As previously mentioned, at the single-cell level, the generation of calcium microdomains and global calcium waves can be ascribed to the interplay of several factors: the highly variable cell morphology and the specific distribution of pumps, channels, and buffers (e.g., proteins, mitochondria), with the typical compartmentalization in lipidic rafts, caveolae and supermolecular complexes (signalplexes or signalosomes)[11,15,16,32,33]. However, the most common laboratory-based techniques used to investigate calcium events, despite providing an impressive amount of data, are affected by many drawbacks and alter the physiology of the cell. Calcium imaging is indeed obtained by the use of fluorescent probes that act as calcium buffers and therefore perturb the physiological intracellular homeostasis of the ion. Also, all the typical limitations intrinsic to fluorescence-based techniques, sensitivity to other ions and pH, and bleaching, must be considered. Moreover, protein-based calcium measures require the cell to be transfected. The patch clamp technique is powerful for investigating biophysical quantitative properties of calcium currents and channels, but intracellular solutions are significantly altered, mostly in whole cell configuration. In addition, single-cell electrophysiological measurements are performed in a limited number of cells[34]. All these limitations can be overcome by computational (in silico) approaches, which can be also used, in a predictive manner, to determine the pitfalls of experimental manipulations, thus providing a useful guide for new in vitro realizations.

In the past 20 years, an increasing number of mathematical models have been used for the study of calcium signaling in different cell types, including ECs[35-37]. They are based on continuous reaction-diffusion (RD) systems, in which the kinetics of channels and pumps (located in the plasma membrane and in specific intracellular compartments such as the endoplasmic reticulum) are typically described by saturating Michaelis-Menten functions, and the propagation of the ion is mainly determined by its diffusion coefficient.

We have recently proposed a similar continuous model, based on RD equations and implemented on Virtual Cell software (already successfully applied by other authors for other cell types[38-40]), for the specific analysis of proangiogenic Ca_i signals in a single EC, stimulated by micromolar concentrations of vascular endothelial growth factor (VEGF) and regulated by the interplay between AA and NO[41]. Beside reproducing Ca_i events in close comparison with the results previously obtained with fluorescent probes[11,27], the mathematical approach has been able to focus on the influence of the spatial localization of the complex and variable cell morphology, highly diverging from a regular spherical shape. We have designed realistic 3D reconstructions of the EC starting from the relative confocal images, that is, with and without filopodium-like processes, and have shown how the maximal accumulation of the ion is restricted in the most peripheral and flat regions of the cell. Moreover, our approach has provided additional insights into the role played in calcium machinery of factors that are difficult or even impossible to control in vitro, such as the concentration and distribution of channels or intracellular buffers. In summary, the use of the proposed computational method has been able to analyze relevant biochemical mechanisms and parameters underlying proangiogenic calcium signature. However, further experimental data are now required to upgrade and improve the model, to reduce the number of assumptions and increase the quantitative constraints. Indeed, this continuous feedback and feedforward between in vitro and in silico approaches is the core principle of a systems biology approach, applied with an increasing frequency and success in biology.

The intrinsic multilevel nature of angiogenic progression and the related calcium signaling requires us to take into account factors characteristic of different spatiotemporal levels of organization, which range from selected biophysical properties of the entire EC population to specific intracellular biochemical dynamics. An accurate study of this complex flow of information makes evident the need for multilevel research and brings many challenging questions, which can be more efficiently addressed by collaboration with applied mathematics.

The literature on mathematical models of vascular progression is mainly focused on the reproduction and analysis of tubulogenic assays and is partitioned into two main streams, according to the type of approach used, that is, continuous or discrete. From a macroscopic point of view, continuous techniques represent the EC population as a density, which satisfies a set of balance laws and/or diffusion equations. In particular, these methods typically assume that the key mechanisms of tubule formation are: (1) a chemotactic force generated by extracellular gradients of specific growth factors (e.g., VEGF, fibroblast growth factor); and (2) interaction with the matrix substrate, considered as a continuum body whose stretch influences cell movement and stabilization[42-44]. These approaches reproduce in detail the early phases of the vasculogenic process (i.e., before tubule maturation), predicting a characteristic length of the capillary-like network, which is dictated by the diffusion coefficient of the exogenous angiogenic factor, in good agreement with the relative experimental observation. However, such continuous techniques overlook the behavior of single cells and fail to describe their mutual and local interactions. They may therefore be unsatisfactory because, to deal with angiogenic progression, what occurs at the level of the single cell is crucial. Moreover, continuous models are usually unable to describe intracellular biochemical dynamics, including calcium signaling.

The second type of models reproducing vessel formation are the discrete techniques, widely known as individual cell-based models (IBMs) or cellular automata. They approach the problem with a mesoscopic-phenomenological point of view, preserving the identity of individual ECs. ECs are represented as one or a set of spatially-extended units (defined according to some underlying discretization of the simulation domain, which can be either regular, such as square or cubic grids, or irregular, as in the case of Voronoi tessellations), with rules that describe their movements and interactions. Compared to continuous methods, IBMs can more naturally capture detailed biophysical properties of single vascular cells, such as morphological changes or intrinsic motility, and are also able to handle local adhesive interactions. Among discrete models, one of the most used is the Cellular Potts Model (CPM), a grid-based Monte Carlo technique that follows an energy minimization philosophy[45-47]. Published CPMs have shown that relatively simple cell-level mechanisms, such as adhesion[48], elongation[49], and contact-inhibited chemotaxis[50], are sufficient to obtain the organization of a dispersed population of vascular cells into a 2D network. However, those do not include molecular-level processes, and thus neglect the importance of calcium signaling that underlies the phenomenology of ECs.

The limitations of both purely continuous and discrete models can be overcome by creating hybrid computational frameworks, which are able to span the entire spatiotemporal level of the process with a sufficient degree of accuracy, offering the advantages brought by the different methods. We have recently constructed such a multilevel approach by nesting an adapted version of the above-mentioned single-cell model in a CPM[51,52]. As a distinct feature of the mathematical environment, the subcellular biochemistry, described by the continuous subcellular model, has been coordinated to deliver realistically cell biophysical properties (i.e., motility, chemical sensitivity and compressibility) and behavior (i.e., shape reorganization, chemotactic and persistent migration, and intercellular adhesion), described by the discrete CPM, ultimately mediating the overall formation of the capillary-like structure (Figure 2).

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Figure 2 Hierarchy of scales and environments, with the corresponding biological mechanisms and modeling approaches, taken into account in the proposed model of tubulogenesis. Our theoretical method spans the multiple levels involved in the process: it incorporates a continuous model of the specific agonist-induced, calcium-mediated intracellular cascades in a discrete cellular potts model, which represents the phenomenological evolution of the endothelial cell (EC) population. In particular, the microscopic dynamics alter the mesoscopic biophysical properties of the ECs, such as motility, compressibility, and chemotactic strength, affecting their behavior and, ultimately, the overall tubule formation. The different spatiotemporal levels, integrated in a hybrid framework, therefore directly affect each other and feedback over the whole simulation. VEGF: Vascular endothelial growth factor.

The resulting hybrid model has reproduced with remarkable accuracy the kinetics and temporal dynamics of the pattern organization, as well as its final configuration characterized by well-defined topological features (i.e., intercapillary distances, branch length and width). Furthermore, the computational analysis has followed the proangiogenic calcium events during the entire process: evoked in the early phases, when the cell population is still not connected in a mature network, they are significantly downregulated in the later phases. Indeed, such a peculiar temporal evolution of responses may play a specific role in transducing information during the different stages of the process, being likely associated with differential membrane channel functionalities, at either the gene or the protein level. Finally, the computational model has allowed us to test specific antiangiogenic therapies. In particular, the efficacy of strategies that are currently in use (such as interference with VEGF machinery) or in trials (blocking of calcium entry) has been confirmed. At the same time, the model has been able to predict the potential success of biomedical interventions inhibiting the mechanisms of cell cytoskeletal remodeling, chemical sensitivity, and adhesive capability. It is useful to emphasize that the large combinatorial space of possible therapies would have been unfeasible to search using only laboratory-based methods, but it has been efficiently analyzed by a mathematical system, confirming the usefulness of close collaboration between experimental and theoretical researchers.