The infection pathway of virus in cytoplasm of a living cell is studied from the viewpoint of diffusion theory. The cytoplasm plays a role of a medium for stochastic motion of the virus contained in the endosome as well as the free virus. It is experimentally known that the exponent of anomalous diffusion fluctuates in localized areas of the cytoplasm. Here, generalizing fractional kinetic theory, such fluctuations are described in terms of the exponent locally distributed over the cytoplasm, and a theoretical proposition is presented for its statistical form. The proposed fluctuations may be examined in an experiment of heterogeneous diffusion in the infection pathway.
Deep Dive into Heterogeneous anomalous diffusion of virus in cytoplasm of a living cell.
The infection pathway of virus in cytoplasm of a living cell is studied from the viewpoint of diffusion theory. The cytoplasm plays a role of a medium for stochastic motion of the virus contained in the endosome as well as the free virus. It is experimentally known that the exponent of anomalous diffusion fluctuates in localized areas of the cytoplasm. Here, generalizing fractional kinetic theory, such fluctuations are described in terms of the exponent locally distributed over the cytoplasm, and a theoretical proposition is presented for its statistical form. The proposed fluctuations may be examined in an experiment of heterogeneous diffusion in the infection pathway.
In recent years, an exotic phenomenon has experimentally been observed by making use of the technique of real-time single-molecule imaging in the infection pathway of adeno-associated viruses in cytoplasm of a living HeLa cell [1,2]. (Here, the adeno-associated virus is a small virus particle, and the HeLa cell is a line of human epithelial cells.) In each experiment, the virus solution of low concentrations, in which the virus is labeled with fluorescent dye molecule, was added to a culture medium of the living cells. Then, the trajectories of the fluorescent viruses in the cytoplasm were observed. The experiments show that the adeno-associated virus exhibits stochastic motion inside the cytoplasm in two different forms: one is in the free form and the other is the form being contained in the endosome (i.e., a spherical vesicle, see Fig. 1).
Let 2 x be the mean square displacement in stochastic motion. In general, it scales for large elapsed time, , t as . (1). However, what is truly remarkable is the fact [1] that, in the case of subdiffusion, α fluctuates between 0.5 and 0.9, depending on localized areas of the cytoplasm. This manifests the heterogeneous structure of the cytoplasm as a medium for stochastic motion. It is noted [1] that this heterogeneity is not due to the forms of existence of the virus (i.e., being free or contained in the endosome). Thus, this phenomenon is in marked contrast to traditional anomalous diffusion [3] discussed for physical systems, such as particle motion in turbulent flow [4], charge carrier transport in amorphous solids [5], the flow of contaminated vortex in fluid [6], chaotic dynamics [7], porous glasses [8], and so on.
The experimental result mentioned above poses a novel interesting problem for the physics of diffusion. On the other hand, in biology, it is essential to understand the virus infection process for both designing antiviral drug and developing efficient gene therapy vectors. It is therefore of obvious importance to investigate the virus infection pathway from the physical viewpoint.
In this paper, we study the infection pathway of the adeno-associated virus in the cytoplasm of the living HeLa cell by generalizing traditional theory of anomalous diffusion. We regard the cytoplasm as a medium for stochastic motions of both the free virus and the virus contained in the endosome. Then, we imaginarily divide the medium into many small blocks. In other words, a block is identified with a localized area of the cytoplasm. This procedure seems to be necessary when the infection pathway of the virus in the entire cytoplasm is considered. The mean square displacement of the virus does not always show normal diffusion and/or subdiffusion with a fixed exponent, since the virus in a given localized area moves to neighboring ones before reaching the nucleus of the cell [1]. Thus, the exponent, , α in Eq. ( 1) locally fluctuates from one block to another in the cytoplasm. Furthermore, we consider that this fluctuation varies slowly over a period of time, which is much longer than the time scale of the stochastic motion of virus in a localized area of the cytoplasm. It is therefore assumed that there is a large time-scale separation in the infection pathway. For the virus in each block, we apply fractional kinetic theory, which generalizes Einstein’s approach to Brownian motion [9]. Generalizing traditional fractional kinetic theory, we describe the local fluctuations of the exponent, .
α We propose the statistical form of fluctuations from the experimental data. Then, we show that the proposed form of fluctuations can be derived by the maximum entropy principle [10].
Let us start our discussion with the motion of the virus in a one-dimensional block (i.e., a segment). To describe it, we consider the following evolution equation based on the scheme of continuous-time random walks [11]: The virus moves from one block to another, and in such a process the exponent α locally fluctuates over the cytoplasm. We shall therefore develop a generalized fractional kinetic theory, in which this fluctuation is incorporated. To do so, it is essential to clarify the statistical property of the fluctuations. According to the experiment [1], the trajectories of 104 viruses are analyzed. 53 trajectories among them exhibit 1 = α in Eq. ( 1), and other 51 show α varying between 0.5 and 0.9. Besides this fact, no further information is available about the weights of ). 9 . 0 , 5 . 0 ( ∈ α Although α for the virus contained in the endosome might be different from that for the free virus in general, we here assume that the exponents found in both the free and endosomal forms differ from each other only slightly. The virus tends to reach the nucleus of the cell. Due to this tendency, the exponent near 0 = α may seldom be realized. On the other hand, normal diffusion is often to be the case. From these considerations, we propose a Poisson-like form of fluctuations:
with a positive constan
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