No general theory of functional rigidity / flexibility in biological netowrks.

 The network approach became a widely used tool to understand the behaviour of complex systems in the last decade. We start from a short description of structural rigidity theory. A detailed account on the combinatorial rigidity analysis of protein structures, as well as local flexibility measures of proteins and their applications in explaining allostery and thermostability is given. We also briefly discuss the network aspects of cytoskeletal tensegrity. Finally, we show the importance of the balance between functional flexibility and rigidity in protein-protein interaction, metabolic, gene regulatory and neuronal networks. Our summary raises the possibility that the concepts of flexibility and rigidity can be generalized to all networks.

http://arxiv.org/abs/1204.6389

Various types of networks (represented by weighted, directed, coloured graphs) became widely applied to describe complex systems in the last decade. Modelling a complex system as a network is appropriate and practical, when it is built up from similar, but distinct objects (represented by nodes/vertices of the graph) and decisively pair-wise interactions (represented by edges/links). However, especially in biological systems, where data coverage often has technical difficulties, not all of the possible interactions are strong enough to take them into account. The complex functions are usually related to network structure. Most self-organized networks (and biological networks in particular) are small worlds with scale-free degree distribution [1, 2]. It is important to note, that for the functional analysis of the complex system, network topology is not enough, but network dynamics has to be also included. In dynamic network models quantities assigned to nodes and/or edges may vary, and/or the background topology itself may also change, where the latter phenomenon is commonly referred as “network evolution”. The fact that biological sciences use more and more network models is not accidental. Biological entities can endure evolutionary selection, if they can cooperatively interact [3]. Therefore, networks form a hierarchy from molecular level to societies [4]. In many of these networks, nodes represent individual physical units, such as amino acids of protein structural networks [5–7]. Other networks are abstract conceptualizations, such as the gene regulatory network [8, 9], where nodes represent DNA segments, and their connections are often indirect representing the interactions between the transcriptional processes of the corresponding genes. Definition of the most important biological networks can be found in several papers and books [4, 10, 11]. However, it is important to note that network definition may represent a rather difficult problem itself. The definition of the entities representing nodes, their connections representing edges and especially the weight of the interaction may not be straightforward. In most cases, edge weights refer to the strength or probability of the interaction [12–14]. However, in numerous studies edge weights are neglected. Rigidity and flexibility Rigidity and flexibility have a lot of meanings in everyday life and in science as well. We can talk about rigidity of materials, frameworks or behaviours. The properties: stability, stiffness, robustness are often used together with the concept of rigidity. In this paper, rigidity and flexibility are used as properties, which characterize the possible states that can be reached by the system under certain type of external influences. This concept implies that flexibility also measures the internal degree of freedom and its distribution in the system. Corresponding to the above definition, two kinds of rigidity/flexibility concepts are used in the literature: 1.) structural and 2.) functional rigidity/flexibility. We talk about the structural rigidity of networks, when the network represents the geometrical structure of a complex system. This means that the network is a framework, which is embedded in a metric space. In simplest cases, this can be studied by the so-called mathematical theory of structural rigidity, which was born with the early paper of Maxwell written on constraint counting in 1864 [15]. This theory is introduced in the next section, followed by its applications studying protein 2 flexibility. Closing the sections on structural rigidity, properties of cytoskeletal network will be reviewed in conjunction with the biotensegrity concept [16], since it plays an important role in the structure and dynamics of the cell. Contrary to structural rigidity/flexibility, in the case of functional rigidity/flexibility, the network has to be dynamic (or has to undergo evolution), but the network need not to be embedded into metric space. Since both network dynamics and the external influence provoking this dynamics depend on the actual problem studied, there is no general theory of functional rigidity/flexibility. 

Key Points

An appropriate balance between rigidity and flexibility is crucial for proper functioning of complex biological systems.
Structural rigidity and functional rigidity lay often rather close to each other in our
everyday thinking but were approached by different methodologies so far.
Combinatorial graph theory proved to be a useful method in the characterization of
structural rigidity/flexibility. However, the use of information and matroid theories may
provide novel generalizations of these key properties.
Dimension-independent generalizations of structural rigidity on weighted networks will be
essential for future progress.
Using the concept that flexibility describes the internal degrees of freedom, structural
rigidity/flexibility and functional rigidity/flexibility may be better connected in the future.
However, currently this poses a large challenge for us in the field.