Question

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.)

Short answer

A liver cell would require 550 μM of free glucose to match the glycogen content. Low/good branching enhances/phosphorylase action; optimal \( g_c \) satisfies cell efficiency constraints.

Step-by-step solution

Understand cellular concentration requirement

To find the cellular concentration of free glucose equivalent to glycogen, note that the typical glycogen particle contains 55,000 glucose units, and the liver concentration of glycogen is 0.01 μM. Since each glycogen particle represents 55,000 glucose molecules, for free glucose, this would translate to a concentration of 0.01 μM × 55,000 due to the breakdown of each glycogen particle into glucose molecules.

Calculate equivalent free glucose concentration

Compute the equivalent free glucose concentration:\[\text{Free glucose concentration} = 0.01 \,\text{μM} \times 55,000 = 550 \, \text{μM}\] This extremely high concentration would lead to osmotic and metabolic problems as it is much higher than typical intracellular glucose concentrations.

Explain low degree of branching

In a hypothetical glycogen chain without branches, the enzyme glycogen phosphorylase can only act on one linear chain, causing slow glucose release. Hence, a low degree of branching results in fewer exposed ends for phosphorylase to work on, diminishing the rate of glucose release.

Explain high degree of branching

Although branching increases end points for phosphorylase action, too many branches lead to closely packed and structurally constrained chains, physically limiting the enzyme's accessibility and slowing down the glucose release.

Derive the formula for unbranched chains,\(C_A\)

Let's derive the formula \(C_A=2^{t-1}\):Each tier of glycogen adds branches exponentially, and the glucose chains follow a binary branching pattern starting with one initial chain, branching into two, and so forth. Hence, in tier \(t\), the number of endpoint chains is \(2^{t-1}\), reflecting the exponential growth pattern in tiers.

Derive the formula for total chains,\(C_T\)

For \(C_T=2^{t}-1\), consider the binary branching structure:- The initial tier (tier 1) has \(1\) chain,- Tier 2 will have \(2\) branches, tier 3 will have \(4\),- Thus, by recursively summing all chains across tiers,- Total chains = \(1 + 2 + 4 + \,\ldots\, + 2^{t-1} = 2^t - 1\).

Apply available glucose formula \(G_{PT}\)

The formula \(G_{PT}=(g_c-4)\times2^{t-1}\) reflects glucose available for phosphorylase action. Since phosphorylase cannot act on chains <5 residues long, subtracting 4 allows for producing split shorter chains from each available tier endpoint, calculated as \(2^{t-1}\).

Derive particle volume formula \(V_s\)

To obtain \(V_s = \frac{4}{3} \pi t^3 (0.12g_c + 0.35)^3\):Each tier has thickness \(0.12 g_c \text{ nm} + 0.35 \text{ nm}\) and contributes to sphere volume, expressing in a volumetric form where the number of tiers \(t\) dictates overall dimensions.

Determine optimum \(g_c\) value

Computing for optimal \(g_c\) with chosen \(t\) value solves \[f = \frac{G_T C_A G_{PT}}{V_s}\]By maximizing this relationship, iterate through values using a spreadsheet to find the optimal \(g_c\), typically corresponding to experimentally verified glycogen structures.

Key concepts

Glycogen Phosphorylase

One of the critical enzymes involved in glycogen breakdown is glycogen phosphorylase. This enzyme is responsible for cleaving glucose residues from glycogen polymer chains. Its action starts at one end of the glycogen chain and proceeds sequentially, releasing glucose molecules one after another.
However, there's a catch! Glycogen phosphorylase can only access and remove glucose from the outer branches of the glycogen molecule. Thus, the structure of glycogen - particularly its branching - plays a significant role in determining how effectively and quickly glucose can be released.
Branches in the glycogen molecule make more ends available for glycogen phosphorylase to work on. This increases the overall rate at which glucose can be mobilized, crucial for quick energy release during high-demand activities like exercise.
Nonetheless, this enzyme is limited in its action. It can only break down glucose chains longer than four residues, meaning branching and chain length directly influence the work efficiency of glycogen phosphorylase.

Glucose Release Rate

The rate at which glucose is released from glycogen is pivotal for energy regulation in cells. This rate largely depends on how readily glycogen phosphorylase can access glucose chains. More branching means more chain ends, which allows glycogen phosphorylase to remove glucose faster.
But, if the degree of branching is too low, the enzyme has fewer endpoints to access, slowing down glucose release. Conversely, too many branches could pack the chains so tightly that they become physically impenetrable for the enzyme, again reducing the glucose release rate.
Thus, there's an optimal degree of branching, which balances accessibility and structural integrity. This ensures maximum glucose release efficiency, providing an immediate energy source during physical exertion. The structural optimization found in naturally occurring glycogen reflects this balance perfectly.

Glycogen Branching

Branching in glycogen is a fascinating structural feature with significant biochemical implications. It refers to the presence of additional chains branching off the main glucose polymer chain. This unique structure results in a spherical shape of glycogen particles, with many terminal ends.
The presence of these branches is crucial because each branch point acts as a new starting point for glycogen phosphorylase. This significantly increases the number of access points for glucose release, enhancing the rapid mobilization of glucose.
A balanced level of branching allows for efficient packing and rapid availability while preventing excessive clustering or density that would impede enzymatic actions. The degree of branching found in physiological glycogen serves as an optimal structure for both storage volume and rapid energy release.

Glycogen Particle Calculations

Glycogen particles are not just random clusters of glucose; they follow a sophisticated structural pattern that affects their function. Each particle can be broken down into tiers, with calculations involving attributes like the number of tiers (\( t \)) and the chain length of glucose residues (\( g_{c} \)).
The number of chains available for glycogen phosphorylase before the intervening debranching is given by \( C_{A} = 2^{t-1} \). This relates to how many chains exist at the particle's surface layer, accessible for cleavage.
Similarly, the total number of chains within a glycogen particle, \( C_{T} \), can be calculated as \( 2^{t} - 1 \). Understanding these calculations allows researchers to predict how efficient a particle is at storing glucose.
To maximize the functional efficiency of glycogen, Meléndez-Hevia and colleagues used these calculations in combination with a formula that incorporates particle volume \( V_{s} \), aiming to find the optimum structure that balances storage and accessibility of glucose within the constraints of cellular space.