Effective Motion of a Virus Trafficking Inside a Biological Cell

Virus trafficking is fundamental for infection success and plasmid cytosolic trafficking is a key step of gene delivery. Based on the main physical properties of the cellular transport machinery such as microtubules, motor proteins, our goal here is to derive a mathematical model to study cytoplasmic trafficking. Because experimental results reveal that both active and passive movement are necessary for a virus to reach the cell nucleus, by taking into account the complex interactions of the virus with the microtubules, we derive here an estimate of the mean time a virus reaches the nucleus. In particular, we present a mathematical procedure in which the complex viral movement, oscillating between pure diffusion and a deterministic movement along microtubules, can be approximated by a steady state stochastic equation with a constant effective drift. An explicit expression for the drift amplitude is given as a function of the real drift, the density of microtubules and other physical parameters. The present approach can be used to model viral trafficking inside the cytoplasm, which is a fundamental step of viral infection, leading to viral replication and in some cases to cell damage.

💡 Research Summary

The paper presents a quantitative mathematical framework for describing the intracellular transport of a virus (or plasmid DNA) from the cell periphery to the nucleus. The authors model the cytoplasm as a two‑dimensional circular domain of radius R with a concentric nuclear disk of radius δ (δ ≪ R). Microtubules are idealized as N uniformly spaced radial filaments that emanate from the cell membrane and converge at a microtubule‑organizing center (MTOC) located near the nucleus. The angular spacing between neighboring microtubules is Θ = 2π/N, defining a wedge‑shaped fundamental domain ˜Ω.

Within ˜Ω the viral particle undergoes two alternating phases: (i) free Brownian diffusion with diffusion coefficient D until it contacts a microtubule, and (ii) deterministic directed motion along the microtubule toward the nucleus with a constant speed V. The binding time on a microtubule is assumed to be exponentially distributed with mean tm, so that the deterministic displacement per binding event is dm = V·tm. After detachment the particle is released at a random angular position on a circle of radius equal to its distance from the nucleus, and the cycle repeats.

The overall trajectory is therefore a sequence of “diffusion‑plus‑drift” steps. The authors introduce the random variables Rk (the radial distance at the beginning of step k) and compute the mean first‑passage time (MFPT) for each diffusion substep, denoted u(Rk). To obtain u(Rk) they solve the Dynkin equation DΔu = −1 in the wedge with mixed boundary conditions: absorbing at the two microtubule sides (θ = 0 and θ = Θ), reflecting at the outer membrane (r = R), and absorbing at the nuclear boundary (r = δ). By separation of variables in polar coordinates and Fourier series expansion, they derive an explicit series solution involving eigenvalues λn = (2n + 1)π/Θ. For small Θ (dense microtubule network) the series is approximated by a simple closed form:

 ū(r) ≈ (r²/4D)