$.$

Many soluble secreted peptides that influence cell behavior, such as the fibroblast

growth factor $($FGF$),$ bind reversibly to components of extracellular matrix, such as

heparin, according to:

Extracellular matrix is a gel, and heparin is fixed within the gel and does not diffuse.

FGF can diffuse freely through the extracellular matrix and bind reversibly to heparin.

Such binding is governed by the equilibrium relationship

$K_d=\frac{C_{\text{B}\text{o}\text{d}}{C}_{\text{h}\text{r}\text{e}}}{C_{r-\text{h}\text{i}\text{n}\text{p}\text{a}\text{n}}}$

Where $K_d$ is the dissociation constant $($M$),C_{FG}$ is the concentration of free $($unbound$)$

FGF$,C_{\text{b}\text{e}\text{p}}$ is the concentration of free FGF binding sites on heparin, and $C_{F-\text{M}\text{a}\text{p}\text{e}\text{i}\text{g}}$ is the

concentration of FGF bound to heparin. The total concentration of heparin binding sites

is the sum of the free heparin binding sites and the complexes,

This problem explores how the binding to heparin might influence transient and steady

state transport properties of FGF in matrix.

A$.$ Consider a situation where a region of extracellular matrix containing FGF at a

constant concentration is suddenly exposed to fluid with no FGF$.$ Note that the total

concentration of FGF in matrix comprises the sum of free and heparin$-$bound FGF$.$

Such a situation might happen in a wound, for example. This situation can be modeled

as a semi$-$inifinite slab with a change in surface concentration at time zero.

Perform a shell balance and show that, provided the binding reaction is very fast

compared to diffusion, the transient diffusion of free FGF in the matrix is given by:

B$.$ Show that the equation in $($A$)$ can be expressed in the more useful form:

What form does this equation take if $A{C}_{\text{h}\text{e}\text{p}\text{w}\text{a}\text{l}}-K_2{C}_{\text{h}\text{e}\text{n}\text{w}\text{a}\text{l}}-10K_d{C}_{\text{h}\text{e}\text{p}\text{w}\text{a}\text{d}}-1000K_d{C}_{mF}$ For this $limit,$ calculate the

leharacteristic diffusion times RELATIVE $TO$ THE $CASE$ WHERE $NO$ HEI'ARIN $IS$

PRFSINT $(i.e.,$ free difusion case$)$ for the following situations;

$C_{\text{h}\text{e}\text{p}\text{w}\text{a}\text{l}}-K_2$

$C_{\text{h}\text{e}\text{n}\text{w}\text{a}\text{l}}-10K_d$

$C_{\text{h}\text{e}\text{p}\text{w}\text{a}\text{d}}-1000K_d$

Note that $in$ order $to$ calculate the relative times, you $do$ not need the $absolute$ values $of$

any parameters. What are the implications for the presence $of$ heparin $on$ how fast $FGF$

might diffuse out $of$ matrix?