Information processing by biochemical networks: a dynamic approach

Abstract

Understanding how information is encoded and transferred by biochemical networks is of fundamental importance in cellular and systems biology. This requires analysis of the relationships between the stochastic trajectories of the constituent molecular (or submolecular) species that comprise the network. We describe how to identify conditional independences between the trajectories or time courses of groups of species. These are robust network properties that provide important insight into how information is processed. An entire network can then be decomposed exactly into modules on informational grounds. In the context of signalling networks with multiple inputs, the approach identifies the routes and species involved in sequential information processing between input and output modules. An algorithm is developed which allows automated identification of decompositions for large networks and visualization using a tree that encodes the conditional independences. Only stoichiometric information is used and neither simulations nor knowledge of rate parameters are required. A bespoke version of the algorithm for signalling networks identifies the routes of sequential encoding between inputs and outputs, visualized as paths in the tree. Application to the toll-like receptor signalling network reveals that inputs can be informative in ways unanticipated by steady-state analyses, that the information processing structure is not well described as a bow tie, and that encoding for the interferon response is unusually sparse compared with other outputs of this innate immune system.

Figure 1.

Illustrative example of a network with two modules. Each circle represents a module, i.e. a group of biochemical species. The network species are grouped into three non-overlapping sets A, B and D as shown, in such a way that A_t↮_Dt B_t —i.e. the trajectories (up to any time t) of A and B are independent given the trajectory of D. Hence A_t and B_t contain no mutual information given D_t, all information transfer between the two modules being conveyed via D_t.

Figure 2.

Kinetic independence graph of the NFκB signalling network. TNF is the tumour necrosis factor ligand, R is the ligand's receptor, IkBa is IκBα, and IkBe is IκBɛ; IKK has three forms (ne neutral, a active, and i inactive); O denotes a generic output protein; the prefix g denotes gene, t transcript, p phosphorylation and n nuclear location. For ease of viewing, input- and output-associated species are coloured red and blue, respectively. Complete details of the reaction network are given in the electronic supplementary material, SM2.

Figure 3.

Overview of the stages of the MIDIA algorithm. The algorithm automatically identifies a modularization of the biochemical network on dynamic, informational grounds (see text) and displays it as a junction tree in which the modules are the nodes of the tree. The undirected version of the KIG, G, is formed simply by replacing all arrows with (‘undirected’) lines. In forming G^T, enough edges are added (but no more) to make the graph triangulated. The cliques of G^T are its maximally complete subgraphs. For these and other concepts in basic graph theory see Cowell et al. [, ch. 4]. The details of the MIDIA algorithm are explained in the electronic supplementary material, SM3.

Figure 4.

Modularization 𝒯_M,I for the NFκB signalling network. The modularization is the one returned after stage 3 of the MIDIA algorithm (using a minimum residual size of 4 (except for the ‘root’ module)), and is based on instantaneous kinetics alone (see text). Each module M_d is labelled with the corresponding residual (M_d\S_d); each edge is labelled with the intersection of the two modules it connects.

Figure 5.

Modularization of the TLR signalling network showing I–O paths. Tree, 𝒯_M, returned by the MIDIA algorithm tailored to analyse the I–O paths of the network (see text). Modules containing external input processes are coloured red and those containing outputs blue. Edges on the path between any input and any output module are uniquely numbered (with the number of species in the edge also shown in parentheses)—the colours of edges match the ones used in figure 6. Species names are as in [28], table S1) where L and LIG both denote a receptor ligand, but with the following exceptions: NOD1I(e) is the external input process for the NOD1 receptor, and similarly for NOD2I(e); NADPH-OX-3P(v) corresponds to PHOX-GTP-3P(v) in ([28], table S1), and similarly for NADPH-OX-8P(v).

Figure 6.

Sequential information processing by the TLR network: the I–O path matrix. Each column corresponds to a species (named as in [28], table S1)) and input processes are labelled on the vertical axis. The upper, central and lower regions of the plot correspond to the outputs AP1-JUN, ISRE-IRF3 and NADPH-OX-3P, respectively. Each row in the matrix indicates the species composition of a given edge on an I–O path, with edges colour-coded (as in figure 5). Rows of the matrix are grouped according to the individual I–O combinations and ordered within a group from top to bottom according to the sequence of the edges in the corresponding I–O path in 𝒯_M in figure 5.