The Origins of Time-Delay in Template Biopolymerization Processes

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.Author Summary: Genetic networks display exceedingly complex and rich behavior which is modulated by multiple mechanisms, including many diverse types of interactions between DNA, mRNA and protein molecules. Mathematical models of gene networks must necessarily consider the essential mechanistic details of the processes involved in order to make reliable predictions. However, even though the description of the process becomes more accurate as more mechanistic details are incorporated into the mathematical model, the added mathematical complexity will make it difficult to parameterize and extract information from such models given the limited amount of experimental data. Protein synthesis is precisely one of the phases in the network machinery where certain mechanistic details are important and should thus be taken into account. Here, we develop a methodology to reduce a mathematical model for protein synthesis by performing approximations on a mechanistic model, retaining the essential details of the process. 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.