A twostep algorithm for sampling of parameter spaces
The algorithm we propose starts from the definition of a viability condition and of a cost function (Figure 1). Depending on the biological model considered, the viability condition may include stability of a specific steady state, bistability [50], oscillations whose period lies in a given interval [20, 24], the production of specific gene expression patterns [22], and many others. The cost function measures how closely the model's behavior matches the viability condition.
The first step of the algorithm consists of a global coarsegrained exploration of the viable space by an outofequilibrium adaptive Monte Carlo (OEAMC) sampling of the entire parameter space (Figure 2). Following a thermodynamic analogy used by simulated annealing [35] and Metropolis Monte Carlo sampling [36–41], we identify the parameter space and the cost function with the state space and the energy, respectively, of a thermodynamic system that is in contact with a thermal bath with variable temperature. The objective of OEAMC is to identify viable regions in the parameter space by adjusting the "temperature" and the length of the jumps through the parameter space. Briefly, OEAMC adapts the "temperature" and jump lengths to force a finite but small frequency of sampled viable parameter points, and a high proportion of accepted transitions to new parameter points. This helps OEAMC not to "get lost" in the parameter space, but at the same time lets it "travel" through nonviable regions where the cost function may have moderately high values. Thus, this procedure allows OEAMC to visit and sample from regions of the viable space that may be poorly connected to each other.
The low frequency of sampled viable parameter points forces OEAMC to explore the viable space at low resolution. To characterize the viable space in greater detail, it is necessary to define its borders more precisely, and to gain insight into its local geometry. In a second step, we therefore carry out a finegrained exploration of the viable regions already identified through OEAMC, using a technique we call multiple ellipsoidbased sampling (MEBS) (Figure 3). This technique performs a local exploration of the parameter space by sampling from ellipsoids (an approach that is widely used in search algorithms, see [47] and references therein) that change their centres and expand or shrink their axes to enclose different regions of the viable space in which viable points are found. To cover locally nonconvex and/or poorly connected viable spaces, different ellipsoid expansions start from parameter points far away from each other (see Methods and Additional File 1).
The end result of OEAMC and MEBS is a set of viable parameter points that can be used for a variety of purposes. One of them is to define the integration domain in which a Monte Carlo integration estimates the volume of the viable space. (Note that the set of viable points obtained by OEAMC and MEBS is not an uniform sample from this space, and cannot be used directly for this purpose). We define this domain as the union of multiple ellipsoids  different from those used in MEBS sampling  that are constructed by grouping the viable parameter points into clusters, and by determining the ellipsoid with minimum volume that encloses the viable points in each of the clusters (Figure 4). This integration domain thus designed can cover nonconvex and high dimensional viable spaces "tightly". That is, the proportion of viable parameter points in this new integration domain is much higher than in the whole parameter space. By sampling viable points uniformly within this domain, we can compute the volume of a viable space. We reasoned that our procedure would allow us to reduce the computational effort in estimating a viable volume substantially. We will show in the next section that this is indeed the case. More generally, the large set of uniformly distributed viable parameter points that our method can generate permits us to characterize not only the size, but also the topology of a viable space. It also allows us to connect the robustness of a biological system to the geometrical properties of its viable space. Furthermore, this large set of viable parameters opens the possibility for a "glocal" analysis [20], in which the global characterization is supplemented by a local analysis around every viable parameter point. Thus, our algorithm can be used together with a local robustness measurement (e.g., that proposed by Dayarian et al.[7]) to get insight into the distribution of a model's robustness in a viable space.
Efficient sampling of highdimensional spaces
In a first test problem, we estimated the volume of a nonconvex region defined by either one single or two tangent multidimensional spherical shells (Figure 5). We chose this study system to analyze the efficiency of our method as a function of the geometry and dimension of a viable space, because here the viable volume can be calculated analytically.
We define the parameter space as Θ ^{d} = Θ_{1} × Θ_{2} × ⋯ × Θd, where Θ _{
i
} = [10, 10], i = 1, 2, ..., d. The cost function and the viability condition are given by
\begin{array}{c}{E}_{n}\left(\theta \right)=\mathsf{\text{mi}}{\mathsf{\text{n}}}_{j}\left\left\right\theta {c}_{j}\left\right\frac{{r}_{e}+{r}_{i}}{2}\right,\phantom{\rule{1em}{0ex}}{E}_{n}\le \frac{{r}_{e}{r}_{i}}{2},\\ j=1,2,\dots ,n,\phantom{\rule{1em}{0ex}}\left\right{c}_{j}{c}_{j1}\left\right=2{r}_{e},\end{array}
(14)
where c_{
j
} is a point in Θ ^{d} and r_{
e
} and r_{
i
} are two scalars that fulfill r_{
e
} > r_{
i
} (in all our numerical tests r_{
e
} = 0.5 and r_{
i
} = 0.3).
When n = 1 (single spherical shell test case), the lines of constant cost are multidimensional spheres centred on c_{1} (Figure 5b). The (degenerate) global minimum of the cost function occurs in the multidimensional sphere centred on c, and with radius \frac{{r}_{e}+{r}_{i}}{2} (Figure 5a, b). The viability condition is fulfilled by the parameter points that lie in the region enclosed by two multidimensional spheres with centre c and radii r_{
i
} and r_{
e
} , respectively.
For n = 2 (two tangent spherical shells test case), the cost function has its degenerate global minimum in two multidimensional spheres centered on c_{1} and c_{2}, respectively, with radius \frac{{r}_{e}+{r}_{i}}{2} (Figure 5c, d); the viable parameter points lie in the inner region of two tangent multidimensional spherical shells with internal radii r_{
i
} , external radii r_{
e
} and centers c_{1} and c_{2}, respectively.
The volume filled by the viable region can be computed analytically as:
\begin{array}{c}\mathsf{\text{Vo}}{\mathsf{\text{l}}}_{v,t}=n{C}_{d}\left({r}_{e}^{d}{r}_{i}^{d}\right),\\ {C}_{d}=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}d=0,\hfill \\ \hfill 2,\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}d=1,\hfill \\ \hfill \frac{2\pi}{d}{C}_{d2},\hfill & \hfill \mathsf{\text{otherwise}},\hfill \end{array}\right.\end{array}
(15)
where C_{
d
} is the volume of a d  dimensional hypersphere with radius 1.
We now compare the performance of (i) MEBS and OEAMC alone, (ii) both of them together, (iii) uniform sampling, and (iv) the method proposed by Hafner et al.[20] based on Gaussian sampling (see the Additional File 1 for details). For the single spherical shell test case, MEBS and OEAMC alone, and the combination of both methods can identify the viable regions and obtain a good estimate of the viable volumes for dimensions up to d = 15 (Figure 6b). Specifically, for all dimensions we studied they sample more than 95 per cent of the whole viable volume before converging. In addition, for this test case MEBS alone is much more efficient than OEAMC or a combination of both (Figure 6a). Specifically, MEBS converges after sampling substantially fewer parameter points, because the frequency of viable points sampled by OEAMC is comparatively small, and OEAMC thus needs more sampling to estimate the viable volume to a given accuracy. For example, to achieve the same accuracy of volume estimation in d = 15 dimensions, MEBS uses 3fold less samples than the OEAMC, and 2fold less samples than the combination of both methods. In this first test case, the viable space, albeit nonconvex, is wellconnected. This permits a ready exploration of the space by ellipsoid expansions  efficient "travel" of ellipsoids inside the viable volume is possible.
MEBS, OEAMC, and their combination are much more efficient than uniform sampling of the parameter space. For instance, at d = 15 dimensions, "brute force" sampling uses 17 orders of magnitude more sampling points to estimate the viable volume (Figure 6a inset).
The Gaussian sampling carried out by Hafner's method et al. does not permit to identify in detail the borders of the viable volume for high dimensional spaces. Therefore, this technique can not estimate viable volumes in high dimensional spaces with precision (Figure 6b). Moreover, in high dimensional spaces the tiny proportion of the whole parameter space filled by the viable volume forces this technique to sample a large number of viable points before converging (Figure 6a). For example in d = 15 dimensions, Hafner's method uses 4fold more samples than MEBS and underestimates the viable volume by 25 percent.
For the test case of two tangent spherical shells, MEBS and Hafner's method often fail to "find" half of the viable volume in high dimensions (Figure 6d). For example, in 14 dimensions, only 25 percent of the explorations carried out by MEBS and Hafner's method find both shells. The two methods share the same limitation: the inability of sampling a point from the second shell, when starting from a random parameter point in the first shell. To find the second shell starting from the first shell, MEBS and Hafner's method must sample from an ellipsoid or from a Gaussian distribution, respectively, both of which must cover viable regions from both shells. However, both also include nonviable parameter points. In high dimensions the fraction of viable points becomes very small, and the probability of finding a viable point from the second shell is very low.
In contrast, OEAMC alone, and the combination of both OEAMC and MEBS sample the viable regions well (Figure 6d). Specifically, for up to d = 15 dimensions, they estimate the viable volume with an error smaller than a 5 percent. Importantly, the combination of both OEAMC and MEBS is more efficient than OEAMC alone (Figure 6c). For instance, to achieve the same accuracy of volume estimation in d = 15 dimensions, the combination of MEBS and OEAMC used approximately 2fold smaller samples than the OEAMC alone (and 17 order of magnitude smaller samples than uniform sampling).
The key for the success of the combination of OEAMC and MEBS is the complementary nature of their individual strengths. OEAMC does not need many sampled points to find two poorly connected regions. For example, in our two shell test case, it always hit both shells before sampling 25000 parameter in d = 15 dimensions. However, its low frequency of sampled viable points forces it to sample excessively many parameter points in order to explore a viable region in detail. In contrast, the bottleneck for the MEBS procedure is the discovery of a viable region  the second spherical shell in our example  that is poorly connected to a region that it already explored. Once such a region has been discovered by OEAMC, MEBS is able to sample from it efficiently, even if the region is nonconvex.
In sum, the combination of OEAMC and MEBS explores nonconvex and poorly connected viable regions in high dimensional parameter spaces more efficiently and accurately than either method alone and than other methods we evaluated. In addition, for both test cases the number of parameter points sampled by the combination of OEAMC and MEBS scales linearly with the number of dimensions (Figure 6a and Figure 6c). This suggests that for a given fixed complexity of the viable space, the computational effort needed by our method scales linearly with the dimensionality of the parameter space. This property makes our method suitable to explore high dimensional viable spaces.
Model of a biochemical oscillator with two feedback loops
The viable space of a realistic model of a biological system is in general unknown. Therefore, it is necessary to get an estimate of the viable volume through uniform sampling in order to check the performance of our method. However, complex models may have tiny and complex viable spaces that make it infeasible to get such an estimate. This hampers the use of biological models with realistic complexity to characterize our algorithm. To illustrate the application of our method and to check its performance with a biological model, we therefore used a very simplified biological model containing only 12 parameters that permits us to compare the results of our method with the uniform sampling of the parameter space.
This model describes a biochemical oscillator introduced by Hafner et al.[51]. It mimics the basic architecture of biological oscillators, such as cardiac pacemaker cells [52], intracellular calcium oscillations [53], cell cycle [27, 54], and circadian clocks [55]. The model comprises two feedback loops (Figure 7) and it contains 12 individual parameters and 5 state variables which correspond to the concentrations of different proteins. Briefly, in this model a protein R is expressed, phosphorylated and degraded. Protein R can also autophosphorylate. In the positive feedback loop, the phosphorylated form R_{
p
} acts as a kinase for protein Z whose active state Z_{
p
} increases the autophosphorylation rate of R. This kind of positive loop is a basic mechanism behind substratedepletion oscillators. An example is the maturation promoting factor (MPF) oscillator involved in the cell division cycle of frog eggs [56]. The negative feedback loop is composed of three steps: R_{
p
} acts as kinase for an intermediate protein X. Its phosphorylated form X_{
p
} phosphorylates a second protein Y, whose phosphorylated state Y_{
p
} increases the degradation rate of R. Such negative feedback has been proposed as a basis for oscillations in many biological systems (see [27, 28] for reviews).
The dynamics of the concentrations of the proteins R and R_{
p
} follow mass action kinetics [57]
\begin{array}{l}[\dot{R}]={\tilde{k}}_{1}p\left(\left[{Z}_{p}\right]\right)[R],\\ [{\dot{R}}_{p}]=p\left(\left[{Z}_{p}\right]\right)[R]n\left(\left[{Y}_{p}\right]\right)\left[{R}_{p}\right],\end{array}
(16)
where p ([Z_{
p
} ]) and n ([Y_{
p
} ]) respectively, reflect the effects of a positive and a negative feedbacks loops
\begin{array}{c}p\left(\left[{Z}_{p}\right]\right)={\stackrel{\u0303}{k}}_{2}+{\stackrel{\u0303}{k}}_{11}\left[{Z}_{p}\right],\\ n\left(\left[{Y}_{p}\right]\right)={\stackrel{\u0303}{k}}_{3}+{\stackrel{\u0303}{k}}_{12}\left[{Y}_{p}\right].\end{array}
(17)
In contrast, the concentrations of X_{
p
} , Y_{
p
} , and Z_{
p
} are governed by MichaelisMenten kinetics [57]
\begin{array}{c}\left[{\u1e8a}_{p}\right]=\frac{{\stackrel{\u0303}{k}}_{4}\left[{R}_{p}\right]\left(\left[{X}_{T}\right]\left[{X}_{p}\right]\right)}{{\stackrel{\u0303}{k}}_{10}+\left(\left[{X}_{T}\right]\left[{X}_{P}\right]\right)}\frac{{\stackrel{\u0303}{k}}_{5}\left[{X}_{P}\right]}{{\stackrel{\u0303}{k}}_{10}+\left[{X}_{P}\right]},\\ \left[{\u1e8e}_{p}\right]=\frac{{\stackrel{\u0303}{k}}_{6}\left[{X}_{p}\right]\left(\left[{Y}_{T}\right]\left[{Y}_{P}\right]\right)}{{\stackrel{\u0303}{k}}_{10}+\left(\left[{Y}_{T}\right]\left[{Y}_{P}\right]\right)}\frac{{\stackrel{\u0303}{k}}_{7}\left[{Y}_{P}\right]}{{\stackrel{\u0303}{k}}_{10}+\left[{Y}_{P}\right]},\\ \left[{\u017b}_{p}\right]=\frac{{\stackrel{\u0303}{k}}_{8}\left[{R}_{p}\right]\left(\left[{Z}_{T}\right]\left[{Z}_{P}\right]\right)}{{\stackrel{\u0303}{k}}_{10}+\left(\left[{Z}_{T}\right]\left[{Z}_{P}\right]\right)}\frac{{\stackrel{\u0303}{k}}_{9}\left[{Z}_{P}\right]}{{\stackrel{\u0303}{k}}_{10}+\left[{Z}_{P}\right]},\end{array}
(18)
where [X_{
T
} ], [Y_{
T
} ], and [Z_{
T
} ] denote the total concentration of X, Y, and Z, respectively. For the sake of simplicity, we normalize all concentrations to one, i.e., [X_{
T
} ] = [Y_{
T
} ] = [Z_{
T
} ] = 1.
The combination of active positive and negative feedback loops creates oscillators with a tunable frequency, and a robust amplitude [30]. These features make the negative plus positive loop oscillator suitable for systems like beating hearts and cell cycles. Here, we focused on oscillations in a narrow range of frequencies such as those produced by circadian clocks, and used the model to study the robustness of the oscillation period to parameter variations.
To explore broad ranges of parameters values we work in a logarithmic domain in which the logarithm of individual parameters are constrained as follows
\begin{array}{c}{k}_{i}=log\left({\stackrel{\u0303}{k}}_{i}\right),\\ {k}_{i}\in \left[4,2\right],\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}i=1,2,\dots ,10,\\ {k}_{i}\in \left[7,2\right],\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}i=11,12.\end{array}
(19)
Together, these ranges define the 12dimensional parameter space Θ^{12} = k_{1} × k_{2} × ⋯ × k_{12}. We use the cost function
{E}_{m}\left(\theta \right)=\left\{\begin{array}{cc}\hfill {\left[\left({T}_{{R}_{p}}\left(\theta \right)1\right)\u22150.1\right]}^{2},\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}{R}_{p}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{oscillates}},\hfill \\ \hfill \infty ,\hfill & \hfill \mathsf{\text{otherwise}},\hfill \end{array}\right.
(20)
where {T}_{{R}_{p}}\left(\theta \right) is the period of the oscillations of R_{
p
} for a parameter point θ = (k_{1}, k_{2}, ..., k_{12}). The minimum of this cost function is attained by parameter vectors for which {T}_{{R}_{p}}\left(\theta \right)=1.
Finally, we introduced the viability condition
meaning that a parameter point θ is viable if it causes R_{
p
} to oscillate with a period in the narrow interval [0.9, 1.1].
To explore the viable space we carried out an OEAMC sampling followed by a MEBS. The viable parameter points obtained during this exploration are shown in Figure 8, which displays the 12dimensional parameter space through six twodimensional projections. The blue and red points, acquired by MEBS and OEAMC, respectively, occur in similar regions of the parameter space. This shows that the MEBS explored in detail the viable regions previously visited by OEAMC, just as for our spherical shells test case. The combination of OEAMC and MEBS revealed the nonconvexity of the viable space and its implications for the model function. Specifically, we note the viable region in Figure 8f, which is composed of two approximately rectangular or barlike regions that, together, form a nonconvex shape resembling an inverted L. Parts of these regions define topologies in which a single feedback loop produces the oscillations. More precisely, the left part of the horizontal bar corresponds to viable parameter points for which k_{12} is large and k_{11} small. In this region, only the negative feedback loop is active. Conversely, the bottom part of the vertical bar consists of viable parameter points for which k_{12} is small and k_{11} high. It corresponds to architectures where only the positive feedback loop is active (see Figure 7).
In a next step, we performed a Monte Carlo integration (see Methods and Additional File 1 for details) to estimate the viable volume. The integration domain is defined by using the viable points obtained by the OEAMC and MEBS explorations. This domain is approximately 630times smaller than the whole parameter space. After uniformly sampling over the integration domain we obtained 3595 viable points, and estimated a viable volume of Vol _{
v
} = 8.3 · 10^{4} ± 2 · 10^{3}. To validate this estimate, we uniformly sampled over the whole parameter space with the same number of points we used in the OEAMC, MEBS, and integration parts of our algorithm. Only 9 of these points were viable, leading to a viable volume estimate of Vol _{
v
} = 8.1 · 10^{4} ± 2.7 · 10^{4}. The two estimates are very similar, but the estimation obtained through uniform sampling has an uncertainty one order of magnitude larger than the one calculated through our method. In addition, we uniformly sampled 4 · 10^{7} points from the whole parameter space to compare the distributions of every single viable parameter. The results showed that the distributions of each of the 12 parameters obtained through our method and the extensive brute force sampling are very similar (Figure S1).
In sum, our method yields an accurate characterization of the viable space for this complex twelvedimensional system at much higher efficiency than bruteforce approaches. Specifically, by using the same number of sampling points it carries out a 13 times more accurate estimation of the viable volume, and obtains 400 times more uniformly distributed viable points.
Robustness of positive and negative feedback loops
The sample of the viable space we obtained suggests a clear distinction between two oscillatory regimes, one driven by a positive and the other driven by a negative feedback loops. We next discuss these regimes, as an illustration of the type of analyses that our method enables.
The many viable parameter points we found allowed us to characterize key properties of model architectures with individual or combined feedback loops via the geometry of the viable space. For this purpose, we classified each of the viable points into one of the following categories:

Essential negative feedback loop: The model keeps fulfiling the viability condition (21) after removing the positive loop, or after substituting this loop with a higher activation rate of R_{
p
}(see Additional File 1).

Essential positive feedback loop: The model keeps fulfiling the viability condition (21) after removing the negative loop or substituting this loop with a higher degradation rate of R_{
p
}(see Additional File 1).

Essential positive and negative feedback loops: No loop can be removed or substituted by a higher activation or degradation rate without violating the viability condition (21).
We found that model architectures for which the negative feedback loop is essential occupy the vast majority (86%) of the viable space we sampled. In contrast, significantly fewer parameter combinations lead to viable oscillations based on an essential positive loop (10%), or on a combination of essential positive and negative feedback loops (4%).
If a single loop is essential, the parameters mainly responsible for this loop will be constrained. These are parameters k_{8}, k_{9}, k_{11} for the positive loop, and parameters k_{4}, k_{5}, k_{6}, k_{7}, k_{12} for the negative loop (Figure 7). Figures 9a and 9b illustrate these constraints. For example, in Figure 9a, black coloring indicates to what extent parameters involved in the negative loop are constrained if this loop is essential, blue coloring indicates these constraints if only the positive loop is essential, and green coloring indicates these constraints if both loops are essential. Clearly, parameters involved in the negative loop can vary to a lesser extent if this loop is essential than when it is not essential. Analogous observations can be made for parameters involved in the positive loop (Figure 9b).
A comparison of Figures 9a and 9b also shows that parameters involved in the negative and positive feedback loops are constrained to different extents. Specifically, negative loop parameters can vary over broader intervals when the negative loop is essential than positive loop parameters can when this loop is essential. In addition, the parameters that do not form part of any loop (k_{1}, k_{2}, k_{3}, k_{10}) are more constrained in architectures with essential positive feedback loop than in topologies with an essential negative feedback loop (Figure 9c).
Taken together, these observations imply that model architectures based on a negative loop fill more of the viable space, and allow individual parameters to vary more broadly than architectures based on positive feedback loops. In other words, model topologies based on an essential negative feedback loop are more robust than topologies with essential positive loops, or topologies with both essential positive and negative loops.
To further explore this aspect of robustness, we used the method proposed by Dayarian et al.[7] which estimates the number of steps that a random walk needs to escape from the viable space. Briefly, we started ten random walks from every viable parameter point. Each new point in a random walk was selected from an independent Gaussian distribution centred on the previous parameter point and with a diagonal covariance matrix with standard deviations σ = 0.01. We followed every random walk until it arrived at a nonviable parameter point, and recorded the number of steps it had taken to reach this nonviable point. We used this number of steps as an indicator of local robustness around such parameter point. The mean number of steps before exiting the viable region was higher if the starting point corresponded to an architecture with a negative loop than to an architecture with an essential positive loop, or to a combination of essential positive and negative loops (Figure 10). Moreover, the distribution of the number of steps for the negative feedback architectures has a long tail (Figure 10a). Specifically, two times more steps may be needed to leave the viable space than for the other two architectures (Figure 10b, c). Hence, also in terms of local properties revealed by this approach, architectures with an essential negative feedback loop are significantly more robust than other topologies.
In addition, we found that adding a positive (not necessarily essential) loop to a model architecture based on a negative feedback loop further increases robustness and the allowable range of parameter variation. Figure 11a already hints at this observation, because it shows that the largest density of viable parameter points occurs in regions of parameter space where both k_{11} and k_{12} are high. These parameters are important for the positive and negative feedback loops, respectively. In regions with the most viable parameter points both feedback loops are active and at least one of these loops is essential.
Further analysis corroborates this observation. In architectures with an essential negative feedback loop, the mean value of the parameter k_{11}, which controls the strength of the positive feedback loop, is significantly higher (pvalue = 2.0 · 10^{27}; Wilcoxon signed rank test) than the centre of the interval in which k_{11} is defined. In other words, the randomly sampled architectures with an essential negative feedback loop preferentially occur in regions of parameter space where a positive loop is also active. Moreover, the density of viable parameter points increases with the value of the parameter k_{11} (Figure 11b). Thus, a higher strength of the positive feedback loop increases the number of parameter combinations that gives rise to viable oscillations.
Taken together, these observations suggest that an added nonessential positive feedback loop gives a negativeloopbased model oscillator access to more viable parameter points. In the Additional File 1 we perform a similar analysis with a more complex model of a mammalian circadian oscillator. For this more realistic model we also observe that the circadian oscillations can be generated by a single negative feedback loop, whereas an additional positive feedback loop increases the robustness of the oscillations.
Connectivity of the viable space
The connectivity of the viable space indicates to what extent different model architectures with the same behavior can change into one another through small changes in individual parameters, as might occur on evolutionary time scales.
To study this connectivity, we chose a set of viable points in which each of the three basic model architectures we consider are represented. For every pair of parameter points, we defined a straight line connecting them, and identified a set of three points that subdivide the line into four equally long segments (we also subdivided the line into 5, 6, 7, and 8 equally long segments, obtaining qualitatively identical results). We then asked whether each of these points was located in the viable space. If so, it may be possible to connect the two parameter points by a straight line that lies entirely in the viable space. Based on this information, we defined a graph whose nodes are the set viable parameter points. Two nodes are connected by an edge if the entire straight line between the nodes does not leave the viable space. Such an edge reflects the existence of potential evolutionary paths from one to the other node (parameter point) that does not leave the viable space. We find that this graph has one large connected component that comprises 95 percent of all nodes. This observation, together with our earlier analysis (Figure 8f) shows that most of the viable space forms a nonconvex connected body with possible evolutionary trajectories that maintain the same behaviour and that connect qualitatively different system topologies through small changes in individual parameters.
The connected component contains nodes associated with all three basic architectures, but these three kinds of nodes are not equally likely to be connected to each other. Specifically, nodes (viable points) corresponding to model topologies with essential negative feedback loops are only connected to themselves, and to nodes with essential positive and negative feedback loops. Similarly, nodes that define topologies with essential positive feedback loops are only connected to themselves and to nodes with essential positive and negative feedback loops. Potential evolutionary trajectories that connect model architectures based on essential positive feedback loop and essential negative feedback loop, need to pass through configurations for which both loops are essential.
Overall, the global geometry of the viable space shows that model topologies based on an essential negative feedback loop are more robust than other architectures. Essential negative feedback allows the individual parameters to span larger intervals than essential positive feedback. Moreover, our local analysis reveals that topologies based on an essential negative feedback loop sustain the most change before losing viability. Successive small parameter changes can transform oscillators with an essential positive feedback loop into oscillators with an essential negative feedback loop, or vice versa. To do so, requires an intermediary stage in which both loops are essential.