Systems-Theoretic Approaches to Design Biological Networks with Desired Functionalities.
TL;DR: This chapter introduces a generic approach inspired by systems theory to deduce the network structures for a particular biological functionality, and adopts the functionality of adaptation, and demonstrates how network topologies that can achieve adaptation can be identified using such a systems-theoretic approach.
Abstract: The deduction of design principles for complex biological functionalities has been a source of constant interest in the fields of systems and synthetic biology. A number of approaches have been adopted, to identify the space of network structures or topologies that can demonstrate a specific desired functionality, ranging from brute force to systems theory-based methodologies. The former approach involves performing a search among all possible combinations of network structures, as well as the parameters underlying the rate kinetics for a given form of network. In contrast to the search-oriented approach in brute force studies, the present chapter introduces a generic approach inspired by systems theory to deduce the network structures for a particular biological functionality. As a first step, depending on the functionality and the type of network in consideration, a measure of goodness of attainment is deduced by defining performance parameters. These parameters are computed for the most ideal case to obtain the necessary condition for the given functionality. The necessary conditions are then mapped as specific requirements on the parameters of the dynamical system underlying the network. Following this, admissible minimal structures are deduced. The proposed methodology does not assume any particular rate kinetics in this case for deducing the admissible network structures notwithstanding a minimum set of assumptions on the rate kinetics. The problem of computing the ideal set of parameter/s or rate constants, unlike the problem of topology identification, depends on the particular rate kinetics assumed for the given network. In this case, instead of a computationally exhaustive brute force search of the parameter space, a topology-functionality specific optimization problem can be solved. The objective function along with the feasible region bounded by the motif specific constraints amounts to solving a non-convex optimization program leading to non-unique parameter sets. To exemplify our approach, we adopt the functionality of adaptation, and demonstrate how network topologies that can achieve adaptation can be identified using such a systems-theoretic approach. The outcomes, in this case, i.e., minimum network structures for adaptation, are in agreement with the brute force results and other studies in literature.
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TL;DR: This paper translates the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, and claims that the necessary structural conditions derived are the strictest among the ones hitherto existed in the literature.
Abstract: Constructing biological networks capable of performing specific biological functionalities has been of sustained interest in synthetic biology. Adaptation is one such ubiquitous functional property, which enables every living organism to sense a change in its surroundings and return to its operating condition prior to the disturbance. In this paper, we present a generic systems theory-driven method for designing adaptive protein networks. First, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as ‘design requirements’ for the underlying networks. We go on to prove that a protein network with different input–output nodes (proteins) needs to be at least of third-order in order to provide adaptation. Next, we show that the necessary design principles obtained for a three-node network in adaptation consist of negative feedback or a feed-forward realization. We argue that presence of a particular class of negative feedback or feed-forward realization is necessary for a network of any size to provide adaptation. Further, we claim that the necessary structural conditions derived in this work are the strictest among the ones hitherto existed in the literature. Finally, we prove that the capability of producing adaptation is retained for the admissible motifs even when the output node is connected with a downstream system in a feedback fashion. This result explains how complex biological networks achieve robustness while keeping the core motifs unchanged in the context of a particular functionality. We corroborate our theoretical results with detailed and thorough numerical simulations. Overall, our results present a generic, systematic and robust framework for designing various kinds of biological networks.
12 citations
TL;DR: The present review provides a qualitative and quantitative study of all three approaches in the light of three well-researched biological phenotypes, namely, oscillation, toggle switching, and adaptation.
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TL;DR: This work presents a generic, systematic, and robust framework for advancing the understanding of complex biological networks and proposes a sufficient condition for imperfect adaptation and shows that adding negative feedback in an IFFLP topology improves the robustness concerning parametric fluctuations.
Abstract: Biological adaptation, the tendency of every living organism to regulate its essential activities in environmental fluctuations, is a well-studied functionality in systems and synthetic biology. In this work, we present a generic methodology inspired by systems theory to discover the design principles for robust adaptation, perfect and imperfect, in two different contexts: (1) in the presence of deterministic external disturbance and (2) in a stochastic setting. In all the cases, firstly, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as design requirements for the underlying networks. Thus, contrary to the existing approaches, the proposed methodologies provide an exhaustive set of admissible network structures without resorting to computationally burdensome brute-force techniques. Further, the proposed frameworks do not assume prior knowledge about the particular rate kinetics, thereby validating the conclusions for a large class of biological networks. In the deterministic setting, we show that unlike the incoherent feed-forward network structures (IFFLP), the modules containing negative feedback with buffer action (NFBLB) are robust to parametric fluctuations when a specific part of the network is assumed to remain unaffected. To this end, we propose a sufficient condition for imperfect adaptation and show that adding negative feedback in an IFFLP topology improves the robustness concerning parametric fluctuations. Further, we propose a stricter set of necessary conditions for imperfect adaptation. Turning to the stochastic scenario, we adopt a Wiener-Kolmogorov filter strategy to tune the parameters of a given network structure towards minimum output variance. We show that both NFBLB and IFFLP can be used as a reduced order W-K filter. Further, we define the notion of nearest neighboring motifs to compare the output variances across different network structures. We argue that the NFBLB achieves adaptation at the cost of a variance higher than its nearest neighboring motifs whereas the IFFLP topology produces locally minimum variance while compared with its nearest neighboring motifs. We present numerical simulations to support the theoretical results. Overall, our results present a generic, systematic, and robust framework for advancing the understanding of complex biological networks.
1 citations
TL;DR: In this article , the authors provide a classification of all possible network topologies capable of generating infinitesimal homeostasis in arbitrarily large and complex multiple-input parameter networks.
Abstract: Homeostasis, also known as adaptation, refers to the ability of a system to counteract persistent external disturbances and tightly control the output of a key observable. Existing studies on homeostasis in network dynamics have mainly focused on ‘perfect adaptation’ in deterministic single-input single-output networks where the disturbances are scalar and affect the network dynamics via a pre-specified input node. In this paper we provide a full classification of all possible network topologies capable of generating infinitesimal homeostasis in arbitrarily large and complex multiple-input parameter networks. Working in the framework of ‘infinitesimal homeostasis’ allows us to make no assumption about how the components are interconnected and the functional form of the associated differential equations, apart from being compatible with the network architecture. Remarkably, we show that there are just three distinct ‘mechanisms’ that generate infinitesimal homeostasis. Each of these three mechanisms generates a rich class of well-defined network topologies – called homeostasis subnetworks. Most importantly, we show that these classes of homeostasis subnetworks provides a topological basis for the classification of ‘homeostasis types’: the full set of all possible multiple-input parameter networks can be uniquely decomposed into these special homeostasis subnetworks. We build on previous work that treated the cases of single-input node and multiple-input node, both with a single scalar input parameter. Furthermore, we identify a new phenomenon that occurs in the multiparameter setting, that we call homeostasis mode interaction, in analogy with the well-known characteristic of multiparameter bifurcation theory.
1 citations
TL;DR: In this paper, a generic system theory-driven method for designing adaptive protein networks is presented, which translates the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which they then map back as 9design requirements9 for the underlying networks.
Abstract: Constructing biological networks capable of performing specific biological functionalities has been of sustained interest in synthetic biology. Adaptation is one such ubiquitous functional property, which enables every living organism to sense a change in its surroundings and return to its operating condition prior to the disturbance. In this paper, we present a generic systems theory-driven method for designing adaptive protein networks. First, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as 9design requirements9 for the underlying networks. We go on to prove that a protein network with different input--output nodes (proteins) needs to be at least of third-order in order to provide adaptation. Next, we show that the necessary design principles obtained for a three-node network in adaptation consist of negative feedback or a feed-forward realization. Interestingly, the design principles obtained by the proposed method remain the same for a network of arbitrary size and connectivity. Finally, we prove that the motifs discovered for adaptation are non-retroactive for a canonical downstream connection. This result explains how complex biological networks achieve robustness while keeping the core motifs unchanged in the context of a particular functionality. We corroborate our theoretical results with detailed and thorough numerical simulations. Overall, our results present a generic, systematic and robust framework for designing various kinds of biological networks.
1 citations
References
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TL;DR: Network motifs, patterns of interconnections occurring in complex networks at numbers that are significantly higher than those in randomized networks, are defined and may define universal classes of networks.
Abstract: Complex networks are studied across many fields of science. To uncover their structural design principles, we defined “network motifs,” patterns of interconnections occurring in complex networks at numbers that are significantly higher than those in randomized networks. We found such motifs in networks from biochemistry, neurobiology, ecology, and engineering. The motifs shared by ecological food webs were distinct from the motifs shared by the genetic networks of Escherichia coli and Saccharomyces cerevisiae or from those found in the World Wide Web. Similar motifs were found in networks that perform information processing, even though they describe elements as different as biomolecules within a cell and synaptic connections between neurons in Caenorhabditis elegans. Motifs may thus define universal classes of networks. This
6,992 citations
TL;DR: In this article, the authors developed analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system's entire dynamics.
Abstract: The ultimate proof of our understanding of natural or technological systems is reflected in our ability to control them. Although control theory offers mathematical tools for steering engineered and natural systems towards a desired state, a framework to control complex self-organized systems is lacking. Here we develop analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system's entire dynamics. We apply these tools to several real networks, finding that the number of driver nodes is determined mainly by the network's degree distribution. We show that sparse inhomogeneous networks, which emerge in many real complex systems, are the most difficult to control, but that dense and homogeneous networks can be controlled using a few driver nodes. Counterintuitively, we find that in both model and real systems the driver nodes tend to avoid the high-degree nodes.
2,889 citations
TL;DR: It is argued that the key properties of biochemical networks should be robust in order to ensure their proper functioning, and it is shown that this applies in particular to bacterial chemotaxis.
Abstract: 1 are responsible for many important cellular processes, including cell-cycle regulation and signal transduction. Here we address the issue of the sensitivity of the networks to variations in their biochemical parameters. We propose a mechanism for robust adaptation in simple signal transduction networks. We show that this mechanism applies in particular to bacterial chemotaxis 2-7 . This is demonstrated within a quantitative model which explains, in a unified way, many aspects of chemotaxis, including proper responses to chemical gradients 8-12 . The adaptation property 10,13-16 is a consequence of the network's connectivity and does not require the 'fine-tuning' of parameters. We argue that the key properties of biochemical networks should be robust in order to ensure their proper functioning. Cellular biochemical networks are highly interconnected: a per- turbation in reaction rates or molecular concentrations may affect numerous cellular processes. The complexity of biochemical net- works raises the question of the stability of their functioning. One possibility is that to achieve an appropriate function, the reaction rate constants and the enzymatic concentrations of a network need to be chosen in a very precise manner, and any deviation from the 'fine-tuned' values will ruin the network's performance. Another possibility is that the key properties of biochemical networks are robust; that is, they are relatively insensitive to the precise values of biochemical parameters. Here we explore the issue of robustness of one of the simplest and best-known signal transduction networks: a biochemical network responsible for bacterial chemotaxis. Bacteria such as Escherichia coli are able to sense (temporal) gradients of chemical ligands in their vicinity 2 . The movement of a swimming bacterium is composed of a series of 'smooth runs', interrupted by events of 'tumbling', in which a new direction for the next run is chosen randomly. By modifying the tumbling frequency, a bac- terium is able to direct its motion either towards attractants or away from repellents. A well established feature of chemoxis is its property of adaptation 10,13-16 : the steady-state tumbling frequency in a homogeneous ligand environment is insensitive to the value of ligand concentration. This property allows bacteria to maintain their sensitivity to chemical gradients over a wide range of attractant or repellent concentrations. The different proteins that are involved in chemotactic response have been characterized in great detail, and much is known about the interactions between them (Fig. 1a). In particular, the receptors that sense chemotactic ligands are reversibly methylated. Biochem- ical data indicate that methylation is responsible for the adaptation property: changes in methylation of the receptor can compensate for the effect of ligand on tumbling frequency. Theoretical models proposed in the past assumed that the biochemical parameters are fine-tuned to preserve the same steady-state behaviour at different ligand concentrations 17,18 . We present an alternative picture in which adaptation is a robust property of the chemotaxis network and does not rely on the fine-tuning of parameters. We have analysed a simple two-state model of the chemotaxis network closely related to the one proposed previously 2,19 . The two- state model assumes that the receptor complex has two functional states: active and inactive. The active receptor complex shows a kinase activity: it phosphorylates the response regulator molecules,
1,567 citations
TL;DR: It is shown that for a class of feedback systems of arbitrary order the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked, and that this open-loop, feedback-blocked system is guaranteed to be bistable for some range of feedback strengths.
Abstract: It is becoming increasingly clear that bistability (or, more generally, multistability) is an important recurring theme in cell signaling. Bistability may be of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or “remember” transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex. Here, we show that for a class of feedback systems of arbitrary order the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this open-loop, feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems and illustrate the method with two examples taken from recent experimental studies of bistable systems: a two-variable Cdc2/Wee1 system and a more complicated five-variable mitogen-activated protein kinase cascade.
976 citations
TL;DR: In this article, the authors search all possible three-node enzyme network topologies to identify those that could perform adaptation, and two major core topologies emerge as robust solutions: a negative feedback loop with a buffering node and an incoherent feed forward loop with proportioner node.
826 citations