Nonideal Solutions

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.

Entropically Driven Long-Range Order in Crowded Solutions

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) state has a high rotational entropy. By comparison the nematic (orientationally aligned) state has a low rotational entropy. At the same time, the translational entropy of the two states is similar. Thus, as expected, the isotropic state is entropically favored. In the lower panel, the particles are sufficiently long or concentrated that the average distance between them is short compared to their length. In this case, particle motions are greatly hindered and the isotropic state has low rotational and translational entropy. By aligning in the nematic state, the particles have little rotational entropy to lose. On the other hand, by rotating out of each others' way, they gain considerable translational entropy (because, in the language of the previous paragraph, they reduce their mutual excluded volume). Thus, under crowded conditions, the nematic state actually has the greater overall configurational entropy. (This is why it is easier to put crayons or toothpicks or tree branches into a container, above a certain packing density, if they are aligned than if they are randomly oriented.)

Figure 1 Orientational ordering of elongated particles (rods or disks seen edge-on). Top: When particles are short compared to the interparticle distance, the orientational entropy lost in alignment exceeds the translational entropy gained. Bottom: When particles are long compared to the interparticle distance, the translational entropy gained in alignment exceeds the orientational entropy lost. Uniaxial particles (round rods and disks) form phases with a single unique direction. The analogous entropically driven alignment of biaxial particles yields biaxial phases.23

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 concentrations^3-9 and substantiated by computer simulations¹⁰ and experimental studies.^11-15 Recent extensions to persistent flexible hard rods^16-22 give similar phenomenology except that, for long rods, ordering acts on persistence length segments rather than on the entire contour length of the particle. In this case, the persistence length, rather than the overall length, becomes the determinant of ordering behavior as a function of concentration.

Crowding is also sufficient to induce positional ordering of hard particles. This was first demonstrated in computer simulations of hard spheres.^24-26 By localizing around lattice points, spheres minimize pockets of wasted space and maximize their overall configurational entropy.²⁷ For spheres, the three translational dimensions are equivalent and there is a single freezing transition with increasing concentration. For elongated parallel particles, richer phase behavior occurs because the three different dimensions generally freeze at different concentrations. Thus theory^28-36and computer simulations^37,38 find that a solution of long parallel rods goes from a fully fluid state to a one-dimensionally ordered lamellar state, to a two-dimensionally ordered columnar state, and finally to a three-dimensional crystalline state with increasing concentration.

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 ordering in columns. In addition to clearly illustrating the trade-offs between different dimensions, the cell construct is a theoretically important tool.^29,30 Since the fluid and ordered degrees of freedom are decoupled by the impenetrable cell boundaries, it is relatively easy to calculate the configurational entropy and determine whether the translational entropy gained in the fluid dimensions upon ordering exceeds the translational entropy lost in the ordered dimensions. Furthermore, this calculation represents a worst case scenario because the positional constraints imposed by impenetrable cell boundaries are artificially severe. Less severe positionally ordered states may have an even greater overall configurational entropy and become stable at lower concentrations. So the cell construct sets an upper bound on the crowding necessary to induce positional ordering of a given symmetry. A disadvantage of the cell construct is that the all-or-none description of positional ordering does not allow for the gradual growth of density waves and thus enforces first-order behavior at all the ordering transitions. Thus, while computer simulations for selected particle axial ratios show the same sequences of phases as predicted by the cell construct, the simulations show ordered phases at lower concentrations and a second-order fluid-to-lamellar transition. Density functional theory, which explicitly considers density waves of variable amplitude, does a better job of describing the details of the transitions but is not useful for obtaining a global picture of the phase behavior.^31-36

Figure 2 Cell model of the translational ordering of elongated particles (rods left and disks right). Small adjustments of particle positions (to one side or the other of a cell boundary) allow particles which are obstructed in the nematic state (N) to move laterally in the lamellar state (L) or axially in the columnar state (C). Spontaneous translational ordering occurs when the translational entropy gained in the fluid dimensions exceeds that lost in the ordered dimensions.

Coupling to Self-Assembly

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 Analogously, some proteins have the functionally important capacity to reversibly aggregate into stiff multistranded filaments of indeterminate length. Normal examples include the proteins that constitute the filamentous skeleton that is responsible for giving most cells their shape and strength.^47,48 Sickle cell hemoglobin constitutes a pathological example in which filaments distort and stiffen the red blood cells causing occlusion of blood vessels and the attendant pain and tissue damage symptomatic of sickle cell disease.⁴⁹ Perhaps the most common self-assembly occurs in surfactant solutions. For rigid surfactants (such as the perfluorinated fatty acids), micelles tend to grow with a disk-like morphology.^50,51 However, greater polymorphism is possible for flexible surfactants, with micelles growing in one dimension to form rod-like particles or two dimensions to form plate-like particles, depending on conditions.

An intrinsic property of self-assembling systems is the broad size distribution of the particles that are formed and its variability with concentration and other conditions. A theory for such systems should preferably make no assumptions about the size distribution but rather derive it, jointly with the spatial distribution, as the distribution that minimizes the free energy. This requires a means for evaluating the effect of excluded volume on the configurational entropy for all possible particle size distributions, as well as the full range of possible spatial distributions. For very crowded solutions, an accurate virial expansion is not practical and a closed-form expression is needed. One approach has been to use lattice combinatorics.⁵² However, although this gives an improvement over Onsager's second virial estimate, the effects of excluded volume are still significantly underestimated at very high concentrations.

A more accurate treatment of hard polydisperse convex particles has been developed using scaled particle theory.^5,53 "Scaling" allows the general difficulty of evaluating many-body interactions to be avoided by interpolating between two extremes that are easy to evaluate. Since, on the one hand, an infinitesimally small particle can be excluded by only one other particle at a time,⁵³ the probability of being able to insert it into a crowded solution is easy to calculate. At the other extreme, the difficulty of inserting a macroscopically large particle into a crowded solution simply corresponds to the pressure-volume work of excavating the necessary hole. The wonder of scaled particle theory is that interpolation between these two extremes, with care to include enough scaling dimensions and preserve thermodynamic consistency, yields remarkably accurate estimates of the configurational entropy as assessed by computer simulations¹⁰ and experimental studies.^11,12

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 In contrast, surfactants generally form minimum spherical micelles at low concentrations and growth into elongated forms occurs gradually at higher concentrations.

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 repulsion^55-57 or a long persistence length⁵⁸ raises the cost of condensation sufficiently to keep the aggregates suspended in solution at high concentrations, as observed experimentally.

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, to keep the aggregates in solution. In both cases, the strength of aggregation is given by a single parameter, , corresponding to the free energy of monomer addition to an aggregate. When is large, aggregation is sufficiently strong that sufficiently elongated aggregates are formed at low concentrations to result in spontaneous orientational ordering (a nematic phase). At higher concentrations, positional ordering also sets in. For rod-like aggregates, the lamellar phase is suppressed because, in that configuration, the variable lengths of the particles would waste a lot of space (see Figure 2). On the other hand, the polydispersity of rod-like aggregates does not present a problem in the columnar phase. Thus for rod-like aggregates the nematic gives way to a columnar phase with increasing concentration. For disk-like aggregates, the situation is the reverse, with polydispersity in diameter problematic in the columnar phase but accommodated well in the lamellar phase (again see Figure 2). Thus, for disk-like aggregates the nematic gives way to a lamellar phase at increasing concentrations. For both rods and disks, a jump occurs in the average aggregate size at each ordering transition, which reflects the coupling between spatial ordering and aggregate growth.⁵⁹

	Figure 3 Theoretically predicted behavior of solutions of a rod-forming solute as a function of the particle volume fraction (vp) and the free energy of monomer association (kT).59 Horizontal tie lines in the two-phase regions connect coexisting isotropic (I), nematic (N), columnar (C), and crystalline (X) phases.
	Figure 4 Theoretically predicted behavior of solutions of a disk-forming solute as a function of the particle volume fraction (vp) and the free energy of monomer association (kT).59 Horizontal tie lines in the two-phase regions connect coexisting isotropic (I), nematic (N), lamellar (L), and columnar (C) phases.

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 aggregates^42,60,61 or rigid surfactants that self-assemble to form disk-like aggregates.^62,63 The robust qualitative correspondence between the experimentally observed phase behavior and the simple theory lends confidence that the important features of these systems have been captured in the model. The opportunities for quantitative comparison are limited in these systems, but the predicted aggregate size at the triple point compares well with experiment.⁵⁹

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 The hard interactions presumably reflect the very compact folding of this globular protein. For deoxygenated sickle cell hemoglobin at low concentrations, the osmotic pressure is the same as for monomeric hemoglobin. Cooperative self-assembly at higher concentrations is signaled by a sharp break in the osmotic pressure curve. However, condensation does not occur; on the other side of a brief transition region, the osmotic pressure increases again with a simple quadratic dependence on concentration.⁶⁵ This is not consistent with a naked hard core repulsion between the filaments. Once the hard-core interactions have been minimized by the growth and alignment of filaments, any concentration dependence of the osmotic pressure reflects the soft part of the repulsions between long parallel filaments.⁵⁷ The softness of sickle cell hemoglobin filaments may reflect some looseness in the structures of the filaments, consistent with the polymorphism that has been observed.^66,67 On the other hand, the increasing strength of the soft repulsions with decreasing temperature from 37 to 3 C⁶⁵ suggests that water structure plays a role.

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 Figure 5 shows the results for surfactants that have no intrinsic preference for rod-like or disk-like assembly. In these systems, the interplay between the entropy of mixing and the excluded volume leads to a succession of states with different micelle morphologies as a function of concentration. At low concentrations the micelles are relatively small and isotropically oriented. When growth occurs at moderate concentrations, aligned rod-like micelles are formed. But growth at high concentrations produces aligned plate-like micelles. This behavior occurs because the entropy of mixing favors small micelles, while packing constraints favor plate-like micelles. Rod-like micelles represent a compromise between the two factors. Figure 5 shows the corresponding phase diagram. These results agree well with typical surfactant phase diagrams and suggest that excluded volume interactions are responsible for the reconstruction of micelles from rod-like to disk-like morphologies that is generally observed with increasing surfactant concentration.

Figure 5 Theoretically predicted phase behavior for a surfactant that forms rod-like and disk-like micelles with the same free energy of monomer addition.68 is the difference between the free energy for monomer addition to elongate an existing micelle and the free energy per monomer for forming a new spherical micelle. The greater , the more elongated the micelles become and the lower the concentration at which the isotropic phase (I) gives way to an ordered phase. When growth occurs at moderate concentrations, an axial phase (A) of aligned rod-like micelles is formed. When growth occurs at high concentrations, a planar phase (P) of aligned disk-like micelles is formed. Horizontal tie lines connect coexisting phases in the two-phase region.

Crowded Mixtures

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 Understanding the behavior of mixtures of sickle and non-sickle hemoglobins is therefore of medical interest.

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 predicts that the increase in the total hemoglobin concentration required to form filaments as the proportion of sickle cell hemoglobin decreases is not very sensitive to the free energy penalty for substituting a non-sickle hemoglobin in the filaments, unless the penalty is extremely small. This is consistent with experimental observations that the "solubilization of sickle cell hemoglobin" is similar for various non-sickle hemoglobins.^72,73 On the other hand, the composition of the filaments that separate from the isotropic solution above the "solubility limit" is predicted to be very sensitive to the value of the penalty for substitution in the filament.^71,74,75 Again this is consistent with measurements of the compositions of hemoglobin filaments obtained in sedimentation experiments.⁷³ These experiments show that normal adult hemoglobin can be accommodated in the filament more easily than fetal hemoglobin. This is reasonable because normal adult hemoglobin is more similar to sickle cell hemoglobin than is fetal hemoglobin. Detailed quantitative analysis of the measured filament compositions indicates that the penalty for substituting normal adult hemoglobin in filaments is much smaller than the free energy required to neutralize the acidic group that distinguishes it from sickle cell hemoglobin. In fact, the penalty is more consistent with the free energy required to protonate a histidine residue.⁷¹ This small penalty is also supported by a detailed quantitative analysis of osmotic pressure data⁷⁶ for mixtures of sickle cell and normal adult hemoglobins. Taken together, the analyses suggest that normal adult hemoglobin is accommodated in the filament by formation of a salt bridge with a nearby protonated histidine residue. Examination of the structure of double strands of sickle cell hemoglobin reveals a histidine residue (77) which is well situated for such an interaction.

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}

Figure 6 Theoretically predicted phase diagram for mixtures of a protein that linearly self-assembles into persistent flexible filaments (A) and a globular protein of the same diameter (B) in solvent (*) when the filament persistence length is 100 times the diameter.58 For low volume fractions of A (right leg), an isotropic phase occurs. Additional A leads to filament growth and alignment. For moderate solute volume fractions, B mixes freely with A whether A is aligned (N) or not (I). At high concentrations dramatic demixing occurs. The steep tie lines show very dense domains of nematically ordered solutions of A coexisting with relatively dilute isotropic solutions of B. In the empty region of the diagram, the behavior is still more extreme and not numerically tractable. The dot represents the composition of a typical cell, comprising 25 vol % protein, of which 10% self-assembles into filaments and 90% does not.

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. These results suggest that, as long as the packing is not extraordinarily tight, the filaments will spontaneously segregate into separate bundles.

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.

Control

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.

Conclusions

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.

Acknowledgment

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.