Nonlinear Dirac equations describe the motion of relativistic spin-1 2 particles in presence of external electromagnetic fields, modelled by an electric and magnetic potential, and taking into account a nonlinear particle self-interaction. In recent years, the construction of numerical splitting schemes for the solution of these systems in the nonrelativistic limit regime, i.e., the speed of light c formally tending to infinity, has gained a lot of attention. In this paper, we consider a nonlinear Dirac equation with Thirring type interaction, where in contrast to the case of the Soler type nonlinearity a classical twoterm splitting scheme cannot be straightforwardly applied. Thus, we propose and analyze a three-term Strang splitting scheme which relies on splitting the full problem into the free Dirac subproblem, a potential subproblem, and a nonlinear subproblem, where each subproblem can be solved exactly in time. Moreover, our analysis shows that the error of our scheme improves from O τ 2 c 4 to O τ 2 c 3 if the magnetic potential in the system vanishes. Furthermore, we propose an efficient limit approximation scheme for solving nonlinear Dirac systems in the nonrelativistic limit regime c 1 which allows errors of order O c −1 without any c-dependent time step restriction.
5The Maxwell-Klein-Gordon equation describes the interaction of a charged particle with 6 an electromagnetic field. Solving this equation in the non-relativistic limit regime, i.e.
7the speed of light c formally tending to infinity, is numerically very delicate as the so-8 lution becomes highly-oscillatory in time. In order to resolve the oscillations, standard 9 numerical time integration schemes require severe time step restrictions depending on 10 the large parameter c 2 .
11The idea to overcome this numerical challenge is to filter out the high frequencies
Quorum sensing (QS) is a widespread mechanism of environment sensing and behavioral coordination in bacteria. At its core, QS is based on the production, sensing and response to small signaling molecules. Previous work with Pseudomonas aeruginosa shows that QS can be used to achieve quantitative resolution and deliver a dosed response to the bacterias density environment, implying a sophisticated mechanism of control. To shed light on how the mechanistic signal components contribute to graded responses to density, we assess the impact of genetic (AHL signal synthase deletion) and/or signal supplementation (exogenous AHL and PQS addition) perturbations on lasB reaction-norms to changes in density. Our approach condenses data from 2,000 timeseries (over 74,000 individual observations) into a comprehensive view of QS-controlled gene expression across variation in genetic, environmental, and signal determinants of lasB expression. We first confirm that deleting either (ΔlasI, ΔrhlI) or both (ΔlasIΔrhlI) signal synthase genes attenuates QS response. In the ΔrhlI background we show persistent yet attenuated density-dependent lasB expression due to native 3-oxo-C12 signaling. We then test if density-independent quantities of signal (3-oxo-C12, C4, PQS or combined) added to the WT either flatten or increase the reaction norm and find that the WT response is robust to all tested concentrations of signal, alone or in combination. We then move to progressively supplementing the genetic knockouts and find that cognate signal supplementation (ΔlasI with 3-oxo-C12, ΔrhlI with C4) is sufficient to restore lasB expression and as well as reactivity to density encoded by the intact signal synthase. We also find that dual supplementation of the double synthase knockout restores expression but does not flatten the reaction norm. Despite adding a density-independent amount of AHL, the double signal synthase can still quantitively sense density. Our results show that a positive reaction norm to density is robust to multiple combinations of gene deletion and density-independent signal supplementation and that while density-independent signal supplementation can increase mean expression, the WT QS still retains the ability to quantitatively respond to density. Our work develops a modular approach to query the robustness and mechanistic bases of the central environmental sensing phenotype of quorum sensing.
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