13\. 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 \(\beta\)-particle contains about 55,000 glucose residues (see Eig_
15-2). 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 \(0.01 \mu
\mathrm{M}\). 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
(see Eig. 15-3). Glycogen chains are branched (see Eig.15-2), 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 Eig_15-2) 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 mole-cule in
Eig.15-2 has five tiers);
\(g_{c}=\) the number of glucose residues in each chain. The \(y\) set out to find
the values of \(t\) and \(g_{c}\) 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 \(\left(C_{A}\right)\)
per unit volume (i.e., number of A 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 \(C_{A}=2^{t-1}\). 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 \(C_{\mathrm{T}}=2^{t}-1\). For purposes of this calculation, consider
the primers to be a single chain. Thus
\(G_{\mathrm{T}}=g_{\mathrm{c}}\left(C_{\mathrm{T}}\right)=g_{c}\left(2^{t}-1\right)\),
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
\(G_{\mathrm{PT}}=\left(g_{e}-4\right)\left(2^{t-1}\right)\). 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 \(0.12 g_{\mathrm{c}} \mathrm{nm}+0.35 \mathrm{~nm}\). Show that the
volume of a particle, \(V_{5}\), is given by the equation
$$
V_{\mathrm{s}}=4 / 3 \pi t^{3}\left(0.12 g_{\mathrm{c}}+0.35\right)^{3}
\mathrm{~nm}^{3}
$$
Meléndez-Hevia and coauthors then determined the optimum values of \(t\) and
\(g_{c}\) - those that gave the maximum value of a quality function, \(f\), that
maximizes \(G_{\mathrm{T}}, C_{A}\), and \(G_{P T}\), while minimizing \(V_{8}:
f=\frac{G_{\mathrm{T}} C_{\mathrm{A}} G \mathrm{PT}}{V_{8}}\). They found that
the optimum value of \(g_{c}\) 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 \(g_{e}\) found in liver
glycogen (see Egg.15-2)? (Hint: You may find it useful to use a spreadsheet
program.)
