Del Vecchio, Domitilla
http://hdl.handle.net/1721.1/85573
2018-11-17T00:33:23ZThe Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version)
http://hdl.handle.net/1721.1/118380
The Number of Equilibrium Points of Perturbed Nonlinear Positive Dynamical Systems (Extended Version)
The number of equilibrium points of a dynamical system dictates important qualitative properties such as the ability of the system to store different memory states, and may be significantly affected by state-dependent perturbations. In this paper, we develop a methodology based on tools from degree theory to determine whether the number of equilibrium points in a positive dynamical system changes due to structured state-dependent perturbations. Positive dynamical systems are particularly well suited to describe biological systems where the states are always positive. We prove two main theorems that utilize the determinant of the system's Jacobian to find algebraic conditions on the parameters determining whether the number of equilibrium points is guaranteed either to change or to remain the same when a nominal system is compared to its perturbed counterpart. We demonstrate the application of the theoretical results to genetic circuits where state-dependent perturbations arise due to fluctuations in cellular resources. These fluctuations constitute a major problem for predicting the behavior of genetic circuits. Our results allow us to determine whether such fluctuations change the genetic circuit's intended number of steady states.
2018-10-05T00:00:00ZGenetic Circuit-Host Ribosome Transactions: Diffusion-Reaction Model
http://hdl.handle.net/1721.1/118144
Genetic Circuit-Host Ribosome Transactions: Diffusion-Reaction Model
Barajas, Carlos; Del Vecchio, Domitilla
Deterministic models of bacterial genetic circuits commonly assume a well-mixed ensemble of species. This assumption results in ordinary differential equations (ODEs) describing the rate of change of the mean species concentration. It is however well known that species are non-homogenously distributed within a bacterial cell, where genes on the chromosome are found mostly at the center of the cell while synthetic genes residing on plasmids are often found at the poles. Most importantly, ribosomes, the key gene expression resource, are also arranged according to a non-homogenous profile. Therefore, when analyzing the effects of sharing gene expression resources, such as ribosomes, among synthetic genetic circuits and chromosomal genes, it may be important to consider the effects of spatial heterogeneity of the relevant species.
In this paper, we use a partial differential equation (PDE) model to
capture the spatial heterogeneity of species concentration. Solutions to the model are gathered numerically and approximations are
derived via perturbation analysis in the limit of fast diffusion.
The solutions are compared to those of the conventional ``well-mixed'' ODE model.
The fast-diffusion approximation predicts
higher protein production rates for all mRNAs in the cell and in some cases,
these rates are more sensitive to the activation of synthetic genes relative to
the well-mixed model. This trend is confirmed numerically using common biological parameters to simulate the full PDE system.
2018-09-18T00:00:00ZDeterministic-like model reduction for a class of multi-scale stochastic differential equations with application to biomolecular systems (Extended Version)
http://hdl.handle.net/1721.1/110896
Deterministic-like model reduction for a class of multi-scale stochastic differential equations with application to biomolecular systems (Extended Version)
Herath, Narmada; Del Vecchio, Domitilla
2017-08-01T00:00:00ZSignaling architectures that transmit unidirectional information
http://hdl.handle.net/1721.1/106135
Signaling architectures that transmit unidirectional information
Shah, Rushina; Del Vecchio, Domitilla
A signaling pathway transmits information from an upstream system to downstream systems, ideally unidirectionally. A key bottleneck to unidirectional transmission is retroactivity, which is the additional reaction flux that affects a system once its species interact with those of downstream systems. This raises the question of whether signaling pathways have developed specialized architectures that overcome retroactivity and transmit unidirectional signals. Here, we propose a general mathematical framework that provides an answer to this question. Using this framework, we analyze the ability of a variety of signaling architectures to transmit signals unidirectionally as key biological parameters are tuned. In particular, we find that single stage phosphorylation and phosphotransfer systems that transmit signals from a kinase show the following trade-off: either they impart a large retroactivity to their upstream system or they are significantly impacted by the retroactivity due to their downstream system. However, cascades of these architectures, which are highly represented in nature, can overcome this trade-off and thus enable unidirectional information transmission. By contrast, single and double phosphorylation cycles that transmit signals from a substrate impart a large retroactivity to their upstream system and are also unable to attenuate retroactivity due to their downstream system. Our findings identify signaling architectures that ensure unidirectional signal transmission and minimize crosstalk among multiple targets. Our results thus establish a way to decompose a signal transduction network into architectures that transmit information unidirectionally, while also providing a library of devices that can be used in synthetic biology to facilitate modular circuit design.
Submitted for review.
2016-12-24T00:00:00Z