Quenched exit times for random walk on dynamical percolation

We consider random walk on dynamical percolation on the discrete torus $\mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(\mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation environment. Here we study exit times in the quenched setting, where we condition on a typical dynamical percolation environment. We obtain an upper bound for all $p$ which for $p<p_c$ matches the known lower bound

Mixing time for random walk on supercritical dynamical percolation

Abstract: We consider dynamical percolation on the d-dimensional discrete torus Znd of side length n, where each edge refreshes its status at rate μ=μn≤1/2 to be open with probability p. We study random walk on the torus, where the walker moves at rate 1 / (2d) along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case ppc(Zd), the mixing time is Θ(n2+1/μ); here the implied constants depend only on d and p. We prove a quenched (and hence annealed) version of this conjecture up to a poly-logarithmic factor under the assumption θ(p)>1/2. When θ(p)>0, we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times. The latter implies sharp (up to poly-logarithmic factors) upper bounds on exit times of large balls throughout the supercritical regime. Our proofs are based on percolation results (e.g., the Grimmett–Marstrand Theorem) and an analysis of the volume-biased evolving set process; the key point is that typically, the evolving set has a substantial intersection with the giant percolation cluster at many times. This allows us to use precise isoperimetric properties of the cluster (due to G. Pete) to infer rapid growth of the evolving set, which in turn yields the upper bound on the mixing time

Photochemical internalisation of chemotherapy potentiates killing of multidrug-resistant breast and bladder cancer cells

Multidrug resistance (MDR) is the major confounding factor in adjuvant solid tumour chemotherapy. Increasing intracellular amounts of chemotherapeutics to circumvent MDR may be achieved by a novel delivery method, photochemical internalisation (PCI). PCI consists of the co-administration of drug and photosensitiser; upon light activation the latter induces intracellular release of organelle-bound drug. We investigated whether co-administration of hypericin (photosensitiser) with mitoxantrone (MTZ, chemotherapeutic) plus illumination potentiates cytotoxicity in MDR cancer cells. We mapped the extent of intracellular co-localisation of drug/photosensitiser. We determined whether PCI altered drug-excreting efflux pump P-glycoprotein (Pgp) expression or function in MDR cells. Bladder and breast cancer cells and their Pgp-overexpressing MDR subclones (MGHU1, MGHU1/R, MCF-7, MCF-7/R) were given hypericin/MTZ combinations, with/without blue-light illumination. Pilot experiments determined appropriate sublethal doses for each. Viability was determined by the 3-[4,5-dimethylthiazolyl]-2,5-diphenyltetrazolium bromide assay. Intracellular localisation was mapped by confocal microscopy. Pgp expression was detected by immunofluorescence and Pgp function investigated by Rhodamine123 efflux on confocal microscopy. MTZ alone (0.1–0.2 μg ml−1) killed up to 89% of drug-sensitive cells; MDR cells exhibited less cytotoxicity (6–28%). Hypericin (0.1–0.2 μM) effects were similar for all cells; light illumination caused none or minimal toxicity. In combination, MTZ /hypericin plus illumination, potentiated MDR cell killing, vs hypericin or MTZ alone. (MGHU1/R: 38.65 and 36.63% increase, P<0.05; MCF-7/R: 80.2 and 46.1% increase, P<0.001). Illumination of combined MTZ/hypericin increased killing by 28.15% (P<0.05 MGHU1/R) compared to dark controls. Intracytoplasmic vesicular co-localisation of MTZ/hypericin was evident before illumination and at serial times post-illumination. MTZ was always found in sensitive cell nuclei, but not in dark resistant cell nuclei. In illuminated resistant cells there was some mobilisation of MTZ into the nucleus. Pgp expression remained unchanged, regardless of drug exposure. Pgp efflux was blocked by the Pgp inhibitor verapamil (positive control) but not impeded by hypericin. The increased killing of MDR cancer cells demonstrated is consistent with PCI. PCI is a promising technique for enhancing treatment efficacy

Mixing time for random walk on supercritical dynamical percolation

Abstract: We consider dynamical percolation on the d-dimensional discrete torus Znd of side length n, where each edge refreshes its status at rate μ=μn≤1/2 to be open with probability p. We study random walk on the torus, where the walker moves at rate 1 / (2d) along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case ppc(Zd), the mixing time is Θ(n2+1/μ); here the implied constants depend only on d and p. We prove a quenched (and hence annealed) version of this conjecture up to a poly-logarithmic factor under the assumption θ(p)>1/2. When θ(p)>0, we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times. The latter implies sharp (up to poly-logarithmic factors) upper bounds on exit times of large balls throughout the supercritical regime. Our proofs are based on percolation results (e.g., the Grimmett–Marstrand Theorem) and an analysis of the volume-biased evolving set process; the key point is that typically, the evolving set has a substantial intersection with the giant percolation cluster at many times. This allows us to use precise isoperimetric properties of the cluster (due to G. Pete) to infer rapid growth of the evolving set, which in turn yields the upper bound on the mixing time

Mixing time for random walk on supercritical dynamical percolation

Abstract: We consider dynamical percolation on the d-dimensional discrete torus Znd of side length n, where each edge refreshes its status at rate μ=μn≤1/2 to be open with probability p. We study random walk on the torus, where the walker moves at rate 1 / (2d) along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case ppc(Zd), the mixing time is Θ(n2+1/μ); here the implied constants depend only on d and p. We prove a quenched (and hence annealed) version of this conjecture up to a poly-logarithmic factor under the assumption θ(p)>1/2. When θ(p)>0, we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times. The latter implies sharp (up to poly-logarithmic factors) upper bounds on exit times of large balls throughout the supercritical regime. Our proofs are based on percolation results (e.g., the Grimmett–Marstrand Theorem) and an analysis of the volume-biased evolving set process; the key point is that typically, the evolving set has a substantial intersection with the giant percolation cluster at many times. This allows us to use precise isoperimetric properties of the cluster (due to G. Pete) to infer rapid growth of the evolving set, which in turn yields the upper bound on the mixing time