Entropically Driven Order in Crowded Solutions: From Liquid Crystals to Cell Biology
Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254-9110
Received April 7, 1995
Solution nonideality is more commonly regarded as an unpleasant fact of life than as a source of interesting phenomenology. Usually represented by simple activity coefficients overlaid on ideal descriptions, nonideality is generally seen as quantitatively important but qualitatively insignificant. This Account will take the opposite point of view, focusing on novel structural features and physical properties that arise from interactions in solution. Applications range from simple binary solutions to biological solutions. In the former case, the state of knowledge is more advanced and current work seeks to understand relatively subtle variations in structure and properties. For biological solutions, the appreciation of the phenomenological significance of nonideality is relatively new. However, these diverse systems have much in common, as will be emphasized here.
In describing the interactions that give rise to nonideality in solution, it is practical to consider the solvent as background. This means that the solvent is generally not described explicitly and the solute is described as a gas of particles. The van der Waals interactions that hold the system together are implicit, and the solute-solute interactions are the effective interactions as mediated by solvent. In poor solvents, there are net short-range attractions between solute particles, leading to immiscibility.1 For charged solutes, the strength of the net long-range electrostatic interactions depends on the dielectric constant and Debye length of the solvent.
However, even for electrically neutral solutes in good solvents, strong nonideality occurs at high volume fractions. This is due to the mutual impenetrability of particles. For example, the volume excluded to the center of one hard sphere by another hard sphere is a sphere of radius equal to the sum of the radii of the two hard spheres. Thus for dilute monodisperse spheres, the excluded volume is 8 times the occupied volume. For elongated particles the excluded volume can be larger, depending on the relative orientations of the particles. In the present Account we will be interested in the behaviors that result from these most minimal interparticle interactions in a continuum solvent background.2,81 The simplest description of such interactions assumes a sharp particle boundary with an infinite step potential (i.e., a hard-core potential). In real life, the boundary is less sharp and, when this is important, a weak short-range repulsive shell, consisting of one or more small step potentials, can be added to soften the excluded volume interaction.
In 1949, Onsager showed theoretically that the
hard-core excluded volume interaction (crowding)
alone suffices to induce orientational alignment of
elongated particles in solution.2 Figure
1 illustrates
the phenomenon for hard, rigid rod-shaped particles.
In the upper panel, the particles are sufficiently
dilute
or short that the average distance between them is
long compared to their length. In this case, the
rotation and translation of the particles are not greatly
hindered and the isotropic (orientationally
random)
The trade-off between rotational and translational
degrees of freedom, as a function of particle axial
ratios or concentration, leads to an entropically-driven
isotropic-nematic ordering transition. This
transition
is first order, and the range of intermediate concentrations over which separation into coexisting isotropic
and nematic phases occurs depends on the axial ratios
of the particles and the degree to which soft attractions
or repulsions between the particles (relative to their
interactions with solvent) modulate the hard-core
excluded volume interaction. Onsager's
result has
been refined by numerous theoreticians using more
sophisticated models suitable for higher concentrations3-9
Crowding is also sufficient to induce positional
ordering of hard particles. This was first demonstrated in computer simulations of hard
spheres.24-26
Figure 2 illustrates the gain in translational
entropy
along the fluid dimensions when particles are constrained by impenetrable cell boundaries in the ordered dimensions. Consider the shaded particle in
each top panel. It cannot move vertically or horizontally due to slight obstructions in each direction.
The
horizontal obstruction is released by ordering in layers, and the vertical obstruction is released by
The foregoing discussion has implicitly assumed
particles of a single specific geometry and noted that
the greater the particle axial ratio, the lower the volume fraction at which spontaneous spatial ordering
sets in (i.e., the lower the volume fraction at which it
becomes entropically advantageous for particles to
stay out of each others' way by orientational or positional alignment). For a given volume fraction, ordering is more advantageous when there are fewer, longer
particles than when there are more, shorter particles.
The reciprocal will also be true if the solute is able
to
reconfigure itself: if particles are staying out of each
others' way by orientational or positional alignment,
then it will be easier for the solute to compose itself
into fewer, longer rods than it would be in the absence
of order. Thus ordering promotes self-assembly in
systems where this degree of freedom exists and all
degrees of freedom, spatial and self-assembling, will
equilibrate jointly to minimize the free
energy.39-41
Reversible self-assembly to form elongated aggregates is a distinctive property of a variety of molecules.
For example, polyaromatic dye and drug molecules
have the capability of reversibly aggregating into
cylindrical stacks of indeterminate
length.42-46
An intrinsic property of self-assembling systems
is
A more accurate treatment of hard polydisperse
convex particles has been developed using scaled
particle theory.5,53
While excluded volume favors aligned elongated aggregates, the entropy of mixing favors dissociation and random spatial distributions. The balance is determined by the free energy changes involved in putting together the aggregates. For purposes of understanding the thermodynamics of these systems, the details of the contacts between assembling monomers are not important. What is important is the dependence of the overall free energy of formation on aggregate size, which can be encompassed satisfactorily by a phenomenological model. In the simplest case of linear aggregation, an n-mer will have (n - 1) contacts between monomers and the free energy of forming these contacts will be proportional to (n - 1). This single proportionality constant suffices to describe the free energy of formation of aggregates of all lengths (with the associated changes in rotational and translational entropy being accounted for in the mixing and configurational free energy terms). Linear assembly may also involve a change in each of the monomers to allow them to adhere to each other. This may be a change to a new conformation or a limitation to a smaller range of conformations than is sampled by the free monomer. In this case the free energy of assembling an n-mer will have a second term that is proportional to n, with its corresponding proportionality constant. The two terms can be rearranged to put all the n-dependence into one term, corresponding to the total free energy of monomer addition to a large aggregate, leaving a constant free energy term that represents the end effects for a finite aggregate. This form can be generalized to multistranded filaments and cylindrical micelles.
The two-term phenomenological free energy for assembly of rod-like aggregates has terms that are zeroth and first order in the aggregation number, n. To imagine a more complicated phenomenological free energy for assembly of rod-like aggregates requires terms of other orders in n. This would require somewhat exotic effects, such as the buildup of strain along the length of the aggregate, for which we know of no reported evidence. Thus, it seems safe to accept the two-term phenomenological free energy of assembly as sufficiently general for most one-dimensional aggregate growth. For plate-like aggregates, a reasonable two-term expression consists of a term that is first order in n, representing the total free energy of monomer addition to a large aggregate, and a term that goes as the square root of n which describes edge effects.
As soon as there are two terms in the free energy of
assembly, there is the possibility of cooperativity: if
ends (or edges) have a cost that is not overcome by
contact free energies until the aggregate reaches a certain size, aggregates will not form until the concentration is high enough to support aggregates above this
critical size. This kind of cooperative assembly is
seen
in the formation of protein filaments.54
While the free energy of aggregation depends only
on the size distribution of the aggregates, we have
seen that the entropy of mixing and the free energy
of interparticle interactions depend on the spatial
distribution as well. The equilibrium size and
spatial
distribution jointly minimizes the total free energy.
As expected, spontaneous alignment occurs at sufficiently high concentrations and the critical concentration for alignment decreases as the free energy of
self-assembly becomes more favorable and therefore
the average aggregate size increases. In addition, if
the aggregates are assumed to be perfectly hard
(infinitely steep interparticle potential) and perfectly
rigid (infinitely long persistence length), scaled particle theory predicts that crowding at high concentrations will induce the condensation of infinitely large,
perfectly aligned aggregates. Because the number of
particles tends to be reduced indefinitely by open-ended aggregation to avoid packing constraints, the
entropy of mixing is not sufficient to keep the particles
in solution. On the other hand, because the aggregates are very long, the thermodynamic behavior
is very sensitive to soft interactions and flexibility.
Even a weak soft repulsion55-57
Figures 3 and 4 show the
dependence of the ordering
symmetries on concentration for the simplest cases of
rod-like assembly with no end costs and disk-like
assembly with no edge costs. Here, the particles were
assumed to be rigid and the hard-core interaction was
softened slightly, by addition of a weak repulsive step
potential,
When decreases, aggregation is weaker and the isotropic-nematic transition is postponed to higher concentrations. Eventually, is so weak that the nematic phase does not occur at all and there is a direct transition to the positionally ordered phase. In between, there is a triple point at which the isotropic, nematic, and columnar/lamellar phases coexist. When is weaker yet, there is less drive toward polydispersity and severe packing constraints at very high concentrations induce transitions to more highly ordered phases of monodisperse particles (i.e., a crystalline phase, X, in the rod-forming system and a columnar phase, C, in the disk-forming system).
The topology of these phase diagrams, in particular
the occurrence of a triple point, has been observed in
laboratory studies of polyaromatic molecules that self-assemble to form rod-like aggregates42,60,61
For self-assembling proteins, the most extensive
experimental data are for deoxygenated sickle cell
hemoglobin. Sickle cell hemoglobin is a highly
soluble
protein (perhaps the most soluble after normal hemoglobin), but unfortunately its aggregation is just strong
enough, at physiological temperature and pH, to occur
at physiological concentrations (~35 vol %) when
oxygen is as depleted as it can be in the microcirculation. Under these conditions normal hemoglobin
remains monomeric. For monomeric hemoglobin, the
osmotic pressure rises very steeply at very high concentrations, in quantitative agreement with scaled
particle theory predictions for hard convex particles
(spheres or cubes) of dimensions approximately corresponding to the hemoglobin crystallographic structure with some water of hydration.57,64
Whereas the structures of proteins, polyaromatic
molecules, and perfluorinated surfactants are such
that their aggregation is essentially restricted to rod-like or disk-like morphologies, the flexible chains of
ordinary surfactants allow them to form both rod-like
and disk-like micelles. So far, the only treatment
allowing for particle distributions of both morphologies
has used lattice statistics to account for excluded
volume.68
In the previous section we have considered the
coupling between the spatial and self-assembly degrees of freedom of a single solute at high concentrations. However, self-assembling proteins normally
occur in nature mixed with other proteins. In
ordinary
cells, 20-30% of the volume is occupied by proteins,
but only about 10% of these are of the filament-forming variety. Red blood cells are unusual in
containing primarily hemoglobin. However, there are
multiple forms of hemoglobin. Heterozygous individuals will produce two forms of adult hemoglobin. If one
is sickle cell hemoglobin and the other is normal adult
hemoglobin, then there are no physiological problems.
In individuals homozygous for sickle cell hemoglobin
the severity of the disease is reduced if the residual
expression of fetal hemoglobin genes is high. This
has
led to clinical trials of drug regimens designed to
increase the expression of normally repressed fetal
hemoglobin genes.69,70
Usually it is assumed that non-sickle hemoglobins
do not participate in the formation of filaments. In
fact, the various hemoglobins are sufficiently similar
that it must be possible to substitute non-sickle hemoglobins in the filaments with a finite penalty, and
crowding might help to overcome this penalty. When
copolymerization is included in the model for crowded
hemoglobin solutions, the theory71
In ordinary cells, the filaments are formed by
distinctive cytoskeletal proteins and copolymerization
of other proteins is not generally plausible. On the
other hand, packing rod-like and globular proteins
together at the ca. 25 vol % typically found in cells is
also problematic. Large pockets of space are wasted
when spherical particles are interspersed among rod-like particles, and above a certain concentration, the
particles gain more in translational entropy by staying
out of each others' way (i.e., demixing) than they lose
in mixing entropy. (The same principle makes it
advantageous to put sticks and stones in separate piles
when cleaning the garden.) Furthermore, the net
entropy gained upon demixing is maximized if most
of the solvent goes with the smaller globular particles.
Thus, the theoretically predicted result is that
tight
bundles of long aligned filaments will form, leaving
behind a relatively dilute solution of globular protein
(as illustrated in Figure 6). These predictions
are
consistent with observations of spontaneous bundling
when globular macromolecules are added to cell-free
solutions of actin filaments.77
78
The cytoskeleton actually comprises several proteins
that form filaments with different properties.
Tubulin
forms hollow microtubules with 13 monomers around
the circumference. This is a relatively thick and
stiff
structure. Actin forms microfilaments comprising a
twisted double strand of monomers. This structure
is relatively thin and flexible. Other proteins form
intermediate filaments. The question then arises
whether the demixing of filaments induced by high
concentrations of globular protein will produce mixed
filament bundles or segregated filament bundles. A
rigorous answer requires consideration of systems
with two filament-forming proteins and globular protein. So far this has not been feasible, but the
behavior of mixtures of two filament-forming proteins
is suggestive.58 At moderate concentrations the
aligned
filaments mix freely in a single phase. But at higher
concentrations the stiff filaments begin to interfere
with the bending of the flexible filaments and the
flexible filaments interfere with the
translational
freedom of the stiff filaments. At these
concentrations
the entropy is maximized by complete segregation,
with the solvent partitioning preferentially with the
thinner, more flexible filaments. At extremely high
concentrations, where even the flexible filaments are
forced to straighten out, the filaments mix once again.
How does this relate to cytoskeletal organization in cells? Segregated filament bundles are often found in cells, but this arrangement has generally been attributed to bundling proteins that form cross-links between the filaments. The assumption has been that cytoskeletal filaments would be randomly dispersed were it not for the bundling proteins. But we see now that the theory for crowded solutions indicates that bundles will form spontaneously without bundling proteins and that the bundling proteins must have a more subtle purpose. A likely role is that they could be responsible for the detailed structure of filament bundles by stabilizing the appropriate polarity or registration of the filaments. The important point is that bundling proteins are fine-tuning the spontaneous bundling that occurs in crowded environments rather than fighting the random dispersal that occurs at low concentrations.
The thermodynamic arguments up to this point depict cytoskeletal organization as a phase separation with random formation of dense domains of aligned filaments. However, cells normally exert exquisite control over cytoskeletal organization, building it up in some regions while dismantling it in others, and forming filament bundles (e.g., stress fibers) in some instances vs gels in others. Careful consideration of the theory of crowding allows the identification of several opportunities for control over cytoskeletal organization:
(1) To begin with, local conditions in the cell (e.g., concentrations of Ca2+ or nucleoside triphosphates) will influence the free energy of monomer addition to filaments and therefore the extent of filament formation.
(2) In addition, the cell manufactures a host of filament accessory proteins. So-called capping proteins bind reversibly to the ends of filaments. The more numerous and the more tenacious these proteins are, the shorter the average filament length will be. This mitigates the rod-sphere packing problem and may lead to the dissolution of filament bundles.79
(3) Cells also contain different types of cross-linking proteins. Whereas bundling proteins stabilize parallel contacts between filaments, other cross-linking proteins stabilize skewed contacts between filaments. If these are sufficiently numerous and tenacious, they would be expected to frustrate bundling80 and lead to the formation of gels instead.
(4) When bundles are formed, their placement is important, and this can be controlled by nucleating phase separation at the desired locations. Thus, the nucleation centers that have been found in cells may serve to control the placement of filament bundles.
The many microscopic degrees of freedom of self-assembling systems give rise to very rich phase behavior under highly crowded conditions. The use of scaled particle theory to describe excluded volume effects in fluid dimensions and a cell model to describe excluded volume in ordered dimensions, combined with phenomenological descriptions of self-assembly and a simple mean field treatment of soft interactions, suffices to explain the experimentally observed behavior. For polyaromatic molecules and rigid surfactants, this includes triple points for isotropic, nematic, and positionally ordered phases. For ordinary surfactants, this includes the changes in micelle morphology with increasing concentration. For proteins, this includes the experimentally observed copolymerization of non-sickle hemoglobins with sickle cell hemoglobin and the spontaneous formation of segregated cytoskeletal bundles in the presence of physiological concentrations of globular proteins. The theory also lends insight into the means by which cells might control cytoskeletal organization.
Much of the work described in this Account was carried out under the auspices of NIH Grant HL-36546 by associates Mark Taylor, Reinhard Hentschke, Thomas Madden, Jining Han, and Daniel Kulp. The author especially thanks Jining Han for his critique of the manuscript and for assistance in gathering references and graphic material. The author is also proud to have been instructed in the use of Aldus Superpaint by her younger daughter, Rachel H. Griffin.
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Judith Herzfeld is a product of a post-Sputnik special science program in New York City public schools and higher education specializing in chemistry and physics at Barnard College and MIT. She was first introduced to biophysics in Ph.D. thesis research with H. Eugene Stanley concerning heterotropic cooperativity in hemoglobin. Her academic career has included faculty positions at Amherst College, Harvard Medical School, and Brandeis University, where she has been professor of biophysical chemistry since 1990 and assumed the chair of the Department of Chemistry in 1995. Her current research interests include statistical thermodynamic studies of self-assembly in crowded solutions (particularly as it affects the morphology, rheology, and locomotion of cells) and solid state NMR studies of molecular mechanisms in membrane proteins (with emphasis on chromophore tuning in retinal pigments, light-driven proton transport in bacteriorhodopsin, and the interfacial properties of gas vacuoles).