Abstract
Time-delays are common in many physical and biological systems and they give rise to complex dynamic phenomena. The elementary processes involved in template biopolymerization, such as mRNA and protein synthesis, introduce significant time delays. However, there is not currently a systematic mapping between the individual mechanistic parameters and the time delays in these networks. We present here the development of mathematical, time-delay models for protein translation, based on PDE models, which in turn are derived through systematic approximations of first-principles mechanistic models. Theoretical analysis suggests that the key features that determine the time-delays and the agreement between the time-delay and the mechanistic models are ribosome density and distribution, i.e., the number of ribosomes on the mRNA chain relative to their maximum and their distribution along the mRNA chain. Based on analytical considerations and on computational studies, we show that the steady-state and dynamic responses of the time-delay models are in excellent agreement with the detailed mechanistic models, under physiological conditions that correspond to uniform ribosome distribution and for ribosome density up to 70%. The methodology presented here can be used for the development of reduced time-delay models of mRNA synthesis and large genetic networks. The good agreement between the time-delay and the mechanistic models will allow us to use the reduced model and advanced computational methods from nonlinear dynamics in order to perform studies that are not practical using the large-scale mechanistic models.
Highlights
Time-Delays in Mathematical Biology Time-delay models are common in mathematical biology, as is demonstrated by the use of mathematical models incorporating time-delays in a wide range of applications
We focus on the mathematical modeling of protein synthesis, which is central to cellular processes and gene networks
The framework is based on systematically approximating a mechanistic mathematical model of ordinary differential equations (ODEs), first derived in [21,22,23], by a continuum model in the form of a partial differential equation (PDE) model and showing rigorously that this PDE model is completely equivalent to a time delay model
Summary
Time-Delays in Mathematical Biology Time-delay models are common in mathematical biology, as is demonstrated by the use of mathematical models incorporating time-delays in a wide range of applications. Incorporating time delays in models of gene networks is often essential to capture the whole range of dynamic behavior. A single self-repressed gene has been observed in experiments to display oscillatory behavior which cannot be captured by models that ignore the time delay required to obtain a finished protein from the expressed gene. This oscillatory behavior is reproduced by a mathematical model in terms of time delayed differential equations [7,8,9]. Our methodology opens up the possibility of utilizing powerful mathematical tools, such as bifurcation analysis, for understanding the complex dynamics displayed by genetic networks and design strategies for metabolic engineering and synthetic biology
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