Response time re-scaling and Weber´s law in adapting biological systems

Systems biology has revealed numerous examples of networks whose dynamic behavior is robust to system perturbations and noise. In many cases, this behavior arises from simple yet fundamental features of the system architecture. A well-studied example is the chemotactic response of Escherichia coli. In various models of this system, it is shown that simple assumptions on the receptor methylation dynamics lead to robust perfect adaptation of chemotactic activity. Recent experimental work has also shown that the transient E. coli chemotactic response is unchanged by a scaling of its ligand input signal; this behavior is called fold change detection (FCD), and is in agreement with earlier mathematical predictions. However, this prediction was based on very particular assumptions on the structure of the chemotaxis pathway. In this work, we begin by showing that behavior similar to FCD can be obtained under weaker conditions on the system structure. Namely, we show that under relaxed conditions, a scaling of the chemotaxis system´s inputs leads to a time scaling of the output response. We propose that this may be a contributing factor to the robustness of the experimentally observed FCD. We further show that FCD is a special case of this time scaling behavior for which the time scaling factor of unity. We then proceed to extend the conditions for output time scaling to more general adapting systems, and demonstrate this time scaling behavior on a published model of the chemotaxis pathway of the bacterium Rhodobacter sphaeroides. This work therefore provides examples of how robust biological behavior can arise from simple yet realistic conditions on the underlying system structure.