Agent-based modeling of morphogenetic systems: Advantages and challenges

Abstract

The complexity of morphogenesis poses a fundamental challenge to understanding the mechanisms governing the formation of biological patterns and structures. Over the past century, numerous processes have been identified as critically contributing to morphogenetic events, but the interplay between the various components and aspects of pattern formation have been much harder to grasp. The combination of traditional biology with mathematical and computational methods has had a profound effect on our current understanding of morphogenesis and led to significant insights and advancements in the field. In particular, the theoretical concepts of reaction-diffusion systems and positional information, proposed by Alan Turing and Lewis Wolpert, respectively, dramatically influenced our general view of morphogenesis, although typically in isolation from one another. In recent years, agent-based modeling has been emerging as a consolidation and implementation of the two theories within a single framework. Agent-based models (ABMs) are unique in their ability to integrate combinations of heterogeneous processes and investigate their respective dynamics, especially in the context of spatial phenomena. In this review, we highlight the benefits and technical challenges associated with ABMs as tools for examining morphogenetic events. These models display unparalleled flexibility for studying various morphogenetic phenomena at multiple levels and have the important advantage of informing future experimental work, including the targeted engineering of tissues and organs.

Fig 1. Overview of the two most popular theories for pattern formation during morphogenesis, reaction–diffusion systems and positional information, as well as common features of agent-based models for morphogenesis.

(A) Archetypal Turing reaction–diffusion system with an activator and inhibitor generating repetitive patterns from differential diffusivities and nonlinear reaction terms. The reaction–diffusion system depends on the concept of a chemical prepattern developing in advance of cell fate decision and emphasizes the ability to induce pattern formation from an allegedly homogeneous initial state. (B) Wolpert’s positional information theory proposes an interpretation step based on concentration thresholds that alleviates the need for a morphogenetic pattern to match the chemical prepattern. In positional information, a cell is capable of multiple fate decisions from a single molecular gradient by discerning subtle variances in concentration along the gradient. (C) Agent-based modeling provides a framework capable of implementing features from both theories. Cell agents can act as the sources of activators (“A”) and inhibitors (“I”), permit localized reactions, and make autonomous decisions in response to their local environment. In addition to the generation of static patterns, agent-based modeling allows for the investigation of dynamic, spatiotemporal patterning.

Fig 2. The three most common physical representations of cells in ABMs.

Lattice models are generally the least complex, given the constraints to movement and interaction, whereas both CPM and lattice-free models possess varying degrees of complexity depending on the features included in the model. Miniature depictions of the respective images will be placed beside each subsection discussing morphogenetic ABMs in later sections to designate the types of models described. 2D, two-dimensional; 3D, three-dimensional; ABM, agent-based model; CPM, Cellular Potts Model.

Fig 3. Classification used to categorize three types of agent-based models for morphogenesis; each model class describes the primary mechanism that induces pattern formation.

(A) Proliferation models depend on the differential division of cells, either between cell types with variable growth rates, between cell generations, or as an alternative to cell death. The illustration depicts volume exclusion of cell types A and B when cell type C has a fast division rate. (B) Migration models concentrate on directional movement, both in a purely migrational sense and in terms of polarized growth. The patterning of migration models is typically centered on overall morphology, such as the branched network of a vascular network. (C) The focus of differentiation models is on patterning that is reliant on fate change, in most cases following a hierarchy of lineage commitments. The interactions between the various cell types are major determinants for the resultant behavior.

Fig 4. Comparison of three crypt model implementations and agent descriptions.

(A) Cells are defined on a continuous 2D lattice such that a cell moving off the right edge of the grid reappears on the left edge. Divide and Die gradients are used to describe the behavior of different cell states in the crypt while movement up the crypt is the result of apoptosis. (B) A centroid model may be employed to investigate the role of crypt geometry on the location of anoikis within a crypt. The proliferative state of a cell is defined by prespecified regions along the crypt, and cell death is solely dependent on the occurrence of anoikis. (C) Centroid model that includes differentiation and dedifferentiation between the four main phenotypes present in an intestinal crypt. Wnt signaling is defined by the position within the crypt, whereas Notch signaling is determined by the phenotype of the cell and its neighbors. 2D, two-dimensional.

Fig 5. Overview of the analysis of spatial features using a combination of network analysis and dimensional reduction techniques.

A set of metrics is calculated or extracted from a series of pattern classes that depict typical cell organizations within the system—here, using the defined pattern classes from [133, 134]. The selected metrics should be equally represented in the simulated and experimental systems. Dimensional reduction techniques allow the multivariate data to be condensed to a few axes, ideally separating the defined pattern classes into distinct regions of the latent space. A transform function trained on the pattern class data can then be applied to metrics calculated from experimental and modeling results, mapping both into latent space and compared to the locations of the pattern classes. PCA, principal component analysis; t-SNE, t-distributed stochastic neighbor embedding.