(Integrates with Chapter 3 .) Fructose is present outside a cell at \(1 \mu M\) concentration. An active transport system in the plasma membrane transports fructose into this cell, using the free energy of ATP hydrolysis to drive fructose uptake. What is the highest intracellular concentration of fructose that this transport system can generate? Assume that one fructose is transported per ATP hydrolyzed; that ATP is hydrolyzed on the intracellular surface of the membrane; and that the concentrations of ATP, ADP, and \(P_{i}\) are \(3 \mathrm{m} M, 1 \mathrm{m} M,\) and \(0.5 \mathrm{m} M,\) respectively. \(T=298 \mathrm{K}\). (Hint: Refer to Chapter 3 to recall the effects of concentration on free energy of ATP hydrolysis.)

ATP (adenosine triphosphate) hydrolysis is a fundamental biological process where energy is released for cellular functions. This process involves breaking the high-energy phosphate bond in ATP to form ADP (adenosine diphosphate) and an inorganic phosphate (\( P_i \)). It's akin to withdrawing money from your energy bank. The energy released during this reaction, often referred to as \( \-\Delta G \) or Gibbs free energy change, powers a vast array of cellular activities, including active transport.

For instance, a transport protein may harness this energy to move substances against their concentration gradient, which is critical for maintaining cellular homeostasis. Understanding ATP hydrolysis is essential as it underpins many physiological processes including muscle contraction, nerve impulse propagation, and, crucially, membrane transport systems.

Membrane transport systems are like the entry and exit doors of a cell, regulating what comes in and goes out. These systems encompass a range of proteins that help transport materials across the cell's plasma membrane. They ensure substances such as nutrients, ions, and waste products are moved efficiently, even in the face of a concentration gradient.

Active transport mechanisms require energy to function, often sourced from ATP hydrolysis. They are designed to move compounds from areas of low concentration to those of high concentration, which is the opposite direction favored by natural diffusion. In the context of the exercise, the fructose uptake mechanism represents a specific type of membrane transport system.

The fructose uptake mechanism is a classic illustration of active transport, where fructose, a sugar, is moved from lower to higher concentration inside a cell. This movement goes against the natural diffusion tendency and therefore requires an external energy source which, as the exercise suggests, is ATP.

The transport protein involved in this scenario couples the hydrolysis of ATP directly to the transport of fructose. Each ATP molecule that is hydrolyzed allows the protein to change its shape, effectively pushing a fructose molecule from the outside to the inside of the cell. This process is essential for cells that need higher concentrations of fructose internally than their environment provides, such as liver and intestinal cells.

The intracellular and extracellular concentration gradients refer to the difference in the concentration of substances inside and outside the cell. These gradients are vital for the direction and force of various transport processes. Think of them as a hill: substances naturally roll down the gradient, just like objects roll down a hill due to gravity.

In active transport, however, cells have to move substances 'uphill,' which is against their concentration gradient. They do this using energy from ATP, mimicking a situation where one would use additional energy to push an object up a hill. Gradients are essential in determining not just the direction but also the rate of transport across the membrane, making them a key concept in understanding how cells maintain balance in their environment.

Free energy change calculation (\( -\Delta G \) calculation) is akin to measuring the 'energy cost' for a particular reaction or process. In the context of cell biology, calculating the free energy change associated with ATP hydrolysis allows us to predict how much work can be performed by the released energy.

In the exercise, this calculation helps to understand the highest possible intracellular fructose concentration that can be achieved by the active transport system. It uses the known concentrations of ATP, ADP, and \( P_i \) and incorporates them into a formula that includes the universal gas constant (\( R \) and temperature (\( T \) in Kelvin. By computing \( -\Delta G \), we can link the energetics of ATP hydrolysis to the potential for fructose transport into the cell, emphasizing the pivotal role of energy in cellular transport mechanisms.