King's College London

Project Abstract

Recently, it has been discovered that eukaryotic cells contain membrane-less assemblies that remain coherent in space and time for minutes and often hours, and facilitate compartmentalisation of specific sets of molecules in a limited region of space [1]. These structures represent liquid-phase condensates, which form via a biologically regulated liquid-liquid phase separation process in the cytosol. This phase separation process is reminiscent of the clustering of receptors or the formation of lipid rafts at the plasma membrane. There is a growing amount of evidence that phase separation process is important for the physiological roles of certain proteins and that mutations can lead to the formation of aggregates associated with neurodegenerative diseases [5]. Phase separation in cells is similar to the way oil and vinegar demix in salad dressings. However, there are a few important distinctions between phase separation in simple fluids and in biological mixtures, which imply that the textbook methods of statistical physics do not straightforwardly apply to the latter.

A first complication is that the cytosol consists of thousands of different macromolecules that mutually interact. Furthermore, chemical reactions between these molecules take place at concentrations far from thermal equilibrium. An approach that has been proposed to study phase separation of complex fluids is to consider a Flory- Huggins model with an infinite number of components subject to (largely unknown) pairwise interactions, which are modelled in terms of a random matrix with independent entries [2,3]. Mathematically, the problem amounts to characterising when a random function on a high-dimensional landscape (the free energy of the mixture) becomes non-convex. The works [2,3] use mainly numerical simulations and simple mean-field estimates.

The purpose of this project is to formulate more realistic random matrix models for the free energy of multi-component mixtures, and to connect the stability properties of the mixture with spectral properties of the matrix of second-virial coefficients. We also aim to study extensions of the model involving non-equilibrium effects, such as phosphorylation of proteins. In parallel, we will use methods from spin glass theory, such as the cavity and the replica method, to analyse when the free energy of the ensemble becomes non-convex. The analytical work will be supplemented with numerical simulations.

Project Aims

How cells organise their biochemistry in space and time is an important question in cell biology. It is well known that eukaryotic cells contain several membrane-bound organelles such as secretory vesicles and the endoplasmic reticulum. In addition, there are a number of organelles (nucleoli, Cajal bodies, P granules...) that instead do not possess an enclosing membrane yet remain coherent structures in space and time, and facilitate compartmentalisation of specific sets of molecules in a limited region of space. These membrane-less assemblies exhibit remarkable liquid-like features: for instance, they typically assume a round shape and coalesce into a single ‘droplet’ upon contact with one another, as well as with wet intracellular surfaces. There is convincing evidence that these structures represent liquid-phase condensates, which form via a biologically regulated liquid-liquid phase separation process. In recent times, this mechanism has acquired the status of a fundamental working hypothesis for how the intracellular space is organised and functions. Consistently with this picture, several membrane-less organelles have been shown to exhibit a concentration threshold for assembly, a hallmark of phase separation. Moreover, more heterogeneous and multilayered structures for the condensates have also been recently discovered, whose architecture is regulated by the complex interplay between liquid surface tensions. Despite their dynamic nature, such condensates can maintain a spatio-temporal coherence for minutes or even hours while exchanging their components with the surrounding cytoplasm or nucleoplasm on time scales of seconds.

Phase-separated liquids are familiar from everyday experience (for instance, water and oil will always ‘demix’ and phase separate, no matter how vigorously shaken): the difference with what happens in the intracellular setting, though, is that such condensates exhibit remarkably diverse composition. For instance, there are dozens of proteins found in P and stress granules, and many hundreds of enriched proteins making up nucleoli. How the droplet composition, biophysical properties, and protein phase behaviour change as the system incorporates more components is an outstanding and very relevant question, which this project plans to address.

One of the most successful theoretical frameworks for understanding phase separation is the so called ‘Flory-Huggins model’, which considers molecules interacting on a lattice. The number of possible distinct ways of placing the molecules on the lattice provides the entropy of mixing, while the enthalpic contribution to the free energy arises from a mean-field interaction term, computed from pairwise interactions randomly assigned to neighbouring particles. For binary and ternary mixtures, the model has enjoyed some success and provided some partial insights into the dynamical behaviour of multi- component systems, however a full treatment (even if anyway limited to numerical simulations) of a realistic number of components in a biophysical setting is nowhere in sight.

A completely different approach was pioneered by Sear and Cuesta [2] and followed more recently by Jacobs and Frenkel [3]: the free energy of a N component mixture is written in terms of a pairwise interaction matrix B_ij, which couples the densities rho_i and rho_j of components i and j in the mixture. B is the matrix of second-virial coefficients, and – in their statistical approach, which is vaguely reminiscent of Wigner’s approach to heavy nuclei spectra in the 50s – is assumed to be random, with independent elements parametrised by their mean and variance. In their words ‘Once we are using random matrices, the fact that we have N>>1 components is a help, not a hindrance.’ The issues related to the stability of the mixture can be settled by considering how the convexity properties of the free energy change upon small perturbations of the density. Technically, this becomes a question related to the location of the lowest eigenvalue of a rank-N deformation of the matrix B, and information related to the composition of the equilibrium mixture is encoded in its associated eigenvector.

This highly simplified setting (further supplemented by a few, rather crude approximations) has the merit of reproducing two distinct types of phase transitions as parameters are varied, namely i) a condensation instability (due to a lone smallest eigenvalue that splits off from the bulk of other eigenvalues), where the densities of all components increase (or decrease) together. This models the incipient formation of one phase enriched in all the components, coexisting with a phase depleted in all the components, and ii) the demixing of two phases, each one enriched in some components and depleted in others. The main technical tools used in [2] and [3] are simple mean-field and scaling estimates, and numerical simulations. Notwithstanding a few features and predictions that are in qualitative agreement with biological data, the current state of the art in this field does not allow to draw quantitatively reliable conclusions for given systems: this has much to do with the highly simplified nature of the assumptions made.

The present project is about formulating a comprehensive theoretical framework – based on Random Matrix Theory and Spin Glass Theory – for phase separation in biological mixtures with a large number of components, taking into account the following improved features: i) introduce correlation between elements of the B matrix. It is indeed well known that some classes of biomolecules, such as intrinsically disordered proteins, are more likely to interact promiscuously. This will obvious imply a more structured interaction matrix than the standard Gaussian ensemble. ii) Lift any prior assumption about the nature of the equilibrium concentrations (e.g. uniform across constituents), which is instead present in [2,3]. Moreover, due attention will be paid to producing sound theoretical results for the statistical behaviour of smallest eigenvalue and associated eigenvector (generalising works by Johansson and Peche’) for matrices of the type A+D (with D diagonal, and A random and ‘structured’). Moreover, the classical tools of spin glass physics (cavity and replica methods) will be used to develop a theory of when and how the convexity properties of a random free energy undergo a critical change. All the analytical results will be corroborated with numerical simulations, and possibly with biophysical data when available. The supervising team jointly enjoys just the right combination of expertise, experience and professional networks to make this project a success, which would be much more difficult to achieve within a single-supervisor framework, and is committed to holding regular joint meetings and being both actively involved in the development of the theory, analytical calculations, and numerical simulations.

Reference 1: C. P. Brangwynne, C. R. Eckmann, D. S. Courson, A. Rybarska, C. Hoege, J. Gharakhani, F. Juelicher, A. A. Hyman, Germline P Granules are Liquid Droplets That Localize by Controlled Dissolution/Condensation, Science 324, 1729-1731 (2009).

Reference 2: William M. Jacobs and Dean Frenkel, Phase Transitions in Biological Systems with Many Components Biophys. J. 2017, 112(4): 683-691.

Reference 3: Richard P. Sear and Jose’ A. Cuesta, Instabilities in Complex Mixtures with a Large Number of Components, Phys. Rev. Lett. 91, 245701 (2003).

Reference 4: Chiu Fan Lee and Jean D. Wurtz, Novel physics arising from phase transitions in biology, Journal of Physics D: Applied Physics 52, 023001 (2019).

Reference 5: Yongdae Shin and Clifford P. Brangwynne, Liquid Phase condensation in cell physiology and disease, Science 22 Sep 2017: Vol. 357, Issue 6357, eaaf4382 DOI: 10.1126/science.aaf4382