224 Chapter 7. Entropic forces at work[[Student version, January 17, 2003]]
the large particle to spend most of its time at the wall of the vesicle. By analyzing how much of its
time it spent there, the experimenters measured the free energy reduction when the particle was
sticking to the wall and quantitatively verified the estimate Equation 7.10 (appropriately modified
for a curved surface).
When we replace the image of sheep by large macromolecules, and sheepdogs by polymer coils
or small globular proteins, we see that the presence of the latter small objects can significantly help
the large macromolecules to find each others’ specific recognition sites. For example, introduction
of bovine serum albumin (BSA, a protein) or polyethylene glycol (PEG, a polymer) reduces the
solubility of deoxyhemoglobin and other large proteins by helping them to stick together; the
magnitude of the effect can be a 10-fold reduction of the solubility. PEG or dextran can also
stabilize complexes against thermal disruption, for instance increasing the melting temperature of
DNA by several degrees (see Chapter 9), or enhancing the association of protein complexes by an
order of magnitude or more. In all these examples we see the general theme that the entropic part
of a reaction’s free-energy change,−T∆S,isfully interchangeable with the other, more obvious
terms in ∆F,and so affects the reaction’s equilibrium point (see Section 6.6.3 on page 197).
Crowding can also speed up reactions, as the sheepdogs jockey the sheep into their best contact.
The presence of a “crowding agent” like PEG or BSA can increase the rate of self-assembly of actin
filaments, or the action of various enzymes, by orders of magnitude. We can interpret this result
in terms of free energy: The entropic contribution toFlowers an activation barrier to assembly
(see Section 6.6.2). Indeed some cellular equipment, for example the DNA replication system of
E. coli,just doesn’t work in vitro without some added crowding agent. As our simple physical
model predicts, it doesn’t matter too much what exactly we choose as our crowding agent—all that
matters are its size relative to the assembly and its number density.
It may seem paradoxical that the drive towarddisorder canassemblethings. But we must
remember that the sheepdogs are much more numerous than the sheep. If the assembly of a few big
macromolecules liberates some space for many smaller molecules to explore, then thetotaldisorder
of the system can go up, not down. In just the same way we will see later how another entropic
force, the hydrophobic interaction, can help direct the exquisitely organized folding of a protein or
the assembly of a bilayer membrane from its subunits.
7.3 Beyond equilibrium: Osmotic flow
The discussion of Section 7.2.1 illustrates the power, the beauty, and the unsatisfying feeling we
get from very general arguments. We found a quantitative prediction, which works in practice (see
Problem 7.2). But we are still left wonderingwhythere should be a pressure drop. Pressure involves
an honest, Isaac Newton–type force. Force is a transfer of momentum. But the argument given in
Section 7.2.1 makes no mention of momentum, and instead just manipulates entropy (or disorder).
Where exactly does the force come from? How does a change inordertransmute into a flow of
momentum?
Wemet an analogous situation in the context of the ideal gas law: The result obtained abstractly
in Section 7.1 would not have been very convincing had we not already given a more concrete, albeit
less general, argument in Chapter 3. We need the abstract viewpoint because it can take us safely
into situations so complicated that the concrete view obscures the point. But whenever possible
weshouldalsoseek concrete pictures, even if they’re very simplified. Accordingly this section will
revisit osmotic pressure, developing a simplified dynamical view of the van ’t Hoff relation. As a