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Edited by: Luca Mesin, Politecnico di Torino, Italy

Reviewed by: Chiara Giverso, Politecnico di Torino, Italy; Trevor R. Cardinal, California Polytechnic State University, United States

This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Sprouting angiogenesis is a necessary process in regeneration and development as well as in tumorigenesis. VEGF-A is the main pro-angiogenic chemoattractant and it can bind to the decoy receptor VEGFR1 or to VEGFR2 to induce sprouting. Active sprout cells express Dll4, which binds to Notch1 on neighboring cells, in turn inhibiting VEGFR2 expression. It is known that the balance between VEGFR2 and VEGFR1 determines tip selection and network architecture, however the quantitative interrelationship of the receptors and their interrelated balances, also with relation to Dll4-Notch1 signaling, remains yet largely unknown. Here, we present an agent-based computer model of sprouting angiogenesis, integrating VEGFR1 and VEGFR2 in a detailed model of cellular signaling. Our model reproduces experimental data on VEGFR1 knockout. We show that soluble VEGFR1 improves the efficiency of angiogenesis by directing sprouts away from existing cells over a wide range of parameters. Our analysis unravels the relevance of the stability of the active notch intracellular domain as a dominating hub in this regulatory network. Our analysis quantitatively dissects the regulatory interactions in sprouting angiogenesis. Because we use a detailed model of intracellular signaling, the results of our analysis are directly linked to biological entities. We provide our computational model and simulation engine for integration in complementary modeling approaches.

Angiogenesis is a pivotal process in various aspects of vertebrate life. In development (Breier,

In sprouting angiogenesis, quiescent endothelial cells (ECs) in an existing vessel adopt a motile tip cell phenotype, release matrix metalloproteases (MMPs) to degrade the extracellular matrix (ECM) around the vessel, and lead a sprout followed by proliferative stalk cells (Logsdon et al.,

VEGF-A induced Delta-Notch signaling drives tip cell selection (Liu et al.,

The decoy receptor VEGF-receptor 1 (VEGFR1, also called Flt-1) binds VEGF-A with a higher affinity than VEGFR2, however leads to negligible activation of downstream targets (Waltenberger et al.,

VEGF-A has been implied as a target in several clinically relevant conditions, e.g., ischemia, arthritis, or obesity (Carmeliet,

A considerable number of computational models of sprouting angiogenesis exist (Merks et al.,

The “memAgent-Spring Model” describes the alignment of membrane patches assigned to specific cells on a pre-defined shape, each patch can have its own dynamics concerning signaling (Bentley et al.,

In Cellular Potts models, cells are described by a collection of nodes on a lattice. Cell shapes and movement arise dynamically from re-assigning nodes to minimize an energy function (Merks et al.,

Agent based models in which single cells are represented by single nodes on a lattice have been used to describe blood vessel formation in bone healing scenarios (Checa and Prendergast,

Computational models have also been used to explore the role of VEGFR1 in angiogenesis. Most of these models focus on the establishment of VEGF-A and VEGF receptor gradients and binding kinetics (MacGabhann and Popel,

Here, we present a computational agent based model to quantitatively dissect the interrelation of the regulatory mechanisms in tip selection and sprouting. We focus on Delta-Notch and VEGF signaling including VEGFR1 and VEGFR2. Our model allows us to assess the experimentally inaccessible interdependencies of intracellular signaling and extracellular conditions. Our analysis shows that VEGFR1 is efficient in guiding sprouts away from existing vessels and it also highlights the importance of Delta-Notch signaling, specifically the degradation of NICD, for angiogenesis. We provide our model in an open and reproducible way, thus facilitating integration into different contexts.

We use a custom simulator, AngioABM, for our agent based modeling, implemented in C++11 using the boost libraries (The Boost Community,

In AngioABM, an agent based model consists of global variables, local variables and agents placed on a discrete 2D grid. Global variables are single numerical values defined for all positions of the grid. Local variables are 2D matrices of the same dimensions as the grid. Agents contain an id, coordinates, internal variables, and update rules. For clarity, we will refer to local variables using

AngioABM uses discrete time steps for time course simulations. At each time step, update and output functions of all local variables and all agents are called iteratively as shown in

Flowchart of the agent based model simulations. Colors indicate processes/decisions associated with simulation time (blue), local variables (yellow), agents' intracellular dynamics (turquoise), and agent movement and proliferation (green).

Local variables represent molecules (in particle numbers per lattice site). Basic functions for diffusion, degradation, and adding molecules are implemented in a base class Localvar. These functions can be used to implement derived classes of local variables that exhibit custom dynamics. AngioABM uses forward integration to compute local variables' temporal and spatial evolution. The simulator assumes Dirichlet boundary conditions.

Agents represent cells. Basic functions for agent movement and interaction with the environment are implemented in a base class Agent. Derived classes representing specific cells can use these functions, lists of internal variables and parameters and can additionally use custom rules or formulas to define cell dynamics. Agents are updated asynchronously in random order. Hence, changes in one agents' state can influence another agents' dynamics immediately. This reduces numerical errors arising from the discrete time intervals between agents' information exchange and induces stochasticity into the simulation. To reduce errors in integration, especially in the computation of local variables, numerical stability is checked and internal steps are fine grained accordingly. Agents' parameters can be specified to refer to a fixed value for all agents or to be sampled from a random distribution for each agent. Drawing agents' parameters from random distributions induces cellular heterogeneity.

The model described here consists of the local variables

We assume the 2D grid considered in simulations to be a subarea of the bottom of a well plate containing culture medium as described in Kappas et al. (

We model the EC agents according to the intracellular signaling model described in the next section. The EC agents extend or retract

^{*},

with threshold parameters ^{*} and ^{*}. A tip cell determines the direction of the

The model describes interactions relevant for tip cell selection and lateral inhibition. It consists of 21 variables and 43 parameters that describe rate laws for binding, transcription, translation, and degradation of molecules, see

Model topology indicating signaling systems in different colors (green: Dll4-Notch1, turquoise: VEGFR1, purple: VEGFR2). _{2} indicates Dll4 from neighboring cells. Symbols adhere to the Systems Biology Graphical Notation format (Novère et al.,

Binding of _{nb}) releases the active Notch1 intracellular domain _{transcription} by using Hill-Equations (reviewed, e.g., in Tummler et al.,

where _{max} is the maximal transcription rate upon full activation, _{0} determines the modifier value at which _{transcription} is half its maximal value and

All other reactions (translation and degradation, protein binding, and dissociation) follow standard Mass-Action kinetics. VEGFR1 is regarded in two isoforms, membrane-bound VEGFR1 (

The model also includes cis-inhibition of Dll4-Notch1 signaling since it has been extensively discussed in literature to be necessary for lateral inhibition (Sprinzak et al.,

We obtained ranges for parameter values from literature (MacGabhann and Popel,

We parameterized the model to reproduce experimental data from Kappas et al. (

To generate the parameterization for the whole model, we first estimated parameters for the intracellular ODE model to generate tip and stalk phenotypes using Copasi (Hoops et al.,

For initial values and parameters related to the general condition of a cell (_{max} of transcription and translation) and the NICD degradation rate (

We analyzed the sensitivity of the simulation results with regard to changes in parameters. We used the percentage of the area occupied by ECs (

To measure branch points, we saved simulation results as image files and analyzed these with ImageJ-MATLAB (Hiner et al.,

The constructed model with the described parameterization reproduces experimental data on mouse embryoid bodies of wild type and

Representative simulation results: extracellular VEGF-A, and VEGFR2 mRNA for VEGFR1 -/- strain

Initial endothelial cells in the model are seeded at two locations on the grid, corresponding to the seeding of multiple embryoid bodies (Kappas et al.,

The transition between tip and stalk cells is smooth. Although the mean values of intracellular variables strongly differ for tip and stalk cells, they overlap: non-tip cells exist for which some intracellular variables have similar values as generally observed in tip cells. Exemplary distributions are given in

To analyze the contribution of the different VEGF-A receptors to sprouting, we evaluated the sensitivity of _{max} parameter of Equation (1) for transcription.

Concerning _{max} of transcription for VEGFR2, we observe that too little VEGFR2 transcription renders cells insensitive to VEGF-A while too much VEGFR2 leads to a hyperactivation of Dll4 and thus too high NICD levels that suppress sprouting. This is true for both VEGFR2 transcription (

Parameter sensitivity: changes in relative vessel area (_{max} for

We have also analyzed the translation parameters of Dll4 and Notch1 (

Parameter sensitivity: changes in relative vessel area (

The translation of Notch1 shows a different pattern: For _{max} of Notch1 translation below half the original value, no

Concerning the interactions in Notch1-Dll4 signaling, we have analyzed the sensitivity to parameters

The amount of NICD, determined by Notch1 activation and NICD degradation, is crucial for balancing VEGFR1/VEGFR2 levels and thus determine the degree of lateral inhibition (as indicated in

For the Notch1-Dll4 signaling interactions, parameter variations lead similar trends for

For most parameters in the Dll4-Notch1 signaling system, the maximal value of

Parameter distributions strongly influence model dynamics. The standard deviation of the parameter distributions were varied from 0.01 to 2.56·μ. Sampled parameters are: _{max} of transcription (tc), translation rates (tl), and the degradation rate of NICD (nicd_deg) or all of the above. Initial values for new agents were sampled accordingly under all scenarios. Model output for 20 simulations of each setting have been recorded as

To assess the influence that the sampling of parameters has on simulation results, we have analyzed simulation runs using distributions with similar means but different standard deviations, as depicted in

For the degradation of NICD, an increased variability leads to a quite stable

The variability of the model output does not increase additively when sampling transcriptional rate parameters, translational rate parameters and the NICD degradation rate (“all” in

Again, the number of branch points by vessel length follows the trends observed in the

Angiogenesis is a multi-scale process where multiple signals must be integrated to enable cellular decision making leading to the sprouting of new vessels. Here, we present an agent based computational model of sprouting angiogenesis that focuses on intracellular signaling. We show that the release of the decoy receptor sVEGFR1 has two distinct effects. By reducing VEGF-A concentrations in the vicinity of stalk cells, VEGFR1 (1) reduces sprouting and (2) directs sprouts to grow orthogonal to existing structures. The second effect leads to a faster vascularization of an avascular area under homogeneous VEGF-A conditions. We observe this impact of VEGFR1 under a wide range of parameters. Our results are consistent with previous findings on the interaction of VEGF and sVEGFR1 in a static environment (Hashambhoy et al.,

Our analysis also reveals that the system dynamics, especially the timing of sprouting, strongly depend on the parameterization of the Dll4-Notch1 signaling system that determines lateral inhibition. Our model predicts that the stability of the notch intracellular domain strongly influences the timing of sprouting without a strong impact on vessel network architecture.

We furthermore show that cellular heterogeneity is an important driver of pattern formation. The case of NICD in our model highlights that varying cell-to-cell variation of a parameter has different effects than varying the magnitude of the same parameter. This also has implications for experimental research, because it is accordingly not sufficient to know the mean value of some parameter in an ensemble of multiple cells but we also need to know its variability specified by the distribution and its moments.

The model presented here is a 2D lattice based model with fixed cell shapes. This means that its predictions are limited to 2D environments and cannot be directly transferred to clinical conditions, e.g., in tumor growth.

Finding the appropriate balance of model detail or resolution is a fundamental challenge in modeling. Any increase in model complexity increases the amount of experimental data necessary to find a reliable parameterization. Here, we have not included all molecules involved in VEGF signaling in sprouting angiogenesis (see e.g., Ferrara et al.,

This parameterization is not guaranteed to be equal to the physiological parameterization. Because we have restricted parameter boundaries based on experimental data and our data reproduces findings reported in literature (Hashambhoy et al.,

Kearney et al. (

The analysis of our model indicates that VEGFR1 suppresses sprouting by reducing the global VEGF-A concentration with increasing cell density (compare the upper and lower row of simulation results in

Diffusion properties and morphogen gradients are notoriously difficult to quantify in tissues experimentally (Bothma et al.,

Also, our analysis indicates that only the complete removal of VEGFR1 transcription abolishes the spatial cues provided by sVEGFR1 (

In contrast to the gradual changes in simulation output caused by changes in VEGFR1 or VEGFR2 production rates, changes in parameters involved in Delta-Notch1 signaling cause stronger and more abrupt changes over certain parameter value ranges (

In the model we present, the base state of a cell is a quiescent stalk-cell like state, and the basic activity of lateral inhibition is relatively high. For sprouting, cells have to overcome this basal lateral inhibition. Cellular heterogeneity through sampled parameters is an important driver of the differentiation of individual cells in our model. Although the sampling is random, a specific cell can be primed for sprouting by the sampled parameter values. Neighboring cells might be assigned parameter values that make them less likely to sprout and the difference between these cells is then further amplified through lateral inhibition. The most important parameter to generate heterogeneity for sprouting is the degradation rate of NICD. Even small stochasticity in NICD degradation increases cellular heterogeneity in the lateral inhibition pathway and significantly improves sprouting efficiency. Because the stability of NICD does not directly affect sprouting but the heterogeneity between neighboring cells, larger stochasticity in NICD degradation does not lead to a breakdown of the system, as observed when sampling transcription or translation parameters from wide distributions (

Beyond this specific finding, our analysis on the role of parameter distributions in computational models of multicellular ensembles also has a general relevance. Assuming a fixed value for NICD degradation for all cells and varying this value showed only a narrow band of parameter values that lead to efficient sprouting (

The agent based model we present here omits cell shape dynamics and ensures vessel connectivity by dividing stalk cells upon tip movement. This neglects cell shape changes and their effect on network formation (Boas and Merks,

The complete model code can be downloaded and reused and we hope that further development of the model by us and other groups will help to further understand the dynamics of sprouting angiogenesis. Future development of the model could focus on the interactions between chemical and mechanical signals or in the cross-talk between different signaling pathways.

In order to generate reliable and clinically relevant predictions of angiogenesis to improve regeneration, enable fine-tuned tissue engineering or ablate tumor-dependent angiogenesis, we need to integrate various mechanisms in a dynamic environment. Moreover, these models need to be parameterized based on dedicated experimental data. By providing our model in an open and reproducible manner, we enable its integration into models that consider cell shape dynamics, interactions with the ECM, or network formation in 3D.

We present an agent based computer model of sprouting angiogenesis focusing on intracellular signaling and its effects on pattern formation. Specifically, we focus on the role of VEGFR1 and its interrelation with VEGFR2 and Delta-Notch signaling. Our findings do not replace published modeling efforts but are complementary because our focus differs from existing literature. Our simulations go beyond comparable models concerning level of detail of intracellular signaling, cell numbers considered, and simulation time.

Our results support the hypothesis that soluble VEGFR1 provides spatial cues to guide sprouts away from their origin and it predicts that this effect is robust over a wide range of parameters. Our model analysis shows that lateral inhibition, and especially the regulation of NICD stability and its variability across cells, is critical for the regulation of sprouting angiogenesis.

This leads us to argue that complementary modeling approaches need to combined and supplemented with dedicated experimental data to generate reliable predictions, to generate models that do not only reproduce training data but are also capable of predicting validation data correctly and function over a wide range of conditions. The advantage of incorporating dedicated signaling models into cell-based models of angiogenesis is that these models provide direct links between experimental perturbations and model dynamics.

Source Code for the simulator and analysis can be found at

CK and SC designed the study, wrote and revised the manuscript, read the manuscript, and approved its content. CK implemented simulator and analysis software, constructed and parameterized models, and analyzed simulation results.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Edda Klipp for providing computers at the Theoretical Biophysics Department of the Humboldt-Universität zu Berlin, and Ivo Maintz for helping with them.

The Supplementary Material for this article can be found online at: