Department of Biochemistry, University of Zurich, Zurich, Switzerland

Department of Biosystems Science and Engineering, ETH Zurich, Zurich, Switzerland

Swiss Institute of Bioinformatics, Lausanne, Switzerland

School of Computer and Communication, EPFL, Lausanne, Switzerland

The Santa Fe Institute, Santa Fe, New Mexico, USA

Department of Biology, University of New Mexico, Albuquerque, New Mexico, USA

Abstract

Background

A biological system's robustness to mutations and its evolution are influenced by the structure of its viable space, the region of its space of biochemical parameters where it can exert its function. In systems with a large number of biochemical parameters, viable regions with potentially complex geometries fill a tiny fraction of the whole parameter space. This hampers explorations of the viable space based on "brute force" or Gaussian sampling.

Results

We here propose a novel algorithm to characterize viable spaces efficiently. The algorithm combines global and local explorations of a parameter space. The global exploration involves an out-of-equilibrium adaptive Metropolis Monte Carlo method aimed at identifying poorly connected viable regions. The local exploration then samples these regions in detail by a method we call multiple ellipsoid-based sampling. Our algorithm explores efficiently nonconvex and poorly connected viable regions of different test-problems. Most importantly, its computational effort scales linearly with the number of dimensions, in contrast to "brute force" sampling that shows an exponential dependence on the number of dimensions. We also apply this algorithm to a simplified model of a biochemical oscillator with positive and negative feedback loops. A detailed characterization of the model's viable space captures well known structural properties of circadian oscillators. Concretely, we find that model topologies with an essential negative feedback loop and a nonessential positive feedback loop provide the most robust fixed period oscillations. Moreover, the connectedness of the model's viable space suggests that biochemical oscillators with varying topologies can evolve from one another.

Conclusions

Our algorithm permits an efficient analysis of high-dimensional, nonconvex, and poorly connected viable spaces characteristic of complex biological circuitry. It allows a systematic use of robustness as a tool for model discrimination.

Background

High-throughput experimental technologies have allowed biology to generate huge amounts of data. The enormity of these data sets permits a systemic view of the cell

Mathematical modeling in biology faces several challenges that arise from uncertainty about relevant parameters. For example, the chemical reactions and the corresponding kinetic equations governing any one biological system are only partially known

The investigation of viable spaces is closely linked to the analysis of robustness in biology. We here define robustness as the persistence, under perturbations, of a behavior that is characteristic for a system

The geometry of viable spaces also plays an important role in a system's robustness. Geometries that permit moderate parameter fluctuations without leaving the viable volume enhance robustness. In evolutionary terms, different ways of performing the same function - for instance, by conserved pathways with homologous yet different proteins

A final motivation to characterize viable spaces comes from model building itself. As we pointed out above, some relevant components and interactions in cellular networks are typically unknown. It follows that the structure of mathematical models describing these networks contains uncertainties. These uncertainties may lead to qualitatively different models that match experimental observations equally well. In this case, robustness can be used as a tool to discriminate between more and less plausible models. Everything else being equal, a model can be considered superior if it is more robust than other plausible models

The use of robustness for model discrimination raises the problem of how to measure robustness. Most robustness analyses in the literature are local (e.g. see

The main challenges for global methods typically result from the fact that parameter spaces can have many dimensions and a complex geometry, about which one has little prior knowledge. To characterize a viable space, some authors perform uniform sampling of the whole parameter space to identify regions where a model displays the desired behavior

Here, we propose an algorithm that overcomes these limitations. Specifically, it can efficiently characterize nonconvex and poorly connected viable spaces. The algorithm consists of two steps, namely a coarse-grained sampling of the viable space, which in turn delivers starting points for a finer-grained exploration. The sampled points also define a domain for subsequent volume computations by Monte Carlo integration, and for acquisition of a large set of uniformly distributed viable points. After describing the algorithm, we analyse a synthetic test problem involving a nonconvex and poorly connected viable space. This analysis will show that in high dimensional spaces our algorithm converges faster and identifies a larger proportion of the viable space than uniform sampling and Hafner's method. Moreover, in contrast to uniform sampling and Hafner's algorithm, whose performances scale exponentially with the number of dimensions, our algorithm's performance scales linearly with the number of dimensions. Subsequently, we illustrate an application of our method to a biochemical circuit. To this end, we focus on a simplified model of biochemical oscillators with positive and negative feedback loops

Methods

Viable regions

Given a model that involves

where Θ_{i }
_{i }
_{1}, _{2}, ..., _{d}
^{d }

Hypothetical cost function and viability condition

**Hypothetical cost function and viability condition**. Contour plot (red curves) of a generic cost function in a 2-dimensional parameter space. Blue areas correspond to the viable space defined by a threshold on the cost function. Some regions in the viable space may have different cost, indicated by different shades of blue in the left panel.

that reflects how well a model produces a behavior under consideration. For a given

A parameter point

that is, if the cost function does not exceed some positive threshold _{0}. For example, ^{d }

Out-of-equilibrium adaptive Monte Carlo sampling

We next describe our coarse-grained, global exploration of the viable space via an out-of-equilibrium adaptive Metropolis Monte Carlo sampling (OEAMC) (Figure

Flowchart representing the basic scheme of the out-of-equilibrium adaptive Monte Carlo (OEAMC) algorithm

**Flowchart representing the basic scheme of the out-of-equilibrium adaptive Monte Carlo (OEAMC) algorithm**. Given an initial parameter point _{0}, covariance matrix ∑ and _{MC }

The Metropolis algorithm was initially introduced to analyse thermodynamic systems ^{d }

This analogy allows us to use an adaptive selection probability with covariance matrix ∑

in order to propose the transitions between parameter points, and Metropolis adaptive acceptance ratios

to accept or not those transitions.

Given _{0}. Then, from the current _{0 }a new _{0}. If _{0}), the new _{1}. In contrast, if _{0}), _{0}))], in which case it becomes _{1}. If _{1 }= _{0}. This scheme is repeated for a predefined number of iterations

After

where _{v }
_{a }
_{v }> f_{0}), _{a }> f_{u}
_{v }
_{a }< f_{l}
_{MC }

Several differences of OEAMC to existing approaches are worth noting. First, OEAMC does not increase

Multiple ellipsoid-based sampling

The OEAMC samples the viable space at low resolution. Thus, it is necessary to introduce a method that uses the viable points already found by OEAMC to explore the viable space in detail. A novel method we call multiple ellipsoid based sampling (MEBS) (Figure

Flowchart for the multiple ellipsoid-based sampling (MEBS) procedure

**Flowchart for the multiple ellipsoid-based sampling (MEBS) procedure**. Given _{MC}

The use of an ellipsoid to bound viable regions in search spaces has been known for decades (see

The _{
v,j
}in an adaptive way (see the Additional File _{
v,j
}(see Additional File _{i}

**Supplementary Information for "Efficient Characterization of High-Dimensional Parameter Spaces for Systems Biology"**. This document shows additional technical information about: • The calculation of minimum volume enclosing ellipsoids involved in OEAMC, MEBS, and the construction of the integration domain. • The determination of the number of clusters involved in the construction of the integration domain. • The acquisition of viable parameter points near the boundary of the viable space involved in the MEBS. • The choice of starting points for new ellipsoid expansions involved in MEBS. • The exploration and volume calculation of spherical shells. • The exploration exploration and volume calculation the viable space associated to biochemical oscillator model. • Characterization of the viable space of a model of the mammalian circadian oscillator with two feedback loops.

Click here for file

The selection of the scaling parameter _{i }

where _{l}
_{u}

The rationale behind equation (7) is as follows: Points in _{0 }
_{
v,j
}, the method then performs a second iteration with _{1 }
_{u}
_{l }

The _{
v,j
}finishes when _{i }
_{
e,j
}, a set of sampled viable points that contains the 2

Then, the MEBS initiates a _{
v,j+1}, is chosen from the set composed by _{MC }
_{
e,k
}, _{
v,j+1 }that is far away from the average of all previous starting points _{
v,k
},

At the end of each ellipsoid expansion, the algorithm determines if MEBS should stop. To do so, the viable parameter points found so far {_{MC}
_{
e,1}, _{
e,2 }..., _{
e,j
}, _{
e,j+1}} are divided into a predefined number of clusters. Then, MEBS calculates the ellipsoids with minimum volume that enclose the points grouped in each cluster and computes the sum of all ellipsoids volumes. The algorithm stops when the sum of the volume of all ellipsoids converges, or when a maximum number of ellipsoid expansions is reached. The final result of MEBS is the set of viable parameter points {_{MC}
_{
e,1}, _{
e,2}, ..., _{
e,j
}, _{
e,j+1}}.

Volume computation and acquisition of a large set of uniformly distributed viable parameter points

The end result of OEAMC and MEBS is a set of viable parameter points that can be used for a variety of purposes. Specifically, this set allows us to obtain simultaneously a large set of uniformly distributed viable points and an estimate of the viable volume Vol_{v}

To calculate Vol_{v }

Given ^{d}

where

Thus, if a high proportion of the ^{d }

This approach is usually sufficient to carry out viable volume estimations in low-dimensional spaces ^{d }

To construct such a subspace (Figure _{t }
_{v }
_{W }
_{t }
_{t }

Flowchart representing the algorithm for viable volume estimation, and the acquisition of a set of viable parameter points

**Flowchart representing the algorithm for viable volume estimation, and the acquisition of a set of viable parameter points**. A set of viable parameter points found by OEAMC and MEBS (uppermost set of blue points in the figure) which are nonuniformly distributed over the whole viable space (area covered by the red curve in the figure) seeds the algorithm. Then, the method groups these points into

In this procedure, the subspace

where _{i }
_{i}

This integrand evaluates the parameter points in the ellipsoid intersections only once. Therefore, by sampling

where _{i }
_{i}

This approach of covering the viable region with ellipsoids can reduce the sampling volume dramatically, and thus increase the proportion of viable parameter points sampled in ^{d}

We caution that in practice, one can never be certain that the whole viable space is contained in the integration domain _{t }

Results and Discussion

A two-step algorithm for sampling of parameter spaces

The algorithm we propose starts from the definition of a viability condition and of a cost function (Figure

The first step of the algorithm consists of a global coarse-grained exploration of the viable space by an out-of-equilibrium adaptive Monte Carlo (OEAMC) sampling of the entire parameter space (Figure

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 fine-grained exploration of the viable regions already identified through OEAMC, using a technique we call multiple ellipsoid-based sampling (MEBS) (Figure

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

Efficient sampling of high-dimensional 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

Single and tangent spherical shells: cost function and viability condition

**Single and tangent spherical shells: cost function and viability condition**. The top-left and bottom-left panels show the cost function for a single and two tangent spherical shells, respectively, in a two-dimensional parameter space. The top-right and bottom-right panels show the contour plots that correspond to the left-side panels. In both cases, the viability condition is fulfilled by all the points enclosed by the two curves for which the value of the cost function is 0.1.

We define the parameter space as Θ^{d }
_{1 }× Θ_{2 }× ⋯ × Θ_{d}
_{i }

where _{j }
^{d }
_{e }
_{i }
_{e }> r_{i }
_{e }
_{i }

When _{1 }(Figure _{i }
_{e}

For _{1 }and _{2}, respectively, with radius _{i}
_{e }
_{1 }and _{2}, respectively.

The volume filled by the viable region can be computed analytically as:

where _{d }

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

Sampling efficiency of the single and tangent spherical shells test cases

**Sampling efficiency of the single and tangent spherical shells test cases**. Panel and inset (a): Number of sampled parameters before convergence as a function of the dimension of the parameter space for the single spherical shell test case. The main panel and the inset show linear and logarithmic scales, respectively. Panel (b): Proportion of sampled viable volume before convergence for the single spherical shell test case. Panel and inset (c): Number of sampled parameters before convergence as a function of the dimension of the parameter space for the two tangent spherical shells test case. The main panel and the inset show linear and logarithmic scales, respectively. Panel (d): Proportion of identified viable volume before convergence for the two tangent spherical shells test case. Red, blue, magenta, green, and black circles represent the results obtained by OEAMC, MEBS, the combination of OEAMC and MEBS, the Hafner's method

MEBS, OEAMC, and their combination are much more efficient than uniform sampling of the parameter space. For instance, at

The Gaussian sampling carried out by Hafner's method

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

In contrast, OEAMC alone, and the combination of both OEAMC and MEBS sample the viable regions well (Figure

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

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

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 _{p }
_{p }
_{p }
_{p }
_{p }

Reaction diagram of the model of a simplified biochemical oscillator with two feedback loops proposed by Hafner

**Reaction diagram of the model of a simplified biochemical oscillator with two feedback loops proposed by Hafner et al. **

The dynamics of the concentrations of the proteins _{p }

where _{p}
_{p}

In contrast, the concentrations of _{p}
_{p}
_{p }

where [_{T }
_{T }
_{T }
_{T}
_{T}
_{T}

The combination of active positive and negative feedback loops creates oscillators with a tunable frequency, and a robust amplitude

To explore broad ranges of parameters values we work in a logarithmic domain in which the logarithm of individual parameters are constrained as follows

Together, these ranges define the 12-dimensional parameter space Θ^{12 }= _{1 }× _{2 }× ⋯ × _{12}. We use the cost function

where _{p }
_{1}, _{2}, ..., _{12}). The minimum of this cost function is attained by parameter vectors for which

Finally, we introduced the viability condition

meaning that a parameter point _{p }

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 _{12 }is large and _{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 _{12 }is small and _{11 }high. It corresponds to architectures where only the positive feedback loop is active (see Figure

Exploration of the viable space for the oscillator with two feedback loops

**Exploration of the viable space for the oscillator with two feedback loops**. Panels show projections of the 12-dimensional parameter space of the oscillator model onto six two-dimensional spaces corresponding to different parameter pairs. Red and blue points correspond to the viable parameter vectors found by OEAMC and MEBS, respectively.

In a next step, we performed a Monte Carlo integration (see Methods and Additional File _{v }
^{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 }
^{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 twelve-dimensional system at much higher efficiency than brute-force 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 _{p }

• 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 _{p }

• 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 _{8}, _{9}, _{11 }for the positive loop, and parameters _{4}, _{5}, _{6}, _{7}, _{12 }for the negative loop (Figure

Distribution of single parameters for model architectures with an essential negative, an essential positive, or essential positive and negative feedback loops

**Distribution of single parameters for model architectures with an essential negative, an essential positive, or essential positive and negative feedback loops**. The top, central, and bottom panels show the distribution of single parameters involved in the negative loop, positive loop, and not involved in any loop, respectively. Black, blue, and green boxplots correspond to parameter points that define architectures based on an essential negative loop, an essential positive loop, or essential positive and negative loop, respectively.

A comparison of Figures _{1}, _{2}, _{3}, _{10}) are more constrained in architectures with essential positive feedback loop than in topologies with an essential negative feedback loop (Figure

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

Local robustness: distribution of the mean number of random walk steps needed to escape from the viable region for different model architectures

**Local robustness: distribution of the mean number of random walk steps needed to escape from the viable region for different model architectures**. Panels (a), (b), and (c) show the distributions of the mean number of steps for architectures based on essential negative, essential positive, as well as essential positive and negative feedback loops, respectively. The mean number of steps averaged over all the viable parameter points that define topologies with an essential negative feedback loop is significantly higher than the mean number of steps for oscillators with essential positive or a combination of negative and positive feedback loops (Wilcoxon rank sum test: ^{-29 }and ^{-20}, respectively).

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 _{11 }and _{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.

Distribution of viable parameter points in the _{11 }_{12 }plane

**Distribution of viable parameter points in the k _{11 }k_{12 }plane**. (a) proportion of viable parameter points found through Monte Carlo integration in every bin of the

Further analysis corroborates this observation. In architectures with an essential negative feedback loop, the mean value of the parameter _{11}, which controls the strength of the positive feedback loop, is significantly higher (^{-27}; Wilcoxon signed rank test) than the centre of the interval in which _{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 _{11 }(Figure

Taken together, these observations suggest that an added nonessential positive feedback loop gives a negative-loop-based model oscillator access to more viable parameter points. In the Additional File

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

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.

Conclusions

In biological systems, the diversity of biochemical parameter values that can lead to similar behavior makes it useful to introduce the concept of a viable space in which a biological system maintains a given function. The algorithm we present here allows an efficient exploration and characterization of such a viable space in systems with many parameters. It involves a global coarse grained identification of viable regions, followed by detailed local explorations of these regions. The global part of our algorithm can find viable regions that may be poorly connected. In the local part, the viable regions discovered in the global part are explored in detail. The exploration of the viable space allows us to identify a (typically nonconvex) subspace of the whole parameter space in which the proportion of viable parameter points is much higher than in the whole space. Knowledge of this subspace can dramatically reduce the number of samples needed to characterize the viable space. It also permits us to acquire a large number of uniformly distributed viable parameter points. The advantages of our method are especially dramatic in high-dimensional parameter spaces. It allows us to explore high dimensional nonconvex and poorly connected viable regions more efficiently and accurately than iterative Gaussian sampling

An intrinsic limitation of our approach is imposed by the potential increase of the viable space's geometric complexity, when the dimension of the parameter space also increases. That is, increasing the dimensionality may cause the emergence of more poorly connected viable regions, which can exponentially increase the minimum number of iterations needed to identify all poorly connected viable regions and to sample them thoroughly. A second potential limitation concerns the identification of unconnected viable regions that are far from each other. The finite sampling frequency of viable parameter points required in the global exploration prevents one from "getting lost" in high dimensional spaces, but it may not allow the algorithm to travel across the wide nonviable region that may separates two viable regions far from each other. A third limitation includes that values for the parameters involved in the global and local explorations steps need to be chosen judiciously. These parameters include the maximum frequency of sampled viable points, bounds for the frequency of accepted iterations, and scaling factors for ellipsoid expansions.

Efficient sampling of the viable space allows one to accurately estimate the viable volume to assess model robustness, to study the topology of the viable space, and to carry out a "glocal" analysis

We showed that the viable space of this oscillator forms a nonconvex connected body in which three classes of parameter points exist. They correspond to model architectures where the negative feedback loop, the positive feedback loop, or both loops are essential for fixed period oscillations. We also found that topologies with an essential negative feedback loop provide more robust fixed period oscillations than those based on an essential positive loop. Moreover, the addition of a nonessential positive feedback loop to a model with an essential negative feedback loop increases the number of parameter combinations that give rise to viable oscillations, and it therefore increases the robustness of fixed period oscillations. In spite of the model's simplicity, these results are consistent with well known structural properties of circadian oscillators: they typically rely on positive and negative feedback loops

In summary, we have introduced an efficient algorithm that explores and characterizes the often tiny regions of a parameter space in which a model displays a desired behavior. We have applied our method to a biological model, but it is not restricted to such systems. It is suitable for all models with many parameters whose values are not well constrained by experimental data. Its spectrum of applications ranges from systems biology

An implementation of our algorithm in MATLAB is available as the package HYPERSPACE from

Authors' contributions

Project planing: EZS, JS, AW. Development of the theory: EZS. Conceived and designed the experiments: EZS, MH, JS, AW. Performed the experiments: EZS. Analyzed the data: EZS, MH, JS, AW. Contributed reagents/materials/analysis tools: EZS, MH, AI. Creation of the figures: EZS, MH. Wrote the paper: EZS, JS, AW. All authors read and approved the final manuscript.

Acknowledgements

We would like to acknowledge support through SNF grants 315200-116814, 315200-119697, and 315230-129708, as well as through the YeastX project of SystemsX.ch. EZS wants to thank Eric Hayden, Karthik Raman, and Adrian Lopez Garcia de Lomana for a careful reading of the manuscript and revealing discussions.