Class 1 integrons facilitate horizontal gene transfer, significantly influencing antibiotic resistance gene (ARG) dissemination within microbial communities. Wastewater treatment plants (WWTPs) are critical reservoirs of ARGs and integrons, yet the integron-mediated dynamics of ARG transfer across different WWTP types remain poorly understood. Here we show distinct ARG profiles associated with class 1 integrons in municipal and industrial WWTPs using a novel approach combining nested-like high-throughput qPCR and PacBio sequencing. Although industrial WWTPs contained higher absolute integron abundances, their relative ARG content was lower (1.27 × 107–9.59 × 107 copies/ng integron) compared to municipal WWTPs (3.72 × 107–1.98 × 108 copies/ng integron). Of the 132,084 coding sequences detected from integrons, 56.8 % encoded antibiotic resistance, with industrial plants showing lower ARG proportions, reduced ARG array diversity, and greater incorporation of non-ARG sequences. These findings suggest industrial WWTP integrons integrate a broader array of exogenous genes, reflecting adaptation to complex wastewater compositions. This work enhances our understanding of integron-driven ARG dynamics in wastewater and offers a robust strategy for environmental integron analysis.
The first nontrivial lower bound of the worst-case approximation ratio for the maxcut problem was achieved via the dual Cheeger problem, whose optimal value is referred to the dual Cheeger constant $h^+$, and later improved through its modification $\widehat{h}^+$. However, the dual Cheeger problem and its modification themselves are relatively unexplored, especially lack of effective approximate algorithms. To this end, we first derive equivalent spectral formulations of $h^+$ and $\widehat{h}^+$ within the framework of the nonlinear spectral theory of signless 1-Laplacian, present their interactions with the Laplacian matrix and 1-Laplacians, and then use them to develop an inverse power algorithm that leverages the local linearity of the objective functions involved. We prove that the inverse power algorithm monotonically converges to a ternary-valued eigenvector, and provide the approximate values of $h^+$ and $\widehat{h}^+$ on G-set for the first time. The recursive spectral cut algorithm for the maxcut problem can be enhanced by integrating into the inverse power algorithms, leading to significantly improved approximate values on G-set. Finally, we show that the lower bound of the worst-case approximation ratio for the maxcut problem within the recursive spectral cut framework can not be improved beyond $0.769$.
Numerical resolution of moderately high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality for the classical numerical methods including finite difference, finite element and spectral methods. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7- D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving moderately high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy