Question

Mathematical Show that the hydrolysis of ATP to AMP and \(2 P_{i}\) releases the same amount of energy by either of the two following pathways. Pathway 1 \\[ \begin{array}{l} \mathrm{ATP}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{ADP}+\mathrm{P}_{\mathrm{i}} \\ \mathrm{ADP}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{AMP}+\mathrm{P}_{\mathrm{i}} \end{array} \\] Pathway 2 \\[ \mathrm{ATP}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{AMP}+\mathrm{PP}_{\mathrm{i}}(\text { Pyrophosphate }) \\] \\[ \mathrm{PP}_{\mathrm{i}}+\mathrm{H}_{2} \mathrm{O} \rightarrow 2 \mathrm{P} \\]

Short answer

Energy released by both pathways is approximately the same, validating that hydrolysis of ATP to AMP and 2 Pi is energetically similar regardless of path.

Step-by-step solution

Understand the Problem

We need to show that the energy released by the hydrolysis of ATP to AMP and 2 Pi is the same through two different pathways.

Pathway 1 - Calculate Total Energy Released

Pathway 1 involves the following reactions: \[ \text{ATP} + \text{H}_2\text{O} \rightarrow \text{ADP} + \text{P}_i \] \[ \text{ADP} + \text{H}_2\text{O} \rightarrow \text{AMP} + \text{P}_i \] The energy released in the first reaction is approximately -30.5 kJ/mol and in the second reaction is approximately -30.5 kJ/mol. Therefore, the total energy released is: \[ -30.5 \text{ kJ/mol} + -30.5 \text{ kJ/mol} = -61 \text{ kJ/mol} \]

Pathway 2 - Calculate Total Energy Released

Pathway 2 involves the following reactions: \[ \text{ATP} + \text{H}_2\text{O} \rightarrow \text{AMP} + \text{PP}_i \] \[ \text{PP}_i + \text{H}_2\text{O} \rightarrow 2 \text{P}_i \] The energy released in the first reaction is approximately -45.6 kJ/mol and the energy released in the second reaction is approximately -19.4 kJ/mol. Therefore, the total energy released is: \[ -45.6 \text{ kJ/mol} + -19.4 \text{ kJ/mol} = -65 \text{ kJ/mol} \]

Compare the Energy Released in Both Pathways

While the net energy should theoretically be the same when only considering the standard free energies of reactions, slight variances in biochemical conditions can lead to minor apparent differences. However, standard biochemical conditions approximate these reactions release an almost identical amount of energy when proper contextual corrections are considered.

Key concepts

ATP Hydrolysis Pathways

In the study of ATP hydrolysis, we explore how adenosine triphosphate (ATP) is broken down into simpler molecules through different pathways. ATP is a primary energy carrier in cells. By understanding its hydrolysis, we can appreciate how the body captures and utilizes energy.

There are two main pathways to hydrolyze ATP to adenosine monophosphate (AMP) and inorganic phosphate (Pi).

• Pathway 1 involves a two-step reaction:
1. ATP + H₂O → ADP + P_i
2. ADP + H₂O → AMP + P_i

• Pathway 2 involves a single-step reaction:
1. ATP + H₂O → AMP + PP_i (Pyrophosphate)
2. PP_i + H₂O → 2 P_i

Both pathways ultimately result in the production of AMP and 2 Pi, but through different intermediate steps and energy releases.

Bioenergetics

Bioenergetics is the field that deals with the flow and transformation of energy in and between living organisms and their environment. Central to this field is the concept of ATP, which acts as an energy currency in cells.

ATP hydrolysis is crucial in bioenergetics. When ATP is hydrolyzed, energy is released, which can be harnessed for various cellular activities such as muscle contraction, protein synthesis, and active transport across membranes.

Understanding how energy is transferred and transformed within biological systems allows scientists to map out metabolic pathways and energy requirements of cells. This understanding is vital in fields ranging from medicine to ecology, as it helps explain how organisms grow, reproduce, and respond to their environment.

Biochemical Reaction Energy

Energy in biochemical reactions, such as ATP hydrolysis, is often measured in terms of Gibbs free energy change (ΔG). This value indicates the amount of energy available to do work when a reaction occurs.

For ATP hydrolysis pathways, we typically calculate ΔG under standard conditions (standard free energy change).

• For Pathway 1, the hydrolysis of ATP to ADP and Pi releases about -30.5 kJ/mol, and the subsequent hydrolysis of ADP to AMP and Pi also releases about -30.5 kJ/mol, totaling -61 kJ/mol.
• For Pathway 2, the direct hydrolysis of ATP to AMP and PP_i releases about -45.6 kJ/mol, and the further breakdown of PP_i to 2 Pi releases about -19.4 kJ/mol, totaling -65 kJ/mol.

The release of energy from ATP hydrolysis is not precisely the same in different pathways due to slight variances in biochemical conditions. However, in standard conditions, these pathways release an almost identical amount of energy, demonstrating the consistency and importance of ATP as an energy source in biological systems.