Experimental validation of polymerization-like self-assembly kinetics and valence-dependent sol–gel–crystal boundaries for spherical, limited-valence DNA-encoded nanoparticles

A 2019 joint theoretical–computational study (Gu, Ma, Zhang and Lin, JACS 2019) predicted that binary mixtures of spherical nanoparticles carrying a small, fixed number of complementary DNA strands (limited valence) self-assemble according to the kinetics and statistics of reversible step-growth polymerization: a single valence-independent assembly rate constant governs chain growth, chain lengths follow the Flory–Schulz distribution, and mean-field Flory–Stockmayer thresholds locate the sol-to-gel transition. These predictions were validated only against coarse-grained simulations and, qualitatively, against experiments on bivalent DNA–protein conjugates. The problem is to test this polymerization-like kinetic description directly on spherical, limited-valence DNA-encoded nanoparticles, to map the valence-, temperature- and concentration-dependent sol–gel boundaries and to examine whether crystalline ordering competes with or supersedes gelation at higher valences, and to quantify the protocol- and molecular-scale effects that produce deviations from the mean-field picture.

Background

DNA-programmed self-assembly is a materials-design strategy in which nanoscale or colloidal particles are grafted with short oligonucleotides so that complementary single-stranded 'sticky ends' hybridize and act as reversible, sequence-specific bonds. When a spherical nanoparticle core (for example a gold nanosphere) carries only a small, controlled number of such DNA linkages, its bonding is said to be of limited valence — the valence being the number of functional DNA bonds a particle can form. Limited valence directs assembly away from the dense crystalline superlattices typical of highly multivalent DNA-coated particles and toward discrete 'colloidal molecules', one-dimensional chains, and open gels. This is the colloidal analogue of molecular step-growth polymerization, the classical reaction scheme in which bifunctional or multifunctional monomers link arbitrarily with no initiator and no chain termination. For ideal step-growth systems, Flory's theory gives simple relations between the fraction $p$ of reacted functional groups and observables: the number-average chain length $M_{n}=1/(1-p)$, a monotonically decaying Flory–Schulz distribution of chain lengths $\mu=p^{X-1}(1-p)^{2}$ (here $X$ is the number of monomers in a chain), and, for monomers of functionality three or more, a critical conversion at which an infinite network first percolates (the Flory–Stockmayer sol–gel threshold). These size and distribution relations concern the finite, soluble clusters: the system-spanning network formed at and above the gel point is not part of the chain-length distribution, and the whole-system number average diverges there. Experimental realizations of step-growth-like colloidal polymerization existed before 2019, but each deviated from the spherical, DNA-bonded, limited-valence setting: polymer-tethered nanorods showed step-growth kinetics with branching and cyclization (Liu, Nie, Zhao, Li, Rubinstein and Kumacheva, Science 2010); supracolloidal polymers of polymer-tethered particles obeyed step-growth kinetics with a diffusion-controlled, concentration-dependent rate coefficient rather than a constant one (Ma, Zhou, Zhang, Lin and Tian, Nanoscale 2018); and bivalent protein–DNA conjugates — β-galactosidase and green fluorescent protein (GFP) modified with one or two oligonucleotides — formed one-dimensional wires and chains with both linear and cyclic products, characterized by gel electrophoresis and cryo-electron microscopy (McMillan and Mirkin, JACS 2018; McMillan, Hayes, Remis and Mirkin, JACS 2018). In this context, Gu, Ma, Zhang and Lin (JACS 2019, 141(41):16408–16415) developed a joint theoretical–computational framework combining coarse-grained molecular dynamics simulations (in which each nanoparticle is a rigid sphere bearing explicit, reversibly hybridizing DNA strands) with reversible step-growth kinetic theory. For binary mixtures of complementary limited-valence DNA-encoded nanoparticles they reported that: the hybridization extent $p=(c_{0}-c)/c_{0}$ (with $c_{0}$ the initial and $c$ the instantaneous free-strand concentration) controls the number-average chain length through $M_{n}=1/(1-p)$ and its time evolution through a reversible step-growth rate law; a single, valence-independent assembly rate constant $k_{a}=0.61\sigma^{3}\tau^{-1}$ (in simulation units) collapses $k_{a}c_{0}$ versus $c_{0}$ across bivalent/trivalent (2/2, 2/3, 3/3) mixtures; equilibrium chain-length distributions follow the Flory–Schulz law; the critical hybridization extents reported for the sol-to-gel transition, obtained within the mean-field gelation treatment, are $p_{c}=0.73$ for the 2/3 and $p_{c}=0.51$ for the 3/3 mixture, separating finite sols from system-spanning gels; and an ellipsoidal rigid-body model of bivalent DNA–protein conjugates reproduces the qualitative behavior of the protein experiments. The authors mapped design variables (strand length, spacers, stoichiometry, temperature, concentration) onto kinetic timescales and final states. Crucially, however, every load-bearing validation of this framework was simulation-based or drawn from the non-spherical protein system: no experimental data on spherical, limited-valence DNA-encoded nanoparticles existed against which the predicted kinetics, size/dispersity, chain-length statistics, or gelation thresholds could be compared, the gel points were not tested quantitatively even in simulation, the phase behavior beyond the mean-field thresholds (in particular the valence range where crystalline superlattices compete with, or supersede, gelation in analogy to the well-known crystal-forming behavior of high-valence DNA-coated particles) was unmapped, and adjacent DNA-mediated experimental systems show documented departures from ideal step-growth behavior — coexisting cyclic and linear products, failure of the constant-reactivity assumption with stagnant oligomer diffusion, and an effective valence above the nominal one — pointing to unquantified protocol, cyclization, unequal-reactivity, and kinetic-trapping effects that the coarse-grained model abstracts away.

Problem Statement

Establish experimentally the polymerization-like self-assembly of spherical, limited-valence DNA-encoded nanoparticles, as follows; parts (1)-(3) below are jointly required and together constitute the experimental test of the framework. (1) Kinetic validation: prepare binary mixtures of spherical nanoparticles with precisely controlled valences (bivalent 2/2, mixed 2/3, and trivalent 3/3, and higher valences where feasible) bearing complementary DNA sticky ends, and measure time-resolved number-average superstructure length $M_{n}(t)$, dispersity, and equilibrium chain-length distributions as functions of temperature and initial strand concentration $c_{0}$; test quantitatively the reversible step-growth predictions of Gu et al. (2019) — the relation $M_{n}=1/(1-p)$ between chain length and hybridization extent, the time-dependent step-growth rate law, the Flory–Schulz form of the chain-length distribution, the existence of a single assembly rate constant $k_{a}$ that is independent of valence and of $c_{0}$ across the 2/2, 2/3 and 3/3 mixtures, and the reported critical hybridization extents $p_{c}=0.73$ (2/3) and $p_{c}=0.51$ (3/3) for the sol-to-gel transition. (2) Phase-boundary mapping: determine the valence-, temperature- and concentration-dependent sol–gel boundaries for these particles and whether the mean-field thresholds locate the experimental gel points, and examine whether crystalline ordering competes with or supersedes gelation as valence increases. (3) Deviation mechanisms: identify and quantify whichever of the protocol and molecular effects — non-equilibrium assembly routes, mixing sequence, kinetic trapping, ring (cyclization) formation, unequal strand reactivity, and sequence-dependent hybridization kinetics — are responsible for observed deviations from the ideal step-growth/Flory–Schulz picture, and establish the limits of applicability of simulation-derived rate constants and scaling relations for real spherical limited-valence DNA-encoded nanoparticles.

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