"
Team:KULeuven/Modelling/Vanillin Receptor
From 2009.igem.org
Vanillin diffusion
intro
Because the VirA protein sensor domain is located in the cytoplasmic region, it senses intercellular vanillin concentration. (see figure) Because the objective of this project is to regulate the extracellular vanillin concentration we investigate the relation between the intercellular and extracellular vanillin concentration. It will be shown that on the time-scale we are interested in, intracellular and extracellular concentration are equal.
We can approximate the relationship with following continuity equation (equilibrium in cytoplasm and extracellular):
<math>d[V_inter]/dt = V_production-k_diffusion(V_inter-V_extra)</math>
V_inter: intercellular concentration V_extra: extracellular concentration V_production: production of vannilin k_diffusion: speed of diffusion in and out cell
rewriting the above equation gives <math>V_extra = V_intra+1/k_diffusion*(d[V_inter]/dt - V_production)</math>
If the k_diffusion is big, on the time-scale we are interested in, we are not interested in phenomena which occur within 1 minute, V_extra and V_intra can be regarded as equal.
simulation
The following more exact treatise will try to determine the time scale on which the vanillin concentration reaches equilibrium conditions (V_extra = V_inter). The process is described by following (more general) continuity equation:
<math> d[V]/dt = V_production+k_degradation*V+d[D_V*d[V]/dx]/dx</math>
V: Vanillin concentration V_production: Vanillin production k_degradation: speed of degradation but will here be considered 0 D_V: diffusion coefficient of vanillin
The cell will be modelled as a sphere, with following characteristics:
Inner Membrane thickness: 8 nm Outer Membrane thickness: 12 nm Periplasm thickness: 10 nm
We neglect the outer membrane because the porins make the membrane permeable (see figure of cell wall). To determine the diffusion coefficient of vanillin in cytoplasm, inner membrane and periplasm we use following equation:
D = kT/f_a
k: constant of Boltzmann T: Temperature °K f_a: friction coefficient for a sphere radius rA.
The radius of the equivalent hydrodynamic sphere of vanillin is 2 A°.
f_a = 6*π*η*rA
The friction coefficient of vanillin
η_water = η_cytoplasm = η_IM = η_PP =
A first approximation is the transfer time of vanillin in the cell.
tau = delta_r^2/D
which was approximately 1 ms.
We neglect the outer membrane because the porins make the membrane permeable.
