Stability Criterion for the Assembly of Core–Shell Lipid–Polymer–Nucleic Acid Nanoparticles

Hybrid core–shell lipid–polycation–nucleic acid nanoparticles (LPNPs) provide unique delivery strategies for nonviral gene therapeutics. Since LPNPs consist of multiple components, involving different pairwise interactions between them, they are challenging to characterize and understand. Here, we propose a method based on fluorescence cross-correlation spectroscopy to elucidate the association between the three LPNP components. Through this lens, we demonstrate that cationic lipid shells (liposomes) do not displace polycations or DNA from the polycation–DNA cores (polyplexes). Hence, polyplexes and liposomes must be oppositely charged to associate into LPNPs. Furthermore, we identify the liposome:polyplex number ratio (ρN), which was hitherto an intangible quantity, as the primary parameter predicting stable LPNPs. We establish that ρN ≥ 1 ensures that every polyplex is enveloped by a liposome, thus avoiding coexisting oppositely charged species prone to aggregation.


S1. Fluorescence Cross-Correlation Spectroscopy background
In the most common experimental implementation of Fluorescence Cross-Correlation Spectroscopy (FCCS), two components of a system are labelled with two spectrally resolved fluorophores (a and b, typically emitting in the green and red spectra) and their fluorescence signals (Fa and Fb) are followed as they diffuse in and out of the two corresponding confocal volumes.The overlap between the two confocal volumes should be as large as possible.The dynamics of the components cause fluorescence intensity fluctuations that can be represented through the normalized correlation function ( ), determined from the time-dependent fluorescent signals: Here and is the lag time.Autocorrelation functions use only one signal (a=b) and represent the dynamics of species containing that signal.The cross-correlation uses both signals (ab) and represents only the dynamics of species containing the two labelled components.FCCS can therefore be used to quantify colocalization between two labelled species.For simplicity the crosscorrelation function will be denoted by .
In the case of free diffusion in three dimensions, the correlation function G can be represented through: where the amplitude A contains information on the number of species, and , which contains information regarding their dynamics, is given by: (S3) Here, is the aspect ratio between the axial and lateral radii of the detection volume ( and , respectively), is the diffusion time, and is the diffusion coefficient.By fitting Eqs.S2-S3 to the experimental autocorrelation function, the diffusion time and amplitudes of the species are obtained.Through , the hydrodynamic diameter can be obtained through the Stokes-Einstein relation.
Figure S2 shows a comparison between sizes obtained with DLS and FCCS.FCCS size data has larger error bars since measurements are quasi-single particle, requiring long measurements or multiple measurements to obtain statistical results identical to DLS.Conversely, DLS measurements are more susceptible to bias towards larger averages when samples are polydisperse.The very good agreement between FCCS and DLS results indicates that LPNP samples have low polydispersity 1 .
The relation between the amplitude of the autocorrelation function of signal a and the number of species contributing to it is given by: (S4a) Here Ni represents the average number of species i in the confocal volume of signal a, and represents the brightness of specie i, also on the fluorescent signal a.Conversely, the cross-correlation amplitude between signals a and b, , is given by: (S4b) therefore, provides quantitative information regarding the association between the species producing signals a and b.

S2. Implementing Fluorescence Cross-Correlation Spectroscopy to quantify formation of DNAcarrying hybrid lipid-polymer nanoparticles
In the present work, polylysine-DNA polyplexes (PPs) are combined with cationic liposomes (Ls) to form hybrid lipid-polymer-DNA nanoparticles (LPNPs).For the simplified case of: (i) one polyplex associating with one cationic liposome (1:1 stoichiometry) to form one LPNP; (ii) no crosstalk between the two channels; and (iii) no changes in the brightness of the species following the association, the equations relating the amplitudes of the correlation curves with the number of species in the confocal volume become straightforward (Eqs.3a-c in the main text): Here and are the amplitudes of the green and red auto-correlation, respectively, and is the amplitude of the cross-correlation., , and , are the average number of free polyplexes, free liposomes and LPNPs in the confocal volumes, respectively.
For the more general case in which one polyplex associates with liposomes to form a LPNP nanoparticle containing liposomes, and, as before, assuming no crosstalk between the two channels and no changes in the brightness of the species following the association, the equations relating the amplitudes of the correlation curves with the number of species in the confocal volume become: The and ratios are now given by: (S6a) and (S6b) where (Eq.4a).Note that for , Eq. S6 reduces to Eq. 4. Regardless of the value of , the fraction of liposomes that associate with polyplexes continues to be given by the ratio, but will no longer be given simply by if , and depends now also on and .The latter term can be removed by combining Eq.S6b with Eq.S6a, and the fraction of coated LPNPs, , can now be obtained by: (S7) Because both and are unknowns, Eq.S7 still cannot be used alone to determine the fraction of coated LnPNPs without knowledge of .However, it can still be used to estimate expected ratios for different scenarios (i.e. for different fractions of coated LPNPs and different ) and, by comparison with the experimentally observed ratios, determine which scenarios are more likely.
S3. Fitting the experimental data with a polyplex:liposome stoichiometry model Since the condition for stability is reached at (Fig. 2K), the number of free liposomes can be estimated from the following expression, valid for and : (S8) Here is defined as the value of at which the condition is met.In this work, for , is fixed at , as determined in Fig. 2K.Eq.S8 allows rearranging the and expressions (Eqs.4 and S6) as a function of to be compared with the experimental data.Substituting in Eq.S6 with Eq.S8, and recalling the assumption, results in: (S9a) and (S9b) For , these expressions reduce to and .As can be seen in Fig. 2I, the agreement with the data is remarkable for , especially taking into consideration that there are no fitting parameters.However, for the discrepancy between the model and the data becomes more noticeable at larger , even though the line is still within the error bars.The increasing discrepancy of the line with the data suggests that, increasing the number of free liposomes, increases the number of collisions with LPNPs, with some of these collisions resulting in an LPNP with an extra liposome.If is the probability of a free liposome enveloping an existing LPNP, the average number of liposomes per LPNP, , can be estimated by: (S10) Combining Eqs.S8, S9 and S10, results in: (S11a) and (S11b) These modified expressions for and account for an increasing when increases and can be readily fitted to the data with as the only fitting parameter.(Recall that here, for , is fixed at , as determined in Fig. 2K).Note also that Eq.S11 reduces back to and if is zero ( ). Figure 2I shows the best fit of Eq.S11 to the experimental data of LPNPs with 10 mol% PEG liposomes.Fig. S3 shows the best fits for the 5 and 0 mol% PEG systems.Unless otherwise noted, the fit is performed by a least-squares minimization of both and expressions simultaneously (Eqs.S11a and S11b).This way, each fit produces a single value for each LPNP system that is a compromise for both and data.For comparison the results with the simple stoichiometry model are also plotted (dashed lines).As discussed in the main text, for the 10 mol% PEG system, the modified model improves the fit to the data, but not to , which worsens.If the data is fit alone, one obtains , which is the value expected for the stoichiometry model.Regarding the 5 and 0 mol% PEG systems, the model improves the fit for both and data, although if the data is fit alone, the values of are also lower (Table S6).Overall, this shows that the stoichiometry model is reasonable as a first order approximation and explains the data fairly well.The deviations observed when the number of free liposomes coexisting with LPNPs becomes significant are partly addressed, although not entirely, by the model described above.Regarding the fitting parameters, we observe an increase of as the degree of PEGylation in the liposomes decreases (this is even more noticeable if the data is fitted alone -Table S6), indicating that liposomes with lower amounts of PEG are more likely to envelop an existing LPNP.This observation agrees with the general understanding that PEGylated particles repel each other more strongly, hence lowering .

S4. Determining the overlap between the green and red excitation volumes using single-and doublelabelled liposomes
The measured cross-correlation amplitude ( ) is limited by the amount of overlap between the green and red excitation volumes.We define the overlap volume correction factor ( ) as the factor that corrects the value of to the value of cross-correlation amplitude ( ) that would be expected if the overlap between the two excitation volumes was perfect, according to Eq. S12, (S12) To determine and correct the cross-correlation we performed a series of measurements with a set of samples consisting of a mixture between three PEGylated liposomes.Two of the liposomes were labelled with just one dye, one liposome type (L1) with 0.1 mol% of Atto-488 (green), and the other (L2) with 0.1 mol% Texas-red.The third liposome type (L3) was labelled with both dyes simultaneously.By gradually replacing liposomes L1 and L2, which are always in equal amounts and provide a non-colocalized signal, by liposomes L3, in which the two fluorescent probes are expected to be perfectly colocalized, can be determined.A related calibration approach was described recently by Werner et al 2 .
The results presented in Fig. S4 show that both the and ratios increase as the single-labelled liposomes are gradually replaced by liposomes labelled with the two dyes.The maximum cross-correlation percentages detected in the sample with only dual-labelled liposomes were below 80%, due to the non-perfect overlap between the green and red confocal volumes.On the other hand, when the fraction of double-labelled liposomes is zero, there is still some amount of crosscorrelation measured, due to the existence of some crosstalk between the dyes.(Note: as an exception to the remaining figures, where cross-correlation amplitudes are always corrected for crosstalk according to the procedure described in Bacia et al 3 , the data in Fig. S4 is not corrected for crosstalk).The expected amplitudes for the auto-and cross-correlation functions in this setting are as follows: where is a factor describing the fraction of signal from the green liposomes (L1) that is detected in the red detector due to crosstalk.Both and values are fitted simultaneously with Eq.S13, allowing a robust determination of .Besides providing the correction factor for the volume overlap, this experiment also validates the suitability of FCCS to determine the colocalization of soft nanoparticles of ca. 100 nm., where the DNA is labelled with YOYO-1.Green autocorrelation curves are shown in green, red auto-correlation curves are shown in red, and crosscorrelation curves are shown in blue.The existence of labelled free polylysine in excess in the system lowers dramatically the amplitude of the green autocorrelation to values where it becomes inaccurate.This hinders the determination of the association between cationic polyplexes and cationic liposomes (C).Hence, in this case ( ), DNA was also labelled with YOYO-1 (D).The very low values of cross-correlation amplitude compared to both the autocorrelation amplitudes shows that cationic polyplexes and cationic liposomes do not associate.

Fig. S4 .
Fig. S4.Determination of the confocal overlap volume.By employing a mixture of liposomes labelled with either Atto-488 or Texas Red, and liposomes labelled with both dyes, the fraction of colocalization in solution can be controlled and compared with the measured cross-correlation.As the fraction of dual-labelled liposomes increases, (green symbols) and (red symbols) also increase, as expected.The estimated , obtained by fitting both curves simultaneously, is 74.6%.The measurements also confirm the suitability of FCCS to measure colocalization in soft self-assembled nanostructures of ca. 100 nm.

Fig. S5 .
Fig. S5.Representative auto-and cross-correlation curves measured for each of the 10 mol% PEG LPNP formulations.The liposomes are labelled with Texas Red.(A-C) LPNPs using polyplex cores with Atto 488-labeled polylysine ( , and ).(D) LPNP formulations prepared employing a cationic polyplex core, where the DNA is labelled with YOYO-1.Green autocorrelation curves are shown in green, red auto-correlation curves are shown in red, and crosscorrelation curves are shown in blue.The existence of labelled free polylysine in excess in the system lowers dramatically the amplitude of the green autocorrelation to values where it becomes inaccurate.This hinders the determination of the association between cationic polyplexes and cationic liposomes (C).Hence, in this case ( ), DNA was also labelled with YOYO-1 (D).The very low values of cross-correlation amplitude compared to both the autocorrelation amplitudes shows that cationic polyplexes and cationic liposomes do not associate.

Table S1 .
Characterization by FCS, DLS and electrophoretic mobility ( ) of polyplexes prepared with different charge ratios.Data are Means ± SD (N=3).

Table S2 .
Characterization by FCS, DLS and electrophoretic mobility ( ) of cationic liposomes prepared with different PEGylation degrees.Data are Means ± SD (N=3).

Table S5 .
Summary of FCCS data for LPNPs.Results shown are obtained after fitting the auto-and cross-correlation curves with Eqs. 3 and 4. Data are Means ± SD (N≥3).Note that for the simplified model of one liposome complexing with one polyplex (1:1 stoichiometry), AG -1 and AR -1 correspond to the total number of polyplexes (NPPf + NLPP) and liposomes (NLf + NLPP) per confocal volume in the sample, respectively.b Note that within the same 1:1 stoichiometry approximation AX/AR and AX/AG indicate the fraction of polyplexes converted to LPNPs (  ) and fraction of liposomes used in LPNPs, respectively. a

Table S6 .
Results for the probability of a free liposome enveloping an existing LPNP ( ), as obtained through fitting Eqs.S11a and S11b to the and data.The second column shows the results obtained by fitting both and data simultaneously.This is the default method, whose results are shown in Figs.2I and S3.The third column shows the results obtained by fitting solely .