Time-delayed model of the unbiased movement of Tetrahymena Pyriformis
- PDF / 174,522 Bytes
- 11 Pages / 595 x 842 pts (A4) Page_size
- 99 Downloads / 163 Views
TIME-DELAYED MODEL OF THE UNBIASED MOVEMENT OF TETRAHYMENA PYRIFORMIS ´fa ´ r1 , L a ´szlo ´ Ko ˝ hidai2 and Andrea Hegedu ˝s3 Orsolya Sa 1
Department of Analysis, Budapest University of Technology and Economics E-mail: [email protected] 2
Department of Genetics, Cell- and Immunology, Semmelweis University
3
Department of Genetics, Cell- and Immunology, Semmelweis University (Received August 31, 2010; Accepted July 10, 2011)
Abstract In our paper we investigate the unbiased movement of the unicellular eukaryotic ciliate Tetrahymena Pyriformis. We use a time-delayed version of the previously known model to describe the specific movement of this species. With the help of semi-discretization, we state analytic results for the model.
1. Introduction The most common principle for modeling self-organizing systems in developmental biology is the law of conservation. With an arbitrary surface ∂Ω enclosing the volume Ω, the rate of change of the amount of the substance inside Ω is equal to the flux across the surface ∂Ω plus the production of material inside Ω. Thus ∂ ∂t
u(t, x)dV = −
Ω
Jds +
∂Ω
f (u, t, x)dV, Ω
where u(t, x) is the amount of material at point x at time t, J is the flux of material and f (u, t, x) is the rate of production of u(t, x). Applying the divergence theorem and taking into account that the volume Ω is arbitrary yields ∂ u(t, x) = −∇J + f (u, t, x) ∂t Assuming there is no cell proliferation, the unbiased motion of the cells is described by Fick’s equation (see [5]): Mathematics subject classification number : 92C17, 34K60. Key words and phrases: eukaryotic ciliate, time-delay, semi-discretization. 0031-5303/2011/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
216
´ AR, ´ ˝ ˝ O. SAF L. KOHIDAI and A. HEGEDUS
∂u(t, x) ∂2 = D 2 u(t, x) (1.1) ∂t ∂x where u(t, x) is the concentration of cells at time t at point x. D > 0 is the Fick constant, which is proportional to the typical displacement of the cells in a given time. To have a unique solution, we have to specify the initial conditions u(0, x) = u0 (x), and boundary conditions (for a closed system, ∂u(t,x) ∂ν =0). The idea that the unbiased movement of the unicellulars can be approximated with the same equation as molecular diffusion is based on the observation that if a system of bacteria is left alone, the cells move fast and randomly. This random bacterial movement can be approximated with the diffusion (in fact, very accurately).
2. The delay Due to the fact that in an average Tetrahymena Pyriformis population, a considerable amount of cells (even up to one third of them, see [2]) is in “rest state” (they do not move or react to chemical compounds), there is a delay in their reaction to the changes of the environment (like the changes of cell density or gradient of a chemotactical compound), while equation (1.1) assumes immediate response. The delay we have to deal with is, however, not constant, since at any given time just a portion of the cells is unresponsive. So the change of the syste
Data Loading...
