Team:IISc-Bangalore/Model-Examples
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Illustration 1: Steady state solution for a column
We assume the boundary conditions to be such that \(C'=0\), \(C_1\) can be calculated by normalising the function over gas vesicles in the column.
\[ \int_{0}^{L} C_1 e^{\frac{-V_s h}{D}} A dh = N \]Where A is the area of the column and N the total number of gas vesicles in the vessel. The normalised equation is,
\[ C(h)=\frac{V_s N}{AD(1-e^{\frac{-V_sL}{D}})} e^{\frac{-V_s h}{D}} \]We take the following parameters to simulate a 1 cm2 cuvette with solution filled up to a height of 2 cm ,
\[V_s \sim \frac{4gR_H^2}{18\eta} (\rho-\rho_{gv})\] \[A \sim 1 cm^2\] \[D = 6\pi \eta R_H \] \[L \sim 2cm\]To take into account the dependence of terminal velocity on hydrodynamic size, we have considered the gas vesicle to be a sphere of a radius equal to the hydrodynamic radius measured in DLS. The steady state profiles for some effective \(R_H\) are given below,
These curves make it quite clear that unaided gas vesicles (\(R_H \sim 100nm\)) have an almost uniform distribution across the length even at the steady state which is achieved after an infinitely long span of time. This was the first insight that made it clear that to make any efficient use of a gas vesicle's buoyant properties, a strategy was required that would increase the \(R_H\) considerably and would allow them to float better.
Illustration 2: Uniform initial profile with exponential boundary conditions.
Assuming the initial mixing is uniform, this is the most natural way in which we expect the concentration function to evolve on the boundaries. The exact initial and boundary conditions are taken such that they satisfy each other nicely. The contour plot below shows how the profile evolves over time. The legend on the right shows the relative magnitude of concentrations present in the plot.
Illustration 3: Exponential initial profile with exponential boundary conditions.
Such a situation can arise in the event the particles have settled down due to gravity initially but start floating instantaneously at t=0. While quite unrealistic, this solution might be useful in cases where the gas vesicles have been reversibly denatured but are allowed to float up after removing the denaturing agent. The plot below shows how the profile evolves over time.
Illustration 4: Linear initial profile with exponential boundary conditions.
This is the most unlikely of the three cases we've dealth with till now. An linear initial profile is almost impossible under normal conditions unless some kind of external agency maintains it. Even then, we've tried to solve the equations to show how the concentrations will evolve for such a case.
