Multicellular Interaction Networks: Principles

• Multicellular Interaction Networks are quantitative models depicting cellular communication, contact, and mechanical coupling across scales.
• They simplify complex cell systems into modular designs using representations like directed graphs, dynamic graphs, and multilayer networks.
• Applications include understanding tissue pattern formation, developmental self-organization, and optimizing intervention strategies in multicellular behaviors.

Multicellular interaction networks are quantitative representations of how cells interact with their environments and with one another through communication, contact, exchange, and mechanically mediated coupling. In this literature, cells or cell types are commonly treated as nodes and interactions as edges, but the formalism extends to dynamic graphs, multilayer networks, tissue contact networks, and coupled intracellular–intercellular models. The central objective is to reduce the apparent complexity of multicellular systems to tractable design principles that relate microscopic interaction rules to macroscopic phenomena such as homeostasis, pattern formation, information transfer, tissue architecture, and developmental self-organization (Olimpio et al., 2016, Smart et al., 2021, Allison, 27 Mar 2026).

1. Foundational concepts and design principles

A recurring starting point is modularity: complex multicellular systems are decomposed into functional and/or spatial modules with some degree of independence. This decomposition is used as a simplification strategy for model construction and for relating local interactions to whole-system behavior. In the same framework, multicellular communication is represented by directed graphs in which nodes denote cells or cell types, edges denote communication, self-loops denote autocrine signaling, and edge states can be positive, negative, or absent. The standard communication classes are autocrine, paracrine, and juxtacrine signaling (Olimpio et al., 2016).

The combinatorial growth of possible architectures is itself a design-principle result. With two cell types $A$ and $B$, the four possible directed edges $A \to A$, $A \to B$, $B \to B$, and $B \to A$ each admit three states—positive, negative, or absent—so there are $3^4 = 81$ possible architectures. More generally,

$N_{\text{networks}} = 3^{n^2},$

where $n$ is the number of cell types. Spatial state spaces are similarly large: if $N$ cells are each ON or OFF, the number of possible configurations scales as $B$0 (Olimpio et al., 2016).

The search for quantitative principles has therefore focused on reduced descriptors. In Maire and Youk’s secrete-and-sense framework, cells secrete a signaling molecule and sense local concentration, often with positive feedback, so spatial patterns such as stripes and islands emerge from feedback and parameter tuning. The number of stable spatial patterns can be summarized by a population entropy,

$B$1

whose dependence on secretion rate, receptor levels, signaling threshold, and signaling length quantifies a spectrum from autonomy to collectiveness. High signaling length is associated with more collective behavior and lower entropy; low signaling length with greater autonomy and higher entropy (Olimpio et al., 2016).

A second foundational principle is that multicellular behavior need not be described solely as a sum of intracellular circuits. In the generalized tissue model of collective gene expression, increasing intercellular signaling strength drives a cascade of transitions from single-cell autonomy to self-organized collective states, while the set of stable tissue phenotypes contracts from an enormous combinatorial space to a relatively small number of compositionally and spatially simple tissue types (Smart et al., 2021). This suggests that interaction structure acts as a constraint that channels many possible microscopic states into a much smaller macroscopic repertoire.

2. Formal representations and mathematical structure

Several mathematical formalisms are used to encode multicellular interaction networks, each emphasizing a different biological scale. In one class of models, a tissue is treated as $B$2 cells on a graph with adjacency matrix $B$3, while each cell contains an intracellular gene network. In the framework of collective gene expression, each cell state is a binary vector $B$4, intracellular interactions are encoded by $B$5, and intercellular signaling by $B$6. The total tissue state is governed by a multicellular Hamiltonian with a tunable signaling-strength parameter $B$7, and the corresponding block interaction matrix takes the form

$B$8

This explicitly embeds within-cell gene regulation and between-cell signaling in a single multilayer interaction network (Smart et al., 2021).

A second class of formalisms treats multicellular organization as a dynamic graph. In this view, the graph at time $B$9 is $A \to A$0, nodes are cells, edges are interactions, and both nodes and edges evolve through cell division, adaptation, and interaction. Pairwise edges encode direct physical or chemical contacts, whereas hyperedges are introduced for interactions that simultaneously involve sets of cells, such as responses to diffusible morphogens. Node states can be updated by local rules, for example

$A \to A$1

and the network itself becomes part of the control logic of development or self-organization (Allison, 27 Mar 2026).

A third class is the formalism of multilayer network science. A general multilayer network can be represented by a rank-4 adjacency tensor $A \to A$2, where $A \to A$3 index nodes and $A \to A$4 index layers. Flattening yields a supra-adjacency matrix in $A \to A$5 whose diagonal blocks encode intra-layer interactions and off-diagonal blocks encode inter-layer couplings. This language is appropriate when the same physical entities participate in multiple distinct interaction modes, contexts, or timescales (Artime et al., 2024).

Representation	Core elements	Representative use
Directed cell–cell graph	Cells or cell types as nodes; autocrine/paracrine/juxtacrine edges	Communication architecture and network counting (Olimpio et al., 2016)
Dynamic graph or hypergraph	$A \to A$6, changing nodes/edges, hyperedges	Division, adaptation, diffusible multicellular interactions (Allison, 27 Mar 2026)
Multilayer or dual graph	Layers, replicas, or coupled cell/junction graphs	Tissue-specific function prediction and morphogenesis learning (Zitnik et al., 2017, Yang et al., 2024)

These representations are not merely notational variants. They determine which biological questions can be posed. Directed graphs are suited to architecture enumeration and feedback analysis; dynamic graphs foreground graph propagation under division and adaptation; multilayer and dual-graph formalisms support cross-scale coupling, including the coupling of cell geometry to junction topology or of tissue-specific molecular networks to tissue hierarchies.

3. Signaling, contact, and information transfer

Cell–cell signaling networks are often studied as excitable, thresholded, or contact-mediated systems. One canonical dynamical reduction is the FitzHugh–Nagumo model,

$A \to A$7

which has been used to describe collective oscillations in systems such as Dictyostelium discoideum cAMP signaling and neuronal action potentials, including threshold behavior for aggregation and desynchronization by external stimuli (Olimpio et al., 2016).

A prominent synthetic realization of juxtacrine signaling is the synNotch multicellular sheet. In this system, Sender cells expressing membrane-bound GFP interact with Receiver or Transceiver cells carrying a GFP-binding synNotch receptor. Upon contact, the intracellular domain $A \to A$8 is released and activates transcription of a fluorescent reporter and, in the transceiver design, the GFP ligand itself, creating a positive-feedback signaling cascade. Signal propagation exhibits three experimentally resolved density-dependent phases: persistent propagation at low or moderate densities ($A \to A$9 confluence, approximately $A \to B$0 cells/mm$A \to B$1), transient propagation at intermediate densities ($A \to B$2 confluence, approximately $A \to B$3 cells/mm$A \to B$4), and no propagation at high densities ($A \to B$5 confluence, approximately $A \to B$6 cells/mm$A \to B$7). The corresponding model incorporates an exponential density attenuation factor $A \to B$8, a transcriptional delay, and logistic growth, producing a phase diagram indexed by initial density and growth rate (Santorelli et al., 2021).

Developmental patterning models extend this local-contact logic by combining global positional information with neighbor communication. In a 1D tissue of $A \to B$9 cells, morphogen concentration is taken as

$B \to B$0

while contact-mediated Notch signaling adaptively regulates selective gene expression in the presence of that global field. In this hybrid picture, local receptor–ligand interactions supplement morphogen-based positional information by altering the effective thresholds of morphogen concentration that determine cell fate (Kuyyamudi et al., 2021).

Information-transfer studies of confluent neuronal monolayers have made this logic explicit at the network level. Under cyclic ATP stimulation in microfluidic devices, Granger inference on single-cell calcium dynamics reconstructs directed causal networks between nearest neighbors. These networks are spatially decentralized and temporally stationary, but their connectivity depends strongly on the temporal profile of the stimulus: short periods, or long periods with small duty fractions, reduce connectivity and yield fractured topology. A communicating-excitable-units model predicts an optimal communication strength that maximizes connectivity, and this prediction is experimentally confirmed (Li et al., 2022).

Related calcium-signaling work in fibroblast monolayers shows that oscillation propensity increases not only with ATP stimulus but also with cell density, implicating increased gap-junction communication. Introducing MDA-MB-231 cancer cells as communication-defective “defects” reduces oscillation propensity in fibroblasts, demonstrating that collective sensory response depends on the integrity of the multicellular communication network rather than on stimulus alone (Potter et al., 2015).

4. Physical, metabolic, and mechanical self-organization

Not all multicellular interaction networks are primarily biochemical. In hybrid models of multicellular adhesion, a 2D lattice contains empty sites and two cell states, and nearest-neighbor interactions are governed by an adhesion-energy matrix $B \to B$1. Cells move by stochastic swapping with a Boltzmann probability derived from the local energy difference, and they also undergo stochastic phenotypic switching. The combination of differential adhesion and switching supports a space of pattern-forming rules that produces spots, stripes, and labyrinths. A common misconception is that such Turing-like patterns require reaction–diffusion; in this model they are obtained without resorting to reaction-diffusion processes (Bonforti et al., 2016).

When that adhesion framework is expanded to include proliferation, death, nutrient diffusion, and toxic waste, the system exhibits multiple phases characterized by regularly spaced patterns, and some organizational modes reach higher population levels than others. Competition between populations with different adhesion matrices further shows that structural organization can improve fitness, with “trabecular” pattern-forming adhesion matrices outcompeting others under ecological stress (Bonforti et al., 2016). Here the interaction network is both spatially embedded and dynamically rewired by cell movement and state switching.

Chemical exchange can play an analogous role. In a dynamical-systems model of isoclonal cells with identical intracellular catalytic networks, cells interact through diffusion of shared chemical components in a well-mixed medium. Under limited resources, strong cell–cell coupling, and nonlinear catalytic reactions, the initially homogeneous state becomes unstable, cells spontaneously differentiate into multiple attractors, and the differentiated aggregate achieves division of labor and a higher growth rate than the unicellular case. Robustness of the differentiated composition depends on feedback through exchanged chemicals; when critical products are effectively exchanged, minority and majority types rebalance, whereas ineffective exchange destabilizes coexistence (Yamagishi et al., 2015).

Mechanical interaction networks have likewise been formalized. Contractile cells on elastic substrates are modeled as force dipoles whose interaction is mediated by substrate deformation. The relevant network measures include percolation, fractal dimension, and local motifs such as junctions, branches, and rings. Both simulations and endothelial-cell experiments show that network formation is substrate stiffness-dependent, being optimal at intermediate stiffness, and that long-range mechanical interactions provide a more robust and efficient route to space-spanning networks than local interactions alone (Noerr et al., 2022).

Recent work on intercellular metabolic exchange adds a further network-theoretic layer. A tiling-based reconstruction method infers pairwise exchange fluxes from single-cell proton/lactate fluxes and cell positions, allowing the dynamic unfolding of exchange networks to be mapped in mammalian co-cultures. These networks evolve from a dense matrix of exchanges to small disconnected clusters. A two-parameter Maximum-Entropy multicellular metabolic model predicts a crossover from a densely interconnected network to a sparse, motif-dominated state as glucose and oxygen consumption shift, with a power-law decay in cluster-size distribution at the critical transition and a mean-field critical line derived from percolation theory (Latoski et al., 2024).

5. Developmental control, multiscale interventions, and “daisy chains”

Developmental theories of multicellular interaction networks treat them not merely as transport or contact structures but as control architectures. In a computational theory of bilateral symmetry, multicellular organisms are governed by executable developmental control networks, or cenes, interpreted by an interpretive-executive system. Bilateral symmetry arises when two daughter founder cells are produced in the same developmental control network state but with opposite orientation along one axis; these orientation states are epigenetically inherited by their progeny, leading to mirror-image development. The same framework is used to explain symmetry breaking, situs inversus, gynandromorphs, inside-out growth, and bilaterally symmetric cancers (Werner, 2012).

A closely related theoretical proposal casts multicellular self-organization as a sequence of dynamic graphs in which gene-network outputs create interaction networks, and those interaction networks become the gene-network inputs for later stages. In this view, multicellular interaction networks act as a distributed, dynamic control layer over gene expression, especially in clonal communities, and developmental propagation occurs through multicellular daisy chains. The stronger claim in this literature is that daisy chains are proposed to be necessary and sufficient for multicellular self-organization (Allison, 27 Mar 2026). Work on Escherichia coli extends the same idea, arguing that observed self-organization indicates that multicellular interactions serve as inputs for key gene networks and are themselves dynamically generated as outputs of gene networks, thereby enabling robust expression of otherwise noisy genes (Allison, 15 Aug 2025).

The importance of this control perspective becomes especially clear in intervention studies. In a T-LGL leukemia survival/apoptosis Boolean network based on Zhang et al. (2008), candidate pro-apoptotic interventions were identified by exhaustive search simulation, stable motif control, and an individual-based mean field approximation, then translated into a continuous-time multicellular agent-based model using the PhysiBoSS framework. Interventions that perform equally well in the implicit-time single-cell setting become separable in the multiscale setting in their effects on population growth and spatial distribution. Stable motif interventions irreversibly commit cells to apoptosis and produce slow but steady decline, whereas target controls may permit recovery after stimulus withdrawal. The time to $B \to B$2 population reduction correlates with network distance from the intervention target to the output node Apoptosis, showing that network topology and internal dynamics determine population-level efficacy (Metzcar et al., 27 Jan 2025).

Taken together, these studies argue against the assumption that cell-level control laws translate directly to tissue-level behavior. In multicellular settings, the causal path from genotype or circuit perturbation to phenotype is filtered by orientation inheritance, spatial embedding, intercellular signaling, and the evolving interaction network itself.

6. Inference, learning, and computational analysis of tissues

Empirical reconstruction of multicellular interaction networks often begins with tissue architecture. In epithelial monolayers, cells are represented by polygons in a planar tiling, cell centroids become graph nodes, and an edge is drawn when two cells share a border. Voronoi diagrams and their dual Delaunay triangulations are standard constructions, and the number of polygon sides becomes the node degree. Because tissue topology changes through cell division, apoptosis, and migration, the resulting contact networks are intrinsically dynamic. This framework has also been coupled to evolutionary game theory to model cooperative and non-cooperative cell–cell communication in tissue homeostasis and tumorigenesis (Csikász-Nagy et al., 2013).

Current single-cell and spatial technologies have pushed network inference beyond contact topology alone. A growing line of work argues that cell taxonomies should be complemented by multicellular analyses based on three descriptive levels: compositional representations, organizational representations, and coordination or multicellular programs. Ligand–receptor analyses such as CellPhoneDB and NicheNet, spatial interaction models, matrix factorization, tensor decomposition, DIALOGUE, and Tensor-cell2cell are used to infer putative multicellular information networks from cross-condition single-cell and spatial datasets. In this setting, a top-down workflow identifies multicellular programs, uses organizational priors as a scaffold, and refines intercellular edges with predictive models analogous to gene regulatory network inference (Flores et al., 2024).

Machine learning models have begun to operate directly on these structured representations. OhmNet constructs a multi-layer network in which each layer is a tissue-specific protein–protein interaction network, and tissues are organized by a hierarchy with 107 leaf tissues and 219 ontology elements. The method jointly learns embeddings that preserve within-tissue neighborhoods while regularizing the same protein across related tissues. In 48 tissues with known tissue-specific cellular functions, it provides more accurate predictions than alternative approaches and can transfer information to functionally uncharacterized tissues (Zitnik et al., 2017).

For developmental morphology, Multicell-Fold represents tissue state with a dual-graph data structure containing cells, vertices, cell–cell adjacency, vertex–vertex junctions, and cell–vertex incidence. This unified graph supports interpretable 4-D morphological sequence alignment and prediction of local cell rearrangements before they occur. In the reported embryogenesis task, the model predicts junction loss or “T1 transitions” minutes ahead of time with $B \to B$3 and accuracy $B \to B$4 one minute ahead, while activation maps and ablation studies indicate that cell geometries and cell junction networks jointly regulate local rearrangement (Yang et al., 2024).

These data-driven approaches close a loop already anticipated by theory. The generalized model of collective gene expression was explicitly framed to align with spatial transcriptomics, because tissue state is represented as spatially arranged gene-expression profiles on a graph. A plausible implication is that multicellular interaction networks are becoming less a metaphorical description of tissue organization and more a directly inferable object, jointly constrained by imaging, spatial omics, and mechanistic models (Smart et al., 2021).