(301e) Nucleation Kinetics from Finite-Size Equilibrium Molecular Simulations: From Crystals to Biomolecular Condensates

Nucleation is the process that accompanies the emergence of a stable phase from a metastable one via a first-order phase transition. The characteristic time and length scales of nucleation events make it challenging to characterize experimentally and an ideal playground for atomistic simulations. Nevertheless, when dealing with computationally tractable atomistic models, where several molecules many orders of magnitude smaller than Avogadro's are assembling into a nucleus, the free energy landscape associated with nucleation is distorted. Such a distortion leads to qualitative and quantitative deviations from the macroscopic conditions of interest in engineering applications. Therefore, understanding and mitigating such deviations is typically required to obtain quantitative insight from atomistic simulations [1].

In this work, we take a different perspective, demonstrating that by embracing the finite-size effects that plague nucleation events in confined small volumes, one can devise an efficient route to compute self-consistent thermodynamic and kinetic parameters associated with nucleation. By embracing the effects of confinement, our methodology circumvents the need for direct simulation of rare nucleation events and enables calculating homogeneous nucleation rates as a function of supersaturation from a limited number of equilibrium molecular simulations [2,3,4].

A crucial advancement over prior work in this area [2,3,4], is the explicit acknowledgement of a state correspondence principle (SCP) connecting open and confined ensembles. In the context of this work the SCP posits that a steady-state cluster in the confined system corresponds to a critical nucleus in a hypothetical macroscopic system. This macroscopic system would have a constant supersaturation that is equal to the instantaneous, local supersaturation experienced by the steady-state cluster in the confined simulation. In this work we show how this principle, discussed originally by Grossier and Veesler [5], is general and can be leveraged to obtain absolute homogeneous nucleation rate estimates from equilibrium simulations.

Here, we validate our approach by computing nucleation rates for notable benchmark systems such as NaCl crystallizing from aqueous solution [1], and liquid Argon condensating from a supersaturated vapour [6]. Then, we apply this approach to obtain estimates of nucleation rates for the nucleation of liquid-like biomolecular condensates of the intrinsically disordered protein Dhh1c and some relevant Dhh1c mutants modelled with the Mpipi coarse-grained potential [7].
We show that our proposed approach efficiently computes phase diagrams and overlays them with absolute nucleation rates, providing insight into the kinetics of the assembly process and locating the limit of solution stability on the density/temperature plane [4].

Moreover, by computing thermodynamics and kinetics of phase separation for mutants of Dhh1c, we draw a connection between sequence and phase separation behaviour, which is critical to develop the quantitative structure/activity relationships necessary to rationally design phase separating peptides.
In conclusion, this study underscores the power and versatility of a unified computational strategy that harnesses finite-size effects in equilibrium simulations to gain quantitative insights into the fundamental process of nucleation across diverse chemical and biological systems.

This framework offers a generalizable and computationally efficient tool for predicting phase behaviour, self-assembly kinetics, and material properties, thereby advancing our understanding and control over these crucial phenomena.

[1] Finney, A. R., & Salvalaglio, M. (2024). Molecular simulation approaches to study crystal nucleation from solutions: Theoretical considerations and computational challenges. Wiley Interdisciplinary Reviews: Computational Molecular Science, 14(1), e1697.
[2] Li, L., Paloni, M., Finney, A. R., Barducci, A., & Salvalaglio, M. (2023). Nucleation of biomolecular condensates from finite-sized simulations. The Journal of Physical Chemistry Letters, 14(7), 1748-1755.
[3] Bachtiger, F., Rahimee, A., Li, L., & Salvalaglio, M. (2025). Solution Thermodynamics of l-Glutamic Acid Polymorphs from Finite-Sized Molecular Dynamics Simulations. Industrial & Engineering Chemistry Research, 64(2), 1309-1318.
[4] Villois, A., Capasso Palmiero, U., Mathur, P., Perone, G., Schneider, T., Li, L., Salvalaglio, M., deMello, A., Stavrakis, S. and Arosio, P., 2022. Droplet Microfluidics for the Label‐Free Extraction of Complete Phase Diagrams and Kinetics of Liquid–Liquid Phase Separation in Finite Volumes. Small, 18(46), p.2202606.
[5] Grossier, R. and Veesler, S., 2009. Reaching one single and stable critical cluster through finite-sized systems. Crystal Growth and Design, 9(4), pp.1917-1922.
[6] Salvalaglio, M., Tiwary, P., Maggioni, G.M., Mazzotti, M. and Parrinello, M., 2016. Overcoming time scale and finite size limitations to compute nucleation rates from small scale well tempered metadynamics simulations. The Journal of chemical physics, 145(21).
[7] Joseph, J.A., Reinhardt, A., Aguirre, A., Chew, P.Y., Russell, K.O., Espinosa, J.R., Garaizar, A. and Collepardo-Guevara, R., 2021. Physics-driven coarse-grained model for biomolecular phase separation with near-quantitative accuracy. Nature computational science, 1(11), pp.732-743.