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.
Deep Dive into 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 pa
1. Introduction. Because cytosolic transport has been identified as a critical barrier for synthetic gene delivery [1], plasmids or viral DNAs delivery from the cell membrane to the nuclear pores has attracted the attention of many biologists. The cell cytosol contains many types of organelles, actin filaments, microtubules and many others, so that to reach the nucleus, a viral DNA has to travel through a crowded and risky environment. We are interested here in studying the efficiency of the delivery process and we present a mathematical model of virus trafficking inside the cell cytoplasm. We model the viral movement as a Brownian motion. However, the density of actin filaments and microtubules, inside the cell, can hinder diffusion, as demonstrated experimentally [2]. In a crowded environment, we will model the virus as a material point. This reduction is simplistic for several reasons: actin filament network can trapped a diffusing object and beyond a certain size, as observed experimentally, a DNA fragment cannot find its way across the actin filaments [2]. Active directional transport along microtubules or actin filaments seems then the only way to deliver a plasmid to the nucleus. The active transport of the virus involves in general motor proteins, such as Kinesin (to travel in the direction of the cell membrane) or Dynein (to travel toward the nucleus). Once a virus is attached to a Dynein protein, its movement can be modeled as a deterministic drift toward the nucleus.
Recently, a macroscopic modeling has been developed to describe the dynamics of adenovirus concentration inside the cell cytoplasm [3]. This approach offers very interesting results about the effect of microtubules, but neglects the complexity of the geometry and cannot be used to describe the movement of a single virus, which might be enough to cause cellular infection. Modeling a virus trafficking imposes to use a stochastic description. We model here the motion of a virus as that of a material point, so the probability of its trapping by actin filaments or microtubules is neglected. In the present approximation, the viral movement has two main components: a Brownian one, which accounts for its free movement, and a drift directed towards the centrosome or MTOC (Microtubules Organization Center), an organelle located near the nucleus. The magnitude of the drift along microtubules depends on many parameters, such as the binding and unbinding rates and the velocity of the motor proteins [4].
In the present approach, we present a method to approximate a time dependent dynamics of virus trafficking by an effective stochastic equation with a radial steady state drift. The main difficulties we have to overcome arise from the time dependent nature of the trajectories which consists of intermittent epochs of drifts and free diffusion. We propose to derive an explicit expression for the steady state drift amplitude. In this approximation, the effective drift will gather the mean properties of the cytoplasmic organization such as the density of microtubules and its off binding rate.
Our method to find the effective drift can be described as follow: first, we approximate the cell geometry as a two dimensional disk and use a pure Brownian description to approximate the virus diffusion step. This geometrical approximation is valid, for any two dimensional cell such as the in vitro flat skin fibroblast culture cells [3]: indeed, due to their adhesion to the substrate, the thickness of these cells can be neglected in first approximation. Second, when the distribution of the initial viral position is uniform on the cell surface, we will estimate, during the diffusing period, the hitting position on a microtubule. By solving a partial differential equation, inside a sliced shape domain, delimited by two neighboring microtubules, we will provide an estimate of the mean time to the most likely hitting point. Finally, the amplitude of the radial steady state drift will be obtained by an iterative method which assumes that, after a virus has moved a certain distance along a microtubule, it is released at a point uniformly distributed on the final radial distance from the nucleus, ready for a new random walk. This scenario repeats until the virus reaches the nucleus surface. Finally, we will compute the mean time, the mean number of steps before a virus reaches the nucleus and the amplitude of the effective drift by using the following criteria: the Mean First Passage Time (MFPT) to the nucleus of the iterative approximation is equal to the MFPT obtained by solving directly an Ornstein-Uhlenbeck stochastic equation. The explicit computation of the effective drift is a key result in the estimation of the probability and the mean time a single virus or DNA molecule takes to reach a small nuclear pore [5].
Modeling stochastic viral movement inside a biological cell. We approximate the cell as a two dimensional geometrical domain Ω, which is here a disk
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