A Mathematical Reconstruction of Endothelial Cell Networks

Anthony Bell; Okezue Bell

arxiv: 2405.09748 · v2 · pith:Z5PYKY5Hnew · submitted 2024-05-16 · 🧬 q-bio.CB · math.CO· math.GR· q-bio.QM

A Mathematical Reconstruction of Endothelial Cell Networks

Okezue Bell , Anthony Bell This is my paper

Pith reviewed 2026-05-25 08:11 UTC · model grok-4.3

classification 🧬 q-bio.CB math.COmath.GRq-bio.QM

keywords endothelial cellspi-graphsnetwork connectivitygraph isomorphismvascular networkstopological invariantsangiogenesisjunction sets

0 comments

The pith

π-graphs represent endothelial cell networks so that π-isomorphism implies but is not implied by standard graph isomorphism.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces π-graphs as abstract objects consisting of endothelial cells and their junction sets to model multi-type junction connectivity in vascular and lymphatic networks. It defines π-isomorphism as the relation that holds when two π-graphs share the same connectivity structure. The authors prove that π-isomorphism of these objects implies isomorphism of the corresponding unnested endothelial graphs, yet the reverse does not hold. The formalism is extended with a temporal dimension to track evolution of topological invariants and a topological framework for spatial embeddings into geometric spaces. If correct, the approach supplies a quantitative language for relating network connectivity to functions such as angiogenesis and vessel permeability.

Core claim

We define π-graphs as abstract objects consisting of endothelial cells and their junction sets, and introduce the key notion of π-isomorphism that captures when two π-graphs have the same connectivity structure. We prove several propositions relating the π-graph representation to traditional graph-theoretic representations, showing that π-isomorphism implies isomorphism of the corresponding unnested endothelial graphs, but not vice versa. We also introduce a temporal dimension to the π-graph formalism and explore the evolution of topological invariants in spatial embeddings of π-graphs.

What carries the argument

π-graphs, defined as abstract objects consisting of endothelial cells and their junction sets, together with the relation of π-isomorphism that identifies equivalent connectivity structures.

If this is right

Quantitative analysis of endothelial network connectivity and its relation to function becomes possible.
Topological invariants can be tracked as π-graphs evolve over time.
Spatial embeddings of the networks can be represented within a topological framework.
Distinctions between networks that appear identical under standard graphs but differ under π-isomorphism can be detected.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Image-derived maps of real vessel junctions could be converted into π-graphs to test whether structural differences predict changes in permeability.
The stricter equivalence relation might be applied to other multi-junction cellular networks such as those in lymphatic drainage or tumor vasculature.
Temporal π-graphs could be used to simulate how junction rearrangements alter overall network invariants during angiogenesis.

Load-bearing premise

The connectivity structure of endothelial networks is adequately captured by abstract objects consisting of endothelial cells and their junction sets.

What would settle it

Finding real endothelial cell junctions that cannot be partitioned into the multi-type junction sets required by the π-graph definition, or two networks whose π-isomorphism status contradicts their unnested graph isomorphism status in a way the propositions forbid.

Figures

Figures reproduced from arXiv: 2405.09748 by Anthony Bell, Okezue Bell.

**Figure 1.** Figure 1: Example of an endothelial π-graph G = (E ,π). (A) Schematic of an EC monolayer with ECs (labeled x1,..., x5) connected by adherens junctions (red), tight junctions (blue), and gap junctions (green). (B) The π-graph representation of the monolayer, with π-incidence displayed as junction sets associated to each cell. For instance, π(x1) = ∼AJ ,∼T J . Proof 1 Let y ∈ E with y ̸= x. If y shares a junction with… view at source ↗

**Figure 2.** Figure 2: Schematic illustration of π-union. (A) Two π-graphs G1 and G2 with overlapping ECs x3, x4 (purple). (B) The π-union G1 ⊔π G2 merges the graphs along the common cells. The π-incidence of each common cell unions those from G1 and G2 (purple sets). Having established the basic definition and properties of π-graphs, we turn next to the notion of π-graph isomorphism to characterize when two π-graphs have the sa… view at source ↗

**Figure 3.** Figure 3: Example of π-isomorphic graphs G1 and G2. The bijection ϕ : E1 → E2 (dashed arrows) preserves the π-incidence structure, i.e. xi ∼ xj in G1 if and only if ϕ(xi) ∼ ϕ(x j) in G2 for each junction type ∼. The coloring of the graph edges and corresponding junction type is consistent with [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Unnested graph construction. (A) π-graph G with ECs x1,..., x5 connected by multiple junction types (colored edges). (B) The unnested graph UG retains the EC vertices but replaces the π-incidences with individual junction edges. Distinct junction types between two ECs yield multiple edges (e.g. the AJ and GJ between x1 and x2). Proposition 5 There exist π-graphs G1 and G2 such that UG1 and UG2 are isomorph… view at source ↗

**Figure 5.** Figure 5: Example of a topological π-graph. (A) Abstract π-graph with vertices x1,..., x5 and junctions represented by colored edges. (B) Topological π-graph embedding into R 2 . Vertices are embedded as discrete points (black dots) and junctions are embedded as line segments connecting vertices (colored lines). The π-incidence is represented by the adjacency of vertex points and junction segments. 1. ϕE : (E ,TE ) … view at source ↗

**Figure 6.** Figure 6: Example of spatial embedding. (A) Topological π-graph with discrete vertex topology and Euclidean junction topology. (B) Spatial embedding into R 2 . Vertices are embedded as points and junctions are embedded as line segments (AJ and GJ) or circular disks (TJ and NJ). Incidence compatibility is depicted by vertex points lying in their incident junction regions. Definition 8 Let G be a topological π-graph a… view at source ↗

**Figure 7.** Figure 7: Examples of topological invariants. (A) Two topologically equivalent embeddings with the same connected component, cycle, and intersection pattern. (B) Two embeddings that are not topologically equivalent, differing by the presence of a cycle (left) versus a void (right) [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: Schematic illustration of a temporal π-graph evolving over time. The network topology, encoded by the π-incidence, changes continuously as junctions form, dissolve, and remodel. 3.8 Evolution of Topological Invariants The spatial embedding framework introduced in Section 3.6 naturally extends to temporal π-graphs. We define a spatiotemporal embedding of a temporal π-graph G = (G, τ) into R d as a pair of c… view at source ↗

**Figure 9.** Figure 9: Evolution of topological invariants of an endothelial network embedded in R 3 . The Betti numbers β0, β1, and β2, which count the number of connected components, cycles, and voids respectively, change over time as the network remodels. To quantify the evolution of topological invariants, we can employ techniques from persistent homology, which provide a multiscale description of the topology of a spatial e… view at source ↗

read the original abstract

Endothelial cells form the linchpin of vascular and lymphatic systems, creating intricate networks that are pivotal for angiogenesis, controlling vessel permeability, and maintaining tissue homeostasis. Despite their critical roles, there is no rigorous mathematical framework to represent the connectivity structure of endothelial networks. Here, we develop a pioneering mathematical formalism called $\pi$-graphs to model the multi-type junction connectivity of endothelial networks. We define $\pi$-graphs as abstract objects consisting of endothelial cells and their junction sets, and introduce the key notion of $\pi$-isomorphism that captures when two $\pi$-graphs have the same connectivity structure. We prove several propositions relating the $\pi$-graph representation to traditional graph-theoretic representations, showing that $\pi$-isomorphism implies isomorphism of the corresponding unnested endothelial graphs, but not vice versa. We also introduce a temporal dimension to the $\pi$-graph formalism and explore the evolution of topological invariants in spatial embeddings of $\pi$-graphs. Finally, we outline a topological framework to represent the spatial embedding of $\pi$-graphs into geometric spaces. The $\pi$-graph formalism provides a novel tool for quantitative analysis of endothelial network connectivity and its relation to function, with the potential to yield new insights into vascular physiology and pathophysiology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces π-graphs and π-isomorphism for endothelial networks but rests on definitions and unshown proofs whose novelty and correctness cannot be checked from the abstract.

read the letter

The core contribution here is a set of new definitions: π-graphs built from endothelial cells and their junction sets, plus π-isomorphism as a stricter equivalence than ordinary graph isomorphism. The authors claim to prove that π-isomorphism implies isomorphism of the underlying unnested graphs but not conversely, and they sketch extensions to time and spatial embeddings. That is the actual new material on offer. The attempt to give a typed, junction-focused language for these networks is reasonable on its face and could be useful for modelers who need to track multi-cell junctions explicitly. Credit for identifying the modeling gap and trying to close it with a formal object rather than ad-hoc diagrams. The main weakness is that the abstract states the existence of proofs without supplying them or even the precise definitions, so it is impossible to verify whether the claimed implication holds for non-trivial reasons or whether π-graphs collapse to existing constructions such as hypergraphs or typed graphs. No examples, no comparison table to prior vascular network representations, and no indication that the formalism has been applied to real data appear in the provided text. The modeling assumption that junction sets alone capture the relevant connectivity is stated but not defended against alternatives like directed edges or weighted contacts. A reader who already works in mathematical cell biology might find the framework worth testing once the full definitions and proofs are available. For anyone else the paper is too preliminary to engage. I would send it to referees only if the full manuscript contains the actual proofs and a clear literature comparison; otherwise it is still too thin for review.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces π-graphs as abstract objects consisting of endothelial cells and their junction sets to model multi-type junction connectivity in endothelial networks. It defines π-isomorphism to capture equivalent connectivity structure, claims to prove that π-isomorphism implies isomorphism of the corresponding unnested endothelial graphs (but not conversely), introduces a temporal dimension to track evolution of topological invariants, and outlines a framework for spatial embeddings of π-graphs into geometric spaces.

Significance. If the stated propositions are valid, the π-graph formalism supplies a specialized graph-theoretic language for endothelial connectivity that distinguishes typed junctions from standard graphs; this could support quantitative analysis of vascular network topology and its functional implications. The work consists entirely of new definitions and internal consistency statements rather than derivations from data or existing models.

major comments (1)

[Abstract] Abstract: the text asserts that 'we prove several propositions' showing π-isomorphism implies (but is not implied by) isomorphism of unnested endothelial graphs, yet supplies none of the definitions, propositions, or derivations. Without these, the central mathematical claim cannot be verified or assessed for internal consistency.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review of our manuscript. We respond to the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the text asserts that 'we prove several propositions' showing π-isomorphism implies (but is not implied by) isomorphism of unnested endothelial graphs, yet supplies none of the definitions, propositions, or derivations. Without these, the central mathematical claim cannot be verified or assessed for internal consistency.

Authors: Abstracts are concise summaries by design and do not contain full definitions or derivations; this is standard practice. The manuscript body supplies the definitions of π-graphs and π-isomorphism, states the propositions relating π-isomorphism to ordinary graph isomorphism (one direction only), and provides their proofs. If the structure of the paper made these difficult to locate, we can revise the abstract to explicitly reference the relevant sections containing the formal statements and derivations. revision: partial

Circularity Check

0 steps flagged

No significant circularity; purely definitional mathematics

full rationale

The paper introduces new definitions for π-graphs and π-isomorphism, then proves internal propositions relating them to standard graph isomorphism. These steps are self-contained within the stated definitions and do not reduce to fitted parameters, external self-citations, or prior results by the same authors. No data, predictions, or ansatzes are involved; the implication statements hold (or fail) strictly by construction of the new objects. This is standard mathematical exposition with no load-bearing circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The contribution rests on the introduction of new abstract objects rather than on external data or prior fitted parameters.

axioms (1)

standard math Standard axioms of set theory and graph theory
The paper relates π-graphs to traditional graph-theoretic representations.

invented entities (2)

π-graph no independent evidence
purpose: Abstract object consisting of endothelial cells and their junction sets to model multi-type connectivity
Defined in the abstract as the central new construct.
π-isomorphism no independent evidence
purpose: Equivalence relation capturing when two π-graphs have the same connectivity structure
Introduced alongside π-graphs in the abstract.

pith-pipeline@v0.9.0 · 5751 in / 1174 out tokens · 34911 ms · 2026-05-25T08:11:44.603742+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

[1]

& Shimokawa, H

Godo, S. & Shimokawa, H. Endothelial functions. Arter. Thromb. Vasc. Biol. 37, DOI: 10.1161/atvbaha.117.309813 (2017)

work page doi:10.1161/atvbaha.117.309813 2017
[2]

L., Gourdou-Latyszenok, V ., Couturaud, F

Pilard, M., Ollivier, E. L., Gourdou-Latyszenok, V ., Couturaud, F. & Lemarié, C. A. Endothelial cell phenotype, a major determinant of venous thrombo-inflammation. Front. Cardiovasc. Medicine9, DOI: 10.3389/fcvm.2022.864735 (2022)

work page doi:10.3389/fcvm.2022.864735 2022
[3]

& Jung, F

Krüger-Genge, A., Blocki, A., Franke, R.-P. & Jung, F. Vascular endothelial cell biology: An update. Int. J. Mol. Sci. 20, DOI: 10.3390/ijms20184411 (2019)

work page doi:10.3390/ijms20184411 2019
[4]

Aird, W. C. Phenotypic heterogeneity of the endothelium: I. structure, function, and mechanisms. Circ. Res. 100, 158–173, DOI: 10.1161/01.res.0000255691.76142.4a (2007)

work page doi:10.1161/01.res.0000255691.76142.4a 2007
[5]

& Torbey, M

Yang, Y . & Torbey, M. T. Angiogenesis and blood-brain barrier permeability in vascular remodeling after stroke.Curr. Neuropharmacol. 18, 1250–1265, DOI: 10.2174/1570159x18666200720173316 (2020)

work page doi:10.2174/1570159x18666200720173316 2020
[6]

Sailem, H. Z. & Al Haj Zen, A. Morphological landscape of endothelial cell networks reveals a functional role of glutamate receptors in angiogenesis. Sci. Reports 10, DOI: 10.1038/s41598-020-70440-0 (2020)

work page doi:10.1038/s41598-020-70440-0 2020
[7]

& Vermeren, S

Zudaire, E., Gambardella, L., Kurcz, C. & Vermeren, S. A computational tool for quantitative analysis of vascular networks. PLoS ONE 6, e27385, DOI: 10.1371/journal.pone.0027385 (2011)

work page doi:10.1371/journal.pone.0027385 2011
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Beter, M. et al. Sproutangio: an open-source bioimage informatics tool for quantitative analysis of sprouting angiogenesis and lumen space. Sci. Reports 13, DOI: 10.1038/s41598-023-33090-6 (2023)

work page doi:10.1038/s41598-023-33090-6 2023
[9]

& Gamba, A

Lamberti, F., Montrucchio, B. & Gamba, A. Quantitative analysis of vascular structures geometry using neural networks. In IEEE Workshop on Signal Processing Systems Design and Implementation, 2005., 378–383, DOI: 10.1109/SIPS.2005. 1579897 (2005)

work page doi:10.1109/sips.2005 2005
[10]

& Jurek, J

Materka, A. & Jurek, J. Using deep learning and b-splines to model blood vessel lumen from 3d images. Sensors 24, 846, DOI: 10.3390/s24030846 (2024)

work page doi:10.3390/s24030846 2024
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Stoewer, P. et al. Neural network based successor representations to form cognitive maps of space and language. Sci. Reports 12, DOI: 10.1038/s41598-022-14916-1 (2022)

work page doi:10.1038/s41598-022-14916-1 2022
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& Vallenet, D

Sorokina, M., Medigue, C. & Vallenet, D. A new network representation of the metabolism to detect chemical transforma- tion modules. BMC Bioinforma. 16, DOI: 10.1186/s12859-015-0809-4 (2015)

work page doi:10.1186/s12859-015-0809-4 2015
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Jin, S. et al. Inference and analysis of cell-cell communication using cellchat. Nat. Commun. 12, DOI: 10.1038/ s41467-021-21246-9 (2021)

work page 2021
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& Pan, J

Ma, Q., Li, Q., Zheng, X. & Pan, J. Cellcommunet: an atlas of cell–cell communication networks from single-cell rna sequencing of human and mouse tissues in normal and disease states. Nucleic Acids Res. 52, D597–D606, DOI: 10.1093/nar/gkad906 (2023)

work page doi:10.1093/nar/gkad906 2023
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Koh, G. C. K. W., Porras, P., Aranda, B., Hermjakob, H. & Orchard, S. E. Analyzing protein–protein interaction networks. J. Proteome Res. 11, 2014–2031, DOI: 10.1021/pr201211w (2012)

work page doi:10.1021/pr201211w 2014
[16]

& McDonald, D

Dejana, E., Orsenigo, F., Molendini, C., Baluk, P. & McDonald, D. M. Organization and signaling of endothelial cell-to-cell junctions in various regions of the blood and lymphatic vascular trees. Cell Tissue Res. 335, 17–25, DOI: 10.1007/s00441-008-0694-5 (2008)

work page doi:10.1007/s00441-008-0694-5 2008
[17]

& Takeichi, M

Meng, W. & Takeichi, M. Adherens junction: Molecular architecture and regulation. Cold Spring Harb. Perspectives Biol. 1, a002899–a002899, DOI: 10.1101/cshperspect.a002899 (2009)

work page doi:10.1101/cshperspect.a002899 2009
[18]

Anderson, J. M. & Van Itallie, C. M. Physiology and function of the tight junction. Cold Spring Harb. Perspectives Biol. 1, a002584–a002584, DOI: 10.1101/cshperspect.a002584 (2009)

work page doi:10.1101/cshperspect.a002584 2009
[19]

Goodenough, D. A. & Paul, D. L. Gap junctions. Cold Spring Harb. Perspectives Biol. 1, a002576–a002576, DOI: 10.1101/cshperspect.a002576 (2009)

work page doi:10.1101/cshperspect.a002576 2009
[20]

& Nakanishi, H

Takai, Y . & Nakanishi, H. Nectin and afadin: novel organizers of intercellular junctions. J. Cell Sci. 116, 17–27, DOI: 10.1242/jcs.00167 (2003)

work page doi:10.1242/jcs.00167 2003
[21]

Dickerson

Moorhead, A. Mal’cev complexes, DOI: 10.48550/ARXIV .2311.02759 (2023)

work page internal anchor Pith review doi:10.48550/arxiv 2023
[22]

& La Rivière, P

Trinkle, S., Foxley, S., Wildenberg, G., Kasthuri, N. & La Rivière, P. The role of spatial embedding in mouse brain networks constructed from diffusion tractography and tracer injections. NeuroImage 244, 118576, DOI: 10.1016/j.neuroimage.2021. 118576 (2021). 14/14

work page doi:10.1016/j.neuroimage.2021 2021

[1] [1]

& Shimokawa, H

Godo, S. & Shimokawa, H. Endothelial functions. Arter. Thromb. Vasc. Biol. 37, DOI: 10.1161/atvbaha.117.309813 (2017)

work page doi:10.1161/atvbaha.117.309813 2017

[2] [2]

L., Gourdou-Latyszenok, V ., Couturaud, F

Pilard, M., Ollivier, E. L., Gourdou-Latyszenok, V ., Couturaud, F. & Lemarié, C. A. Endothelial cell phenotype, a major determinant of venous thrombo-inflammation. Front. Cardiovasc. Medicine9, DOI: 10.3389/fcvm.2022.864735 (2022)

work page doi:10.3389/fcvm.2022.864735 2022

[3] [3]

& Jung, F

Krüger-Genge, A., Blocki, A., Franke, R.-P. & Jung, F. Vascular endothelial cell biology: An update. Int. J. Mol. Sci. 20, DOI: 10.3390/ijms20184411 (2019)

work page doi:10.3390/ijms20184411 2019

[4] [4]

Aird, W. C. Phenotypic heterogeneity of the endothelium: I. structure, function, and mechanisms. Circ. Res. 100, 158–173, DOI: 10.1161/01.res.0000255691.76142.4a (2007)

work page doi:10.1161/01.res.0000255691.76142.4a 2007

[5] [5]

& Torbey, M

Yang, Y . & Torbey, M. T. Angiogenesis and blood-brain barrier permeability in vascular remodeling after stroke.Curr. Neuropharmacol. 18, 1250–1265, DOI: 10.2174/1570159x18666200720173316 (2020)

work page doi:10.2174/1570159x18666200720173316 2020

[6] [6]

Sailem, H. Z. & Al Haj Zen, A. Morphological landscape of endothelial cell networks reveals a functional role of glutamate receptors in angiogenesis. Sci. Reports 10, DOI: 10.1038/s41598-020-70440-0 (2020)

work page doi:10.1038/s41598-020-70440-0 2020

[7] [7]

& Vermeren, S

Zudaire, E., Gambardella, L., Kurcz, C. & Vermeren, S. A computational tool for quantitative analysis of vascular networks. PLoS ONE 6, e27385, DOI: 10.1371/journal.pone.0027385 (2011)

work page doi:10.1371/journal.pone.0027385 2011

[8] [8]

Beter, M. et al. Sproutangio: an open-source bioimage informatics tool for quantitative analysis of sprouting angiogenesis and lumen space. Sci. Reports 13, DOI: 10.1038/s41598-023-33090-6 (2023)

work page doi:10.1038/s41598-023-33090-6 2023

[9] [9]

& Gamba, A

Lamberti, F., Montrucchio, B. & Gamba, A. Quantitative analysis of vascular structures geometry using neural networks. In IEEE Workshop on Signal Processing Systems Design and Implementation, 2005., 378–383, DOI: 10.1109/SIPS.2005. 1579897 (2005)

work page doi:10.1109/sips.2005 2005

[10] [10]

& Jurek, J

Materka, A. & Jurek, J. Using deep learning and b-splines to model blood vessel lumen from 3d images. Sensors 24, 846, DOI: 10.3390/s24030846 (2024)

work page doi:10.3390/s24030846 2024

[11] [11]

Stoewer, P. et al. Neural network based successor representations to form cognitive maps of space and language. Sci. Reports 12, DOI: 10.1038/s41598-022-14916-1 (2022)

work page doi:10.1038/s41598-022-14916-1 2022

[12] [12]

& Vallenet, D

Sorokina, M., Medigue, C. & Vallenet, D. A new network representation of the metabolism to detect chemical transforma- tion modules. BMC Bioinforma. 16, DOI: 10.1186/s12859-015-0809-4 (2015)

work page doi:10.1186/s12859-015-0809-4 2015

[13] [13]

Jin, S. et al. Inference and analysis of cell-cell communication using cellchat. Nat. Commun. 12, DOI: 10.1038/ s41467-021-21246-9 (2021)

work page 2021

[14] [14]

& Pan, J

Ma, Q., Li, Q., Zheng, X. & Pan, J. Cellcommunet: an atlas of cell–cell communication networks from single-cell rna sequencing of human and mouse tissues in normal and disease states. Nucleic Acids Res. 52, D597–D606, DOI: 10.1093/nar/gkad906 (2023)

work page doi:10.1093/nar/gkad906 2023

[15] [15]

Koh, G. C. K. W., Porras, P., Aranda, B., Hermjakob, H. & Orchard, S. E. Analyzing protein–protein interaction networks. J. Proteome Res. 11, 2014–2031, DOI: 10.1021/pr201211w (2012)

work page doi:10.1021/pr201211w 2014

[16] [16]

& McDonald, D

Dejana, E., Orsenigo, F., Molendini, C., Baluk, P. & McDonald, D. M. Organization and signaling of endothelial cell-to-cell junctions in various regions of the blood and lymphatic vascular trees. Cell Tissue Res. 335, 17–25, DOI: 10.1007/s00441-008-0694-5 (2008)

work page doi:10.1007/s00441-008-0694-5 2008

[17] [17]

& Takeichi, M

Meng, W. & Takeichi, M. Adherens junction: Molecular architecture and regulation. Cold Spring Harb. Perspectives Biol. 1, a002899–a002899, DOI: 10.1101/cshperspect.a002899 (2009)

work page doi:10.1101/cshperspect.a002899 2009

[18] [18]

Anderson, J. M. & Van Itallie, C. M. Physiology and function of the tight junction. Cold Spring Harb. Perspectives Biol. 1, a002584–a002584, DOI: 10.1101/cshperspect.a002584 (2009)

work page doi:10.1101/cshperspect.a002584 2009

[19] [19]

Goodenough, D. A. & Paul, D. L. Gap junctions. Cold Spring Harb. Perspectives Biol. 1, a002576–a002576, DOI: 10.1101/cshperspect.a002576 (2009)

work page doi:10.1101/cshperspect.a002576 2009

[20] [20]

& Nakanishi, H

Takai, Y . & Nakanishi, H. Nectin and afadin: novel organizers of intercellular junctions. J. Cell Sci. 116, 17–27, DOI: 10.1242/jcs.00167 (2003)

work page doi:10.1242/jcs.00167 2003

[21] [21]

Dickerson

Moorhead, A. Mal’cev complexes, DOI: 10.48550/ARXIV .2311.02759 (2023)

work page internal anchor Pith review doi:10.48550/arxiv 2023

[22] [22]

& La Rivière, P

Trinkle, S., Foxley, S., Wildenberg, G., Kasthuri, N. & La Rivière, P. The role of spatial embedding in mouse brain networks constructed from diffusion tractography and tracer injections. NeuroImage 244, 118576, DOI: 10.1016/j.neuroimage.2021. 118576 (2021). 14/14

work page doi:10.1016/j.neuroimage.2021 2021