Self-Assembly of Biomolecular Condensates with Shared Components

Biomolecular condensates self-assemble when proteins and nucleic acids spontaneously demix to form droplets within the crowded intracellular milieu. This simple mechanism underlies the formation of a wide variety of membraneless compartments in living cells. To understand how multiple condensates with distinct compositions can self-assemble in such a heterogeneous system, which may not be at thermodynamic equilibrium, we study a minimal model in which we can “program” the pairwise interactions among hundreds of species. We show that the number of distinct condensates that can be reliably assembled grows superlinearly with the number of species in the mixture when the condensates share components. Furthermore, we show that we can predict the maximum number of distinct condensates in a mixture without knowing the details of the pairwise interactions. Simulations of condensate growth confirm these predictions and suggest that the physical rules governing the achievable complexity of condensate-mediated spatial organization are broadly applicable to biomolecular mixtures.

Figure 1

Encoding compositionally distinct droplets in a multicomponent mixture. (a) We aim to encode $n$ target phases, each enriched in a specific subset of the components, that can be selectively assembled from a homogeneous mixture via nucleation and growth. (b) To achieve this, each target phase must be a local minimum of the grand potential $\mathrm{\Omega }$. We sketch $\mathrm{\Omega }$ along a linear path between the homogeneous and target-phase concentrations ${\overrightarrow{\varphi }}^{(0)}$ and ${\overrightarrow{\varphi }}^{(\alpha )}$; orthogonal directions of concentration space ${\overrightarrow{\varphi }}_{\perp }$ are indicated by dashed curves.

Figure 2

Mean-field predictions of optimal target encoding. (a) The probability that a set of $n$ randomly generated targets can be stably encoded in the mean-field model. Inset: The probability of successful encoding roughly follows a master curve when scaled by $n_{1/2}$, the number of targets that can be encoded with 50% probability. (b) For $N\gg M$, numerical calculations of $n_{1/2}$ obtained via quadratic programming (QP) agree well with those obtained by semidefinite programming (SDP), showing that $n_{1/2}$ follows a power law as a function of the number of components. (c) The scaling exponent $\gamma$ increases with the number of species enriched in each target $M$ as predicted by graph-theoretical arguments (see text).

Figure 3

Lattice simulations with finite-range interactions follow the mean-field predictions. (a) Bulk phases are considered stable if the composition of the lattice fluctuates near the target composition such that $\cos {\theta }^{(\alpha )}\simeq 1$. Inset: Illustration of first- and second-nearest neighbor shells on the three-dimensional lattice. (b) Growth from a metastable gas phase is considered successful if the composition of the droplet matches that of the initial seed at time $t=0$. Components colored red are enriched in the target phase, while blue components are depleted. (c) The values of $n_{1/2}$ obtained by simulating stability and growth approach the mean-field predictions when $N\gg M$. However, short-range interactions (i.e., $l_{\mathrm{nbr}}=1$) slightly reduce the number of targets than can be stably encoded and then assembled via droplet growth.

Figure 4

Encoding 250 compositionally distinct target phases, each enriched in five species, using only 200 components. (a), left panel: Projections of the grand potential on an $L\times L\times L$ lattice with $L=6$ and $l_{\mathrm{nbr}}=1$. Each curve shows the free-energy surface along a linear path in concentration space, parameterized by $\mathrm{\Delta }{\varphi }_{\alpha \beta }$, between a pair of target phases (blue) or between the homogeneous phase and a target phase (red). (a), right panel: The corresponding distribution of barrier heights, i.e., saddle points on the grand potential landscape, between pairs of phases ${\mathrm{\Omega }}_{\alpha \beta }^{‡}$. (b) Snapshots from $l_{\mathrm{nbr}}=1$ growth simulations, seeded one at a time by each of the 250 target phases. Species belonging to the target phase are colored red or yellow, while those depleted in the target phase are colored blue or cyan. (c) The number of successfully grown target phases as a function of the mean interaction energy within each target phase ${\epsilon }_{\text{target}}$ and the interaction range $l_{\mathrm{nbr}}$; $z=6$ is the lattice coordination number.