Optimal Glycogen Structure Muscle cells need rapid access to large amounts of glucose during heavy exercise. This glucose is stored in liver and skeletal muscle in polymeric form as particles of glycogen. The typical glycogen particle contains about 55,000 glucose residues (see Fig. $15-35$ b). Meléndez-Hevia, Waddell, and Shelton (1993) explored some theoretical aspects of the structure of glycogen, as described in this problem.
(a) The cellular concentration of glycogen in liver is about $0.01 \mu$ M. What cellular concentration of free glucose would be required to store an equivalent amount of glucose? Why would this concentration of free glucose present a problem for the cell?
Glucose is released from glycogen by glycogen phosphorylase, an enzyme that can remove glucose molecules, one at a time, from one end of a glycogen chain. Glycogen chains are branched (see Figs $15-28$ and $15-35$ b), and the degree of branching - the number of branches per chain-has a powerful influence on the rate at which glycogen phosphorylase can release glucose.
(b) Why would a degree of branching that was too low (i.e., below an optimum level) reduce the rate of glucose release? (Hint: Consider the extreme case of no branches in a chain of 55,000 glucose residues.)
(c) Why would a degree of branching that was too high also reduce the rate of glucose release? (Hint: Think of the physical constraints.)
Meléndez-Hevia and colleagues did a series of calculations and found that two branches per chain (see Fig. $15-35$ b) was optimal for the constraints described above. This is what is found in glycogen stored in muscle and liver.
To determine the optimum number of glucose residues per chain, Meléndez-Hevia and coauthors considered two key parameters that define the structure of a glycogen particle: $t$
$=$ the number of tiers of glucose chains in a particle (the molecule in Fig. $15-35$ b has five tiers $) ; g_{\mathrm{c}}=$ the number of glucose residues in each chain. They set out to find the values of $t$ and $g_{\mathrm{c}}$ that would maximize three quantities: (1) the amount of glucose stored in the particle $\left(G_{\mathrm{T}}\right)$ per unit volume; (2) the number of unbranched glucose chains $\left(C_{\mathrm{A}}\right)$ per unit volume (i.e., number of chains in the outermost tier, readily accessible to glycogen phosphorylase); and (3) the amount of glucose available to phosphorylase in these unbranched chains $\left(G_{\mathrm{PT}}\right)$
(d) Show that $C_{\mathrm{A}}=2^{t-1}$. This is the number of chains available to glycogen phosphorylase before the action of the debranching enzyme.
(e) Show that $C_{\mathrm{T}},$ the total number of chains in the particle, is given by $C_{\mathrm{T}}=2^{t}-1$
Thus $G_{\mathrm{T}}=g_{\mathrm{c}}\left(\mathrm{C}_{\mathrm{T}}\right)=g_{\mathrm{c}}\left(2^{t-1}\right),$ the total number of glucose residues in the particle.
(f) Glycogen phosphorylase cannot remove glucose from glycogen chains that are shorter than five glucose residues. Show that $G_{\mathrm{PT}}=\left(g_{\mathrm{c}}-4\right)\left(2^{t-1}\right) .$ This is the amount of glucose readily available to glycogen phosphorylase.
(g) Based on the size of a glucose residue and the location of branches, the thickness of one tier of glycogen is $0.12 \mathrm{g}_{\mathrm{c}} \mathrm{nm}+0.35 \mathrm{nm} .$ Show that the volume of a particle, $V_{\mathrm{s}},$ is given by the equation $V_{\mathrm{s}}=4 / 3 \pi \mathrm{t}^{3}\left(0.12 g_{\mathrm{e}}+0.35\right)^{3} \mathrm{nm}^{3}$
Meléndez-Hevia and coauthors then determined the optimum values of $t$ and $g_{\mathrm{c}}-$ those that gave the maximum value of a quality function, $f,$ that maximizes $G_{\mathrm{T}}, C_{\mathrm{A}},$ and $G_{\mathrm{PT}}$ while $V_{\mathbf{8}}: f=\frac{G_{\mathrm{T}} C_{\mathrm{A}} G_{\mathrm{PT}}}{V_{\mathrm{L}}} .$ They found that the optimum value of $g_{\mathrm{c}}$ is independent of $t$ .
(h) Choose a value of $t$ between 5 and 15 and find the optimum value of $g_{\mathrm{c}} .$ How does this compare with the $g_{\mathrm{c}}$ found in liver glycogen (see Fig. $15-35$ b)? (Hint: You may find it useful to use a spreadsheet program.)