Small-world networks of neuroblastoma cells cultured in three-dimensional polymeric scaffolds featuring multi-scale roughness

Abstract

Understanding the mechanisms underlying cell-surface interaction is of fundamental importance for the rational design of scaffolds aiming at tissue engineering, tissue repair and neural regeneration applications. Here, we examined patterns of neuroblastoma cells cultured in three-dimensional polymeric scaffolds obtained by two-photon lithography. Because of the intrinsic resolution of the technique, the micrometric cylinders composing the scaffold have a lateral step size of ~200 nm, a surface roughness of around 20 nm, and large values of fractal dimension approaching 2.7. We found that cells in the scaffold assemble into separate groups with many elements per group. After cell wiring, we found that resulting networks exhibit high clustering, small path lengths, and small-world characteristics. These values of the topological characteristics of the network can potentially enhance the quality, quantity and density of information transported in the network compared to equivalent random graphs of the same size. This is one of the first direct observations of cells developing into 3D small-world networks in an artificial matrix.

Chinese Library Classification No. R459.9; R319; Q2

Introduction

In tissues and organs, and especially the brain, the interaction between a large number of cells determines the emergence of functions that are not explicable in terms of individual cells taken individually (Hopfield, 1982; Fano, 1992; Koch and Laurent, 1999; Ravasz and Barabási, 2003; Bassett and Bullmore, 2006; Bullmore and Sporns, 2012; Melnattur et al., 2014; Trabesinger, 2016; Onesto et al., 2017). In similar systems, organization, communication, and cooperation between elements are decisive for the correct functioning and optimal performance of the systems themselves. Network theory is a practical way to examine biological systems on a quantitative basis (Strogatz, 2001; Barabási and Oltvai, 2004; Rubinov and Sporns, 2010; Sun et al., 2011; Sporns, 2013). Networks are groups of nodes (or vertices) and edges (or links) that connect those nodes – in which the nodes represent the elements of the systems and the edges the interactions between them (Gunduz et al., 2004; van Steen, 2010; Chartrand and Zhang, 2012; Barabási, 2016). Networks can be described through three sole parameters. The degree of a network k is the average number of links per node. The local clustering coefficient cc_ι of a node is the proportion of active established links to the number of possible connections in the neighborhood of that node (the global clustering coefficient cc is determined by averaging cc_ι over the nodes of the network). The characteristic path length cpl is the average number of steps that separate two nodes randomly chosen in the grid (Gunduz et al., 2004; van Steen, 2010; Chartrand and Zhang, 2012; Barabási, 2016). Among the great variety of network categories with different features, small-world networks have recently attracted attention because it is believed that networks with small-world attributes can transmit signals (or instructions) more efficiently than periodic or unstructured graphs of the same size (Watts and Strogatz, 1998; Lago-Fernández et al., 2000; Latora and Marchiori, 2001; Watts, 2003; Takahashi et al., 2010). For the same reason, a system with a small-world architecture can divide tasks between its components, coordinate activities, and optimize processes more efficiently than ordinary systems without structure. In a small-world network, the distance (i.e., cpl) between the elements of the network raises less rapidly than its size (N), such that cpl~log(N): this indirectly implies that – typically – nodes of the network form few, highly connected clusters, with short paths between them (Watts and Strogatz, 1998; Lago-Fernández et al., 2000; Latora and Marchiori, 2001; Watts, 2003; Takahashi et al., 2010). In such context, materials science combined with nanofabrication methods provide ways to drive the organization of living cells into artificial structures engineered according to a desired design. In previous works, some of the authors of this paper have shown that cells, cultured on nano-patterned surfaces, evolve to form networks with small-world characteristics, where the nano-scale roughness of the substrate represents the external factor that forces the cells to collapse into energetically favorable configurations (Onesto et al., 2017). A similar behavior has been observed for neuroblastoma cells on mesoporous silicon substrates (Marinaro et al., 2015), as wells as for neuronal cells on rough silicon substrates (Onesto et al., 2017) and arrays of zinc oxide nanowires (Onesto et al., 2019). Nonetheless, these results are limited to bi-dimensional geometries.

Here, we examined the topology of 3D networks of neuroblastoma cells cultured in polymeric scaffolds fabricated by two-photon lithography, focusing on the data reported in reference (Accardo et al., 2017). Due to the fabrication process characteristics, the scaffold presents topography details over multiple scales: the micrometer-scaled cylinders of the scaffold have been patterned with a lateral step size of ~200 nm that, in turn, results in a surface roughness of ~20 nm. Upon networks analysis, we found that cells form small-world networks, with higher clustering and smaller paths than in equivalent random graphs with the same size: the hierarchical structure of the scaffold being the possible cause of cell response. This is one of the first direct observations of cells developing into 3D small-world networks in an artificial matrix.

This study was not focused on optimizing performance nor examining in detail all the possible combinations of geometrical characteristics of the scaffold, mechanical properties, surface functionalization, cell lines characteristics, which may affect cell behavior. The main objective of this preliminary study was to demonstrate the feasibility of using network analysis and the small-world-network model to describe the spatial organization of mammalian cells in 3D architectures. Data presented in the paper and analysis thereof represent a preliminary reference for future studies. More sophisticated test campaigns that will be performed over time will verify the combined effect of geometry, nano-topography, and mechanical properties of the scaffold on the collective behavior and self-assembly of primary hippocampal neurons, which are more representative of the central nervous systems.

Materials and Methods

Fabrication of the polymeric scaffolds

Briefly, the 3D scaffolds were fabricated by exploiting a two-photon direct laser writing approach where the slicing (minimal distance between two adjacent planes) and hatching (lateral distance of two adjacent lines within a layer) distances were set at 300 and 200 nm, respectively, as described in reference (Accardo et al., 2017).

Scanning electron microscopy imaging of the samples

Scanning electron microscopy (SEM) imaging was performed on the samples (metal-coated with 15 nm of sputtered gold) by using a Hitachi S-4800 microscope (Hitachi, Tokyo, Japan) with an acceleration voltage ranging from 0.8 to15 kV.

Culturing mouse neuroblastoma N2A cells in the scaffold

Prior to cell culture, the 3D scaffold was first sterilized for 1 hour under ultraviolet (UV) radiation (254 nm), then coated with 0.01% poly-l-lysine (Sigma-Aldrich), to favor the electrostatic interactions between the negatively charged ions of the cellular membrane and the employed polymer, washed twice with sterile water and dried for 2 hours. Then, laminin (a protein of the extracellular matrix, involved as well in the mechanisms of cell adhesion) solution (40 μg/mL) (Invitrogen, Bordeaux, France) was applied for 3 hours. The fast-growing mouse neuroblastoma cell line, N2A, was obtained from the American type culture collection (Molsheim, France). Dulbecco’s modified Eagle’s medium (DMEM) containing glutamax, pyruvate (Invitrogen, Bordeaux, France) and 10% fetal bovine serum was used for the N2A cells. 100,000 cells/cm² were inoculated onto the scaffold and left in culture for 3 days in vitro. Cells were incubated in an atmosphere containing 5% CO₂ at 37°C.

Two-photon confocal imaging of cells in the scaffolds

The two-photon confocal imaging experiments were performed using an AxioImager upright microscope LSM 7MP (Carl Zeiss, La Rochelle, France). Z-stack acquisitions were performed with a 20× W-Plan Apochromat water immersion objective with 1.0 N.A. (numerical aperture), with the laser excitation wavelength tuned to 820 nm, giving a resolution on the x/y-axis of 430 nm and on the z-axis of 1.3 μm, as described in reference (Accardo et al., 2017).

Cluster analysis

Cluster centers were determined using a density based clustering algorithm reported in a previous study (Rodriguez and Laio, 2014). The algorithm determines the number of groups into which the elements of the set are partitioned, and assigns each element to a group. The algorithm classified elements into categories on the basis of their similarity. Cluster centers were determined as those points in the set with higher density than their neighbors and by a relatively large distance from points with higher densities. For each cell i in the original distribution, the algorithm:

• (i)Determines the density of i, ρ(i), as the number of points that falls within a cut off distance δ_co from i, divided by the total number of points in the set.
• (ii)Finds the subset s ∈ S of points in the dataset with densities ρ(s) > ρ(i).
• (iii)Finds the point a ∈ S with minimum distance to i, this distance is δ_min/ρ(i): the minimum distance of i from points with higher densities than i.
• (iv)Derives a diagram where the density ρ is reported against δ_min/ρ per each element in the data set. Points in the set with higher density than their neighbors and by a relatively large distance from points with higher densities emerge as singularities in the diagram, an example of which is reported in Figure 1. These points are the cluster centers.
• (v)Assigns each point in the set to different clusters on the basis a minimum distance criterion: a point b is assigned to a cluster G_i if the minimum distance of b to G_i is the smaller among all the minimum distances calculated with the remaining clusters. Thus clusters are constructed per accumulation. The cluster centers represent the seeds of the clusters. In Figure 1, points are colored according to the cluster of group to which they are assigned.

Figure 1.

The density distribution of cells within the scaffold, derived from the position of the cells within the initial volume (i.e., cube).

The density distribution is here projected onto either the lateral surfaces or the edges of the cube. Variations of the values of frequency are indicative of the non-uniformity of cells within the scaffold. Values of local density vary up to three times within the considered volume.

Wiring cells to form networks: the Waxman algorithm adapted to 3D spaces

In order to establish the nodes connections, we first need to derive the distance between the nodes. Being each node in the network described by three coordinates (x, y, z), the distance matrix d is obtained calculating the Euclidean distance between each node pair in the 3D space. If there are n elements in the network, the distance matrix is a symmetric two-dimensional array having size nxn. No matter how many dimensions we have in a Euclidean space, once derived the distance matrix, the calculation of the nodes connections is the same of the case of the 2D spaces. The Waxman model (Waxman, 1988) is used to establish the connections between the nodes, whereby the probability of being a link between two nodes exponentially decreases with the Euclidean distance d between those nodes. For a given set of two nodes u and v, the link probability, P(u,v) is defined as:

P(u,v)=αe^-d(u,v)/βl      (1)

where l is the largest possible Euclidean distance between two nodes of the grid. In the equation, α and β are the Waxman model parameters and, upon tuning these, the graph may be more or less dense. α and β should be chosen between 0 and 1. Selecting smaller values of these parameters results in a smaller number of links. For the present configuration, these parameters were set to α = 1 and β = 0.1. The probability P varies between 0 for a pair of nodes with an ideally infinite distance, and 1 for a pair of nodes with an ideally zero distance. The information about the connections among the nodes in a graph is contained in the adjacency matrix A = a_ij, where the indices i and j run through the number of nodes n in the graph; a_ij = 1, if there exists a connection between i and j, a_ij = 0 otherwise. In the analysis, reciprocity between nodes is assumed, and thus if information can flow from i to j, it can reversely flow from j to i. In the framework of graph theory, we call a similar network an undirected graph. Notice that this property translates into symmetry of A being a_ij = a_ji. We showed above how to derive the distances between nodes d_ij in the networks. On the basis of d, we may decide whether a pair of nodes is connected, we use at this end the formula:

αe^-d_i,j/βL – P ≥ 0      (2)

in which P is a constant that we have chosen being between 0.3–0.1 that the probability of being a connection is q = 1–P = 0.7–0.9.

Network analysis of 3D graphs

We quantified some network parameters including the clustering coefficient, the characteristic path length and the small-world-ness. In graph theory, the clustering coefficient (C_c) is a measure of the degree to which nodes in a graph tend to cluster together. C_c ranges from 0 (none of the possible connections among the nodes are realized) to 1 (all possible connections are realized and nodes group together to form a single aggregate). The clustering coefficient is defined as:

where k is the number of neighbors of a generic node i, E_i is the number of existing connections between those, k(k–1)/2 being the maximum number of connections, or combinations, that can exist among k nodes. Notice that the clustering coefficient C_i is defined locally, and a global value, C_c, is derived upon averaging C_i over all the nodes that compose the graph. The characteristic path length (Cpl) is defined as the average number of steps along the shortest paths for all possible pairs of network nodes. We shall call the minimum distance between a generic couple of nodes the shortest path length (Spl), which is expressed as an integer number of steps. With these premises, we show now how to calculate the Spl for a couple of nodes n_l and n_m. In A, a_l,i and a_i,m account for all the pairs of nodes which are connected to n_l and n_m respectively. The sum of a_l,i and a_i,m over all the nodes in A, is stored in a new matrix A₂ = ∑a_l,i a_i,m1 for all the l and m and A₂ has the same dimension of A. Now multiplicate A₂ and A repeatedly A₂ = A₂ A, until all the terms of A₂ are non-zero and those terms in position ij will be the Spl between node i and node j. Finally, the characteristic path length Cpl is calculated like the average of Spl over A₂. Once obtained the C_c and Cpl values, we defined a precise measure of ‘small-world-ness’, the ‘small-world-ness’ coefficient (SW), based on the tradeoff between high local clustering and short path length. A network G with n nodes and m edges is a small-world network if it has a similar path length but greater clustering of nodes than an equivalent Erdos-Rényi (E–R) random graph with the same m and n (an E–R graph is constructed by uniquely assigning each edge to a node pair with uniform probability) (Strogatz, 2001; Watts, 2003). Let Cpl_u and cc_u be the mean shortest path length and the mean clustering coefficient for the E–R random graphs, obtained meaning the Cpl and the Cc of 20 uniform distributions, and Cpl_graph and cc_graph the corresponding quantities for the graphs derived using the methods described above. We can calculate:

Thus, the ‘small-world-ness’ coefficient is

The categorical definition of small-world network above implies λ ≥ 1, γ >> 1 which, in turn, gives SW > 1 (small-world-ness higher than one).

Results

Three-dimensional polymeric scaffolds

The scaffolds where cells were seeded is an ordered network of polymeric cylinders (Figure 2a), with a diameter D = 20 μm, and a spacing between structures of d₁ = 20 μm and d₂ = 60 μm in the vertical and horizontal direction, respectively (Figure 2b). The entire scaffold is embedded within a cube with edge length of l = 300 μm. While the overall volume of the scaffold is such to contain a sufficiently high number of cells, the openings within the scaffold enable cells to develop without geometrical constraints. SEM inspection of the scaffold reveals that, due to the fabrication process resolution (methods), the final structures are discretized into smaller blocks with a lateral step size of s~200 nm (Figure 2c). 2D photo polymerization of structures with similar values of lateral step size results, generally, in a surface roughness that is not zero. The surface roughness R_a can be estimated by previous works (Guo et al., 2006), where R_a has determined as a function of the lateral step size of three dimensional polymerized structures. For the present configuration R_a is estimated as R_a~20 nm (Guo et al., 2006). Structures in the scaffold have therefore a multiscale roughness and details organized in a hierarchical architecture, from the micro scale level (cylinder size) to the high (lateral step size) and low (surface roughness) nano meter range. We have generated in Figure 2d the surface state of a cylinder with a similar structure and, by Fourier analysis of the surface topography (Gentile et al., 2011), we have derived the Power Spectrum (PS) density function of the scaffold (Figure 2e). The PS describes how the information content of an image (Q) varies as a function of its spatial frequency (ν) – it reports a change of density (of information) as a change of scale. Therefore, the slope β of a PS can be correlated to the fractal dimension D_f of a structure as (Gentile et al., 2011) D_f = (8–β)/2. For the present configuration, it results D_f ~2.7. Moderate values of surface roughness (R_a~20 nm) and high values of fractal dimension (D_f ~2.7) are agents that may force cells to cluster into few groups, with many elements per group, as previously reported for 2D geometries (Marinaro et al., 2015; Onesto et al., 2017, 2019).

Figure 2.

Scaffold details and characteristics.

The scaffold for cell culture growth and organization is a 3D network of regularly spaced polymeric cylinders (a), where the size and spacing of the cylinders is in the micro-meter range (b). Due to the fabrication process characteristics, the final structures are discretized into blocks with a lateral step size of 200 nm (c). The surface roughness resulting from a similar step size is approximately 20 nm – we have recreated the surface topography of the scaffold cylinder with a similar value of surface roughness (R_a) (d). The power spectrum associated to the structures can be used the derive the fractal dimension of the scaffold - D_f~2.7. High value of D_f reveals that the final prototype has details over multiple scales (e).

As regarding the mechanical features of the scaffold: the hierarchical structure was fabricated from liquid polymer, i.e. the negative photoresist IP-Dip. Nanomechanical characterization of IP-Dip (Meza et al., 2015) yielded for this material the following values of average Young’s modulus E = 2.1 ± 0.3 GPa and an average yield strength under compressive loads σ_y = 67.2 ± 4.7 MPa. Nevertheless, experimental measurements of the effective Young modulus E_eff of periodic lattices governed by bending effects was found to scale as E_eff = E_s ρ², where E_s is the bulk Young modulus of the constitutive material (IP-Dip), ρ the relative density of the scaffold (Fleck et al., 2010). Our scaffold exhibits a relative density close to 20%. This gives an effective young modulus of the produced scaffold E_s~80 MPa. N2A cells thus contact a stiff material (IP-Dip) of young modulus in the GPa range, but they are embedded inside a deformable scaffold of effective stiffness around 80 MPa. Remarkably, the stiffness found for IP-Dip is, even considering the reduction in the strength of E due to the lattice geometry, ~40 times larger than the average Young’s modulus of PDMS (E = 2 ± 0.1 MPa), that has been used in previously reported experiments(du Roure et al., 2005) to determine the average traction forces exerted by differentiated Madin-Darby canine kidney epithelial cells on a substrate, being approximately F ~2nN/μm². This, in turn, implies that the deformation of the scaffold caused by N2A cells is – for this configuration – negligible, and the behavior of cells and the characteristics of the networks that they form can be ascribed to geometry solely.

Cell distribution

Neuroblastoma N2A cells were seeded in the scaffolds using the protocols described in the Methods. Neuroblastoma N2A cells, then, were imaged 3 days after seeding using light sheet microscopy and two-photon confocal microscopy as described in reference (Accardo et al., 2017). N2A cells were not differentiated – i.e. no differentiation medium or methods were employed at any point of study. Light sheet microscopy enables qualitative analysis of the cells (Figure 3a), that appear unevenly distributed within the entire volume of the scaffold (here represented as a cube with an edge of l~300 μm). SEM analysis reveals that cells preferentially align along the nanometric grooves of the scaffold (Figure 3b), that therefore exert significant influence over the cell behavior. Surface roughness of the scaffold, organized over different dimensional scales, seems to cause cells to precipitate into aggregates (Figure 3c). Two-photon confocal microscopy was employed to obtain quantitative localization of individual cells within the scaffold. The total number of cells in the volume of interest is N = 233. Using a 3D scatter plot representation of the cells (Figure 4a), with the z axis aligned along the vertical edge of the scaffold, one can observe that cells are preferentially distributed around the upper (high z’s) and lower (small z’s) parts of the scaffold (Figure 4b and c), and generally along its perimeter (Figure 4d). This is also evidenced by rotating clockwise the scatter plot of cells through successive angles θ around the z axis, with θ = (0–π) (Additional Figure 1 ^{(786.8KB, tif)} ), that provides complete vision of the cell distribution within the three-dimensional volume. We then projected the positions of the center of the cells, initially distributed over the entire volume of the scaffold, on the three orthogonal planes xy, xz, yz. This enabled to derive the quantitative profile of cell distribution in the plane that were in turn projected along the edges of the cube to obtain the frequency distributions f along the x, y and z directions (Figure 1). From the distributions, one can notice that f is non-uniform over the planes of projection and along the principal directions of the scaffold, with the number of cells per line micrometer varying between a minimum of f~0.2 cells/μm and a maximum of f~1.2 cells/μm, with a more than six times increase. Notice that the integral of the f’s over the edge of the cube is 233, i.e. the number of cells contained in the scaffold.

Figure 3.

Imaging N2A cells cultured in the scaffold.

Light sheet microscopy enables to derive qualitative representation of cell distribution in the scaffold (a). Scanning electron microscopy of the cells suggest that their behavior and organization is guided by the nanometric details on the surface of the scaffolds (b, c).

Figure 4.

Quantitative cell distribution in the scaffold.

Two-photon microscopy was employed for precise, quantitative localization of cells within the volume of the scaffold (a). Front (b), side (c) and top (d) views of the scatter plot of the cells reveal that cells are unevenly distributed in the scaffold.

Cluster analysis of cells

We performed unsupervised cluster analysis of cells to examine whether a non-uniform cell distribution results in the emergence of separate groups in the scaffold. We used a density based clustering algorithm, originally developed by Rodriguez and Laio in 2014, to determine cluster centers as those cells in the set with higher density than their neighbors and by a relatively large distance from cells with higher densities (methods). From the analysis (Figure 5a), it results that there are at least seven separate cluster centers in the original set, as reported in Figure 5b. The remaining cells are then assigned to specific clusters on the basis of their Euclidian distance to the clusters. In Figure 5c, cells are colored according to the cluster to which they are assigned. The discovery of a finite number of sub-groups in the cell distribution suggests that cells can form networks with high clustering and short paths.

Figure 5.

Cluster analysis of cells.

Unsupervised cluster analysis of cells (a) enabled to estimate the severn cluster centers (numbers 1–7) in the originating distribution of cells (b) and attribute cells to specific clusters (c). Cluster centers are highlighted in the original distribution, where cells appear as black dots. In the clustered distribution, cells are colored according to the cluster of reference. Units are expressed in microns.

Network analysis of cells

The two-photon confocal imaging technique used in reference (Accardo et al., 2017) enabled to determine the position of the cells in the scaffold but not their connections. We used the Waxman algorithm (Waxman, 1988) to create, artificially, the topological networks associated to the coordinates of neuroblastoma cells in the space. The Waxman algorithm makes a decision on whether couples of cells are connected or not, based on their distance. It states that the probability p of two nodes (u,v) of being connected is:

p(u,v)=αe^-d(u,v)/βl      (6)

where d(u,v) is the Euclidean distance between u and v, l is the maximum distance between pairs of nodes, and α and β are parameters of the model that are here set as α = 1, β = 0.1 (Additional Figure 2 ^{(896KB, tif)} ). Thus, the larger the distance between two cells, the smaller p. By comparing p to an arbitrary threshold probability P = 1 – q, one can decide if nodes establish a connection (p > P) or not (p < P). We used different values of q and P (q = 0.7 – 0.95→P = 0.3 – 0.05) to construct the topological graphs associated to the real biological prototype (Figure 6). The larger q (the smaller P), the denser the graphs, in the limiting case q = 1 nodes of the graph establish all possible connections between them, producing a complete graph (at the opposite extreme, q = 0, the graph is null, with no connections between nodes).

Figure 6.

Network formation through the Waxman algorithm.

To determine the topological parameters of cell distributions, cells were wired using the Waxman algorithm, using different probabilities of wiring (q). q was varied between 0.7 and 0.9. The larger q, the higher of connections per node. Units are expressed in microns.

We then computed the graphs parameters using the algorithms reported in reference (Onesto et al., 2017) and the methods. For specific values of q (Figure 7a) we determined the degree of the graph k, the clustering coefficient cc, the characteristic path length cpl, the small world coefficient SW (Figure 7b–e). The graph parameters exhibit a very high sensitivity to q. The degree of the network k varies between k~3 for q = 0.7 and k~16 for q = 0.95 (Figure 7b). The degree of a network represents the average number of connections per node, i.e. the number of synapses per neuron in a biological interpretation of the results.

Figure 7.

Network analysis.

After cell wiring (a), we determined the topological parameters of the networks as a function of the connectivity q. The degree of the network, k, increases with the cubic power of q (b). Values of the clustering coefficient, cc, (c) and the characteristic path length, cpl, (d) are above 0.7 (cc) and below 3.5 (cpl) for large intervals of q. The small world coefficient (SW) is greater than one for any q > 0.7, indicating that cell networks exhibit small world characteristics (e). In (a), units are expressed in microns.

Even if N2A cells cannot differentiate into fully functional neurons, still they can develop multiple neuritic connections per single cell, although rarely more than 5, and each connection can develop sub-branches which may or may not link with other cells. In previous experimental reports, it has been observed that the number of neuritic extensions per N2A cell varies from approximately 2–3 for un-patterned SiO₂ (Béduer et al., 2012), to around 3 for free-standing photo‐crosslinked poly(ethylene glycol) diacrylate hydrogel architectures (Accardo et al., 2018), to 3–5 for surfaces patterned with arrays of carbon nanotubes(Béduer et al., 2012). Numbers of neuritic extension/cell greater than 5 are observed with lower probability values.

On modulating q between the considered interval, resulting values of k oscillate around the characteristic value k = 4, stretching below and above it. This choice of q’s is thus founded on a biological basis. The maximum theoretical value of k = 16, while significantly larger than the maximum determined number of neuritic extension per cell, ~5, still is sufficiently large to cover possible cases in which, owing to specific conditions and geometries, the number of neuritic extensions that cells develop rises above the limits heretofore registered.

Best fit of data and graphical representation of k(q) indicate that k is a cubic function of q: k = –571 + 2240 q – 2928 q² + 1286 q³, perhaps indicating the fact that the network is immersed in a 3D-space and, for a variation in q (i.e. the probability of connectivity), the nodes of the systems tend to create new connections along three independent directions. We calculate a r-squared statistic r² to test whether the data in several different bands are consistent with the matching template. Values of r² near unity and of estimated variance σ² near zero indicate that the signal is consistent with the model (r²~0.997, σ²~0.04).

Differently from k, and with the exception of the first (q = 0.7) and last (q = 0.95) values of q, the clustering coefficient shows a low sensitivity to q, with values of cc oscillating around the average (cc~0.71), and in any case greater than cc~0.70 (Figure 7c). Recalling that the clustering coefficient varies between 0 and 1, in the considered range of q’s, the cells of the network exhibit a very high inclination to cluster together. Vice versa, in the same domain of variation for q, the characteristic path length cpl of the network is generally low, the number of steps separating two nodes of the graph being smaller than 3.5 for each considered configuration (Figure 7d). Combined together, the clustering coefficient cc and the characteristic path length cpl enable to determine the small-world coefficient of a graph. We used the topological measure small-world-ness SW, defined in reference (Humphries and Gurney, 2008) and the methods to examine whether cultured cell graphs exhibit small-world attributes. Calculated values of SW range between SW~1 and SW~1.4 for q > 0.72, while SW~0.85 for q = 0.70 (Figure 7e). For the large majority of the considered possible values of q the graphs surpass the small-worldness test (SW values are greater than one) and, analyzing their degree distribution, for large κ the probability of finding a highly connected vertex decreases exponentially with κ (power-lawness test was negative –Additional Figure 3 ^{(422KB, tif)} ), therefore we are confident to claim that these networks obey a small-world configuration (Moore and Newman, 2000).

Discussion

The central and peripheral nervous system could be seen as a very large array of computational units linked together in a structure with some degree of order. The organization of the brain affects, in turn its efficiency, and its ability to performing tasks – such as image and speech recognition, object classification, unstructured problem-solving – that are inherently difficult in conventional Von Neumann architectures, on which much of modern computers are based. In Von Neumann architectures, a core logic operates sequentially on data fetched from memory. In contrast, biological computing distributes both computation and memory among an enormous number of relatively simple neurons, each communicating with hundreds or thousands of other neurons through synapses. The spatial distribution of neurons over hierarchal scales is the factor that may possibly explain the enhanced computational power, increased versatility, and reduced low energy consumption of neuromorphic systems compared to the von Neumann chips. Many scientists are skeptical about the idea of the brain as a computer, representing an oversimplification of an intrinsically irreducible problem. The cognitive processes underpinning human language and consciousness, intuition and creativity, are perhaps excessively complex to be decomposed into a sequence of more fundamental instructions, or a list of causes and effects. Nevertheless, this view is of some utility if one wants to analyze the nervous system mathematically, and translate problems of the neuroscience into tractable mathematical formulations, which may provide insight into the originating application. In representing the brain as a network of logic gates (the neurons), one can study the nervous systems, or some of its parts, using the methods and the variables typical of topological theories and the theory of information, including the clustering coefficient, the characteristic path length, the small-world coefficient, and the Shannon information entropy (Takahashi et al., 2007, 2010; Cossell et al., 2015).

Results of the paper indicate that N2A nerve cells in 3D geometries evolve to form networks with small-world characteristics. In other words, in a cascade of biological processes, including adhesion and migration, cells transit from an initial, random, configuration to a structure with a strong correlation between its internal parts. The high values (SW > 1) of the small-world coefficient being indicative of such a correlation. Thus, evolution shapes an ensemble of relatively simple elements into a structure that (because of specific topologies) can more efficiently and more rapidly elaborate and transmit information: that is the own function of the brain. While nerve cells are genetically programmed to form similar information-efficient structures, still they may be restricted to do this, unless they are not fueled by a proper external force. In artificial scaffolds that support cell growth and proliferation, this force is generated by the surface roughness, organized on hierarchical levels, that at the single cell level is in the order of some tens of nanometers – similarly to what precedently observed for bi-dimensional geometries (Marinaro et al., 2015; Onesto et al., 2017, 2019). Multi scale nano-topography is the endogenous factor that sets off cell condensation. Understanding how surface topography, cell network topology and information are interconnected, is of fundamental importance in the design of scaffolds for neuronal tissue engineering and neuro-regenerative medicine, where the ultimate goal is replacing, engineering or regenerating nerve cells to restore the normal functions of the cells themselves, i.e. elaborate information.

The networks that we analyzed are the results of a numerical rewiring of the positions of cells tracked with two-photon confocal microscopy techniques. They are not a representation of a real neuronal network, but are partly based on an estimate on how N2A cells may form connections if placed on specific points of the scaffold determined by experiments. The Waxman algorithm that we used to model cell connections is based on the assumption that the strength of a connection decays exponentially with cell-cell distance, that is a reasonable hypothesis validated by independent observations (Ercsey-Ravasz et al., 2013; Cossell et al., 2015), and on three model parameters, i.e. α, β, q. The choice that we made on α, β, q was not arbitrary: α and β were chosen to maximize the sensitivity of the probability of connection to the cell-cell distance, while q was tuned to modulate the average number of connections per node in the resulting networks. In the array of different configurations that we obtained by changing q, the number of connections per node varies from k~3 to k~16. Calculated values of SW are greater than one for each of the configurations in this interval, indicating that results presented in the paper are general in nature and robust to a change in the model parameters. Values of k outside this interval were not considered (i) either because resulting networks are excessively sparse (k < 3) or (ii) because k would is excessively higher the number of connections-per-cell normally found in N2A systems, that is around 3–4 (k > 16). The neuroblastoma N2A cells model that we used in this study is a simplified version of primary neuronal cultures. Compared to these, N2A cells do not form as many neuritic extensions and cannot generate mature neurites. Nevertheless, among the characteristics found in N2A there are: the expression of neuro-filaments, the synthesis of neurotransmitter biosynthetic enzymes, differentiation, the elaboration of neuritic processes that are ultra-structurally and electro-physiologically similar to normal neurons (Thiele, 1998). The evolution and development of N2A cells in a conditioned system can therefore approximate and reflect some of the most salient features and biologically relevant aspects of primary neurons in the same system. Results of this simplified experimental model, while they represent a preliminary reference for the study of the organization of neuronal cells in three dimensional architectures, have to be investigated even further with additional experiments with primary neuronal cells and tissues in even more complex geometries. Some of these experiments may involve the reconstruction of the real connections formed in groups of neurons with techniques as those described, for example, in references (Stetter et al., 2012; Dorkenwald et al., 2017), to achieve maximum adherence of the model to the real physical prototype.

In more sophisticated experiments that will be performed over time, we will assess cell connectivity in networks of primary neuronal cells, and will verify to which extent the small-world-network analogy is relevant in systems of neurons developing multiple, free standing neuritic connections, in rigid as well as in soft materials, where the contribution of scaffold deformations cannot be neglected (Accardo et al., 2018).

Similarly, we will design an experimental campaign to verify the effects of drugs (chemotherapeutic agents or other therapeutic agents) on the 3D/spatial organization of N2A cells, neuronal cells, or other cell lines. We will verify whether and to which extent the delivery of therapeutics and the nano-topography on the scaffold surface combine to facilitate or prevent cell adhesion, proliferation, and networking. More importantly than this: we will examine whether the physiological/pathological conditions of cell and of a system of cells may be reflected by the topological parameters of that system. May the clustering coefficient, the characteristic path length, and a combination of these, quantitatively reflect the health status of systems of cells, i.e. organs, tissues, organoids?

Lastly, since the role of the glial cells cannot be neglected [using recently validated isotropic fractionator, it has been observed that in the brain the glia:neuron ratio is about 1:1, with a total number of less than 100 billion glial cells in the human brain (von Bartheld et al., 2016)], we will design experiments to examine whether co-cultures of neurons/glial cells may affect cell networking.

In summary, our results indicate that cells in 3D scaffolds form non-homogeneous, non-uniform systems, with cells forming few groups with a great many of elements per group. Cells used in this study are neuroblastoma N2A cells. While they share most of the characteristics of primary neuronal cells (see the Discussion of the paper) and give a preliminary indication of the evolution of a system of nerve cells under the influence of surface nano-topography, nonetheless additional experiments with primary neurons are needed to examine the phenomenon on a more rigorous basis. The small-world-network model implemented in the paper describes convincingly the topology of the clusters of cells in the scaffold. Since small-world networks are theoretically believed to receive, elaborate, and transmit information (i.e. signals) more efficiently than equivalent random or periodic networks of the same size, this also suggests that the main drivers for nerve cell condensation are energy minimization and information maximization, these two criteria being perhaps equivalent. That is to say – biological systems of cells are shaped by evolution to keep the energy of the system at a minimum and, simultaneously, the information exchanged in the system at a maximum. Optimization of these cost functions would, to cite a few, explain the low energy consumption of the brain and the formation in the cerebral cortex of structures with a finite size like the cortical mini-columns, that are the basic computational units of the brain. The nano-topographical details that decorate the surface are the factors that trigger cell condensation in artificial scaffolds. Understanding the role of cell topology and surface nano-topography on the organization of nerve cells into complex structure may help to design strategies for tissue engineering, nerve repair, neuronal regeneration, faster and more efficiently.

Additional files:

Additional file: Open peer review report 1 ^{(77.6KB, pdf)} .

Additional Figure 1 ^{(786.8KB, tif)} : 3D plot of cells within the scaffold for different values of the viewpoint.

Additional Figure 1

3D plot of cells within the scaffold for different values of the viewpoint.

The scatter plot of cells in the scaffold is reported for different values of the viewpoint, rotated clockwise through successive angles θ, with θ (0-π). This provides complete vision of the cell distribution within the three-dimensional volume.

Additional Figure 2 ^{(896KB, tif)} : Guidance for the choice of α and β in the Waxman model.

Additional Figure 2

Guidance for the choice of α and β in the Waxman model.

The choice of these values for α and β is not arbitrary. p is the probability of two nodes of being connected. It represent a connection strength. Since we want that the maximum probability is 1, then we set α as α = 1. β operates directly on the distance d. For a fixed α = 1, for large β(β~1) there will be a probability different from zero (and generally high) for two cells of being connected, even if the distance between them is maximum (i.e., the cells are the opposite vertices of the cube). Viceversa, for low values of β (β = 0.01), p vanishes steeply with d – and, for any d larger than the 5% of l (i.e. larger than ~23 μm). cells won't be connected. β = 0.1 is a compromise betweeen small (0.01) and large (1) values of β, which both correspond to improbable biological states, either because cells in the network are poorly interconnected (β = 0.01), because they would form an overabundance of links at longer distances with high metabolic costs (β = 1) (A). Notice also that β influences the number of synapses per neuron (connection density).

There is a second argument that justifies the use of β = 0.1. For different values of β, the change of p for change of d, i.e. the sensitivity s= ∂p/∂d, does also change. We can derive an explicit expression for S = e^d/lβ/lβ. The plot of S as a funtion of β for different values of d (B), indicates that one can achieve high sensitivity in the neighborhood of β = 0.1.

Additional Figure 3 ^{(422KB, tif)} : Evaluating the level of power-lawness of the distribution of the degree of the graph κ.

Additional Figure 3

Evaluating the level of power-lawness of the distribution of the degree of the graph κ.

Here, we show results of a power-law test that demonstrate that the distribution of the degree of the graphs κ is not power-law, but has an exponential decay. We used the test for power-lawness described in a study of Onesto et al. (2017). The power-lawness is a local measure that offers a confidence level to claim whether power-law node distribution governs a particular network or not (Onesto et al., 2017). To ensure power law as a plausible hypothesis, P-value is calculated. If the calculated P-value ≥ 0.1, the power law hypothesis is accepted for the network, otherwise rejected. For this particular configuration, P-value = 0.