Enrico
Lattuada
*,
Debora
Caprara
,
Vincenzo
Lamberti
and
Francesco
Sciortino

Physics Department, Sapienza University, P.le Aldo Moro 5, 00185, Rome, Italy. E-mail: enrico.lattuada@uniroma1.it

Received
27th June 2020
, Accepted 16th October 2020

First published on 12th November 2020

Taking advantage of the base-pairing specificity and tunability of DNA interactions, we investigate the spontaneous formation of hyperbranched clusters starting from purposely designed DNA tetravalent nanostar monomers, encoding in their four sticky ends the desired binding rules. Specifically, we combine molecular dynamics simulations and Dynamic Light Scattering experiments to follow the aggregation process of DNA nanostars at different concentrations and temperatures. At odds with the Flory–Stockmayer predictions, we find that, even when all possible bonds are formed, the system does not reach percolation due to the presence of intracluster bonds. We present an extension of the Flory–Stockmayer theory that properly describes the numerical and experimental results.

In recent years, the relationship between reversible self-assembly of patchy colloidal particles and irreversible aggregation of chemical units is receiving a renewed interest.

For the case of colloidal particles with f patches, there is evidence that the range of validity of the FS predictions becomes wider and wider upon decreasing f.^{7,8} For binary mixtures of very small average “valence” 〈f〉, the FS predictions provide a quite accurate description of the cluster size distribution, except for bond probabilities very close to the percolation threshold (which is properly predicted theoretically).^{1} For this reason, patchy colloidal particles have become a test ground for revisiting old FS predictions under proper equilibrium conditions. Furthermore, they provided a way to access the role of the bonding loops (i.e. close paths of bonds), which are commonly neglected both in FS and Wertheim theories.

An interesting aggregation phenomenon takes place in one-component systems made of AB_{f−1} monomers‡ (i.e. where each monomer is composed of one reactive group of type A and f − 1 of type B), where A condenses with B, but reactions between identical functional groups (AA and BB) are forbidden. The clusters resulting from this aggregation process are commonly known as hyperbranched polymers, a term introduced by Kim and Webster in their works on the synthesis of highly branched polymers.^{9,10} In the last decades, the interest towards the synthesis and understanding of these materials has continuously grown, representing a challenge for innovative applications. Hyperbranched polymers constitute an appealing alternative to dendrimers, owing to their facile synthesis and high tunability.^{11,12} Similar to other branched polymers, they are characterized by high exposure of functional groups, three-dimensional globular structure, low viscosity, and good solubility.^{13,14} Potential applications include surface coating,^{15} composite material filling to increase the thermal and mechanical stability,^{16} drug and gene delivery,^{17–21} nanoparticle grafting for diagnostic imaging to reduce the toxicity,^{22,23} and their use in sensors.^{24–26}

From the theoretical standpoint, the aggregation of AB_{f−1} units is particularly interesting for several reasons: (i) it is analytically tractable (neglecting the formation of closed bond loops); (ii) the cluster size distributions for branched polymers are requested as an intermediate step in the evaluation of several polyfunctional condensation processes; and (iii) it gives rise, according to the FS theory, to an aggregation phenomenon in which the fully bonded case (when all A groups have reacted) corresponds to the percolation transition. Therefore, hyperbranched polymers do not have a gel phase but only a sol one.

Despite the interest towards hyperbranched polymers, an accurate comparison of the FS theoretical predictions with numerical and experimental results has rarely been attempted.^{27,28} Actually, the hypothesis of the absence of intramolecular reactions is expected to get progressively worse for large degrees of polymerization. In fact, an unreacted A will most likely interact with one of the (nearby) B sites belonging to the same cluster, hence forming a closed loop. As a result, the cluster size distribution of hyperbranched polymers may not follow the FS predictions.

In line with the conceptual correspondence between patchy colloids and reactive monomers, we present here a combined numerical and experimental study of a colloidal analog of the AB_{f−1} hyperbranching condensation. In particular, we design specific DNA oligomers able to self-assemble into bulk quantities of identical four-armed (i.e. f = 4) AB_{3} particles which can interact in a controlled way.^{29–32} We exploit molecular dynamics (MD) simulations based on the oxDNA2 coarse-grained interaction potential^{33,34} to follow the particle aggregation process and to compare the numerical results with theoretical predictions. This numerical study allows us to estimate the role of intramolecular binding and how the presence of closed loops modifies the cluster size distribution. We then demonstrate that a cluster-based thermodynamic treatment, which also includes intracluster bonds, can be developed to extend the FS theory in order to accurately describe the simulation data and to provide a significant guide to the experimental results. Finally, we realize the same system in the laboratory and follow the aggregation process via Dynamic Light Scattering (DLS) to provide evidence that the presence of intracluster bonds prevents the formation of a percolating state, even in the limit of full bonding.

Most of the previous works on DNA NSs focused on the A_{4} case, tetra-armed NSs with identical self-complementary sticky sequences, originating all identical AA bonds.^{29–31,39–47} The A_{4} system shows the analog of the gas–liquid phase separation at low temperatures, driven by the association of the self-complementary sticky ends. For concentrations larger than the “liquid” coexistence value, the system forms a reversible equilibrium gel, which exhibits the static and dynamic features expected for colloidal particles of valence four.^{29,30}

With the specific design presented here, an individual NS possesses one sticky end of type A and three of type B in order to mimic a tetravalent AB_{3} monomer. The A and B sticky regions have been adequately selected to allow only AB bonds between different NSs (Fig. 1c and e).

The temperature response of the system is strictly related to the number of nucleotides in the complementary sequences, whose length allows to distinguish different hierarchical self-assembly processes. As shown in Fig. S1 of the ESI,† above T_{NS} ≈ 77 °C; the sample is composed of single and freely diffusing DNA strands. Around T_{NS}, the complementary sequences comprising the double-stranded arms start to self-assemble, giving rise to a solution of unbonded NSs. On further cooling, around T_{b} ≈ 42 °C, the sticky ends start to pair forming inter-, and possibly intra-, NS AB bonds. At ambient temperature and below, essentially all possible AB bonds are formed.

Initial configurations are generated by randomly placing copies of an already assembled DNA tetramer in the simulation box, provided there is no overlap between the nucleotides (for more details, see section B of the ESI†). The volume is computed for the different values of N and concentrations (c = 2 mg ml^{−1}, 10 mg ml^{−1}, and 20 mg ml^{−1}, equal to those experimentally investigated), using a tetramer molecular weight of M_{w} = 63893 g mol^{−1}. We perform MD simulations in the NVT ensemble with systems consisting of N = 300, 1000, and 2000 DNA NSs of AB_{3} type. The largest system corresponds to ∼4 × 10^{5} interaction sites. The temperature in the simulation, kept at T = 45 °C, is enforced using an Anderson-like thermostat that emulates a Brownian motion.^{51} For the selected temperature, we are able to follow the equilibration of the system for up to 10^{10} MD time steps (corresponding to ∼30 μs of real time and six months of continuous computation per run), taking advantage of the computational power of modern Graphic Processing Units (GPU).

We experimentally investigate three samples at different NS concentrations: 2 mg ml^{−1}, 10 mg ml^{−1}, and 20 mg ml^{−1}. The samples are prepared in borosilicate glass capillaries (inner diameter 2.4 mm, Hilgenberg GmbH). We dilute the NS batch suspension with a NaCl solution at a proper molarity, in order to obtain 30 μl of each sample at a final NaCl concentration of 250 mM. Finally, we cover the suspension with 20 μl of silicone oil and seal the capillary using UV resin to avoid sample evaporation.

DLS measurements are carried out at a fixed angle θ = 90° with a custom-made setup consisting of a 633 nm He–Ne Laser (17 mW, Newport Corp.) and a multi-tau digital correlator (Brookhaven Inst.) connected to a multi-mode optical fiber. The scattered light impinging on the fiber is spatially filtered, resulting in a high coherence factor (β ≃ 0.8). Samples are immersed in a water bath connected to a thermostat. The actual temperature of the bath near the sample is measured using a thermocouple probe with a ΔT = ±0.05 °C accuracy.

For each selected temperature, within the interval 10 °C ≤ T ≤ 50 °C every ΔT ≃ 5 °C, the sample is thermalized for 40 minutes before starting the acquisition. Each measurement lasts 10 minutes. The autocorrelation functions of the scattered intensity g_{2}(t) are calculated from the correlator output and converted into the field correlation functions g_{1}(t) using the Siegert relation.^{52}

Similar to other studies, we consider a cluster as an object formed by NSs joined together via AB bonds. Each cluster has at the most one unreacted A site. This A site can bind to any of the unreacted B sites belonging to the same cluster (hence forming an intracluster bond, also indicated as a loop) or to an unreacted B site of another cluster, thus merging the two clusters. The FS mean-field theory^{5,54} provides a prediction for the number of clusters N_{c}(n) of size n

N_{c}(n) = N(1 − p_{b})F(n,p_{b}), | (1) |

(2) |

(3) |

The FS theory, which is formally equivalent to a constrained maximization of the combinatorial entropy,^{55} is based on the evaluation of the number of distinct cluster arrangements, with the restriction that the N monomers are connected by Np_{b} bonds to form polydisperse loopless clusters. In eqn (1), the term N(1 − p_{b}) is equal to the total number of clusters and it clearly reveals the mean-field approximation, being the number of clusters equal to the number of particles minus the number of bonds (Np_{b}).

Fig. 2a and 3a show the cluster size distribution N_{c}(n)/N at three different values of p_{b} (three different times during the simulation) for two different NS concentrations (c = 20 mg ml^{−1} and c = 2 mg ml^{−1}, respectively). The insets show the corresponding time evolution of p_{b}. As can be seen, the FS theoretical predictions, with no fit parameter, become incapable of representing the numerical data when p_{b} ≳ 0.5. This is very clear for the data at 20 mg ml^{−1}, for which the FS predictions underestimate the distribution of small clusters by more than a factor of two. The disagreement between theoretical predictions and numerical data at finite times could originate from kinetic pathways and/or due to the presence of closed loops, neglected in the FS approach. At long times, however, when thermodynamic equilibrium is approached, disagreement can only be ascribed to the presence of intracluster bonds.

Fig. 2 The main graphs show the number distribution of clusters (symbols, panel a) and the number distribution of clusters in the FS subset (panel b) of size n for different simulation times (i.e. different bonding probabilities) for the simulation at c = 20 mg ml^{−1}, N = 2000. The lines are the theoretical predictions given by eqn (1) using p_{b} (panel a) and p^{FS}_{b} (panel b), respectively, whose evolution over the simulation time is displayed in the insets. Symbols and colors correspond to the time step relative to the curves in the main graphs. Legend: Black squares, t ≃ 0.06 μs; red circles, t ≃ 0.30 μs; blue triangles, t ≃ 1.52 μs. |

Fig. 3 Similar to Fig. 2, the main graphs show the number distribution of clusters (symbols, panel a) and the number distribution of clusters in the FS subset (panel b) for different simulation times for the simulation at c = 2 mg ml^{−1}, N = 1000. The lines are the theoretical predictions given by eqn (1) using p_{b} (panel a) and p^{FS}_{b} (panel b), respectively, whose evolution over the simulation time is displayed in the insets. Symbols and colors correspond to the time step relative to the curves in the main graphs. Legend: Black squares, t ≃ 0.30 μs; red circles, t ≃ 0.91 μs; blue triangles, t ≃ 3.03 μs. |

To double-check the role of intracluster bonds, facilitated by the flexibility of the NS and by the growing density of B sites upon increasing the cluster size, we calculate the number of clusters with no reactive A sites, which is equal to the number of intracluster bonds. We also separate the clusters into two groups: the proper FS loopless clusters and the ones with intracluster bonds. The latter violate the hypothesis of the FS theory, which assumes that each cluster has one and only one reactive A site. For the clusters with no loops (NL), we calculate the total number of existing bonds #^{NL}_{b} and the total number of particles composing these clusters N_{NL}. The ratio between these two numbers provides the bond probability p^{FS}_{b} = #^{NL}_{b}/N_{NL} for the subset of clusters satisfying the FS hypotheses. The size distributions of the FS-compatible NL clusters for different simulation times are shown in Fig. 2b and 3b and compared with the FS predictions from eqn (1), identifying N with N_{NL} and p_{b} with p^{FS}_{b}. The quality of the agreement confirms that for the NL clusters, where the FS hypotheses hold by default, the mean-field predictions properly represent the data, suggesting that the time evolution of the aggregation process is sufficiently well-described by the equilibrium solutions.^{56}

Additionally, we provide a quantification of the structural properties of the clusters. We evaluate the mean squared radius of gyration

(4) |

(5) |

Going back to the cluster size distribution, we next provide an extension of the FS theory to account for intracluster bonds. We recall that, formally, for weakly interacting clusters, the probability of formation of a cluster of size n in equilibrium is proportional^{55,57} to its partition function Q_{n} multiplied by an (concentration-dependent) activity z^{n}. As shown in section D of the ESI,† the FS equation can indeed be recast in this ideal gas of clusters thermodynamic formalism as

N^{NL}_{c}(n) = Q^{NL}_{n}z^{n}, | (6) |

(7) |

To include the possibility of intracluster bonds, we sum the partition functions associated with the configurations without and with loops and we write the cluster size distribution of the entire system as

N_{c}(n) = (Q^{NL}_{n} + Q^{L}_{n})z^{n}. | (8) |

The partition function Q^{L}_{n}, compared to Q^{NL}_{n}, must include two terms: (i) an additional factor , which accounts for the presence of the intracluster extra bond, and (ii) a model-dependent factor g(n,β), which quantifies the free-energy gain of forming an intracluster bond. The factor g(n,β) includes the relative number of microscopic configurations with an intracluster loop (with respect to a loopless cluster). It may also include the free-energy cost of bringing the selected B site close to the unbonded A site, and thus it could, in principle, (weakly) depend on T if the monomer arms are not quite flexible. The partition function Q^{L}_{n} can then be written as

(9) |

The unknown g(n,β) can be estimated by evaluating the ratio between the number of clusters of size n with and without loops from the simulated configurations at long times, when equilibrium has been reached,

(10) |

It is worth noting that the ratio N^{L}_{c}(n)/N^{NL}_{c}(n) depends only on the temperature but not on the concentration.

Fig. 5 compares the predictions of eqn (8) and (9) with the numerical data from the simulations at c = 2 mg ml^{−1} (panel a) and c = 20 mg ml^{−1} (panel b) at long times, close to (if not at) equilibrium. In the comparison, g(n,β) and are the only fit parameters (β is fixed), identical for all the densities. The values of z are fixed by the concentration of loopless monomers. Despite the intrinsic noise of the data, the theoretical predictions well represent the numerical values at all densities. The fit suggests that g(n,β) is essentially constant already for n ≳ 2 (i.e. for clusters composed of two monomers or more). This confirms that the unbonded A site essentially binds with a B site on the same particle or, at most, with one of its neighbor monomers. To provide additional support for this statement, we investigate the distribution of loop sizes,§ confirming that the average loop size is quite small (≃1.7 ± 1).

Fig. 5 Comparison of the prediction of eqn (8) and (9) (lines) with the simulation equilibrium data (symbols). Data are obtained by averaging the cluster size distribution from two simulations at the same simulation time. Panel a refers to the simulation at c = 2 mg ml^{−1} (t ≃ 6.9 μs). Panel b refers to c = 20 mg ml^{−1} (t ≃ 2.85 μs). Legend: Circles, N_{c}(n)/N; squares, N^{L}_{c}(n)/N; triangles, N^{NL}_{c}(n)/N. |

Even at the coarse-grained level of the oxDNA model, simulations are still too demanding to access lower temperatures than the one we have studied. However, the previous model allows us to predict the expected cluster size distribution at low T, when the driving force for bonding becomes quite strong and all A sites have reacted. Under these conditions, Q^{NL} is negligible compared to Q^{L} for all n. The cluster size distribution will coincide with the distribution of the clusters with an intracluster bond and, therefore, will be given by

(11) |

(12) |

The term [V_{ref}N_{c}(1)/Vg(1,β)] acts as a renormalized activity. Its value can be tuned to fix the average concentration. The predicted low-T cluster size distributions for the three different investigated densities are shown in Fig. 6. The insets of the same figure show the associated mean cluster size (MCS) and the relative radius of gyration of the mean cluster R^{(MC)}_{g} as a function of the NS concentration, respectively. From these results, we can formulate three important considerations. First, eqn (12) shows that the temperature (apart from the weak dependence entering in g(n,β)) does not play any role: once all possible bonds are formed, the equilibrium distributions are the ones that maximize the entropy. Second, the same equation shows that the NS concentration modulates the cluster size distribution, at odds with the FS predictions, which suggest the formation of an infinite percolating cluster incorporating all monomers. Third, and more importantly, the cluster size distribution remains finite at all physical values of the NS concentrations. Hence, the chance to form intracluster bonds eliminates the possibility to approach the percolation transition. In a more physical way, eqn (12) tells us that, when particles can satisfy all their bonds within the same cluster, the fully bonded (low-T) configuration is not the percolating one. Rather, the equilibrium low-T state exploits the entropic gain provided by the exploration of the system volume by a multiplicity of clusters, modulated by a slightly modified – by g(n, β) – FS combinatorial term.

Fig. 6 Main: Predicted number distribution of clusters N_{c}(n)/N in the limit of the fully bonded system (eqn (12)). Top inset: Mean cluster size as a function of the NS concentration. The line is obtained by analytically computing the mean cluster size from the low temperature cluster distribution of eqn (12). Bottom inset: Radius of gyration of the mean cluster R^{(MC)}_{g} as a function of the NS concentration. The line is obtained by combining the MCS dependence with the fit of the radius of gyration shown in Fig. 4. |

To test the numerical findings, we realize the very same system in the laboratory and examine it via DLS. With experiments, we are not limited to the investigation of one single temperature. Instead, by changing T, we can probe different p_{b} values and even explore the T-window where all bonds are formed (p_{b} = 1 for T ≲ 20 °C), as shown by the melting profile of the sticky sequences reported in section A of the ESI.† In addition, experiments allow to probe the equilibrium properties of the system. Samples are left to equilibrate for several minutes (∼40 min), a time sufficiently long to break and reform several bonds between the A and B sticky ends. We checked that all results are reproducible upon increasing and decreasing T scans and are not affected by aging nor by the previous history.

Fig. 7a–c show the autocorrelation functions of the scattered field g_{1}(t) for the three investigated NS concentrations (c = 2 mg ml^{−1}, 10 mg ml^{−1}, and 20 mg ml^{−1}) and for the explored Ts. For all samples and temperatures, the correlation functions decay to zero within the experimentally accessible time window (10 s), confirming the sample ergodicity. For all studied concentrations, the system is far from a percolation transition, consistent with the numerical simulations and the proposed theoretical extension of the FS theory.

To quantify the slowing down of the dynamics and to extract a typical (slow) relaxation time, the correlation curves are fitted to a double stretched exponential function (see section E of the ESI† for comparison to the fit function)

g_{1}(t) = (1 − A_{s})exp(−t/τ_{f}) + A_{s}exp(−t/τ_{s})^{βs}, | (13) |

(14) |

The insets of Fig. 7a–c show the values of β_{s}, which are associated with the slow relaxation process. For all the measurements, the values lie within the range 0.4 ≲ β_{s} ≲ 0.6. The values of the slow relaxation time are shown in Fig. 8 for all the concentrations and temperatures. To eliminate the trivial effect of the temperature dependence of the solvent viscosity, the times are rescaled to the viscosity η_{solv} of the 250 mM NaCl solvent at the highest investigated temperature (T_{ref} ≃ 48.5 °C) as

(15) |

Fig. 8
T-Dependence of the (viscosity rescaled) slow decay time for samples at the three different concentrations: 20 mg ml^{−1} (blue), 10 mg ml^{−1} (orange), and 2 mg ml^{−1} (magenta). The lines are the two estimated bond lifetimes, calculated according to eqn (16) using α = 0.5 (red) and α = 1 (black), to provide support to the assumption that the relaxation process takes place in a fixed bonding pattern. |

In general, two distinct processes – which can only be disentangled in particular conditions – contribute to the relaxation process probed by DLS: cluster restructuring and cluster diffusion.

The cluster restructuring times, i.e. the times associated with the cluster fracturing and/or the coalescence of different aggregates, are related to the breaking of bonds. Hence, the bond-breaking times control the cluster restructuring. In our DNA design, explicitly selected to generate a fully bonded system for T ≲ 20 °C, 9 base pairs take part to form the AB bond, resulting in an enthalpic ΔH and entropic ΔS contribution to the binding free-energy ΔG equal to ΔH = −61.5 kcal mol^{−1} and ΔS = −178.3 cal mol^{−1} K^{−1}, evaluated using the web application DINAMelt^{58} based on SantaLucia.^{53} A reasonable estimate of the bond lifetime is^{41}

τ = τ_{0}e^{αΔG/RT}, | (16) |

At the lowest density (c = 2 mg ml^{−1}, Fig. 7a), where the hypothesis of independent clusters is more realistic, the relaxation process is quite fast and it does only show a weak hint of two-step relaxation, suggesting that the experiments are probing the free diffusion of limited-size clusters. In Fig. S3d of the ESI,† we compare the experimental data at c = 2 mg ml^{−1} with both the simple and double stretched exponential (eqn (13)) fits. We also notice that, even at low c, the sample is not monodisperse but composed of small aggregates (with a polydispersity lower than the one for the larger concentrations). The relaxation time grows only by a factor of ≈5 when T varies from 50 °C to 10 °C (corresponding to p_{b} from ≈0 to ≈1). At c = 10 mg ml^{−1} (Fig. 7b), a weak additional relaxation process appears, signalling the onset of interactions between clusters. The respective image from the simulation (Fig. 7e), which we can use as a visual aid to interpret the DLS data, suggests that it may become appropriate to tentatively interpret the fast decay as originating from the cluster-free diffusion, while the slow one comes from the confinement induced by the presence of nearby clusters. This hypothesis is supported by the data at c = 20 mg ml^{−1} (Fig. 7c) – and again by the simulation snapshot in Fig. 7f –, where the excluded-volume interactions between clusters are significant, as illustrated by the pronounced two-step decay of the correlation functions.

We then intend to compare the simulation results with the experiments for the sample at c = 2 mg ml^{−1}. We evaluate the intensity-averaged¶ hydrodynamic radius from the cluster size distribution obtained from the simulations at T = 45 °C (cf.Fig. 5) and from the low-T theory (cf.Fig. 6). For each cluster of size n, we calculate the hydrodynamic radius following the recipe described in ref. 60, which provides the equivalent ellipsoid associated with the cluster. The actual hydrodynamic radius is then computed from the known formulae.^{61} Finally, as done for the radius of gyration, we evaluate the cluster size dependence of the ensemble average of R_{h}. The power law fit to the data gives 〈R_{h}〉(n) ≃ R_{0}n^{γ,} with

(17) |

(18) |

For further details, see section F of the ESI.† At high temperature, equal to the one numerically investigated (T = 45 °C), we obtain from the simulations an intensity-averaged hydrodynamic radius of 〈R_{h}〉_{z} ≃ 9.3 nm (using the radius of gyration, one obtains 〈R_{g}〉_{z} ≃ 8.1 nm). By analyzing the correlation time associated with the fast relaxation τ_{f}, we estimate a mean hydrodynamic radius R_{h} ≃ 5.5 nm. At low temperature, the numerical prediction (see Fig. 6) provides an expected radius 〈R_{h}〉_{z} ≃ 14.9 nm (〈R_{g}〉_{z} ≃ 13.2 nm), to be compared to the experimental value of R_{h} ≃ 22.5 nm.

The discrepancy between the experimental values and the ones estimated from the simulations – which is about 50% at low temperature and grows to a factor of ∼0.7 at T = 45 °C – can be mainly ascribed to the approximate model we used for the calculation of the hydrodynamic radius. Another possible source of error can be related to the numerical noise of the data used in the fit to obtain g(n,β).

• It shows that DNA-made NSs with precise binding topologies can be nowadays produced in bulk quantities and exploited as highly precise model systems to tackle the physics of aggregating particles, either directional colloids or functional polymers. Here, we studied the aggregation behavior of AB_{3} monomers made of DNA NSs with the aim of clarifying the behavior of hyperbranched aggregation when intracluster bonds are present.

• It reports an extensive MD simulations study, based on a high-quality coarse-grained potential for the DNA interactions, to investigate the equilibrium cluster size distribution at a fixed temperature and at different initial monomer concentrations. We found that the mean cluster size increases with the particle concentration, from a suspension of monomers and small clusters, at low c, to a highly polydisperse solution at large concentrations. Simulations also suggested that the presence of intracluster bonds eliminates the possibility to reach the percolation transition.

• It reports a novel theoretical way to include the presence of intracluster bonds in the formalism. First, the FS approach is translated in an ideal gas of clusters formalism, which is then extended to include also the clusters with intracluster bonds. Including a model-dependent (but concentration-independent) quantity g(n,β) – which can be determined as a fit parameter – it becomes possible to predict the cluster size distribution without limiting assumptions. The theory explains why percolation is avoided when intracluster bonds are possible. Comparison with MD simulation data supports the quality of the modelling.

• It reports DLS measurements of the same system studied numerically and theoretically to provide evidence that, at odds with the FS predictions, percolation is not encountered in this system. Indeed, by decreasing the temperature, and exploiting the reversibility of the sticky-end pairing, it is possible to investigate the effect of bonding, up to the point where all bonds in the system are formed. The data we collected strongly support the idea that the polydispersity of the aggregates remains finite at low temperatures. Additionally, it shows that the low-T equilibrium cluster size distribution only depends on the initial monomer concentration.

We believe that these results will be also valuable for the community interested in the biotechnological applications of hyperbranched polymers, due to the biocompatibility and versatility of DNA as a building block for complex and innovative materials.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/D0NR04840B |

‡ The AB_{f−1} monomer can be also indicated as A−R−B_{f−1} or ARB_{f−1}. |

§ The loop size is defined as the number of monomers in a cluster that form a closed cycle of bonds. |

¶ The intensity average for the simulation data is obtained by performing the average weighted by the distribution n^{2}N_{c}(n). |

This journal is © The Royal Society of Chemistry 2020 |