Consider a cell composed of a cell body to which is attached a long, thin, tubular "process" $($e$.$g$.,$ the axon of a

neuron or the flagellum of a sperm cell$)$ of length l cm $($Figure on right$).$ Suppose some substance n $($e$.$g$.,$ a

metabolite$)$ is generated in the cell body and diffuses along the process, all along which the substance is

consumed at a constant uniform rate, $$ mol$/$s per unit length. The continuity equation for the substance

thus becomes $$

$$

$=$

$\backslash$Phi $$

$$

$$

$$

$,$ where A is the $($assumed constant$)$ cross$-$sectional area of the process.

a$.$ Combine the modified continuity equation given above with Fick$$s law to obtain a modified form of the

diffusion equation that must be satisfied by $.$

b$.$ Show that a solution of this equation in the steady state $($

$=$

$\backslash$Phi $$

$$

$=0)$ is $$ x $=$

$$

$2$

$$

$2+0+0$ and

find values of the constants a$0$ and b$0$

corresponding to the boundary conditions $=0$ and $\backslash$Phi $=09\_1$ Consider a cell composed of a cell body to which is attached a long, thin, tubular "process" $($e$.$g$.,$ the axon of a

neuron or the flagellum of a sperm cell$)$ of length lcm $($Figure on right$).$ Suppose some substance $n($e$.$g$.,$ a

metabolite$)$ is generated in the cell body and diffuses along the process, all along which the substance is

consumed at a constant uniform rate, ${}_{n}mo\frac{l}{s}$ per unit length. The continuity equation for the substance

thus becomes $\frac{del{c}_n}{\text{d}\text{e}\text{l}\text{t}}=-\frac{\text{d}\text{e}\text{l}{}_n}{\text{d}\text{e}\text{l}\text{x}}-\frac{{}_n}{A},$ where $A$ is the $($assumed constant$)$ cross$-$sectional area of the process.

a$.$ Combine the modified continuity equation given above with Fick's law to obtain a modified form of the

diffusion equation that must be satisfied by $c_n.$

process

b$.$ Show that a solution of this equation in the steady state $(\frac{del{c}_n}{\text{d}\text{e}\text{l}\text{t}}=\frac{\text{d}\text{e}\text{l}{}_n}{\text{d}\text{e}\text{l}\text{t}}=0)$ is $c_n(x)=\frac{{}_n}{2DA}{x}^2+a_{0}x+b_0$ and

find values of the constants $a_0$ and $b_0$ corresponding to the boundary conditions $c_n(o)=C_0$ and ${}_n(l)=0.$