A natural class of robust networks
- *The James Franck Institute and ‡Institute for Biophysical Dynamics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637
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Edited by Peter G. Wolynes, University of California at San Diego, La Jolla, CA, and approved May 1, 2003 (received for review November 6, 2002)
Abstract
As biological studies shift from molecular description to system analysis we need to identify the design principles of large intracellular networks. In particular, without knowing the molecular details, we want to determine how cells reliably perform essential intracellular tasks. Recent analyses of signaling pathways and regulatory transcription networks have revealed a common network architecture, termed scale-free topology. Although the structural properties of such networks have been thoroughly studied, their dynamical properties remain largely unexplored. We present a prototype for the study of dynamical systems to predict the functional robustness of intracellular networks against variations of their internal parameters. We demonstrate that the dynamical robustness of these complex networks is a direct consequence of their scale-free topology. By contrast, networks with homogeneous random topologies require fine-tuning of their internal parameters to sustain stable dynamical activity. Considering the ubiquity of scale-free networks in nature, we hypothesize that this topology is not only the result of aggregation processes such as preferential attachment; it may also be the result of evolutionary selective processes.
Complex protein and genetic networks are large systems of interacting molecular elements that control the propagation and regulation of various biological signals (1–3). They constitute essential classes of biological computations reflecting vital cellular processes such as the regulation of the cell cycle and gene expression. Because some intracellular processes are crucial for the survival of the cell they need to be achieved with reliability. For example, the variation of the concentration of network components in a metabolic network may affect numerous processes. The intricate architecture of such networks raises the question of the stability of their functioning. How can such complex dynamical systems achieve important cellular tasks and remain stable against the variations of their internal parameters?
Three decades ago, Savageau (4, 5) hypothesized that robustness was an essential property of some genetic networks whose functioning would be preserved even if some of their components were produced in various quantities. More recently, several compelling theoretical (6, 7) and experimental (8, 9) studies demonstrated that key processes of specific intracellular networks exhibited a robust behavior to variations of biochemical parameters. Robustness has emerged as a fundamental concept for the characterization of the dynamical stability of biological systems.
Are there universal design principles that would determine whether a given class of networks achieves important intracellular processing with reliability (10)? It would then be possible to predict dynamical properties underlying cell functions without a full knowledge of the molecular details. In particular, it would be important to characterize dynamical robustness as the ability for the network to perform a sequence of biological tasks in the presence of perturbations (3). Driven by such considerations, we analyzed when dynamical robustness may be a direct consequence of the network's architecture.
As a test-bed for the study of the dynamical properties of complex
biological networks we chose to model molecular activities by a simple
two-state model closely related to the one proposed by Kauffman
(11,
12). This model was originally
created for the study of the dynamics of genetic networks, but was later
applied to evolution and social models and became a prototype for the study of
dynamical systems. In this model, the network's architecture follows a random
topology in which every element interacts, on average, with K other
elements. Each network element has two functional states, active and inactive.
The state of a given element is determined by its interactions with other
elements of the network (see Appendix). A generic biochemical
parameter for the whole system is the probability ρ that an arbitrary
element of the network becomes active after interacting with other elements
(13). This probability can be
inferred in principle from the reaction rates and the relative concentration
of each element. The nature of the dynamical process performed by a given
network is determined by its topological parameter K and its
biochemical parameter ρ. Under the simple assumption of a random network
topology, a very rich and unexpected dynamical behavior of the network was
found (11,
12,
14). During the time evolution
of the system, the network elements pass through different states until they
reach a cyclic behavior. Different cycles are possible. Each cycle represents
a variety of intracellular tasks. Two regimes of network activity exist:
chaotic and robust. In the chaotic regime a perturbation in the state of a
single element can make the system jump from one cycle to another. In the
robust regime all such perturbations die out over time. The transition from
one regime to the other is controlled by the parameters K and ρ
and is determined by the equation
Fig. 1a defines the
two regimes of network activity with drastically different dynamical
properties. In the chaotic regime, which entirely dominates the parameter
space (K-ρ), small variations of the initial concentration of
only one component would ruin the network function. Conversely, the robust
regime would give rise to a reliable dynamics for which the network activity
would be insensitive to perturbations in the initial state of the network
elements. With a random network topology this regime is attained only for a
narrow range of the parameters K and ρ.
Dynamical robustness of a Boolean network with random topology. (a) The network dynamics exhibit both chaotic and robust behaviors, depending on the value of the parameters K and ρ. The parameter space K-ρ is then divided into two distinct regimes: chaotic (white) and robust (gray). The curve separating these two regimes is given by the solutions of Eq. 1. For a random network topology (Inset), the parameter space K-ρ is totally dominated by the chaotic regime. Therefore, it is necessary to fine-tune the parameters K and ρ to achieve robust dynamics. (Inset) A typical realization of the random network topology, generated by using a Poisson distribution with K = 2. All elements in the network (•) have approximately the same number of input elements (○). (b) The dynamical robustness R of the network is defined as the fraction of the interval (0,1) for which the parameter ρ gives rise to networks with a robust behavior. This quantity is a function of the mean connectivity K and decreases rapidly a K increases. For a moderate connectivity K = 20, the dynamical robustness is ≈R = 5%.
Unfortunately, this model is inadequate to account for the functioning of biological networks that have heterogeneous architecture. Such topological heterogeneity is present in the human genome. For example, the expression of the β-globin gene is controlled by >20 regulatory proteins. The fibroblast or the platelet-derived growth factor activation induces the activation of >60 other genes (15). Also, the zinc-finger protein Sp1 controls the expression of >300 genes (16). Under an assumption of a random homogeneous architecture with a mean connectivity K = 20, robust behavior would be virtually attained only for ≈5% of the possible values of ρ. Fig. 1b shows the fraction of the interval (0,1) for which the parameter ρ gives rise to networks with a robust behavior. It is apparent that for a random topology only the fine-tuning of the parameters K and ρ in a narrow interval would allow networks to operate with a robust behavior. An alternative approach is to substitute the random network topology for a more realistic one (17).
In recent years, the analyses of the topology of large intracellular networks have revealed a common architecture (18–20). In these natural networks not all elements are equivalent. A small, but significant, fraction of the elements are highly connected, whereas the majority of the elements are poorly connected. This architecture is called scale-free topology and was found to be ubiquitous in networks as diverse as social (21), ecological (22), and genetic, protein networks (18, 23), and the World Wide Web (24, 25). Although the effect of deleterious perturbations on the topology of these networks has been thoroughly studied (23, 26), their dynamical properties need to be explored.
In view of the ubiquity of scale-free networks in nature, we chose to implement the scale-free topology into the two-state model. This topology is defined such that the probability P(k) that an arbitrary element of the networks interacts with exactly k other elements follows the power law P(k) = [Z(γ)k γ]– 1. The parameter γ is called the scale-free exponent, and Z(γ) is the normalization factor. When γ increases both the fraction of highly connected elements and the mean connectivity of the network decrease. In the limit γ → ∞ the network becomes a homogeneous random network with K = 1.
Our approach aims to characterize how the network architecture shapes its dynamical properties and how one particular topology favors a class of networks with robust dynamical behavior (10). Although the average connectivity K is a relevant parameter to characterize the homogeneous topology of random networks, it becomes irrelevant to describe the highly heterogeneous scale-free topology. Thus, the only relevant parameter that characterizes the architecture of scale-free networks is the scale-free exponent γ.
For networks with scale-free topology, the transition from chaotic to
robust dynamics is determined by the equation (see Appendix)
The values of γ and ρ that satisfy this transcendental equation are
plotted on Fig. 2a.
The results of our model reported in the diagram reveal a large class of
robust networks. This class is defined for networks with topological
parameters γ > 2. These networks can operate with a robust dynamical
behavior for which the mean biochemical parameter ρ spans over a large
range of values. In particular, networks with a parameter γ > 2.5
would display a robust dynamics for any value of ρ. The transition from
the chaotic to the robust regimes occurs for values of γ in the interval
[2,2.5] and different values of ρ. Fig.
2a reveals another class of networks with a topological
parameter γ < 2 that exhibit chaotic behavior. This dynamical
behavior is characterized by an extreme sensitivity to small variations of the
initial states of the network elements. Our model shows that the emergence of
a large class of robust networks is a direct consequence of the network
architecture.
Dynamical robustness of a Boolean network with scale-free topology. (a) The mean connectivity K is not a relevant parameter to characterize the network topology for a scale-free network. The dynamics is then characterized in terms of the parameters γ and ρ, where γ is the scale-free exponent. The parameter space γ-ρ is divided in two distinct regimes: chaotic (white) and robust (gray). The transition between these two regimes, given by Eq. 2, is represented here by the solid curve. The chaotic regime does not longer dominate the parameter space. (Inset) A typical realization of the scale-free topology, generated by using a power-law distribution with λ = 2.5. The majority of the elements only have a few connections. But there are two elements connected to almost all other elements in the system. (b) Dynamical robustness R as a function of the scale-free exponent γ. The transition from the chaotic to the robust regime occurs in the interval [2, 2.5]. For γ > 2.5 the dynamics is robust for any value of ρ. The scale-free topology does not require fine-tuning of the parameters γ and ρ to achieve stability. (c) Histogram of 46 scale-free exponents reported for a wide collection of scale-free networks. This collection includes not only biological networks, but also social, ecological, and informatics networks (23, 30–33). It is interesting to note that the majority of the exponents belong to the interval (2, 2.5) where the transition from a robust to a chaotic behavior occurs.
It is interesting to note that experimental values of γ, extracted from real intracellular networks, range in the interval [2, 2.5]. To illustrate this general remark, we indiscriminately report on a histogram 46 published values of scale-free exponents (Fig. 2c). These values were obtained not only from real biological systems but also from many other scale-free networks (20, 23, 25). Remarkably, the majority of these exponents fall in the predicted interval where the transition from chaotic to robust behavior occurs.
This observation leads us to explore the dynamical response of the network to perturbations in the three different regimes: chaotic, critical, and robust. In a scale-free network not all elements are equivalent. We do not expect that perturbing highly connected elements have the same effect on the network dynamics than perturbing poorly connected elements. To test this idea we analyzed numerically how a perturbation produced on only one element affects the dynamics of the network. We computed the overlap (see Appendix) between two trajectories. One trajectory is computed with one element σi with k i connections being perturbed at every time step of the temporal evolution of the network. The second trajectory is computed from the same initial condition but with no perturbation. Such overlap is a measure of the dynamical stability of the system under sustained perturbations. Fig. 3 shows that the stability of the network decreases with the connectivity k i of the perturbed element σi. Remarkably, we found that networks operating in the robust regime are sensitive to perturbations applied to the highly connected elements. In contrast, perturbations produced on arbitrary elements, which in general are poorly connected, will have little effect on the network functioning. The heterogeneity of scale-free networks operating in the robust regime allows different dynamical responses to perturbations depending on the particular element being perturbed. This behavior cannot be observed in homogeneous random networks in which all of the elements are equivalent.
Dynamical stability of scale-free networks. The overlap between a perturbed trajectory and an unperturbed trajectory is computed. Both trajectories start out from the same initial configuration. In the perturbed trajectory one element σi with k i connections randomly takes the values 0 and 1 with the same probability, regardless of the configuration of its input elements. All of the other elements for this trajectory are updated according to Eq. A.1. For the unperturbed trajectory all of the elements follow the dynamics given by Eq. A.1. The graph shows the overlap x between the perturbed and the unperturbed trajectories as a function of the connectivity k i of the perturbed element σi. The different curves correspond to the three different regimes: chaotic (γ = 1.1), critical (γ = 2.5), and robust (γ = 4). In these three cases ρ = 0.5. The simulation was carried out for networks with N = 20 elements. Each point represents the average over 20,000 network realizations.
For a homogeneous network topology, stable behavior was achieved only for very restricted values of the parameters ρ and K. Conversely, the scale-free topology does not require a fine-tuning of the system parameters to attain robustness. This topology allows the existence of robust dynamics in heterogeneous networks. Furthermore, the dynamics of scale-free networks can change if the highly connected elements are perturbed. Thus, scale-free networks operating in the robust regime exhibit both, robustness and sensitivity to perturbations. The above results suggest that the dynamical robustness of a wide variety of biological processes may be a direct consequence of the network architecture. The ubiquity of scale-free networks in nature with a topological parameter 2 < γ < 3 could be the result of evolutionary processes (27). Such a large class of networks, whose key dynamical processes are robust, may have an essential evolutionary advantage in possessing a larger parameter space to adapt to new environmental conditions (27, 28). In our opinion, it would be surprising if nature did not exploit this convenient dynamical property of the scale-free topology.
This work describes the dynamics of networks where complex molecular activities are idealized with a two-state model (29). There exists a fair number of intracellular regulatory systems with components obeying the rules of a two-state model. For example, in signal transduction networks, the signal is processed through phosphorylation cascades where signaling molecules can be active or inactive upon phosphorylation. It is also common to describe regulatory transcription networks in this simple language: A transcription factor regulates positively or negatively the transcription of a given gene. Moreover, available data on the activity of molecular components from large intracellular networks are produced by high-throughput experiments such as two-hybrid assays and C-DNA microarrays. Because of technical constraints, the readout of these experiments is fashioned accordingly to a two-state format that facilitates the applicability of our approach.
The identification of a relationship between topology and dynamical properties of large biological networks primarily motivated our work. But it seems appropriate to think that in view of the significance of scale-free networks in other fields such as sociology and economics our results could be of some interest beyond the scope of biology.
Acknowledgments
We thank Leo P. Kadanoff, Leo Silbert, Tao Pan, Tamara Griggs, and Calin Guet for useful comments. This work was supported by the Materials Research Science and Engineering Center program of the National Science Foundation under Award 9808595 and National Science Foundation Grant DMR 0094569. M.A. also acknowledges the Santa Fe Institute of Complex Systems for partial support through the David and Lucile Packard Foundation Program in the Study of Robustness.
Footnotes
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↵ † To whom correspondence should be addressed. E-mail: maximino{at}control.uchicago.edu.
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This paper was submitted directly (Track II) to the PNAS office.
- Copyright © 2003, The National Academy of Sciences
