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In this paper, we aim to develop a new methodology to model and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using multiple timescale networks (MTN). Fast reactions constitute a positive feedback-loop network (PFN), while slow reactions consist of a cyclic feedback-loop network (CFN), in MTN. Multiple timescales are exploited to simplify models according to singular perturbation theory. We show that a MTN has no stable equilibrium but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of MTNs with only one PFN, and then generalise the result to MTNs with multiple PFNs. Finally, we design a biologically plausible gene regulatory network by the cI and Lac genes, to demonstrate the theoretical results. Since there is less restriction on the network structure of a MTN, it can be expected to apply to a wide variety of areas on the modelling, analysing and designing of biological systems.

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We discuss natural limitations on motor performance caused by the time delay required for feedback signals to propagate within the human body or mechanical control systems. By considering a very simple delayed linear servomechanism model, we show there exists a best possible speed-accuracy trade-off similar to Fitts' law that cannot be exceeded when delay is present. This is strictly a delay effect and does not occur for the ideal case of instantaneous feedback. We then examine the performance of the vector integration to endpoint (VITE) circuit as a model of human movement and show that when this circuit is generalized to include delayed feedback the performance may not exceed that of the servomechanism with an equal delay. We suggest the existence of such a limitation may be a ubiquitous consequence of delay in motor control with the implication that the index of performance in Fitts' law cannot arbitrarily large.

In this paper, we develop a new methodology to analyze and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using negative cyclic feedback systems. We show that negative cyclic feedback networks have no stable equilibria but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of the biological networks composed of cyclic feedback loops, and then extend our results to general cyclic feedback network with less restriction, thereby making our theoretical analysis and design of oscillators easy to implement, even for large-scale systems. Finally, we use one circadian network formed by a period protein (PER) and per mRNA, and one biologically plausible synthetic gene network, to demonstrate the theoretical results. Since there is less restriction on the network structure, the results of this paper can be expected to apply to a wide variety of areas on modelling, analyzing and designing of biological systems.

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