Operon dynamics with state dependent transcription and/or translation delays

The dynamic-process characterization and prediction of synthetic gene circuits by dynamic delay model

Differential equation models are widely used to describe genetic regulations, predict multicomponent regulatory circuits, and provide quantitative insights. However, it is still challenging to quantitatively link the dynamic behaviors with measured parameters in synthetic circuits. Here, we propose a dynamic delay model (DDM) which includes two simple parts: the dynamic determining part and the doses-related steady-state-determining part. The dynamic determining part is usually supposed as the delay time but without a clear formula. For the first time, we give the detail formula of the dynamic determining function and provide a method for measuring all parameters of synthetic elements (include 8 activators and 5 repressors) by microfluidic system. Three synthetic circuits were built to show that the DDM can notably improve the prediction accuracy and can be used in various synthetic biology applications.

A Nonlinear Delay Model for Metabolic Oscillations in Yeast Cells

We introduce two time-delay models of metabolic oscillations in yeast cells. Our model tests a hypothesis that the oscillations occur as multiple pathways share a limited resource which we equate to the number of available ribosomes. We initially explore a single-protein model with a constraint equation governing the total resource available to the cell. The model is then extended to include three proteins that share a resource pool. Three approaches are considered at constant delay to numerically detect oscillations. First, we use a spectral element method to approximate the system as a discrete map and evaluate the stability of the linearized system about its equilibria by examining its eigenvalues. For the second method, we plot amplitudes of the simulation trajectories in 2D projections of the parameter space. We use a history function that is consistent with published experimental results to obtain metabolic oscillations. Finally, the spectral element method is used to convert the system to a boundary value problem whose solutions correspond to approximate periodic solutions of the system. Our results show that certain combinations of total resource available and the time delay, lead to oscillations. We observe that an oscillation region in the parameter space is between regions admitting steady states that correspond to zero and constant production. Similar behavior is found with the three-protein model where all proteins require the same production time. However, a shift in the protein production rates peaks occurs for low available resource suggesting that our model captures the shared resource pool dynamics.

Complex dynamics in an unexplored simple model of the peroxidase-oxidase reaction

A previously overlooked version of the so-called Olsen model of the peroxidase-oxidase reaction has been studied numerically using 2D isospike stability and maximum Lyapunov exponent diagrams and reveals a rich variety of dynamic behaviors not observed before. The model has a complex bifurcation structure involving mixed-mode and bursting oscillations as well as quasiperiodic and chaotic dynamics. In addition, multiple periodic and non-periodic attractors coexist for the same parameters. For some parameter values, the model also reveals formation of mosaic patterns of complex dynamic states. The complex dynamic behaviors exhibited by this model are compared to those of another version of the same model, which has been studied in more detail. The two models show similarities, but also notable differences between them, e.g., the organization of mixed-mode oscillations in parameter space and the relative abundance of quasiperiodic and chaotic oscillations. In both models, domains with chaotic dynamics contain apparently disorganized subdomains of periodic attractors with dinoflagellate-like structures, while the domains with mainly quasiperiodic behavior contain subdomains with periodic attractors organized as regular filamentous structures. These periodic attractors seem to be organized according to Stern-Brocot arithmetics. Finally, it appears that toroidal (quasiperiodic) attractors develop into first wrinkled and then fractal tori before they break down to chaotic attractors.