A group theoretic approach to model comparison with
simplicial representations
Sean T. Vittadello222School of Mathematics and Statistics and School of
BioSciences, The University of Melbourne, Parkville, Victoria 3010,
Australia,111Corresponding author<EMAIL_ADDRESS>and Michael
P.H. Stumpf1

The complexity of biological systems, and the increasingly large amount of
associated experimental data, necessitates that we develop mathematical models
to further our understanding of these systems. Because biological systems are
generally not well understood, most mathematical models of these systems are
based on experimental data, resulting in a seemingly heterogeneous collection
of models that ostensibly represent the same system. To understand the system
we therefore need to understand how the different models are related to each
other, with a view to obtaining a unified mathematical description. This goal
is complicated by the fact that a number of distinct mathematical formalisms
may be employed to represent the same system, making direct comparison of the
models very difficult. A methodology for comparing mathematical models based
on their underlying conceptual structure is therefore required. In previous
work we developed an appropriate framework for model comparison where we
represent models, specifically the conceptual structure of the models, as
labelled simplicial complexes and compare them with the two general
methodologies of comparison by distance and comparison by equivalence. In this
article we continue the development of our model comparison methodology in two
directions. First, we present a rigorous and automatable methodology for the
core process of comparison by equivalence, namely determining the vertices in
a simplicial representation, corresponding to model components, that are
conceptually related and the identification of these vertices via simplicial
operations. Our methodology is based on considerations of vertex symmetry in
the simplicial representation, for which we develop the required mathematical
theory of group actions on simplicial complexes. This methodology greatly
simplifies and expedites the process of determining model equivalence. Second,
we provide an alternative mathematical framework for our model-comparison
methodology by representing models as groups, which allows for the direct
application of group-theoretic techniques within our model-comparison
methodology.

††footnotetext: Key words and phrases: Model comparison, model similarity,
model equivalence, simplicial complex, group action, orbit space

##  1 Introduction

Describing biological systems mathematically remains a challenge. We can
collect data at unprecedented scales [1, 2, 3, 4], but we still lack the
models required to integrate the often heterogeneous data into a single
analytical and predictive framework. Mathematical models can achieve this, but
also much more. They are of fundamental importance in developing new knowledge
of these complex systems for multiple reasons: analysis and interpretation of
large data sets; simulations of the biological system, or a subsystem thereof,
to develop new hypotheses; and, guiding new experimental design [5, 6, 7, 8,
9, 10, 11, 12, 13].

Formally, a _model_ is an abstraction of an observable phenomenon [14]. Models
may assume many different forms, and here we consider two general classes,
conceptual models and mathematical models [9]. A _conceptual model_ is a
qualitative representation of the reference system under investigation,
consisting of the relevant concepts which correspond to observable objects or
phenomena, and the interconnections between these concepts. A conceptual model
is obtained by integrating our knowledge of a system into a descriptive
framework, with a range of possible levels of conceptual detail depending on
our knowledge of the system. For example, in a developmental-patterning system
two possible concepts are a _morphogen_ and _diffusion of the morphogen_ ,
which are dependent concepts in the system since the morphogen undergoes
diffusion. Some developmental-patterning systems may have a second morphogen,
which is unrelated to the first morphogen since they are independent concepts.
A _mathematical model_ is a quantitative representation of a conceptual model,
and hence a representation of the reference system, in the formalism of
mathematical concepts. Mathematical models are based on conceptual models
through specification of the relevant mathematical concepts and their
relationships. Mathematical models allow for the calculation of quantifiable
concepts of the system, for example the spatial distribution of the morphogen
concentration as a function of time.

Comparison of mathematical models based on their underlying conceptual
structure is of fundamental importance for understanding the corresponding
biological systems. Many aspects of complex biological systems are not well
understood, so the development of mathematical models of such systems
necessarily follows an inductive process employing experimental data which
provides limited constraint for the range of possible conceptual models,
resulting in non-uniqueness of the mathematical models [15, 16]. While
mathematical models make conceptual models more specific and precise, such
specificity may provide room for many alternative mathematical
representations. Indeed, there are many possible formulations of a
quantitative mathematical model from a qualitative conceptual model, depending
on the particular quantitative description required, and alternative
mathematical models can appear very distinct and unrelated since the
underlying concepts may be obfuscated by the mathematical formalism or may be
interpretable in different ways. Further, while the mathematical model is
intended to be in direct correspondence with the conceptual model, and
therefore the reference system, the particular formalism of the mathematical
model introduces many implicit connections between the mathematical concepts
in the mathematical model. Some of these interconnections may not be intended,
and may be inconsistent with the underlying conceptual model; therefore the
mathematical model may not correspond directly to the conceptual model. It is
therefore essential to consider carefully whether the underlying conceptual
model of the mathematical model represents the reference system. Indeed, since
the mathematical formalism may introduce conceptual interconnections unrelated
to the original system, care must be taken to ensure that the model is not a
weak or incorrect representation of the system of interest.

Because of the possible multiplicity, different mathematical formalisms, and
misrepresentation of mathematical models associated with a biological system,
a general methodology for model comparison is required in order to further
understanding of the system underlying the different types of models [17, 18].
Model comparison is therefore of fundamental importance in understanding the
similarities and differences between models and between data sets, and
consequently in understanding biological systems. While there are many
approaches to compare the quantitative outputs of mathematical models, see for
example [19] and [20], such comparisons provide little insight into the
underlying conceptual structure of the mathematical models and hence of the
corresponding biological systems. Neither can we rely on model selection [21]
or robustness analysis, as this also relies on a comparison of the outputs of
different models. These approaches cannot be used _a priori_ to determine
conceptual similarities, equivalences, or, indeed, differences between
mathematical models.

Model comparison can simplify the landscape of mathematical modelling by
organising models into categories of models that are closely related. For
example, models in the same category may be incremental variations of each
other, but share some fundamental qualitative characteristic. We are
interested in this fundamental level of agreement: the conceptual basis of a
model. Or alternatively, the set of design principles [22, 23] shared by all
models in the same category. The mathematical models are merely tools for
quantitative computations, and the mathematical formalism can in fact be a
hindrance to the model comparison at the conceptual level.

In Vittadello and Stumpf [24] we present a framework for model comparison
where models are represented as labelled simplicial complexes. Models are then
compared in terms of their simplicial representations using two general
methodologies: the first is comparison by distance, where the difference
between models is measured in terms of the differences between the
corresponding simplicial representations; the second is comparison by
equivalence, where we identify any equivalences between the components of the
simplicial representations of different models, and then employ particular
operations on the complexes to transform one into the other — such
equivalence, when it exists, reveals common underlying characteristics of the
models. Our methodology for model comparison allows for any model to be
represented and compared within our framework, irrespective of formalism,
granularity, and level of abstraction. To construct the simplicial
representation of a model, we must first define the components of the model
that we would like to represent, and for this we can fall back on relevant
domain expertise. Depending on the level of detail required, the simplicial
representation of a model may not be unique. One main outcome of our model-
comparison methodology is for models of developmental-pattern formation [25,
26], where we demonstrate that the Turing-pattern activator-inhibitor model is
equivalent to the positional-information annihilation model from a significant
conceptual perspective.

In this article we continue the development of our methodology for model
comparison [24] in two respects. First, we develop a rigorous and automatable
methodology for the main process when comparing models with respect to
equivalence: the determination of the vertices that are conceptually related
in a simplicial representation and the subsequent identification of these
vertices via simplicial operations. We say that vertices are conceptually
related when their labels, corresponding to model components, are conceptually
related. Without automatability, establishing which vertices are conceptually
related in a simplicial representation is challenging if not unfeasible, even
for relatively small models. This methodology is based on considerations of
vertex symmetry in the simplicial representation, and we establish the
necessary mathematical theory for group actions on simplicial complexes. The
methodology provides a major step toward simplifying and facilitating the
process of determining model equivalence, which in turn can assist in the
unification of models for a biological system and the discovery of design
principles for synthetic biology. Second, we present a concise description of
an alternative mathematical framework for our model-comparison methodology
where we represent models as groups. This allows for the direct application of
group-theoretic techniques within our model-comparison methodology, in
addition to the current algebraic-topological techniques associated with
simplicial complexes.

The remainder of this article is organised as follows. In Section 2 we provide
an overview of the required background material: Subsection 2.1 outlines the
required background in algebraic topology; we then describe our methodology
for model comparison by equivalence in Subsection 2.2. We present and discuss
our new results in Section 3. In particular, in Subsection 3.1 we provide a
descriptive overview of our approach for identifying equivalent components in
simplicial representations through group actions, and then develop this
mathematical theory in Subsection 3.2; we present a step-by-step methodology
for determining model equivalence in Subsection 3.3, which employs our
automatable theory from Subsection 3.2; we then apply our methodology to a
collection of Turing-pattern and positional-information models (see the
Appendix in Section 5 for the model details) in Subsection 3.4; in Subsection
3.5 we give an incisive description of an alternative mathematical framework
for our model-comparison methodology by representing models as groups, which
we call $G$-representations. Finally, in Section 4, we conclude with a
discussion of the utility of our new theory for comparing models.

##  2 Background

In this section we provide a brief and informal description of the necessary
background in simplicial algebraic topology; further details are available
from standard references [27, 28, 29]. We also discuss our methodology for
model comparison by equivalence as described in Vittadello and Stumpf [24],
and developed further in this article.

###  2.1 Algebraic topology

Geometrically a _$p$ -simplex_ is a generalisation of a filled triangle to an
arbitrary dimension $p$, whereby a point is a 0-simplex, a line segment is a
1-simplex, a filled triangle is a 2-simplex, a filled tetrahedron is a
3-simplex, and so forth. We may regard 0-simplices as vertices and 1-simplices
as edges, so that we consider a simple graph as a collection of 0-simplices
and 1-simplices. A _$k$ -face_ of a simplex is a $k$-dimensional subsimplex,
and the 0-faces (or vertices) of a simplex _span_ the simplex. If the simplex
$\tau$ is a face of the simplex $\sigma$ then we write $\tau\leq\sigma$. A
face $\tau$ of a simplex $\sigma$ is _proper_ if $\tau\neq\sigma$.

A _simplicial complex_ is a generalisation of a simple graph, whereby
simplicies of dimension higher than one represent higher-dimensional
interactions between the vertices. Specifically, a simplicial complex $K$
consists of a set of simplices such that: if $\sigma$ is a simplex in $K$ then
every face of $\sigma$ is also in $K$; and, the nonempty intersection of any
two simplices in $K$ is a simplex in $K$. We denote the set of vertices of $K$
as $\operatorname{Vert}(K)$. A _simplicial subcomplex_ $L$ of a simplicial
complex $K$ is a collection $L\subseteq K$ that is also a simplicial complex.
A simplicial subcomplex $L$ of a simplicial complex $K$ is a _full subcomplex_
if every simplex of $K$ that has all its vertices in $L$ is also a simplex of
$L$. The _$k$ -skeleton_ of a simplicial complex $K$ is the subcomplex
$K^{(k)}$ consisting of the simplices in $K$ with dimension at most $k$. In
particular, the 0-skeleton $K^{(0)}$ is the set of vertices and the 1-skeleton
$K^{(1)}$ is the _underlying graph_ of the simplicial complex $K$.

In some contexts it is useful to define the _empty simplex_ , denoted
$\emptyset$, as a $(-1)$-simplex, and then the corresponding simplicial
complex is the _empty simplicial complex_ , denoted $\\{\emptyset\\}$, which
has dimension $-1$. In this work we exclude the empty simplex, as it is not
required. We include, however, the _void simplicial complex_ , which is the
simplicial complex consisting of the empty collection of sets. We denote the
void simplicial complex by $\emptyset$, which has the assigned dimension of
$-\infty$. Further details on these degenerate cases are found in [30, Page 8,
Remark 2.3].

We shall move freely between the interpretations of simplicial complexes as
both geometric and abstract objects, since they are equivalent and both
contain the same combinatorial information; and we have no need for the
topological aspects of geometric simplicial complexes. Notationally, if
$v_{1},v_{2},\ldots,v_{n}$ are vertices that span a simplex $K$ then we denote
$K$ as $\\{v_{1},v_{2},\ldots,v_{n}\\}$ when we want to emphasise the spanning
vertices. Note that vertices are therefore denoted as both $v_{i}$ and
$\\{v_{i}\\}$.

###  2.2 Model comparison by equivalence

We refer to a specific conceptual detail of a given model as a _component_ of
the model. Every model consists of a finite number of components and the
conceptual interconnections between these components. For example, a Turing-
pattern system of reaction-diffusion equations is a model of a developmental-
patterning process, and includes components such as morphogens, diffusion,
reactions, boundary conditions, and a morphogen gradient, along with the
interconnections between particular components such as a morphogen and its
diffusion.

To compare a collection of models we first determine the set of all model
components, at the required level of conceptual detail, associated with the
models in the collection. The model components are the scientific concepts
represented within the model, interpreted with knowledge of the relevant
scientific domains.

######  Definition 2.1 (Model components, generated model).

Let ${\mathcal{C}}$ be the set of all required components from the collection
of models under consideration for comparison. We say that each model in the
collection is _generated_ by a subset of components from ${\mathcal{C}}$.

We represent each model in the collection for comparison as a labelled
simplicial complex, which we call a _simplicial representation_ , where each
model component is represented as a labelled 0-simplex. The label associated
with a 0-simplex corresponds to the relevant concept in the model, and for
simplicity and ease of automation could have a numerical label:

######  Notation 2.2 (Representations of model components).

Let $\operatorname{Ord}\colon{\mathcal{C}}\to\\{1,2,\ldots,|{\mathcal{C}}|\\}$
be a bijection that specifies an arbitrary order for the categorical data
elements in ${\mathcal{C}}$. We represent each model component from
${\mathcal{C}}$ as a 0-simplex that is labelled with the name of the model
component or the corresponding positive integer under $\operatorname{Ord}$.

######  Definition 2.3 (Simplicial representation).

Let ${\mathcal{C}}$ be the set of all required components from the collection
of models under consideration for comparison. To each model generated by a
subset of components from ${\mathcal{C}}$ we associate a labelled simplicial
complex, which we call a _simplicial representation_ , where the labelled
0-simplices correspond to components of the model, and the one- and higher-
dimensional simplices correspond to the conceptual interconnections between
the model components as determined by the model.

The ordering function $\operatorname{Ord}$ provides for efficient labelling of
simplices. For example, a 2-simplex can be labelled as $\\{1,2,3\\}$, where
the 0-simplices are $\\{1\\}$, $\\{2\\}$, and $\\{3\\}$, and the 1-simplices
are $\\{1,2\\}$, $\\{1,3\\}$, and $\\{2,3\\}$.

For a particular level of component detail of a given model we associate a
simplicial representation consisting of labelled 0-simplices, which represent
the model components, along with the one- and higher-dimensional simplices
that represent the conceptual interconnections between the components: the
dyadic interactions are 1-simplices, the triadic interactions are 2-simplices,
and so forth as required. The labelling of the 0-simplices induces a labelling
of the higher-dimensional simplices through their spanning 0-simplices. Since
the models for comparison are generated by the same set of components
${\mathcal{C}}$, a particular labelled simplex representing specific
components and interconnections can be identified within different simplicial
representations associated with the models.

Our definition of the equivalence of two models is in terms of a formal
equivalence between the two corresponding simplicial representations. The
formal equivalence is based on partial operations on the relevant set of
simplicial representations, which we make explicit here.

######  Definition 2.4 (Partial operation on a set).

Let ${\mathcal{X}}$ be a set. A _partial operation_ on ${\mathcal{X}}$ is a
partial unary function $f\colon{\mathcal{X}}\rightharpoonup{\mathcal{X}}$.
Denoting the domain of definition of $f$ as
${\mathcal{D}}\subseteq{\mathcal{X}}$, we say that $f$ is _invertible_ if
there exists a partial operation
$g\colon{\mathcal{X}}\rightharpoonup{\mathcal{X}}$ such that $(g\circ
f)|_{{\mathcal{D}}}=\operatorname{id}_{{\mathcal{D}}}$ and $(f\circ
g)|_{f({\mathcal{D}})}=\operatorname{id}_{f({\mathcal{D}})}$.

Note that the domain of definition of a partial operation may have cardinality
one.

######  Definition 2.5 (Equivalence of simplicial representations).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{S}}$ be a
collection of simplicial representations corresponding to models generated by
subsets from ${\mathcal{C}}$. Further, let ${\mathcal{O}}$ be a set of partial
operations on ${\mathcal{S}}$. Two simplicial representations $K$,
$L\in{\mathcal{S}}$ are _equivalent_ with respect to ${\mathcal{O}}$ if and
only if there exists a (possibly empty) sequence of invertible partial
operations $(f_{i})_{i=0}^{n}$ in ${\mathcal{O}}$ such that
$f_{n}\circ\cdots\circ f_{1}\circ f_{0}(K)=L$.

######  Remark 2.6.

Note that the equivalence of simplicial representations is always relative to
the set ${\mathcal{O}}$ of partial operations on ${\mathcal{S}}$. The choice
of the set ${\mathcal{O}}$ determines which model components will be
considered as conceptually equivalent. We employ partial operations on
${\mathcal{S}}$ that we call _admissible_ , meaning that they only relate
model components that we consider to be conceptually equivalent. We discuss
this further below when we define the operations that we employ.

######  Remark 2.7.

It follows from our notion of equivalence of simplicial representations in
Definition 2.5 that the set
$\\{\,(K,L)\in{\mathcal{S}}\times{\mathcal{S}}\mid\text{$K$ and $L$ are
equivalent}\,\\}$ is an equivalence relation on ${\mathcal{S}}$.

######  Remark 2.8.

Note that the equivalence of two simplicial representations $K$,
$L\in{\mathcal{S}}$ can also be defined as follows: $K$ and $L$ are
_equivalent_ if and only if there exist two (possibly empty) sequences of
invertible partial operations $(f_{i})_{i=0}^{m}$ and $(g_{i})_{i=0}^{n}$ on
${\mathcal{S}}$ such that $f_{m}\circ\cdots\circ f_{1}\circ
f_{0}(K)=g_{n}\circ\cdots\circ g_{1}\circ g_{0}(L)$. In this work we use the
fact that this form of equivalence of simplicial representations implies the
form in Definition 2.5.

We previously defined five partial operations that we use to establish
equivalence of simplicial representations, namely: (Operation 1) Adjacent-
vertex identification, (Operation 2) Nonadjacent-vertex identification,
(Operation 3) Vertex split, (Operation 4) Inclusion, and (Operation 5) Vertex
substitution. These five partial operations are each induced by a map between
simplicial complexes, and are all invertible for suitable domains of
definition: an adjacent-vertex identification is mutually inverse with a
corresponding vertex split; a nonadjacent-vertex identification is mutually
inverse with an inclusion; and, a vertex substitution is mutually inverse with
another vertex substitution. From the perspective of Remark 2.8 we will only
explicitly consider Operations 1, 2, and 5, while Operations 3 and 4 will be
implicit.

For the equivalence of simplicial representations to be conceptually
meaningful we need to ensure that the employed partial operations are
themselves conceptually meaningful, or _admissible_ , meaning that the
operations preserve the conceptual representation of the general physical
system. Importantly, an established equivalence between models is based on the
components of the models that we choose to identify as equivalent, and
therefore the corresponding partial operations that we consider admissible.
The equivalence of two models is therefore determined by the formal
requirements in terms of partial operations, along with tight domain-specific
constraints.

We now state Operations 1, 2, and 5. Further details are in Vittadello and
Stumpf [24]. Recall that two distinct vertices in a simplicial complex are
_adjacent_ if they belong to the same simplex, and for a vertex $u$ in a
simplicial complex $K$ we denote the set of all vertices adjacent to $u$ as
$V_{K}(u)$, noting that $u\notin V_{K}(u)$. Further, let ${\mathcal{C}}$ be
the set of all components of models under consideration, and let $K$ and $L$
be two simplicial representations with labels from ${\mathcal{C}}$.

######  Definition 2.9 (Operation 1: Adjacent-vertex identification).

Let $u$ and $v$ be a pair of adjacent vertices in $K$ such that the following
all hold:

  * •

$V_{K}(u)\setminus\\{v\\}=V_{K}(v)\setminus\\{u\\}$.

  * •

For any nonempty subset $W\subseteq V_{K}(u)\setminus\\{v\\}$, the vertices
$W\cup\\{u\\}$ span a simplex in $K$ if and only if the vertices
$W\cup\\{v\\}$ span a simplex in $K$.

A simplicial map $\pi_{1}\colon K\to L$ is an _adjacent-vertex identification_
if $\pi_{1}$ is surjective, and is injective and label preserving on every
vertex except at the pair of vertices $u$ and $v$ which are mapped to a single
vertex $c\in L^{(0)}$.

######  Definition 2.10 (Operation 2: Nonadjacent-vertex identification).

Let $u$ and $v$ be a pair of nonadjacent vertices in $K$ such that the
following all hold:

  * •

$V_{K}(u)=V_{K}(v)$.

  * •

For any nonempty subset $W\subseteq V_{K}(u)$, the vertices $W\cup\\{u\\}$
span a simplex in $K$ if and only if the vertices $W\cup\\{v\\}$ span a
simplex in $K$.

A simplicial map $\pi_{2}\colon K\to L$ is a _nonadjacent-vertex
identification_ if $\pi_{2}$ is surjective, and is injective and label
preserving on every vertex except at the pair of vertices $u$ and $v$ that are
mapped to a single vertex $c\in L^{(0)}$.

######  Definition 2.11 (Operation 5: Vertex substitution).

A simplicial map $\pi_{5}\colon K\to L$ is a _vertex substitution_ if
$\pi_{5}$ is bijective and preserves all labels except for one whereby the
labelled vertex $u\in K^{(0)}$ is mapped to the labelled vertex $c\in
L^{(0)}$.

We may extend each partial operation to effect, for example, multiple vertex
identifications or substitutions by extending the assumptions in Operations 1,
2, and 5.

######  Remark 2.12.

We note here that our model-comparison methodology is designed to be flexible,
so that the partial operations can be chosen based on specific requirements.
Different operations may produce different model equivalences, however
operations that correspond to very weak conceptual similarity may yield
conceptual equivalences that are not scientifically meaningful.

The specific partial operations on simplicial representations that we employ
(vertex identifications and substitution) are chosen to represent our notion
of conceptual similarity for the models and underlying biological systems that
we consider here. Of primary interest for our work are the vertex-
identification operations, which identify vertices in a simplicial
representation corresponding to conceptually-related model components; the
identification of such vertices in a simplicial representation in a rigorous
and automatable manner is the subject of Section 3. Additional partial
operations may be useful for grouping models based on equivalence, however the
operations that we consider provide much flexibility while preserving
conceptual equivalence.

##  3 Results and discussion

Here we have two main objectives. The first is to develop an automatable
methodology for identifying equivalent components in models. The second is to
provide a representation of models as groups, as an alternative to simplicial
representations, with which models can be compared in a manner equivalent to
that with simplicial representations. We begin with an overview of our
approach for formalising the identification of equivalent model components
with group actions.

###  3.1 Overview

The most difficult aspect of establishing the equivalence of two simplicial
representations is in determining whether two adjacent or two nonadjacent
vertices in one simplicial representation can be identified conceptually with
a vertex in the other simplicial representation, either via a vertex
identification or vertex split operation. Identifiable vertices in a
simplicial representation must be in positions that are conceptually
equivalent, and in particular in symmetric positions within the corresponding
unlabelled simplicial complex. We can therefore employ group actions on
simplicial complexes to greatly simplify and expedite the process of
determining the existence of identifiable vertices, while also providing for
the process to be automated. The existence of a relevant nontrivial group
action then induces a vertex-identification operation.

Given a simplicial representation, our goal is to determine whether any pairs
of vertices are in symmetrical positions in the simplicial complex, and so are
candidates for a possible vertex-identification operation. We therefore need
to determine whether the simplicial complex has any symmetries of a certain
class. By a symmetry of a simplicial complex we mean a permutation on the set
of vertices of the complex that acts simplicially, so sends simplices to
simplices, and that sends the whole complex to itself. We are interested in a
class of symmetries, which we may call _exchange symmetries_ , of the complex
that exchange two vertices and leave the other vertices fixed, since these
symmetries reveal the vertices that are candidates for a possible vertex-
identification operation. From a set of exchange symmetries of the complex we
then obtain a new simplicial complex in which the exchangeable vertices are
identified by a vertex-identification operation; this process can be iterated
until a simplicial complex is obtained for which no exchange symmetries exist,
and only relabelling of the vertices need be considered by a vertex-
substitution operation.

###  3.2 Identification of equivalent model components through group actions

We begin by establishing the relevant symmetries of simplicial complexes,
which are described by a group action. Recall the definition of a permutation
of a set:

######  Definition 3.1 (Permutation, Transposition).

A _permutation_ of a set $X$ is a bijection from $X$ onto itself, and the set
of all permutations of $X$ is the symmetric group on $X$ which we denote by
$\operatorname{Sym}(X)$. A _transposition_ is a permutation that exchanges two
elements of $X$ and leaves the other elements fixed.

######  Notation 3.2.

We write permutations in both function notation and cycle notation, as is
convenient, and we compose permutations from right to left. In cycle notation
a transposition is a 2-cycle, written $(x\;y)$ for $x$, $y\in X$.

######  Remark 3.3.

Herein our convention is to view permutations in the active sense.

Formally we refer to a symmetry of a simplicial complex as an automorphism.
Recall the following standard definition:

######  Definition 3.4 (Automorphism of a simplicial complex, Automorphism
group).

An _automorphism_ , or symmetry, of a simplicial complex $K$ is a permutation
on the set of vertices $\operatorname{Vert}(K)$ that acts simplicially. The
_automorphism group_ of $K$ is the set of all automorphisms of $K$, denoted
$\operatorname{Aut}(K)$, which is isomorphic to a subgroup of
$\operatorname{Sym}\big{(}\operatorname{Vert}(K)\big{)}$.

Herein we denote the set of positive integers from 1 to $n$ as $[n]$.

######  Notation 3.5.

Let $\pi$ be an automorphism of a simplicial complex $K$, and denote
$\operatorname{Vert}(K)=\\{v_{i}\\}_{i=1}^{n}$. If $\sigma:=\\{v_{i}\\}_{i\in
I}$ is a simplex in $K$ for some subset $I\subseteq[n]$, then we denote the
action of $\pi$ on $\sigma$ by $\pi(\sigma)=\\{\pi(v_{i})\\}_{i\in I}$. We
will usually denote vertices as singleton sets of positive integers, in which
case $\operatorname{Vert}(K)=\big{\\{}\\{a_{i}\\}\big{\\}}_{i=1}^{n}$ where
each $a_{i}$ is a positive integer, and we then denote the action of $\pi$ on
$\sigma$ by $\pi(\sigma)=\big{\\{}\\{\pi(a_{i})\\}\big{\\}}_{i\in I}$.

The automorphisms of a simplicial complex are described by the action of a
group on the complex. Our definition of a (left) group action on a simplicial
complex is standard [31, Chapter III, Page 115].

######  Definition 3.6 (Group action).

Let $G$ be a group with identity $e$, and let $X$ be a set. An _action_ of the
group $G$ on $X$ is a map $\Theta\colon G\times X\to X$ such that:

  1. 1.

$\Theta(gh,x)=\Theta(g,\Theta(h,x))$ for all $g$, $h\in G$ and $x\in X$;

  2. 2.

$\Theta(e,x)=x$ for all $x\in X$.

If $K$ is a simplicial complex then, setting $X=K$, we also assume that:

  1. 3.

For each $g\in G$ the map $\theta_{g}\colon K\to K$, given by
$\theta_{g}(\sigma)=\Theta(g,\sigma)$ for $\sigma\in K$, acts simplicially on
$K$.

We say that each $g\in G$ acts simplicially on $K$, and each such action of a
group $G$ is called a _$G$ -action_.

######  Remark 3.7.

Equivalently, an action of the group $G$ on the set $X$ is a group
homomorphism $\Phi\colon G\to\operatorname{Aut}(X)$. In particular, recalling
that an automorphism of a simplicial complex $K$ is simplicial, a _simplicial
action_ of the group $G$ on the simplicial complex $K$ is a group homomorphism
$\Phi\colon G\to\operatorname{Aut}(K)$.

######  Notation 3.8.

For an action of the group $G$ on the set $X$, and for $g\in G$ and $x\in X$,
we denote $\Theta(g,x)=\Phi(g)(x)$ by $g\cdot x$.

While conditions of _regularity_ [31, Chapter III, Page 116, Definition 1.2]
are generally assumed for $G$-actions, such conditions are neither required
nor desirable for our purposes since we are interested in identifying labelled
simplices that, upon permutation of the spanning vertices, produce an
invariant simplicial representation. Indeed, a _regular_ simplicial action of
the group $G$ on the simplicial complex $K$ is a simplicial action that
satisfies the following condition for the action of each subgroup of $G$ [31,
Chapter III, Page 115, Statement (B)]: Condition (A) — if
$g_{1},g_{2},\ldots,g_{m}\in G$, and $\\{v_{1},v_{2},\ldots,v_{m}\\}$ and
$\\{g_{1}\cdot v_{1},g_{2}\cdot v_{2},\ldots,g_{m}\cdot v_{m}\\}$ are both
simplices of $K$, then there exists an element $g\in G$ such that $g\cdot
v_{i}=g_{i}\cdot v_{i}$ for all $i\in[m]$. Condition (A) implies the following
Condition (B) [31, Chapter III, Page 116, Statement (A′) and the following two
paragraphs]: Condition (B) — if $g\in G$, $v$ is a vertex in $K$, and $v$ and
$g\cdot v$ belong to the same simplex, then $v=g\cdot v$. Condition (B)
illustrates that regular actions fix vertices that are sent to the same
simplex. To the contrary, we require nonregular actions that allow for the
permutation of the vertices of a given simplex, as illustrated in the
following simple example.

######  Example 3.9.

Let $K$ be the 1-simplex $\big{\\{}\,\\{1\\},\\{2\\},\\{1,2\\}\,\big{\\}}$,
regarded as a simplicial representation. Since the vertices $\\{1\\}$ and
$\\{2\\}$ are adjacent and in symmetrical positions in the complex, they are
conceptually equivalent so may be identified by an adjacent-vertex
identification. While we can directly observe from $K$ that the two vertices
are conceptually equivalent, this is very difficult for more complicated
simplicial representations, and the use of group actions simplifies the
process. In this case, let $G=\big{\langle}(1\;2)\big{\rangle}$ be the
permutation group on the set $\\{1,2\\}$ generated by the transposition
$(1\;2)$, that is, $G=\big{\\{}e,(1\;2)\big{\\}}$. Then $G$ acts simplicially
on $K$ by permuting the vertices of $K$ (we discuss such actions in detail
below). Now, $\\{1,2\\}$ and $\big{\\{}e\cdot 1,(12)\cdot 2\big{\\}}=\\{1\\}$
are both simplices of $K$, however there is no $g\in G$ such that
$g\cdot\\{1,2\\}=\\{1\\}$. So $G$ does not satisfy Condition (A), and hence is
not a regular action. Such a group action is, however, a symmetry of interest
in our work here, since the permutation of the vertices of the simplex $K$
results in an invariant complex.

To reveal any conceptually-equivalent vertices in a simplicial representation
we require a specific class of simplicial automorphisms, which we call
exchange automorphisms (see [32, Section 3.2, Page 322, Definition 3.20]).

######  Definition 3.10 (Exchange automorphism, Exchangeable vertices).

Let $K$ be a simplicial complex, and let $u$, $v\in\operatorname{Vert}(K)$. An
_exchange automorphism_ for the vertices $u$ and $v$ is an automorphism
$\phi\in\operatorname{Aut}(K)$ such that $\phi(u)=v$, $\phi(v)=u$, and
$\phi(w)=w$ for all $w\in\operatorname{Vert}(K)\setminus\\{u,v\\}$. If an
exchange automorphism exists for vertices $u$ and $v$ then we say that $u$ and
$v$ are _exchangeable_ in $K$.

Note that if $K$ is a simplicial complex and $u$, $v\in\operatorname{Vert}(K)$
are exchangeable then the associated exchange automorphism on $K$ is unique.

######  Remark 3.11.

Let $K$ be a simplicial complex. The binary relation $\\{\,(u,v)\mid\text{$u$,
$v\in\operatorname{Vert}(K)$ are exchangeable}\,\\}$ over the set of
exchangeable vertices in $K$ is an equivalence relation. Reflexivity follows
from the identity automorphism on $K$, and symmetry follows from the
definition of exchangeable vertices. For transitivity we observe that if
$\phi\in\operatorname{Aut}(K)$ is an exchange automorphism for the vertices
$u$ and $v$, and $\psi\in\operatorname{Aut}(K)$ is an exchange automorphism
for the vertices $v$ and $w$, then
$\phi\circ\psi\circ\phi\in\operatorname{Aut}(K)$ is an exchange automorphism
for the vertices $u$ and $w$.

We employ group actions of a particular class:

######  Definition 3.12 (Group action by exchangeable vertices).

A _group action by exchangeable vertices_ on a simplicial complex $K$ consists
of an action of a group $G$ on $K$ such that $G=\langle M\rangle$ where:

  * •

$M\subseteq\operatorname{Sym}\big{(}\operatorname{Vert}(K)\big{)}$ is a set of
transpositions $(u\;v)$ where $u$ and $v$ are exchangeable in $K$; and,

  * •

the action of each $(u\;v)\in M$ is the exchange automorphism on $K$ that
exchanges $u$ and $v$.

Once identifiable vertices are found in a simplicial representation through a
group action, we can obtain a new simplicial complex, called the orbit space,
with these vertices identified as single vertices. The definition of the orbit
space of an action of a group on a simplicial complex follows [31, Chapter
III, Page 117, Paragraph 2].

######  Definition 3.13 (Orbit, Orbit space).

Let $K$ be a simplicial complex, and suppose that the group $G$ acts on $K$.
The _orbit_ , or _$G$ -orbit_, of an element $\sigma\in K$ is the set
$G\cdot\sigma:=\\{\,g\cdot\sigma\mid g\in G\,\\}$. The _orbit space_ , or
_quotient_ , of an action of a group $G$ on $K$, denoted $K/G$, is the set
consisting of the orbits $\operatorname{Vert}(K/G):=\\{\,G\cdot v\mid
v\in\operatorname{Vert}(K)\,\\}$, along with the finite subsets $\\{G\cdot
v_{i}\\}_{i=1}^{n}$ of $\operatorname{Vert}(K/G)$ for which
$\\{u_{i}\\}_{i=1}^{n}$ spans a simplex in $K$, where each $u_{i}$ is a
representative of $G\cdot v_{i}$, and the existence of such a simplex in $K$
is not required for all systems of representatives of the orbits $G\cdot
v_{i}$.

######  Example 3.14.

Let $K$ be the simplicial 2-complex
$\big{\\{}\\{1\\},\\{2\\},\\{3\\},\\{1,2\\},\\{1,3\\},\\{2,3\\},\\{1,2,3\\}\big{\\}}$,
which is geometrically a filled triangle. Let
$G:=\big{\langle}(1\;2)\big{\rangle}\cong{\mathbb{Z}}/2{\mathbb{Z}}$ be the
cyclic subgroup of $S_{3}$, the symmetric group of degree 3, generated by the
transposition $(1\;2)$. Then $g\cdot K=K$ for all $g\in G$. The action of $G$
on $K$ is illustrated geometrically in Figure 1(a), where $(1\;2)$ is a
reflection.

Figure 1: Geometric illustration of the action of the group $G$ on the
simplicial complex $K$: (a) The action of the transposition $(1\;2)$ on $K$;
(b) The orbit space $K/G$ is the 1-simplex with vertices
$\big{\\{}\\{3\\}\big{\\}}$ and $\big{\\{}\\{1\\},\\{2\\}\big{\\}}$. Geometric
illustration of the action of the group $H$ on the simplicial complex $K$: (c)
The action of the permutation $(1\;2\;3)$ on $K$; (d) The orbit space $K/H$ is
the 0-simplex $\big{\\{}\\{1\\},\\{2\\},\\{3\\}\big{\\}}$.

We have $G\cdot\\{1\\}=G\cdot\\{2\\}=\big{\\{}\\{1\\},\\{2\\}\big{\\}}$ and
$G\cdot\\{3\\}=\big{\\{}\\{3\\}\big{\\}}$, hence
$\operatorname{Vert}(K/G)=\big{\\{}G\cdot\\{1\\},G\cdot\\{3\\}\big{\\}}$.
Since $\\{1,3\\}$, or alternatively $\\{2,3\\}$, is a simplex in $K$,
$\big{\\{}G\cdot\\{1\\},G\cdot\\{3\\}\big{\\}}$ is a simplex in $K/G$.
Therefore
$K/G=\Big{\\{}G\cdot\\{1\\},G\cdot\\{3\\},\big{\\{}G\cdot\\{1\\},G\cdot\\{3\\}\big{\\}}\Big{\\}}$
is a 1-simplex, as illustrated in Figure 1(b), resulting from the
identification of the symmetric vertices $\\{1\\}$ and $\\{2\\}$ in $K$.

Now let
$H:=\big{\langle}(1\;2\;3)\big{\rangle}\cong{\mathbb{Z}}/3{\mathbb{Z}}$ be the
cyclic subgroup of $S_{3}$ generated by the cyclic permutation $(1\;2\;3)$.
Then $h\cdot K=K$ for all $h\in H$. The action of $H$ on $K$ is illustrated
geometrically in Figure 1(c), where $(1\;2\;3)$ is a counter-clockwise
rotation. We have
$H\cdot\\{1\\}=H\cdot\\{2\\}=H\cdot\\{3\\}=\big{\\{}\\{1\\},\\{2\\},\\{3\\}\big{\\}}$,
and it follows that $K/H=\big{\\{}H\cdot\\{1\\}\big{\\}}$ is a 0-simplex, as
illustrated in Figure 1(d), resulting from the identification of the three
vertices in $K$.

It is a standard observation that the orbit space $K/G$ is a simplicial
complex [31, Chapter III, Page 117, Paragraph 2], however we provide a proof
for completeness.

######  Proposition 3.15.

Let $K$ be a simplicial complex, and let $G$ be a group action on $K$. Then
the orbit space $K/G$ is a simplicial complex with vertex set
$\operatorname{Vert}(K/G)$.

###### Proof.

By definition, $K/G$ consists of the set of vertices
$\operatorname{Vert}(K/G):=\\{\,G\cdot v\mid v\in\operatorname{Vert}(K)\,\\}$
along with the subsets $\\{G\cdot v_{i}\\}_{i\in I}$ of
$\operatorname{Vert}(K/G)$, for index set $I$, for which there exists
$(u_{i})_{i\in I}\in\prod_{i\in I}G\cdot v_{i}$ such that $\\{u_{i}\\}_{i\in
I}$ spans a simplex in $K$.

To show that $K/G$ is a simplicial complex we need to show that for each
subset $\\{G\cdot v_{i}\\}_{i\in I}$ of $\operatorname{Vert}(K/G)$ that spans
a simplex in $K/G$, every nonempty subset of $\\{G\cdot v_{i}\\}_{i\in I}$
also spans a simplex in $K/G$. So let $\\{G\cdot v_{j}\\}_{j\in J}$ be a
subset of $\\{G\cdot v_{i}\\}_{i\in I}\in K/G$, where $J\subseteq I$ is
nonempty. Since $\\{G\cdot v_{i}\\}_{i\in I}\in K/G$, there exists a simplex
$\\{u_{i}\\}_{i\in I}\in K$ with $(u_{i})_{i\in I}\in\prod_{i\in I}G\cdot
v_{i}$. Then $\\{u_{j}\\}_{j\in J}$ is a subsimplex of $\\{u_{i}\\}_{i\in I}$
in $K$, with $(u_{j})_{j\in J}\in\prod_{j\in J}G\cdot v_{j}$, hence $\\{G\cdot
v_{j}\\}_{j\in J}$ is a simplex in $K/G$, as required. ∎

We now show that there is a canonical map from a simplicial complex onto the
orbit space corresponding to a $G$-action. This simplicial map induces a
vertex-identification operation between the two simplicial complexes.

######  Proposition 3.16.

Let $K$ be a simplicial complex, and let $G$ be a group action on $K$. Then
the canonical vertex map $v\mapsto G\cdot v$ extends to a surjective
simplicial map $p\colon K\to K/G$.

###### Proof.

Let $\psi\colon\operatorname{Vert}(K)\to\operatorname{Vert}(K/G)$ be the
canonical vertex map such that $v\mapsto G\cdot v$. Define the map $p\colon
K\to K/G$ by $p\big{(}\\{v_{i}\\}_{i\in
I}\big{)}:=\big{\\{}\psi(v_{i})\big{\\}}_{i\in I}$ for a simplex
$\\{v_{i}\\}_{i\in I}\in K$, noting that the $\psi(v_{i})$ may be equal for
distinct indices in $I$. Then $p$ is simplicial since, letting $J\subseteq I$
be a subset of smallest cardinality such that
$\big{\\{}\psi(v_{j})\big{\\}}_{j\in J}=\big{\\{}\psi(v_{i})\big{\\}}_{i\in
I}$, we have that $\\{v_{j}\\}_{j\in J}$ is a simplex in $K$ and hence, by the
definition of $K/G$, $\big{\\{}\psi(v_{j})\big{\\}}_{j\in J}$ is a simplex in
$K/G$. Surjectivity of $p$ follows from the surjectivity of $\psi$, and $p$
extends $\psi$ since $p|_{\operatorname{Vert}(K)}=\psi$. ∎

######  Notation 3.17.

We refer to the canonical map $p\colon K\to K/G$ in Proposition 3.16 as the
_projection map_.

When we identify conceptually-equivalent vertices in a simplicial
representation $K$ to give an orbit space, we would expect that the orbit
space is isomorphic to a simplicial subcomplex of $K$ as a consequence of the
underlying symmetry identified with the group action. We now confirm that this
is indeed the case, beginning with two lemmas.

######  Lemma 3.18.

Let $G$ be a group acting on a simplicial complex $K$ by exchangeable
vertices, and let $u\in\operatorname{Vert}(K)$. If $v$, $w\in G\cdot u$ then
the transposition $(v\;w)$ is in $G$. In particular, $v$ and $w$ are
exchangeable in $K$.

###### Proof.

Since $v$, $w\in G\cdot u$ there exists $g\in G$ such that $g\cdot v=w$, where
$g=t_{n}t_{n-1}\cdots t_{2}t_{1}$ is a product of transpositions $t_{i}\in G$
for $i\in[n]$. Without loss of generality we may assume that there is no
proper subsequence of the sequence of transpositions $(t_{i})_{i=1}^{n}$ whose
product sends $v$ to $w$. In particular: $v$ is an element of $t_{1}$ only;
$w$ is an element of $t_{n}$ only; and the transpositions are mutually
disjoint unless they are successive in the sequence, whereby their
intersection is a singleton. We then have $t_{1}=(v\;x_{1})$,
$t_{i+1}=(x_{i}\;x_{i+1})$ for $i\in[n-2]$, and $t_{n}=(x_{n-1}\;w)$, for some
$\\{x_{i}\\}_{i=1}^{n-1}\subseteq\operatorname{Vert}(K)$. It follows that
$(v\;w)=t_{1}t_{2}\cdots t_{n-1}t_{n}t_{n-1}\cdots t_{2}t_{1}\in G$. ∎

######  Lemma 3.19.

Let $G$ be a group acting on a simplicial complex $K$ by exchangeable
vertices. Suppose that $\\{G\cdot v_{i}\\}_{i\in I}$ spans a simplex in $K/G$,
for some index set $I$, and $\\{u_{i}\\}_{i\in I}$ spans a simplex in $K$ with
$(u_{i})_{i\in I}\in\prod_{i\in I}G\cdot v_{i}$. Then for each $(w_{i})_{i\in
I}\in\prod_{i\in I}G\cdot v_{i}$ we have that $\\{w_{i}\\}_{i\in I}$ spans a
simplex in $K$, and there is a $g\in G$ such that, for each $i\in I$, $g\cdot
u_{i}=w_{i}$, $g\cdot w_{i}=u_{i}$, and $g\cdot v=v$ for all
$v\in\operatorname{Vert}(K)\setminus\big{(}\\{u_{i}\\}_{i\in
I}\cup\\{w_{i}\\}_{i\in I}\big{)}$.

###### Proof.

Suppose $(w_{i})_{i\in I}\in\prod_{i\in I}G\cdot v_{i}$. It follows from Lemma
3.18 that for each $i\in I$ the transposition $t_{i}:=(u_{i}\;w_{i})$ is in
$G$. So define $g\in G$ as the product $g:=\prod_{i\in I}t_{i}$, where the
order in which the $t_{i}$ are multiplied is unimportant since these group
elements mutually commute: indeed, two vertices from distinct $G$-orbits are
not exchangeable. Then $g\in G$ is the required group element, and it follows
that the action of $g$ maps the simplex spanned by $\\{u_{i}\\}_{i\in I}$ to
the simplex spanned by $\\{w_{i}\\}_{i\in I}$. ∎

######  Definition 3.20 (Fundamental domain).

Let $K$ be a simplicial complex with an action of the group $G$ on $K$. A
_fundamental domain_ for the action of $G$ on $K$ is a full subcomplex $L$ of
$K$ that intersects each vertex $G$-orbit exactly once.

Note that a fundamental domain is not necessarily unique, and the group action
transforms between the different fundamental domains of the complex. We now
show that a fundamental domain exists for a group action on a simplicial
complex by exchangeable vertices, and that the corresponding orbit space is
isomorphic to the fundamental domain. From this result one infers that all
fundamental domains are isomorphic for a given group action on a simplicial
complex by exchangeable vertices.

######  Theorem 3.21.

Let $G$ be a group acting on a simplicial complex $K$ by exchangeable
vertices. Then a fundamental domain exists for the action of $G$ on $K$, and
the orbit space $K/G$ is isomorphic to the fundamental domain.

###### Proof.

We first show that a fundamental domain exists for $G$ on $K$. Let
$\operatorname{Vert}(K/G):=\\{\,G\cdot v_{i}\,\\}_{i\in I}$ be the set of
distinct vertex orbits where $I$ is the index set, choose $(w_{i})_{i\in
I}\in\prod_{i\in I}G\cdot v_{i}$, and let $L$ be the full subcomplex of $K$
with vertex set $\\{w_{i}\\}_{i\in I}$. Then $L$ is a fundamental domain.

It remains to show that $K/G$ is isomorphic to $L$. Let $\lambda\colon K/G\to
L$ be the map such that if $\sigma$ is a simplex in $K/G$ spanned by the
vertices $\\{\,G\cdot v_{j}\,\\}_{j\in J}$, where $J\subseteq I$, then
$\lambda(\sigma)$ is the simplex in $L$ spanned by the vertices
$\\{w_{j}\\}_{j\in J}$. The map $\lambda$ is well defined, since if the
simplex $\sigma\in K/G$ is spanned by $\\{\,G\cdot v_{j}\,\\}_{j\in J}$ then
there exists a simplex in $K$ spanned by a set of vertices $\\{u_{j}\\}_{j\in
J}$ with $(u_{j})_{j\in J}\in\prod_{j\in J}G\cdot v_{j}$, so by Lemma 3.19 the
vertices $\\{w_{j}\\}_{j\in J}$ span a simplex in $K$, which is also in the
full subcomplex $L$. The map $\lambda$ is then a simplicial bijection, and
hence a simplicial isomorphism. ∎

Since the orbit space may have exchangeable vertices that were either not
accounted for or not present in the original complex, we can obtain the orbit
space of the orbit space, and so on for a finite number of steps until an
orbit space is obtained which has no further exchangeable vertices. To see
this more clearly, we now show how a group action on a simplicial complex by
exchangeable vertices relates to the full group action on the corresponding
orbit space by exchangeable vertices. Note that we say that a group action on
a simplicial complex by exchangeable vertices is a _full group action_ when
the group action gives all possible exchange automorphisms of the complex. We
first require a lemma.

######  Lemma 3.22.

Let $G$ be a group action on a simplicial complex $K$ by exchangeable
vertices, and let $t$ be a transposition, not necessarily in $G$, from the
full group action on $K$ by exchangeable vertices that acts to exchange the
vertices $u$, $w\in\operatorname{Vert}(K)$. Further, let $H$ be the full group
action on $K/G$ by exchangeable vertices, and let $\\{G\cdot v_{i}\\}_{i\in
I}$ be the set of distinct orbits in $K/G$ which partition
$\operatorname{Vert}(K)$. Then there exists a transposition $\overline{t}\in
H$ that acts to exchange the vertices $G\cdot v_{j}$, $G\cdot
v_{k}\in\operatorname{Vert}(K/G)$ where $j$, $k\in I$, $u\in G\cdot v_{j}$,
and $w\in G\cdot v_{k}$. Further, $\overline{t}$ is the identity transposition
if and only if $t\in G$.

###### Proof.

Suppose that $t\in\operatorname{Sym}\big{(}\operatorname{Vert}(K)\big{)}$
transposes the vertices $u$, $w\in\operatorname{Vert}(K)$, and that $(u,w)\in
G\cdot v_{j}\times G\cdot v_{k}$ for some $j$, $k\in I$. The permutation
$\pi_{t}$ on the vertices $\\{G\cdot v_{i}\\}_{i\in I}$ of $K/G$ which
transposes $G\cdot v_{j}$ and $G\cdot v_{k}$ and fixes all other vertices
induces a map $\phi_{t}\colon K/G\to K/G$ which sends a simplex $\tau\in K/G$
spanned by $\\{G\cdot v_{i}\\}_{i\in J}$ for some $J\subseteq I$ to the
simplex $\phi_{t}(\tau)$ spanned by $\big{\\{}\pi_{t}(G\cdot
v_{i})\big{\\}}_{i\in J}$ where $G\cdot v_{j}$ and $G\cdot v_{k}$ are
exchanged, if required.

To show that $\phi_{t}$ is well defined we need to establish that
$\phi_{t}(\tau)$ is a simplex in $K/G$. Since $\\{G\cdot v_{i}\\}_{i\in J}$
spans the simplex $\tau\in K/G$, there exists $(u_{i})_{i\in J}\in\prod_{i\in
J}G\cdot v_{i}$ such that $\\{u_{i}\\}_{i\in J}$ spans a simplex $\sigma\in K$
by the definition of $K/G$. By Lemma 3.19 we may assume without loss of
generality that $u_{j}=u$ if $j\in J$ and $u_{k}=w$ if $k\in J$. Then the
image of $\sigma$ under the action of $t$ is the simplex $t\cdot\sigma\in K$
spanned by $\\{t\cdot u_{i}\\}_{i\in J}$ where $u$ and $w$ are exchanged, if
required. Now, if $i\neq j$ or $k$ then $t\cdot u_{i}=u_{i}\in G\cdot
v_{i}=\pi_{t}(G\cdot v_{i})$, if $i=j$ then $t\cdot u_{i}=t\cdot
u_{j}=u_{k}\in\pi_{t}(G\cdot v_{j})$, and if $i=k$ then $t\cdot u_{i}=t\cdot
u_{k}=u_{j}\in\pi_{t}(G\cdot v_{k})$. It follows that the simplex
$t\cdot\sigma\in K$ satisfies $(t\cdot u_{i})_{i\in J}\in\prod_{i\in
J}\pi_{t}(G\cdot v_{i})$, hence $\phi_{t}(\tau)$ is a simplex in $K/G$. So
$\phi_{t}$ is a well-defined simplicial map.

Since $\phi_{t}$ is self-inverse it is bijective and therefore an
automorphism. It follows that $\phi_{t}$ is an exchange automorphism that
exchanges the vertices $G\cdot v_{j}$ and $G\cdot v_{k}$, so the transposition
$\overline{t}:=(G\cdot v_{j}\;G\cdot v_{k})$ is in $H$.

Finally, $t\in G$ if and only if $u$ and $w$ are in the same $G$-orbit if and
only if $G\cdot v_{j}=G\cdot v_{k}$ if and only if $\overline{t}$ is the
identity transposition, where the first equivalence uses Lemma 3.18. ∎

######  Proposition 3.23.

Let $G$ and $F$ be two group actions on a simplicial complex $K$ by
exchangeable vertices, and let $H$ be the full group action on $K/G$ by
exchangeable vertices. Then there is a group homomorphism $\alpha\colon F\to
H$ such that $\alpha(t_{n}\cdots
t_{2}t_{1})=\overline{t}_{n}\cdots\overline{t}_{2}\overline{t}_{1}$ for each
element $t_{n}\cdots t_{2}t_{1}\in F$, where the $t_{i}$ for $i\in[n]$ are
generating transpositions in $F$, and $\ker(\alpha)=F\cap G$.

###### Proof.

We first show that $\alpha\colon F\to H$ is a well-defined map. Let $f\in F$,
and suppose that $f=t_{m}\cdots t_{2}t_{1}$ where the $t_{i}$, for $i\in[m]$,
are generating transpositions in $F$. By Lemma 3.22, each $\overline{t}_{i}$
for $i\in[m]$ is a transposition in $H$, so
$\overline{f}:=\overline{t}_{m}\cdots\overline{t}_{2}\overline{t}_{1}\in H$.
Now, if we also have that $f=s_{n}\cdots s_{2}s_{1}$ where the $s_{i}$, for
$i\in[n]$, are generating transpositions in $F$, then the corresponding
permutations of $\operatorname{Vert}(K/G)$, namely
$\overline{t}_{m}\cdots\overline{t}_{2}\overline{t}_{1}$ and
$\overline{s}_{n}\cdots\overline{s}_{2}\overline{s}_{1}$, are equal. It
follows that $\alpha$ is well defined.

To show that $\alpha$ is a homomorphism, let $t_{m}\cdots t_{2}t_{1}$ and
$s_{n}\cdots s_{2}s_{1}$ be two permutations in $F$, in terms of the
generating transpositions. Then $\alpha\big{(}(t_{m}\cdots
t_{2}t_{1})(s_{n}\cdots
s_{2}s_{1})\big{)}=(\overline{t}_{m}\cdots\overline{t}_{2}\overline{t}_{1})(\overline{s}_{n}\cdots\overline{s}_{2}\overline{s}_{1})=\alpha(t_{m}\cdots
t_{2}t_{1})\alpha(s_{n}\cdots s_{2}s_{1})$, and the result follows.

It remains to show that the kernel of $\alpha$ is equal to $F\cap G$. For
this, let $\\{G\cdot v_{i}\\}_{i\in I}$ be the set of distinct vertices in
$K/G$, which partition $\operatorname{Vert}(K)$. Suppose that
$f\in\ker(\alpha)$, so that $\alpha(f)=\overline{f}$ is the identity in $H$.
Then $\overline{f}$ fixes each vertex $G\cdot v_{i}$ of $K/G$, so the
definition of $\overline{f}$ implies that $f\cdot(G\cdot v_{i})\subseteq
G\cdot v_{i}$ for $i\in I$. We can write $f$ as a product of disjoint
permutations, $f=\prod_{i\in I}p_{i}$, where each such permutation $p_{i}$
permutes only elements in $G\cdot v_{i}$ with the same action as $f$ on
$G\cdot v_{i}$, and $p_{i}$ fixes all elements in
$\operatorname{Vert}(K)\setminus G\cdot v_{i}$. Then each $p_{i}$ can be
expressed as a product of transpositions in $G$, hence $f\in G$. It follows
that $\ker(\alpha)\subseteq F\cap G$. Conversely, if $f\in F\cap G$ then,
since $f\in G$, $\alpha(f)=\overline{f}\in H$ fixes the $G$-orbits of
$\operatorname{Vert}(K)$, which are the vertices of $K/G$, so $\alpha(f)$ is
the identity in $H$, hence $f\in\ker(\alpha)$. Therefore $F\cap
G\subseteq\ker(\alpha)$. ∎

In Subsection 3.3 we employ the theory developed here to provide a methodology
for determining model equivalence.

###  3.3 Methodology for determining model equivalence

To compare the simplicial representations corresponding to two models, with
respect to equivalence, we can compare the two corresponding orbit spaces that
have no further exchangeable vertices, which we refer to as the _final_ orbit
spaces. These orbit spaces, which may be smaller than the original simplicial
complexes, allow for the identification of subsets of equivalent components
within each model, and in turn the equivalence of components between the two
models.

Employing the theory in Subsection 3.2 for identifying equivalent components
within simplicial representations, we now provide a general methodology for
determining model equivalence in terms of Operations 1, 2, and 5. There is not
necessarily a unique order with which Operations 1, 2, and 5 should be applied
to a simplicial representation. Given two simplicial representations, however,
the easiest approach is to first reduce the simplicial representations by
applying vertex identifications (Operations 1 and 2) when possible, and
finally relabelling one of the complexes by a vertex substitution (Operation
5). We will assume that all possible vertex identifications for a given
simplicial representation are applied simultaneously. We could apply a proper
subset of the possible vertex identifications to the simplicial
representation, and then apply the remaining identifications to the
corresponding orbit space (see Proposition 3.23), however applying them all at
once minimises the number of required simplicial maps.

The following steps describe the process of finding model equivalences using
simplicial representations. For the purpose of automation on computer we can
relabel the vertices of the simplicial representations, which correspond to
model components, with positive integers so that the simplices of the
simplicial representations are subsets of these positive integers. Assume that
$K$ and $L$ are two simplicial representations corresponding to two models.

  1. Step 1.

Find all pairs of exchangeable vertices of $K$ and $L$: We describe the
process for $K$, and the process for $L$ is similar. Two vertices $u$ and $v$
in $\operatorname{Vert}(K)$ are _exchangeable_ if exchanging the labels of the
two vertices leaves $K$ unchanged (Definition 3.10). All exchangeable vertices
of $K$ can be found by exchanging all pairs of vertices of $K$ in succession,
however this process can be made more efficient in a number of ways: for
example, exchangeable vertices must have the same number of adjacencies, so
two vertices with different numbers of adjacencies are not exchangeable. Given
a pair of exchangeable vertices $u$ and $v$ in $\operatorname{Vert}(K)$ there
is a corresponding transposition
$(u\;v)\in\operatorname{Sym}\big{(}\operatorname{Vert}(K)\big{)}$, and the set
of all such transpositions corresponding to exchangeable vertices in $K$
generates a subgroup $G_{1}$ of
$\operatorname{Sym}\big{(}\operatorname{Vert}(K)\big{)}$. We then have an
action of $G_{1}$ on $K$ by automorphisms, from which we construct the orbit
space $K/G_{1}$ with vertices the $G_{1}$-orbits of the vertices in $K$
(Definition 3.13). If $K$ has no exchangeable vertices then $G_{1}$ is the
trivial subgroup which acts as the identity automorphism of $K$, so we can
then take $K/G_{1}$ to be $K$ since these complexes are isomorphic.

A transposition $(u\;v)\in G_{1}$ acts as an exchange automorphism of $K$ that
exchanges the vertices $u$, $v\in K$, and induces a corresponding
adjacent-/nonadjacent-vertex identification operation in terms of a surjective
simplicial map $p\colon K\to K/\big{\langle}(u\;v)\big{\rangle}$ which sends
both $u$ and $v$ to a single vertex in $K/\big{\langle}(u\;v)\big{\rangle}$
and fixes all other vertices in $K$. Note that $p$ is the projection map
(Proposition 3.16). Applying all permutations in $G_{1}$ simultaneously
corresponds to $G_{1}$ acting on $K$, and therefore induces a finite number of
vertex-identification operations in terms of the projection map $p_{1}\colon
K\to K/G_{1}$.

  2. Step 2.

Find the sequences of orbit spaces corresponding to $K$ and $L$: Again, we
describe the process for $K$, and it is similar for $L$. Finding the
exchangeable vertices of $K/G_{1}$ follows a process analagous to that for $K$
in Step 1, except the vertices of $K/G_{1}$ are subsets of vertices of $K$.
Given a pair of exchangeable vertices $u$ and $v$ in
$\operatorname{Vert}(K/G_{1})$ there is a corresponding transposition
$(u\;v)\in\operatorname{Sym}\big{(}\operatorname{Vert}(K/G_{1})\big{)}$, and
the set of all such transpositions corresponding to exchangeable vertices in
$K/G_{1}$ generates a subgroup $G_{2}$ of
$\operatorname{Sym}\big{(}\operatorname{Vert}(K/G_{1})\big{)}$. We then have
an action of $G_{2}$ on $K/G_{1}$ by automorphisms, from which we construct
the orbit space $(K/G_{1})/G_{2}$ with vertices the $G_{2}$-orbits of the
vertices in $K/G_{1}$. The action of $G_{2}$ on $K/G_{1}$ induces a finite
number of vertex-identification operations in terms of the projection map
$p_{2}\colon K/G_{1}\to(K/G_{1})/G_{2}$.

Continuing in this manner until we obtain an orbit space with no exchangeable
vertices, namely the final orbit space, results in the following sequence of
simplicial complexes and projection maps $p_{i}$ representing vertex
identifications,

$K\xrightarrow{\enspace p_{1}\enspace}K/G_{1}\xlongrightarrow{\enspace
p_{2}\enspace}(K/G_{1})/G_{2}\xlongrightarrow{\enspace
p_{3}\enspace}\cdots\xlongrightarrow{\enspace
p_{m}\enspace}\Big{(}\cdots\big{(}(K/G_{1})/G_{2}\big{)}\cdots\Big{)}/G_{m},$
(1)

where the final orbit space in the sequence, which we denote by $\widehat{K}$,
has no exchangeable vertices. Similarly, for the simplicial representation $L$
we have a sequence of subgroups $H_{i}$ generated by transpositions giving the
following sequence of simplicial complexes and simplicial maps $q_{i}$
representing vertex identifications,

$L\xrightarrow{\enspace q_{1}\enspace}L/H_{1}\xrightarrow{\enspace
q_{2}\enspace}(L/H_{1})/H_{2}\xrightarrow{\enspace
q_{3}\enspace}\cdots\xrightarrow{\enspace
q_{n}\enspace}\Big{(}\cdots\big{(}(L/H_{1})/H_{2}\big{)}\cdots\Big{)}/H_{n},$
(2)

where the final orbit space in the sequence, which we denote by $\widehat{L}$,
has no exchangeable vertices.

Note that if any complex from the sequence in Equation (1) is isomorphic to a
complex from the sequence in Equation (2), as unlabelled simplicial complexes,
then $\widehat{K}$ and $\widehat{L}$ are isomorphic as unlabelled simplicial
complexes. Therefore, it suffices to determine whether or not $\widehat{K}$
and $\widehat{L}$ are isomorphic, which are the simplest complexes in their
respective sequences.

With regard to the sequences in Equations (1) and (2) note that the number of
vertices in an orbit space is reduced, relative to the previous complex in the
sequence, by the number of distinct and nontrivial transpositions in the
associated group action. This observation provides a measure of the reduction
in the number of possible isomorphisms between the orbit spaces of each
simplicial complex.

  3. Step 3.

Determine all isomorphisms of $\widehat{K}$ and $\widehat{L}$: An isomorphism
between $\widehat{K}$ and $\widehat{L}$ is a bijective simplicial map from
$\widehat{K}$ onto $\widehat{L}$, that is, a bijective vertex map from
$\operatorname{Vert}(\widehat{K})$ onto $\operatorname{Vert}(\widehat{L})$
that sends a simplex in $\widehat{K}$ to a simplex in $\widehat{L}$. We begin
by finding all bijective vertex maps from $\operatorname{Vert}(\widehat{K})$
onto $\operatorname{Vert}(\widehat{L})$, noting that there may be none. If
there are multiple bijective vertex maps then we may reduce the number of
these maps by considering properties that must be preserved by a simplicial
isomorphism, such as the dimension of simplices and the number of edge
adjacencies of vertices. Further, if $\widehat{K}$ and $\widehat{L}$ each have
a vertex label containing the same model component then we would generally
ensure that the corresponding vertices must be identified by any bijective
vertex map, based on the requirement for conceptual equivalence. We could omit
the preservation of model components for the bijective vertex maps, however
this will likely only increase the number of bijective vertex maps to consider
without yielding any new meaningful equivalences between the model concepts.
It should not be the case that a vertex label in $\widehat{K}$ has model
components in common with two distinct vertices in $\widehat{L}$, as this
would indicate a conceptual inconsistency with at least one of the simplicial
representations.

Given a bijective vertex map, which we can describe by a correspondence
between the positive-integer labels of the vertices in $\widehat{K}$ and
$\widehat{L}$, we then try to extend it to a bijection from $\widehat{K}$ onto
$\widehat{L}$ by relabelling $\widehat{K}$ with the corresponding vertex
labels of $\widehat{L}$ in accordance with the bijective vertex maps. We then
check whether the relabelled complex $\widehat{K}$ is the same as
$\widehat{L}$, and if so then the bijective vertex map extends canonically to
a simplicial isomorphism, so that $\widehat{K}$ and $\widehat{L}$ are
isomorphic. If no simplicial isomorphism exists between $\widehat{K}$ and
$\widehat{L}$ then they are not isomorphic as complexes.

  4. Step 4.

Equivalence of $K$ and $L$: If $K$ and $L$ are equivalent then $\widehat{K}$
and $\widehat{L}$ are isomorphic; therefore, if $\widehat{K}$ and
$\widehat{L}$ are not isomorphic then $K$ and $L$ are not equivalent. If
$\widehat{K}$ and $\widehat{L}$ are isomorphic then for each simplicial
isomorphism we can consider, for each pair of vertices identified between
$\widehat{K}$ and $\widehat{L}$ via the isomorphism, whether the set of model
components associated with the vertex in $\widehat{K}$ is conceptually related
to the set of model components associated with the vertex in $\widehat{L}$.
Since there may be more than one isomorphism between $\widehat{K}$ and
$\widehat{L}$ we consider the vertex identifications for each such
isomorphism. Importantly, an isomorphism between $\widehat{K}$ and
$\widehat{L}$ can reveal novel conceptual relations between $K$ and $L$, and
therefore the corresponding models.

If there is an isomorphism between $\widehat{K}$ and $\widehat{L}$ that
provides a conceptually meaningful relationship between the concepts of the
two models then we can use a vertex-substitution operation induced by the
isomorphism to transform $\widehat{K}$ into $\widehat{L}$, and then we
conclude that $K$ and $L$, and hence the corresponding models, are equivalent.
Otherwise, if no isomorphism between $\widehat{K}$ and $\widehat{L}$ provides
a conceptually meaningful relationship between the concepts of the two models
then we conclude that $K$ and $L$, and hence the corresponding models, are not
equivalent.

  5. Step 5.

Restricting conceptual equivalence: A simplicial isomorphism between the final
orbit spaces $\widehat{K}$ and $\widehat{L}$ provides for the maximum
conceptual equivalence between the components of the two models. It may
happen, however, that we regard a certain equivalence for a model,
corresponding to a vertex identification, as invalid. In this case, we don’t
consider $\widehat{K}$ and $\widehat{L}$ as equivalent. There may, however, be
equivalent simplicial subcomplexes of $K$ and $L$ with respect to a restricted
set of vertex identifications corresponding to the valid conceptual
equivalences of model components. To determine whether equivalent subcomplexes
exist in this restricted sense, we can work backwards from $\widehat{K}$ and
$\widehat{L}$ to remove all of the invalid vertex identifications, while
retaining the isomorphic correspondence. If we can remove corresponding vertex
identifications stepwise from both $\widehat{K}$ and $\widehat{L}$, where each
vertex identification is either adjacent for both complexes or nonadjacent for
both complexes, and arrive at isomorphic complexes with only valid
equivalences between model components then $K$ and $L$ are equivalent, and
otherwise not.

######  Remark 3.24.

Here we briefly discuss some computational aspects of our model comparison
methodology.

  1. 1.

Orbit spaces: Constructing the orbit space of a simplicial complex $K$
requires first determining the exchange automorphisms, which only involves
exchanging pairs of vertices in $K$ and checking that $K$ is unchanged. Once
all relevant exchange automorphisms are found, the orbit space is constructed
from the vertex orbits and the known simplices in $K$. Isomorphism of
simplicial complexes: We need to determine whether the final orbit spaces are
isomorphic. In general, checking isomorphism of simplicial complexes is
computationally intensive, however our simplicial complexes are labelled (with
sets of model components, or corresponding positive integers) and we are
usually interested in isomorphisms that preserve model components. This
restriction reduces the number of possible isomorphisms to consider.
Automation: Our methodology for model comparison by equivalence consists of
three main parts. Given a model, the first part requires the construction of
the simplicial representation. This requires a translation of the model into
its underlying concepts and their interrelations. In general this is very
difficult to automate, as it requires the ability to identify the model
concepts and interrelations for a model with any formalism, as well as
equivalences between model concepts. The latter is also dependent on context
and individual requirements.

  2. 2.

The second part of our methodology, which we discuss in this article, is
completely automatable up to and including the determination of whether the
final orbit spaces are isomorphic. This provides for possible equivalences
between model concepts. The second part is the most challenging of all three
parts, since establishing whether vertices in a simplicial representation are
in symmetrical positions is difficult for even small models.

  3. 3.

The third part requires determining whether the possible equivalences between
model concepts are valid, which returns to part one, and is difficult to
automate in general.

###  3.4 Example

In this example we apply our model-comparison methodology to the two main
categories of models for developmental pattern formation, namely Turing-
pattern (TP) models and positional-information (PI) models. We have described
models from these two categories previously [24]: four TP models, namely
activator-inhibitor, substrate depletion, inhibition of an inhibition, and
modulation; five PI models, namely linear gradient, synthesis-diffusion-
degradation (SDD), opposing gradients, annihilation, and induction-contraction
(active modulation). We also discuss these models in the Appendix (Section 5).
Interestingly, we found that the TP activator-inhibitor model is equivalent to
the PI annihilation model from a significant conceptual perspective. This
finding was obtained by visual inspection of the two simplicial
representations, quite by chance as it is generally difficult to compare
simplicial representations visually. Employing the automatable aspect of our
methodology for model equivalence as described in this article, however,
provides simple and rigorous comparison of simplicial representations, and we
now compare all nine models for equivalence.

Of the nine models for developmental pattern formation we consider here,
consisting of four TP models and five PI models, three of the models have
final orbit spaces that are not isomorphic to the final orbits spaces of any
of the other eight models, so each of these three models is not equivalent to
any of the other eight models. These three models are PI induction-
contraction, TP inhibition of an inhibition, and TP modulation. We therefore
consider the remaining six models. Note that the labelling for these
simplicial representations, both as positive integers and model components, is
described in Vittadello and Stumpf [24]. For each of these simplicial
representations the 0- and 1-simplices are specified by the model, and the
higher-dimensional simplices are obtained by forming cliques, where possible,
incrementally in dimensions two and higher. For simplicity, in Figures 2 and 3
we show only the 1-skeletons of the simplicial representations.

The three PI models linear gradient, synthesis-diffusion-degradation, and
opposing gradients, all have final orbit spaces that are a 0-simplex, so
consist of a single vertex (Figure 2).

Figure 2: 1-skeletons of the simplicial representations and orbit spaces for
the (a) PI linear gradient, (b) PI synthesis-diffusion-degradation, and (c) PI
opposing gradients models.

The unlabelled final orbit spaces are therefore isomorphic, so we can compare
the corresponding sets of model components to determine model equivalence.

The orbit space for the PI linear gradient model has the label
$\big{\\{}\\{1,2,40,43\\},\\{6,7\\}\big{\\}}$, and the orbit space for the PI
SDD model has the label $\big{\\{}\\{1,2,40\\},\\{3,6\\}\big{\\}}$.
Equivalence of these two models requires that we consider the model components
in $\\{1,2,40,43\\}$ and $\\{6,7\\}$ as equivalent to the model components in
$\\{1,2,40\\}$ and $\\{3,6\\}$, respectively. That is, we need the following:
7 is equivalent to 3; since 1, 2, 40, and 43 are conceptually equivalent, any
one of the vertex identifications where 1 and 43 are identified as 1, or 2 and
43 are identified as 2, or 40 and 43 are identified as 40. The equivalence of
7 (Outflux 1) and 3 (Degradation 1) may be reasonable, since they both
represent removal of Morphogen 1 from the system. The identification of 43
(Global scale-invariance) with one of 1 (Morphogen 1), 2 (Diffusion 1), or 40
(Monotonic gradient), however, assumes that we are not considering the
property of scale-invariance of the morphogen gradient for any model for
comparison, which would depend on the required level of conceptual detail. For
our purposes, scale-invariance of the morphogen gradient is important for
establishing developmental patterning, so is a necessary conceptual detail in
the models, and we therefore conclude that the PI linear gradient model is not
equivalent to the PI SDD model.

While the PI linear gradient model and PI SDD model each have a single
morphogen, the PI opposing gradients model has two morphogens which we regard
as an important conceptual detail, so we do not consider the PI opposing
gradients model as equivalent to the PI linear gradient model or PI SDD model.
Indeed, the label for the orbit space of the PI opposing gradients model shows
that the model concepts associated with 1 (Morphogen 1), that is
$\big{\\{}\\{1,2\\},\\{3,6\\}\big{\\}}$, are identified with the model
concepts associated with 9 (Morphogen 2), that is
$\big{\\{}\\{9,10\\},\\{11,13\\}\big{\\}}$, to effectively reduce to a single
morphogen.

In Figure 3 we consider the PI annihilation model, the TP activator-inhibitor
model, and the TP substrate depletion model.

Figure 3: 1-skeletons of the simplicial representations and orbit spaces for
the (a) PI annihilation, (b) TP activator-inhibitor, and (c) TP substrate
depletion models.

These three models have isomorphic unlabelled final orbit spaces, so we can
compare the corresponding sets of model components to determine model
equivalence. Note that there are two possible isomorphisms between any two of
these three orbit spaces when unlabelled, however only one of these
isomorphisms preserves the labelling when the same label is in both complexes.
First we consider the equivalence of the PI annihilation model and the TP
activator-inhibitor model, which we established in Vittadello and Stumpf [24],
however now we employ the automatable approach using orbit spaces. We regard
the model components 6, 13, 26, and 40 as conceptually equivalent to the model
components $\\{5,27\\}$, 12, $\\{28,29\\}$, and 41, respectively. In
particular, we consider vertex identifications where 5 and 27 are identified
as 6, and 28 and 29 are identified as 26. We conclude that the PI annihilation
model and the TP activator-inhibitor model are equivalent from a significant
conceptual perspective. For further discussion of this example, including a
more detailed consideration of the required partial operations, see [24,
Subsubsection 3.2.4].

Now consider the equivalence of the TP activator-inhibitor model and the TP
substrate depletion model, which would require that the model components 9
(Morphogen 2), 10 (Diffusion 2), 11 (Degradation 2), 12 (Basal production 2),
and $\\{28,29\\}$ (Activation of Morphogen 2 by Morphogen 1, Inhibition of
Morphogen 1 by Morphogen 2) are conceptually equivalent to the model
components 18 (Substrate 1), 19 (Diffusion of Substrate 1), 20 (Degradation of
Substrate 1), and 21 (Basal production of Substrate 1), 34 (Depletion of
Substrate 1 by Morphogen 1), respectively. We consider that the agent
Morphogen 2 in the TP activator-inhibitor model is conceptually equivalent to
the agent Substrate 1 in the TP substrate depletion model. Further, the two
reactions given corresponding to 28 and 29 form an inhibitory cycle in the TP
activator-inhibitor model, which we consider to be conceptually equivalent to
the reaction corresponding to 20 in the TP substrate depletion model (compare
with the discussion in Vittadello and Stumpf [24] regarding the equivalence of
the PI annihilation model and the TP activator-inhibitor model). We conclude
that the TP activator-inhibitor model and the TP substrate depletion model are
equivalent from a significant conceptual perspective, which is not surprising
since the TP substrate depletion model is based very closely on the TP
activator-inhibitor model. Since the relation of model equivalence is
transitive, it also follows that the PI annihilation model is equivalent to
the TP substrate depletion model.

###  3.5 $G$-representations

In this subsection we provide an overview of an alternative mathematical
framework for our model-equivalence methodology by associating a group to a
simplicial representation. While the group-theoretic framework is closely
related to the simplicial-complex framework for model equivalence, the group-
theoretic approach provides an alternative mathematical perspective that
allows for the application of group-theoretic techniques for model comparison.
While the mathematical details of the two frameworks differ, the underlying
methodology is the same, so we provide only the minimal details required to
develop the group-theoretic framework for model equivalence.

We first describe a standard construction of a group from a given set.

######  Proposition 3.25.

Let $S$ be a set and let $2^{S}$ be the power set of $S$. With the closed
binary operation of symmetric difference, $\triangle\colon 2^{S}\times
2^{S}\to 2^{S}$, the pair $(2^{S},\triangle)$ is a group.

###### Proof.

Under the operation $\triangle$, $2^{S}$ is closed, the identity is the empty
set $\emptyset$, and each element is self-inverse. It remains to show
associativity of the operation, so let $A$, $B$, and $C\in 2^{S}$. We show
that $(A\triangle B)\triangle C=A\triangle(B\triangle C)$. Setting some
notation, for a set $D\in 2^{S}$ we denote by $\mathbbm{1}_{D}\colon
S\to\\{0,1\\}$ the characteristic function of $D$ such that, for $x\in S$,
$x\mapsto 1$ if and only if $x\in D$. Further, denote by $\oplus$ the
associative operation such that for any two characteristic functions of sets
$D$, $E\in 2^{S}$ we set
$(\mathbbm{1}_{D}\oplus\mathbbm{1}_{E})(x):=\mathbbm{1}_{D}(x)+\mathbbm{1}_{E}(x)\pmod{2}$.
Associativity of $\triangle$ then follows from

$\mathbbm{1}_{(A\triangle B)\triangle C}=\mathbbm{1}_{A\triangle
B}\oplus\mathbbm{1}_{C}=(\mathbbm{1}_{A}\oplus\mathbbm{1}_{B})\oplus\mathbbm{1}_{C}=\mathbbm{1}_{A}\oplus(\mathbbm{1}_{B}\oplus\mathbbm{1}_{C})=\mathbbm{1}_{A}\oplus\mathbbm{1}_{B\triangle
C}=\mathbbm{1}_{A\triangle(B\triangle C)}.\qed$

To a simplicial representation we now associate a group as in Proposition
3.25, which we call a $G$-representation to indicate that the representation
has the structure of a group. While we could construct the group without any
reference to the simplicial representation, the group encodes the same model
structure in terms of model components and their interconnections, so it is
natural to indicate the relationship between the group and the corresponding
simplicial representation.

######  Definition 3.26 ($G$-representation).

Let $K$ be a simplicial representation of a model. The _$G$ -representation_
of the model is the group $G_{K}:=(2^{K},\triangle)$.

For a simplicial representation $K$, the elements of $G_{K}$ are subsets of
simplicies from $K$. These subsets are of three types: a set containing a
single simplex from $K$ is identified as a simplex; a set containing one or
more simplices from $K$ is a simplicial subcomplex of $K$ if it is closed
under taking faces, in particular a set containing just a vertex is both a
simplex and a simplicial complex; a set containing two or more simplices from
$K$ that is not a simplicial subcomplex of $K$ is a set of simplices.

######  Notation 3.27.

Let $K$ be a simplicial representation of a model, with corresponding
$G$-representation $G_{K}$. The subset
$\operatorname{Vert}(G_{K}):=\Big{\\{}\,\big{\\{}\\{v\\}\big{\\}}\mid
v\in\operatorname{Vert}(K)\,\Big{\\}}$ of $G_{K}$ is referred to as the
vertices of $G_{K}$. Further, we denote by $\rho_{K}\colon
2^{\operatorname{Vert}(G_{K})}\setminus\emptyset\to K$ the map such that
$\rho(S)=\big{\\{}\cup_{i\in I}\\{s_{i}\\}\big{\\}}$ for
$S=\Big{\\{}\big{\\{}\\{s_{i}\\}\big{\\}}\Big{\\}}_{i\in I}\in
2^{\operatorname{Vert}(G_{K})}\setminus\emptyset$, which sends a set of
vertices in $G_{K}$ to the simplex spanned by the corresponding vertices in
$K$.

Since simplicial representations are labelled simplicial complexes, the
elements of $G$-representations are also labelled, and there is a canonical
bijection $K\mapsto G_{K}$ from the set of simplicial representations of
models onto the set of $G$-representations of models.

######  Example 3.28.

Let $K$ be the 1-simplex with vertices labelled 1 and 2, that is,
$K=\big{\\{}\\{1\\},\\{2\\},\\{1,2\\}\big{\\}}$. Then
$G_{K}=\Big{\\{}\emptyset,\big{\\{}\\{1\\}\big{\\}},\big{\\{}\\{2\\}\big{\\}},\big{\\{}\\{1,2\\}\big{\\}},\big{\\{}\\{1\\},\\{2\\}\big{\\}},\big{\\{}\\{1\\},\\{1,2\\}\big{\\}},\big{\\{}\\{2\\},\\{1,2\\}\big{\\}},\big{\\{}\\{1\\},\\{2\\},\\{1,2\\}\big{\\}}\Big{\\}}$.
Note that $\big{\\{}\\{1\\}\big{\\}}$ and $\big{\\{}\\{2\\}\big{\\}}$
correspond to both simplices and simplicial subcomplexes of $K$,
$\big{\\{}\\{1,2\\}\big{\\}}$ corresponds to a simplex of $K$,
$\big{\\{}\\{1\\},\\{2\\}\big{\\}}$ and
$\big{\\{}\\{1\\},\\{2\\},\\{1,2\\}\big{\\}}=K$ correspond to simplicial
subcomplexes of $K$, while each of the elements
$\big{\\{}\\{1\\},\\{1,2\\}\big{\\}}$ and
$\big{\\{}\\{2\\},\\{1,2\\}\big{\\}}$ are not closed under taking faces so are
sets of simplices from $K$.

While the group $G_{K}$ contains subsets of $K$ that are not subcomplexes, we
can describe $G_{K}$ in terms of the simplices in $K$. Recall that for a prime
$p$, a $p$-group is a group in which every element has order a power of $p$.
Since every nontrivial element of $G_{K}$ has order 2, $G_{K}$ is therefore an
elementary commutative 2-group. Further, since $G_{K}$ is a finite $p$-group
it follows from the Burnside Basis Theorem [33, Chapter 5, Page 124, Theorem
5.50] that any two minimal generating sets of $G_{K}$ have equal cardinality,
here equal to $\lvert K\rvert$, and are therefore generating sets of smallest
cardinality.

######  Proposition 3.29.

Let $K$ be a simplicial representation of a model. Then
${\mathcal{K}}:=\big{\\{}\,\\{\sigma\\}\mid\sigma\in K\,\big{\\}}$ is a
minimal set of generators for $G_{K}$.

###### Proof.

Let $S\in G_{K}$ be nontrivial. Then $S\subseteq{\mathcal{K}}$ by the
definition of $G_{K}$, so by associativity of the symmetric difference we have
$S=\triangle_{\sigma\in S}\\{\sigma\\}$, and it follows that ${\mathcal{K}}$
generates $G_{K}$ as a group. Further, ${\mathcal{K}}$ is minimal since, for
each $\\{\sigma\\}\in{\mathcal{K}}$,
$\\{\sigma\\}\notin\big{\langle}{\mathcal{K}}\setminus\\{\sigma\\}\big{\rangle}\subseteq
G_{K}$. ∎

Since $G_{K}$ is an elementary commutative 2-group it can be simply expressed
as an internal direct sum of cyclic subgroups of subcomplexes of $K$, or
isomorphically as an external direct sum of copies of
${\mathbb{Z}}/2{\mathbb{Z}}$.

######  Proposition 3.30.

Let $K$ be a simplicial representation of a model. Then
$G_{K}\cong\bigoplus_{\sigma\in
K}\big{\langle}\\{\sigma\\}\big{\rangle}\cong\bigoplus_{\sigma\in
K}{\mathbb{Z}}/2{\mathbb{Z}}$.

###### Proof.

By the fundamental theorem of finite commutative groups (see [33, Chapter 6,
Page 128, Theorem 6.5]), $G_{K}$ is a direct sum of primary cyclic groups.
Since ${\mathcal{K}}$ is a minimal set of generators for $G_{K}$ by
Proposition 3.29, we therefore have $G_{K}\cong\bigoplus_{\sigma\in
K}\big{\langle}\\{\sigma\\}\big{\rangle}$. Finally, each
$\\{\sigma\\}\in{\mathcal{K}}$ has order 2 in $G_{K}$, so
$\big{\langle}\\{\sigma\\}\big{\rangle}\cong{\mathbb{Z}}/2{\mathbb{Z}}$. ∎

Viewing $G_{K}$ as the direct sum $\bigoplus_{\sigma\in
K}\big{\langle}\\{\sigma\\}\big{\rangle}$ highlights the structure of the
group, and allows for easier characterisation of, for example, homomorphisms
between such groups.

Following from the definition of distance between two simplicial
representations $K$ and $L$ of two models [24], we can define the distance
between the two $G$-representations $G_{K}$ and $G_{L}$ as the cardinality of
the symmetric difference of $K$ and $L$. Indeed, this distance between
$G$-representations can be formalised as a metric space, similar to the metric
space of simplicial representations, and is also isometric to the latter.

The group $G_{K}$ associated with a simplicial representation $K$ is itself a
representation of the model corresponding to $K$. The group elements of
$G_{K}$ have labels that are induced by the labelled simplices contained in
the group elements, which in turn have labels induced by the component labels
of the model. The labels of the elements of $G_{K}$ must be accounted for by
group homomorphisms when required. We therefore introduce our concept of a
_label-preserving group homomorphism_.

######  Definition 3.31 (Label-preserving group homomorphism).

Let $G_{K}$ and $G_{L}$ be two $G$-representations associated with two
simplicial representations $K$ and $L$ of two models. A group homomorphism
$\phi\colon G_{K}\to G_{L}$ is _label preserving_ if whenever the group
elements $g\in G_{K}$ and $h\in G_{L}$ have the same label then $\phi(g)=h$.

Our definition of the equivalence of two $G$-representations of models is as
follows.

######  Definition 3.32 (Equivalence of $G$-representations).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{G}}$ be a
collection of $G$-representations $G_{K}$ associated with simplicial
representations which correspond to models generated by subsets from
${\mathcal{C}}$. Two $G$-representations $G_{K}$, $G_{L}\in{\mathcal{G}}$ are
_equivalent_ if and only if there exist two (possibly empty) sequences of
invertible partial operations $(f_{i})_{i=0}^{m}$ and $(g_{i})_{i=0}^{n}$ on
${\mathcal{G}}$ such that $f_{m}\circ\cdots\circ f_{1}\circ
f_{0}(G_{K})=g_{n}\circ\cdots\circ g_{1}\circ g_{0}(G_{L})$.

Note that the set
$\\{\,(G_{K},G_{L})\in{\mathcal{G}}\times{\mathcal{G}}\mid\text{$G_{K}$ and
$G_{L}$ are equivalent}\,\\}$ is an equivalence relation on ${\mathcal{G}}$.
For the equivalence of $G$-representations to be conceptually meaningful, just
as for the equivalence of simplicial representations, we need to ensure that
the employed partial operations are themselves conceptually meaningful. We
consider specific operations on $G$-representations that are analogous to
those for simplicial representations. For the $G$-representations, however,
the operations are now induced by label-preserving group homomorphisms.

For brevity we only outline the operations on $G$-representations, as they
correspond closely with the operations for simplicial operations. Note that
for $G$-representations the action of an adjacent/nonadjacent-vertex
identification does not send two vertices to the same vertex as for simplicial
representations, but rather removes all group elements associated with one of
the two vertices. This action is essentially the same as that for simplicial
representations, and allows for a description in terms of group homomorphisms.
Further, we omit definitions for the vertex split operation and the inclusion
operation, both of which can be defined in terms of canonical injections
between the direct-sum decompositions of appropriate $G$-representations,
since we explicitly describe the equivalence of $G$-representations with the
operations of adjacent/nonadjacent-vertex identifications and vertex
substitution, while the vertex split and inclusion operations are employed
implicitly. Similar to the operations for simplicial representations, all five
of these partial operations on a set of $G$-representations are invertible for
suitable domains of definition, and the equivalence of $G$-representations
must be based on the admissibility of the partial operations.

For all operations, let ${\mathcal{C}}$ be the set of all components of models
under consideration, and let $G_{K}$ and $G_{L}$ be $G$-representations
associated with the two simplicial representations $K$ and $L$ with labels
from ${\mathcal{C}}$.

Adjacent-vertex identification:

Suppose $\big{\\{}\\{u,v\\}\big{\\}}\in G_{K}$, where $u$,
$v\in\operatorname{Vert}(K)$, and that assumptions hold for $u$ and $v$
corresponding to those in Definition 2.9. A label-preserving group
homomorphism $\phi_{1}\colon G_{K}\to G_{L}$ is an _adjacent-vertex
identification_ if $G_{L}=\bigoplus_{\\{\sigma\in K\mid
v\notin\sigma\\}}\big{\langle}\\{\sigma\\}\big{\rangle}$ and
$\phi_{1}\colon\bigoplus_{\sigma\in
K}\big{\langle}\\{\sigma\\}\big{\rangle}\to\bigoplus_{\\{\sigma\in K\mid
v\notin\sigma\\}}\big{\langle}\\{\sigma\\}\big{\rangle}$ is the canonical
projection.

Nonadjacent-vertex identification:

Suppose $\big{\\{}\\{u\\}\big{\\}}$, $\big{\\{}\\{v\\}\big{\\}}\in G_{K}$,
$\big{\\{}\\{u,v\\}\big{\\}}\notin G_{K}$, and that assumptions hold for $u$
and $v$ corresponding to those in Definition 2.10. A label-preserving group
homomorphism $\phi_{2}\colon G_{K}\to G_{L}$ is a _nonadjacent-vertex
identification_ if $G_{L}=\bigoplus_{\\{\sigma\in K\mid
v\notin\sigma\\}}\big{\langle}\\{\sigma\\}\big{\rangle}$ and
$\phi_{2}\colon\bigoplus_{\sigma\in
K}\big{\langle}\\{\sigma\\}\big{\rangle}\to\bigoplus_{\\{\sigma\in K\mid
v\notin\sigma\\}}\big{\langle}\\{\sigma\\}\big{\rangle}$ is the canonical
projection.

Vertex substitution:

Suppose $u\in\operatorname{Vert}(K)$, $v\in\operatorname{Vert}(L)$, and
$\operatorname{Vert}(K)\setminus\\{u\\}=\operatorname{Vert}(L)\setminus\\{v\\}$.
Let $\pi\colon\operatorname{Vert}(K)\to\operatorname{Vert}(L)$ be a map that
is injective except that $\pi(u)=v$. A label-preserving group isomorphism
$\phi_{5}\colon G_{K}\to G_{L}$ is a _vertex substitution_ if, for each
$\sigma\in K$, $\phi_{5}(\\{\sigma\\})=\big{\\{}\\{\,\pi(w)\mid
w\in\sigma\,\\}\big{\\}}$.

To determine whether there are conceptually equivalent vertices in a
$G$-representation $G_{K}$, we can employ group actions on $G_{K}$ that
correspond to exchange automorphisms for simplicial representations. Recall
that an action of a group $H$ on a group $G$ is a group homomorphism
$\Theta\colon H\to\operatorname{Aut}(G)$. While we required orbit spaces for
vertex identifications when considering simplicial representations, for
$G$-representations we have no need for orbit spaces and can employ
$G$-representations directly. The methodology for determining model
equivalence detailed in Subsection 3.3 is easily adapted for
$G$-representations.

##  4 Conclusion

In this article we develop a rigorous and automatable methodology for
determining whether vertices in a simplicial representation are conceptually
related, corresponding to conceptually-related model components, and then
identifying these vertices using simplicial operations. This process is the
main consideration of comparison by equivalence as defined in Vittadello and
Stumpf [24], and exploits the symmetry associated with conceptually equivalent
vertices in a simplicial representation by constructing group actions on the
simplicial representations. We develop the required mathematical theory for
group actions on simplicial complexes, and demonstrate how this methodology
greatly simplifies and facilitates the process of determining model
equivalence by reducing the initial simplicial representations to simplicial
complexes with no conceptually equivalent vertices. Our approach to model
comparison by equivalence provides a rigorous and efficient framework for
comparing the underlying conceptual structure of mathematical models of a
particular biological system, as well as mathematical models of different
biological systems.

The important point to note is that our approach, like our previous one,
differs from the overwhelming majority of model comparison approaches by being
independent of outputs: it is solely based on the underlying conceptual
structure of the model. Our definition of model structure is also more
inclusive than, for example, motif-based definitions[34], as these are known
to fall short of constraining dynamics [35]. By including model properties
such as boundary conditions and symmetry relationships, for example, as
components of the simplicial complexes, our approach here is capable of
capturing even subtle differences in the dynamics. Our methodology is
applicable to all models irrespective of mathematical formalism, and allows
for comparison from different conceptual perspectives.

Model comparison is essential for managing the seemingly heterogeneous and
possibly numerous models associated with a system. We conclude this analysis
with a discussion of an alternative mathematical framework for our model
comparison methodology by representing models as groups, which provides for
the direct application of techniques from group theory within our model-
comparison methodology. That we can represent our model comparison methodology
with two distinct mathematical frameworks shows the flexibility of our
approach, which is important for the utilisation of a broad range of
mathematical techniques when working on the difficult problem of model
comparison.

Modelling in biology cannot rely on the same fundamental principles — such as
symmetries and conservation laws — that have been so successful in physics.
Instead, statistical methods [21], robustness analysis [36], or algebraic
methods [37] are required to narrow down the vast ‘universe’ of potential
mathematical models in order to identify those models that can reasonably
explain a given scenario. The approach developed here is capable of taking
these models, including large sets of good candidates [26], and distilling
design principles from this analysis. Design principles, or the conceptual
bases for mathematical models describing a given system, are arguably our best
chance to gain conceptual mechanistic insights into biological systems.

##  5 Appendix

Here we provide the details of the mathematical models that we use in the
example in Subsection 3.4. We also provide a brief discussion on representing
the models as simplicial complexes.

These models are from the two main classes of models for developmental pattern
formation, namely Turing-pattern (TP) models and positional-information (PI)
models. We consider the patterning to occur within a two-dimensional
rectangular domain, with zero-flux boundary conditions on the two sides
parallel to the morphogen concentration gradient. We assume that the
velocities of the cytoplasm and the growing tissue are negligible; we
therefore assume no advection. Further details of the models and the
associated simplicial representations, including the vertex numbers associated
with the simplicial representations, are in Vittadello and Stumpf [24]. Note
that the boundary conditions for the TP models may be Dirichlet, Neumann,
Robin, or periodic [38, 39, 40].

###  5.1 Positional-information: Linear gradient

A linear concentration profile of a morphogen results when the production and
degradation of the morphogen occur outside and on opposite sides of the tissue
domain, and the morphogen undergoes passive diffusion along the domain from
the side where it is produced to the side where it is degraded [41, 42, 43,
44]. Mathematically, the steady-state morphogen concentration in this system
satisfies Laplace’s equation, which yields global scale-invariant positional
information. Specifically, if the tissue length is $L$ with initial position
at $x=0$ then the morphogen concentration $m(x,t)$ can be modelled as

$\frac{\partial m}{\partial t}=D\frac{\partial^{2}m}{\partial x^{2}},$ (3)

where $D$ is the morphogen diffusivity. The steady-state solution of Equation
3 is the linear equation $m(x)=ax+b$, for arbitrary constants $a$,
$b\in{\mathbb{R}}$. The boundary conditions are influx at one end and outflux
at the other end, and we may assume without loss of generality that influx
occurs at $x=0$ and outflux at $x=L$. The original Dirichlet boundary
conditions specify the constant concentrations $m(0,t)=m_{0}>0$ and
$m(L,t)=m_{L}\geq 0$, with $m_{0}>m_{L}$, so the solution is
$m(x)=\big{(}(m_{L}-m_{0})/L\big{)}x+m_{0}$. Monotonically decreasing linear
gradients can also be achieved with Neumann boundary conditions at one
boundary and Dirichlet boundary conditions at the opposite boundary.

The simplicial representation of the linear model is shown in Figure 2(a).
Note that the simplicial complex is 4-dimensional, however we only show the
1-skeleton of the simplicial complex for simplicity.

###  5.2 Positional-information: Synthesis-diffusion-degradation (SDD)

In the SDD model a morphogen gradient forms by morphogen production from a
localised source at the boundary combined with morphogen diffusion and uniform
degradation throughout the tissue [45, 46, 44]. Mathematically, the morphogen
concentration $m(\bm{x},t)$ can be modelled as

$\frac{\partial m}{\partial t}=D\nabla^{2}m-km,$ (4)

where $D$ is the morphogen diffusivity, and $k$ is the morphogen degradation
rate which is independent of morphogen concentration. The boundary conditions
are Neumann for the influx and zero outflux. The steady-state morphogen
gradient is an exponential function, so this system is not scale invariant
unless $D$, $k$, and the influx vary with the tissue length $L$ in a very
specific manner [46].

The simplicial representation of the SDD model is shown in Figure 2(b). Note
that the simplicial complex is 3-dimensional, however we only show the
1-skeleton of the simplicial complex for simplicity.

###  5.3 Positional-information: Opposing gradients

There are two mechanisms whereby two opposing morphogen gradients provide size
information for developmental patterning [47]: the gene expression depends on
the relative concentrations of the two morphogens. One is the annihilation
model, and the other is the scaling-by-opposing-gradients model.

For the opposing gradients model the sources of each morphogen are at opposite
ends of the domain. The combination of the two morphogen gradients provides
effective local scaling for the boundaries of individual target genes, however
it does not produce global scaling across the entire domain [42, 48, 47, 44].
The degree of scaling for the opposing gradients mechanism can be increased if
the morphogens are irreversibly inactivated upon binding to each other [47].
Here we assume no direct interaction between the two morphogens.
Mathematically, the two morphogen gradients with concentrations $m(\bm{x},t)$
and $c(\bm{x},t)$ can be modelled as

$\displaystyle\frac{\partial m}{\partial t}$
$\displaystyle=D_{m}\nabla^{2}m-k_{m}m,$ (5) $\displaystyle\frac{\partial
c}{\partial t}$ $\displaystyle=D_{c}\nabla^{2}c-k_{c}c,$ (6)

where $D_{m}$ and $D_{c}$ are diffusivities, and $k_{m}$ and $k_{c}$ are
degradation rates. The boundary conditions for each morphogen are Neumann at
both boundaries, with flux at the source and zero flux at the opposite
boundary.

The simplicial representation of the opposing gradients model is shown in
Figure 2(c). Note that the simplicial complex is 4-dimensional, however we
only show the 1-skeleton of the simplicial complex for simplicity.

###  5.4 Positional-information: Induction-contraction

The largest degree of scaling across the patterning domain is obtained when
the biophysical properties of the morphogen are influenced by an accessory
modulator that senses the size of the domain [45, 49, 46, 44]. The modulator
molecules with concentration $c(\bm{x},t)$ may influence the diffusivity,
degradation rate, or influx rate, of the morphogen with concentration
$m(\bm{x},t)$. If the kinetics and source of the modulator are dependent on
the morphogen concentration then the mechanism is _active modulation_. An
example of active modulation is the _induction-contraction_ model, which
scales globally [50, 51]. In this case, the morphogen induces the production
of a contractor molecule, here the modulator with high diffusivity once again,
which contracts the range of the morphogen gradient through a decrease in the
morphogen diffusivity or an increase of the morphogen degradation rate. Note
that while the amplitude and shape of the morphogen gradient is globally
scale-invariant, the same is not necessarily true of the modulator since the
modulator level reflects the domain size and therefore increases or decreases
accordingly.

The scaling by modulation model with the induction-contraction mechanism,
which can be represented mathematically as

$\displaystyle\frac{\partial m}{\partial t}$
$\displaystyle=D_{m}(c)\nabla^{2}m-k_{m}(c)m,$ (7)
$\displaystyle\frac{\partial c}{\partial t}$
$\displaystyle=D_{c}\nabla^{2}c-k_{c}c+\rho(m),$ (8)

where $D_{m}(c)$ and $D_{c}$ are diffusivities, $k_{m}(c)$ and $k_{c}$ are
degradation rates, and $\rho(m)$ is the localised production source for the
modulator. Note that $D_{m}(c)$ is a decreasing function of $c$, $k_{m}(c)$ is
an increasing function of $c$, and $\rho(m)$ is an increasing function of $m$.
Further note that ‘degradation’ in this context refers to not only physical
destruction, but to any mechanism that effects the removal of the morphogen
from the patterning system, including irreversible complex formation.
Morphogen boundary conditions are Neumann for the influx and zero flux at the
opposite end of the domain. For the modulator there are various possibilities
for the boundary conditions, such as the same type of boundary conditions as
the morphogen when the modulator source is outside the domain or, as we
consider here, zero flux at both boundaries when the modulator source is
within the domain.

###  5.5 Positional-information: annihilation model

The annihilation model, and also the opposing gradients model are mechanisms
whereby two opposing morphogen gradients provide size information for
developmental patterning. In the annihilation model, the target gene responds
to the concentration of Morphogen 1, to which Morphogen 2 irreversibly binds
and thereby inhibits the action of Morphogen 1 on activity of transcription,
so that the gradient of Morphogen 2 provides size information to the
concentration field of Morphogen 1 [47].

The sources of each morphogen are at opposite ends of the domain, and the
morphogens interact by an annihilation reaction with rate $k$ that results in
global scale-invariant patterning [52, 47]. Mathematically, the two morphogen
gradients with concentrations $m(\bf{x},t)$ and $c(\bf{x},t)$ can be modelled
as

$\displaystyle\frac{\partial m}{\partial t}$
$\displaystyle=D_{m}\nabla^{2}m-k_{m}m-kmc,$ (9) $\displaystyle\frac{\partial
c}{\partial t}$ $\displaystyle=D_{c}\nabla^{2}c-k_{c}c-kmc,$ (10)

where $D_{m}$ and $D_{c}$ are diffusivities, and $k_{m}$ and $k_{c}$ are
degradation rates. The boundary conditions for each morphogen are Neumann at
both boundaries, with influx at the source and zero flux at the opposite
boundary.

The simplicial representation of the annihilation mode is shown in Figure
3(a). Note that the simplicial complex is 4-dimensional, however we only show
the 1-skeleton of the simplicial complex for simplicity.

###  5.6 Turing-pattern: Activator-inhibitor model

The activator-inhibitor system [53, 54, 55] consists of two diffusible
morphogens, an autocatalytic activator with concentration $m(\bf{x},t)$ and a
rapidly diffusing inhibitor with concentration $c(\bf{x},t)$. This model can
be represented mathematically as

$\displaystyle\frac{\partial m}{\partial t}$
$\displaystyle=D_{m}\nabla^{2}m+\frac{\rho
m^{2}}{c(1+\mu_{m}m^{2})}-k_{m}m+\rho_{m},$ (11) $\displaystyle\frac{\partial
c}{\partial t}$ $\displaystyle=D_{c}\nabla^{2}c+\rho m^{2}-k_{c}c+\rho_{c},$
(12)

where $D_{m}$ and $D_{c}$ are diffusivities, $\rho_{m}$ and $\rho_{c}$ are
basal production rates, $k_{m}$ and $k_{c}$ are degradation rates, and
$\mu_{m}$ is a saturation constant. The parameter $\rho$ is the _source
density_ , which measures the general ability of the cells to perform the
autocatalytic reaction. The patterning arises through local self-enhancement
of the activator, activation of the inhibitor, and long-range inhibition of
the activator. We assume that the boundary conditions are zero flux at both
boundaries.

The simplicial representation of the activator-inhibitor model is shown in
Figure 3(b). Note that the simplicial complex is 5-dimensional, however we
only show the 1-skeleton of the simplicial complex for simplicity.

###  5.7 Turing-pattern: Substrate depletion

While long-range inhibition of an autocatalytic activator can occur with an
inhibitor, as in the activator-inhibitor model, the antagonistic effect can
also arise from the depletion of a substrate that is consumed during activator
production. In this system, the production of an autocatalytic activator with
concentration $m(\bm{x},t)$ results in either direct or indirect depletion of
a substrate with concentration $s(\bm{x},t)$ [54]. This model can be
represented mathematically as

$\displaystyle\frac{\partial m}{\partial t}$
$\displaystyle=D_{m}\nabla^{2}m+\rho sm^{2}-k_{m}m+\rho_{m},$ (13)
$\displaystyle\frac{\partial s}{\partial t}$
$\displaystyle=D_{s}\nabla^{2}s-\rho sm^{2}-k_{s}s+\rho_{s},$ (14)

where $D_{m}$ and $D_{s}$ are diffusivities, $\rho_{m}$ and $\rho_{s}$ are
basal production rates, $k_{m}$ and $k_{s}$ are degradation rates, and $\rho$
is the source density for the autocatalytic reaction of the activator, similar
to the activator-inhibitor model. The diffusivity $D_{s}$ of the substrate
must be much faster than the diffusivity $D_{m}$ of the activator, and it is
assumed that the substrate is produced uniformly throughout the domain. We
assume that the boundary conditions are zero flux at both boundaries.

The simplicial representation of the substrate depletion model is shown in
Figure 3(c). Note that the simplicial complex is 5-dimensional, however we
only show the 1-skeleton of the simplicial complex for simplicity.

###  5.8 Turing-pattern: Inhibition of an inhibition

Dynamics analogous to the activator-inhibitor model can be realised through an
inhibition of an inhibition mechanism [54]. In this case, two morphogens with
concentrations $a(\bm{x},t)$ and $c(\bm{x},t)$ inhibit the production of each
other, thereby forming a switching system in which one of the morphogens
becomes fully activated similar to being autcatalytic. In order that pattern
formation occur, a third morphogen with concentration $b(\bm{x},t)$ acts as a
long-range signal that disrupts the indirect self-enhancement of either $a$ or
$c$. Morphogen $b$ is rapidly diffusing, is produced under control of $a$, and
inhibits the inhibition of $c$ production by $a$. therefore acting as
inhibitor. This model can be represented mathematically as

$\displaystyle\frac{\partial a}{\partial t}$
$\displaystyle=D_{a}\nabla^{2}a+\frac{\rho_{a}}{\kappa_{a}+c^{2}}-k_{a}a,$
(15) $\displaystyle\frac{\partial b}{\partial t}$
$\displaystyle=D_{b}\nabla^{2}b+k_{b}(a-b),$ (16) $\displaystyle\frac{\partial
c}{\partial t}$
$\displaystyle=D_{c}\nabla^{2}c+\frac{\rho_{c}}{\kappa_{c}+a^{2}/b^{2}}-k_{c}c,$
(17)

where $D_{a}$, $D_{b}$, and $D_{c}$ are diffusivities, $\rho_{a}$ and
$\rho_{c}$ are production rates, $k_{a}$ and $k_{c}$ are degradation rates,
$k_{b}$ is both the rate of production and degradation of $b$, and
$\kappa_{a}$ and $\kappa_{c}$ are saturation constants that limit the
production rate if the concentrations of $a$ or $c$ become too low. We assume
that the boundary conditions are zero flux at both boundaries.

###  5.9 Turing-pattern: Modulation

The diffusion of a morphogen may be inhibited by adsorption on negatively-
charged extracellular matrix (ECM) components, resulting in modulated
diffusion and a smaller effective diffusivity for the morphogen [56]. The
corresponding modulation model [56] is an extension of the activator-inhibitor
model [53, 54, 55]. Our description of the modulation model is based on the
version of the activator-inhibitor model described above. The modulation model
consists of two diffusible morphogens, an autocatalytic activator and an
inhibitor, along with available binding sites on the ECM onto which the
activator adsorbs. The inhibitor does not bind to the ECM, and the free
activator and the inhibitor have equal diffusivities $D$. It is assumed that
autocatalysis and activation of the inhibitor by the activator occurs in both
the free and adsorbed states. It is also assumed that the degradation rate of
the activator is equal in both the free and adsorbed states. Free binding
sites on the ECM therefore appear due to both desorption and degradation of
the activator. Such modulation allows for stable dissipative morphogens
gradients. Mathematically, the free activator has concentration $a(\bm{x},t)$,
the bound activator has concentration $b(\bm{x},t)$, the inhibitor has
concentration $c(\bm{x},t)$, and the concentration of available binding sites
on the ECM is $s(\bm{x},t)$:

$\displaystyle\frac{\partial a}{\partial t}$
$\displaystyle=D\nabla^{2}a+\frac{\rho(a+b)^{2}}{c(1+\mu_{a}(a+b)^{2})}-k_{a}a+\rho_{a}-k_{1}sa+k_{-1}b,$
(18) $\displaystyle\frac{\partial c}{\partial t}$
$\displaystyle=D\nabla^{2}c+\rho(a+b)^{2}-k_{c}c+\rho_{c},$ (19)
$\displaystyle\frac{\partial s}{\partial t}$
$\displaystyle=-k_{1}sa+(k_{-1}+k_{a})b,$ (20)

where $D$ is the diffusivity, $\rho_{a}$ and $\rho_{c}$ are basal production
rates, $k_{a}$ and $k_{c}$ are degradation rates, $\mu_{a}$ is the saturation
constant, $\rho$ is the source density, $k_{1}$ is the rate of adsorption of
the activator onto the ECM, and $k_{-1}$ is the rate of desorption of the
activator from the ECM. We assume that the boundary conditions are zero flux
at both boundaries.

###  5.10 Construction of the simplicial representations

The simplicial representations of the four Turing-pattern and five positional-
information models are constructed by first determining the set of components
${\mathcal{C}}$ on which all of the the models under consideration are based.
In this case the general components are: agents, namely morphogens,
modulators, and substrates; reactions involving the agents, such as self-
activation, activation, inhibition and annihilation; agent degradation; influx
and outflux boundary conditions; agent diffusion; profile of the morphogen
gradient; and scale invariance of the morphogen gradient. The ordered set of
these components is shown in Figure 3 in [24].

The reference complex is the simplex spanned by the complete set of components
for all nine models. To construct the simplicial representation of each
Turing-pattern and positional-information model, we first specify the
0-simplices, which represent the model components, and the 1-simplices, which
represent direct interconnections between the components. While the 0- and
1-simplices are specified by the model, to give a combinatorial graph, for
these models we obtain higher-dimensional simplices by forming cliques where
possible, incrementally in dimensions 2 and higher. These higher-dimensional
simplices indicate higher-dimensional interactions between the corresponding
model components.

## Author contributions

S.T.V. conceived, planned, and conducted this research; S.T.V. drafted the
manuscript; S.T.V. and M.P.H.S. reviewed, edited, and approved the final
version.

## Funding

The authors gratefully acknowledge funding through a “Life?” programme grant
from the Volkswagen Stiftung. M.P.H.S. is funded through the University of
Melbourne Driving Research Momentum program.

## Competing interests

We declare we have no competing interests.

## Acknowledgments

We thank members of the Theoretical Systems Biology group at the University of
Melbourne and Imperial College London, Heike Siebert (FU Berlin), James
Briscoe (The Francis Crick Institute) and Mark Isalan (Imperial College
London) for helpful discussions on Turing patterns and model discrimination.

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