ENERGETICS OF SELF-ASSEMBLY
If A represents a monomer of a multimeric assembly, the reaction n A An, when considered in its spontaneous direction, will have a negative delta G. From delta G = delta H - T delta S it is clear that this negativity is the result of the signs and magnitudes of the delta H and delta S for the reaction and of the temperature. A negative delta H (a loss of potential energy, as when two opposite charges get closer) contributes to a negative delta G. So does a positive delta S (a loss of order in the system). Since, in the assembly, the protein subunits are in a certain order, the delta S for its formation should be expected to be negative. In such a case, the only way to get a negative delta G is to have a negative delta H of such a magnitude that adding it to the positive value of -T delta S yields a negative delta G. This negativity of delta H results from the formation of weak bonds. It happens, though, that measurements of delta H for the self-assembly of many proteins and large assemblies yields positive values, which means that, rather than forming bonds, the assemblage is either breaking bonds or bringing equal electrical charges closer to each other. Then, the only way to get a negative delta G is to have a positive delta S of such a magnitude that -T delta S added to the positive delta H will yield a negative delta G. This may be summarized as follows: in the first case when delta H is negative (enthalpy driven process), |delta H| > -T delta S, and in the second case when delta H is positive (entropy driven process), delta H < |-T delta S|.
Since we had concluded before that the delta S of assembly for the protein is negative, we must look for another component of the system that may show a large positive delta S. This component is water, whose inability to form hydrogen bonds with hydrophobic parts of the protein renders it highly structured in these environments. On the other hand, in bulk fluid, the H-bonds between adjacent water molecules have a half life of about 10-10 sec. Any one molecule is constantly breaking its H-bonds and reforming them with other nearby molecules. The detailed structure of the system is constantly changing from one state to another, and the entropy content (disorder) is large. Since the number of bonds breaking is equal to the number of those forming, the delta H for this process is zero and since the disorder remains the same, so is delta S.
If the hydrophobic region of a molecule is introduced in the water, the adjacent water molecules cannot reform any broken H-bonds with the hydrophobic surface. Therefore, the process of breaking them to allow rotation of the molecules would have a large positive delta H and no entropic gain, so it would not be spontaneous. What happens then is that the energy spent by sources external to the system (the surroundings) in introducing the hydrophobic surface (as when a mixture of oil and water is shaken) is used by the system in lowering its entropy content. As soon as the external input of energy ceases, the state of the system becomes unstable and, if a process by which the system may reduce its G content is available, it will happen. This implies that the system will seek to reduce its G content either by reducing its energy (negative delta H), increasing its entropy (positive delta S) or both. In the case of hydrophobic molecules in water, the association of the molecules will squeeze water away from the hydrophobic surfaces thereby increasing water entropy and, consequently, the entropy of the system.
The temperature dependance of self-assembly processes usually indicates to which of the two kinds (entropy driven or enthalpy driven) belongs the one under study. If an increase in temperature shifts the equilibrium towards disassociation, the process is enthalpy driven because the more energetic thermal agitation of the molecules tends to destabilize the bonds that keep the molecules together. If an increase in temperature shifts the equilibrium towards association, the process is entropy driven because the product T delta S increases in magnitude and drives delta G to a larger negative value.
Example 1.
Consider again the reaction nA An . In this case the aggregation happened through weak bond formation and the following values are obtained: delta H = -5KJ mol-1, delta S = -10J mol-1deg-1. At T = 310 deg K,
delta G = -5000 J mol-1 - 310 deg K x (-10 J mol-1 deg K-1) = -1900 J mol-1
At T = 290 deg K,
delta G = -5000 J mol-1 - 290 deg K x (-10 J mol-1 deg K-1) = -2100 J mol-1
Example 2.
In this example the aggregate is stabilized through hydrophobic interactions and the following values are obtained: delta H = 5 KJ mol-1, delta S = 17 J mol-1 deg K-1.
At T = 310 deg K,
delta G = 5000 J mol-1 - 310 deg K x (17 J mol-1 deg K-1) = -270 J mol-1
At T = 290 deg K.
delta G = 5000 J mol-1 - 290 deg K x (17 J mol-1 deg K-1) = 70 J mol-1.
Since at this temperature the forward reaction has a positive delta G, the backward reaction will have a negative delta G and be spontaneous. Consequently, the aggregate dissociates at low temperature.
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