The Energetics of Life: Thermodynamics and Free Energy in Biological Systems

3.1 Free Energy

Thermodynamic Systems

Thermodynamics provides the framework for understanding energy changes in biological systems. A system is any part of the universe chosen for study, while everything else is the surroundings. Systems are characterized by their composition, temperature, pressure, and volume.

• Closed systems: Exchange energy (heat or work) but not matter with surroundings.

• Open systems: Exchange both matter and energy with surroundings (e.g., living cells).

• Isolated systems: Exchange neither matter nor energy with surroundings.

• Living systems generally operate at constant temperature and pressure.

The First Law of Thermodynamics and Enthalpy

The first law of thermodynamics states that energy is conserved. In a closed system, energy can be exchanged with the surroundings as work or heat. Enthalpy (H) is the heat exchanged at constant pressure, a typical condition for living systems.

• Mathematically, enthalpy change ($\Delta H$) is the heat ($Q$) at constant pressure.

Reversible and Irreversible Processes

• Reversible process: Occurs near equilibrium; forward and reverse rates are equal (e.g., ice melting at 0°C).

• Irreversible process: Proceeds far from equilibrium toward equilibrium (e.g., burning paper).

The Second Law of Thermodynamics and Entropy

The second law of thermodynamics states that the entropy of an isolated system tends to increase to a maximum value. Entropy (S) measures the degree of randomness or disorder in a system.

• High entropy: More disorder (e.g., water vapor).

• Low entropy: More order (e.g., ice).

Lower Entropy	Higher Entropy
Ice, at 0°C	Water, at 0°C
Water, at 10°C	Water vapor, at 10°C (fog)
Unblended yogurt, bananas, honey, strawberries	Fruit smoothie (same ingredients blended)

3.2 Free Energy: The Second Law in Open Systems

Cellular Systems as Open Systems

Living systems are open, not isolated. The second law for open systems involves both enthalpy and entropy:

• $-T\Delta S_{universe} = -T\Delta S_{system} + \Delta H_{system}$

• Where $T$ is absolute temperature in Kelvin.

The Gibbs free energy (G) is defined as:

• $G = H - TS$

• $\Delta G = \Delta H - T\Delta S$

$\Delta G$ represents the energy available to do useful work at constant temperature and pressure.

Free Energy and Process Direction

If $\Delta G$ is...	Free energy is...	The process is...
Negative	Available to do work	Thermodynamically favorable (exergonic)
Zero	Zero	At equilibrium (reversible)
Positive	Required to do work	Thermodynamically unfavorable (endergonic)

Interplay of Enthalpy and Entropy

Whether a process is favorable depends on both $\Delta H$ and $\Delta S$:

• Example: Ice melting at 263 K ($\Delta H = +5630$ J/mol, $\Delta S = +20.6$ J/K·mol)

• $\Delta G = \Delta H - T\Delta S = 5630 - (263 \times 20.6) = +212$ J/mol (unfavorable)

$\Delta H$	$\Delta S$	Low T	High T
+	+	Not favored	Favored
+	-	Not favored	Not favored
-	+	Favored	Favored
-	-	Favored	Not favored

3.3 Relationships Between Free Energy, Equilibrium, and Nonequilibrium Concentrations

Chemical Reactions at Equilibrium

For a general reaction $aA + bB \rightleftharpoons cC + dD$:

• The equilibrium constant (K) is $K = \frac{[C]^c[D]^d}{[A]^a[B]^b}$

Chemical Reactions at Any State

The reaction quotient (Q) is defined similarly to $K$ but for any set of concentrations:

• $Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}$

Changes in Concentration and $\Delta G$

• $\Delta G = \Delta G^\circ + RT \ln Q$

• $\Delta G^\circ$ is the standard free energy change (all solutes at 1 M).

• At equilibrium, $Q = K$ and $\Delta G = 0$.

• Thus, $\Delta G^\circ = -RT \ln K$ and $K = e^{-\Delta G^\circ/RT}$

Example: Glucose-6-phosphate to Fructose-6-phosphate

• G6P $\rightarrow$ F6P, $\Delta G^\circ = +1.7$ kJ/mol

• Calculation of $K$ using $K = e^{-\Delta G^\circ/RT}$

Equilibrium versus Homeostasis

• Homeostasis: Maintenance of constant conditions (temperature, pH, ion concentrations) in living systems.

• In both equilibrium and homeostasis, concentrations are constant, but in homeostasis, they are maintained far from equilibrium values (requires energy input).

• In homeostasis, $Q$ is maintained far from $K$ so $\Delta G \neq 0$.

Value of Q	Value of $\Delta G$	Favored Direction
< K	< 0	Forward reaction (products form)
= K	= 0	Neither (equilibrium)
> K	> 0	Reverse reaction (reactants form)

Coupling Unfavorable and Favorable Reactions

• Cells couple favorable ($\Delta G^\circ < 0$) and unfavorable ($\Delta G^\circ > 0$) reactions to drive necessary processes forward.

• Example: $A \rightarrow B$ ($\Delta G^\circ = +10$ kJ/mol, unfavorable), $C \rightarrow D$ ($\Delta G^\circ = -30$ kJ/mol, favorable), overall $A + C \rightarrow B + D$ ($\Delta G^\circ = -20$ kJ/mol, favorable).

• This coupling is essential for driving chemical reactions, active transport, nerve impulses, and muscle contraction.

3.4 Free Energy in Biological Systems

The Biochemical Standard State

• Biochemical reactions often involve $H_2O$ and $H^+$, which are not at 1 M in cells.

• Standard biochemical conditions ($\Delta G'^\circ$): $[H^+] = 10^{-7}$ M (pH 7), activity of $H_2O$ defined as 1.

• For ATP hydrolysis: $\Delta G = \Delta G'^\circ + RT \ln \left( \frac{[ADP][HPO_4^{2-}][H^+]}{[ATP][H_2O]} \right)$

Calculation Example: ATP Hydrolysis

• Given: [ATP] = 5 mM, [ADP] = 0.1 mM, [HPO$_4^{2-}$] = 35 mM, pH 7.4, 25°C

• $\Delta G'^\circ$ for ATP hydrolysis = -32.2 kJ/mol

• Calculation: $\Delta G = -32.2 + 2.478 \ln \left( \frac{(0.0001)(0.035)(0.398)}{0.005} \right) = -52.5$ kJ/mol

• The actual $\Delta G$ is much more negative than $\Delta G'^\circ$ due to cellular concentrations.

ATP as the Common Energy Currency

• ATP (adenosine triphosphate) is the universal energy currency in cells.

• Hydrolysis of ATP, ADP, and AMP releases energy used to drive cellular processes.

Hydrolysis of Cellular Phosphate Compounds

Different phosphate compounds have varying standard free energies of hydrolysis ($\Delta G'^\circ$):

Hydrolysis Reaction	$\Delta G'^\circ$ (kJ/mol)
Phosphoenolpyruvate + H2O → pyruvate + Pi	-61.9
1,3-Bisphosphoglycerate + H2O → 3-phosphoglycerate + Pi	-49.4
ATP + H2O → AMP + PPi + H+	-45.6
ATP + H2O → ADP + Pi + H+	-32.2
Glucose-6-phosphate + H2O → glucose + Pi	-13.8

Free Energy and Phosphoryl Group Transfer Potential

• Some compounds (e.g., phosphoenolpyruvate, PEP) have higher phosphoryl group transfer potential than ATP.

• PEP hydrolysis is highly exergonic due to resonance stabilization, charge repulsion, and tautomerization of products.

• High transfer potential compounds can be used to synthesize ATP from ADP in coupled reactions.

Free Energy and Concentration Gradients Across Membranes

• The free energy change for moving a solute across a membrane is:

$\Delta G = RT \ln \left( \frac{[A]_2}{[A]_1} \right)$

• If $[A]_1 < [A]_2$, $\Delta G$ is negative (favorable transfer from region 1 to 2).

• If $[A]_1 > [A]_2$, $\Delta G$ is positive (unfavorable transfer).

• If $[A]_1 = [A]_2$, $\Delta G = 0$ (equilibrium).

Chapter Summary

• Thermodynamics explains how living systems extract and use energy.

• Processes are spontaneous if $\Delta G$ is negative (exergonic).

• Cells maintain homeostasis, operating far from equilibrium, with $\Delta G = \Delta G^\circ + RT \ln Q$.