Protein microparticles visualize the contact network and rigidity onset in the gelation of model proteins

The study addresses how protein aggregates evolve into a mechanically rigid gel network and what microscopic mechanism underlies this transition. Protein aggregation is crucial in biology and food science but is hard to model due to complex interactions and multistep pathways (preaggregation to primary aggregates, then network formation). Cold-set gelation enables decoupling of activation (denaturation) from later aggregation. Using well-controlled spherical protein microparticles as models for primary aggregates, the authors directly image gelation in 3D and track network rigidity by microrheology. They investigate whether gelation follows spinodal decomposition, equilibrium percolation, or a nonequilibrium percolation scenario, focusing on the role of particle coordination (Z) as an order parameter and on the onset of rigidity.

The paper situates protein gelation among general gelation phenomena in polymers, proteins, and colloids, where various routes to arrest and rigidity exist depending on attraction strength, range, and density. At high attraction, irreversible sticking leads to open, ramified structures; at lower attraction near equilibrium, compact structures can form and gelation may proceed via spinodal decomposition followed by glass-like arrest. Other proposed mechanisms include equilibrium percolation and rigidity percolation. Prior work on globular proteins reports fractal aggregation with dimensions 1.7–2.3, similar to colloids, particularly near the isoelectric point where interactions are strong and irreversible. The authors reference recent findings on weakly attractive colloids showing nonequilibrium percolation-like critical behavior, providing a framework to test with protein microparticles.

• Materials and particle synthesis: Whey protein isolate (WPI) microparticles were prepared by a modified two-step emulsification (microemulsification) method. An aqueous WPI stock (25% w/w) was emulsified in sunflower oil containing 5% (w/w) PGPR using a rotor-stator homogenizer (5000 rpm), then heated at 80 °C for 20 min to form heat-induced WPI microgel beads. Oil was removed by centrifugation (36,000 × g, 1 h), beads were washed/extracted with acetone overnight, filtered, dried to yield a powder. Composition of WPI: ~50% β-lactoglobulin, ~20% α-lactalbumin, remainder BSA, IgG, proteose peptones.
• Particle properties and sample preparation: Microparticles were spherical with mean diameter 2r0 = 1.9 µm and ~15% polydispersity; fluorescently labeled with rhodamine. The aqueous solvent contained 60% (w/w) sucrose to index- and density-match the particles, minimizing sedimentation and slowing diffusion (single-particle diffusion time ~60 s per radius). Typical particle volume fraction φ = 6% (range 3–6% tested, no qualitative change). Aggregation was induced by 0.36% (w/w) glucono-δ-lactone (GDL), slowly acidifying toward the isoelectric point pH ≈ 4.7 after ~370 min. Temperature T = 20 °C.
• pH monitoring: Separate vial measurements recorded pH every 30 s over 700 min for both microparticle and regular protein suspensions to track acidification.
• Macroscopic rheology: Bulk rheology measured storage (G′) and loss (G″) moduli during acidification for both proteins and microparticles to identify gelation (G′ surpasses G″).
• Confocal microscopy and 3D particle tracking: Laser-scanning confocal imaging (Zeiss 5 Live) recorded 3D stacks (108 µm × 108 µm × 40 µm; 200 images with 0.2 µm spacing) every 5 min, and fast 2D time series (5000 frames at 24 s−1) at 20 µm height to assess dynamics. Particle positions were located with trackpy, with estimated localization accuracy ~20 nm (xy) and ~35 nm (z). Bonds were defined for particle pairs separated by less than the inflection point of the pair correlation function g(r). Bonded particles were grouped into clusters; local coordination number Z (number of bonded neighbors) was computed. Cluster reconstructions colored by Z visualized structural evolution.
• Microrheology: Two-point cross-correlation of particle displacements as a function of separation was used to extract frequency-dependent viscoelastic moduli (following Mason-Weitz approach). Evolution of G′ and G″ versus time and versus mean coordination 〈Z〉 was analyzed, including low-frequency plateau emergence.
• Dynamics analysis: Mean-square displacements (MSDs) were computed before and after subtracting cluster center-of-mass motion to show particle arrest within clusters and diffusive motion of entire clusters.
• Critical scaling analysis: The order parameter was 〈Z〉. The fraction of particles in the largest cluster fp(〈Z〉), cluster correlation length ξ(〈Z〉) computed via ξ = sqrt(2 Σi Rgi^2 Ni / Σi Ni^2), and degree of polymerization pm (total bonded particles) were measured versus 〈Z〉. Power-law divergences approaching critical 〈Zc〉 were fitted to obtain exponents α, ν, and γ. Cluster mass distributions P(n) were fitted to P ∝ n^−3/2 exp(−n/ns) and analyzed at different normalized times t/tg. Fractal dimension df was extracted from cluster mass versus radius of gyration scaling.

• Irreversible aggregation: Particles that join clusters become immobilized relative to the cluster; clusters themselves diffuse, demonstrating broken detailed balance and irreversible bonding.
• Percolation-driven transition: As mean coordination 〈Z〉 increases, the fraction of particles in the largest cluster fp diverges with fp ∼ (Zc − 〈Z〉)^α, with α = 1.60 ± 0.15 and Zc = 3.3 ± 0.1. The largest cluster grows rapidly while the second-largest declines as percolation is approached.
• Diverging length and connectivity scales: The cluster correlation length diverges as ξ ∼ (Zc − 〈Z〉)^ν with ν = 0.8 ± 0.1. The average degree of polymerization (total bonded particles) diverges as pm ∼ (Zc − 〈Z〉)^γ with γ = 0.8 ± 0.1.
• Mechanical rigidity onset: Microrheology shows G′ rises above G″ with the development of a low-frequency plateau. Plotting G′ versus 〈Z〉 indicates a power-law increase with exponent ≈1 near Zc = 3.4 ± 0.1, consistent with random networks dominated by bond-bending interactions.
• Cluster statistics and structure: Cluster-mass distributions evolve from exponential cutoff to a power law P(n) ∼ n^−3/2 near gelation. Aggregates exhibit a constant fractal dimension df = 2 across growth stages. The hyperscaling relation α = ν df holds (α ≈ 1.6, ν ≈ 0.8, df ≈ 2).
• Network morphology: Coordination number analysis shows a significant fraction of Z = 2 with bond angles near π, indicating thin single-particle strands and ramified structures, attributed to strong attractions (~30 kBT near isoelectric point). Critical coordination numbers are lower than in weakly attractive colloids (Zc ≈ 3.3 vs ~5.5), reflecting more open structures.

Direct, particle-resolved observations link the emergence of mechanical rigidity in protein microparticle gels to a nonequilibrium continuous percolation transition controlled by connectivity 〈Z〉. Irreversible bonding drives kinetic, nonequilibrium percolation distinct from equilibrium phase-separation scenarios. Mechanical data suggest rigidity arises from bond-bending-dominated networks, compatible with slender strands and strong interparticle bonds. Critical exponents (α ≈ 1.6, ν ≈ 0.8, γ ≈ 0.8), cluster-size power-law (−3/2), and df = 2 match percolation predictions and align with results in weakly attractive colloids, suggesting universality across short-range attractive particulate systems. The lower Zc here reflects more ramified structures due to stronger attractions. These insights likely extend to protein systems that form spherical primary aggregates before gelation, informing design and control of gel microstructure and mechanics in food and biomaterial applications.

The study provides a microscopic, quantitative framework for protein gelation using model protein microparticles: gelation proceeds via a nonequilibrium percolation process with connectivity-controlled criticality and bending-dominated rigidity. Structural (fp, ξ, pm) and mechanical (G′) observables exhibit consistent power-law behavior with exponents matching percolation theory and prior colloidal studies, implying universality. The approach decouples activation from aggregation, enabling direct visualization of contact networks and rigidity emergence. Future work should explore broader protein chemistries, interaction strengths and ranges, volume fractions, and larger fields of view, coupled with large-scale simulations, to generalize and test the universality and to connect microscopic bond properties to macroscopic gel mechanics.

• Affiliation of findings to a single protein system (WPI) and specific preparation route (microgel microparticles) may limit generalizability to other proteins or native aggregates.
• Finite imaging volume and field of view could bias percolation measurements near criticality; boundary effects and finite-size scaling were not exhaustively analyzed.
• Particle size and polydispersity (~1.9 µm, 15%) and solvent modifications (60% sucrose) alter diffusion and hydrodynamics relative to protein-only systems.
• Acidification protocol (0.36% GDL) sets a specific pH-time trajectory; other kinetics or ionic conditions were not surveyed.
• Mapping from model microparticle networks to molecular-scale protein gels assumes similar interaction topology; direct molecular-scale validation is pending.
• Exponent estimates carry uncertainties from bond definition (g(r) inflection), tracking accuracy, and fitting ranges; alternative bond criteria were not reported.