Construction of the spinodal and binodal (coexistence) phase boundaries. An illustrative example is shown for the procedure described in the text and in ref. [18]. (a) Free energy in units of V k B T , f (ρ), is given by the blue curve for one selected temperature. The f (ρ) = 0 inflection points for the spinodal phase boundary (Eq. 5) are marked by the two open circles. The common tangent (inclined dashed line) of the two coexistence phases for binodal phase separation is determined by the two equalities in Eq. 10 to obtain the coexisting concentrations ρ α and ρ β marked by the two filled circles. (b) The spinodal (dashed blue curve) and binodal (solid blue curve) phase boundaries are constructed by joining the ρ values of the open and filled circles in (a), respectively, at different temperatures, as indicated by the vertical dashed lines between (a) and (b). Here T is absolute temperature. The shaded regions represent the spinodal (darker) and binodal (lighter) regimes of the phase diagram. Results in (a) and (b) are those of a FH model with f (ρ) = (ρ/N ) ln ρ + (1 − ρ) ln ρ + χρ(1 − ρ), N = 3, and T = 1/χ (same as that in Fig. 1 of ref. [10]); the f (ρ) curve in (a) is for χ = 1.7. (c) and (d): Cartoons of spinodal decomposition (c) and droplet formation in binodal phase separation (d), wherein solute is depicted in gold and solvent in cyan. Microscopic images of IDP spinodal decomposition and binodal phase separation can be found, e.g., for Ddx4 in Fig. 7 of ref. [18] and Fig. 3 of ref. [9] respectively.

Contexts in source publication

Context 1

... the case of incompressible two-component systems that satisfy ρ+ρ w = 1 (without loss of generality, the unit for concentration is chosen such that the maximum concentration is unity, see example in Fig. 1), there is only one independent concentration variable and thus the Hessian matrixˆHmatrixˆ matrixˆH reduces to a single second derivative. It follows that the boundary condition for spinodal instability, detˆHdetˆ detˆH = 0, takes the form ...

Context 2

... ρ to is allowed to vary under some-but not all-fixed environmental conditions (temperature, hydrostatic pressure, etc. [11]), there are two ρ's satisfying the spinodal boundary condition Eq. 5, one on the dilute (relatively smaller-ρ) side and the other on the condensed (relatively larger-ρ) side of the free energy function (Fig. 1a, open circles). When environmental conditions such as temperature that affect the interaction strength (symbolized as u) are varied, the free energy function f also varies accordingly. At a certain interaction strength u cr at which the ρ values for the two spinodal boundaries merge into a single ρ = ρ cr , the u cr and ρ cr are recognized as the ...

Context 3

... numerical techniques described below. Formulations of three-component LLPS have recently been applied to study systems with two polyampholyes with different charge patterns (as model IDPs) [24] and models of natural IDP solutions with salt [28]. In this connection, it is worth noting that short-chain (small N ) FH models similar to those used in Fig. 1 and Fig. 2 for illustration can be useful for modeling LLPS of folded proteins as well [57,58]. A step-by-step practical guide to the calculations in these simple FH models are discussed in Sect. 2.5 ...

Context 4

... object PolySol was created on an M = 32 lattice with ∆x = ¯ a = b/ √ 6, v = 0.0068b 3 , l B = 0.38b and charge sequence sv20. The model polyampholyte sequence sv20 is one of the thirty overall neutral "sv" sequences in ref. [22]. Here we use several "sv" sequences as well as the as1, as4 model polyampholyte sequences from ref. [30] as examples in Figs. 5-10. The field variables, represented by w →w and ψ →psi, were initially set to random complex values using the PolySol.set fields function. The fields were then evolved with time-step ∆t = 0.01 in CL time using the CL step SI function, which implements the semi-implicit integration scheme. At every 50th step, the polymer chemical ...

Context 5

... produced by this fitting procedure. As further illustrations of the simulated phase behaviors of the model LAF-1 system in Fig. 9, snapshots of simulated configurations generated by the VMD package [104,105] (https://www.ks.uiuc.edu/Research/vmd/) are provided in Fig. 9d and f. Additional snapshots generated using the same package are provided in Fig. 10 to illustrate two recent applications of the "slab" simulation protocol: Figure 10a-e shows condensed-phase demixing of two polyampholyte species with significantly different sequence charge patterns, supporting predictions by RPA [24] and FTS [25] (see also Fig. 8). Figure 10f-j shows a polyampholyte-rich droplet in explicit solvent ...

Context 6

... provided in Fig. 9d and f. Additional snapshots generated using the same package are provided in Fig. 10 to illustrate two recent applications of the "slab" simulation protocol: Figure 10a-e shows condensed-phase demixing of two polyampholyte species with significantly different sequence charge patterns, supporting predictions by RPA [24] and FTS [25] (see also Fig. 8). Figure 10f-j shows a polyampholyte-rich droplet in explicit solvent as observed in a recent study of dielectric effects in polyampholyte condensates [35]. ...

Context 7

... snapshots generated using the same package are provided in Fig. 10 to illustrate two recent applications of the "slab" simulation protocol: Figure 10a-e shows condensed-phase demixing of two polyampholyte species with significantly different sequence charge patterns, supporting predictions by RPA [24] and FTS [25] (see also Fig. 8). Figure 10f-j shows a polyampholyte-rich droplet in explicit solvent as observed in a recent study of dielectric effects in polyampholyte condensates [35]. ...

Context 8

... the case of incompressible two-component systems that satisfy ρ+ρ w = 1 (without loss of generality, the unit for concentration is chosen such that the maximum concentration is unity, see example in Fig. 1), there is only one independent concentration variable and thus the Hessian matrixˆHmatrixˆ matrixˆH reduces to a single second derivative. It follows that the boundary condition for spinodal instability, detˆHdetˆ detˆH = 0, takes the form ...

Context 9

... ρ to is allowed to vary under some-but not all-fixed environmental conditions (temperature, hydrostatic pressure, etc. [11]), there are two ρ's satisfying the spinodal boundary condition Eq. 5, one on the dilute (relatively smaller-ρ) side and the other on the condensed (relatively larger-ρ) side of the free energy function (Fig. 1a, open circles). When environmental conditions such as temperature that affect the interaction strength (symbolized as u) are varied, the free energy function f also varies accordingly. At a certain interaction strength u cr at which the ρ values for the two spinodal boundaries merge into a single ρ = ρ cr , the u cr and ρ cr are recognized as the ...

Context 10

... numerical techniques described below. Formulations of three-component LLPS have recently been applied to study systems with two polyampholyes with different charge patterns (as model IDPs) [24] and models of natural IDP solutions with salt [28]. In this connection, it is worth noting that short-chain (small N ) FH models similar to those used in Fig. 1 and Fig. 2 for illustration can be useful for modeling LLPS of folded proteins as well [57,58]. A step-by-step practical guide to the calculations in these simple FH models are discussed in Sect. 2.5 ...

Context 11

... object PolySol was created on an M = 32 lattice with ∆x = ¯ a = b/ √ 6, v = 0.0068b 3 , l B = 0.38b and charge sequence sv20. The model polyampholyte sequence sv20 is one of the thirty overall neutral "sv" sequences in ref. [22]. Here we use several "sv" sequences as well as the as1, as4 model polyampholyte sequences from ref. [30] as examples in Figs. 5-10. The field variables, represented by w →w and ψ →psi, were initially set to random complex values using the PolySol.set fields function. The fields were then evolved with time-step ∆t = 0.01 in CL time using the CL step SI function, which implements the semi-implicit integration scheme. At every 50th step, the polymer chemical ...

Context 12

... produced by this fitting procedure. As further illustrations of the simulated phase behaviors of the model LAF-1 system in Fig. 9, snapshots of simulated configurations generated by the VMD package [104,105] (https://www.ks.uiuc.edu/Research/vmd/) are provided in Fig. 9d and f. Additional snapshots generated using the same package are provided in Fig. 10 to illustrate two recent applications of the "slab" simulation protocol: Figure 10a-e shows condensed-phase demixing of two polyampholyte species with significantly different sequence charge patterns, supporting predictions by RPA [24] and FTS [25] (see also Fig. 8). Figure 10f-j shows a polyampholyte-rich droplet in explicit solvent ...

Context 13

... provided in Fig. 9d and f. Additional snapshots generated using the same package are provided in Fig. 10 to illustrate two recent applications of the "slab" simulation protocol: Figure 10a-e shows condensed-phase demixing of two polyampholyte species with significantly different sequence charge patterns, supporting predictions by RPA [24] and FTS [25] (see also Fig. 8). Figure 10f-j shows a polyampholyte-rich droplet in explicit solvent as observed in a recent study of dielectric effects in polyampholyte condensates [35]. ...

Context 14

... snapshots generated using the same package are provided in Fig. 10 to illustrate two recent applications of the "slab" simulation protocol: Figure 10a-e shows condensed-phase demixing of two polyampholyte species with significantly different sequence charge patterns, supporting predictions by RPA [24] and FTS [25] (see also Fig. 8). Figure 10f-j shows a polyampholyte-rich droplet in explicit solvent as observed in a recent study of dielectric effects in polyampholyte condensates [35]. ...