Automatic Actin Filament Quantiﬁcation and Cell Shape Modeling of Osteoblasts on Charged Ti Surfaces

Featured Application: In medical implantology the human biosystem interacts with the artiﬁcial biomaterial. For permanent implants, e.g. titanium, it is of importance that cells immediately accept and occupy the material surface. To get deeper insights into molecular mechanisms of the cell behavior at the interface is of great importance. The future aim is to ﬁnd an optimum in material surface design for tissue-integrated, long living implants. Abstract: Surface charges at the cell–biomaterial interface are known to determine cellular functions. Previous ﬁndings on cell signaling indicate that osteoblastic cells favor certain moderately positive surface charges, whereas highly positive charges are not tolerated. In this study, we aimed to gain deeper insights into the inﬂuence exerted by surface charges on the actin cytoskeleton and the cell shape. We analyzed surfaces with a negative, moderately positive, and highly positive zeta ( ζ ) potential: titanium (Ti), Ti with plasma polymerized allylamine (PPAAm), and Ti with a polydiallyldimethylammonium chloride (PDADMA) multilayer, respectively. We used the software FilaQuant for automatic actin ﬁlament quantiﬁcation of osteoblastic MG-63s, analyzed the cell edge height with scanning ion conductance microscopy (SICM), and described the cellular shape via a mathematical vertex model. A signiﬁcant enhancement of actin ﬁlament formation was achieved on moderately positive (+7 mV) compared with negative ζ -potentials ( − 87 mV). A hampered cell spreading was reﬂected in a diminished actin ﬁlament number and length on highly positively charged surfaces (+50 mV). Mathematical simulations suggested that in these cells, cortical tension forces dominate the cell–substrate adhesion forces. Our ﬁndings present new insights into the impact of surface charges on the overall cell shape and even intracellular structures. ﬂuctuations around the mean value and are seen as a point cloud. The model suggests that substrate charges inﬂuence the cell shape in an actin-dependent manner.


Introduction
In bone tissue engineering, chemical [1][2][3] and topographical [4][5][6] surface modifications have been proven to be a powerful tool in optimizing implant designs for clinical applications in dental and orthopedic surgery in terms of interaction with osteoblastic cells. Biomaterial surface properties such as chemistry, roughness, stiffness, wettability, voltage is below the typical cell membrane potential of 60 to 80 mV [27] and should not disrupt cellular membrane or kill cells in the tissue [26]. In physiological solution, there are no free electrons to carry the current. Thus, electric currents are carried by charged ions [18]. As mentioned above, in the extracellular matrix of tissues and around cells, the glycosaminoglycan hyaluronan molecules possess carboxyl groups (C(= O)OH) that are negatively charged.
Thus, the development of new surface modification strategies for implant materials can benefit from these physiological phenomena of endogenous fields and charges of the biosystem.
Bioactivation of biomaterial surfaces: Due to the aforementioned electrostatic forces between cells and materials at the interface, it was imaginable that positive surface charges could trigger the cell adhesion. Indeed, a chemical surface modification with a nanometer thin plasma polymer layer [28] is sufficient to push the cells' initial adhesion (already after 2 min), as this nanolayer provides positively charged amino groups. This effect is to a greater extent than binding of the ECM protein collagen to the material surface [29].
The early stages of cell-substrate interaction involve cellular attachment, adhesion, and spreading [30], whereby the cell shape changes from spherical to elongated, resulting in a decreased cell height and an increased cell contact area with strengthened adhesion force [31]. This process is critical for normal cell function and successful osseointegration [32], as further cell activities are triggered thereafter.
Our previous research showed that material surface charges at the interface, deduced from the ζ-potential, possess a remarkable influence on diverse cellular functions, among them spreading [33], morphology, membrane integrity, proliferation, and intracellular calcium signaling [34]. According to the cell behavior, three categories of ζ-potential were described: negative, moderately positive, and highly positive [34]. Positive charges at the material surface provide unique cues that initiate an intensive cell response. However, it was revealed that these cell reactions occur in a so-called dose-dependent manner. Cells favor only a certain range of moderately positive material surface charges and cannot tolerate high charges around +50 mV [34]. Figure 1 summarizes our current knowledge on cell spreading and cell shape as a response to material surface charge. Here, the cellular behavior on 12 differently charged surfaces is averaged and depicted (see further information in the Supplementary Material for all layers used). For the first cell encounter, electrostatic forces between the material surface and the cells play an important role. These forces originate from the negatively charged pericellular hyaluronan coat around the osteoblasts, resulting in an overall net negative cell charge of −15.6 mV [15]. Therefore, rapid cell attachment can be mediated especially by a positive material surface charge. Our results show a strong correlation between the material surface charge and cell spreading within 24 h, which is optimal between +1 and +10 mV (ζ-potential). As the cell physiology is adversely affected on highly positively charged material surfaces [33,34], cells exhibit on these surfaces also an impaired cell spreading course and retain their circular morphology.
In this context, it is of interest to take a closer look at the cell shape. For this, the present study aimed to focus on the actin cytoskeleton as a stabilizer of the cell shape. Actin organization is essential for adhesion, cell contact formation and migration, as well as for directed cytoplasmic trafficking, such as signal transduction [35][36][37]. In particular, it is reported that the arrangement of the actin cytoskeleton is pivotal for cellular spreading as for the cell length control [38,39]. Several studies highlight a surface topography-dependent shift in actin filament formation in osteoblasts [40,41]. Hence, micropillar structures induce actin formation on the plateau of the pillars in a mimicry fashion. Is the charge of a surface also capable of affecting the actin arrangement and thus cell shape? To answer this question, we used three differently charged titanium surfaces, one from each ζ-potential category based on Gruening et al. [34]. We quantified the actin filament number and length of MG-63 osteoblasts after growing 1 and 24 h on the substrates in an automated approach using our FilaQuant software, which is freely available (version 2011, University of Rostock, Institute of Mathematics, Mathematical Optimization, Rostock, Germany) [40]. Additionally, we wanted to gain a deeper understanding of the influence that surface charges exert on the cells morphology. To this end, we measured the cell edge height via scanning ion conductance microscopy (SICM) and developed a mathematical vertex model to explain how actin rearrangement, caused by different surface charges, influences the cell shape.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 20 cell shape? To answer this question, we used three differently charged titanium surfaces, one from each ζ-potential category based on Gruening et al. [34]. We quantified the actin filament number and length of MG-63 osteoblasts after growing 1 and 24 h on the substrates in an automated approach using our FilaQuant software, which is freely available (version 2011, University of Rostock, Institute of Mathematics, Mathematical Optimization, Rostock, Germany) [40]. Additionally, we wanted to gain a deeper understanding of the influence that surface charges exert on the cells′ morphology. To this end, we measured the cell edge height via scanning ion conductance microscopy (SICM) and developed a mathematical vertex model to explain how actin rearrangement, caused by different surface charges, influences the cell shape. Figure 1. Summary of our current knowledge of ζ-potential-dependent change in cell area and shape over time on functionalized Ti surfaces. Our previous research showed that surface charges at the interface deduced from the ζ-potential possess a remarkable influence on cellular spreading [33] and morphology [34]. Overall, 12 different top layers of the categories of amino polymers, extracellular matrix proteins/peptide motifs, polyelectrolyte multilayers, and metals were deposited on Ti substrates and analyzed corresponding to surface charge and spreading behavior. (a) Cell area: According to the cell spreading, these surfaces were categorized into three groups of ζ-potential, and the respective cell area values were averaged: negative (blue), moderately positive (green), and highly positive (orange). Note that cells favor only a certain range of moderately positive surface charges and cannot tolerate high charges of around +50 mV. Osteoblasts on the four moderately positive surfaces RGD, APTES, PPAAm, and PEI (+1 to +10 mV, green box) exhibit the highest cell area after 1 h (1791 ± 275 µm 2 ). This cell area changes only slightly within 24 h (1669 ± 66 µm 2 ). Cells on the five negatively charged surfaces Au, PSS, Ti, Matrigel, and Col I (−120 to −3 mV, blue box) initially have a smaller cell area after 1 h (838 ± 114 µm 2 ) but spread within 24 h to reach the same level of cell area as on moderately positive surfaces (1513 ± 108 µm 2 after 24 h, arrow). Cells on the three highly positively charged surfaces PPI-G2, PPI-G4, and PDADMA (+50 to +55 mV, orange box) show initially significantly increased spreading (1321 ± 89 µm 2 ), but the cell area decreases dramatically within 24 h (510 ± 65 µm 2 after 24 h, arrow). Plot: Microscopic images of PKH26 stained cells after 1 h on Ti, PPAAm, and PDADMA as representatives for each category. Cell area after 1 and 24 h in mean ± sem, ζ-potentials in mean ± SD, Mann-Whitney-U test, p < 0.05; significances after 1 h: − vs. + and − vs. ++; significances after 24 h: − vs. ++ and + vs. ++. (b) Circularity: A perfect circle is defined by a value of 1.0, and an increasingly elongated polygon is represented by a value converging toward 0. While the cell shape on negatively and moderately positively charged surfaces changes from circular to a more elongated shape within 24 h, cells on highly positively charged surfaces exhibit an impaired cell spreading and retain their circular morphology. Plot: ζ-potentials in mean; circularity after 1 and 24 h in medians (see Figure 1. Summary of our current knowledge of ζ-potential-dependent change in cell area and shape over time on functionalized Ti surfaces. Our previous research showed that surface charges at the interface deduced from the ζ-potential possess a remarkable influence on cellular spreading [33] and morphology [34]. Overall, 12 different top layers of the categories of amino polymers, extracellular matrix proteins/peptide motifs, polyelectrolyte multilayers, and metals were deposited on Ti substrates and analyzed corresponding to surface charge and spreading behavior. (a) Cell area: According to the cell spreading, these surfaces were categorized into three groups of ζ-potential, and the respective cell area values were averaged: negative (blue), moderately positive (green), and highly positive (orange). Note that cells favor only a certain range of moderately positive surface charges and cannot tolerate high charges of around +50 mV. Osteoblasts on the four moderately positive surfaces RGD, APTES, PPAAm, and PEI (+1 to +10 mV, green box) exhibit the highest cell area after 1 h (1791 ± 275 µm 2 ). This cell area changes only slightly within 24 h (1669 ± 66 µm 2 ). Cells on the five negatively charged surfaces Au, PSS, Ti, Matrigel, and Col I (−120 to −3 mV, blue box) initially have a smaller cell area after 1 h (838 ± 114 µm 2 ) but spread within 24 h to reach the same level of cell area as on moderately positive surfaces (1513 ± 108 µm 2 after 24 h, arrow). Cells on the three highly positively charged surfaces PPI-G2, PPI-G4, and PDADMA (+50 to +55 mV, orange box) show initially significantly increased spreading (1321 ± 89 µm 2 ), but the cell area decreases dramatically within 24 h (510 ± 65 µm 2 after 24 h, arrow). Plot: Microscopic images of PKH26 stained cells after 1 h on Ti, PPAAm, and PDADMA as representatives for each category. Cell area after 1 and 24 h in mean ± sem, ζ-potentials in mean ± SD, Mann-Whitney-U test, p < 0.05; significances after 1 h: − vs. + and − vs. ++; significances after 24 h: − vs. ++ and + vs. ++. (b) Circularity: A perfect circle is defined by a value of 1.0, and an increasingly elongated polygon is represented by a value converging toward 0. While the cell shape on negatively and moderately positively charged surfaces changes from circular to a more elongated shape within 24 h, cells on highly positively charged surfaces exhibit an impaired cell spreading and retain their circular morphology. Plot: ζ-potentials in mean; circularity after 1 and 24 h in medians (see detailed values in Table S1). Abbreviations: Au (gold), PSS (polystyrene sulfonate), Ti (titanium), Matrigel (basement membrane matrix), Col I (collagen type I), RGD (arginine-glycine-aspartate sequence), APTES (3-aminopropyltriethoxysilane), PPAAm (plasma polymerized allylamine), PEI (polyethyleneimine), PPI-G4 (polypropyleneimine dendrimer generation 4), PDADMA (polydiallyldimethylammonium chloride), and PPI-G2 (polypropyleneimine dendrimer generation 2).

Applied Titanium Arrays and Their Properties
Planar silicon arrays sputtered with 100 nm titanium (Ti) were obtained from the Center for Microtechnologies (ZFM, Norbert Zichner, University of Technology Chemnitz, Germany) at 1 × 1 × 0.075 cm in size (length × width × depth). Ti is covered by a natural oxide layer. The Ti specimens exhibited a negative ζ-potential of −87.5 mV at pH 7.4, a water contact angle (WCA) of 87.4 • , and a surface free energy (SFE) of 37.6 mN/m [34]. On the one hand, these Ti substrates were coated with the plasma polymerized allylamine (PPAAm) nanolayer by a low-pressure plasma reactor [34,42] and were acquired from the Leibniz Institute for Plasma Science and Technology e.V. (INP). They displayed a moderately positive ζ-potential in the physiological range (+7.1 mV), a WCA of 66.9 • , an SFE of 46.2 mN/m, and a layer thickness of 24.5 nm [34]. On the other hand, the substrates were modified by a polyelectrolyte multilayer (consisting of 10.5 polycation/polyanion pairs) terminated with polydiallyldimethylammonium chloride (PDADMA) by dip coating oppositely charged polyelectrolytes with a deposition robot (Riegler & Kirstein, Potsdam, Germany) [34,43]. They featured the following characteristics: +51.4 mV ζ-potential at pH 7.4 (0.15 bar), 38.7 • WCA, 62.0 mN/m SFE, and 38.0 nm layer thickness [34]. The positive charge originates from weakly bound polycation chains, which can protrude into the solution under suitable conditions [43]. For a detailed description of surface preparations and characteristics, please refer to Gruening et al. [34]. Briefly, concerning characteristics, the ζ-potentials of the Ti samples were determined with the SurPASS TM system (Anton Paar, Ostfildern, Germany) by analyzing the streaming potentials at pH 6.5-8.0 in a 1 mM KCl solution. The exact values at pH 7.4 (physiological conditions for the cells in vitro) were calculated via a linear regression (GraphPad Prism version 6.05) [34]. The WCA and the SFE of the substrate/air interface were determined by the sessile drop method (distilled water and diiodomethane) using the Drop Shape Analyzer-DSA25 (Krüss, Hamburg, Germany) [34].

Plasma Membrane Staining and Cell Area Quantification
The membranes of the MG-63 cells were stained with the PKH-26 General Cell Linker Kit (Sigma-Aldrich Chemie, Taufkirchen, Germany) for 5 min at 37 • C in suspension [45]. Then, cells were seeded for 1 h on the substrates (50,000 cells/cm 2 ) and fixed with 4% paraformaldehyde (PFA, 10 min, room temperature (RT)) after cultivation. Cell areas after 24 h were determined by vital fluoro-3-acetoxymethyl ester-stained cells (fluo-3, Life Technologies Corporation, Eugene, Oregon, US) [34]. Microscopic images were obtained using a confocal laser scanning microscope LSM780, a C-Apochromat 40x/1.20 W Korr M27 objective, and the ZEN black software 2011 SP4 (Carl Zeiss, Jena, Germany). The cell area in µm 2 and the circularity of 45 cells per surface was assessed using the "measurement report" of the software Photoshop CC 2017 (Adobe, San Jose, CA, USA) (n = 3 independent experiments). The software calculates the circularity C according to Equation (1): where A is the cell area and P is the cell perimeter. A perfect circle is described by a value of 1.0 and an increasingly elongated polygon by a value converging to 0. MG-63s (20,000 cells/cm 2 ) were cultured for 1 h on the PPAAm-modified Ti arrays as previously described [34]. Briefly, after fixing with 2.5% glutardialdehyde (Merck, Darmstadt, Germany) at 4 • C overnight and rinsing with 0.1% sodium phosphate buffer (according to Sørensen, Merck, Darmstadt, Germany), the samples were dehydrated through an ethanol series of 30, 50, 75, 90, and 100% (for 5, 5, 15, 10, and 10 min, respectively) and dried in a critical point dryer (K 850, EMITECH, Taunusstein, Germany). The samples were sputtered with gold for 50 s (15 nm, SCD 004, BAL-TEC, Wetzlar, Germany). Field emission scanning electron microscopy (FE-SEM, ZEISS Merlin VP compact, Carl Zeiss, Oberkochen, Germany) was conducted with a high efficiency secondary electron detector HE-SE2 at magnifications of 2000× and 5000× (5 kV, 5.1 mm working distance).
FilaQuant analysis: Contrast and brightness settings were adjusted identically for all LSM images. Subsequent actin filament quantification via mathematical image processing was conducted by the software FilaQuant 2011 (University of Rostock, Institute of Mathematics, Mathematical Optimization) [46]. The main algorithm is based on a relaxation of the analytical criterion for a ridge, which is expressed in terms of a ridgeness measurement (line detection and line tracking). The parameters filament number, total filament length, maximum filament length, and mean filament length were evaluated for single cells (n = 10 cells per surface). Software settings were as follows: top-hat radius 10-12, noise penalizer 0.2-0.3, min. edge-length 4.4-5, max. edge-length 30-34.7.

Edge Height Measurement via Scanning Ion Conductance Microscopy
First, 4000 MG-63 cells/cm 2 were cultured on pristine glass cover slides (Thermo Fisher Scientific, Waltham, MA, USA) as well as plasma polymer (PPAAm)-coated cover slides for 1 and 3 h. After fixation with 4% PFA, cells were washed thrice with PBS before measuring their nanomorphology in PBS with a Scanning Ion Conductance Microscope (SICM) (NX-bio, Park Systems, Korea) [47]. The measurements were performed with borosilicate glass nanopipettes exhibiting opening diameters below 80 nm. They were pulled from capillary tubes (inner diameter 0.58 mm) using a CO 2 laser puller (Sutter P-2000, Sutter Instrument, Novato, CA, USA) with the following parameters: Heat: 260, Filament: 4, Velocity: 50, Delay: 225, Pull: 140. Topography images were acquired in approach-retract-scheme mode (ARS mode), which is often referred to as dynamic or hopping mode. The ion current setpoint was 98% of the base current (i.e., current far away from the sample, corresponding to ca. 1 nA). For each cell (n = 6 cells per surface after 1 h, n = 18 cells per surface after 3 h), 3 topography images (10 × 10 µm 2 , 128 × 128 pixels) at randomly chosen spots at the cell edges were taken. Cell edge heights were exemplarily measured with the open-source software Gwyddion (gwyddion.net, accessed on 17.03.2021) at 5 representative locations for each topography image.

Mathematical Model for the Description of Cell Shape Changes
In order to study the influence of actin filament dynamics on the shape of cells placed on differently charged substrates, we simulated a mathematical vertex model for cell shapes [48][49][50]. The two-dimensional cross-section of a cell in this model is represented as a polygon specified by surrounding vertices (Figure 2a) [49,51,52]. The cell mechanical properties in stationary conditions are described by an energy function whose functional form is given by: where A is the cross-sectional area and P is the perimeter of the cell. A 0 is the preferred area that the cell strives to attain such that any deviation from it, i.e., A 0 , corresponds to cellular pressure, the magnitude of which depends on the deviation (A − A 0 ) and the constant K A , which denotes the strength of resistance to cell area changes. The second term represents the tensile properties of the cell membrane and consists of two terms, one quadratic in perimeter and the other linear in perimeter. The term that is quadratic in perimeter describes bond tension due to the actomyosin ring in the subcellular cortex. In contrast, the term that is linear in perimeter describes an effective cell membrane tension due to focal adhesions arising from cell-substrate interaction. The net contribution to cell membrane tension comes from cortical tension, due to actin filaments, that resists an increase in cell perimeter, and cell-substrate adhesion that tends to increase the cell perimeter by growing actin filaments. K P captures the effective resistance offered by the cell to any changes in the cell perimeter length. By introducing L, which is the typical diameter of an osteoblastic cell and has the dimension of length (µm), and dividing the whole equation by K P L 2 , Equation (2) can be expressed in the non-dimensional form as: where E = E/(K P L 2 ), K A = K A L 2 K P , A = A/L 2 , A 0 = A 0 /L 2 , P = P/L, and P 0 = P 0 /L. Various parameters of the model, as described by Equation (3), and their meaning are listed in Table 1. The shape of the cell is determined by quasi-static simulations in which, for a given set of parameter values of the energy function given by Equation (3), the system is relaxed to a force-balanced state using a Monte Carlo-based minimization algorithm. Table 1. Summary of all the model parameters, in dimensionless form, and their meaning.

Model Quantities Meaning
A Dimensionless area of the cell A 0 Dimensionless preferred area of the cell P Dimensionless perimeter of the cell P 0 Dimensionless preferred perimeter of the cell

Dimensionless strength of resistance to area changes
The vertex models described by Equation (2) have been successful in predicting cell shapes due to mechanical interactions between cells in a confluent tissue [48][49][50].
Here, we use this model for studying shape changes in single cells. In our model, an osteoblastic cell is represented by a polygon with six sides, as shown in Figure 2a. A dimensionless shape index parameter can be defined as p 0 = P 0 / √ A 0 . Earlier studies of the vertex model for confluent tissues given by Equation (2) showed that a transition takes places at a critical value of p 0 = p * 0 . When p 0 < p * 0 , the stable configuration of a polygon with six sides is given by the ground state and has the shape of a regular hexagon, which is characterized by a higher circularity value (Figure 2b). However, when p 0 > p * 0 , the ground state is degenerate, and the polygon has an irregular shape which is characterized by a lower circularity value. For a regular hexagon, this transition has been shown to occur at p * 0 ∼ 3.71 [53][54][55]. Based on this knowledge, we use the vertex model to understand what changes in the cell mechanical properties can induce experimentally observed changes in the morphology of single osteoblastic cells seeded on substrates with different charge properties.
vertex model for confluent tissues given by Equation (2) showed that a transition takes places at a critical value of = * . When < * , the stable configuration of a polygon with six sides is given by the ground state and has the shape of a regular hexagon, which is characterized by a higher circularity value (Figure 2b). However, when * , the ground state is degenerate, and the polygon has an irregular shape which is characterized by a lower circularity value. For a regular hexagon, this transition has been shown to occur at * ~3 .71 [53][54][55]. Based on this knowledge, we use the vertex model to understand what changes in the cell mechanical properties can induce experimentally observed changes in the morphology of single osteoblastic cells seeded on substrates with different charge properties.

Figure 2.
(a) In the vertex model, an osteoblastic cell is represented as a polygon with six sides and is specified by the surrounding vertices, where the length of the side between two neighboring vertices and is . The cell shape is determined by quasi-static simulations in which a force-balanced state is achieved using an algorithm that performs a Monte Carlo-based minimization of Equation (3). In the final force-balanced state, the shape of the cell is defined by its area and perimeter . (b) Earlier studies of the vertex model have shown that when the shape index factor = /√ is below a critical value * , the stable configuration is the ground state and has the shape of a regular polygon, i.e., with higher circularity value. However, when the value of is higher than * , the ground state is degenerated and the polygon has an irregular shape, which is characterized by a lower circularity value.

Statistics
Non-parametric Kruskal-Wallis followed by Dunn's multiple comparisons test and Mann-Whitney-U test with the software GraphPad Prism version 6.05 for Windows (GraphPad Software, La Jolla, CA, United States) were conducted on the p-values < 0.05. Results are presented in mean ± sem (standard error of the mean) for actin filament (Table 2) and cell area values ( Figure 1). An unpaired t-test was performed for cell edge height values (Section 3.3) presented in mean ± SD (standard deviation).  Figure 2. (a) In the vertex model, an osteoblastic cell is represented as a polygon with six sides and is specified by the surrounding vertices, where the length of the side between two neighboring vertices i and j is l ij . The cell shape is determined by quasi-static simulations in which a force-balanced state is achieved using an algorithm that performs a Monte Carlo-based minimization of Equation (3). In the final force-balanced state, the shape of the cell is defined by its area A 0 and perimeter P 0 . (b) Earlier studies of the vertex model have shown that when the shape index factor p 0 = P 0 / √ A 0 is below a critical value p * 0 , the stable configuration is the ground state and has the shape of a regular polygon, i.e., with higher circularity value. However, when the value of p 0 is higher than p * 0 , the ground state is degenerated and the polygon has an irregular shape, which is characterized by a lower circularity value.

Statistics
Non-parametric Kruskal-Wallis followed by Dunn's multiple comparisons test and Mann-Whitney-U test with the software GraphPad Prism version 6.05 for Windows (Graph-Pad Software, La Jolla, CA, United States) were conducted on the p-values < 0.05. Results are presented in mean ± sem (standard error of the mean) for actin filament (Table 2) and cell area values ( Figure 1). An unpaired t-test was performed for cell edge height values (Section 3.3) presented in mean ± SD (standard deviation).

Results
As the major component of the cellular cytoskeleton, actin is known to play a crucial role in diverse cellular processes which control the cell shape and thus the functional behavior of the cells. Cellular spreading requires the extension of the plasma membrane in association with the assembly of actin in filaments [56,57]. As shown in Figure 3, membrane protrusions occur at the growth zone of the cell, the lamellipodia, as thin finger-like structures called filopodia that sense their surroundings. They extend outward from the perimeter of the cell with a stabilizing and forward-growing effect pushed by actin filaments.

Results
As the major component of the cellular cytoskeleton, actin is known to play a crucial role in diverse cellular processes which control the cell shape and thus the functional behavior of the cells. Cellular spreading requires the extension of the plasma membrane in association with the assembly of actin in filaments [56,57]. As shown in Figure 3, membrane protrusions occur at the growth zone of the cell, the lamellipodia, as thin finger-like structures called filopodia that sense their surroundings. They extend outward from the perimeter of the cell with a stabilizing and forward-growing effect pushed by actin filaments.

Actin Cytoskeleton and Quantification of Filaments
We focused on the quantification of the cellular actin filament organization on the three differently charged surfaces: Ti (−), PPAAm (+), and PDADMA (++). Confocal microscopy was applied to visualize the actin cytoskeleton after 1 and 24 h and the quantification was carried out by the software FilaQuant (Figure 4).
When regarding actin distribution after 1 h, considerable rearrangements were revealed, in particular on PPAAm. Here, the cells are extremely well developed with numerous, long, filopodial protrusions in comparison with uncoated Ti. Compared with

Actin Cytoskeleton and Quantification of Filaments
We focused on the quantification of the cellular actin filament organization on the three differently charged surfaces: Ti (−), PPAAm (+), and PDADMA (++). Confocal microscopy was applied to visualize the actin cytoskeleton after 1 and 24 h and the quantification was carried out by the software FilaQuant (Figure 4).
When regarding actin distribution after 1 h, considerable rearrangements were revealed, in particular on PPAAm. Here, the cells are extremely well developed with numerous, long, filopodial protrusions in comparison with uncoated Ti. Compared with PPAAm, filopodia on PDADMA surfaces are considerably shorter. In detail, the resulting data quantification via FilaQuant revealed a significantly shorter mean filament length on PDADMA (4.3 µm, compared with 6.2 and 6.6 µm on Ti and PPAAm, respectively), whereas the total filament length on PPAAm surfaces (1050.0 µm) tripled as a result of the threefold increase in the actin filament number (148.3) after 1 h in contrast to Ti (filament number: 62.3, total filament length: 386.5 µm) and PDADMA (filament number: 51.4, total filament length: 241.6 µm) ( Figure 5, Table 2). The advantage of this moderately positively charged surface for cell occupation is clearly evident.
After 24 h, actin is organized typically as a network of well-defined stress fibers, spanning the entire cell body on negatively charged Ti and moderately positively charged PPAAm surfaces (Figure 4). On highly positively charged PDADMA surfaces, the actin formation is still confined to the outer cell edge and is presented in a clustered form. Here, the cells could neither spread nor form stress fibers, remaining round in shape, and having also shrunk in size. The total filament length shortened by about 6 times with respect to cells on Ti and PPAAm ( Figure 5, Table 2). Similarly, the mean and maximum filament length decreased significantly by approximately 4.5 and 6 times, respectively. on PDADMA (4.3 µm, compared with 6.2 and 6.6 µm on Ti and PPAAm, respectively), whereas the total filament length on PPAAm surfaces (1050.0 µm) tripled as a result of the threefold increase in the actin filament number (148.3) after 1 h in contrast to Ti (filament number: 62.3, total filament length: 386.5 µm) and PDADMA (filament number: 51.4, total filament length: 241.6 µm) ( Figure 5, Table 2). The advantage of this moderately positively charged surface for cell occupation is clearly evident.
After 24 h, actin is organized typically as a network of well-defined stress fibers, spanning the entire cell body on negatively charged Ti and moderately positively charged PPAAm surfaces (Figure 4). On highly positively charged PDADMA surfaces, the actin formation is still confined to the outer cell edge and is presented in a clustered form. Here, the cells could neither spread nor form stress fibers, remaining round in shape, and having also shrunk in size. The total filament length shortened by about 6 times with respect to cells on Ti and PPAAm ( Figure 5, Table 2). Similarly, the mean and maximum filament length decreased significantly by approximately 4.5 and 6 times, respectively.    Table 2, n = 10 cells per surface).

Mathematical Modeling of the Cell Shape
Experiments show that the changes in the cell shape, characterized by cell area and circularity measurements, are associated with changes in the length of cortical actin filaments. In order to understand how actin dynamics affects the shape of single cells placed on differently charged substrates, we developed a vertex-based model as described in Section 2.3. Here, we focus only on surfaces with PPAAm (moderately positive charges) and PDADMA (highly positive charges).
Experiments show that irrespective of the substrate charge, cells initially, i.e., 1 h after seeding on the substrate, have a higher circularity, i.e., closer to 1. Therefore, in our  Table 2, n = 10 cells per surface).

Mathematical Modeling of the Cell Shape
Experiments show that the changes in the cell shape, characterized by cell area and circularity measurements, are associated with changes in the length of cortical actin filaments. In order to understand how actin dynamics affects the shape of single cells placed on differently charged substrates, we developed a vertex-based model as described in Section 2.3. Here, we focus only on surfaces with PPAAm (moderately positive charges) and PDADMA (highly positive charges).
Experiments show that irrespective of the substrate charge, cells initially, i.e., 1 h after seeding on the substrate, have a higher circularity, i.e., closer to 1. Therefore, in our simulations, for the initial state of the cell, we used the values for parameters listed in Table 1 such that the cell shape has a circularity close to 1 (Figure 6a). We chose a high ratio of K A L 2 /K P , where L, as described in Section 2.3, is the typical diameter of an osteoblastic cell, so that the cell cross-sectional area at the initial time point is fixed at the value A 0 . In our simulations, we did not consider the full time-dependent evolution of the cell shape, but only the final force-balanced state resulting from a change in the model parameter values. The cell morphology in the final state is defined by its area and circularity. Cell circularity is calculated using the relation given by Equation (1). Table 1 such that the cell shape has a circularity close to 1 (Figure 6a). We chose a high ratio of / , where , as described in Section 2.3, is the typical diameter of an osteoblastic cell, so that the cell cross-sectional area at the initial time point is fixed at the value . In our simulations, we did not consider the full time-dependent evolution of the cell shape, but only the final force-balanced state resulting from a change in the model parameter values. The cell morphology in the final state is defined by its area and circularity. Cell circularity is calculated using the relation given by Equation (1). , compared with the cell area and circularity at 1 h. In model ++, the cell state at 24 h was obtained by changing only K P . In this case, K P was increased to a much higher value, compared with the initial state. All the other parameter values were kept the same as for the cell state at 1 h. This state is characterized by (b) a significant drop in the cell area (value after 1 h shown as red circle and after 24 h shown as blue diamond) and (c) an insignificant change in the value of the cell circularity (value after 1 h shown as red circle and after 24 h shown as blue diamond), compared with the cell area and circularity at 1 h. Each data point in the (b,c) is the average of 10 independent simulation runs. (d) When P 0 < 83.18, the cell shape is characterized by a circularity close to 1. In this case, circularity values obtained from 10 independent simulation runs show a very small, insignificant difference and are seen aligned as a line. However, when P 0 ≥ 83.18, the ground state of the cell is degenerate, and therefore, although the circularity is much lower than 1, the values obtained from 10 independent simulation runs show noticeable fluctuations around the mean value and are seen as a point cloud. The model suggests that substrate charges influence the cell shape in an actin-dependent manner.

Model +: Cell Morphology on the Substrate with Moderately Positive Surface Charge
Studies on the rigidity transition observed in vertex models show that the cell shape change from a regular polygon, with circularity close to 1, to an irregular elongated polygon with no significant change in the cell area can be obtained by changing the parameter p 0 , as defined in Section 2.3 (Figure 6b,c). In our simulations, the experimentally observed shape changes are obtained when P 0 ≥ 83.18, since we chose A 0 = 500. By increasing the value of P 0 = 90, our simulations show that the circularity drops from an initial value of 0.9 to 0.7, whereby the area is conserved, i.e., A ∼ = 500 in the final state (Figure 6b,c). This suggests that a moderately positively charged substrate promotes cell-substrate adhesion such that adhesive forces dominate the cell cortical tension.

Model ++: Cell Morphology on a Substrate with Highly Positive Surface Charge
Experimentally observed cell shape changes in this case can be reproduced by keeping the initial value of P 0 , i.e., 20 and increasing only the value K P . By increasing the value of K P to 48, 000, our simulations show that the circularity remains at the initial value of 0.9, but the cell area reduces to A ∼ = 300 in the final state, similar to experiments (Figure 6b,c). This suggests that in the case of highly positively charged substrates, cell cortical tension dominates cell-substrate adhesion and tends to shrink the cells but keep the overall initial circularity.
Multiple independent simulation runs of the model show that the variation in the value of cell circularity is quite insignificant when P 0 < 83.18, which corresponds to the case of cells on a moderately positively charged substrate, i.e., PPAAm (Figure 6d). However, the circularity value shows a significant spread when P 0 ≥ 83.18, which corresponds to the case of cells on a highly positively charged substrate, i.e., PDADMA (Figure 6d). Similar behavior of a higher variation in the circularity values of individual cells on negatively and moderately positively charged substrates compared with cells on highly positively charged surfaces was observed in the experiments (Figure 1b).

Edge Height of Cells on Moderately Positively Charged PPAAm
The edge height at the border of cells can be helpful to judge adhesion performance. At the edge of the cell, the height inheres in strongly curved or folded membrane regions and adjustment of the cleft distance between the basal membrane and the materials surface. An interesting parameter is the cell height, which represents the sum of the cleft and the curved region of the membrane, including the extracellular matrix (ECM). This height is directly accessible by SICM, a scanning probe microscopy method, largely avoiding forces on cells. As we pointed out earlier [47], the cell edge height is a strongly varying quantity (between 100 nm and 1 µm) when measuring around the whole perimeter of the cell. Therefore, we took average values for each cell. For cells after only 1 h culturing on the substrate, the average cell edge heights were 227 nm ± 18 nm on glass and 164 nm ± 7 nm on PPAAm, i.e., the cell edge is significantly ≈30% smaller on PPAAm, indicating a higher cell-substrate contact (Figure 7). Upon culturing for 3 h on the surfaces, this difference has already vanished (on glass 171 nm ± 11 nm, on PPAAm 150 nm ± 8 nm). However, if we consider regions of peripheral ruffles, we find again a significant difference between 246 nm ± 17 nm on glass and 174 nm ± 10 nm on PPAAm. Note: This type of ruffle is a loosened and folded-back membrane and it can originate from membrane excess, which was more pronounced on glass [47]. The ruffles need to be excluded for this analysis, because they would lead to overestimation of the edge height in the case of glass.

Discussion
Most clinical implant failures occur during the early healing phase [58]. Therefore, the development of new bioactive coatings for dental and orthopedic implants, for example, is fundamental to promote biological interfacial responses. In order to produce suitable coatings, it is important to know how the cells interact with the surface properties.
Our previous research revealed that osteoblasts are sensitive to the surface charge of a biomaterial and exhibit a specific physiological response, involving individual, phenotypical traits of viability, intracellular calcium ion signaling, and morphology depending on the materials´ charge level [33,34]. However, more detailed investigations on the formation of the cell shape in response to material surface charges are hardly found in the literature. Therefore, in this study, our focus was to elucidate the role of the surface charge on the osteoblastic actin cytoskeleton organization and cell shape. We summarized our previous findings concerning the cell area and circularity during the spreading course of 24 h and could show an overall relation of material surface charge and morphological cell changes. In particular, osteoblasts that were grown on highly positively charged surfaces displayed hampered cell spreading and a maintenance of the round cell shape, which is accompanied by functional alterations as shown by Gruening et al. [34].
Cellular spreading is a complex process involving various biological components such as the integrin adhesion receptors and the actin cytoskeleton, which are connected via adapter proteins in functional units called focal adhesions. In cultured cells, beside actin stress fibers throughout the cell, the actin cytoskeleton is presented in lamellipodia, the growth zone of the cell, and filopodia, which protrude from the lamellipodial actin network serving as anchor points for actin stress fibers spanning between two cell-matrix contacts [59][60][61][62]. Through its connection with the focal adhesion complexes of the cell, the actin cytoskeleton mediates the transduction of extracellular signals and mechanical forces into the cell [63,64]. The multitude of functions renders the actin filament network a highly significant and central key component in cell shape, anchorage, and motility [59,65]. In consequence, we assumed a rearrangement of the actin cytoskeleton to be the cause of the priorly observed individual phenotypic changes dependent on the ζ-potential [33,34].
As expected, we could find a well-spread cell morphology on negatively charged Ti surfaces, which is dominated by prominent actin stress fibers after 24 h. The more filigree appearance of the actin cytoskeleton with long filopodia on moderately positively charged PPAAm after 1 h can be explained by a higher level of cellular activity [34,66].

Discussion
Most clinical implant failures occur during the early healing phase [58]. Therefore, the development of new bioactive coatings for dental and orthopedic implants, for example, is fundamental to promote biological interfacial responses. In order to produce suitable coatings, it is important to know how the cells interact with the surface properties.
Our previous research revealed that osteoblasts are sensitive to the surface charge of a biomaterial and exhibit a specific physiological response, involving individual, phenotypical traits of viability, intracellular calcium ion signaling, and morphology depending on the materials' charge level [33,34]. However, more detailed investigations on the formation of the cell shape in response to material surface charges are hardly found in the literature. Therefore, in this study, our focus was to elucidate the role of the surface charge on the osteoblastic actin cytoskeleton organization and cell shape. We summarized our previous findings concerning the cell area and circularity during the spreading course of 24 h and could show an overall relation of material surface charge and morphological cell changes. In particular, osteoblasts that were grown on highly positively charged surfaces displayed hampered cell spreading and a maintenance of the round cell shape, which is accompanied by functional alterations as shown by Gruening et al. [34].
Cellular spreading is a complex process involving various biological components such as the integrin adhesion receptors and the actin cytoskeleton, which are connected via adapter proteins in functional units called focal adhesions. In cultured cells, beside actin stress fibers throughout the cell, the actin cytoskeleton is presented in lamellipodia, the growth zone of the cell, and filopodia, which protrude from the lamellipodial actin network serving as anchor points for actin stress fibers spanning between two cell-matrix contacts [59][60][61][62]. Through its connection with the focal adhesion complexes of the cell, the actin cytoskeleton mediates the transduction of extracellular signals and mechanical forces into the cell [63,64]. The multitude of functions renders the actin filament network a highly significant and central key component in cell shape, anchorage, and motility [59,65]. In consequence, we assumed a rearrangement of the actin cytoskeleton to be the cause of the priorly observed individual phenotypic changes dependent on the ζ-potential [33,34].
As expected, we could find a well-spread cell morphology on negatively charged Ti surfaces, which is dominated by prominent actin stress fibers after 24 h. The more filigree appearance of the actin cytoskeleton with long filopodia on moderately positively charged PPAAm after 1 h can be explained by a higher level of cellular activity [34,66]. However, after 24 h, cells on PPAAm are also well expanded and display stress fibers.
There was no difference in actin filament number between Ti and PPAAm at this time point, as was also indicated by Kunz et al. [67] and Matschegewski et al. [45]. In contrast, highly positive surface charges on PDADMA resulted in a drastically diminished filament number and length after 24 h. The remaining high cell circularity and decrease in cell area was accompanied by a clustered actin formation still confined to the outer cell edge. Similar to this cell behavior on highly positively charged surfaces, studies with respect to surface topography as an external stimulus demonstrated phenotypic changes of actin organization such as clustering of the actin filaments, which in turn resulted in functional changes in the cell [45,68]. Osteoblasts with a reorganization of the actin cytoskeleton on pillar [40] and groove structures [40,69] exhibited a decreased cell area after 24 h as well.
In addition, the cell edge height is lower on moderately positively charged PPAAm compared with negatively charged borosilicate glass after 1 h. Since we measured the sum of both cleft and cell height at the cell edge, we cannot directly compare with work addressing the cleft only. However, our results are compatible with reported cleft thicknesses of about 60 nm, as measured by metal-induced energy transfer [70] and about 40 nm by interferometric photo-activated localization microscopy [71].
Interestingly, after 3 h, there is no significant difference in cell edge height on PPAAm and glass. However, taking into account regions of peripheral ruffles, we observe a higher average cell edge height on glass. This indicates that either the region of peripheral ruffles is higher on glass i.e., specifically at the edges, the cell is not well attached to the substrate, or that if the edges fold back, they fold back further, leading to longer peripheral ruffles and therefore higher edge heights.
In the initial phase of adhesion, physicochemical interactions dominate over the slow cellular responses requiring biosynthesis. The difference in ζ-potentials of material surface (−60 mV for glass and +7 mV for PPAAm [34]) and cell surface (for MG-63 osteoblasts −16 mV [15]) amounts to ≈(−60 mV) − (−16 mV) = −44 mV for glass and to ≈(+7 mV) − (−16 mV) = +23 mV for PPAAm substrates. Thus, electric fields or Coulomb forces, if not fully screened by the double layer, would be stronger with respect to the glass substrate. The interaction of medium or ECM protein with the surface might be of relevance. Due to its amine content, protein tends to exhibit slightly positive ζ-potentials. Upon adsorption on a surface, the delicate protein folding will be least disturbed or best preserved on surfaces exhibiting low ζ-potentials in terms of absolute value [72] e.g., −1 mV or +1 mV (just immobilizing, hardly deforming). Hence, probably just the difference to the ζ-potential of the cell membrane is decisive. It is conceivable that cells prefer sites with adsorbed intact protein or with weak field gradients, leaving their ECM protein structure largely intact. Otherwise, MG-63 cells exhibit negative charges on their outer surface due to their hyaluronan coat; this is responsible for the cells' attachment on positively charged material surfaces via electrostatic forces [42].
To further study the changes in the shape of single cells placed on charged substrates, we adopted a vertex-based model originally developed to investigate cell shape changes in confluent tissues. The model shows that different cell shapes, characterized by cell area and circularity, as observed in experiments, arise due to changes in cell mechanical properties, such as area elastic modulus governing cell area compressibility, bond tension due to actomyosin contractility, and mechanics of cell-substrate adhesion. Here, we used the knowledge of the well-known rigidity transition studied in vertex models for confluent tissues to understand cell shape changes due to intracellular actin cytoskeleton remodeling on differently charged surfaces. On negatively and moderately positively charged substrates, cell shape change is associated with a drop in the value of cell circularity but no significant change in the cell area. The choice of model parameter values for which the experimental observations could be reproduced suggest that negatively and moderately positively charged substrates promote cell-material adhesion after a longer time for growth (24 h). In the case of cells seeded on a highly positively charged substrate, the choice of model parameter values suggests that the forces due to cell cortical tension dominate the forces due to cell-substrate adhesion. As a result, cells decrease their cross-sectional area but retain their initial, higher circularity value. The model also shows that the circularity value of cells placed on a moderately positively charged substrate is more dispersed than in the case of cells placed on highly positively charged surface, as also observed in experiments. Taken together, our analysis indirectly suggests that the charge of the substrate exerts an influence on the cell mechanical properties, causing a transition in the mechanical behavior of the cells from a fluid-like to a more solid-like state.
The measured changes in cell shape and actin cytoskeleton could be explained by the following ideas: Gruening et al. [34] reported that in the presence of highly positive surface charges, the membrane integrity was compromised; i.e., the membranes displayed holes. This could be attributed to the incomplete formation of small membrane vesicles around the high charges, as described before for dendrimer structures by Mecke et al. [73]. It might be possible that the high charges increase the cortical membrane tension, as suggested by the model.
For micrometer-sized pillar structures, it is known that osteoblasts try to phagocytize them [74]. Since nanoparticles with highly positive surface charges are commonly used for drug delivery due to their strong interaction with the membrane's proteoglycans, it could be speculated that the cells also try to take up the interfering high surface charges of biomaterials, resulting in clustered actin filaments. As for the reduced cell viability on highly positively charged material coatings [34], a great cytotoxicity has been also described for these nanoparticles [75].
A question still open involves exactly how negative and positive charges at the material interface affect the actin filament growth. Here, further in vitro experiments with Cytochalasin D causing the disruption of actin filaments and inhibition of actin polymerization could be helpful. In the study by Kunz et al. [67], the application of 1 µM Cytochalasin D to cells on Ti surfaces induced a decrease in cell area after 3 h compared with the untreated control. Controversially, the cells were able to overcome this decrease on PPAAm in most parts. The authors suggested a more membrane-driven cell spreading. However, the substrate-generated electric fields apparently play a role and regulate intracellular signaling pathways, subsequently influencing the actin dynamics. It could be potentially interesting to investigate the Rho signaling pathways, as changes in the balance of the Rho protein family activity are known to be sufficient to achieve rearrangements in the actin cytoskeleton and thus link the cell shape and function [59,76]. Lamellipodia are induced via Rac, filopodia are induced via Cdc42, and actin stress fibers are induced via Rho [59,77]. As Moerke et al. [78] indicated, the integrin downstream signaling of RhoA/ROCK plays a major role in the recognition of microtopographies and hence, the adaption of actin filaments to a surface by osteoblasts. The authors revealed an attenuated actin clustering on pillar structures by the inhibition of ROCK (Rho-associated kinase).
In this study, surface charges were shown to profoundly affect the organization of the actin cytoskeleton and the resulting cell shape. Highly positive surface charges caused a significant decline in cell area and actin filament number and length, which underlines the importance of an intact actin cytoskeleton for changes in cell shape during the spreading process. Using the mathematical model, it was shown that different organizational behaviors of the actin cytoskeleton give rise to different cell shapes, as observed in the experiments. However, in order to understand how the surface charges regulate actin cytoskeleton organization in the first place, a mathematical multi-scale model linking actin dynamics and the electric field generated from the surface charges with the overall cell shape is needed.

Conclusions
Osteoblasts are sensitive to the surface charge of a biomaterial and exhibit a specific physiological and morphological response depending on the charge level. In this study, we focused on the influence of negative, moderately and highly positive surface charges on the osteoblastic actin cytoskeleton organization and cell shape. We could show an overall relation of surface charges to morphological cell changes. While cells on moderately positively charged PPAAm surfaces exhibited a high number of long actin filaments and a minor cell edge height after 1 h, promoting cell-material adhesion, osteoblasts on highly positively charged PDADMA surfaces displayed a drastically diminished filament number and length after 24 h. Our mathematical simulation suggests that in these cells, forces due to cell cortical tension dominate the forces due to cell-substrate adhesion.
Supplementary Materials: The following are available online at https://www.mdpi.com/article/10 .3390/app11125689/s1, Figure S1: Spreading course of MG-63s on modified Ti surfaces dependent on the ζ-potential, Table S1: Cell spreading data of 1 and 24 h dependent on the ζ-potential.