In addition to membrane-bound organelles, eukaryotic cells feature various membraneless compartments, including the centrosome, the nucleolus and various granules. Many of these compartments form through liquid–liquid phase separation, and the principles, mechanisms and regulation of their assembly as well as their cellular functions are now beginning to emerge. Biomolecular condensates are micron-scale compartments in eukaryotic cells that lack surrounding membranes but function to concentrate proteins and nucleic acids. These condensates are involved in diverse processes, including RNA metabolism, ribosome biogenesis, the DNA damage response and signal transduction. Recent studies have shown that liquid–liquid phase separation driven by multivalent macromolecular interactions is an important organizing principle for biomolecular condensates. With this physical framework, it is now possible to explain how the assembly, composition, physical properties and biochemical and cellular functions of these important structures are regulated.

Biomolecular condensates: organizers of cellular biochemistry

Cells are organized on length scales ranging from angstrom to micrometres. However, the mechanisms by which angstrom-scale molecular properties are translated to micrometre-scale macroscopic properties are not well understood. Here we show that interactions between diverse synthetic, multivalent macromolecules (including multi-domain proteins and RNA) produce sharp liquid-liquid-demixing phase separations, generating micrometre-sized liquid droplets in aqueous solution. This macroscopic transition corresponds to a molecular transition between small complexes and large, dynamic supramolecular polymers. The concentrations needed for phase transition are directly related to the valency of the interacting species. In the case of the actin-regulatory protein called neural Wiskott-Aldrich syndrome protein (N-WASP) interacting with its established biological partners NCK and phosphorylated nephrin, the phase transition corresponds to a sharp increase in activity towards an actin nucleation factor, the Arp2/3 complex. The transition is governed by the degree of phosphorylation of nephrin, explaining how this property of the system can be controlled to regulatory effect by kinases. The widespread occurrence of multivalent systems suggests that phase transitions may be used to spatially organize and biochemically regulate information throughout biology.

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Phase transitions in the assembly of multivalent signalling proteins

Light scattering and the absolute characterization of macromolecules

Using lattice statistical mechanics, we develop theory to account for the folding of a heteropolymer molecule such as a protein to the globular and soluble state. Folding is assumed to be driven by the association of solvophobic monomers to avoid solvent and opposed by the chain configurational entropy. Theory predicts a phase transition as a function of temperature or solvent character. Molecules that are too short or too long or that have too few solvophobic residues are predicted not to fold. Globular molecules should have a largely solvophobic core, but there is an entropic tendency for some residues to be "out of place", particularly in small molecules. For long chains, molecules comprised of globular domains are predicted to be thermodynamically more stable than spherical molecules. The number of accessible conformations in the globular state is calculated to be an exceedingly small fraction of the number available to the random coil. Previous estimates of this number, which have motivated kinetic theories of folding, err by many tens of orders of magnitude.

Theory for the folding and stability of globular proteins.

The entanglement concept in polymer rheology

The current status of the statistical thermodynamics of dilute solutions of chain polymers is reviewed, with particular attention to the interactions between single chain molecules and solvents and to the mutul interactions of two chains.



With respect to the problems of one and two chains, the following topics are considered: the potential of mean force between chain segments and its dependence on the slevent; relationship between latice and continuum theories; comparison between machine calculations and analytical theories of chain dimensions; exact series and approximate or semi-empirical theories of the osmotic second virial coefficient, and comparison with the experimental data; effects of polydispersity; behavior in mixed solvents. Higher virial coefficients and phase separation are also briefly discussed.



Der heutige Stand der statistischen Thermodynamik der verdunnten Hochpolymerlosungen wird betrachtet. Folgende Probleme werden diskutiert: das Potential der Durchschnittskraft zwischen Kettenelementen und seine Abhangigkeit von dem Losungsmittel; die Beziehungen zwischen Gittermodell und Kontinuum-Modell; ein Vergleich zwischen Monte-Carlo-Rechnungen und analytischen Theorien der Dimensionen von Kettenmolekulen; verschieden Theorien des zweiten osmotischen Virialkoeffizienten; der Einflus der Molekulagewichtsverteilung; das Verhalten in Mischungen von Losungsmitteln; der dritte Virialkoeffizient und die Phasentrennung.

Problems of the statistical thermodynamics of dilute polymer solutions

In an earlier era of high polymer chemistry, branching not infrequently served as a whipping boy to explain deviations from an expected physical behavior * 9 often without any independent or clear-cut evidence for the occurrence of chemical reactions leading to branched molecular structures. Today, it is clear from the general nature of the chain transfer reaction that high-conversion vinyl polymers (at least if prepared by free-radical initiation) contain more or less highly branched molecules. A sounder and more quantitative attack on the question of branching is therefore mandatory on both experimental and theoretical fronts. From the theoretical point of view, two general problems related to branching then confront us. The one concerns the mechanism and kinetics of the branchproducing reactions and ultimately the distribution of molecular weights and degrees of branching in samples prepared under specified conditions, and is illustrated by other papers3v4 in this symposium. The other concerns the effect of branching on observable physical properties, and forms the subject of this paper, It will be apparent from the previous contributions that in experimental studies of branching these two problems are not so easily separable. Our remarks are confined almost entirely to the properties of dilute solutions. It is not intended thereby to imply that the properties of polymers in bulk are insensitive to branching; on the contrary, the known or possible influences of branching on bulk properties furnish a strong practical impetus for thorough study of the subject. I t seems likely, however, that a firm theoretical understanding can be more easily and quickly attained for the properties of dilute polymer solutions than for those of the bulk materials. The properties to be discussed in the following sections are derivable from measurements of osmotic pressure, light scattering, sedimentation, diffusion and viscosity. The type of molecular structure considered is that of rather long branches, such as might occur through chain transfer reactions or copolymerization with polyfunctional monomers. Shorter, regularly spaced branches on a main backbone, such as the alkyl side groups in polyvinyl stearate, will not be treated.

Dilute solutions of branched polymers

The dilute solution properties of copolymers are briefly discussed in relation to those of the parent homopolymers. It is shown that copolymer molecules are usually more expanded in solution than would be expected from the averaged behavior of the pure polymers, because of repulsive interactions between the unlike monomer units. A thermodynamic parameter χAB characterizing these interactions can be derived from measurements of the dilute solution properties of copolymers. In favorable cases this parameter can be independently evaluated from studies of ternary systems composed of the two parent homopolymers and a solvent, thus allowing prediction of the behavior of the copolymer. Light scattering and viscosity measurements on fractions of approximately equimolal copolymers of styrene and methyl methacrylate are presented and analyzed. The values of χAB deduced from the results in two solvents agree satisfactorily with each other, but are somewhat larger than those earlier obtained from measurements on ternary systems.

Copolymers in dilute solution. I. Preliminary results for styrene‐methyl methacrylate

The theory of molecular distribution in condensation polymers has been extended to include systems of a rather general type. Equations for the distribution and the weight-average molecular weight are given.

Molecular distribution in condensation polymers

Weight-average molecular weights and intrinsic viscosities in dimethyl formamide at 25° have been measured for four unfractionated low-conversion acrylonitrile polymers covering the molecular weight range 30 to 250 thousand. The results follow the equation:

Walter H. Stockmayer

Papers

Problems of the statistical thermodynamics of dilute polymer solutions

Dilute solutions of branched polymers

Copolymers in dilute solution. I. Preliminary results for styrene‐methyl methacrylate

Molecular distribution in condensation polymers

An intrinsic viscosity‐molecular weight relation for polyacrylonitrile