Designing enhanced entropy binding in associative polymers

Lorenzo Rovigatti

Physics Department, Sapienza University of Rome

IntCha24, April 23rd, 2024

A foreword

1. "The $N!$ factor accounts for the indistinguishability of the particles"
2. "Entropy is a measure of disorder"

Alternative title for today: "Harnessing entropy to drive phase separation"

The classic, molecular "gas-liquid" picture

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]

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?

• Monomers can form both intra- and inter-molecular bonds
• I will focus on the fully-bonded state: all bonds are formed, only entropy plays a role
• 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

• The conversion of intra- to inter-molecular bonds is continuous
• SCNPs start to bind to each other and form transient polymer networks

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: longer loop $\to$ higher cost
• Larger distance between "compatible" monomers $\to$ longer loops

Effective (two-body) chain-chain interaction

• 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

Robustness checks

Theoretical confirmation

• We extend the Semenov and Rubinstein theory[1]
• We confirm the numerical qualitative trends with number of attractive species
• We find a quantitative match, perhaps serendipitously

An interesting biological context where this can be relevant

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)

Some evidence for the role of entropy?

• These organelles are made of (bio)polymers that can bind to themselves or to others
• There is 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!

• L. Rovigatti and F. Sciortino, Phys. Rev. Lett. (2022)
• L. Rovigatti and F. Sciortino, SciPost Phys. (2023)
• S. Chiani, F. Sciortino, and L. Rovigatti, in preparation

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	$-\epsilon$	$-2 \epsilon + \lambda\epsilon$	$\epsilon$

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 $U_{\rm eff}(R) = -k_B T \log(g(r)_{\rm unbias})$

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

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

Phase diagram of the (ABAB)$_6$ model