Design of block-copolymer-based micelles for active and passive targeting

A self-consistent field study is presented on the design of active and passive targeting block-copolymeric micelles. These micelles form in water by self-assembly of triblock copolymers with a hydrophilic middle block and two hydrophobic outer blocks. A minority amount of diblock copolymers with the same chemistry is taken to coassemble into these micelles. At the end of the hydrophilic block of the diblock copolymers, a targeting moiety (TM) is present. Assuming that the rotation of the micelle towards the target is sufficiently fast, we can elaborate a single gradient cell model, wherein the micelle is in the center and the receptor (R) substrate exists on the outer plane of the spherical coordinate system. The distribution function of the targeting moiety corresponds to a Landau free energy with local minima and corresponding maxima. The lowest minimum, which is the ground state, shifts from within the micelle to the adsorbing state upon bringing the substrate closer to the micelle, implying a jumplike translocation of the targeting moiety. Equally deep minima represent the binodal of the phase transition, which is, due to the finite chain length, first-order like. The maximum in-between the two relevant minima implies that there is an activation barrier for the targeting moiety to reach the receptor surface. We localize the parameter space wherein the targeting moiety is (when the micelle is far from the target) preferably hidden in the stealthy hydrophilic corona of the micelle, which is desirable to avoid undesired immune responses, and still can jump out of the corona to reach the target quick enough, that is, when the barrier height is sufficiently low. The latter requirement may be identified by a spinodal condition. We found that such hidden TMs can still establish a TM-R contact at distances up to twice the corona size. The translocation transition will work best when the affinity of the TM for the core is avoided and when hydrophilic TMs are selected.

Figure 1

Equilibrium radial density profiles. (a) The equilibrium radial density profile of ${\mathrm{PLGA}}_{60}{\mathrm{PEO}}_{60}{\mathrm{PLGA}}_{60}$ in a spherical coordinate system. The profiles for the apolar segments PLGA, the polar segments PEO, and the solvent $W$ (water) are given. Parameters ${\chi }_{{\text{LGA-H}}_2\text{O}}=1.6, {\chi }_{\text{LGA-EO}}=1.0$, and ${\chi }_{{\text{EO-H}}_2\text{O}}=0.4$. The aggregation number $g=237$. The grand potential $\mathrm{\Omega }=10k_{B}T$ [19]. (b) The equilibrium radial density profile of the brush made of ${\mathrm{PLGA}}_1{\mathrm{PEO}}_{60}{\mathrm{PLGA}}_1$ where the A units are pinned to be next to the particle surface, with radius $R=20$ (lattice units). The particle is made of segments of type A. In line with the aggregation number of the micelle of panel (a), the grafting density $\sigma =0.047$. Here, the radial coordinate is given in lattice sites units. For the radial coordinate, we have used a lattice site dimension (which corresponds to 0.8 nm).

Figure 2

The equilibrium radial density profile of the end point (TM) of the minority chain in the corona of the micellelike particle given by Fig. 1 in log-lin coordinates and lattice site units. The minority chain length is modified from $N_{\text{B}}=20$ to 40 as indicated. The curve for the chain length 30 is dashed. The dashed gray line represents the hydrodynamic diameter ($D_h$).

Figure 3

Schematic two-dimensional representation of (left panel) a micelle with a minority chain adsorbing with a targeting moiety (TM) to a receptor (R) site on an external planar substrate (e.g., a cell wall with receptor). Right panel: a central micelle surrounded by a spherical “substrate” as considered in the cell model; the TM of the minority chain adsorbed on this substrate, which is expected to be covered by receptors (not indicated). The core of the micelle is red, the corona blue [a few corona chains (loops) are indicated]. The minority chain is green and has a TM (represented by TM).

Figure 4

The dimensionless Landau free energy $F\left(r\right)$ (in units of $k_{B}T$), where $r$ is the distance to the center of the core for a probe chain ${\mathrm{A}}_1{\mathrm{B}}_{25}{\mathrm{TM}}_5$ in lattice site units for various values of the affinity of the TM for the receptor ${\chi }_{\text{TM-R}}$ as indicated, for the position of the receptor surface. (a) Position of the receptor surface $r=32$. (b) Position of the receptor surface $r=35$. The Landau free energy is normalized such that $F\left(16.8\right)=0$. Note that in panel (a) the curves for $r<27$ overlap. The same happens in panel (b) for $r<30$. The vertical (dashed black) line represents the upper limit of the cell and where the receptor surface is present.

Figure 5

Landau free energy $F\left(r\right)$ (in units of $k_{B}T$), $r$ as a function of the distance to the center of the core for a probe chain ${\mathrm{A}}_1{\mathrm{B}}_{25}{\mathrm{TM}}_5$ in lattice site units (reminder 1 lattice unit = 0.8 nm) for various values of the affinity of the TM for the core (composed of units A) ${\chi }_{\text{TM-LGA}}$ as indicated, for the position of the receptor surface (dashed black line) $r_{\text{R}}=35$ and ${\chi }_{\text{TM-R}}=-3$. The Landau free energy is normalized such that $F\left(r_{\text{R}}\right)=0$. Note that all curves overlap for $r>25$.

Figure 6

(a) State diagram in the ($N_{\text{B}}, \mathrm{\Delta }r$) coordinates. The solid curve is the binodal condition and the dashed curve represents the spinodal. (b) Height of the energy barrier $U^{*}$ (in units of $k_{B}T$) at the binodal as a function of the chain length $N_{\text{B}}^{*}$. The solid curve in panel (b) indicates that the TM (in absence of a receptor) is within the hydrodynamic diameter and the dotted curve indicates that the TM is outside the hydrodynamic diameter. Affinity of the TM for the core: ${\chi }_{\text{TM-LGA}}=1.0$, for the receptor ${\chi }_{\text{TM-R}}=-3$, and ${\chi }_{{\text{TM-H}}_2\text{O}}=0.4, N_{\text{TM}}=3$. Other parameters: see Sec. 2c.

Figure 7

(a) Collection of state diagrams in the ($N_{\text{B}},\mathrm{\Delta }r$) coordinates. The solid curves are the binodal conditions and the dashed curves represent the spinodals. (b) The height of the energy barrier $U^{*}$ at the binodal as a function of the chain length $N_{\text{B}}^{*}$. The solid curves in panel (b) indicate that the TM (in the absence of a receptor) is within the hydrodynamic diameter and the dotted curves indicate that the TM is outside the hydrodynamic diameter. The colors represent different parameters, namely, black: ${\chi }_{{\text{TM-H}}_2\text{O}}=0.1$, red: ${\chi }_{{\text{TM-H}}_2\text{O}}=0.4$, blue: ${\chi }_{{\text{TM-H}}_2\text{O}}=1.0$, and green: ${\chi }_{{\text{TM-H}}_2\text{O}}=3.0$ (spinodal curve is for $\mathrm{\Delta }r<0$ and therefore not plotted). Other parameters as in Fig. 6.

Figure 8

(a) Various state diagrams in the $N_{\text{B}}\mathrm{\Delta }r$ coordinates. The solid curves are the binodal conditions and the dashed curves represent the spinodals. (b) The height of the energy barrier $U^{*}$ at the binodal as a function of the chain length $N_{\text{B}}^{*}$. The solid curves in panel (b) indicate that the TM is within the hydrodynamic diameter (when the receptor is absent) and the dotted curves indicate that the TM is outside the hydrodynamic diameter. The colors represent different lengths of the TM, namely, black: $N_{\text{TM}}=1$, red: $N_{\text{TM}}=3$, blue: $N_{\text{TM}}=5$, and green: $N_{\text{TM}}=10$. Other parameters as in Fig. 6.

Figure 9

(a) Collection of state diagrams in the $N_{\text{B}}-\mathrm{\Delta }r$ coordinates. The solid curves are the binodal conditions, the dashed curves represent the spinodals, and the dotted curves are drawn for the $U=3k_{B}T$ condition. (b) The height of the energy barrier $U^{*}$ at the binodal as a function of the chain length $N_{\text{B}}^{*}$. The solid curves in panel (b) indicate that the TM is (when the receptor is absent) within the hydrodynamic diameter and the dotted curves indicate that the TM is outside the hydrodynamic diameter. The colors represent changes in TM-receptor affinity, namely, red: ${\chi }_{\text{TM-R}}=-3$, black: ${\chi }_{\text{TM-R}}=-6$, blue: ${\chi }_{\text{TM-R}}=-12$, and green: ${\chi }_{\text{TM-R}}=-18$. Other parameters as in Fig. 6.

Figure 10

State diagrams in the $N_{\text{TM}}\mathrm{\Delta }r$ coordinates for TM situated within the hydrodynamic diameter. The dotted curves are the practical spinodals $U=3k_{B}T$ and the dashed curves represent the spinodals. The colors represent different parameters, namely, red: ${\chi }_{\text{TM-R}}=-3$, black: ${\chi }_{\text{TM-R}}=-6$, blue: ${\chi }_{\text{TM-R}}=-12$, and green: ${\chi }_{\text{TM-R}}=-18$. The other calculation parameters are similar to the default system (cf. Fig. 6). Here, we have chosen for the longest possible minority chains, that is, we have implemented the constraint $N_{\text{TM}}+N_{\text{B}}=35$.