Mass Transfer and Surface Kinetics of Cellular Glucose Uptake.

Cells continually take up glucose as a nutrient for a variety of cellular functions. Glucose

must diffuse from the bulk extracellular space to the cellular surface. We can

approximate a typical cell as a sphere of radius $R$ for the purposes of mass transfer

analysis. We will assume steady state and spherical symmetry. There are no

homogeneous reactions involving glucose in the bulk fluid.

$($known glucose concentration far from cell$)$

$x_A=x_A,$ as $r$

at surface of cell $N_{Ar|r}=R=-kc{x}_{A|r}=R$

$(x_A$ is the mole fraction of $A$;$c=$ total concentration; $k$ is a first$-$order glucose

uptake rate constant; $R$ is the cell radius$)$

$($a$).$ Beginning with an appropriate equation below, simplify the differential mass

conservation equation for component A $($glucose$)$ for this system. Make sure to show

specifically how your assumptions are used to make these simplifications.

Rectangular $,\frac{del{C}_i}{\text{d}\text{e}\text{l}\text{t}}=-(\frac{del{N}_{i_x}}{\text{d}\text{e}\text{l}\text{x}}+\frac{del{N}_{i_y}}{\text{d}\text{e}\text{l}\text{y}}+\frac{del{N}_{i_z}}{\text{d}\text{e}\text{l}\text{z}})+R_i$

Cylindrical $,\frac{del{C}_i}{\text{d}\text{e}\text{l}\text{t}}=-(\frac{1}{r}\frac{\text{d}\text{e}\text{l}(r{N}_{i_r})}{\text{d}\text{e}\text{l}\text{r}}+\frac{1}{r}\frac{del{N}_{i_{}}}{\text{d}\text{e}\text{l}}+\frac{del{N}_{i_z}}{\text{d}\text{e}\text{l}\text{z}})+R_i$

Spherical $,\frac{del{C}_i}{\text{d}\text{e}\text{l}\text{t}}=-(\frac{1}{r^2}\frac{\text{d}\text{e}\text{l}(r^2{N}_{i_r})}{\text{d}\text{e}\text{l}\text{r}}+\frac{1}{rsin}\frac{\text{d}\text{e}\text{l}(N_{i}sin)}{\text{d}\text{e}\text{l}}+\frac{1}{rsin}\frac{del{N}_{i_{}}}{\text{d}\text{e}\text{l}})+R_i$

$($b$)$ For this system component $B$ is taken to be the extracellular fluid $($everything minus

the glucose$),$ and it is assumed to be stagnant. Derive the appropriate general form of

the

expression for the flux $N_{AC}($refer to this equation:

$N_i=-D_{ij}grad{C}_i+C_i{v}_s$ and

$C_{v^{**}}={\displaystyle \underset{i=1}{\overset{n}{}}}{N}_i$

and

$\{$:$N_i=-C{D}_{ij}grad{x}_i+x_i{\displaystyle \underset{i=1}{\overset{n}{}}}{N}_i)$ for this system.

$($c$)$ After substituting the flux expression for this system, indefinitely integrate the

differential mass balance equation from Part $($a$)$ to obtain an expression for the mole

fraction $x_A$ in terms of two unknown integration constants.

$($d$)$ Obtain the two integration constants by applying the boundary conditions under the

additional assumption that the surface uptake is finite but rapid, i$.$e$.,$ you may

approximate the natural log at the surface as $ln(1+x_{A|r}=R)$~~$x_{A|r}=R.$

where $x_{Al}=R$ is the unknown surface concentration of glucose. Note that it is

inappropriate to use this assumption anywhere else in the diffusion path. Importantly,

note that the determination of $x_{Al}=R$ is part of the solution to this problem.