A Bond-Fluctuation Model of Translational Dynamics of Chain-like Particles through Mucosal Scaffolds

Abstract

Mucus scaffolds represent one of the most common barriers in targeted drug delivery and can remarkably reduce the outcome of pharmacological therapies. An efficient transport of drug particles through a mucus barrier is a precondition for an efficient drug delivery. Understanding the transport mechanism is particularly important for treatment of disorders such as cystic fibrosis. These are characterized by an onset of high-density mucus scaffolds imposing an increased steric filtering. In this study, we employed the bond-fluctuation model to analyze the effect of steric interactions on slowing the translational dynamics of compound chain-like particles traversing through scaffolds of different configurations (regular isotropic and anisotropic versus irregular random). The model, which accounts for both the geometry-imposed steric interaction as well as the intrachain steric interaction between the chain subunits, yields a transient subdiffusive motional pattern persists between the short-time and long-time Gaussian diffusion limits. The motion is analyzed in terms of a mean-squared displacement, diffusion coefficient, and radius of gyration. With higher levels of restriction or larger particles, the subdiffusive motional regime persists longer. The study also demonstrates that an important feature of the motion is also geometry-induced chain accommodation. The presented model is generic and could also be applied to studying the translational dynamics of other particles with more complex architecture such as dendrites or chain-decorated nanoparticles.

Introduction

The outcome of a drug-based treatment is critically dependent on the biochemical efficacy of the drug, bioaccessibility, and efficient drug delivery to the target tissue. The delivery can be particularly slow to mucus-covered tissues that protect the organs exposed to the external environment (such as the pulmonary airways, gastrointestinal tract, genitourinary tract, and ocular surface). Mucus is a complex viscoelastic hydrogel consisting of a semiflexible glycoprotein scaffold (mainly interconnected mucin molecules), of which the pores are filled by interstitial fluid. Mucus acts as a protective layer that prevents the ingress of pathogens and pollutants to the epithelium interior (1). The filtering action of mucus is twofold: it can entrap particles by attractive forces to the particles (interaction filtering), and it can prevent transport of the particles with sizes comparable to the mucus pore size or larger (steric filtering). In physiological conditions, the mucociliary motion (2) efficiently removes the entrapped particles (3). In the pathophysiological condition, however, the motion is impaired, which in turn can result in an increased mucus thickness and viscosity as well as in an accumulation of potentially harmful particles (4, 5).

The structure and rheology of a mucus layer is altered in cystic fibrosis (CF), which is an incurable, ultimately fatal inherited disorder. CF is associated with a loss-of-function mutation of a CF transmembrane conductance regulator, i.e., a chloride cell-membrane protein channel (6). In CF, ion channels are misfolded and thus biochemically degraded before their integration in the cell membrane, resulting in a significantly reduced cell-membrane permeability. The ion concentration imbalance further results in a reduced amount of interstitial fluid in mucus and consequently in mucin-scaffold shrinkage with, on average, 2.4-fold pore-size reduction (down to 140 ± 50 nm) (7, 8). The developments in CF treatment are focused on two different treatment approaches: corrective and symptomatic. The corrective treatment aims to correct the folding, trafficking, and gating defects of the CF transmembrane regulator (9) or to supplement the missing ion channels by surrogates using different gene transfer techniques (10), whereas the symptomatic treatment aims to invasively eradicate proliferating bacteria that were entrapped in hyperviscous mucus (11). With both treatment approaches, overcoming the mucus barrier still remains one of the main challenges (1, 12).

There is a great research interest in developing new drug-delivery vehicles, viral (13) and nonviral (1), with a high mucus-penetrating potential. In comparison to viral gene-delivery vectors, nonviral vectors are characterized by a lower immunogenicity and higher ability of incorporating larger genes. They are also more easily produced on a larger scale (14). For example, Lai et al. developed large polymer nanoparticles that exhibit rapid transport properties in fresh undiluted human mucus (15). To overcome the limitations associated with particle size distribution and stability of the cationic component of the nanoparticles, Zhang et al. employed the atom-transfer radical polymerization technique to synthesize protein-polymer conjugates for nonviral gene delivery (16). As demonstrated by Craparo et al. (17), PEGylation of ibuprofen-containing nanoparticles can remarkably increase the mucus-penetration potential. In a recent study, the motional pattern of nanoparticles in porcine pulmonary mucus was found to depend on the mode of nanoparticle administration (mechanical mixing versus aerosol deposition). This phenomenon is not fully understood yet and demands further research on the complex particle-mucus interaction mechanism (18).

In addition to experimental studies, of which the ultimate goal is to develop novel mucus-penetrating treatment strategies, theoretical and numerical simulation studies of translational motion of drug/gene vectors through cage-like hydrogel scaffolds can also help elucidate the mechanical and transport phenomena of these vectors. The phenomena are typically modeled within the frame of coarse-grained molecular dynamics simulations (19) or stochastic Monte Carlo (MC) simulations (20). These approaches enabled studying a diffusive character of penetrating nanoparticles as well as time-dependent density profiles of nanoparticles taken up across the mucus barrier, respectively. Here, subdiffusion refers to anomalous (non-Gaussian) diffusion, usually found in media with microscopic structures, in which the mean-square displacement (MSD) of diffusive particles increases with time as $MSD\left(t\right)\propto t^{\alpha }$ with α < 1, as opposed to normal (Gaussian) diffusion, in which $MSD\left(t\right)\propto t$ (21). Moreover, the coarse-grained molecular dynamics approach revealed that rotational motion of rod-like particles facilitates their transport through scaffolds (22). Another numerical model to study the effects of a confined geometry on the translational motion of point-like tracer particles is based on solving the Langevin equation in the massless Smoluchowski approximation. This model was successfully applied to characterize the dynamics in various tissues. For example, Tourell et al. (23) and Momot (24) analyzed diffusional anisotropy in a regularly/irregularly shaped articular cartilage fibrous model, whereas Ernst et al. recently applied the model to explain the experimentally observed transient subdiffusive regime that had been experimentally observed in CF mucus samples (25). In the study, it was found that diffusion is normal for short and long timescales and is subdiffusive between these timescales. The same conclusions were also drawn in other studies of anomalous diffusion dealing with different restrictive geometries (26). The timescale range of subdiffusion depends on the shape and size of diffusive particles and of obstacles in the medium. A similar model was also applied to simulate NMR detection of the translational motion of spin-bearing particles via motion-related spin phase discord (27, 28). In the model, a random particle displacement at a given simulation time step is determined by the particle diffusion coefficient that is, according to the well-known Einstein-Stoke relation, inversely proportional to the particle hydrodynamic radius. The radius is time-independent for small and rigid particles with a well-defined stable structure. However, the hydrodynamic radius is effectively dependent on time for compound shape-changing particles, such as polymer-decorated nanoparticles (29) and biocompatible dendritic polymers (30). The dynamics of chain-like particles was thoroughly studied in the field of polymer translocation, e.g., translation of a single polymer chain through a wall pinhole and solid-wall channels (31), polymer chains in polymer melts (32), or polymer melts filled with solid nanoparticles (33).

An efficient computational approach to simulate the dynamics of larger macromolecules is the bond-fluctuation model (BFM) (34). The model is basically a stochastic self-avoiding MC model that, unlike the self-avoiding walk model, preserves ergodicity. A dynamical system is ergodic when it exhibits the same behavior averaged over time or averaged over the system states. The self-avoiding walk model is not ergodic because of possible onsets of nonevolving configurations. In the BFM model, macromolecules are composed of monomers of a finite size. The monomers are connected by MC-driven fluctuating bonds that are allowed to attain only a few selected bond lengths. In the last decade, the model was intensively studied by Sommer et al., who applied the model to analyze the effect of interparticle interactions on translocation of homopolymers (35) or triblock copolymers (36) through a selectively permeable lipid bilayer. In addition, the model was successfully applied to study the dynamics of adsorbed/grafted (37) and dendritic polymer structures (38). However, the BFM model has not yet been applied to study targeted drug delivery by translational motion of drug macromolecules through a scaffold.

Motivated by the recent work of Ernst et al., in which the translational motion of noninteracting particles through the mucus-like semipermeable geometry was modeled in the massless Smoluchowski approximation (25), the aim of this study was to employ the BFM model to simulate the translational motion of sterically interacting chain-like molecules of different sizes through different geometries (regular isotropic, regular anisotropic, and random). The motion was characterized by means of time-dependent MSD, diffusion coefficient, and radius of gyration. The results of the study confirm that the motion of chains traversing through the geometries is transiently subdiffusive and consists also of geometry-induced chain accommodations. The presented BFM model is generic and thus also applicable for studying the translational dynamics of arbitrarily structured particles of different levels of complexity.

Methods

In the study, BFM simulations were performed in two dimensions (2D). Scaffold geometry was approximated either by arrangement of 3 × 3 solid circles with radius R (regular isotropic geometry), of 3 × 3 ellipses with semiaxes a and b (regular anisotropic geometry), or of randomly distributed solid obstacles (random geometry). For each geometry, a restriction level Φ of scaffold geometry, defined by the volume fraction of obstacles in the geometry, was calculated as the ratio between the obstacle area and the area of the computational box (2D square with N × N sites). A chain consisting of N_m monomers (each of 2 × 2 sites) was randomly inserted into the interobstacle space (interconnected pores). The BFM chain dynamics was calculated according to the 2D BFM scheme (39). In this scheme, the attempted move of a selected monomer with a step size equal to unity in a Cartesian lattice was accepted according to the Metropolis MC algorithm (40). For the sake of simplicity, in the algorithm, only the steric interaction $\left(E=\infty \right)$ was considered. Another BFM criterion for the acceptance of the MC-attempted move was the BFM bond length. In 2D, the allowed lengths between the adjacent monomers are equal to either 2, $\sqrt{5}$, $2\sqrt{2}$, 3, $\sqrt{10}$, or $\sqrt{13}$ (39). Large-scale simulations were enabled by employing periodic boundary conditions (25).

Each BFM simulation was initiated with an insertion of one chain into the scaffold geometry. The insertions became hardly feasible with high restriction levels Φ and larger chain sizes N_m. In these cases, multiple insertion tries were performed until the entire chain fitted into the interobstacle space. After the chain insertion, a simulation run consisting of $N_{MC}\times N_m$ BFM dynamic simulation time steps (MC attempted moves) was performed (35). The simulation run was repeated Q times for each chain size N_m to enable further ensemble averaging. The ensemble-averaged BFM chain dynamics was quantified by calculating the time-dependent radius of gyration

$$R_g^2\left(t\right)=\frac{1}{Q}\sum _{q=1}^Q\frac{1}{N_m}\sum _{n=1}^{N_m}{\left({\mathit{r}}_{n,q}\left(t\right)-{\mathit{r}}_{cm,q}\left(t\right)\right)}^2$$ (1)

and the corresponding MSD

$$MSD\left(t\right)=\frac{1}{Q}\sum _{q=1}^Q{\left({\mathit{r}}_{cm,q}\left(t\right)-{\mathit{r}}_{cm,q}\left(0\right)\right)}^2,$$ (2)

whereas the time-dependent diffusion coefficient was calculated as (24):

$$D_{\mu \nu }\left(t\right)=\frac{1}{Q}\sum _{q=1}^Q\frac{1}{2t}\left(r_{cm,q}^{\mu }\left(t\right)-r_{cm,q}^{\mu }\left(0\right)\right)\left(r_{cm,q}^{\nu }\left(t\right)-r_{cm,q}^{\nu }\left(0\right)\right).$$ (3)

Here, ${\mathit{r}}_{n,q}\left(t\right)$ is a position of the n^th monomer of the chain in the q^th simulation run, ${\mathit{r}}_{cm,q}\left(t\right)=\frac{1}{N_m}{\sum }_{n=1}^{N_m}{\mathit{r}}_{n,q}\left(t\right)$ is the corresponding center-of-mass trajectory, and $r_{cm,q}^{\mu }\left(t\right)$, $\mu ,\nu =x,y$ is the respective trajectory component. In addition, average diffusivity was calculated using the relation $D\left(t\right)=\left(1/2\right)\left(D_{xx}\left(t\right)+D_{yy}\left(t\right)\right)$. The quantities were acquired at $N_{aq}$ equidistantly distributed simulation times. For regular confined geometries, the simulated time-dependent diffusion coefficient was further modeled using the following function:

$$D\left(t\right)=D_{\infty }+\left(D_0-D_{\infty }\right)\frac{1-e^{-t/\tau }}{t/\tau }.$$ (4)

The origin of this equation and how it can be transformed to the corresponding diffusion spectrum is explained in more detail in Eqs. S1 and S2.

All simulated variables can be expressed in physical units composed of a spatial unit Δx (which is identical to Δy) and a time unit Δt. The spatial unit corresponds to the physical dimension of a lattice cell, whereas the time unit corresponds to N_m (chain size) simulation steps. The latter is a consequence of the BFM model, in which in the physical time unit, all chain monomers for all chain sizes must have an identical probability for fluctuation. With introduction of the base units, physical units of the simulated variables are the following: physical time equivalent $t_{BFM}\left[\mathit{\Delta }t\right]$, mean-square displacement $MSD\left[\mathit{\Delta }{x}^2\right]$, diffusion coefficient $D\left[\mathit{\Delta }{x}^2/\mathit{\Delta }t\right]$, and radius of gyration $R_g\left[\mathit{\Delta }x\right]$.

The 2D BFM simulations were implemented within the MATLAB programming environment (The MathWorks, Natick, MA). The environment enabled an elegant initialization of the simulations, whereas a calculation of the time-consuming BFM dynamics was performed by employing a C/C++ -based MATLAB executable (MEX) approach that significantly accelerated software performance. The following simulation parameters were used: N = 66, $N_{MC}={10}^6$, Q = 700, $N_{aq}=1000$, $N_m=1\cdots 30$, $R=0\cdots 16$, a =$0\cdots 16$, and b = 5. With these parameters, the restriction level Φ was in the range between 0 and 65%, whereas the computational time per simulation run was approximately equal to 70 h (on a single CPU core). In the postprocessing, 2D BFM snapshots were visualized by using the PovRAY rendering program (Persistence of Vision, Williamstown, Victoria, Australia).

Results

Fig. 1 shows BFM simulation snapshots for two selected chain sizes (N_m = 8, 16) and for the three restrictive geometry configurations (regular isotropic, regular anisotropic, and random), all with the restriction level Φ = 19%. In the snapshots, the initial chains (orange: with isotropic, cyan: with anisotropic, and green: with random geometries) and the final chains (red: with isotropic, blue: with anisotropic, and dark green: with random geometries) are interconnected by the corresponding center-of-mass trajectories. As can be seen by visual inspection, the isotropic scaffold geometry imposes moderate motional restriction and entraps moving chains, particularly larger ones, in the interobstacle space. With the anisotropic scaffold geometry, a preferential chain motion is established along the direction of anisotropy with the minimal steric confinement for the chain motion. With the random scaffold geometry, however, the motion is most restricted because of the disordered nature of the confining geometry, characterized by a locally increased density of the obstacles. The confining geometry could result in an entrapment of the chains and thus also in a further quenching of the hopping-like BFM dynamics. This effect is most apparent with longer chains that in turn also exhibit the smallest center-of-mass displacements. The chain trajectories also contain information on the time-dependent diffusion coefficient, i.e., the finer and shorter the trajectory, the smaller the corresponding diffusion coefficient.

Figure 1.

Selected simulation snapshots of traversing chains in the initial and final translational states, interconnected by the center-of-mass trajectories. The snapshots are shown for two selected chain sizes (N_m = 8, 16) and for different restrictive scaffold geometries (regular isotropic, regular anisotropic, and random) with restriction level Φ = 19%. A structure of a representative chain along with the self-restrictive 2 × 2 box-like objects, indicating the reach of the steric interaction of individual monomers (right side inset), is shown. To see this figure in color, go online.

Translational motion in the restrictive geometry can be quantitatively characterized by calculating the ensemble-averaged time courses of the MSD(t), diffusion coefficient D(t), and radius of gyration R_g(t), as depicted in Fig. 2. The results are displayed for all three scaffold geometries, for various chain sizes N_m, and for two different restriction levels Φ. Gray-to-black curves correspond to a moderate restriction level Φ = 19% (isotropic, anisotropic, random), whereas gray-to-red curves correspond to a higher restriction level Φ = 64% (isotropic) and $\mathit{\Phi }\approx$ 35% (anisotropic, random). In the simulations, translational motion exhibits three regimes depending on the timescale, i.e., a short-time motional regime with normal (Gaussian) diffusion, in which the chains traverse too short distances too collide with the confining obstacles; an intermediate-time motional regime, in which the chains already underwent multiple collisions with the obstacles so that the motion is subdiffusive; and a long-time motional regime, in which translational motion reverts to normal diffusion associated with transport between pores. The existence of the three motional regimes can be best seen with higher restriction levels (Φ = 64%, isotropic) and shorter chains. For longer chains, the existence of the third motional regime can be seen in Fig. S3, in which the simulation time is extended by a factor of 40. In the intermediate time (subdiffusive) regime, MSD curves flatten because of transient pore entrapment of the chains. This can also be seen as a significant decrease of the corresponding diffusion coefficients with time. As expected, diffusion coefficients, on average, decrease with an increasing chain size N_m, as shown in Fig. 2 by the gray-to-black and gray-to-red arrows. Another interesting feature of the BFM translational motion can be seen in the R_g(t) curves. The curves are mostly independent on time with small N_m; however, the curves exhibit fluctuations with larger N_m. The fluctuations are associated with chain accommodations imposed by the surrounding obstacles in the scaffold and by the intrachain steric interactions. These fluctuations can be observed with all three scaffold geometries. With the higher restriction level (e.g., isotropic Φ = 64%) and with larger N_m, the R_g(t) curves are characterized by a sudden decrease of R_g (denoted by the asterisk in Fig. 2). This phenomenon is associated with an entrapment of larger chains, which were initially spread over the interconnected neighboring pores. The entrapped chains never completely unfolded back to the initial state. The Supporting Materials and Methods give the inverse curves to the presented MSD(t) curves, i.e., t = t(MSD) (Fig. S1), as well as the diffusion spectra $D_{\omega }\left(\omega \right)$ calculated with the higher restriction level for the isotropic scaffold geometry (Fig. S2).

Figure 2.

Quantitative analysis of translational motion of chains as obtained by BFM simulations: time courses of mean square displacement MSD(t), diffusion coefficient D(t), and radius of gyration R_g(t). The time courses are displayed for the three restrictive scaffold geometries, for two different restriction levels (black (upper) curves indicating a moderate restriction level of Φ = 19% versus red (lower) curves indicating higher restriction levels of Φ = 34, 35, 64%) and for various chain sizes N_m. In the graphs, darker colors correspond to higher values of N_m. The asterisk denotes the curve with a sudden decrease of R_g associated with an entrapment of larger chains, which were initially spread over the interconnected neighboring pores. To see this figure in color, go online.

Fig. 3 shows short-time and long-time diffusion coefficients, $D_0=\underset{t\to 0}{\lim }D\left(t\right)$ and $D_{\infty }=\underset{t\to t_{MC}}{\lim }D\left(t\right)$, respectively, as a function of chain size N_m and restriction level Φ for all three scaffold geometries. Coefficients $D_0$ and $D_{\infty }$ were determined as model parameters of the best fit between the model in Eq. 4 and the simulated time-dependent diffusion coefficient data. In the short-time regime, the diffusion coefficient $D_0$ exhibits a power-law dependence on N_m as well as a moderate dependence on Φ. The Φ-dependence is most apparent with the isotropic scaffold geometry, by which high values of Φ can be reached, and with the random scaffold geometry, in which the uneven distribution of confining obstacles impedes the dynamics of the entrapped chains. In the long-time regime, the diffusion coefficient $D_{\infty }$, for all Φ values and for small N_m, also exhibits a power-law dependence. In Fig. 3, the solid lines in the range 1 ≤ N_m ≤ 5 denote best-fit power-law curves $D_0\propto N_m^{-c_0}$ and $D_{\infty }\propto N_m^{-c_{\infty }}$, where $c_0$ and $c_{\infty }$ are the respective exponents in the power-law model. With higher values of N_m and Φ, the translational motion is highly restricted, and consequently the power-law dependence becomes inaccurate.

Figure 3.

A dependence of short-time $D_0$ and long-time $D_{\infty }$ diffusion coefficients on the restrictive scaffold geometry, chain size N_m and the restriction level Φ. The results are plotted for $0\le \mathit{\Phi }\mathrm{\lesssim }64$ % (isotropic), $0\le \mathit{\Phi }\mathrm{\lesssim }25$ % (anisotropic), and $0\le \mathit{\Phi }\mathrm{\lesssim }35$ % (random). To see this figure in color, go online.

Fig. 4 depicts a correlation between the hydrodynamic radius in the long-time regime $R_{H,\infty }$, which is, according to the Einstein-Stokes relation, inversely proportional to the respective diffusion coefficient $D_{\infty }$ and the corresponding ensemble- and time-averaged radius of gyration R_g. By using the Einstein-Stokes relation, hydrodynamic radii $R_{H,\infty }\left(N_m,\mathit{\Phi }\right)$ as a measure of an inverse diffusion coefficient were calculated from the relation $R_{H,\infty }\left(N_m,\mathit{\Phi }\right)D_{\infty }\left(N_m,\mathit{\Phi }\right)=const=R_{H,\infty }\left(\mathrm{1,0}\right)D_{\infty }\left(\mathrm{1,0}\right)$. The graphs indicate a power-law behavior of $R_{H,\infty }\left(R_g\right)$ for the nonrestricted translational motion and a strong deviation from this behavior for the restricted translational motion.

Figure 4.

Correlations between the hydrodynamic radii and the corresponding radii of gyration, i.e., R_H(R_g), as a function of restriction level Φ for all three restrictive scaffold geometries. To see this figure in color, go online.

The power-law model exponents $c_0$ and $c_{\infty }$ in relations between the short-time and long-time diffusion coefficient on the chain size, i.e., $D_0\propto N_m^{-c_0}$and $D_{\infty }\propto N_m^{-c_{\infty }}$, are for different restriction levels Φ plotted in Fig. 5, a and b, respectively. The plotted exponents correspond to best-fit curves of Fig. 3 with 1 ≤ N_m ≤ 5. It can be seen that with low restriction levels (in the limit of $\mathit{\Phi }\to 0$), the exponents are independent on Φ and approximately equal to $c_0\left(\mathit{\Phi }\to 0\right)\approx c_{\infty }\left(\mathit{\Phi }\to 0\right)\approx 1.5$ for both the short-time and long-time motional regime. The corresponding Einstein-Stokes relations thus become equal to $D_0\propto N_m^{-1.5}$ and $D_{\infty }\propto N_m^{-1.5}$. Therefore, the hydrodynamic radii of small unconfined chains (1 ≤ N_m ≤ 5) scale as $R_{H,0}\propto N_m^{1.5}$ and $R_{H,\infty }\propto N_m^{1.5}$. With larger values of Φ, the exponents increase significantly and reach values $c_{\infty }\left(\mathit{\Phi }=35\text{ \%}\right)\approx 2.5$ and $c_{\infty }\left(\mathit{\Phi }=64\text{ \%}\right)\approx 3.2$ with the random and isotropic scaffold geometry, respectively. Moreover, as can be seen from Fig. 5 c, an ensemble- and time-averaged radius of gyration is another quantity that scales with the chain size: $R_g\propto N_m^c$, where c is the fitting exponent. The graph in Fig. 5 c shows the c(Φ) dependence for all three scaffold geometries. Independently on the scaffold geometry, the radius of gyration scales as $R_g\propto N_m^{0.75}$ for Φ < 40%. With higher Φ values (Φ > 40%), which in the study are only obtained with the isotropic scaffold geometry, the pore interconnections are gradually reduced down to the monomer size ($\mathit{\Phi }\approx$ 56%) or below it ($\mathit{\Phi }\approx$ 64%). This results in a formation of globular structures. With the isotropic scaffold geometry, an inverted sigmoidal curve is fitted to the c(Φ) data, indicating a transition between the plateau at $c\approx 0.75$ lower values with Φ > 40%. The graph in Fig. 5 d demonstrates representative size distributions of ensemble-time-averaged R_g (blue-to-white vertical stripes) as a function of N_m that are calculated with Φ = 0. The size distributions overlap with the size-averaged R_g values, through which the best-fit $R_g\propto N_m^c$ curve is plotted. The graph in Fig. 5 d depicts three additional characteristic curves, i.e., the first curve corresponding to a straight chain, $R_g\left(N_m\right)={\left(\left(13/12\right)N_m^2+\left(1/6\right)\right)}^{1/2}$, as well as the other two R_g(N_m) curves calculated by the MC approach by either maximizing or minimizing R_g.

Figure 5.

The exponents of the power-law dependences $D_0\propto N_m^{-c_0}$, $D_{\infty }\propto N_m^{-c_{\infty }}$, and $R_g\propto N_m^c$ as a function of restriction level: $c_0\left(\mathit{\Phi }\right)$(a), $c_{\infty }\left(\mathit{\Phi }\right)$(b), and c(Φ) (c), as well as R_g distributions as a function of the chain size N_m (d). In the simulations, geometrical restrictions were excluded (Φ = o). To see this figure in color, go online.

Fig. 6 shows the results of unfolding dynamics of chains. Each chain was initially completely folded around one of its ends into a regular spiral and then subjected to the BFM motion. The motion is governed by translocations of individual chain monomers which ultimately leads to the chain unfolding. In Fig. 6, the unfolding motion is characterized by two surface plots, i.e., ensemble-averaged D(t,N_m)/D(0,1) in a log format and normalized ensemble-averaged R_g(t,N_m)/N_m, that are supplemented by the initial/final chains of five selected chain sizes N_m = 5, 20, 35, 50, 65. The unfolding dynamics is simulated in $N_{MC}\times N_m$ simulation time steps with $N_{MC}=2\mathrm{\times }{10}^4$. From the log (D(t,N_m)/D(0,1)) surface plot, it can be seen that the short-time diffusion coefficient decreases with an increasing chain size. With an increasing simulation time, smaller chains with $N_m\mathrm{\lesssim }20$ relatively rapidly undergo the transition from the folded state into the unfolded one; in the unfolded state, the fixed end prevents further unrestricted-like motion. This is demonstrated by a significant decrease of the respective diffusion coefficient. On the contrary, the larger chains with $N_m\mathrm{\gtrsim }35$ exhibit a time-independent diffusion coefficient that corresponds to the short-time diffusion limit. The BFM dynamics can be additionally elucidated by analyzing the corresponding R_g(t,N_m)/N_m surface plot: the dynamics progresses in a form of irregular R_g fluctuations that are associated, on average, with unfolding/refolding alternations. At the timescales applied in the simulations, the refolding $\left(R_g/N_m\to 0\right)$ is most apparent with the intermediate-sized chains; the smallest chains tend to be mostly unfolded, whereas the largest chains unfold only slightly. In the length units given by the normalized ensemble-averaged R_g (t,N_m)/N_m, the smallest chains $\left(N_m\mathrm{\lesssim }10\right)$ are rapidly unfolded to lengths of up to 0.7, larger chains $\left(10\mathrm{\lesssim }{N}_m\mathrm{\lesssim }20\right)$ attain lengths of up to 0.5, and the largest chains $\left(60\mathrm{\lesssim }{N}_m\mathrm{\lesssim }70\right)$ attain only a slightly loosened spiral shape, far from being unfolded, with lengths below 0.2.

Figure 6.

Unfolding dynamics of spirally wrapped and anchored chains presented by log (D(t,N_m)/D(0,1)) and by normalized ensemble-averaged R_g(t,N_m)/N_m surface plots along with the selected chains of various chain sizes. To see this figure in color, go online.

Discussion

The aim of this numerical simulation study was to model translational dynamics of chain-like particles in an arrangement of rigid obstacles mimicking mucosal scaffold. In the simulations, three different 2D restrictive-scaffold geometries (isotropic, anisotropic, and random) and different restriction levels were chosen and individually populated by chain-like particles in the interobstacle space. The chain dynamics was calculated in 2D within the frame of the MC-based BFM that was intensively elaborated by Sommer et al. (35, 41, 42). The BFM model preserves ergodicity and also efficiently accounts for steric interactions. Therefore, the model was also appropriate to address the challenging steric filtering problem in targeted drug delivery. The principal finding of this study is that steric interactions play a crucial role in the translational dynamics of chains in the confining geometries. The interactions have a twofold effect on chains, namely, they impede chain penetration into the confining geometries as well as preventing overlap of chain subunits. The effects were quantified by analyzing the center-of mass dynamics (characterized by the center-of-mass diffusion coefficient) as well as by studying the relative motion of the chain subunits with respect to the center-of-mass (characterized by the radius of gyration).

Recently, Ernst et al. proposed a minimalistic diffusion model to explain an experimentally observed transient subdiffusive regime of nanoparticles traversing through the human sputum taken from CF patients (25). The model is essentially based on solving the Langevin equation in the massless Smoluchowski approximation and on modeling the sputum compartmentalization by introducing semipermeable/semireflective membranes. The model, which was reduced to one dimension without loss of generality, yielded a transient regime in the timescale range 0.1 s < t < 2 s for relatively large nanoparticles (diameter of 200 nm) residing in compartments with a size of 250 nm. The agreement between the predicted and observed transient regime encouraged further calculation of particle dynamics also beyond the observable timescales. Consistent with the experimental observations, the model correctly predicted unrestricted (Gaussian) diffusion regime in the short-time (t < 0.1 s) and long-time (t > 2 s) limits. The MC-based model presented in this study aims to upgrade those simulations by accounting for shape alterations of the traversing particles. These alterations were, in our model, most apparently demonstrated by the fluctuations in R_g curves. As expected, the BFM model yielded unrestricted motional behavior in the short-time regime, whereas at the later times, when the number of simulation time steps was approaching to $N_{MC}\times N_m$, the motion was subdiffusive because of multiple wall collisions. However, by taking a closer look at the MSD curves with Φ = 64% (Fig. 2, isotropic), the timescales used in the study were in fact sufficiently long to capture a turning point from the transient subdiffusive motional regime back to normal (Gaussian) diffusion. The turning point is best expressed with smaller chains. The recovery of normal diffusion is tightly associated with the pore-to-pore hopping dynamics. Thus, in predicting the transient subdiffusive regime, the presented BFM model is in an agreement with the results of Ernst et al. (25). The BFM simulation results with the isotropic configuration and prolonged simulation times $\left(N_{MC}\to 40N_{MC}\right)$ are for five selected chain sizes $N_m=1\cdots 4,10$ and for two selected restriction levels Φ = 19, 64% shown in Fig. S3. As can be seen from Fig. S3, the onset and duration of the transient subdiffusive regime (43) highly depends on the chain size N_m as well as on the intrapore connectivity (44) that was controlled by different restriction levels.

In the study, restricted translational dynamics of chains was quantified in terms of three important observables, i.e., MSD(t), diffusion coefficient D(t), and radius of gyration R_g(t). Among these, MSD(t) is most relevant for studying targeted drug delivery; its inverse function t = t(MSD) namely defines the time needed for the drug particles to reach the target tissue (e.g., mucus-covered epithelial cells). The t(MSD) curves clearly show significantly slowed translational dynamics, which can be explained by an increased residence time of particles at the spatial scales corresponding to the characteristic sizes of the pores. The BFM model is essentially a lattice MC method with no explicitly defined physical units. However, by a proper mesh-size calibration, physical timescales of drug delivery could be obtained. Diffusion coefficient curves provide an alternative approach of the microrheological description of partially restricted translational dynamics. In this study, diffusion coefficients are given in both the time and the frequency domain, i.e., D(t) and $D_{\omega }\left(\omega \right)$, respectively. These two quantities can be measured by means of NMR by employing diffusion sensitizing magnetic field gradients (45, 46). The method is promising in noninvasive detection of translational dynamics in various mucosal tissues (47, 48). Finally, the third quantity, R_g(t), gives an insight into the confinement-enhanced chain-accommodation dynamics that cannot be probed by rigid spherical or point-like particles. With an increasing restriction level, the R_H − R_g correlation starts deviating from the power-law dependence because of confinement effects. By using random walk simulations, Uehara et al. recently demonstrated that a complex R_H − R_g relation exists also for unconfined and entangled polymer chains with high topological complexity (49).

As the main goal of the study was to address the steric filtering phenomena arising in targeted drug delivery, the presented computational approach accounts only for steric interactions, whereas the short- and long-range interactions are not included in the model. It is expected that an inclusion of repulsive short-range interactions would effectively extend the range of highly repulsive steric interactions and therefore enable more efficient modeling of optimal solvent conditions. On the contrary, nonzero attractive interactions would impose poor solvent conditions by collapsing chains into quenched droplet-like structures (50). Similarly, the quality of the solvent could be additionally reduced by employing an explicit-solvent BFM model, i.e., by populating the simulation box with additional solvent monomers (35). Another important limitation of this study was also a relatively small number of chains in the ensemble because of the limited computational capacity, which resulted in noisier data. Nevertheless, it was still possible to resolve the characteristic motional patterns. In this study, the simulations were performed in 2D to significantly reduce the computational load while preserving system complexity. In three dimensions (3D), MSD is at any time 50% larger than in 2D. Moreover, restrictions to particle motion are expected to be lower because of an additional degree of freedom that results in a reduced probability of interchain collisions and an increased connectivity between the pores. The study could be further improved by considering BFM particles with a more complex 3D architecture, which remains a challenging topic for the future studies.

Conclusions

Among the processes relevant for targeted drug delivery, steric filtering has an exceptional importance. However, the process is not yet well understood and still remains one of the most important challenges in understanding targeted drug delivery. This study aims to elucidate the steric filtering process by performing a series of BFM model simulations of translational dynamics of chain-like particles in different arrangements of rigid obstacles mimicking mucus scaffold. By analyzing parameters such as the MSD, diffusion coefficient, and radius of gyration, it was found that the translational dynamics of the particles is to a large extent governed by the steric interactions. The presented model was analyzed in 2D for specific geometry of obstacles and particles, though the model is generic and could be applied also to other, structurally more complex particles and to other porous systems; the dimensionality of the problem could be extended to 3D as well. Potentially, the development of new biophysical theoretical models for targeted drug delivery could help in developing advanced nanodelivery vehicles with higher mucus-penetration potential and thus improve treatment of diseases such as CF.

Author Contributions

F.B. conceived the study. F.B. developed the simulations. F.B. and I.S. analyzed the data. F.B. and I.S. wrote the manuscript. F.B. and I.S. edited the manuscript.

Editor: Alexander Berezhkovskii.