Who am I?

Some pictures

Polymer networks

Polymeric objects (microgels)

All-DNA materials

DNA nanotechnology

This is not a scientific talk at a conference, it is a lesson

Please do interrupt me to ask questions

Harnessing entropy to drive phase separation

Lorenzo Rovigatti

Physics Department, Sapienza University of Rome

Biological and Soft Matter, December 14th, 2023

The classic, molecular "gas-liquid" picture

Tuning the phase diagram

Example: the effect of the range of a spherical attraction

$\approx$ particle size

Fraction of particle size

P. Camp, Phys. Rev. E (1997), Muschol and Rosenberger, J. Chem. Phys. (1997)

The soft-matter picture

• Particle-particle interactions can be tuned to a large extent
• The phase behaviour can (at least in principle) finely tuned
• A classic example: the depletion interaction
  • Mixture of small (depletants) and big (colloids) particles
  • There is only excluded volume: energy is 0 or $\infty$ $\to$ only entropy counts
  • Two colloids get close $\to$ depletants have more space $\to$ higher entropy

Different mechanisms, similar phase diagrams

• The "molecular" phase diagram[1]
• A depletion phase diagram[2]

Tuning the phase behaviour of soft-matter systems

Self-assembly vs. phase separation

• At low temperature, Janus particles form stable micelles that stabilise the gas[1]
• Chain formation in network fluids stabilises the liquid[2]
• The two mechanisms together can produce a second critical point[3]

Increasing complexity: flexibility & deformability

• Polymers (chains, rings, star polymers, etc.)
• Micron-sized polymer networks (microgels)
• DNA-based particles (DNA nanostars, DNA origami, etc.)

Associative polymers

• Polymers decorated with attractive monomers
• Diluted $\to$ single-chain nanoparticles (SCNPs)[1]
• Concentrated $\to$ transient polymer networks (useful as e.g. model biosystems[2])
• Strong but exchangeable bonds $\to$ covalent adaptable networks (e.g. vitrimers[3])

What contributes to the thermodynamics of associative polymers?

• I will focus on models that can form both intra- and inter-molecular bonds
• At low concentrations polymers don't meet: intra-molecular bonds are more likely
• Forming an intra-molecular bond costs entropy
• All things being equal, inter-molecular bonds are combinatorially favoured

The thermodynamics of associative polymers - results from literature

• According to theory, "simple" associative polymers do not phase separate[1]
  • The conversion of intra- to inter-molecular bonds is continuous
  • SCNPs start to bind to each other and form transient polymer networks
• Experiments[2] and simulations[3] confirm this picture

Associative polymer models

• Equispaced reactive monomers along the chain
• Three models: one, two or three types of (alternating) reactive monomers
• Only like-type reactive monomers can bind to each other

The low-density fully-bonded limit

• In all three cases the energy is the same
• To satisfy all bonds, polymers have to "fold"
• Closing a loop costs entropy: the longer the loop, the higher the cost
• The largest the distance between "compatible" monomers, the longer the loop
• The cost per reactive site is $\approx -4\, k_B$ for $(AAAA)_6$ and $\approx -5\, k_B$ for $(ABCD)_6$

Effective potentials from two-chain simulations

• Add a contribution depending on the chain-chain distance $R$ to $H$, $H_{\rm bias}(R)$
• Run simulations to obtain biased averages, $\langle \cdot \rangle_{\rm bias}$
• Any observable that depends on $R$ can be obtained from biased simulations as $$ \langle O(R) \rangle_{\rm unbias} = \langle O(R) \rangle_{\rm bias} e^{\beta H_{bias}(R)} $$
• If necessary, run simulations with different biases and "glue" the results together, recalling that $U_{\rm eff}(R) = -k_B T \log(g(r)_{\rm unbias})$

Effective potentials from two-chain simulations

• All bonds are formed $\to$ the only contribution is entropic
• Only intra-molecular bonds $\to$ strong repulsion
• Intra- and inter-molecular bonds $\to$ from weakly repulsive to strongly attractive
• In the $(AAAA)_6$ system, $\beta V \approx 0$ $\to$ attraction $\approx$ repulsion

Equations of state in the fully-bonded limit

• The $(AAAA)_6$ behaves like an ideal gas in a large density range
• Both "designed" systems display a non-monotonic equation of state

Signature of phase separation!

• In the $(AAAA)_6$ system, the interfaces melt
• In the designed systems, the density remains heterogeneous

Phase diagram of the (ABAB)$_6$ model

Robustness checks

An interesting biological context where this can be important

Organisms are organised hierarchically

1 m	$\approx 10\, \mu$m	$\approx 1\, \mu$m	$\approx 10$ nm
			McGuffee and Elcock, PLoS Comput. Biol. 2010

Proteins and nucleic acids are polymers!

Organelles

• Perform specialised tasks
• Well-separated from the rest of the cell
• Membrane-bound
• Maintain their "identity" over time
• Communicate through molecular signals

More organelles?

Droplets $\to$ phase separation?

"Liquid-liquid" phase separation (LLPS)

A (gas-like) highly-diluted phase coexist with a (liquid-like) dense phase

Membraneless organelles

E. Gomes and J. Shorter, J. Mol. Bio. 2018

• Often called (sometimes improperly) biomolecular condensates
• Liquid-like (they can and do flow, cfr. Brangwynne et al)
• Made of multivalent (intrinsically-disordered) proteins and/or RNA
• Act as reservoirs of biomolecules or as microreactors
• The mechanisms behind their formation are linked to the pathogenesis of several diseases (e.g. Alzheimer's, ALS)

On-the-fly compartmentalisation

• Phase separation relies on a subtle interplay between entropy and enthalpy
• Slightly changing some conditions can suppress/enhance phase separation
• Condensates can quickly adapt to environmental changes!

A visual example

• We take a multi-component mixture $\to$ we need to set many different interactions (red-blue, red-green, green-blue, etc.)
• For this particular choice the resulting system is homogeneous
• We change a single interaction $\to$ phase separation

What are these things made of?

• There seems to be many different membraneless organelles in cells
• Composition and behaviour vary greatly from type to type
• Made of proteins interacting through intrinsically disordered regions or lock-and-key mechanisms, and/or RNA
• The main structure is called scaffold and is (usually) formed by phase-separating biomacromolecules
• Other macromolecules interacting with it are called clients

Some evidence?

• The scaffold parts are often (bio)polymers that can bind to themselves or to others
• There is also flexibility, torsional stiffness, different interaction strengths, thermal energy, ...
• Simulations on condensate-forming RNA suggests that "our" entropy plays a role

Conclusions & perspectives

• We find a way to controllably tune the behaviour of associative polymers
• The resulting phase behaviour is fully determined by entropy
• The effect may be in play in biological contexts
• Future work: experimental realisation (with DNA?) and dynamics

Thank you!

Model & methods - details

• Implicit-solvent, coarse-grained description: polymer connectivity (Kremer-Grest) + attraction between the reactive monomers
• Molecular dynamics simulations to compute two-body (effective interactions) and many-body (e.g. pressure) quantities
• A three-body potential to enforce a single-bond-per-site constraint and enable a bond-swapping mechanism to speed-up the dynamics

The three-body potential

Configuration
V2	$-\epsilon$	$-2\epsilon$	$-\epsilon$
V3	$0$	$\lambda\epsilon$	$0$
V2 + V3	$-\epsilon$	$-2 \epsilon + \lambda\epsilon$	$\epsilon$

Weak dependence of the # of bonds and $\beta V$ on the spacing

The bond-bond autocorrelation function decays to $\approx 0$