 1
Mutual information MI(c,t) between clock state c and time t is only affected by the variance of the clock state distribution p(c|t) at a given time t along the direction of motion and not orthogonal to it.

In this toy example, we assume the distribution p(c|t) to be supported on a rectangle of size ax and ay in a 2d clock state space. The clock state moves at a speed u in the x-direction. Time telling quality is affected by how much the population at different times overlap with each other. Consequently, clocks with large ax and small ay (bottom) have lower mutual information MI(c,t) relative to clocks with small ax and large ay (top). Consequently, we use the population variance along the direction of motion as an instantaneous measure of time-telling ability in the paper.

https://doi.org/10.7554/eLife.37624.015
Appendix 5—figure 1
The population variance of clock states is reduced by dusk and can be computed geometrically.

(a) A population of clocks near state θ on the day cycle is mapped to the neighborhood of state ϕ on the night cycle by the dusk transition. We define ϕ=P(θ) to be the map relating the clock state θ on the day cycle just before dusk to its eventual position ϕ on the night cycle after dusk (assumed greater than the relaxation time). (b) This map can be analytically computed for circles of size R with centers separated by length L. (c) For a given R/L = 2 , we obtain P(θ) shown here. Since θ=π/2 corresponds to the dusk time of the entrained trajectory, the slope s1=dP/dθ at θ=π/2 determines the change in population variance of clock states at dusk. (d,e) The variance drop s2 at dusk, defined as σ2σ2/s2 at dusk, seen in both the external (averaging over weather) and internal noise (averaging over Langevin noise) simulations agree well with the geometrically computed s(R/L), especially at large R/L. We find that s21L/R for large-R/L limit cycles.

https://doi.org/10.7554/eLife.37624.017
Appendix 5—figure 2
Increase in population variance due to random weather conditions can be estimated from the phase shifts ΔΦ due to dark pulses (i.e., the Phase Response Curve).

(a) A single dark pulse administered during the day shifts the phase of a clock (purple) relative to a clock that experiences no such dark pulse (black). (b) We can compute the phase shift ΔΦ due to such a dark pulse geometrically by computing the deviation in trajectory. Assuming a dark pulse of length τ, the clock evolves for a time τ according to the night cycle dynamics. At the end of such a pulse, we switch back to the day limit cycle and compute the resulting phase shift ΔΦ. (c) The resulting phase shift ΔΦ due to a pulse of length τ=2.4 hrs, depends on the time θ when it is administered but is generally smaller for larger R/L. (d) We find that ΔΦ2 for a specific τ=2.4 hrs dark pulse administered at the same time (8 AM) falls as (L/R)2 for large-R/L limit cycles. This trend matches the variance gain σclouds2 seen in stochastic simulations that average over random weather conditions (pulses of different length, intensity and time of application). The broken brown curve shows a theoretical prediction for such an average ΔΦ2, obtained by sampling the curve shown in (c) at different points of application and differing intensity. Despite the presence of a variance-reducing zero around mid-day in (c), σclouds2 drops as (L/R)2, much as ΔΦ2 for any particular pulse. (Brown theory curves translated together using one fitting parameter).

https://doi.org/10.7554/eLife.37624.018

Data availability

The code and data used in the simulations are available via GitHub at https://github.com/WeerapatP/elife_tradeoff_clocks

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