Biology makes use of the principles of self-assembly, where well defined structures are formed without detailed atom-by-atom control of the fabrication. In this sense, proteins would not be classed as a self-assembled system, since the residual groups on the protein chain are synthesized and chemically bound deterministically and one by one. A handful of proteins (principally actin and tubulin) are the components of the cell cytoskeleton, the structure that provides mechanical integrity to cells. These filaments are formed by the polymerisation of single proteins that join up in a linear structure. This process is dynamic (i.e. at any time a filament may be growing or shrinking, and the detailed balance depends on temperature, chemical environment and physical confinement).

Other examples of self-assembly are related to molecules (mostly lipids, but also some specific proteins) that have one hydrophilic end and one hydrophobic end. These are called amphiphilic, due to the tendency to partition at the interface between an aqueous and an organic phase. Molecules with these characteristics will tend to aggregate spontaneously, and may give rise to a variety of aggregate shapes, depending on the molecular structure and chemical environment. An everyday example of this is the protein β-casein, present in milk, which assembles to form compact, spherical micelles. Its amphiphilic character also makes it effective at coating particles of fat, enabling small fat globules to remain stable in suspension.

This lecture is about the self-assembly of micelles (Slide 1): aggregates of molecules that form spheres, cylinders, sheets or filaments. Their formation and concentration can be described in terms of statistical thermodynamic functions, such as chemical potential and free energy.

Slide 1 The formation of micelles and bilayers can be described in terms of statistical thermodynamic functions, such as chemical potential and free energy. (Copyright E Perez, Laboratoire IMRCP, Toulouse.)

### 3.1 Amphiphiles

Micelles form from short "amphiphilic" chain molecules (Slide 2) having a hydrophilic ("water-loving") polar head group and a hydrocarbon chain tail, which is hydrophobic ("water hating"); amphiphilic means "both loving". These amphiphiles have low solubility in both water and oil, but will readily adsorb to an interface between water and oil (or water and air), orienting themselves to optimise interactions; molecules that behave in this way are known as surfactants. Naturally occurring amphiphiles are usually called lipids (Slide 3). There are also many synthetic surfactants, such as industrial detergents.

Slide 3 Schematic, chemical and physical structures of a phospholipid molecule. Copyright © 1994 From Molecular Biology of the Cell by Bruce Alberts, et al. Reproduced by permission of Garland Science/Taylor & Francis Books, Inc.

The basic driving force behind self-assembly processes is the minimisation of the free energy, by minimising the degree of mixing between the hydrophobic tails with water, while keeping the hydrophilic head groups in water, forming either a micelle or a bilayer (Slide 4). The lipid bilayer is the basic structure of a cell membrane.

An animation that models the formation of micelles can be seen at the Concord Consortium website:

http://mw.concord.org/modeler1.3/mirror/biology/micelle.html

For the lipid molecules to organise into a micelle or bilayer implies a significant loss of entropy. This can only occur if other changes in the system lead to an overall decrease in free energy. It might be thought that this arises from enthalpic interactions, but in fact this is not so, or at least it is not dominant. The driving force arises from the entropy of the solvent molecules. In the unassociated state, the lipid molecules reduce the entropy of the solvent molecules as water organises around each individual lipid.

From the schematic diagram on Slide 5, it can be seen how grouping solute molecules can increase the entropy of solvent molecules. This entropic driving force due to a gain in water molecules is known as the hydrophobic force. It is very important, not just for lipids but for protein structures too. It can therefore be seen as a key component of self-assembly in biological systems.

### 3.2 Free energy for surfactants

As the concentration of an amphiphile in solution increases, the solution will tend to separate into phases due to the low solubility of the amphiphiles. However, there is an alternative to complete phase separation. The amphiphiles can associate to form micelles, which are supramolecular aggregates in which the amphiphiles are packed to maximise the number of favourable interactions. The shape and size of the "aggregate" depends on the details of the molecule. This aggregation happens only above a threshold concentration, called the critical micellar concentration (CMC). What determines the CMC?

Consider a solution of *N* molecules, and assume that these can form associations whose aggregation number is α, i.e. they are composed of a number α of molecules (Slide 6). *N*_{α} is the number of molecules belonging to aggregates of size α and *n*_{α} the number of micelles of size α,

*n*_{α} = *N*_{α}/α

Equation 1

With this notation,

Equation 2

The partition function (see Lecture 1) of a system of volume V of these micelles is

Equation 3

where

is the volume occupied by one micelle of size α, and

The resulting Helmholtz free energy, F, is

Equation 4

where ƒ_{α}^{int} is the internal free energy of each α micelle (Slide 7),

Equation 5

The chemical potential, μ_{α}, of a surfactant molecule belonging to an α-micelle is

Equation 6

where ε'_{α} is the internal free energy per surfactant molecule belonging to a micelle of aggregation number α (see Slide 8),

Equation 7

It is then useful to rewrite Equation 6 by considering the total number Ni>N_{T} of molecules in the system, both water and surfactant (Slide 9). Then by *x*_{α} we denote the mole fraction of molecules in an α-aggregate,

Equation 8

The chemical potential (Equation 6) then becomes

Equation 9

where

Equation 10

with *v* being the mean volume of molecules in solution,

that is, the volume of a water molecules in Equation 9 in the first term is the translational entropy of the micelle, and the term ε_{α} is the free energy change when a molecule is taken from the bulk and put into an aggregate of α molecules.

The condition for equilibrium between aggregates of different size is that their chemical potential is the same. Calling μ this constant value of chemical potential, this means (using Equation 9) that the mole fraction of molecules in an aggregate *x*_{α} is given by

Equation 11

Since

the mean aggregation number is

Equation 12

In Equation 12, note that by definition *x*_{1} < 1, and physically

ε_{1}–ε_{α} > 0,

since the assembly of molecules reduces the energy.

**3.2.2 Critical Micellar Concentration**

At very low surfactant concentrations, the chemical potential μ is small and lower than all of the ε_{α}. Therefore,

and the system does not show aggregation. As the surfactant concentration is increased, μ increases and becomes comparable to some of the ε_{α}, and aggregates form with mean size α. The CMC is conventionally defined as the concentration at which half of the molecules are in aggregates and half remain in monomer form. This can be expressed as

Equation 13

During the formation of micelles, upon further increase in surfactant concentration, the chemical potential does not change and remains pinned to a critical value μ_{c}. This means that the molar fraction of monomers *x*_{1} also remains constant, and all of the added surfactant goes into the micelles.

### 3.3 Aggregation and CMC

By considering aggregation as a chemical reaction in which the free surfactant is in equilibrium with the aggregates, we can work out the equilibrium constant of the reaction, in terms of the microscopic interaction energies (Slide 10). (See the supplementary boxes "Chemical reaction kinetics and equilibrium" for a basic review of this topic.)

If *x*_{1} and *x*_{N} are the mole fractions of molecules in monomer form and micelles of size *N*, respectively, then we can write

rate of association (forward reaction) = *k*_{1}*x*_{1}^{N}

Equation 14

rate of dissociation

Equation 15

where *k*_{N} is the rate constant for the *N*th order association process. Hence using the result from Equation 11c,

Equation 16

In equilibrium, the backward and forward rates will be equal, giving a relationship between *k*_{1} and *k*_{N}; the equilibrium constant *K*_{c} (the ratio of the association and dissociation rates) is given by

Equation 17

The following discussion is based on Israelachvili (2010). No *x*_{N} can exceed unity from Equation 16, so we can write

Equation 18

Thus there comes a point when the number of monomers cannot increase, and molecules must be involved in aggregates. This is the CMC. The concentration of monomers at all higher concentrations is given by the equality in Equation 18 (see the graph in Slide 11).

**3.3.1 CMC for spherical micelles**

In practice, it is found that spherical micelles are reasonably monodisperse, that is, we do not simply have a random collection of spherical micelles of all sizes (Slide 12, and see Lecture 4). This is a consequence of how conically shaped molecules can optimally pack.

As noted in Section 3.2, CMC is defined as the concentration at which half of the molecules are assembled into aggregates. If the peak of aggregates is very sharp at a size α* (Slide 13), then at the CMC we have

Equation 19

We have already seen that

Equation 11

If we assume that CMC is small and a is large, we can then write

Equation 20

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