The Design Space of the Embryonic Cell Cycle Oscillator

Abstract

One of the main tasks in the analysis of models of biomolecular networks is to characterize the domain of the parameter space that corresponds to a specific behavior. Given the large number of parameters in most models, this is no trivial task. We use a model of the embryonic cell cycle to illustrate the approaches that can be used to characterize the domain of parameter space corresponding to limit cycle oscillations, a regime that coordinates periodic entry into and exit from mitosis. Our approach relies on geometric construction of bifurcation sets, numerical continuation, and random sampling of parameters. We delineate the multidimensional oscillatory domain and use it to quantify the robustness of periodic trajectories. Although some of our techniques explore the specific features of the chosen system, the general approach can be extended to other models of the cell cycle engine and other biomolecular networks.

Introduction

Starting from the early 1990s, computational models of the embryonic cell cycle have led the way in the systems-level analysis of biomolecular networks (1, 2, 3). The modeling approach to the cell cycle and other biochemical networks starts by converting the information about the established components and interactions into a dynamical system. Usually, at this stage, the aim is to build a model that qualitatively predicts the desired behavior(s). This is followed by analytical or computational analysis of the model, with the minimal requirement to find a vector of model parameter values that yields the desired behavior, such as periodic dynamics with the right period and waveform. As a rule, this vector is not unique and belongs to a set of parameter vectors that are equally successful in describing the data (4, 5). Ideally, one would like to characterize this entire set, which would make it possible to explore functional capabilities of the model only on the basis of the included components and processes.

Since even the simplest models of biochemical networks contain a large number of parameters, characterizing their design spaces is a nontrivial task, and most early modeling studies were limited to providing one or several parameter sets consistent with the desired dynamics. Note that even this step might be quite challenging, although more and more algorithms are being proposed for this purpose (5, 6). At the same time, rapid improvements in computational power and numerical methods make it possible to systematically sample the design space of a model and enable the types of analyses that had previously been very difficult or impossible (7, 8, 9). For instance, one might be interested in determining the parameter vectors that can withstand the maximal possible perturbations without disrupting the desired dynamics. In this article, we try to accomplish these tasks by probing the design space of a mathematical model of the embryonic cell cycle, aiming to highlight the issues that might be relevant for a broad class of models.

Our model is a simplified version of earlier descriptions of cell cycles in Xenopus egg extracts, one of the leading experimental systems for studies of cell cycle regulation (10). The first model of this system was proposed in 1993 by John Tyson and Bela Novak, shortly after the elucidation of the core processes responsible for the periodic dynamics in the activity of the cyclin-dependent kinase (CDK) (11, 12). Their model accounts for steady synthesis of cyclin, association with CDK, reversible activation of the cyclin-CDK complex by phosphorylation, and proteolytic degradation. The model has 10 variables, which could be reduced to two under certain assumptions, and about 20 parameters that correspond to the rates of protein synthesis, strengths of protein-protein interactions, and rate constants of enzymatic reactions. The authors found a set of parameter values that successfully described the periodic activity of CDK and made several predictions that have been confirmed by subsequent experiments (13). Starting from 2003, James Ferrell and his group initiated a systematic effort aimed at quantifying the main regulatory interactions in this model. These studies led to updated models with two variables and ∼10 parameters (14, 15, 16, 17). Note that none of the current models can be viewed as a “first-principles” description of the real system, which is still only partially understood. For example, a recent study investigated the role of the dynamic control of cyclin synthesis, which was assumed to be constant in the first generation of models (18).

One of the key insights of the original Tyson-Novak model and its descendants is that the observed CDK dynamics can be viewed as relaxation oscillations, a periodic trajectory with fast and slow motions (19). Furthermore, quantitative changes in the values of model parameters can lead to qualitative changes in the long-term dynamics, from oscillations to either unique steady states or bistability. Here, we have coarse-grained interactions considered in previous models of the cell cycle, producing a model that is amenable to analytical treatment. We show how the design space of this model can be delineated using a combination of analytical and computational tools and used to explore the robustness of the oscillatory regime. Our results provide new insights into the design principles leading to robust oscillations and can be extended to more complex models of cell regulation systems.

Methods

Numerical continuation of bifurcation points

All numerical continuations (see Figs. 3 and 5) were performed using Matcont (20), a numerical continuation software for MATLAB (The MathWorks, Natick, MA).

Figure 3.

Analytical and numerical two-parameter bifurcation diagrams. (A) Comparison of the numerically-computed and analytically-derived regions of oscillations in the (γ_a,γ_d) plane, for the same parameters as in Fig. 2. The light gray patch is the region of oscillations determined by numerical continuation, and the green and purple lines are the analytically-derived boundaries in the limit that ε→0. At this small value of ε, the results are indistinguishable. The upper boundary (green line) comes from the constraint that the w-nullcline intersects the a-nullcline at its upper extremum, and similarly for the lower boundary (purple line). These conditions are shown graphically in the insets. See also Fig. S1. (B) Numerically-calculated two-parameter bifurcation diagram showing the regions of oscillations and bistability in the (γ_a,γ_d) plane for the same parameter set. The light gray patch is again the region of oscillations, the dark gray lines are Hopf bifurcation points, and the black lines are saddle-node bifurcation points. The insets without borders show nullclines and steady states for selected parameter sets in the 2-parameter diagram. In these insets, solid red lines are the nullclines for a, solid black lines are the nullclines for w, filled dots are stable steady states, and open dots are unstable steady states. The inset bordered by a dashed black line zooms in on the bottom-left portion of the diagram where there is a small region of bistability. To see this figure in color, go online.

Figure 5.

Robustness of oscillations to perturbations in the rate of cyclin synthesis, and analysis of the most robust parameter vector. (A) Histogram of the measure of robustness, $\delta$, to perturbations of the synthesis rate, $s$, for 1000 points in the domain of oscillations. $\delta \equiv \left(s^{\text{max}}/s^{\text{min}}\right)=\left(f^{\text{max}}/f^{\text{min}}\right)$ is the ratio of the largest and smallest values of $s$ that generate oscillations for each parameter set. (B) One-parameter continuation diagram in $f$ for the parameter vector with the largest value of $\delta$ (Table S1). Note that the synthesis rate, $s$, is inversely proportional to $f$. The solid lines denote stable steady states, the dashed line denotes unstable steady states, and black dots denote stable limit cycles. The amplitude of oscillations in $a$ are roughly constant (see also Fig. S4). (C) Period, T, versus the continuation parameter, $f$, for the parameter vector with the largest value of $\delta$. At smaller $f$, the period is relatively insensitive to the value of $f$, but it increases as $f$ increases. Additionally, at the bifurcation points, the period diverges, corresponding to infinite-period bifurcations. Note that in (B) and (C), $f$ was scaled such that $f^{\text{min}}=1$.

Parameter sampling

The model contains eight dimensionless parameters, defined in Table 1. Table 2 summarizes the ranges and scales used to generate samples from the parameter space. Since we were interested in the location and shape of the region of oscillations, ranges were chosen to exclude as much of the non-oscillatory parameter space as possible. In some cases, taking a parameter to a limiting value of zero or infinity would preserve oscillations. In these cases, parameter ranges were chosen such that the model outputs were still somewhat sensitive to the parameters. Finally, whereas all other dimensionless parameters were sampled on a logarithmic scale, Hill coefficients were sampled on a linear scale, since their experimentally measured values are typically of order 1.

Table 1.

Dimensionless Parameter Definitions

Dimensionless Parameter	Definition
$\mu$	$k_{\text{a}}^{-}\text{/}{k}_{\text{i}}$
${\beta }_{\text{a}}$	${\Delta }_{\text{a}}\text{/}{k}_{\text{i}}$
${\beta }_{\text{d}}$	${\Delta }_{\text{d}}\text{/}{k}_{\text{d}}^{-}$
${\gamma }_{\text{a}}$	${\theta }_{\text{a}}\text{/}\left(s\text{/}{k}_{\text{d}}^{-}\right)$
${\gamma }_{\text{d}}$	${\theta }_{\text{d}}\text{/}\left(s\text{/}{k}_{\text{d}}^{-}\right)$
$n_{\text{a}}$	$n_{\text{a}}$
$n_{\text{d}}$	$n_{\text{d}}$
$\epsilon$	$k_{\text{d}}^{-}\text{/}{k}_{\text{i}}$

Table 2.

Ranges and Scales Used for Parameter Sampling

Parameter	Scale	Minimum	Maximum
$\mu$	log	0.01	10
${\beta }_{\text{a}}$	log	0.01	100
${\beta }_{\text{d}}$	log	0.1	100
${\gamma }_{\text{a}}$	log	0.01	10
${\gamma }_{\text{d}}$	log	0.01	10
$n_{\text{a}}$	linear	0.01	15
$n_{\text{d}}$	linear	0.01	15
$\epsilon$	log	0.0001	15

Shown are the parameter ranges used for sampling and the scale on which sampling was performed uniformly for each parameter (i.e., μ was sampled uniformly in logarithm between 0.01 and 10, whereas $n_a$ was sampled uniformly on a linear scale between 0.01 and 15).

For each sample, the first step was to find a steady state by a local method and check the linear stability of that steady state. Steady states were found by first solving for $w$ in $\left(dw/dt\right)=0$ and then plugging that expression into the equation $\left(da/dt\right)=0$. The value of $\text{log}\left(a\right)$ (to ensure positive solutions) that satisfied this expression was found using MATLAB’s fzero function, with an initial guess of $a=0.1$.

The linear stability of the steady state was checked by finding the two eigenvalues of the Jacobian matrix, $J_{ij}=\left(\partial {\dot{X}}_i/\partial X_j\right)$, evaluated at the steady state, where $i,j=1,2$, $X_1=a$, and $X_2=w$, and the dot indicates a time derivative.

If the found steady state was stable, corresponding to the real parts of both eigenvalues being negative, the parameter set was immediately thrown out, because it could not produce oscillations. If an unstable steady state was found, further tests were performed to check whether the parameter set generated oscillations.

If the steady state was unstable, the system was integrated in time using MATLAB’s ode15s until either 1) the system time reached 1000 units, or 2) the time course had exhibited three peaks. If the system time reached 1000 units before producing three peaks, then it reached a stable steady state. This happened when there were three steady states—either two unstable and one stable or two stable and one unstable—and the local solver happened to find an unstable one. Alternatively, if the system generated three peaks before reaching 1000 time units, it could be due to either sustained oscillations or damped oscillations. To check which, the local search for a steady state was repeated from the last point of time integration. If the resulting steady state was stable, then the oscillations were damped, and the parameter set was discarded. If the steady state was unstable, then the oscillations were sustained.

Finally, if the parameter set corresponded to sustained oscillations, we found the converged limit cycle by numerically solving

$$\stackrel{\rightharpoonup }{F}\left(a_0,w_0,T\right)=\left[\begin{array}{c}{a}_0-a\left(T\right)\\ w_0-w\left(T\right)\end{array}\right]=0,$$ (14)

where $a_0$ and $w_0$ were initial points on the limit cycle, $T$ was the period, $a_0$ was fixed, and $w_0$ and $T$ were varied.

Code is available upon request.

Density estimation

Univariate and bivariate marginal distributions were estimated in MATLAB using Z. Botev’s code kde and kde2d, both available on The MathWorks website: https://www.mathworks.com/matlabcentral/fileexchange/14034-kernel-density-estimator?s_tid=srchtitle and https://www.mathworks.com/matlabcentral/fileexchange/17204-kernel-density-estimation. (accessed 12/16/16) (21).

Results

Model description and nondimensionalization

We consider a model with two species, $I$ and $A$, corresponding to the inactive and active forms, respectively, of the cyclin-CDK1 complex (Fig. 1 A). Similar to previous models (11, 15, 16), the complex appears at a constant rate in the active form, reflecting synthesis of cyclins, their rapid association (22) with a large pool of CDK1 (23, 24), and rapid phosphorylation of the cyclin-CDK complex by the CDK-activating kinase. The inactive form is converted into the active form in a process that is promoted by $A$, reflecting cyclin-CDK’s double-negative feedback loop with Wee1/Myt1 (25, 26, 27, 28, 29, 30, 31) and positive feedback loop with Cdc25 (32, 33, 34, 35, 36, 37, 38). $A$ is converted back to $I$ with first-order kinetics. Both forms are degraded in a process that is also promoted by $A$, reflecting activation of the anaphase-promoting complex (APC) by active cyclin-CDK and subsequent degradation of cyclins by the APC (39, 40, 41, 42, 43, 44).

Figure 1.

Model reaction diagram and functional forms of regulatory rate constants. (A) Reaction diagram depicting the synthesis of active cyclin-CDK, $A$, conversion into the inactive form, $I$, inactivation of the active form, and degradation of both forms. The rate constants of activation and degradation depend nonlinearly on the concentration of the active form. (B) Schematic showing the dependence of the activation and degradation rate constants, $k_{\text{a}}\left(A\right)$ and $k_{\text{d}}\left(A\right)$ on $A$. The parameters $k_{\text{a}}^{-}$ and $k_{\text{d}}^{-}$ are the basal values for activation and degradation rate constants when the concentration of $A$ is low. The rate constants reach their half-maximal values when $A={\theta }_{\text{a}}$ or $A={\theta }_{\text{d}}$, respectively, and the maximal increases are denoted by ${\Delta }_{\text{a}}$ and ${\Delta }_{\text{d}}$ (Eqs. 3 and 4).

These processes are modeled by the following system of differential equations:

$$\frac{dI}{dt}=-k_{\text{d}}\left(A\right)I-\left(k_{\text{a}}\left(A\right)I-k_{\text{i}}A\right),$$ (1)

$$\frac{dA}{dt}=s-k_{\text{d}}\left(A\right)A+\left(k_{\text{a}}\left(A\right)I-k_{\text{i}}A\right),$$ (2)

where $s$ is the rate of synthesis of $A$, $k_{\text{d}}\left(A\right)$ is the function describing the rate constant for complex degradation, $k_{\text{a}}\left(A\right)$ is the corresponding function for complex activation, and $k_{\text{i}}$ is the rate constant for complex deactivation. Following methods in experimental studies, the functional forms of $k_{\text{d}}\left(A\right)$ and $k_{\text{a}}\left(A\right)$ are modeled as Hill nonlinearities (14, 15, 16), each of which is characterized by a basal value, threshold, sharpness, and gain (Fig. 1 B):

$$k_{\text{d}}\left(A\right)\equiv k_{\text{d}}^{-}+{\Delta }_{\text{d}}\frac{A^{n_{\text{d}}}}{{\theta }_{\text{d}}^{n_{\text{d}}}+A^{n_{\text{d}}}}$$ (3)

$$k_{\text{a}}\left(A\right)\equiv k_{\text{a}}^{-}+{\Delta }_{\text{a}}\frac{A^{n_{\text{a}}}}{{\theta }_{\text{a}}^{n_{\text{a}}}+A^{n_{\text{a}}}}.$$ (4)

To begin the analysis, we first make the problem dimensionless. Rescaling $A$ and $I$ by $s/k_{\text{d}}^{-}$ gives the dimensionless variables $a$ and $i$, and rescaling time such that $\tau =k_{\text{d}}^{-}t$ gives the dimensionless time. The dimensionless equations take the form

$$\frac{di}{d\tau }=-\left(1+{\beta }_{\text{d}}f\left(a\right)\right)i-\frac{1}{\epsilon }\left(\left(\mu +{\beta }_{\text{a}}g\left(a\right)\right)i-a\right),$$ (5)

$$\frac{da}{d\tau }=1-\left(1+{\beta }_{\text{d}}f\left(a\right)\right)a+\frac{1}{\epsilon }\left(\left(\mu +{\beta }_{\text{a}}g\left(a\right)\right)i-a\right).$$ (6)

In these equations, $f\left(a\right)\equiv \left(a^{n_{\text{d}}}/{\gamma }_{\text{d}}^{n_{\text{d}}}+a^{n_{\text{d}}}\right)$ and $g\left(a\right)\equiv \left(a^{n_{\text{a}}}/{\gamma }_{\text{a}}^{n_{\text{a}}}+a^{n_{\text{a}}}\right)$ are the rescaled Hill functions. After rescaling, eight dimensionless parameters remain, which are defined in Table 1. Adding Eqs. 5 and 6 gives the equation for the dynamics of the total amount of complexes, $w=a+i$:

$$\frac{dw}{d\tau }=1-\left(1+{\beta }_{\text{d}}f\left(a\right)\right)w.$$ (7)

Going forward, we will analyze the ($a$, $w$) system, since $i$ is determined by mass conservation.

Phase-plane analysis in the limit of strong timescale separation

The first insights into the dynamics are provided by phase-plane analysis in the limit of strong timescale separation, when the interconversion between $i$ and $a$ occurs much faster than changes in the total amount of complexes. In this limit, $\epsilon \ll 1$, $a$ becomes a fast variable, and Eq. 6 becomes

$$\epsilon \frac{da}{d\tau }=\left(\mu +{\beta }_{\text{a}}g\left(a\right)\right)\left(w-a\right)-a.$$ (8)

The nullcline for $a$, found by setting the time derivative to zero, is

$$a=\frac{\mu +{\beta }_{\text{a}}g\left(a\right)}{1+\mu +{\beta }_{\text{a}}g\left(a\right)}w.$$ (9)

When the shape of the Hill nonlinearity in $g\left(a\right)$ approaches a step function, this nullcline can be approximated by a piecewise linear function (Fig. 2 A). When $a<{\gamma }_{\text{a}}$, $g\left(a\right)\approx 0$, and $a\approx \mu /\left(1+\mu \right)w$. On the other hand, when $a>{\gamma }_{\text{a}}$, $g\left(a\right)\approx 1$ and $a\approx \left(\mu +{\beta }_{\text{a}}\right)/\left(1+\mu +{\beta }_{\text{a}}\right)w$. The two straight lines are separated by a discontinuity at $a={\gamma }_{\text{a}}$. A similar analysis leads to a piecewise linear approximation for the second nullcline, $w\approx 1,$ when $a<{\gamma }_{\text{d}}$, and $w\approx 1/\left(1+{\beta }_{\text{d}}\right)$ when $a>{\gamma }_{\text{d}}$ (Fig. 2 C).

Figure 2.

Phase plane analysis and the emergence of oscillations. (A) Nullcline for $a$ in the limit that $\epsilon$ goes to zero and $n_a$ is large. Arrows depict the direction of change in $a$ over time on each side of the nullcline. (B) Exact nullcline for $a$, with the same parameter values as in (A). The limit points of the $a$-nullcline are marked with arrows. (C) Nullcline for $w$ in the limit that $\epsilon$ goes to zero and $n_d$ is large. Arrows depict the direction of change in $w$ over time on each side of the nullcline. (D) Exact nullcline for $w$, with the same parameter values as in (C). (E) When the nullclines for $a$ and $w$ intersect in the transition regions of their respective Hill functions, the system has a unique, unstable steady state and exhibits sustained oscillations. The thick red line is the $a$-nullcline, and the thick black line is the $w$-nullcline. The thin black lines are trajectories started from different initial conditions, which all approach a limit cycle at long times. (F) The time course of oscillations on the limit cycle in (E). Here, $\epsilon$ is very small, generating a large separation of time scales and saw tooth-shaped oscillations. To see this figure in color, go online.

For certain parameter values, the $a$-nullcline takes the form of an S-shaped curve, where three distinct steady values of activity $\left(a\right)$ are possible for a particular range of total complex concentrations $\left(w\right)$. Phase-plane analysis shows that all steady states located between the extrema of this curve are unstable. At the same time, the nullcline for the slow variable $w$ always remains a single-valued function. In the limit of strong timescale separation, oscillations in our model exist when the two nullclines intersect only once and when this intersection is located between the limit points of the $a$-nullcline (Fig. 2, E and F; Table 3); (19, 45).

Table 3.

Parameter Values Used to Generate Figs. 2 and 3

Dimensionless Parameter	Value
$\mu$	1/3
${\beta }_{\text{a}}$	2/3
${\beta }_{\text{d}}$	7
${\gamma }_{\text{a}}$	1/8, varied
${\gamma }_{\text{d}}$	1/8, varied
$n_{\text{a}}$	10
$n_{\text{d}}$	10
$\epsilon$	varied

Delineating the oscillatory domain

As mentioned above, oscillations exist when the nullclines intersect only once and between the limit points of the $a$-nullcline. This criterion can be expressed as two inequalities: the steady state value of $a$ $\left(a_{\text{ss}}\right)$ must be less than the value of $a$ at the upper limit point $\left(a_{+}\right)$ and greater than that at the lower one $\left(a_{-}\right)$ (see Fig. 2 B), or $a_{-}<a_{\text{ss}}<a_{+}$. We derived analytical expressions for these inequalities only in terms of parameters for the limit of $\epsilon \ll 1$.

Although the full derivation can be found in the Supporting Material, the outline of the derivation is as follows. We need to translate the above conditions on the state variables to conditions on the parameters. First, we derive the locations of the extrema in the $a$-nullcline in terms of parameters and derive conditions for when these extrema exist. Then, we want the steady state of the system to lie between these extrema. When the steady state lies exactly on an extremum, the system is at the boundary between oscillatory and nonoscillatory behavior. We find these boundaries by making the $w$-nullcline intersect the $a$-nullcline at a limit point, giving one equation for each limit point, each of which can be expressed in terms of parameters only. These equalities define two hypersurfaces that bound the domain of oscillations (i.e., $a_{\text{ss}}=a_{-}$ and $a_{\text{ss}}=a_{+}$).

In terms of parameters, $a_{-}<a_{\text{ss}}<a_{+}$ is satisfied when ${\gamma }_{\text{d},-}<{\gamma }_{\text{d}}<{\gamma }_{\text{d},+}$, where

$${\gamma }_{d,\pm }={\gamma }_a{\left(\frac{g_{\pm }}{1-g_{\pm }}\right)}^{1/n_{\text{a}}}{\left(\frac{\left(1+{\beta }_{\text{d}}\right)n_{\text{a}}{\beta }_{\text{a}}{\gamma }_{\text{a}}{g}_{\pm }\left(1-g_{\pm }\right){\left(\frac{g_{\pm }}{1-g_{\pm }}\right)}^{1/n_{\text{a}}}-{\left(\mu +{\beta }_{\text{a}}{g}_{\pm }\right)}^2}{{\left(\mu +{\beta }_{\text{a}}{g}_{\pm }\right)}^2-n_{\text{a}}{\beta }_{\text{a}}{\gamma }_{\text{a}}{g}_{\pm }\left(1-g_{\pm }\right){\left(\frac{g_{\pm }}{1-g_{\pm }}\right)}^{1/n_{\text{a}}}}\right)}^{1/n_{\text{d}}}.$$ (10, a and b)

In this expression, $g_{\pm }=g\left(a_{\pm }\right)$, and

$$g_{\pm }=\frac{-\left(1-n_{\text{a}}+2\mu \right)\pm \sqrt{\frac{n_{\text{a}}}{{\beta }_{\text{a}}}\left({\beta }_{\text{a}}\left(n_{\text{a}}+\frac{1}{n_{\text{a}}}-2\right)-4\mu \left(\mu +{\beta }_{\text{a}}+1\right)\right)}}{2\left({\beta }_{\text{a}}+n_{\text{a}}\right)}.$$ (11, a and b)

The expressions in Eq. 10, a and b, define two seven-dimensional hypersurfaces in the eight-dimensional parameter space that encloses the domain of oscillations, and any ${\gamma }_{\text{d}}$ between them will produce oscillations. These equations fully characterize the oscillatory domain in the small-$\epsilon$ regime.

What does the region of oscillations in the small-$\epsilon$ regime look like? We are limited to cross sections of the domain to gain insights by visualization. Holding the values of all parameters but ${\gamma }_{\text{a}}$ and ${\gamma }_{\text{d}}$ fixed, we used Eq. 10, a and b to trace out the boundaries of the oscillatory region in a two-parameter bifurcation diagram (Fig. 3 A). In this cross section, we can see that the region of oscillations is a single simply-connected domain. The accuracy of the above analytical expression, valid in the limit $\epsilon \ll 1$, can be evaluated by comparing with the results of numerical bifurcation analysis, which traces the domains of oscillatory and other types of solutions. The numerically-calculated and analytically-derived boundaries show good agreement when $\epsilon$ is small (Fig. 3 A) and deviate as $\epsilon$ increases above 0.01 (Fig. S1). Furthermore, numerical continuation reveals that in this cross section, there are two disjoint regions in which the system is bistable (Fig. 3 B).

Exploring the full parameter space by sampling

In this section, we numerically explore the parameter space by randomly sampling and show how this analysis can produce parameter relationships that improve the probability of generating oscillations. In the two-parameter cross section of the domain of oscillations (Fig. 3), we noticed that the two threshold parameters bounding the oscillatory domain, ${\gamma }_{\text{a}}$ and ${\gamma }_{\text{d}}$, appeared to be highly correlated—oscillations were unlikely when ${\gamma }_a/{\gamma }_d$ was far from 1. To find other correlations among model parameters relevant to oscillations, we sampled from the eight-dimensional parameter space of the full model (Eqs. 6 and 7). Details about how the sampling was performed can be found in Materials and Methods. This screen found one oscillatory parameter set in every 1000 samples, ultimately producing ∼75,000 oscillatory parameter sets.

Inspecting the one- and two-dimensional marginal distributions of the parameter samples (Figs. S2 and S3), we first verified features we expected from theory. First, oscillations were more likely when $\epsilon$ was small, meaning there is a strong separation of timescales between the rates of cyclin activation/deactivation and the rate of cyclin degradation. We also verified that the correlation between ${\gamma }_{\text{a}}$ and ${\gamma }_{\text{d}}$ along the boundaries, observed in the two-parameter cross section, was a general feature of the model’s oscillatory domain. From the bivariate distribution of ${\gamma }_{\text{a}}$ and ${\gamma }_{\text{d}}$, oscillations are likely to emerge when ${\gamma }_{\text{d}}\approx {\gamma }_{\text{a}}$ and disappear when the two threshold values differ significantly (Fig. 4 A). This should come as no surprise: when ${\gamma }_{\text{a}}$ and ${\gamma }_{\text{d}}$ are similar, the extrema of the $a$-nullcline and the transition of the $w$-nullcline occur at roughly the same value of $a$, almost guaranteeing that the $w$-nullcline will intersect the $a$-nullcline between the extrema. When these are equal, the remaining parameters can vary considerably while still maintaining the oscillatory function.

Figure 4.

Selected bivariate distributions of the sampled oscillatory parameter sets. The color denotes the probability density at each point, with lighter colors being higher in value. At pairs of parameter values where the density is higher, a larger space of the remaining 6 parameters generates oscillations. See also Figs. S2 and S3. (A) The joint distribution of ${\text{log}}_{10}\left({\gamma }_a\right)$ and ${\text{log}}_{10}\left({\gamma }_d\right)$ shows that oscillations are more likely when the two are very close in value. (B) The bivariate distribution of $n_a$ and $n_d$ shows that increasing $n_d$ makes larger values of $n_a$ capable of generating oscillations. (C) The joint distribution of ${\text{log}}_{10}\left({\gamma }_a\right)$ and ${\text{log}}_{10}\left(\mu \right)$ shows that oscillations are also more likely when the threshold parameter ${\gamma }_a$ is close in value to the basal activation rate $\mu$. The same trend was observed between ${\gamma }_d$ and $\mu$. (D) The joint distribution of $n_a$ and ${\text{log}}_{10}\left(\mu \right)$. The white dashed line is the boundary separating the regions where the $a$-nullcline can and cannot have extrema, in the limits that ${\beta }_a\to \infty$ and $\epsilon \to 0$ (Equation S1.19). Although this boundary was derived for small $\epsilon$, no sampled parameter vectors crossed it, even those with arbitrary values of $\epsilon$. To see this figure in color, go online.

The marginal distributions also provided insights into how to choose the remaining parameters in a way that would tend to generate oscillations (Figs. 4, S2, and S3). For example, $n_{\text{d}}$, the exponent for the degradation rate constant, could be arbitrarily large, corresponding to a sharp, horizontal transition in the $w$-nullcline. However, the univariate distribution of $n_{\text{a}}$, the exponent in the activation rate constant, peaked at about $n_{\text{a}}=3$. The bivariate distribution showed dependence between $n_{\text{a}}$ and $n_{\text{d}}$ (Fig. 4 B): larger values of $n_{\text{d}}$ were necessary for larger values of $n_{\text{a}}$ to generate oscillations, whereas smaller values of $n_{\text{a}}$ could generate oscillations for most values of $n_{\text{d}}$. We also noticed correlations between $\mu$ and the two threshold parameters, ${\gamma }_{\text{a}}$ and ${\gamma }_{\text{d}}$ (Fig. 4 C). If $\mu$ is small relative to ${\gamma }_{\text{a}}$, then the $a$-nullcline will intersect the $w$-nullcline below the first extremum of the $a$-nullcline, producing a stable steady state. If $\mu$ is large relative to ${\gamma }_{\text{a}}$, the extrema of the $a$-nullcline occur at smaller values of $w$, and fewer values of the remaining parameters cause the nullclines to intersect between the extrema. The correlation between $\mu$ and ${\gamma }_{\text{d}}$ is the result of both parameters being correlated with ${\gamma }_{\text{a}}$. Finally, a nonlinear relationship was found between $\mu$ and $n_{\text{a}}$ (Fig. 4 D). This relationship could be explained by the conditions under which the $a$-nullcline has extrema, which are necessary for oscillations and were derived in the process of deriving Eq. 10, a and b. Overlaying the boundary between the regions where the $a$-nullcline does and does not have extrema (in the limits that ${\beta }_{\text{a}}\to \infty$ and $\epsilon \to 0$) (Eqs. S1.18 and S1.19), it becomes clear that this is the source of the relationship between $\mu$ and $n_{\text{a}}$.

The high density of oscillatory parameter sets at small values of $\epsilon$ suggested that our analytical approximations made in the previous section would be good predictors of oscillations. Of the 75,000 samples, 96.7% satisfied the analytical conditions that the nullclines intersect between the extrema of the $a$-nullcline, which was derived for small $\epsilon$ (Eq. 10, a and b). This meant that a large portion of the oscillatory domain lay inside the analytically derived domain. However, containment does not imply equivalence. It could be that a large portion of the analytical domain contains parameter sets that do not generate oscillations. To determine how well the analytically derived domain approximates the true oscillatory domain, we sampled from the analytically derived domain and determined whether each sample actually generated oscillations or not. Of ∼13,000 samples, 38.4% actually generated oscillations. This demonstrated that the analytical domain delineated by Eq. 10, a and b contains almost all of the true oscillatory domain but overestimates its volume by a factor of ∼2.5 over the range of parameter values considered here.

Exploring the robustness of oscillatory solutions

With many samples from the domain of oscillations in hand, we were then able to ask questions about functions over this set. In particular, to which perturbations is the oscillatory behavior most robust? As mentioned above, the domain of oscillations appears to be narrow and extended along the $\left({\gamma }_{\text{a}},{\gamma }_{\text{d}}\right)$ direction, suggesting that the oscillatory behavior may withstand considerable perturbations along this direction. Recall also that ${\gamma }_{\text{a}}\equiv {\theta }_{\text{a}}/\left(s\text{/}{k}_{\text{d}}^{-}\right)$ and ${\gamma }_{\text{d}}\equiv {\theta }_{\text{d}}/\left(s\text{/}{k}_{\text{d}}^{-}\right)$ are the only dimensionless parameters that depend on the rate of cyclin synthesis, $s$, so perturbations in the $\left({\gamma }_{\text{a}},{\gamma }_{\text{d}}\right)$ direction are equivalent to perturbations in the synthesis rate. Robustness of oscillations to variations in cyclin synthesis would be desirable, given that the assumption of constant synthesis rate has been shown to be a simplification (18).

Dividing the rate of synthesis by a perturbation factor, $f$, we asked what are the largest and smallest values of $f$ for which each parameter set produces oscillations. Note that dividing $s$ by $f$ is equivalent to multiplying both thresholds by $f$. Taking a subsample of 1000 parameter sets from the point cloud generated above, we calculated, via the bisection method, the upper and lower bounds of $f$ for which oscillations still existed. We defined a measure of robustness to this perturbation as

$$\delta \equiv \frac{s^{\text{max}}}{s^{\text{min}}}=\frac{f^{\text{max}}}{f^{\text{min}}},$$ (12)

where $s_0$ was the “initial” value of $s$ for the sampled parameter set, $s^{\text{max}}\equiv s_0/f^{\text{min}}$ was the largest value of $s$ that generated oscillations and $s^{\text{min}}\equiv s_0/f^{\text{max}}$ was the smallest. This measure of robustness does not depend on the particular value of $s_0$; rather, it measures the “width” in log space of the oscillatory domain along the $\left({\gamma }_{\text{a}},{\gamma }_{\text{d}}\right)$ direction at the sampled point. Additionally, although $f^{\text{min}}$ and $f^{\text{max}}$ depend on the parameter vector from which the bisection method was started, $\delta$ does not. Therefore, $f^{\text{min}}$ and $f^{\text{max}}$ can be rescaled such that $f^{\text{min}}=1$, without changing the value of $\delta$. Fig. 5 A shows the histogram of $\delta$ values among the subsampled points.

We then asked what is the most “robust” set of parameter values according to this measure (Table S1), and how do the period and amplitude of oscillations of the active cyclin-CDK complex depend on $f$? For this parameter set, $\delta \sim 52.8$, indicating that the largest value of synthesis rate $s$ that generates oscillations is ∼53 times larger than the smallest one. The parameter vector also exhibited many of the heuristic properties that, from sampling, were found to increase the likelihood of oscillations: ${\beta }_{\text{d}}$ was large, $n_{\text{a}}$ was close to 3, $n_{\text{d}}$ was large, $\mu$ and the thresholds were of similar value, and $\epsilon$ was small. This suggested that the parameter vector would be robust to other parameter variations, as well.

Performing numerical continuation in $f$ for this parameter vector, we found that the amplitude of oscillations in the active component, $a$, remained roughly constant over the range of viable values of $f$ (Fig. 5 B). Since this parameter vector fell in the small-$\epsilon$ regime, we were able to use an analytical argument to explain this invariance to $s$ (see Supporting Material). In this regime, the limit cycle nearly traces the shape of the $a$-nullcline, with jumps at the limit points (Figs. 2 E and S5). Using this observation and our expressions for the positions of the limit points (Eq. S2.4, a and b), we derived an approximate expression for the amplitude of oscillations in $a$:

$$\Delta a={\gamma }_{\text{a}}{n}_{\text{a}}{\beta }_{\text{a}}\left(\frac{\mu +{\beta }_{\text{a}}}{1+\mu +{\beta }_{\text{a}}}\frac{g_{-}\left(1-g_{-}\right){\left(\frac{g_{-}}{1-g_{-}}\right)}^{\frac{1}{n_{\text{a}}}}}{{\left(\mu +{\beta }_{\text{a}}{g}_{-}\right)}^2}-\frac{\mu }{1+\mu }\frac{g_{+}\left(1-g_{+}\right){\left(\frac{g_{+}}{1-g_{+}}\right)}^{\frac{1}{n_{\text{a}}}}}{{\left(\mu +{\beta }_{\text{a}}{g}_{+}\right)}^2}\right),$$ (13)

where $\Delta a$ is the amplitude of oscillations in $a$ and $g_{\pm }$ are given by Eq. 11, a and b. Eq. 13 is linear in ${\gamma }_{\text{a}}$, which along with $a$ is the only term scaled by $s$ that appears in the expression. As a result, this equation predicts that variations in $s$ should cancel out, leaving the amplitude insensitive to the rate of cyclin synthesis. Fig. S4 shows that Eq. 13 is a good predictor of the amplitude of oscillations in $a$ for most of the oscillatory parameter vectors found by sampling.

Finally, the period of oscillations, T, for the most robust parameter vector was relatively insensitive to the value of $f$ at smaller $f$, but increased at larger values of $f$. At the bifurcation points, the period diverged, corresponding to saddle-node infinite period bifurcations (46), which have been previously observed in cell-cycle models ((11, 16); Fig. 5 C). The loss of limit cycles at these bifurcations is due not to a change in stability of the unique steady state (a Hopf bifurcation), but rather to the appearance of a new, stable steady state elsewhere in the phase space. These calculations indicate that oscillations of robust period and amplitude can be achieved at higher synthesis rates (smaller $f$) for this parameter set.

Discussion

Studies of the embryonic cell cycle are prime examples of how mathematical models of biochemical networks are valuable for summarizing current understanding, testing assumptions and guiding experiments, and exploring the functional capabilities of networks (11, 12, 14, 15, 16, 17). Typically, specifying the interactions in the model is not sufficient to generate a particular functional behavior, which also depends on the values of the system parameters. Ideally, one could characterize the parameter domains with desired functions to explore a model’s functional behaviors and understand under what conditions they arise.

Here, we analyzed a minimal set of interactions, based on biochemical experiments, that describes the embryonic cell cycle and is capable of all of the relevant qualitative behaviors—monostability, bistability, and oscillations. The model’s simplicity allowed us to derive analytical expressions for the boundaries of the domain of oscillations in the model’s parameter space, which are valid in the limit of strong separation of timescales. We then explored the design space of oscillations by sampling the parameter space and screening for oscillations. The marginal distributions of the resulting samples revealed heuristics for choosing parameter vectors that were highly likely to generate oscillations. The bivariate distributions, in particular, effectively acted as two-parameter diagrams averaged over the entire oscillatory domain, as opposed to the cross-sectional view provided by conventional numerical continuation. Sampling also allowed us to check the global validity of our approximate boundaries for the domain of oscillations. Finally, with a large collection of samples from the oscillatory domain in hand, we investigated the robustness of oscillations to variations in the rate of cyclin synthesis. The most robust parameter set satisfied many of the heuristics found from the marginal distributions of oscillatory parameter sets, suggesting that it lay deep in the domain of oscillations. Observing that the amplitude of oscillations in the active component was roughly insensitive to changes in the rate of cyclin synthesis led us to derive an approximate analytical expression for the amplitude that explained this finding. Hence, through a combination of analytical and computational techniques, we characterized the design space of this model and a measure of robustness over this region of parameter space.

The fact that our model contained only two dependent variables allowed geometric analysis of the nullclines and an analytical derivation of the domain of oscillations in the limit of strong separation of timescales. We expect the methods demonstrated here to be applicable to any system reducible to two dependent variables (47, 48). Although these features do not generalize to other models, numerical continuation and sampling of the parameter space are viable methods for exploring a model’s functional behaviors and characterizing their parameter domains (7, 8, 9).

Experimental studies have identified several features contributing to robust oscillations in Xenopus embryos. These include bistability in the steady-state response of active cyclin-CDK to the amount of total cyclin (14, 49), ultrasensitivity in the positive feedback loop (50, 51), similar threshold parameters in the Hill functions of the positive and negative feedback loops (16), and a large Hill exponent for cyclin-CDK degradation (16). Our analysis reveals that these features lead to robust oscillations in the Xenopus system, which is a point in the parameter space, because they are also features of the widest parts of the oscillatory region, as measured by the high fraction of oscillatory points that share these features. For example, having similar thresholds, ${\gamma }_{\text{a}}$ and ${\gamma }_{\text{d}}$, in the Hill functions describing the positive and negative feedback loops and a large Hill exponent for cyclin-CDK degradation, $n_{\text{d}}$, leaves the remaining parameters with a large degree of flexibility. It is only by considering the entire oscillatory region that this becomes clear.

Author Contributions

H.H.M. performed the computational analyses. M.S. and H.H.M. derived the analytical expressions. S.Y.S. designed the research. H.H.M., M.S., and S.Y.S. wrote the manuscript.

Editor: Ruth Baker.