The interplay between biomolecular assembly and phase separation
- Max Planck Institute for the Physics of Complex Systems, Germany
- Faculty of Mathematics, Natural Sciences, and Materials Engineering: Institute of Physics, University of Augsburg, Germany
- ETH Zurich, Institute of Biochemistry, Switzerland
Figures
Figure 1
Illustration of assembly reaction scheme and classification.
(a) Illustration of the chemical reaction network associated with the formation of assemblies with size . (b) Identification of three classes based on assembly dimension: d=1, 2, 3. (c) Classification of assemblies based on the scaling of their Flory-Huggins interaction propensity.
Figure 2
Phase diagram and assembly size distributions for different classes and assembly strengths.
(a) Phase diagram as a function of and rescaled temperature (with ) in the regime of high assembly strength, i.e., . The green line is the volume fraction threshold at which intermediate-sized assemblies start to appear, which in this regime precedes the binodal (coloured curve). As indicated by the colour code, the monomer fraction mildly varies in the two phases. (b) Size distributions and c pictorial representations corresponding to different regions of the phase diagram, defined by the relative position of the binodal and the assembly threshold. In region ‘i,’ the system is homogeneous and composed of monomers only. Increasing the total volume fraction of assemblies beyond the assembly threshold , the system enters region ‘ii’ where intermediate assemblies appear. Here, the sizes corresponding to the maximum and the average of the distribution scale with , see Appendix 3. Finally, once exceeds the binodal, the system enters region ‘v’ and demixes in two phases, both rich in intermediate assemblies. In (d-f), we focus on the low assembly strength regime, i.e., . In phase diagram d, the binodal now precedes in the assembly threshold. e In region ‘iv,’ the system phase separates but in both phases monomers dominate the size distribution, while in region ‘v’ the -rich phase becomes populated by intermediate-sized assemblies. Progressively lowering the temperature allows switching between these regions, as depicted in (f). (g, h) Behaviour of dilute mixtures as a function of assembly strength, for the two different classes. Notably, assembly below saturation becomes much more accessible for class 2, as can be seen by comparing the green regions ‘ii’ in (g and h).
Figure 3
Gelation transition in phase-separating systems.
(a) Phase diagram for planar (d=2) and three-dimensional (d=3) assemblies in the limit , as a function of and the rescaled temperature (with ). The coloured curve represents the binodal associated with the free energy , which accounts for the emergence of an infinite assembly. The colour code of the binodal line depicts the monomer fraction in the phases, and the black cross ‘x’ indicates the triple point. In the region labelled as ‘sol-sol,’ the system demixes into two phases both populated mainly by monomers, see panel (b), with . In the region labelled as ‘sol-gel,’ on the other hand, a phase (the ‘sol’), obeying , coexists with a phase (the ‘gel’) that is a macroscopic assembly, containing no solvent (). The latter scenario is represented in panel (b), right side. (c) Lowering the temperature allows transitions from the ‘sol-sol’ to the ‘sol-gel’ region, which manifest with a jump in the total volume fraction of the protein-rich phase.
Figure 4
Assembly kinetics at phase equilibrium.
Assuming that the relaxation to phase equilibrium is fast compared to assembly kinetics, we study the slow relaxation to assembly equilibrium in a compartmentalised system. (a) In the sketch, starting from an initial state composed of monomers and solvent only, assemblies selectively appear in phase I, increasing its volume and total volume fraction . (b, c) For Class 1, as time proceeds, the total macromolecule volume fraction in the two phases, , changes inducing the growth of phase I. In (d and e). we show the time evolution of the full-size distribution in phase II and I, respectively. (f, g) For Class 2, as time proceeds, changes in total macromolecule volume fraction in the two phases cause a shrinkage of phase I. This is reminiscent of recent experimental findings that quantify droplet volume changes along with droplet ageing (Jawerth et al., 2020). (h, i) Time evolution of assembly volume fractions in phase II and I, respectively. Time is measured in units of the discretisation time step , where the rate is introduced in Equation 40.
Figure 5
Identification of shrinkage and growth regions for different classes.
Here, we study phase-separating systems initially composed of monomers only and we monitor phase volume changes as they relax to thermodynamic equilibrium. (a) For linear assemblies (d=1) belonging to class 1, the final binodal line (coloured curve) is wider than the initial one (black curve), corresponding to monomers and solvent only (black curve). Areas in orange and light blue correspond to growth and shrinkage of the -rich phase (phase I), respectively. (b) The behaviour of linear assemblies (d=1) belonging to class 2 is remarkably different. Since, in this class, the interaction with the solvent is screened, the final binodal is shrunk compared to the initial one. As a consequence of the shrinkage, the domain corresponding to phase I growth (light blue area) precedes in the shrinkage domain (orange area), for class 2.
Appendix 3—figure 1
A volume fraction threshold separates two assembly regimes in homogeneous systems.
(a) Illustration of assemblies belonging to Class 1 with different spatial dimensions. (b) Assembly size distribution at low total macromolecular volume fraction: . Disregarding assembly dimension, d, the macromolecules are mainly in the monomer state, i.e., . (c) For , the monomer volume fraction saturates at and big assemblies begin to populate the system. For linear assemblies (corresponding to in Equation 8), the distribution becomes peaked at an intermediate value and then exponentially cut off. For planar and three-dimensional assemblies, , the distribution becomes bimodal, with peaks at and , the maximum assembly size (). This bimodal behaviour hints at the emergence of a gelation transition in the limit . In the insets, we show the scaling of concentrations with assembly size. For , and above the threshold, deviations from the classical exponential decay are present. Here , , , .
Appendix 4—figure 1
Gel-sol free energies.
(a) The coloured curve indicates the binodal obtained with the Maxwell construction for , together with the estimate (black line, defined in Equation 31) for the transition between homogeneous and gel states. The black cross ‘x’ indicates the triple point. (c-e) Maxwell construction for three different temperature values, the coloured and black, dashed curves represent convex and concave branches of , respectively. Parameters are the same as those in Figure 3; see Appendix 1—table 1.
Appendix 5—figure 1
The influence of assemblies on the system phase behaviour.
(a) Focusing on systems with belonging to class 1, we compare three binodals corresponding to assembly strength (coloured curves) and the reference binary mixture composed of monomers and solvent only (black curve). The latter can be associated with the limit . The region enclosed by the binodal, corresponding to phase separation, expands even for assemblies with no assembly energy . This can be explained by the entropic advantage caused by size polydispersity. (b) Dependence of the critical volume fraction and critical temperature on the assembly strength . The presence of assemblies causes and to deviate from the reference values (black dashed lines) corresponding to a binary mixture with monomers and solvent only (). In particular, for Class 1, making assemblies more energetically favourable, i.e., decreasing , induces an increase in and a decrease in , in turn making phase separation more accessible. Here , . (c) Comparison between three binodal lines corresponding to systems belonging to class 2 and , with assembly energies (coloured curves) and the reference binary mixture composed of monomers and solvent only (black curve). (d) For Class 2, decreasing , causes and to decrease, overall hindering phase separation. This is caused by the interaction propensity screening in monomers at the bulk of assemblies belonging to class 2; see Equation 10. Here ,
Appendix 5—figure 2
The influence of phase separation on assembly size.
(a) Comparison between the size distribution in a homogeneous system and in the corresponding phase-separated system (averaged in both compartments). Here, we consider linear assemblies (), , , and . We note that the presence of compartments can favour assembly formation, even when the corresponding homogeneous mixture is populated mainly by monomers. The difference in distributions can be quantified utilizing the functional distance, defined in Equation 35. (b) The magnitude of this distance depends on the droplet size and the temperature chosen. The volume corresponding to the maximum distribution distance shifts towards lower values with decreasing temperature . The distributions separated by the maximum distance, for , are the ones displayed in a. , , .
Appendix 6—figure 1
Kinetic trajectory in the multicomponent phase diagram.
Illustration of the assembly kinetics at phase equilibrium for systems corresponding to and initially composed of monomers only.
Videos
Appendix 1—video 1
Shows the time evolution of the assembly volume fractions in both phases, for Class 1.
Parameters are the same as Figure 4b-e.
Appendix 1—video 2
Shows the time evolution of the assembly volume fractions in both phases for Class 2.
Parameters are the same as Figure 4f-i.
Tables
Appendix 1—table 1
Parameters corresponding to the figures in the main text.
We made use of the temperature scale .
| Figure | ||||||
|---|---|---|---|---|---|---|
| Figure 2a | −5 | −2 | - | ∞ | - | - |
| Figure 2b | –0.5 | −2 | - | ∞ | - | - |
| Figure 2c | - | −2 | - | ∞ | - | 0.6 |
| Figure 2d | - | −2 | 0.2 | ∞ | - | 0.25 |
| Figure 3d | −2 | −2 | - | ∞ | - | - |
| Figure 4b-e | −1 | −2 | - | 15 | 0.3 | 0.45 |
| Figure 4f-i | −1 | −2 | 0.2 | 15 | 0.2 | 0.35 |
| Figure 5a | −1 | −2 | - | 15 | - | - |
| Figure 5b | −1 | −2 | 0.2 | 15 | - | - |
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The interplay between biomolecular assembly and phase separation
eLife 13:RP93003.
https://doi.org/10.7554/eLife.93003.3
