Abstract
In dynamical systems theory, a system which can be described by differential equations is called a continuous dynamical system. In studies on genetic oscillation, most deterministic models at early stage are usually built on ordinary differential equations (ODE). Therefore, gene transcription which is a vital part in genetic oscillation is presupposed to be a continuous dynamical system by default. However, recent studies argued that discontinuous transcription might be more common than continuous transcription. In this paper, by appending the inserted silent interval lying between two neighboring transcriptional events to the end of the preceding event, we established that the running time for an intact transcriptional event increases and gene transcription thus shows slow dynamics. By globally replacing the original time increment for each state increment by a larger one, we introduced fractional differential equations (FDE) to describe such globally slow transcription. The impact of fractionization on genetic oscillation was then studied in two early stage models – the Goodwin oscillator and the Rössler oscillator. By constructing a “dual memory” oscillator – the fractional delay Goodwin oscillator, we suggested that four general requirements for generating genetic oscillation should be revised to be negative feedback, sufficient nonlinearity, sufficient memory and proper balancing of timescale. The numerical study of the fractional Rössler oscillator implied that the globally slow transcription tends to lower the chance of a coupled or more complex nonlinear genetic oscillatory system behaving chaotically.
Highlights
The design and construction of genetic circuits are of great importance to the nascent field of synthetic biology [1]
Considering that in real world applications, the evolution of a general dynamical system governed by the principle of causality is apriori timeirreversible, we use the left-side integral/derivative operators and the initial time t0~0 throughout this paper
By inserting silent intervals into consecutive transcriptional events and globally replacing dt by a larger incrementl, we make the early stage models change from ordinary differential equations (ODE) to fractional differential equations (FDE)
Summary
The design and construction of genetic circuits are of great importance to the nascent field of synthetic biology [1]. One may expect to describe a basic genetic circuit by using a minimal dynamical model (with as few equations as possible), for the purpose of simplicity. The variables in such model represent the quantities of several key products in the circuit. A minimal model (sometimes with only a single equation) often lacks power to cover such complex intermediate processes in a regular timescale This can be seen from the case that a Goodwin oscillator which requires an unrealistic high Hill coefficient (larger than 8) for the destabilization of a fixed point and generating limit cycle oscillations [5,6]
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