Stochastic Model of Virus–Endosome Fusion and Endosomal Escape of pH-Responsive Nanoparticles

Abstract

In this paper, we set up a stochastic model for the dynamics of active Rab5 and Rab7 proteins on the surface of endosomes and the acidification process that govern the virus–endosome fusion and endosomal escape of pH-responsive nanoparticles. We employ a well-known cut-off switch model for Rab5 to Rab7 conversion dynamics and consider two random terms: white Gaussian and Poisson noises with zero mean. We derive the governing equations for the joint probability density function for the endosomal pH, Rab5 and Rab7 proteins. We obtain numerically the marginal density describing random fluctuations of endosomal pH. We calculate the probability of having a pH level inside the endosome below a critical threshold and therefore the percentage of viruses and pH-responsive nanoparticles escaping endosomes. Our results are in good qualitative agreement with experimental data on viral escape.

An exact analytical solution to the non-stationary coagulation equation describing the endosomal network dynamics

“…Let us especially underline that 𝑛(𝑥)Δ𝑥 represents the number of Rab5-positive endosomes per cell for which the LDL fluorescence intensity lies between 𝑥 and 𝑥 + Δ𝑥 13 . This equation describes the endosome network dynamics and characterizes the degradative cargo distributed into a network of early Rab5-positive endosomes that can transform into late Rab7-positive endosomes and lysosomes 13,14,15 . Here the term cargo plays the role of viruses or nanoparticles that are capable to penetrate inside the living cell through its membrane.…”

An exact analytical solution to the nonstationary coagulation equation describing the endosomal network dynamics

“…15 This equation describes the endosome network dynamics and characterizes the degradative cargo distributed into a network of early Rab5-positive endosomes that can transform into late Rab7-positive endosomes and lysosomes. [15][16][17] Here the term "cargo" plays the role of viruses or nanoparticles that are capable to penetrate inside the living cell through its membrane. The first two contributions on the right-hand side (r.h.s.)…”

“…Let us especially underline that $$$ n(x)\Delta x $$$ represents the number of Rab5‐positive endosomes per cell for which the LDL fluorescence intensity lies between $$$ x $$$ and $$$ x+\Delta x $$$ 15 . This equation describes the endosome network dynamics and characterizes the degradative cargo distributed into a network of early Rab5‐positive endosomes that can transform into late Rab7‐positive endosomes and lysosomes 15–17 . Here the term “cargo” plays the role of viruses or nanoparticles that are capable to penetrate inside the living cell through its membrane.…”

Analysis of Smoluchowski’s Coagulation Equation with Injection

“…This is usually the case in the latter stages of a phase transformation in systems of various physical nature (see, among others, Refs. [1][2][3][4][5][6][7][8][9]) ranging from aerosols in the atmosphere [1,2] and crystals in supercooled liquids and supersatureted solutions [6,7] to endosomes inside living cells [8][9][10]. Fundamental papers on particle coagulation theory were the works of Marian Smoluchowski [11,12] dealing with coagulation in hydrosols.…”