Abstract
BackgroundAnalysis and design of complex systems benefit from mathematically tractable models, which are often derived by approximating a nonlinear system with an effective equivalent linear system. Biological oscillators with coupled positive and negative feedback loops, termed hysteresis or relaxation oscillators, are an important class of nonlinear systems and have been the subject of comprehensive computational studies. Analytical approximations have identified criteria for sustained oscillations, but have not linked the observed period and phase to compact formulas involving underlying molecular parameters.ResultsWe present, to our knowledge, the first analytical expressions for the period and amplitude of a classic model for the animal circadian clock oscillator. These compact expressions are in good agreement with numerical solutions of corresponding continuous ODEs and for stochastic simulations executed at literature parameter values. The formulas are shown to be useful by permitting quick comparisons relative to a negative-feedback represillator oscillator for noise (10× less sensitive to protein decay rates), efficiency (2× more efficient), and dynamic range (30 to 60 decibel increase). The dynamic range is enhanced at its lower end by a new concentration scale defined by the crossing point of the activator and repressor, rather than from a steady-state expression level.ConclusionAnalytical expressions for oscillator dynamics provide a physical understanding for the observations from numerical simulations and suggest additional properties not readily apparent or as yet unexplored. The methods described here may be applied to other nonlinear oscillator designs and biological circuits.
Highlights
Analysis and design of complex systems benefit from mathematically tractable models, which are often derived by approximating a nonlinear system with an effective equivalent linear system
Analytical expressions for dynamical systems are useful for mapping underlying parameters to observed properties
The Results show that the analytical results are accurate over a wide range of parameter space, compared with the numerical solutions to corresponding ODEs and a stochastic simulation at the original literature values [10]
Summary
Analysis and design of complex systems benefit from mathematically tractable models, which are often derived by approximating a nonlinear system with an effective equivalent linear system. Electrical, and atomic systems, analysis proceeds by reducing a complicated system to tractable linear system, more often than not involving coupled harmonic oscillators. The input β(t) can often be modeled as an on-off toggle, β(t) = 0 in the repressed state and β(t) = β for full activation. This behavior arises naturally from multimeric binding of transcription factors, giving a sigmoidal Hill function as a function of transcription factor concentration
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