Turing Patterning Using Gene Circuits with Gas-Induced Degradation of Quorum Sensing Molecules

Abstract

The Turing instability was proposed more than six decades ago as a mechanism leading to spatial patterning, but it has yet to be exploited in a synthetic biology setting. Here we characterize the Turing instability in a specific gene circuit that can be implemented in vitro or in populations of clonal cells producing short-range activator N-Acyl homoserine lactone (AHL) and long-range inhibitor hydrogen peroxide (H2O2) gas. Slowing the production rate of the AHL-degrading enzyme, AiiA, generates stable fixed states, limit cycle oscillations and Turing patterns. Further tuning of signaling parameters determines local robustness and controls the range of unstable wavenumbers in the patterning regime. These findings provide a roadmap for optimizing spatial patterns of gene expression based on familiar quorum and gas sensitive E. coli promoters. The circuit design and predictions may be useful for (re)programming spatial dynamics in synthetic and natural gene expression systems.

Model reduction enables Turing instability analysis of large reaction–diffusion models

“…While this system is very similar to the four species model studied in [18], there are some minor differences, but we nevertheless still find that Turing instabilities arise. These differences arise because we arrived at equation (3.14) by reducing a full mass-action system equation (3.13), while the system in [18] is not obtained systematically from a more complex model.…”

“…In [18], it is shown that a model containing five variables (but analysis over four variables due to the presence of a conservation law) can give rise to DDI for certain parameter choices. Already, analysis of Turing instability is made challenging by virtue of there being more than two essential dependent variables.…”

“…Here, we show that the bifurcation properties of models of the synthetic gene circuit in [18] are preserved across models of varying complexity. To demonstrate this, we start by considering a model described by elementary chemical reactions, as follows:…”

“…We reduce the full system of equations to one of intermediate complexity consisting only of AHL, H 2 O 2 , AHL.luxR and aiiA (as considered in [18]), whose concentrations we write as L , H , P and A , respectively. The reduced ODEs are…”

Pattern Formation over Multigraphs

“…For instance, we demonstrated that considering different interconnection structures for different species in reaction-diffusion networks can be leveraged to decrease the often prohibitively large difference in diffusion coefficients required for pattern formation. This result can provide insight when studying self-organization in developmental biology [45] as well as guide the design of population-level behavior in synthetic biology [46]. Additionally, we illustrated not only that random networks with the same density can behave drastically differently in terms of patterning, but more importantly, that this difference can be understood by focusing on the dominant eigenvalue of the interconnection matrix.…”