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A control theoretic paradigm for cell signaling networks: a simple complexity for a sensitive robustness
A control theoretic paradigm for cell signaling networks: a simple complexity for a sensitive robustness Robyn P Araujo and Lance A Liotta The fields of molecular biology and cell biology are being flooded with complex genomic and proteomic datasets of large dimensions. We now recognize that each molecule in the cell and tissue can no longer be viewed as an isolated entity. Instead, each molecule must be considered as one member of an interacting network. Consequently, there is an urgent need for mathematical models to understand the behavior of cell signaling networks in health and in disease. Addresses Center for Applied Proteomics and Molecular Medicine, George Mason University, 10900 University Boulevard, MS 4E3, Manassas, Virginia, 20110, USA Corresponding author: Araujo, Robyn P ([email protected])
Current Opinion in Chemical Biology 2006, 10:81–87 This review comes from a themed issue on Proteomics and genomics Edited by Garry P Nolan and Emanuel F Petricoin Available online 18th January 2006 1367-5931/$ – see front matter # 2005 Elsevier Ltd. All rights reserved. DOI 10.1016/j.cbpa.2006.01.002
Introduction: why model cell signaling networks? The completion of the human genome project in 2003 issued several significant challenges to the field of molecular biology, including the need to identify and study the functions of the myriad proteins encoded by the genome [1]. The final protein products of gene expression represent the cell’s workforce, monitoring the intracellular and extracellular milieux, relaying and processing information generated by internal, microenvironmental and systemic cues, binding or modifying their signaling partners and undertaking crucial synthetic and recognition activities, all via a magnificent orchestration of exquisitely regulated interaction networks — the protein ‘interactome’ [2]. The growing interest in elucidating the ‘organization and dynamics of the metabolic, signaling and regulatory networks through which the life of the cell is transacted’ [3] has spawned powerful proteomic technologies that now make it possible ‘to generate quantitative protein expression data on a scale and sensitivity comparable to that achieved at the genetic level’ [3]. Protein microarrays, in particular, now enable low-abundance proteins to be studied in a reproducible, high-throughput format, prowww.sciencedirect.com
viding crucial information about the functional state of the cell’s signaling pathways [4–6]. In the face of the ‘sheer complexity of signaling, backed up by a flood of raw data’ [7], a recurring theme in the literature today is the necessity to develop sophisticated mathematical models to manage, interpret and understand this information, and to draw the data together into coherent paradigms [8]. The potential benefits of a mechanistic understanding of signal transduction through the protein interactome are many. Since dysregulated signaling networks are the basis of most human diseases [9], mathematical models of cell signaling could furnish insights into the most fundamental mechanisms of pathogenesis and disease progression. These insights may prove crucial to the study of complex signaling disorders such as cancer which, unlike diseases such as cystic fibrosis or muscular dystrophy, are born of a multiplicity of signaling aberrations rather than a single defect [10,11]. Most significantly, these network models will constitute a crucial accessory in the quest for new therapeutic targets, the design of new molecular-targeted agents and the development of more effective therapeutic paradigms for the treatment of cell signaling diseases. Here, we review the current field of network modeling, emphasizing that successful models of the future must use control theory to explain the how a cell’s signaling network can be very stable in the face of unwanted perturbations, while simultaneously responding in a highly sensitive and specific manner to desired stimuli.
Current trends in network modelling Review articles on the modeling of cell signaling networks have been entering the literature at an everincreasing rate over the past two years [7,12,13,14,15], providing a running commentary on the progress in this field and attesting to the growing interest in mathematical approaches to the study of signaling networks. Amongst the wide variety of network models published over this time frame is the ‘scale-free’ organizational paradigm [16–18], arising from the application of statistical mechanics principles to the study of large-scale networks. One of the most significant contributions this framework has made to cell signaling research is a crucial insight into network fragility: scale-free networks are robust against ‘accidental failures’ but very vulnerable to a coordinated attack at a ‘hub’ [19]. Several alternative approaches have been developed in tandem with these statistical models, Current Opinion in Chemical Biology 2006, 10:81–87
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including stochastic simulations [20,21] and mechanistic ‘bottom-up’ approaches [22,23] comprising both largescale computational models [24,25] and modularized network descriptions [13,26]. Within the domain of mechanistic modeling, the growing realization that complex systems have ‘emergent’ properties [27] — having behavior that ‘cannot be understood or predicted simply by analyzing the structure of their components’ [28] — has sparked a renewed interest in ‘systems biology’, a merger of systems theory and cell biology [29,30]. Systems biology is arguably one of the most influential ideas in cell signaling research today, promising to yield a ‘more global and in-depth’ [31] understanding of biological systems, furnishing insights into the root causes of complex diseases such as cancer, contributing to the drug discovery process and facilitating a variety of aspects of health care including treatment evaluation and predictive medicine [31]. In the present article, we emphasize a distinct subdiscipline within systems biology: the application of control theory to the study of cell signaling networks. Although still a relatively new idea, this fresh paradigm is already showing incredible promise and appears poised to define a new era in cellular network modeling.
Cellular network design: a strange harmony of contrasts Recent investigators have painted a portrait of the protein interactome as a sophisticated control system, and have identified intriguing parallels between the functioning of the cell and many modern engineering control systems [29,32,33] (Figure 1). This new concept has emerged from a more encompassing and functional description of the processing and control of intracellular signals, thereby establishing a new focus for the proteomics of the future. The recent review by Araujo et al. [33], for example, has unified various disparate themes in the prevailing understanding of intracellular control theory to describe the design of the protein interactome as ‘a strange harmony of contrasts’. Indeed, the formation of an ‘intelligent’ control circuitry from a potentially vast network of protein–protein interactions appears to unite two intriguing paradoxes of cell signaling. The first of these paradoxes resides in the cell’s ingenious use of the almost unassailable complexity of its signaling networks. It is partly the complexity of these networks that renders the cell such a versatile and sophisticated organism, able to produce a wide variety of responses to a wide variety of possible stimuli. In fact, complexity
Figure 1
A cell signaling pathway as a control system. The example depicted here is a simple negative feedback loop in which a three-tiered kinase cascade is embedded. Feedback (and feedforward) control in intracellular signaling is, of itself, an intriguing phenomenon because a series of protein–protein interactions constitutes both the signal and its own controller. The simple caricature model illustrated here could represent the {Ras ! Raf ! Mek ! Erk –j Ras} pathway, for example. Activated cytosolic Erk (c*), at the terminal end of this control module forms two pools — the ‘output’, which translocates to the nucleus to continue signal transmission, and the ‘sensor’, which then translocates to the membrane to interact with the ‘controller’. In this case, the ‘controller’ is phosphorylated SOS (d*), which promotes the dissociation of the Grb2–SOS complex, thereby downregulating the activation of Ras (‘input’). Current Opinion in Chemical Biology 2006, 10:81–87
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manifests itself on almost every level of the network’s functional structure, on the macroscale and the microscale, and across timescales. Even on the level of a single signaling molecule, the interplay between activators and inhibitors, and between inhibitors and ‘activators of inhibitors’, can introduce complexity through the potential to create toggle switches, signal amplification and lateral signal propagation [34]. Nowhere is complexity more evident than on the macroscale of signaling networks, with myriad interconnections linking different signaling cascades together via junctions and integrators [35], further entwined with manifold control interactions such as feedback and feedforward loops, and operating in concert with still other features such as redundancy and spatiotemporal modularization [2]. It is amid such overwhelming internal complexity that the cell is required to make sense of the signals it encounters, and function stably in making precise and well-defined connections between stimuli and responses.
combination of activation and inhibition [33]. One manifestation of this fundamental building block is the coordination of procaspase cleavage, with the competitive binding of active caspase by a stoichiometric inhibitor (such as a member of the inhibitor of apoptosis (IAP) family, say). The sigmoidality generated by this mechanism is referred to as ‘stoichiometric inhibitor ultrasensitivity’ [37]. By far the most common form of the sigmoidal response element, however, is the covalent modification cycle formed by a kinase and an opposing phosphatase. The seminal work of Goldbeter and Koshland in the early 1980s [38] showed that partial saturation of these interconverting enzymes converted the signal-response curve for this network module from a rectangular hyperbola to a more switch-like sigmoidal shape, a phenomenon they termed ‘zero-order ultrasensitivity’. Sauro and Kholodenko [39] have noted that these sigmoidal response elements are analogous to the behavior of transistors in an electronic circuit.
This apparent paradox is resolved through the subtleties of the fundamental nature of this complexity. Indeed, in view of the cell’s spartan capacity for adaptation to change and ‘intelligent’, precisely regulated responses to stimuli, the complexity of the protein interactome is certainly not a chaotic one: it must be an ordered, structured, ‘simple’ complexity. If, for example, a cell is to undergo a ‘pointof-no-return’ behavior such as apoptosis, say, then overarching the large number of requisite interactions must be a well-defined control mechanism to generate a one-way switch. Such is the ‘simplicity within complexity’ paradox of cell signaling [33]. For this reason, many investigators are beginning to advocate a ‘coarse-grained’ [36] level of description for signaling networks, focusing on ‘the system behavior of the network while neglecting molecular details wherever possible’ [36].
This interplay between activation and inhibition is the universal theme of cell signaling, one that is mirrored at every level and scale of the organization of protein networks. Feedback and feedforward loops are the chief mechanisms by which this interplay is realized on larger network scales, allowing precise and carefully regulated control over entire pathways and enabling the cell to portray complex and sophisticated modes of behavior. Tyson et al. [40] give an excellent overview of all the major classes of feedback and feedforward loops germane to cellular control. Negative feedback loops, for example, are conducive to homeostasis, or even oscillations within certain parameter regimes. Negative feedforward loops, on the other hand promote adaptation of the network response to a stimulus. Thus, it is the capacity for negative regulation of protein activity that arms the cell with the indispensable properties of stability, robustness, adaptation and homeostasis. Well-defined examples of negative control loops abound in the literature today; indeed, it is reasonable to suppose that every pathway in a cell’s signaling network possesses some kind of negative control mechanism, if not a multiplicity of negative controls, to limit the amplification and spread of signals and to promote adaptation to microenvironmental changes.
Perhaps the most remarkable aspect of cell signaling, however, is the apparent paradox of its robustness in the face of unwanted perturbances, while responding specifically and sensitively to relevant inputs — the ‘sensitivity within robustness’ paradox [33]. Thus endowed with remarkable regulation and an extraordinary versatility, the cell may respond decisively or maintain its equipoise as appropriate, filtering out signal noise, resisting harmful assaults and orchestrating specific and coordinated responses in a virtuosic portrayal of functional chiaroscuro. In this way, a ‘strange harmony of contrasts’ is born of the union of two pivotal paradoxes — a ‘simple complexity’ that enables the cell to exhibit a ‘sensitive robustness’.
The orchestration of activators and inhibitors: theme and variations The morpheme of cell signaling is the sigmoidal or ‘ultrasensitive’ response element, generated by the local control of a single signaling molecule by a coordinated www.sciencedirect.com
By contrast, positive feedback loops are required for the all-important switching mechanisms that allow decisive responses to stimuli. Although long under-appreciated in control theory because of their generally destabilizing influence, the ‘one-way’ and toggle switches made possible by this positive regulation now receive much attention from theoreticians and experimentalists alike. Indeed, this control mechanism represents the basis of ‘point-of-no-return’ events such as apoptosis and oocyte maturation, for example [41,42], and permits sustained responses from transient stimuli [43]. Current Opinion in Chemical Biology 2006, 10:81–87
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Figure 2
The trade-off between sensitivity and robustness: positive and negative regulation working together. A feedback funambulism is a composite control motif that represents a kind of ‘balancing act’ between positive and negative feedback loops. The regulated signaling molecule at the terminus of the two opposing loops is the ‘funambulist’, F, whereas the molecule at the source of the control loops is the ‘base protein’, B. This control motif could comprise a dormant adaptor or oscillator (due to negative feedback) and a dormant one-way switch (due to positive feedback), for example. The strength of each of these two opposing influences is regulated by a variety of context-specific cues delivered via other signaling pathways, thereby determining the appropriate behavior for the network in a given situation.
The juxtaposition of these two opposing control archetypes within signaling networks offers a captivating glimpse into the functional basis of the ‘sensitivity within robustness’ paradox of cell signaling [33]. Robust, adaptive, homeostatic mechanisms coexist with sensitive switching mechanisms — negative controls with positive controls, inhibition with activation — with the strength of each of the opposing influences regulated by an array of external and internal cues via other pathways, permitting a contextual assessment of the appropriate balance and the appropriate response (Figure 2).
Figure 3
This all-important co-existence of opposing controls points to a crucial role for ‘signaling funambulisms’ in the regulation of cell signaling networks [33]. A funambulism is a type of ‘balancing act’ in which a signaling molecule is regulated both positively and negatively by either an upstream node (a feedforward funambulism) or a downstream node (a feedback funambulism), as illustrated in Figure 3.
Control-oriented modeling of cell signaling networks Control-oriented modeling of cell signaling networks represents a new era in the theoretical elucidation of cell signaling networks. While a modularized view of signaling Current Opinion in Chemical Biology 2006, 10:81–87
Funambulisms in cell signaling. Two common examples of funambulisms in cell signaling: the Ras-Raf feedforward funambulism, and the IRS-1-Akt feedback funambulism. Funambulists are shown as rectangles and base proteins are shown as triangles. Activation is represented by an arrow, whereas inhibition is represented with a barred line. www.sciencedirect.com
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networks has emerged in recent years [44–46], controloriented modeling refines this concept further to emphasize the use of ‘minimal modules’ [33] — the ‘less is more’ approach [36]. According to this paradigm of cell signaling, many network interactions should be condensed through a judicious choice of ‘effective’ kinetic parameters to reveal the minimum number of interactions required to fully characterize the module’s control capabilities. Exploiting the ‘simplicity within complexity’ paradox of cell signaling in this way, the overarching control principles may be considered in bold relief against the canvas of the associated complex network. Two recent examples of control-oriented models of cell signaling networks are noteworthy. Brandman et al. [47] considered the composite control motif of two interlinked positive feedback loops. While positive feedback is a ubiquitous control motif to convert graded inputs into discontinuous switches, these authors identify many biological systems that appear to rely on the interconnection of a multiplicity of these loops (see [47] and references therein). A mathematical model of this control motif revealed the fascinating finding that the combination of the two loops can make switching faster while reducing signal noise: a slow loop fosters robustness of the alternative steady states, whereas a fast loop enhances the speed of switching between the two states. Bornhold
[36], reviewing the model, notes that this control motif, with its rapid yet robust switching function, is reminiscent of the robustly designed electronic building blocks used to build modern computers. Araujo et al. [33], on the other hand, consider the feedback funambulism exemplified by the interactions of the scaffold protein insulin receptor substrate-1 (IRS-1) with its downstream effectors Akt and mTOR/p70S6K. This mathematical model emphasizes the potential of this particular control motif to contribute to tumor development, since an imbalance in the two opposing feedback loops in favor of the positive loop could create a one-way switch (Figure 4). This mechanism is thought to provide an explanation for the frequently observed constitutive activation of funambulist proteins such as IRS-1 in human tumors [48,49]. Most significantly, the development of a control-oriented model permits a critical assessment of the different classes of therapeutic targets within the network subset under consideration. Indeed, the model revealed that, ‘depending on the concentration of the treating agent and the location of the target within the cellular circuitry, a given treatment can be ineffective, or worse. . .it may actually accelerate the neoplastic process’ [33]. For this reason, control theoretic paradigms appear to be the key to developing effecting treatment strategies of the future.
Figure 4
The convergence of feedback loops: a precarious balance. Adapted from the results presented by Araujo et al. [33]. Feedback loops are thought to play a critical role in carcinogenesis, as suggested by several recent experimental [48,49] and theoretical [33] studies. If the positive and negative feedback loops, as regulated by the base protein, B, are appropriately balanced, then the funambulist, F, will be held in a state of equipoise (a). This permits a controlled, graded response to changes in stimulus. On the other hand, an imbalance in favor of the positive feedback loops could contribute to a malignant phenotype, since the emerging one-way switch could cause the funambulist to be constitutively active (b). Note from the stimulusresponse curve in (b) that once the switch is tripped at the critical stimulus-strength, S*, the system remains at the higher activation state even in the event of a complete withdrawal of the stimulus. www.sciencedirect.com
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Conclusions In the face of the sheer complexity of the information flow within cells and organisms, it is a wonder that living systems remain so robust and adaptable. Mathematical modeling is now providing a glimmer of insight into the sophisticated feedback control and network regulation operating within cells. The interplay between activation and inhibition (e.g. kinases versus phosphatases) is the universal theme of cell signaling, one that is mirrored at every level and scale of the organization of gene and protein networks. Feedback and feedforward loops are the chief mechanisms by which this interplay is realized on larger network scales, enabling the cell to exhibit complex and sophisticated modes of behavior. In this short review we have shown how even simple cellular feedback systems can generate complex non-linear behavior including one-way switches and toggle switches. These complex responses to an input may now begin to explain how a cell can exhibit an apparent paradox of robustness in the face of unwanted perturbances while responding specifically and sensitively to relevant inputs. Control theory will play a critical role in the next era of systems biology network modeling.
References and recommended reading Papers of particular interest, published within the annual period of review, have been highlighted as: of special interest of outstanding interest
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provides a captivating glimpse into the parallels between the control of cell signaling networks and electrical circuits and other modern engineering systems.
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33. Araujo RP, Petricoin EF, Liotta LA: Illuminating the cancer cell’s control circuitry: paving the way to individualized therapeutic strategies. Curr Signal Transduct Ther 2006, in press. These authors introduce the concept of the ‘intelligent’ functioning of the cell being evoked at the intersection of two pivotal paradoxes — ‘simplicity within complexity’ and ‘sensitivity within robustness’. A mathematical model of a ‘feedback funambulism’ is presented emphasizing a ‘minimalist modularization’ approach to network modeling. The concept of ‘control-oriented therapeutic strategies’ is presented, outlining why a control theoretic paradigm is the key to developing effective treatment strategies against diseases of signal transduction such as cancer.
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37. Ferrell JE: Tripping the switch fantastic: how a protein kinase cascade can convert graded inputs into switch-like outputs. Trends Biochem Sci 1996, 21:460-466. 38. Goldbeter A, Koshland DE: An amplified sensitivity arising from covalent modification in biological systems. Proc Natl Acad Sci USA 1981, 78:6840-6844. 39. Sauro HM, Kholodenko BN: Quantitative analysis of signaling networks. Prog Biophys Mol Biol 2004, 86:5-43. 40. Tyson JJ, Chen KC, Novak B: Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr Opin Cell Biol 2003, 15:221-231. This is an outstanding review of all the major control archetypes germane to the regulation of gene and protein networks, illustrating how physiological regulatory systems may be understood in mathematical terms. Some composite control motifs are also presented, including activatorinhibitor oscillators and substrate-depletion oscillators. The article
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Current Opinion in Chemical Biology 2006, 10:81–87
Current Opinion in Chemical Biology 2006, 10:81–87 This review comes from a themed issue on Proteomics and genomics Edited by Garry P Nolan and Emanuel F Petricoin Available online 18th January 2006 1367-5931/$ – see front matter # 2005 Elsevier Ltd. All rights reserved. DOI 10.1016/j.cbpa.2006.01.002
Introduction: why model cell signaling networks? The completion of the human genome project in 2003 issued several significant challenges to the field of molecular biology, including the need to identify and study the functions of the myriad proteins encoded by the genome [1]. The final protein products of gene expression represent the cell’s workforce, monitoring the intracellular and extracellular milieux, relaying and processing information generated by internal, microenvironmental and systemic cues, binding or modifying their signaling partners and undertaking crucial synthetic and recognition activities, all via a magnificent orchestration of exquisitely regulated interaction networks — the protein ‘interactome’ [2]. The growing interest in elucidating the ‘organization and dynamics of the metabolic, signaling and regulatory networks through which the life of the cell is transacted’ [3] has spawned powerful proteomic technologies that now make it possible ‘to generate quantitative protein expression data on a scale and sensitivity comparable to that achieved at the genetic level’ [3]. Protein microarrays, in particular, now enable low-abundance proteins to be studied in a reproducible, high-throughput format, prowww.sciencedirect.com
viding crucial information about the functional state of the cell’s signaling pathways [4–6]. In the face of the ‘sheer complexity of signaling, backed up by a flood of raw data’ [7], a recurring theme in the literature today is the necessity to develop sophisticated mathematical models to manage, interpret and understand this information, and to draw the data together into coherent paradigms [8]. The potential benefits of a mechanistic understanding of signal transduction through the protein interactome are many. Since dysregulated signaling networks are the basis of most human diseases [9], mathematical models of cell signaling could furnish insights into the most fundamental mechanisms of pathogenesis and disease progression. These insights may prove crucial to the study of complex signaling disorders such as cancer which, unlike diseases such as cystic fibrosis or muscular dystrophy, are born of a multiplicity of signaling aberrations rather than a single defect [10,11]. Most significantly, these network models will constitute a crucial accessory in the quest for new therapeutic targets, the design of new molecular-targeted agents and the development of more effective therapeutic paradigms for the treatment of cell signaling diseases. Here, we review the current field of network modeling, emphasizing that successful models of the future must use control theory to explain the how a cell’s signaling network can be very stable in the face of unwanted perturbations, while simultaneously responding in a highly sensitive and specific manner to desired stimuli.
Current trends in network modelling Review articles on the modeling of cell signaling networks have been entering the literature at an everincreasing rate over the past two years [7,12,13,14,15], providing a running commentary on the progress in this field and attesting to the growing interest in mathematical approaches to the study of signaling networks. Amongst the wide variety of network models published over this time frame is the ‘scale-free’ organizational paradigm [16–18], arising from the application of statistical mechanics principles to the study of large-scale networks. One of the most significant contributions this framework has made to cell signaling research is a crucial insight into network fragility: scale-free networks are robust against ‘accidental failures’ but very vulnerable to a coordinated attack at a ‘hub’ [19]. Several alternative approaches have been developed in tandem with these statistical models, Current Opinion in Chemical Biology 2006, 10:81–87
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including stochastic simulations [20,21] and mechanistic ‘bottom-up’ approaches [22,23] comprising both largescale computational models [24,25] and modularized network descriptions [13,26]. Within the domain of mechanistic modeling, the growing realization that complex systems have ‘emergent’ properties [27] — having behavior that ‘cannot be understood or predicted simply by analyzing the structure of their components’ [28] — has sparked a renewed interest in ‘systems biology’, a merger of systems theory and cell biology [29,30]. Systems biology is arguably one of the most influential ideas in cell signaling research today, promising to yield a ‘more global and in-depth’ [31] understanding of biological systems, furnishing insights into the root causes of complex diseases such as cancer, contributing to the drug discovery process and facilitating a variety of aspects of health care including treatment evaluation and predictive medicine [31]. In the present article, we emphasize a distinct subdiscipline within systems biology: the application of control theory to the study of cell signaling networks. Although still a relatively new idea, this fresh paradigm is already showing incredible promise and appears poised to define a new era in cellular network modeling.
Cellular network design: a strange harmony of contrasts Recent investigators have painted a portrait of the protein interactome as a sophisticated control system, and have identified intriguing parallels between the functioning of the cell and many modern engineering control systems [29,32,33] (Figure 1). This new concept has emerged from a more encompassing and functional description of the processing and control of intracellular signals, thereby establishing a new focus for the proteomics of the future. The recent review by Araujo et al. [33], for example, has unified various disparate themes in the prevailing understanding of intracellular control theory to describe the design of the protein interactome as ‘a strange harmony of contrasts’. Indeed, the formation of an ‘intelligent’ control circuitry from a potentially vast network of protein–protein interactions appears to unite two intriguing paradoxes of cell signaling. The first of these paradoxes resides in the cell’s ingenious use of the almost unassailable complexity of its signaling networks. It is partly the complexity of these networks that renders the cell such a versatile and sophisticated organism, able to produce a wide variety of responses to a wide variety of possible stimuli. In fact, complexity
Figure 1
A cell signaling pathway as a control system. The example depicted here is a simple negative feedback loop in which a three-tiered kinase cascade is embedded. Feedback (and feedforward) control in intracellular signaling is, of itself, an intriguing phenomenon because a series of protein–protein interactions constitutes both the signal and its own controller. The simple caricature model illustrated here could represent the {Ras ! Raf ! Mek ! Erk –j Ras} pathway, for example. Activated cytosolic Erk (c*), at the terminal end of this control module forms two pools — the ‘output’, which translocates to the nucleus to continue signal transmission, and the ‘sensor’, which then translocates to the membrane to interact with the ‘controller’. In this case, the ‘controller’ is phosphorylated SOS (d*), which promotes the dissociation of the Grb2–SOS complex, thereby downregulating the activation of Ras (‘input’). Current Opinion in Chemical Biology 2006, 10:81–87
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manifests itself on almost every level of the network’s functional structure, on the macroscale and the microscale, and across timescales. Even on the level of a single signaling molecule, the interplay between activators and inhibitors, and between inhibitors and ‘activators of inhibitors’, can introduce complexity through the potential to create toggle switches, signal amplification and lateral signal propagation [34]. Nowhere is complexity more evident than on the macroscale of signaling networks, with myriad interconnections linking different signaling cascades together via junctions and integrators [35], further entwined with manifold control interactions such as feedback and feedforward loops, and operating in concert with still other features such as redundancy and spatiotemporal modularization [2]. It is amid such overwhelming internal complexity that the cell is required to make sense of the signals it encounters, and function stably in making precise and well-defined connections between stimuli and responses.
combination of activation and inhibition [33]. One manifestation of this fundamental building block is the coordination of procaspase cleavage, with the competitive binding of active caspase by a stoichiometric inhibitor (such as a member of the inhibitor of apoptosis (IAP) family, say). The sigmoidality generated by this mechanism is referred to as ‘stoichiometric inhibitor ultrasensitivity’ [37]. By far the most common form of the sigmoidal response element, however, is the covalent modification cycle formed by a kinase and an opposing phosphatase. The seminal work of Goldbeter and Koshland in the early 1980s [38] showed that partial saturation of these interconverting enzymes converted the signal-response curve for this network module from a rectangular hyperbola to a more switch-like sigmoidal shape, a phenomenon they termed ‘zero-order ultrasensitivity’. Sauro and Kholodenko [39] have noted that these sigmoidal response elements are analogous to the behavior of transistors in an electronic circuit.
This apparent paradox is resolved through the subtleties of the fundamental nature of this complexity. Indeed, in view of the cell’s spartan capacity for adaptation to change and ‘intelligent’, precisely regulated responses to stimuli, the complexity of the protein interactome is certainly not a chaotic one: it must be an ordered, structured, ‘simple’ complexity. If, for example, a cell is to undergo a ‘pointof-no-return’ behavior such as apoptosis, say, then overarching the large number of requisite interactions must be a well-defined control mechanism to generate a one-way switch. Such is the ‘simplicity within complexity’ paradox of cell signaling [33]. For this reason, many investigators are beginning to advocate a ‘coarse-grained’ [36] level of description for signaling networks, focusing on ‘the system behavior of the network while neglecting molecular details wherever possible’ [36].
This interplay between activation and inhibition is the universal theme of cell signaling, one that is mirrored at every level and scale of the organization of protein networks. Feedback and feedforward loops are the chief mechanisms by which this interplay is realized on larger network scales, allowing precise and carefully regulated control over entire pathways and enabling the cell to portray complex and sophisticated modes of behavior. Tyson et al. [40] give an excellent overview of all the major classes of feedback and feedforward loops germane to cellular control. Negative feedback loops, for example, are conducive to homeostasis, or even oscillations within certain parameter regimes. Negative feedforward loops, on the other hand promote adaptation of the network response to a stimulus. Thus, it is the capacity for negative regulation of protein activity that arms the cell with the indispensable properties of stability, robustness, adaptation and homeostasis. Well-defined examples of negative control loops abound in the literature today; indeed, it is reasonable to suppose that every pathway in a cell’s signaling network possesses some kind of negative control mechanism, if not a multiplicity of negative controls, to limit the amplification and spread of signals and to promote adaptation to microenvironmental changes.
Perhaps the most remarkable aspect of cell signaling, however, is the apparent paradox of its robustness in the face of unwanted perturbances, while responding specifically and sensitively to relevant inputs — the ‘sensitivity within robustness’ paradox [33]. Thus endowed with remarkable regulation and an extraordinary versatility, the cell may respond decisively or maintain its equipoise as appropriate, filtering out signal noise, resisting harmful assaults and orchestrating specific and coordinated responses in a virtuosic portrayal of functional chiaroscuro. In this way, a ‘strange harmony of contrasts’ is born of the union of two pivotal paradoxes — a ‘simple complexity’ that enables the cell to exhibit a ‘sensitive robustness’.
The orchestration of activators and inhibitors: theme and variations The morpheme of cell signaling is the sigmoidal or ‘ultrasensitive’ response element, generated by the local control of a single signaling molecule by a coordinated www.sciencedirect.com
By contrast, positive feedback loops are required for the all-important switching mechanisms that allow decisive responses to stimuli. Although long under-appreciated in control theory because of their generally destabilizing influence, the ‘one-way’ and toggle switches made possible by this positive regulation now receive much attention from theoreticians and experimentalists alike. Indeed, this control mechanism represents the basis of ‘point-of-no-return’ events such as apoptosis and oocyte maturation, for example [41,42], and permits sustained responses from transient stimuli [43]. Current Opinion in Chemical Biology 2006, 10:81–87
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Figure 2
The trade-off between sensitivity and robustness: positive and negative regulation working together. A feedback funambulism is a composite control motif that represents a kind of ‘balancing act’ between positive and negative feedback loops. The regulated signaling molecule at the terminus of the two opposing loops is the ‘funambulist’, F, whereas the molecule at the source of the control loops is the ‘base protein’, B. This control motif could comprise a dormant adaptor or oscillator (due to negative feedback) and a dormant one-way switch (due to positive feedback), for example. The strength of each of these two opposing influences is regulated by a variety of context-specific cues delivered via other signaling pathways, thereby determining the appropriate behavior for the network in a given situation.
The juxtaposition of these two opposing control archetypes within signaling networks offers a captivating glimpse into the functional basis of the ‘sensitivity within robustness’ paradox of cell signaling [33]. Robust, adaptive, homeostatic mechanisms coexist with sensitive switching mechanisms — negative controls with positive controls, inhibition with activation — with the strength of each of the opposing influences regulated by an array of external and internal cues via other pathways, permitting a contextual assessment of the appropriate balance and the appropriate response (Figure 2).
Figure 3
This all-important co-existence of opposing controls points to a crucial role for ‘signaling funambulisms’ in the regulation of cell signaling networks [33]. A funambulism is a type of ‘balancing act’ in which a signaling molecule is regulated both positively and negatively by either an upstream node (a feedforward funambulism) or a downstream node (a feedback funambulism), as illustrated in Figure 3.
Control-oriented modeling of cell signaling networks Control-oriented modeling of cell signaling networks represents a new era in the theoretical elucidation of cell signaling networks. While a modularized view of signaling Current Opinion in Chemical Biology 2006, 10:81–87
Funambulisms in cell signaling. Two common examples of funambulisms in cell signaling: the Ras-Raf feedforward funambulism, and the IRS-1-Akt feedback funambulism. Funambulists are shown as rectangles and base proteins are shown as triangles. Activation is represented by an arrow, whereas inhibition is represented with a barred line. www.sciencedirect.com
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networks has emerged in recent years [44–46], controloriented modeling refines this concept further to emphasize the use of ‘minimal modules’ [33] — the ‘less is more’ approach [36]. According to this paradigm of cell signaling, many network interactions should be condensed through a judicious choice of ‘effective’ kinetic parameters to reveal the minimum number of interactions required to fully characterize the module’s control capabilities. Exploiting the ‘simplicity within complexity’ paradox of cell signaling in this way, the overarching control principles may be considered in bold relief against the canvas of the associated complex network. Two recent examples of control-oriented models of cell signaling networks are noteworthy. Brandman et al. [47] considered the composite control motif of two interlinked positive feedback loops. While positive feedback is a ubiquitous control motif to convert graded inputs into discontinuous switches, these authors identify many biological systems that appear to rely on the interconnection of a multiplicity of these loops (see [47] and references therein). A mathematical model of this control motif revealed the fascinating finding that the combination of the two loops can make switching faster while reducing signal noise: a slow loop fosters robustness of the alternative steady states, whereas a fast loop enhances the speed of switching between the two states. Bornhold
[36], reviewing the model, notes that this control motif, with its rapid yet robust switching function, is reminiscent of the robustly designed electronic building blocks used to build modern computers. Araujo et al. [33], on the other hand, consider the feedback funambulism exemplified by the interactions of the scaffold protein insulin receptor substrate-1 (IRS-1) with its downstream effectors Akt and mTOR/p70S6K. This mathematical model emphasizes the potential of this particular control motif to contribute to tumor development, since an imbalance in the two opposing feedback loops in favor of the positive loop could create a one-way switch (Figure 4). This mechanism is thought to provide an explanation for the frequently observed constitutive activation of funambulist proteins such as IRS-1 in human tumors [48,49]. Most significantly, the development of a control-oriented model permits a critical assessment of the different classes of therapeutic targets within the network subset under consideration. Indeed, the model revealed that, ‘depending on the concentration of the treating agent and the location of the target within the cellular circuitry, a given treatment can be ineffective, or worse. . .it may actually accelerate the neoplastic process’ [33]. For this reason, control theoretic paradigms appear to be the key to developing effecting treatment strategies of the future.
Figure 4
The convergence of feedback loops: a precarious balance. Adapted from the results presented by Araujo et al. [33]. Feedback loops are thought to play a critical role in carcinogenesis, as suggested by several recent experimental [48,49] and theoretical [33] studies. If the positive and negative feedback loops, as regulated by the base protein, B, are appropriately balanced, then the funambulist, F, will be held in a state of equipoise (a). This permits a controlled, graded response to changes in stimulus. On the other hand, an imbalance in favor of the positive feedback loops could contribute to a malignant phenotype, since the emerging one-way switch could cause the funambulist to be constitutively active (b). Note from the stimulusresponse curve in (b) that once the switch is tripped at the critical stimulus-strength, S*, the system remains at the higher activation state even in the event of a complete withdrawal of the stimulus. www.sciencedirect.com
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Conclusions In the face of the sheer complexity of the information flow within cells and organisms, it is a wonder that living systems remain so robust and adaptable. Mathematical modeling is now providing a glimmer of insight into the sophisticated feedback control and network regulation operating within cells. The interplay between activation and inhibition (e.g. kinases versus phosphatases) is the universal theme of cell signaling, one that is mirrored at every level and scale of the organization of gene and protein networks. Feedback and feedforward loops are the chief mechanisms by which this interplay is realized on larger network scales, enabling the cell to exhibit complex and sophisticated modes of behavior. In this short review we have shown how even simple cellular feedback systems can generate complex non-linear behavior including one-way switches and toggle switches. These complex responses to an input may now begin to explain how a cell can exhibit an apparent paradox of robustness in the face of unwanted perturbances while responding specifically and sensitively to relevant inputs. Control theory will play a critical role in the next era of systems biology network modeling.
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