cells
Communication

Signal Amplification in Highly Ordered Networks Is Driven
by Geometry
Éva S. Vanamee 1, * , Gábor Lippner 2 and Denise L. Faustman 1, *1

2*Immunobiology Department, Massachusetts General Hospital, Harvard Medical School,
Boston, MA 02129, USA
Department of Mathematics, Northeastern University, Boston, MA 02115, USA; glippner@gmail.com
Correspondence: eva.vanamee@gmail.com (É.S.V.); faustman@helix.mgh.harvard.edu (D.L.F.)

Abstract: Here, we hypothesize that, in biological systems such as cell surface receptors that relay
external signals, clustering leads to substantial improvements in signaling efficiency. Representing
cooperative signaling networks as planar graphs and applying Euler’s polyhedron formula, we canshow that clustering may result in an up to a 200% boost in signaling amplitude dictated solely by
the size and geometry of the network. This is a fundamental relationship that applies to all clusteredsystems regardless of its components. Nature has figured out a way to maximize the signaling
amplitude in receptors that relay weak external signals. In addition, in cell-to-cell interactions,clustering both receptors and ligands may result in maximum efficiency and synchronization. The
importance of clustering geometry in signaling efficiency goes beyond biological systems and caninform the design of amplifiers in nonbiological systems.
Keywords: TNF superfamily signaling; hexagonal clustering; Euler polyhedron formula

Citation: Vanamee, É.S.; Lippner, G.;
Faustman, D.L. Signal Amplificationin Highly Ordered Networks Is
Driven by Geometry. Cells 2022, 11,272. https://doi.org/10.3390/
cells11020272
Academic Editor: Juan M. Zapata
Received: 23 November 2021
Accepted: 7 January 2022
Published: 13 January 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims inpublished maps and institutional affiliations.

Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms andconditions of the Creative Commons
Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

1. Introduction
Membrane receptors allow cells to respond to external stimuli, and earlier models havepredicted the homogeneous arrangement of receptors on the cell surface for optimal ligand
binding and activation. However, this is in contrast to a growing body of experimental
evidence suggesting that signaling efficiency can be greatly improved by the clustering ofreceptors [1,2]. Furthermore, recent super-resolution optical microscopy data have shownthat many receptors preferentially organize in nanoclusters on the cell surface even in theabsence of activation or ligand binding [3,4]. Nanoclustering of receptors may, therefore,
be a common feature of the plasma membrane. The factors that regulate and stabilize
nanoclusters are of considerable interest. In lipid-linked and transmembrane protein
clusters, receptors are primarily stabilized by cis interactions between the ectodomains andcytosolic domains or via lipid anchors. Examples include members of the TNF receptor
superfamily (TNFRSF) that cluster via their pre-ligand assembly domain (PLAD) [5–7].
There is mounting evidence that TNF receptors and their downstream signaling partners
organize into hexagonal lattices [7–12]. Other examples of hexagonal clustering include
chemo- or phototaxis receptors [2,13,14]. These represent examples of receptors that are able
to self-assemble on the cell surface on the basis of their inherent symmetry, although thisdoes not rule out the participation of other proteins in the process. Interestingly, pentamericgamma-aminobutyric acid A receptor (GABA(A)R) and glycine receptor (GlyR), which
are unable to form 2D lattices by themselves, can also organize into hexagonal clusters on
the cell surface by binding to a hexagonal scaffold formed under the cell membrane bygephyrin [15–17]. Further scaffolding examples include the spectrin–ankyrin hexagonal–
triangular clusters [18] that form a network under the cell membrane of eukaryotic cells
and control the surface display and localization of proteins in the plasma membrane [19].
Hexagonal networks are special for several reasons. Regular hexagons are the highest-order
polygons that can be tiled or tessellated in a regular pattern by themselves, where each

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https://www.mdpi.com/journal/cells

Cells 2022, 11, 272

2 of 9hexagon is surrounded by six other hexagons, repeated indefinitely in any direction withno gaps in between. Hexagonal tiling uses the smallest perimeter to enclose a particulararea in space [20]. Therefore, the use of a hexagonal network is the most economical way tobuild a network.
On the basis of the growing experimental evidence for clustering, more recent theoretical models have proposed that the optimal solution to the biological information-processing
problem has to balance two opposing objectives: on the one hand, to reduce noise as thereceptors need to concentrate locally; on the other hand, to provide broader coverage as
the receptors need to be spread out [21]. To our knowledge, to date, no one has explored
what role the clustering geometry might play in signaling. Here, we hypothesize thatclustering may result in signal amplification that depends on cluster size and geometry andillustrate how ordered nanoclustering can provide an optimal solution to the biologicalinformation-processing problem
2. Methods
Hexagonal networks play an important role in many applications, and methods tocalculate the number of edges and vertices were described earlier [22]. The equations
to calculate the e/n ratio for the specific examples shown in Figure 2 are detailed inAppendix A.
3. Results and Discussion
We first consider receptors on a cell surface that are arranged randomly, independentof each other. Receptors can either be inactive (0 state) or activated by their ligands(1 state). The signal in such a random disconnected network travels vertically without
synchronization between the receptors, and the overall amplitude of a signal (i.e., the sumof all the ones and zeros) generated by such a system is low, as, at any given time, onlya small percentage of receptors will be active. In addition, because of the asynchronous
nature of the signal, it will be distributed broadly over time. Next, we consider two receptor
complexes (A and B) physically connected to create a new output signal, wherein theconnection (AB) is only made when both receptors A and B are activated by their input
signal, the ligand (AB = 1 if A and B = 1, and AB = 0 if A or B = 0). In essence, this is alogical AND gate and requires the synchronization between two input signals. An example
is downstream tumor necrosis factor receptor-associated factor (TRAF) dimerization upon
TNF receptor 2 (TNFR2) activation of two adjacent receptor trimers (Figure 1a). The
connection creates a single output from two input signals. AB can only reach a maximum of
50% of the signal strength of the original signals A and B since there is a single connection
between the two input signals. However, if six signaling units were connected in a closed
loop, the signal strength of the input signal can be maintained (Figure 1b). That is because
closed loops have an equal number of edges (e) and vertices or nodes (n). We can achieve
further gains in efficiency by simultaneously activating the receptors by their ligands,
which eliminates the stochasticity. In the case of TNF receptors, simultaneous activation
requires saturating the receptor sites with soluble ligands [23] or using localized ligands
arranged in the same geometry as the receptors (Figure 1d). Most TNF ligands are type
II transmembrane proteins expressed in a membrane-bound form and can be cleaved
to produce soluble ligands. The membrane-bound ligands are known to signal more
effectively via their receptors than the soluble ligands [24]. This has been demonstrated
experimentally in a model system of the apoptosis inducing receptor CD95 (Fas), anothermember of the TNF receptor superfamily, and its ligand CD178 (FasL). Using a DNA
origami platform, the FasL ligands were immobilized and arranged in different geometries
to test the effect of clustering on apoptosis efficiency in HeLa cells overexpressing the Fas
receptor. Hexagonally arranged ligands signaled orders of magnitude better than soluble
ligands [25]. Furthermore, ligands arranged in a hexagon with the right geometry doubled
the rate of apoptosis compared to ligands arranged in pairs [25]. All of this indicates that
membrane-bound ligands achieve more efficient signaling by having the same clusteringCells 2022, 11, x FOR PEER REVIEW

Cells 2022, 11, 272

3 of 93 of 9[25]. Furthermore, ligands arranged in a hexagon with the right geometry doubled therate of apoptosis compared to ligands arranged in pairs [25]. All of this indicates that
membrane-bound ligands achieve more efficient signaling by having the same clusteringgeometryon
onthe
thecellcellsurface
surfaceasasthe
thereceptors.
receptors.As
Aswewedescribed
describedearlier,earlier,members
membersofofthe
theTNF
TNF
geometry
receptorsuperfamily
superfamilyare
arefurther
furtherarranged
arrangedininaahexagonal
hexagonallattice
latticeon
onthe
thecellcellsurface
surface[7,26].
[7,26].
receptor

Figure 1. Illustration of cooperative biological signaling systems. (a) The TNF/TNFR2/TRAF
signaling
complex. The
input signalbiologicalis represented
by TNF
ligand(a)(magenta)
binding to TNFR2(blue).
Figure1. Illustrationof cooperative
signalingsystems.
The TNF/TNFR2/TRAF
signalReceptor
activation
initiates(green) recruitment
and dimerization
via the to
TRAF2
N-terminal
ing
complex.
The input
signalTRAF2
is represented
by TNF ligand(magenta) binding
TNFR2(blue).
Receptor
activation
initiates(green)recruitment
anddimerization
the TRAF2
N-terminal
RING domains(cyan)
that TRAF2
represents
the output
signal.
Connecting
twoviasignalingunits
introduces
RING
domains (cyan)
represents
thesignal
outputConnecting
two signaling
units introduces
cooperativity,
but thethat
overall
outputis signal.
only halfof the original
input signals.
(b) Whencooperativity,
theareoverall
outputis only
halfof thetheoriginal
input
signals.
Whensixbe
six signaling but
units
connected
insignal
a hexagon(topview),
strengthof theinput(b)signalcansignaling units are connected in a hexagon (top view), the strength of the input signal can be mainmaintained. (c) The hexagonal unit can further cluster into an ordered hexagonal signaling network.
tained. (c) The hexagonal unit can further cluster into an ordered hexagonal signaling network. (d)(d) In cell-to-cell interactions, clustered ligands attached to the membrane allow the simultaneous
activation of clustered receptors with the same geometry, resulting in maximum signaling efficiency.
(e) Chemotaxis MCP receptor baseplate. The MCP receptor trimers are connected by the CheA kinase
dimers (blue) to introduce cooperativity. The triangular unit cell is shown in red.

Cells 2022, 11, 272

4 of 9Beyond building a stable network, clustering with a specific geometry has otherimportant benefits, as illustrated below.
Cooperative signaling networks can be represented as planar graphs, i.e., a G planargraph with n nodes (degrees: d1 , d2 , d3 , . . . , dn ). The degree of a node represents thenumber of edges meeting at that node.
e edges (e =

1 n
d ).
2 ∑ j=1 j(1)This graph has f faces (face sides: s1 , s2 , s3 , . . . , sf ). Each face contributes sj facesides or edges; however, because each edge is shared by two faces, the sum total has to be
divided by 2.
f

e=

1
sj .
2 j∑
=1

3f
.
2
Replacing f with Euler’s polyhedron formula [27],(2)

For sj ≥ 3, e ≥(3)

f = e − n + 2,(4)

e ≤ 3n − 6.

(5)yieldsIn a “simple” graph with no self-loops or multiple edges, for each planar graph withsj ≥ 3. and n  1,
e
≤3
(6)
n
For clustered signaling systems, illustrated by tiled polygons, the e/n ratio represents
the ratio of the output signal over the input signal. According to Equation (6), the e/nratio can never exceed 3. In practical terms, what this means is that the input signal in aclustered network is amplified, and a maximum of three times the original amplitude or a
200% gain in amplification can be achieved. The actual number depends on the geometry
and size of the cluster.
As we described earlier, the clustering of TNF receptors on the membrane surface canbe illustrated by tiled regular hexagons. The input signal is represented by the vertices ornodes, and the output signal is represented by the edges of the hexagon. Figure 2a illustrates
two scenarios of hexagonal tiling. As the cluster size grows, the e/n ratio increases andreaches a plateau (Figure 2b). This implies that, in addition to the maximum amplification,
there is also an optimum cluster size for amplification beyond which the input/output
ratio does not increase considerably. For hexagonal clusters, 90% of the maximum signal
amplification can be achieved in a cluster of 100 signaling units and 400 units to reach
95% signal amplification. It takes 200 and 800 signaling units to reach the 90% and 95%
e/n levels, respectively, in a triangular cluster (Figure 2b). This is important because
experimental data indicate that signaling proteins tend to form small nanoclusters on thecell surface around 300–500 nm in diameter [4,15,28]. Optimal cluster size may also dependon the size of the cell-to-cell interface. According to the numbers presented here, a clusterof 100–200 receptors is sufficient to reach 90% of the maximum level of amplification. The
signal amplitude also depends on the geometry of the cluster. Tiling the hexagons in a moreor less symmetrical fashion in each direction is the most efficient, leading to the highest e/nor input/output signal ratio compared to hexagons tiled in a linear fashion. This is because
the e/n ratio is maximized when most hexagons are surrounded by other hexagons or, inother words, when the number of dj nodes with maximum connections is the highest.

Cells 2022, 11, x FOR PEER REVIEW

Cells 2022, 11, 272

5 of 9to the highest e/n or input/output signal ratio compared to hexagons tiled in a linear fashion. This is because the e/n ratio is maximized when most hexagons are surrounded5 ofby9other hexagons or, in other words, when the number of dj nodes with maximum connections is the highest.

Figure2. Illustration
Illustration ofof Euler’s
Euler’s formula
2D polygons.
polygons. (a)The e/ne/n ratio
calculated forfor several
severalFigure 2.
formula forfor 2D(a) Theratio isis calculated
examplesofregulartiled
polygons.
(b)
The
e/n
ratio
plottedagainst
n
for
the
tiled
polygonal
systems
examples of regular tiled polygons. (b) The e/n ratio plotted against n for the tiled polygonal
shown(a). It represents
signal amplification
and reaches
maximumlevel withlevel
tiledwithtriangles
systemsinshownin (a). It represents
signal amplification
anditsreaches
its maximum
tiled
but can never exceed 3. A 90% maximum signal amplification can be achieved with a cluster of ~100
triangles but can never exceed 3. A 90% maximum signal amplification can be achieved with asignaling units in a hexagonal cluster depending on geometry, but the same level of amplification
cluster of ~100 signaling units in a hexagonal cluster depending on geometry, but the same level
requires a cluster of ~200 signaling units in a triangular cluster. Calculations are provided in Apof amplification
requires a cluster of ~200 signaling units in a triangular cluster. Calculations are
pendixA.
provided in Appendix A.

In general, in the case of polygonal tiling in 2D, dj in Equation (1) is not constant;
In general, in the case of polygonal tiling in 2D, d in Equation (1) is not constant;
therefore, we can introduce dave, which can be calculatedj as
therefore, we can introduce dave , which can be calculated as𝒏𝟏n𝒅.
𝒅𝒂𝒗𝒆 =1
(7)
dave
= 𝒏
dj . 𝒋
(7)
𝒋 𝟏
n j∑
=1
Equation (1) can then be rewritten as
Equation (1) can then be rewritten as

and𝒏 𝒅𝒂𝒗𝒆,𝒆 =n dave
𝟐
e=
,
2

(8)
(8)

e
dave
=
.
n
2

(9)

Cells 2022, 11, 272

6 of 9For nodes of polygons on the inside of the cluster, d = dmax , whereas nodes on theoutside of the cluster have a d value smaller than dmax , leading to dave < dmax . As n
increases, dave converges on dmax . For tiled hexagons, a maximum of three edges can bejoined together at each node leading to a dmax of 3, translating to a maximum amplitudeof 1.5 times the original signal in hexagonally clustered receptors. The e/n ratio is also
inversely correlated with the degree of the polygon, and it is highest for tiled triangles witha dmax of 6 (Figure 2b).
From our analysis, the hexagonal lattice of TNF receptors is optimized for signaltransduction. It provides the most economical way to build a stable scaffold. Beyond
simply building the most efficient network, however, clustering not only maintains the
input signal but also boosts it by its specific geometry. The growing number of hexagonal
biological systems indicates that this may be a common arrangement of signaling networks
in general. In addition to TNF receptors and their downstream signaling partners, chemoor phototaxis receptors also cluster into hexagonal core complexes, consisting of trimers of
dimers that further assemble to form large hexagonal arrays [2,13,14]. Signal amplification
has also been observed in these receptors, and it has been proposed that the amplification is
the result of cooperativity in the clustered arrays [14,29,30]. Interestingly, in the case of E. coli
methyl-accepting chemotaxis protein (MCP) receptors, the histidine kinase (CheA) elements
responsible for cooperativity are arranged to create triangular repeats within the hexagonallattice, potentially to maximize signal amplification (Figure 2e) [2]. We hypothesize that, in
chemo- and photoreceptors that sense weak external signals, the geometry is optimized toachieve maximum amplification, which may be crucial to generate an appropriate cellularresponse. Mimicking natural receptor clustering, artificial two-dimensional scaffolds havenow been developed that utilize hexagonal lattices to modulate cell responses [31].
4. Conclusions
In summary, signaling efficiency can be greatly improved in cooperative signaling
networks by clustering receptors on the cell surface. We can simplify such cooperative networks as planar graphs, where the input signal is represented by the vertices or nodes, theoutput signal is represented by the edges, and the relationship between them is determinedby Euler’s polyhedron formula. As the cluster size grows, the output signal is amplified
with respect to the input signal, and the maximum level of amplification, represented in
Equation (9), is determined by cluster size and geometry. Plotting the e/n ratio against
the number of signaling units or cluster nodes illustrates that there is an optimal cluster
size beyond which the relative increase in amplitude becomes negligible. Maximum signal
amplification can be achieved in triangular clusters. The TNF receptor signalosome is
optimized for maximum synchronicity by clustering both the receptors and ligands withthe same geometry. This allows simultaneous activation of the receptors and results ina sharp digital-like output signal. In chemo- and photoreceptors that sense random diffuse signals the geometry seems to be optimized for maximum amplification. Despite its
simplicity, Euler’s polyhedron formula is a deep theorem with numerous applications ingeometry, topology, dynamical systems, and medicine [32]. We have now shown it to be
essential to describe signal amplification in biological signaling networks. How broadly thisrelationship will apply remains to be seen. Signaling systems can deliver both activating
and inhibitory signals, and our model can apply to both scenarios. Another importantquestion is how the process of clustering and cluster size is regulated. It has been proposedthat the dynamic modeling and remodeling of skeletal proteins such as actin regulates thespatiotemporal organization of cell surface molecules [33,34]. In addition, molecules may
be actively transported to the cell surface by other proteins. An example is the display andlocalization of CD45 by the spectrin–ankyrin skeleton [35]. Cluster size maybe determined
simply by the entropic cost of growing the cluster or by skeletal proteins that function
as a “fence” to inhibit cluster growth beyond a certain size [3]. In addition, coregulatory
proteins may aid or interfere with the clustering of other proteins and, therefore, modulate
the amplitude of the output signal. For instance, members of the TNF receptor superfamily

Cells 2022, 11, 272

7 of 9directly interact with members of the immunoglobulin superfamily (IgSF), which function
as coregulatory proteins to modulate cluster formation [36]. As we discussed earlier, nanoclusters of 100–200 signaling units, depending on geometry, are sufficient to reach 90% of
maximum signal amplification. Homogeneously distributing small nanoclusters on the cellsurface instead of individual molecules may, therefore, provide an optimum solution to
the information-processing problem posed earlier. We believe that many other signaling
systems will be identified in the future where clustering with a specific geometry is necessary to improve signaling efficiency. Top candidates include proteins of the cytoskeleton
that may also play a functional role in signal transduction [37,38]. The importance of thesefindings goes beyond biological systems and could influence the design of amplifiers inelectronics and in light sensors, among other systems.
Author Contributions: Conceptualization, É.S.V. and D.L.F.; methodology, É.S.V. and G.L.; originaldraft preparation, É.S.V.; writing—review and editing, É.S.V., D.L.F. and G.L. All authors have readand agreed to the published version of the manuscript.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.

Appendix A
In the following examples the e/n ratio is calculated for tiled regular polygons withdifferent geometry:
Example A1. Hexagonal tiling with k number of hexagons arranged in a line shown in Figure 2a:
e = 6 + (k − 1) × 5,
n = 6 + (k − 1) × 4,
When n  6, ne ≈ 45 .
Example A2. Hexagonal tiling where a and b are the number of horizontally and vertically tiled
hexagons, respectively:e=1e
3ab + 2a + 2b − 2
3 a+b+2= −.
(6ab + 4a − 2 + 4b),n = (a + 1) · (2b + 2) − 2, =2n2ab + 2b + 2a
2n
If a = 2b − 1, then


3
3
1
e
3
≈ − √ =1− √ .
n2 2 n
2nWhen n  1, ne ≈ 32 .
Example A3. Square tiling shown in Figure 2a where k is the number of square edges on each side:e = 2k (k + 1),√
n = (k + 1)2 ⇒ k = n − 1,

√
2 n−1
e
2k(k + 1)
2√=== 2− √ .
n(k + 1)(k + 1)nnWhen n  1, ne ≈ 2.
Example A4. Triangle tiling illustrated in Figure 2a where k is the number of tiles on each side:Cells 2022, 11, 272

8 of 9hie = 21 [3k + 3(1 + 3 + 5 + · · · + 2k + 1)] = 32 k2 + k = 32 k (k + 1),(k+1)·(k+2),
2

n = 1+2+···+k+1 =
e
3k
√6 .
n = k+2 ≈ 3 −2nen ≈ 3.

When n  1,In each example, as n increases, the e/n ratio approaches the value of dmax
2 .
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