Problem 17 - Biochemistry: Concepts and Connections

Question

The \(\mathrm{Na}^{+} /\)glucose symport transports glucose from the lumen of the small intestine into cells lining the lumen. Transport of 1 glucose molecule is directly coupled to the transport of \(1 \mathrm{Na}^{+}\)ion into the cell. $$ 1 \mathrm{Na}_{\text {out }}^{+}+1 \text { glucose }_{\text {out }} \rightarrow 1 \mathrm{Na}^{+} \text {in }+1 \text { glucose }_{\text {in }} $$ Assume the following conditions at \(37{ }^{\circ} \mathrm{C}:\left[\mathrm{Na}^{+}\right]_{\text {in }}=12 \mathrm{mM},\left[\mathrm{Na}^{+}\right]_{\text {out }}=\) \(145 \mathrm{mM}\), [glucose \(]_{\text {out }}=28 \mu \mathrm{M}\), and \(\Delta \psi=-72 \mathrm{mV}\) (inside negative). (a) What is \(\Delta G\) for transport of \(\mathrm{Na}^{+}\)from outside to inside under these conditions? (b) What is the upper limit for [glucose \(]_{\text {in }}\) under these conditions? (c) Which of the two hypothetical symports shown below (A or \(\mathbf{B}\) ) would achieve the highest concentration of [glucose] in under the conditions described above? Briefly explain your choice. A: \(1 \mathrm{Na}_{\text {out }}^{+}+2\) glucose \(_{\text {out }} \rightarrow 1 \mathrm{Na}^{+}\)in \(+2\) glucose \(_{\text {in }}\) B: \(2 \mathrm{Na}_{\text {out }}^{+}+1\) glucose \(_{\text {out }} \rightarrow 2 \mathrm{Na}^{+}\)in \(+1\) glucose \(_{\text {in }}\)

Step-by-Step Solution

Answer

(a) Calculate \( \Delta G_{total} \) with both chemical and electrical potential terms; (b) Use equilibrium condition for glucose; (c) Symporter B achieves higher \([\text{glucose}]_{\text{in}}\).

Step 1: Determine the chemical potential for Na+

Calculate the chemical potential for Na+ moving into the cell. Use the formula: \[ \Delta G_{chem} = RT \ln \left( \frac{[\text{Na}^+]_{\text{in}}}{[\text{Na}^+]_{\text{out}}} \right) \] where \( R \) is the gas constant (8.314 J/mol·K), \( T \) is the temperature in Kelvin (310 K), \([\text{Na}^+]_{\text{in}} = 12 \text{ mM}\) and \([\text{Na}^+]_{\text{out}} = 145 \text{ mM}\).

Step 2: Calculate the electrical potential contribution

The electrical potential contribution to \( \Delta G \) can be calculated using: \[ \Delta G_{elec} = ZF\Delta \psi \] where \( Z \) is the charge of Na+ (+1), \( F \) is Faraday's constant (96485 C/mol), and \( \Delta \psi = -72 \text{ mV} = -0.072 \text{ V}\).

Step 3: Sum chemical and electrical potential

Combine the chemical potential and electrical potential to find \( \Delta G \) for Na+: \[ \Delta G_{total} = \Delta G_{chem} + \Delta G_{elec} \]. Calculate both contributions and find the sum.

Step 4: Determine conditions for glucose equilibrium

For glucose transport, at equilibrium \( \Delta G = 0 \). Use the equation: \[ RT \ln \left( \frac{[\text{glucose}]_{\text{in}}}{[\text{glucose}]_{\text{out}}} \right) = -\Delta G_{total} \] to solve for \([\text{glucose}]_{\text{in}}\) using the previously calculated \( \Delta G_{total} \) and \([\text{glucose}]_{\text{out}} = 28 \text{ \mu M}\).

Step 5: Analyze symporters A and B

Symporter B transports more Na+ ions per glucose molecule, utilizing the larger negative \( \Delta G_{total} \) to drive a higher \([\text{glucose}]_{\text{in}}\). Therefore, \([\text{glucose}]_{\text{in}}\) can be maximized using symporter B. Compare the driving power of each symporter to rationalize why symporter B is more efficient.

Key Concepts

Chemical Potential

The concept of chemical potential is a fundamental idea in biology, especially when discussing the movement of ions like sodium (Na⁺). In essence, the chemical potential acts as a driving force for the movement of substances across cell membranes. For sodium ions moving from outside to inside the cell, their concentration outside is much higher than inside. This difference creates a potential energy difference, pushing the sodium ions to move inward.

To quantify this chemical potential for Na⁺, we use the formula:

• \[\Delta G_{chem} = RT \ln \left( \frac{[\text{Na}^+]_{\text{in}}}{[\text{Na}^+]_{\text{out}}} \right)\]

Where \(R\) is the gas constant (8.314 J/mol·K), and \(T\) is the temperature in Kelvin (310 K for 37°C). This equation helps us calculate the energy change associated with moving sodium ions from outside the cell, where they are more concentrated, to inside the cell, where their concentration is lower.

Such a calculation gives insight into the potential energy that can be harnessed during the process of active transport via symporters.

Electrical Potential

The electrical potential across a cell membrane, often described as the membrane potential, plays a crucial role in ion transport. In our scenario, Na⁺ ions experience an electrical potential difference as they move from outside the cell to inside. This membrane potential is typically negative on the inside, contributing an attractive force for positive ions like Na⁺ to enter the cell.

The electrical potential energy change can be calculated with:

• \[\Delta G_{elec} = ZF\Delta \psi\]

In this equation, \(Z\) represents the charge of the ion (+1 for Na⁺), \(F\) is Faraday's constant (96485 C/mol), and \(\Delta \psi\) is the membrane potential difference (-0.072 V or -72 mV).

Combining the effect of chemical and electrical potential is essential in understanding the total energy change, \(\Delta G_{total}\), which dictates whether the overall transport process requires or releases energy.

Glucose Transport

The transport of glucose into cells via the Na⁺/glucose symporter is an important example of secondary active transport. Here, the energy stored in the sodium ion gradient is used to "pull" glucose molecules into the cell against their concentration gradient.

The symporter exploits the favorable \(\Delta G_{total}\) (both chemical and electrical potentials of Na⁺) to assist glucose transport. At equilibrium, the energy change for glucose is zero, meaning the inward and outward flow is balanced. We can find the internal concentration of glucose using:

• \[RT \ln \left( \frac{[\text{glucose}]_{\text{in}}}{[\text{glucose}]_{\text{out}}} \right) = -\Delta G_{total}\]

This equation allows us to determine the maximum [glucose] inside the cell that can be maintained using the energy from sodium transport. Understanding this balance is critical, especially when comparing different symporter models, like hypothetical symporters A and B.

Symporter B, which utilizes a higher number of Na⁺ ions per glucose, can drive higher glucose concentrations inside due to a larger contribution from \(\Delta G_{total}\), making it more efficient under the given conditions.