Modularity-based mathematical modeling of ligand inter-nanocluster connectivity for unraveling reversible stem cell regulation

Abstract

The native extracellular matrix is continuously remodeled to form complex interconnected network structures that reversibly regulate stem cell behaviors. Both regulation and understanding of its intricate dynamicity can help to modulate numerous cell behaviors. However, neither of these has yet been achieved due to the lack of designing and modeling such complex structures with dynamic controllability. Here we report modularity-based mathematical modeling of extracellular matrix-emulating ligand inter-cluster connectivity using the graph theory. Increasing anisotropy of magnetic nano-blockers proportionately disconnects arginine-glycine-aspartic acid ligand-to-ligand interconnections and decreases the number of ligand inter-cluster edges. This phenomenon deactivates stem cells, which can be partly activated by linearizing the nano-blockers. Remote cyclic elevation of high-anisotropy nano-blockers flexibly generates nano-gaps under the nano-blockers and augments the number of ligand inter-cluster edges. Subsequently, integrin-presenting stem cell infiltration is stimulated, which reversibly intensifies focal adhesion and mechanotransduction-driven differentiation both in vitro and in vivo. Designing and systemically modeling extracellular matrix-mimetic geometries opens avenues for unraveling dynamic cell-material interactions for tissue regeneration.

Fig. 1. Schematic overview of the modeling of interconnected ligand nanocluster for stem cell behavior regulation.

a The liganded GNPs on the material surface, which act as ligand nodes constituting interconnected ligands, are homogeneously arranged with equal inter-distances. Due to equal inter-distances, ligand nodes are connected in a way to form equilateral triangles, resulting in a ligand network model with the edges. b In the graph theory, the network is partitioned into clusters by the Louvain algorithm based on modularity. Intra-cluster edges were marked by thin green lines, and inter-cluster edges were marked by highlighted white lines. The average number of edges between neighboring clusters is referred to as “# inter-cluster edges”. c The presence of anisotropic (“aniso.”) nano-blockers between the ligand nodes obstruct ligand-to-ligand interconnections, thereby reducing the overall # ligand inter-cluster edges. The lowest anisotropy group (“Low aniso.”) exhibits the highest # inter-cluster edges, which promotes the focal adhesion and mechanotransduction of stem cells, and thereby their differentiation, both in vitro and in vivo. In contrast, the highest anisotropy group (“High aniso.”) exhibits the lowest # inter-cluster edge, which inhibits the adhesion and differentiation of stem cells.

Fig. 2. Remote control of high-anisotropy nano-blockers for reversible tuning of the average number of ligand inter-cluster edges.

Schematic illustrations of remotely controlling high-anisotropy nano-blockers on the interconnected ligand-displaying materials, which reversibly modulates the average number of ligand inter-cluster edges (referred to as “# ligand inter-cluster edges”). a Irreversible linearization (“Lin.”) of randomly arranged high-anisotropy nano-blockers facilitates local interconnection between the ligand clusters by ordering them, which partly enhances # ligand inter-cluster edges. b The ligand nodes are initially disconnected by the nano-blockers (“NE.”), which are reconnected via cyclic reversible elevation (“E.”) of anisotropic nano-blockers, thereby escalating # ligand inter-cluster edges. The optimization of minimal polymer linker density used to graft the anisotropic nano-blockers to the material allows the cells to sense the ligands as reconnected under the anisotropic nano-blockers in the elevated state and infiltrate through the nano-gap.

Fig. 3. In situ phase transformation enables the formation of reversibly controllable anisotropic nano-blockers.

a Schematic illustration of the annealing conditions for the in situ phase transformation in the anisotropic nano-blocker precursor to the anisotropic nano-blocker, along with corresponding in situ transmission electron microscopy (TEM), in situ high-resolution TEM (HR-TEM), in situ FFT, in situ selected area electron diffraction (SAD) pattern images, and b in situ SAD pattern intensity analysis. c X-ray diffraction (XRD) analysis exhibiting the distinctive crystalline planes and d vibrating sample magnetometry (VSM) analysis showing the hysteresis loops of the anisotropic nano-blockers and their precursors after normalization to their respective dry weights. The average d-spacing between the successive lattice planes of the akaganeite phase (5.4 Å) and magnetite phase (3.1 Å) are labeled in the HR-TEM, distinctive bright spots of the akaganeite phase [(200) plane] and the magnetite phase [(220) plane] are labeled in the fast Fourier transform (FFT); distinctive rings corresponding to each akaganeite phase [(103), (211), (310), and (411)] and magnetite phase [(220), (311), (511), and (440)] are labeled in the SAD pattern images. In in situ SAD pattern images in (a), red dotted lines were drawn to emphasize the disappearance of (310) and (411) planes of the akaganeite phase after annealing. In in situ SAD pattern analysis in (b), rectangular light red boxes were drawn to emphasize the gradual disappearance of (310) and (411) planes of the akaganeite phase after annealing. Distinctive planes of the akaganeite phase [(103), (211), (310), (411), and (521)] and magnetite phase [(220), (311), and (400)] are labeled in the XRD analysis. Scale bars: 100 nm (TEM), 5 nm (HR-TEM), 5 nm^-1 (FFT), and 2 nm^-1 (SAD). Source data are provided as a Source Data file.

Fig. 4. Reducing the anisotropy or linearizing the nano-blockers independently escalates # ligand inter-cluster edges.

a High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images with elemental energy dispersive spectroscopy (EDS) mapping (Fe from the magnetite phase in the nano-blocker core, and Si from the silica envelop) images marked with white dotted lines at the midsection for b the elemental EDS line profiling of the anisotropic nano-blockers (“Low aniso. nano-blocker”, “Moder. aniso. nano-blocker”, and “High aniso. nano-blocker”). c Computation of the projected area of the nano-blocker (10³ nm²) (n = 10). d Scanning electron microscopy (SEM) images of the materials displaying tuned # ligand inter-cluster edges depending on the anisotropy or arrangement of the nano-blockers [“Low aniso.”, “Moder. aniso.”, “High aniso.”, and “High aniso. (Lin.)”] with the representative calculation of # ligand inter-cluster edges. The edges (green lines) within the ligand network formed by nodes in the SEM image were determined by Delaunay triangulation in Python. Each cluster was partitioned using the Louvain algorithm and distinguished by different colors. e Corresponding computation of the inter-distance of the liganded GNPs (n = 10 edges), the surface density of the anisotropic nano-blockers (n = 10 nano-blockers), # ligand inter-cluster edges (n = 3 biological replicates; ***p < 0.001), and the average inter-angle between anisotropic nano-blockers (in absolute value) (n = 10 nano-blockers; *p < 0.05, **p < 0.01). High-anisotropy nano-blockers were linearized on the materials via magnetic annealing during their grafting to the material surface. Scale bars: 200 nm (HAADF-STEM and SEM). Data are shown as the means ± standard errors. Statistical analysis was performed using one-way ANOVA along with the Tukey–Kramer post hoc test. N.S. denotes no statistically significant difference. Source data are provided as a Source Data file.

Fig. 5. Schematic of cluster partition method in the network model by maximizing modularity with the Louvain algorithm.

Modularity helps to quantify the interconnectivity of clusters within a network. Factors used in such quantification in the modularity formula include the total number of edges (m), the presence of an edge between the two nodes (A_ij), the number of edges from each node (k_i or k_j), and the cluster coincidence of node pair [$\delta$(C_i, C_j)]. The modularity is maximized in the optimal cluster partition where the number of intra-cluster (within the cluster) edges is maximized while the number of inter-cluster (between the clusters) edges is minimized. The optimized cluster partition of a given network can be found using the Louvain algorithm in Python, where this formula is included.

Fig. 6. Cyclic elevation of anisotropic nano-blockers escalates # ligand inter-cluster edges that reversibly stimulate integrin recruitment.

a Schematic illustration of the cyclic elevation (“E.”) and non-elevation (“NE.”) of high-anisotropy nano-blockers grafted to the materials (“High aniso.”) that can reversibly escalate # ligand inter-cluster edges. b In situ atomic force microscopy (AFM) images of the cyclic elevation of high-anisotropy nano-blockers for reversible tuning of # ligand inter-cluster edges repeated over two cycles marked with white dotted lines at the midsection for height analysis and c the subsequent computation of peak height changes of the nano-blockers (n = 5; ***p < 0.001). d Schematic illustration of utilizing L-cysteine to compute the number of polymer linkers coupled to the nano-blocker surfaces using Ellman’s assay for (e) computation of the reacted L-cysteine molarity on the anisotropic nano-blockers either coupled with a low (used in this study) or high density of polymer linker (n = 4; ***p < 0.001). f Scanning electron microscopy (SEM), elemental energy dispersive spectroscopy (EDS) mapping (Fe from the magnetite phase of the nano-blocker core), and overlay images of immuno-GNP (IGNP)-based tagging of integrin in stem cells adhered to the materials displaying tunable # ligand inter-cluster edges. g Schematic illustration and SEM images of the IGNP-tagged integrin in stem cells that could infiltrate through nano-gaps (at low polymer linker density showing the image in Fig. 5f) or were blocked (at high polymer linker density) when the anisotropic nano-blockers were elevated. h Computation of the average number of IGNP-tagged integrin in stem cells at the cell boundary per unit area (μm²) (n = 3 gold nanoparticles; **p < 0.01; ***p < 0.001) and inter-distance of the liganded GNPs after cell culturing (n = 10 edges). A piece of permanent magnet (295 mT) was employed (“E.”) or non-employment (“NE.”) above the materials for the cyclic elevation of high-anisotropy nano-blockers. High-anisotropy nano-blockers were linearized (“Lin.”) on the materials via magnetic annealing during their grafting to the material surface. In the SEM images, stem cells and IGNPs are each colored green and white, respectively. Scale bars: 200 nm (AFM) and 500 nm (SEM and EDS). Data are shown as the means ± standard errors. Statistical analysis was performed using one-way ANOVA along with the Tukey–Kramer post hoc test. N.S. denotes no statistically significant difference. Source data are provided as a Source Data file.

Fig. 7. Remote cyclic tuning of # ligand inter-cluster edges reversibly regulates stem cell behaviors.

a Immunostained fluorescent images of paxillin co-stained with F-actin and nuclei (DAPI) of adherent stem cells after 24 h, 48 h, or 72 h of culturing in growth medium and RUNX2 co-stained with F-actin and nuclei after 72 h of culturing in differentiation medium on materials displaying a cyclically tunable # ligand inter-cluster edges. b Computation of the DAPI-positive cell density (n = 6–9 biological replicates; *p < 0.05; **p < 0.01; ***p < 0.001), focal adhesion number (n = 6 cells; ***p < 0.001), actin-positive cell area (n = 6 cells; ***p < 0.001), and nuclear/cytoplasmic intensity ratio of RUNX2 (n = 6 cells; *p < 0.05; **p < 0.01; ***p < 0.001) along with western-blot analysis of RUNX2 and ALP protein expression (normalized to GAPDH) of adherent stem cells. The employment and non-employment of a piece of permanent magnet (295 mT) above the materials were either switched or maintained every 24 h for up to 72 h (“NE.-NE.-NE.”, “NE.-E.-NE.”, “E.-NE.-E.”, and “E.-E.-E.”). Scale bars: 50 µm (confocal microscopy). Data are shown as the means ± standard errors. Statistical analysis was performed using one-way ANOVA along with the Tukey–Kramer post hoc test. N.S. denotes no statistically significant difference. Source data are provided as a Source Data file.

Fig. 8. Dynamic stem cell behaviors regulated by remote tuning of # ligand inter-cluster edges is driven by the adhesion-relevant molecular machinery.

a Immunostained fluorescent images of YAP co-stained with F-actin and nuclei of adherent stem cells after 48 h of culturing in stem cell growth medium or RUNX2 co-stained with F-actin and nuclei after 72 h of culturing in stem cell differentiation medium on materials displaying remotely tunable # ligand inter-cluster edges. In each case, the medium was supplemented with ML9 (a myosin II-inhibitor), Swinholide A (Swinh. A; an actin polymerization-inhibitor), or Y27632 (a ROCK-inhibitor) or without any of the inhibitors (the control). b Computation of nuclear/cytoplasmic intensity ratio of YAP (n = 6 cells; ***p < 0.001) and RUNX2 (n = 6 cells; *p < 0.05; ***p < 0.001) expression in adherent stem cells. Scale bar: 50 µm (confocal microscopy). Data are shown as the means ± standard errors. Statistical analysis was performed using one-way ANOVA along with the Tukey–Kramer post hoc test. N.S. denotes no statistically significant difference. Source data are provided as a Source Data file.

Fig. 9. Time-resolved tuning of # ligand inter-cluster edges regulates stem cell behaviors in vivo.

a Schematic illustration of the time-resolved elevation (“E.”) and non-elevation (“NE.”) control of high-anisotropy nano-blockers to tune # ligand inter-cluster edges in vivo for the regulation of injected stem cells on the implanted materials. b Immunostained fluorescent images of paxillin or YAP co-stained with F-actin and nuclei (DAPI), and HuNu co-stained with RUNX2 and nuclei (DAPI) of adherent stem cells after 6 h of injection onto the subcutaneously implanted material displaying a reversibly tunable # ligand inter-cluster edges. c Computation of the HuNu-positive cell density (n = 6 biological replicates; *p < 0.05; ***p < 0.001), the focal adhesion number (n = 6 cells; *p < 0.05; ***p < 0.001), and the nuclear/cytoplasmic intensity ratios of YAP and RUNX2 expression (n = 6 cells; *p < 0.05; ***p < 0.001) in adherent stem cells after 6 h of injection. d Hematoxylin and eosin (H&E) stained images of subcutaneous tissue near the implanted site and major organs (liver, heart, spleen, and kidney) of mice before and 7 d after material implantation to assess the toxicity of implanted materials locally and systemically, respectively. A piece of permanent magnet was carefully coupled to the backs of the mice to direct the elevation of high-anisotropy nano-blockers in vivo. The coupling and uncoupling of the magnet were either switched or maintained after 3 h for 6 h (“NE.-NE.”, “NE.-E.”, “E.-NE.”, and “E.-E.”). Scale bars: 50 µm (confocal microscopy) and 200 μm (optical microscopy). Data are shown as the means ± standard errors. Statistical analysis was performed using one-way ANOVA along with the Tukey–Kramer post hoc test. Source data are provided as a Source Data file.