Robustness in simple biochemical networks

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,