January 1, 2018;
Delay models for the early embryonic cell cycle oscillator.
Time delays are known to play a crucial role in generating biological oscillations. The early embryonic cell cycle in the frog Xenopus laevis is one such example. Although various mathematical models of this oscillating system exist, it is not clear how to best model the required time delay. Here, we study a simple cell cycle model that produces oscillations due to the presence of an ultrasensitive, time-delayed negative feedback loop. We implement the time delay in three qualitatively different ways, using a fixed time delay, a distribution of time delays, and a delay that is state-dependent. We analyze the dynamics in all cases, and we use experimental observations to interpret our results and put constraints on unknown parameters. In doing so, we find that different implementations of the time delay can have a large impact on the resulting oscillations.
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Fig 1. The underlying network of interacting proteins generates periodic cell cycle oscillations in the Xenopus embryo.A and B) Timing of surface contraction waves for fertilized (A) and parthenogenetically activated (B) Xenopus eggs, and cell division timing for fertilized eggs. Surface contraction waves are indicated by green triangles, cell division timings by red squares. Time is expressed relative to the timing of the first surface contraction wave. A) Top: images of a fertilized egg in the one cell (a), two cell (b), four cell (c), eight cell (d) and sixteen cell stage (e). Middle: kymograph of this fertilized egg: the intensity along the dotted line in (a) is plotted as a function of time. Bottom: timings of the cell divisions and surface contraction waves for ten fertilized eggs. Full lines indicate the average timing of these events. B) Top: images of a parthenogenetically activated egg. Middle: kymograph of this egg. Bottom: timing of the surface contraction waves for six parthenogenetically activated eggs. C) Images of nuclear envelope breakdown and reformation in a cycling extract prepared from Xenopus eggs, supplemented with demembranated sperm nuclei and GFP-NLS. The extract was visualized in a Teflon tubing with an inner diameter of 300 μm and a length of approximately 10 mm, submerged in mineral oil. Time is expressed relative to the point when the extract was warmed to 24 °C. D) Schematic view of some of the proteins that regulate the cell cycle. Cdk1-cyclin B is the main kinase that phosphorylates substrates in mitosis. Kinases are red, phosphatates are indicated in green. Full lines indicate phosphorylation, dashed lines indicate dephosphorylation. The symbol → means activation and ⊣ means inhibition. E) Simplified model: only the core oscillator is retained, the interactions are summarized by including high ultrasensitivity and a time delay in the response of APC/C.
Fig 2. Oscillations exist when the response is steep and the time delay is long enough.A) Steady state response of APC/C to Cdk1 activity. Higher m corresponds to a steeper response. B) When Cdk1 is suddenly activated, APC/C follows after a fixed time in the model with one discrete delay. C) Phase diagram for parameters c=ksbdegK and τ, for different values of m. Increasing m corresponds to an larger region of oscillations. D) Fixed point location. The dots show the APC/C activity in steady state, for different c. The fixed point can be found as the intersection of the APC/C response curve and the dashed lines, which are derived by putting the right hand side of Eq (1) to zero. E) Phase diagram for parameters m and τ, with period in color. The points correspond to parameter values used for the timeseries in G and H. F) Phase diagram for parameters ks/K and bdeg with period in color. G) Time series (sinusoidal) for m and τ denoted by point G. H) Time series (relaxation-like) for m and τ denoted by point H. Other parameters for G and H: ks = 1.28 nM/min, bdeg = 0.1 min−1. I) The two timeseries from G and H plotted in a plane. Note that APC/C is not an independent variable, but is a time-delayed function of Cdk1. The dashed line denotes the steady-state reponse of APC/C to Cdk1. The oscillations occur around the threshold value.
Fig 3. An approximation for high ultrasensitivity allows to study the period of the oscillations analytically.A) Time series for increasing values of m and for m = ∞. B) Duration of S phase and M phase for a low value of c as function of delay time. Other parameters: ks = 0.64 nM/min, bdeg = 0.1 min−1 C) Duration of S phase and M phase for a high value of c as function of delay time. Other parameters: ks = 2.28 nM/min, bdeg = 0.05 min−1 D) Duration of S phase and M phase as function of c. Equality of S and M phase duration is achieved roughly around c = 1/2. E) Period as function of bdeg and τ for ks = 1.25 nM/min with realistic region indicated in white. The m dependence lies mostly in the existence of oscillations (the phase boundary, in black), but not in the period.
Fig 4. A distributed delay makes it less likely for the system to oscillate.A) Gamma distribution. The parameters N and a influence both width and position of the peak. B) When the average is fixed and N increases, the Gamma distribution becomes more peaked and converges to a Dirac delta distribution, in which the delay is a fixed value τavg. C) Response of APC/C to a jump in Cdk1 activity for distributed delay. Compared with a model in which the delay is fixed, the response is much smoother. D) Phase diagram for low c. The region on the upper right is the region in which oscillations exist. E) Phase diagram for high c. The region on the upper right is the region in which oscillations exist. F) A linear chain of N reactions gives rise to a Gamma distribution. The number of steps and the rate of each step determine the average time delay. G) The model used by Yang and Ferrell  to model APC/C activation. Active Cdk1 catalyzes all steps in the cascade. The last step is made cooperative (parameter γ), in order to obtain an ultrasensitive response. H) When fitting a gamma distribution to the response of APC/C in the paper by Yang and Ferrell , we obtain an average delay of about 40 minutes. I) Using the model from Panel G but starting from a partially activated state, the resulting delay is much shorter.
Fig 5. The importance of activation/inactivation delays depends on the accumulation and degradation rates.A) Response of APC/C to a sudden activation and inactivation of Cdk1. Two different time delays can be modeled using state-dependent delay equations. The response is smooth because APC/C is a separate variable, but still quite sharp if β is large. B) Time series for the model with state-dependent delay. The time delay switches rapidly between τ1 = 10 and τ2 = 5. Other parameters: ks = 1 nM/min, bdeg = 0.1 min−1, m = 15, p = 5. C) Phase diagram when one of either τ1 or τ2 is zero. For very low and high values of c the phase boundary coincides with the boundary for the fixed delay model. D) Period as function of τ1 and τ2 for a low value of c. The indicated area denotes parameter values for which the period is between 20 and 30 minutes. The line denotes parameters where the M phase and S phase are equally long. Other parameters: ks = 1.2 nM/min, bdeg = 0.125 min−1, m = 20, β = 5 min−1, p = 5. E) Same as D, but for a high value of c. Other parameters: ks = 1 nM/min, bdeg = 0.0625 min−1, m = 20, β = 5 min−1, p = 5. F) Period as function of bdeg and τ1. The sum of τ1 and τ2 is fixed at 15 minutes and ks = 1.25 nM/min, m = 20, β = 5 min−1, p = 5. The white area shows parameter values that give a realistic period, the line shows which values give an equal length of M and S phase. The results suggest that bdeg lies in the region around 0.1 min−1.
Desynchronizing Embryonic Cell Division Waves Reveals the Robustness of Xenopus laevis Development.