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Potential unsatisfiability of cyclic constraints on stochastic biological networks biases selection towards hierarchical architectures.

Smith C, Pechuan X, Puzio RS, Biro D, Bergman A - J R Soc Interface (2015)

Bottom Line: Constraints placed upon the phenotypes of organisms result from their interactions with the environment.We show that such network architectures possessing cycles in their topology, in contrast to those that do not, may be subjected to unsatisfiable constraints.Our results identify a constraint that, at least in isolation, would contribute to a bias in the evolutionary process towards more hierarchical -modular versus completely connected network architectures.

View Article: PubMed Central - PubMed

Affiliation: Department of Systems and Computational Biology, Albert Einstein College of Medicine, 1301 Morris Park Avenue, Bronx, NY 10461, USA.

ABSTRACT
Constraints placed upon the phenotypes of organisms result from their interactions with the environment. Over evolutionary time scales, these constraints feed back onto smaller molecular subnetworks comprising the organism. The evolution of biological networks is studied by considering a network of a few nodes embedded in a larger context. Taking into account this fact that any network under study is actually embedded in a larger context, we define network architecture, not on the basis of physical interactions alone, but rather as a specification of the manner in which constraints are placed upon the states of its nodes. We show that such network architectures possessing cycles in their topology, in contrast to those that do not, may be subjected to unsatisfiable constraints. This may be a significant factor leading to selection biased against those network architectures where such inconsistent constraints are more likely to arise. We proceed to quantify the likelihood of inconsistency arising as a function of network architecture finding that, in the absence of sampling bias over the space of possible constraints and for a given network size, networks with a larger number of cycles are more likely to have unsatisfiable constraints placed upon them. Our results identify a constraint that, at least in isolation, would contribute to a bias in the evolutionary process towards more hierarchical -modular versus completely connected network architectures. Together, these results highlight the context dependence of the functionality of biological networks.

No MeSH data available.


Related in: MedlinePlus

Abstract influence (AI) representation of biological networks. (a) The SBGN is capable of representing arbitrary biological networks including processes that involve metabolites, signalling molecules, genes and enzymes [11]. Only a fragment of the SBGN language, where all nodes have equivalent types, is indicated here. (b) We abstract from the SBGN representation of a biological network to a graph representing the AI graph indicating coupling among a subset of the entities present in a biological network. (c) For economy of representation, we use a short hand (SH) hypergraph to denote the AI graph. The topology of the AI and SH graphs are equivalent and this is what we refer to as network architecture.
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RSIF20150179F1: Abstract influence (AI) representation of biological networks. (a) The SBGN is capable of representing arbitrary biological networks including processes that involve metabolites, signalling molecules, genes and enzymes [11]. Only a fragment of the SBGN language, where all nodes have equivalent types, is indicated here. (b) We abstract from the SBGN representation of a biological network to a graph representing the AI graph indicating coupling among a subset of the entities present in a biological network. (c) For economy of representation, we use a short hand (SH) hypergraph to denote the AI graph. The topology of the AI and SH graphs are equivalent and this is what we refer to as network architecture.

Mentions: Most studies of biological networks focus on one type of variable in isolation. For example, many studies focus on one of metabolic networks, protein–protein interaction networks, signalling networks, gene-regulatory networks, or population and community dynamics in the context of ecological networks. A true biological network involves all of these acting together to produce biological phenomena at all scales. Models that integrate information about biological networks, rather than focusing exclusively on particular types of molecules, will likely become more common in the near future [8–10]. The systems biology graphical notation (SBGN) supports the ability to express many of these networks within the context of a single formalism ([11], figure 1). Even when the different types of biological variables are combined into a single network, it is impossible to study all variables simultaneously. As a result, it is always the case that a subnetwork is selected for investigation and the remainder of the network is treated as an environment or context. In figure 1, we show the SBGN process form of six simple examples of biological networks. In each case, we have selected a subset of variables that form a subnetwork as an example of how one might proceed in the investigation of a particular biological system. Once such a subnetwork is chosen, it is possible to abstract away the variables that are not part of the subnetwork. This is represented by the abstract influence (AI) network for each simple example on figure 1b. The transformation from SBGN to the AI network is given simply by collapsing the disconnected components of the ancestors of each node in the focal subnetwork into single AI nodes. This results in a bipartite graph that captures the dependencies among the environmental factors as experienced by the subnetwork and nothing more.Figure 1.


Potential unsatisfiability of cyclic constraints on stochastic biological networks biases selection towards hierarchical architectures.

Smith C, Pechuan X, Puzio RS, Biro D, Bergman A - J R Soc Interface (2015)

Abstract influence (AI) representation of biological networks. (a) The SBGN is capable of representing arbitrary biological networks including processes that involve metabolites, signalling molecules, genes and enzymes [11]. Only a fragment of the SBGN language, where all nodes have equivalent types, is indicated here. (b) We abstract from the SBGN representation of a biological network to a graph representing the AI graph indicating coupling among a subset of the entities present in a biological network. (c) For economy of representation, we use a short hand (SH) hypergraph to denote the AI graph. The topology of the AI and SH graphs are equivalent and this is what we refer to as network architecture.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4528583&req=5

RSIF20150179F1: Abstract influence (AI) representation of biological networks. (a) The SBGN is capable of representing arbitrary biological networks including processes that involve metabolites, signalling molecules, genes and enzymes [11]. Only a fragment of the SBGN language, where all nodes have equivalent types, is indicated here. (b) We abstract from the SBGN representation of a biological network to a graph representing the AI graph indicating coupling among a subset of the entities present in a biological network. (c) For economy of representation, we use a short hand (SH) hypergraph to denote the AI graph. The topology of the AI and SH graphs are equivalent and this is what we refer to as network architecture.
Mentions: Most studies of biological networks focus on one type of variable in isolation. For example, many studies focus on one of metabolic networks, protein–protein interaction networks, signalling networks, gene-regulatory networks, or population and community dynamics in the context of ecological networks. A true biological network involves all of these acting together to produce biological phenomena at all scales. Models that integrate information about biological networks, rather than focusing exclusively on particular types of molecules, will likely become more common in the near future [8–10]. The systems biology graphical notation (SBGN) supports the ability to express many of these networks within the context of a single formalism ([11], figure 1). Even when the different types of biological variables are combined into a single network, it is impossible to study all variables simultaneously. As a result, it is always the case that a subnetwork is selected for investigation and the remainder of the network is treated as an environment or context. In figure 1, we show the SBGN process form of six simple examples of biological networks. In each case, we have selected a subset of variables that form a subnetwork as an example of how one might proceed in the investigation of a particular biological system. Once such a subnetwork is chosen, it is possible to abstract away the variables that are not part of the subnetwork. This is represented by the abstract influence (AI) network for each simple example on figure 1b. The transformation from SBGN to the AI network is given simply by collapsing the disconnected components of the ancestors of each node in the focal subnetwork into single AI nodes. This results in a bipartite graph that captures the dependencies among the environmental factors as experienced by the subnetwork and nothing more.Figure 1.

Bottom Line: Constraints placed upon the phenotypes of organisms result from their interactions with the environment.We show that such network architectures possessing cycles in their topology, in contrast to those that do not, may be subjected to unsatisfiable constraints.Our results identify a constraint that, at least in isolation, would contribute to a bias in the evolutionary process towards more hierarchical -modular versus completely connected network architectures.

View Article: PubMed Central - PubMed

Affiliation: Department of Systems and Computational Biology, Albert Einstein College of Medicine, 1301 Morris Park Avenue, Bronx, NY 10461, USA.

ABSTRACT
Constraints placed upon the phenotypes of organisms result from their interactions with the environment. Over evolutionary time scales, these constraints feed back onto smaller molecular subnetworks comprising the organism. The evolution of biological networks is studied by considering a network of a few nodes embedded in a larger context. Taking into account this fact that any network under study is actually embedded in a larger context, we define network architecture, not on the basis of physical interactions alone, but rather as a specification of the manner in which constraints are placed upon the states of its nodes. We show that such network architectures possessing cycles in their topology, in contrast to those that do not, may be subjected to unsatisfiable constraints. This may be a significant factor leading to selection biased against those network architectures where such inconsistent constraints are more likely to arise. We proceed to quantify the likelihood of inconsistency arising as a function of network architecture finding that, in the absence of sampling bias over the space of possible constraints and for a given network size, networks with a larger number of cycles are more likely to have unsatisfiable constraints placed upon them. Our results identify a constraint that, at least in isolation, would contribute to a bias in the evolutionary process towards more hierarchical -modular versus completely connected network architectures. Together, these results highlight the context dependence of the functionality of biological networks.

No MeSH data available.


Related in: MedlinePlus