Bioenergetics and Enzymes: Free Energy Changes

Gibbs Free Energy (G) and Free Energy Change ($\Delta G$)

• Gibbs Free Energy (G): Represents the energy available to do work within a system.

  • Equation: $G = H - TS$

    • $H$: Enthalpy (chemical energy).

    • $T$: Absolute Temperature (in Kelvin).

    • $S$: Entropy (a measure of disorder).

• Free Energy Change ($\Delta G$): More often, we are interested in the change in Gibbs free energy associated with a process, rather than its absolute value.

  • Equation: $\Delta G = \Delta H - T\Delta S$

    • $\Delta H$: Change in enthalpy.

    • $\Delta S$: Change in entropy.

• Significance of $\Delta G$: This value is crucial because it indicates:

  1. Distance from Equilibrium: The magnitude of $\Delta G$ tells us how far a system is from equilibrium.

  2. Direction of Progress: It predicts what will need to occur for the system to return to equilibrium.

• Applicability: $\Delta G$ can be calculated for various processes, including molecule movements and chemical reactions, though the specific formulas may vary.

Energetically Favorable vs. Unfavorable Processes

Processes are categorized based on their $\Delta G$ value:

Positive $\Delta G$ (Endergonic) - "Uphill" Process

• Definition: The free energy at the end of a process is greater than the free energy at the beginning (G{final} > G{initial}).

• Characteristics:

  • Indicates a positive $\Delta G$.

  • Requires energy input into the system.

  • Also termed endergonic (from Greek "ender" meaning "in").

  • Energetically unfavorable.

  • Non-spontaneous: Will not occur without an external energy input.

• Graphical Representation: An energy profile where the products/final state are at a higher energy level than the reactants/initial state.

Negative $\Delta G$ (Exergonic) - "Downhill" Process

• Definition: The free energy at the end of a process is less than the free energy at the beginning (G{final} < G{initial}).

• Characteristics:

  • Indicates a negative $\Delta G$.

  • Energy is released from the system (products have less energy, so the excess energy leaves).

  • Also termed exergonic (from Greek "exerg" meaning "out").

  • Energetically favorable.

  • Spontaneous: Will occur without any additional energy input because it inherently releases energy. This does not mean it will happen instantly or randomly, but rather that no external energy boost is required.

• Graphical Representation: An energy profile where the products/final state are at a lower energy level than the reactants/initial state.

$\Delta G$ in Molecule Movements Across a Membrane

When considering the movement of molecules (e.g., ions, solutes) across a cellular membrane, the $\Delta G$ reflects the free energy change associated with that movement, primarily driven by concentration differences and entropy.

• Context: This calculation does not typically involve the $\Delta H$ term as it's not a chemical bond-breaking/forming reaction, but focuses on entropy changes due to concentration gradients.

• Equation for Molecule Movement into the Cell: $\Delta G<em>{in} = RT \ln \left( \frac{[X]</em>{in}}{[X]_{out}} \right)$

  • $R$: Gas constant.

  • $T$: Absolute temperature (in Kelvin).

  • $[X]_{in}$: Concentration of molecule $X$ inside the cell.

  • $[X]_{out}$: Concentration of molecule $X$ outside the cell.

Scenarios:

1. System at Equilibrium ($\Delta G_{in} = 0$):

   • Occurs when concentrations are equal inside and outside the cell: $[X]<em>{in} = [X]</em>{out}$.

   • Thus, $\frac{[X]<em>{in}}{[X]</em>{out}} = 1$. The natural logarithm of 1 is 0 ($\ln(1) = 0$).

   • Result: $\Delta G_{in} = 0$. No free energy is available to do work, and there is no net movement.

2. System Not at Equilibrium:

   • Movement Towards Equilibrium (Favorable, Negative $\Delta G$):

     • Example (Blue Rectangles): If [X]{out} > [X]{in}, movement of molecules into the cell brings the system closer to equilibrium.

       • E.g., if $[X]<em>{in} = 1$ and $[X]</em>{out} = 4$, then $\frac{[X]<em>{in}}{[X]</em>{out}} = \frac{1}{4}$. $\ln(1/4)$ is a negative number, resulting in a negative $\Delta G_{in}$.

       • Energy is released; this process can be harnessed by the cell to do work.

   • Movement Away From Equilibrium (Unfavorable, Positive $\Delta G$):

     • Example (Red Circles): If [X]{in} > [X]{out}, movement of molecules into the cell moves the system further away from equilibrium.

       • E.g., if $[X]<em>{in} = 4$ and $[X]</em>{out} = 1$, then $\frac{[X]<em>{in}}{[X]</em>{out}} = \frac{4}{1}$. $\ln(4)$ is a positive number, resulting in a positive $\Delta G_{in}$.

       • Energy input is required for this movement to occur.

Rules of Thumb for Molecule Movements:

• Molecule movements towards equilibrium conditions are favorable (negative $\Delta G$).

• Molecule movements away from equilibrium conditions are unfavorable (positive $\Delta G$).

• Magnitude of $\Delta G$: The further a system is from equilibrium, the greater the magnitude of the $\Delta G$ for movements that would bring it towards equilibrium (more energy released) or away from equilibrium (more energy required).

  • Example: If the concentration gradient of red circles is even steeper ($[X]<em>{in}$ much greater than $[X]</em>{out}$), trying to move red circles into the cell becomes even more unfavorable (a larger positive $\Delta G_{in}$) because it goes against an even stronger