Fig 1.
Characteristic phases in the stickers and spacers formalism.
(a) Dispersed solution phase where the polymers are uniformly mixed in solution. (b) Percolated fluid wherein the polymer chains form a percolated, system-spanning network through physical crosslinks among stickers results. (c) Droplet wherein network formation also causes the polymers to form condensed phases. (d) Two-dimensional representation of the LASSI architecture. The beads with arms denote stickers where arms denote that the monomers are capable of orientational interactions, and the curved lines connecting the monomers represent phantom tethers, which are allowed to freely overlap (implicit spacer model). Different colors denote different sticker and spacer species respectively. Note that the physical bonds are allowed to overlap (dashed circle). For the rest of this work, physical bonds will not be labeled and will only be depicted as overlapping orientational arms.
Fig 2.
Considerations that go into designing a coarse-grained model.
As discussed in the text, the choice of a coarse-grained model has at least three ingredients. These include the type of conformational space (lattice or off-lattice), the nature of the interactions among entities that are represented in the coarse-grained description (isotropic, anisotropic or fluctuating fields), and the parameterization approach. LASSI, as described here, is based on a lattice model that uses anisotropic interactions and a phenomenological model.
Fig 3.
Distribution functions used for calculation of density inhomogeneity.
The data shown are obtained from 5 independent simulations for the An-Bn system with total protein concentration c = 6.89×10−5 voxels-1 and reduced temperature T* = 0.383. Error bars indicate standard deviations. (a) Pair distribution functions P(2)(r) and P0(2)(r), where the former is from the interacting system and the latter from the non-interacting system with chain connectivity (prior pair distribution function). Note that P(2)(r) shows two peaks, the first of which indicates dense phase formation. (b) Radial distribution function . This captures the droplet formation by a sharp and broad peak in the beginning. The inset shows rb where
intersects the line corresponding to
line, delineating between the dense and solution phases. The global density inhomogeneity measure,
, is obtained by integration of absolute deviation of
from 1.
Fig 4.
Assessment of finite size effects in simulations of ideal, non-interacting chains.
(a) Pair distribution functions computed in terms of the spatial separation between chain units. The distributions are maximal at r = L/2, where L is the size of the simulation cell for a given system. Note that L increases as the number of polymers in the system increases. With the exception of the smaller systems, the ideal chains show self-similar behavior for different system sizes. (b) The data plotted in panel (a) are re-plotted in terms of the scaled variable where Li is the size of the simulation cell for boxes with i molecules.
Fig 5.
Assessment of finite size effects in simulations with real interacting chains.
Panels (a), (b), and (c), respectively are the real chain equivalents of panel (a) in Fig 4 computed for three different simulation temperatures that represent three different quench depths of the system into its two-phase regime. Panels (d), (e), and (f), respectively are the real chain equivalents of panel (b) in Fig 9 computed for three different simulation temperatures that represent three different quench depths of the system into its two-phase regime.
Table 1.
Simulation parameters for system description.
Fig 6.
Architecture of the linear multivalent systems.
(a, b) Cartoons to depict the An+Bn and An-Bn systems, respectively. Different colors of beads denote different species of stickers. Note that An-Bn can be simply considered as An+Bn where the two different sections of the proteins are joined together. (c) Linker lengths involved in the architecture (see also Table 1). Each sticker has a neighboring spacer bead that is 1 lattice site apart whereas the neighboring spacer beads are 4 lattice sites apart. This means that consecutive stickers are 6 lattice sites apart and also that the linkers connecting the two have a positive effective solvation volume.
Fig 7.
Phase behavior of the linear multivalent systems.
(a, b) Phase diagrams for the An+Bn and An-Bn systems, respectively. The purple line is a 2-dimensional linear interpolation for = 0.025, and the area encapsulated by the purple line are where the systems have large density inhomogeneities and are thus considered to be phase separated. The green line is a 2-dimensional linear interpolation for ϕc = 0.5 and thus is the proxy for the percolation line. (c, d)
and ϕc curves as a function of concentrations at T* = 0.383 (solid lines in (a) and (b)). (e) Width of the two-phase regime, w(T*), as a function of the reduced temperature. Not only does the An-Bn system have a higher critical temperature (T* ~ 0.6 vs. T* ~ 0.4), but also has a wider two-phase regime than the An+Bn system.
Table 2.
Move frequencies according to their types.
They are normalized to the sum of all frequencies used in each simulation.
Fig 8.
Analysis of acceptance ratios for different move sets.
Curves with different colors indicate acceptance ratios of different types of moves. The dashed lines show the saturation concentrations. The data are obtained from simulations with T* = 0.383. (a) Acceptance ratio data for the An+Bn system. (b) Acceptance ratio data for the An-Bn system.
Fig 9.
Importance of cluster translation moves.
(a) and (b) ϕc curves for the An+Bn (purple) and An-Bn systems (green) at T* = 0.383. The solid lines are identical with the curves in panels (c) and (d) of Fig 7. The dotted lines show the simulation results under the same system conditions but the frequency for cluster translation moves is set to zero. Not only do the systems phase separate and percolate at higher saturation concentrations, but also we can see that both percolation and separation are suppressed highly. Furthermore, errors are generally higher, due to the systems being highly dependent on the initial conditions of the system.
Fig 10.
Architecture of an archetypal branched multivalent system.
(a) Schematic to depict the overall architecture. The pentamer with 10 orange stickers represents the N130 molecule where the gray central oligomerization domain is modeled as a neutral spacer monomer, and the rpL5 peptide is modeled as a linear molecule with 2 blue stickers. (b) Relevant length scales for the architecture (see also Table 1). For the rpL5 molecule a linker length of 3 was chosen between the two stickers, and for the N130 molecule the first sticker (modeling the A1 tract) is 1 lattice site away from the hub spacer whereas the second sticker (modeling the A2 tract) is 3 lattice sites away from the first sticker.
Fig 11.
Phase behaviors of the branched multivalent systems for T* = 0.25.
(a) Full phase diagram, where the purple line denotes the proxy for the binodal and the green line is the proxy for the percolation line (see also the caption for Fig 7). The phase-separated region has an elliptical shape and we have a closed loop, which demonstrates re-entrant phase behavior, whereas the percolation line has a conical shape extending into much higher densities. The solid black lines denote contours of constant total concentration where L1 is the lowest concentration and L3 is the highest concentration. Note that both axes are represented in the log scale. (b, c) and ϕc curves as a function of relative stoichiometric ratio of N130 and rpL5 along the constant-concentration contours. (d) Plot of Λ vs. the apparent stoichiometry along lines L1, L2, and L3.
Fig 12.
Slopes of tie lines within elliptic binodals are important for systems that undergo phase separation via obligate heterotypic interactions.
The ellipse is drawn to fit the locus of points based on the value that meets our criteria for a density transition (see main text). Data for constructing the ellipse were taken from simulations of the N130 + rpL5 system–see Fig 11(A). This ellipse is used to assess the impact of slopes of tie lines for a two-component system comprising of macromolecule A that undergoes phase separation via obligate heterotypic interactions with macromolecule B. (a) Ellipse with nearly horizontal tie lines. The vertical lines shown in green, grey, and blue correspond to fixed values for [B] along the abscissa. As [A] increases, the system traverses across the two-phase regime, delineated by the ellipse, starting outside the ellipse, crossing the ellipse, and exiting the ellipse at high concentrations of A. (b) For each fixed value of [A], the plot shows how [A]Sol varies with [A]. The red points on each curve were extracted from within the two-phase regime, whereas the black points lie outside the two-phase regime. Clearly, [A]Sol does not stay fixed as [A] increases. (c) Equivalent plot to that shown in panel (a) for the tie lines that we obtain for the N130 + rpL5 system. (d) Equivalent plot to that shown in panel (b). Note the non-linear variation of [A]Sol as [A] increases. (e) Ellipse annotated with vertical tie lines. In this case, phase separation of [A] does not depend on obligate heterotypic interactions with B, but B can bind to A and has a choice of binding preferentially to A in either the dense or dilute phase. Here, A becomes the macromolecule and B the ligand. (f) Preferential binding of the ligand to the macromolecule in its dilute phase will shift the saturation concentration, assessed in terms of [A]Sol, upward and this shift will depend on [B]. Accordingly, the plateau value of [A]Sol in the two-phase regime shifts to higher values for higher values of [B].