^{1}

^{2}

^{2}

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Edited by: Felix Sadyrbaev, University of Latvia, Latvia

Reviewed by: Yunjiao Wang, Texas Southern University, United States

Paolo Milazzo, University of Pisa, Italy

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This study addresses a problem of correspondence between dynamics of a parameterized system and the structure of interactions within that system. The structure of interactions is captured by a signed network. A network dynamics is parameterized by collections of multi-level monotone Boolean functions (MBFs), which are organized in a parameter graph

The concept of a network plays a central role in systems biology, where it encodes interactions between the molecular species. Each directed edge has a sign, which represents either a monotonically increasing or monotonically decreasing effect of the source on the target. The restriction to monotone interactions suggests a possibility that there is a relationship between structure of the network and its emergent dynamics.

There are different types of dynamics that can be associated to a network. Some, like those generated by Boolean functions wherein each node can take on a value of 0 or 1, are very tightly linked to the structure of the network. On the other hand, the dynamics generated by ordinary differential equations (ODE) models with an interaction structure given by the network strongly depends on choice of non-linearities and parameters. Within the class of ODE network models, the question of limitations on dynamics imposed by network structure is much more difficult. In particular, one has to carefully define what constitutes the “same” dynamics to compare the dynamics of different networks. Traditional definitions used in comparison of dynamical systems include the concept of conjugate dynamics, or conjugate dynamics on a recurrent set Ω [

In this study, we address the question of the relationship between network structure and its dynamics in the context of multi-valued Boolean systems in which each vertex can take on a set of integer values based on its number of out-edges. The approach is based on [_{u ∈ V}

There is a correspondence [

The connection between parameterization of continuous dynamics of switching systems and collections of multi-valued Boolean models of the same network was also described in Abou-Jaoudé and Monteiro [

With a finite characterization of network dynamics, it is natural to ask whether homomorphisms of signed directed graphs preserve network dynamics

The second homomorphism we study is inclusion of a network as subgraph in another network. This is a very important example in systems biology where gene regulatory networks cannot be assumed to be in their final form as new experimental evidence may reveal additional genes (nodes) and edges. Since the basis of any scientific approach is to study small problems first before using that knowledge to tackle larger problems, we must understand if, and how, the dynamics

The second homomorphism is strongly motivated by systems biology. Gene regulatory networks do not exist in isolation and interact with other networks. It is important to understand if and under what conditions the dynamics of subnetworks persist within a larger network. As a motivating example, consider [

A further motivating example comes from examining the subnetworks of the cell cycle itself, which support multiple phenotypes. The dual view of the cell cycle as either a biochemical “clock” vs. a set of “dominos”, that is, a series of switches where the completion of one step is required for the completion of the next step, has been discussed for at least 35 years [

Apart from addressing a general question of comparison of dynamic repertoires of networks, the work presented here provides the capability to answer the rigorously central hypothesis of motif theory [

We illustrate our work on a small network

We represent all possible ways in which the network can function by enumerating a collection of monotone Boolean functions (MBF) (Section 2.1). These describe how the concentration levels of the input edge(s) of each node activate the output edges. Node

Three monotone Boolean functions in

0 | 0 | 0 | 1 |

1 | 0 | 1 | 1 |

In

Six monotone Boolean functions in

11 | 01 | 0 | 0 | 0 | 1 | 1 | 1 |

10 | 00 | 0 | 0 | 0 | 0 | 0 | 1 |

01 | 11 | 0 | 1 | 1 | 1 | 1 | 1 |

00 | 10 | 0 | 0 | 1 | 0 | 1 | 1 |

Function II corresponds to logical OR function and function V corresponds to a logical AND function.

Logic parameters of

Y | ||||||

0 | 0 0 | 0 0 | 1 0 | 0 0 | 1 0 | 1 1 |

1 | 0 0 | 1 0 | 1 0 | 1 1 | 1 1 | 1 1 |

Each logic parameter corresponds to two monotone Boolean functions

As a consequence of Theorem 5.5, there are two subgraphs of

The organization of the study is as follows. In Section 2, we review the basic concepts of DSGRN: the parameter graph, state transition graph and Morse graph. In Section 3, we describe homomorphisms of signed graphs and a particular example of switching homomorphism. In Section 4, we show that the switching homomorphism preserves dynamics by inducing a graph isomorphism of the parameter graph that preserves dynamics. In Section 5, we start focusing on the network embedding as our second example of signed graph homomorphism. We formulate our main results as Theorem 5.4 about correspondence of dynamics at input inessential parameters and Theorem 5.5 at output inessential parameters. These Theorems are then proved in Sections 6 and 7, respectively. Section 8 contains discussion, while some proofs of the more technical results are delegated to the Section 8 and a detailed analysis of network

A

respectively.

The parameterization of the dynamics of a network depends on the choice of model. We discuss two different types of models and briefly review literature on how the parameterizations of these two types of models are related. Boolean models compatible with regulatory network _{u} = 1, then this activated state, in turn, activates all edges from _{u} compatible with edge signs between _{u}.

An alternative and richer set of models starts with the original work of Thomas et al. [

As mentioned in the introduction, our work [_{u} as a collection of monotone Boolean functions, each describing which Boolean inputs ^{S(u)} activate a particular node

We now start with rigorous description of the set of parameters associated with a regulatory network

Let _{u}: = |_{u}:_{u}} which defines an ordering of the out-edges of _{θu)u ∈ V} is an _{u ∈ V} Θ (

Let 𝔹: = ({0, 1};0 ≺ 1}) be a Boolean lattice with natural order 0 ≺ 1 and let 𝔹^{n} be a lattice of Boolean

To define the logic parameter, we need the following definition.

Definition 2.1. A function ^{n} → 𝔹 is a ^{1} ≺ ^{2} implies ^{1}) ≼ ^{2}).

A

satisfying the

for all ^{S(u)}. The purpose of the function _{u}(_{fu)u ∈ V} is a

The set of parameters is the product of logic and order parameters

In Section 2.3, we show that the _{u} can be equivalently described by a single multi-valued Boolean function _{u}. Such a description is used in Abou-Jaoudé and Monteiro [

Two

_{u} = _{u} and the values of the order parameters θ_{u} and ϕ_{u} are exchanged on a single pair of neighboring entries on which the logic parameters agree. Explicitly, there is an adjacent transposition π of {1, …, _{u}} such that θ_{u} = π°ϕ_{u} and _{u} = _{u} together imply

_{u} = ϕ_{u} and the _{u} and _{u} differ in a single input. Explicitly, there is unique _{u}} and unique ^{0} ∈ 𝔹^{S(u)} such that ^{n} and for ^{0} we require

The _{u ∈ V}

We illustrate the parameter graph construction on network

First consider node _{z} with value θ_{y}(

We list all MBFs at node

Each node in the factor parameter graph _{z}, θ_{z}) where _{z} is a logic parameter and θ_{z} is an order parameter. Since there are two targets of

The multi-valued Boolean dynamics associated with a network

The dynamics occurs on the state space

We call _{xu)u ∈ V} is mapped to an input to a node

where

Note that for activating edge _{u} is below (above) the activating threshold θ_{u}(

An equivalent description of the _{u} uses a single multi-valued Boolean function _{u}:_{u} → _{u} defined by

Clearly, for every logic parameter _{u}; given _{u} and the order parameter θ_{u}, one can reconstruct the collection _{u}. Logic parameter description using a single multi-valued function _{u} is used in Abou-Jaoudé and Monteiro [

Definition 2.2. The dynamics for network

The

This map is also called the

The

If

For any

satisfies

The map

The maps

The multi-level Boolean dynamics

The recurrent dynamics of

If

If _{u} =

If

Following Naserasr et al. [

Definition 3.1. Let

A _{0} ⊸ _{1} ⊸ ⋯ ⊸ _{n}. It is a _{n} = _{0}. The sign of the walk is given by

A _{0} ⊸ _{1} ⊸ ⋯ ⊸ _{n} is positive in _{0}) ⊸ _{1}) ⊸ ⋯ ⊸ _{n}) is positive in

The switching homomorphism between networks is a homomorphism of the underlying directed graphs with the additional constraint of preserved signs of closed walks. A switching homomorphism is distinct from a stricter notion of homomorphisms of signed graphs that require edge signs to be preserved [

Definition 3.2. Let ^{U}(

Since this operation preserves both nodes ^{U}(δ).

The

U-switch of network ^{U}(

For any choice of

Proposition 3.3. 1. If ^{U} is the identity: σ^{U} =

2. For all ^{U} is an involution: σ^{U} ∘ σ^{U} =

3. Given ^{U} and σ^{W} commute: σ^{U} ∘ σ^{W} = σ^{W} ∘ σ^{U}.

4. Given _{1}, …, _{n}} ⊂

In this section, we examine the relationship between the dynamics of network ^{U}(^{U} fixes the nodes

Before we formulate the main result, we outline the main idea. Define bijection λ^{U}:

That is, λ^{U} reflects the components _{u} for each _{u} for ^{U} is an involution. Then, given ^{U} is graph automorphism; that is, it maps not only nodes to nodes but also edges to edges.

Theorem 4.1. Given the U-switch σ^{U}, there is an induced isomorphism ^{U}:

In other words, the dynamics of ^{U}. The map λ^{U} is a graph isomorphism between the ^{U}(

A consequence of Proposition 3.3(4) is that we need only to demonstrate that the diagram

Let _{u} is defined to be _{u} = (_{u}, ϕ_{u}) where

and _{u} = |_{u} is the dual parameter to _{u} is because

Define _{u}.

Lemma 4.2. Given any _{u} is well-defined. Moreover, _{u} is an involution: _{u} ∘ _{u} =

_{u} is a valid order parameter. We now show that _{u} satisfies the ordering condition (_{u} satisfies the ordering condition, we have _{u} satisfies the ordering condition. Next, we show that ^{1} ≺ ^{2}, which implies ¬^{1} ≻ ¬^{2}. Since

This shows

The second part of the Lemma follows by inspection. □

We are now ready to define the parameter graph isomorphism corresponding to σ^{U}. Since σ^{U} fixes the nodes

Let

Proposition 4.3. The ^{U} is a graph automorphism of

_{u}, is an involution on ^{U} is either the identity on that component or given by _{u}, ^{U} is an involution on ^{U} preserves parameter graph adjacency.

Let ^{U}((_{u} and

First suppose _{u} and

If _{u} = θ_{u} and _{u} and

where τ(_{u} + 1 − π(_{u} + 1 − _{u} and

Finally, suppose _{u} and _{u} = _{u} and _{u} and

If _{0} be the unique input such that

and for _{0},

Similarly for

We conclude that _{u} + 1 − _{0} is the unique input such that

Since ^{U} is invertible and preserves both order and logical adjacency, ^{U} is an automorphism of

□

The difference between the target point map ^{U}(^{u} (see the beginning of Section 2.3) on the edge signs δ. The following lemma relates the input map ^{u}(·;θ, δ) for ^{u}(·;ϕ, σ^{U}(δ)) for σ^{U}(

Lemma 4.4. Let, ^{{u}}((^{{u}}(δ). Then for all

For

Consequently,

for

^{{u}}.

First consider _{w} = _{w} = θ_{w}, we have _{w} < θ_{w}(_{w}(_{w}) < ϕ_{w}(

Next, we consider the case

So, if _{u} < θ_{u}(_{u} ≥ ϕ_{u}(

Now, suppose _{w} = _{w} = θ_{w}, and _{u} < θ_{u}(

Since

^{U}(δ). Let

Note that for _{v}(_{v}) = _{v}, _{v} = _{v}, and, by Lemma 4.4 also ^{v}(^{v}(λ(

Consequently, if we denote

For node

Consequently, with

We have shown that λ^{U} verifies the conjugacy of ^{U} is a graph isomorphism between ^{U}(^{U}(^{U}(

Remark 4.5. We want to point out that when ^{V}:^{V} commutes with network projections defined in the next section.

In this section, we address the question of when the dynamics of a subnetwork

Definition 5.1. Let

Alternatively, the map

Since the cut operation only removes edges from

We outline the main ideas of this section. Our goal is to relate the dynamics of network

However, there are important differences between these two types of edges. At a parameter where we remove only input inessential edges from

Maps of edges in ^{e}. The gray regions indicate ^{e}(

On the other hand, at parameters where we remove output inessential edges, we find that the result can be strengthened in two important directions. First, the dynamics at

The (_{u}(_{v}(_{1}}, _{1}, _{2}}, _{2}, _{3}}, and

We now proceed with precise description of inessential edges.

Definition 5.2. Let

The edge

The edge _{u}} and a ^{S(u)} such that ^{w} ∈ 𝔹^{S(u)} is defined by

Otherwise, the edge

An edge

While the output inessential edges

In parameter factor graph

Finally, in parameter graph

In the following exposition, it will be easier to consider the cutting of input inessential and output inessential edges independently, that is,

We define the set of parameter nodes in

Definition 5.3. Let

Let

Let

each edge

each edge

for every

Note that

We define in Section 5.1 two projections. First,

that relates parameters with the same dynamics when we cut a single input inessential edge

that relates parameters with the same dynamics when we cut a collection of output inessential edges. Importantly, as we show in Proposition 8.4, these two projections commute:

The following two theorems are the main results of this section, establishing a correspondence between the dynamics of the networks

Theorem 5.4. Let ^{in}(·;

There is a surjective map μ^{e}:^{e}(

The Morse graph ^{in}(^{e}(

There is a one-to-one correspondence between the Morse nodes labeled

The semi-conjugacy of the dynamics is strengthened to conjugacy when every cut edge is output inessential so that

Theorem 5.5. Let ^{Q} ∈ Θ(^{R} ∈ Θ(

The target point map

Consequently, the map ρ^{OFF,ON} is an embedding of the state transition graph

The Morse graph

In the remainder of the section, we will build the machinery for the proofs of Theorems 5.4, 5.5, to be found in Sections 6, 7, respectively.

In this section, we define the projection maps

where the set of cut edges is

Given a subset of edges

to be the set of source and target nodes of

To define either of the projections Φ^{in}, Φ for given parameter

The idea for the construction of the order parameter ϕ_{u} from θ_{u} is simple: We define ϕ_{u} so that the ordering of the uncut edges is the same as the ordering given by θ_{u}. Explicitly, for (

to be the number of out-edges of _{u}(

For (^{in}((

To construct a

That is, _{u} then to _{u}. So, for a given input to _{u}, its value is computed by evaluating _{u} on the same input with missing inputs replaced with β^{u}. For each

The (

Assume now ^{in}(·;

Remark 5.6. An astute reader will notice that the definition of Φ^{in} and Φ only differs in choice of fixed input β; in

that is, the Φ^{in}(

Remark 5.7. Observe that although we designated the domain of Φ as the set of parameters in ^{in} is also defined on

In Section 8, we provide some algebraic properties of Φ. In particular, in Theorem 8.1, we show that the ^{V} that changes the signs of all edges incident to all nodes, commutes with Φ when the identities of

We also prove that Φ^{in}(·;^{V}.

Recall that

to be the set of states which border the threshold corresponding to _{u} with one of two values. See the gray double rectangles in the right column of

Given an edge ^{e}:^{e}(

An illustration of this collapse is given by comparing the single gray rectangles in the left column of ^{e}(

In Proposition 8.3, we show that μ^{e} commutes with λ^{V}, the reflection bijection defined at the beginning of Section 4, as is suggested by Theorem 8.1 which says Φ^{in} and ^{V} commute.

^{e}(

For _{w} = _{w} and

Next, we consider node ^{u} to node _{w} = ϕ_{w} so that _{u}

If _{u} and _{u}. By assumption, _{u} since _{u} and _{u} agree,

If ^{e}(

as desired.

Finally, we consider node _{w} = θ_{w}, so that _{v}(_{u}, we have

which implies

□

Theorem 5.4 (2) follows from the semi-conjugacy: For any path ^{e}(

To prove Theorem 5.4 (3), we need the following Proposition that shows that most of the edges in ^{e}(^{in}(

Proposition 6.1. Consider a single edge ^{in}(^{e}(

On the other hand, if

If ^{1}, ^{2} ∈ ^{e}(^{1}) = μ^{e}(^{2}) = ^{1} → ^{2}, ^{2} → ^{2} ∈ ^{2} → ^{1}, ^{1} → ^{1} ∈

If ^{e}(

If ^{e}(^{i}) =

If ^{e}(

_{w} and it is strictly monotone in that it satisfies

for η ∈ {±1}. Furthermore,

Since μ^{e} is injective on ^{e}(

Using

which implies

To complete the first claim of the proof, now suppose ^{e} is injective on ^{e}(

Since the target point ^{e} is injective on

implying

Next, we consider the four cases of

^{1}, ^{2}} = (μ^{e})^{−1}(

Since ^{w}(^{1}; ^{w}(^{2};

so that either ^{1} → ^{2}, ^{2} → ^{2} ∈ ^{2} → ^{1}, ^{1} → ^{1} ∈

^{e}(

^{1}, ^{2}} = (μ^{e})^{−1}(

Then, for both

Using

^{e}(^{i}) =

Let

^{2}, and ^{1}. Noting that

Theorem 5.4 (3) is obtained from Proposition 6.1 as follows. When the Morse graph

In this section, we begin the proof of Theorem 5.5 with the proof of point (1). Recall that an assumption of the theorem is that the

Definition 7.1. Let

where for each network node

using

Note that _{u}(_{u} and the target point map _{u}(_{v}(

As suggested by the name, for ^{u}:^{S(u)} introduced at the beginning of Section 2.3.

Proposition 7.2. Let

For each

Since

The idea of the proof of Theorem 5.5 (^{OFF,ON}, which appears in the commutative diagram (

Definition 7.3. The relabeling map

We now show that the value of the input map for ^{OFF,ON}(

Lemma 7.4. Let

where we suppressed the dependency

_{w}(_{w}. Consequently, by the definition of

If _{w}(_{w}. Consequently, by the definition of

This completes the proof for the case

Now consider an edge _{w}(

Since _{w} < θ_{w}(

We are now ready to prove the main result of this subsection.

^{OFF,ON} and

First note that by Lemma 7.4 and the definition of

for each

where the second equality follows from the fact that

Since the target point maps are conjugate and the asynchronous dynamics

□

Finally, we derive consequences of this result for the Morse graphs. Theorem 5.5 (1) implies that the only recurrent set of

In this section, we prove Theorem 5.5 (3) and (4) by constructing a collection of right inverses, {Ψ} to each projection Φ, each of which embeds the parameter graph

Given a parameter ^{OFF} ∈ Θ(^{ON} ∈ Θ(^{OFF} and ζ^{ON}:

We next discuss how to construct a logic parameter

We start by listing conditions we impose on map Ψ in order for it be the right inverse of Φ and to preserve parameter graph adjacency. Let ^{1} is an input vector with entries corresponding to the edges that exist in both networks and ^{2} is an input with entries for edges that exist only in

Whenever

Condition (

For each logic parameter

Then, this implies

implies

To see that the condition (^{1} and unique ^{2} ≠ β^{u},

In general, there are many embeddings of ^{OFF}, ζ^{ON}) because the values of

Because the anchoring logic ^{1}, ^{2}) where ^{2} is incomparable to β^{u}.

Definition 7.5. Given the anchor type (

for each

The first condition in each line of ^{2} ≺ β^{u} and ^{2} ≻ β^{u} is required by constraint (3). The case ^{2} = β^{u} is a consequence of our choice of anchor type, ĝ.

Given an anchoring logic

We define

where (_{u}, θ_{u}) is defined as follows.

The _{u} is defined so that the edges in _{u} and ordered according to _{u} and ordered according to _{u}. Explicitly,

The _{u} is defined so that the edges in

We illustrate our construction of Ψ in

Embedding of factor graphs. ^{S(u)} with each circle containing an input _{v}_{w}. Empty (filled) circles indicate inputs where the value of the logic parameter

Our first result on Ψ is that the constraints on the set of anchoring logics

Proposition 7.6. For any choice of anchoring logic

Let ^{1}, ^{2}) and ^{1}, ^{2}) be elements of

If ^{2} = β^{u} = ^{2} then ^{1} ≺ ^{1}. Since

If both ^{2} ≠ β^{u} and ^{2} ≠ β^{u} then, since the anchoring logic

Let ^{2} = β^{u} and ^{2} ≠ β^{u}. Then ^{2} ≺ ^{2} implies β^{u} ≺ ^{2} so that

Similarly, if ^{2} ≠ β^{u} and ^{2} = β^{u}, then ^{2} ≺ ^{2} implies ^{2} ≺ β^{u} so that

This shows that

Next, we show that _{u} satisfies the ordering condition (^{2} ≠ β^{u}

and the ordering condition for _{u} is satisfied. Suppose ^{2} = β^{u}. Then

Since _{u} satisfies the ordering condition, this implies _{u} satisfies the ordering condition at (^{1}, ^{2}).

It is straightforward to check that θ satisfies (

We next establish that the collection of co-domains ^{OFF} or ζ^{ON} produce distinct sets of order parameters Θ(^{OFF}, ζ^{ON}) follows directly from

Proposition 7.7. 1. The set of anchoring logics

2. If

is a valid anchoring logic in

(2) Suppose ^{1}, ^{2}). Since ^{2} and β^{u} are incomparable, and hence ^{2} ≠ β^{u}. Let ^{2} ≠ β^{u} and

so that

We are now ready to prove Theorem 5.5 (3) which states that Ψ is a right inverse of Φ. It is implied immediately from the following theorem which also identifies

Theorem 7.8. Fix ζ^{OFF} ∈ Θ(^{ON} ∈ Θ(

Let

By

Since θ_{u} orders the edges in _{u}(_{u}(_{u}(_{u}(_{u}(

This holds for each _{u}, so

Next, we verify that

where the first equality follows from

Having shown Ψ is a right inverse, we now show that Ψ is a left inverse when Φ is restricted to

First, we show that

because #_{u}(

because by definition of

because by definition θ orders the edges in ^{OFF}. This covers all types of target nodes

Next, we show that

Now consider ^{2} = β^{u}, we have

where the first equality follows from ^{2} ≠ β^{u}, we have

where the first equality follows from

Since

Finally, we prove Theorem 5.5 (4) which states that Ψ is a graph embedding of

^{OFF}, ζ^{ON}, and anchoring logic

Let _{u} and

To see that _{u} and _{u} and

satisfies _{u} and

Next, suppose _{u} and

On the other hand, for ^{1} ≠

On the input (^{1}, ^{2} ≠ β^{u}), the value of _{u} and

That is, (^{u}) is the unique input so that

Next we show that for _{u} and _{u}(

and for

On the other hand, if

We conclude that θ_{u}(_{u} and

Since Ψ is invertible and adjacency of

Theorem 8.1. (a) Let ^{V} commutes with the projection map Φ(·;

^{V} commutes with the projection map Φ^{in}(·;

We will show that

First, we show that _{u}(

where we now explicitly include the dependency on the parameter

We therefore need to show that

Note that ^{V}(_{u} = |

as desired.

Now we show

Since

where in the first equality, we have used

Since

We have already shown that ^{u} in

To prove the second part, we note that the only difference between Φ and Φ^{in} is that in the definition of logic parameter

Since the projection map Φ and the ^{OFF,ON}, ^{V} commute as well. In the following two propositions, we show that this is indeed the case.

The first proposition relates attractors ^{V}.

Proposition 8.2. Let

^{V}(·;

Since

and therefore

Next, we show

Since λ^{V} is an involution and ρ is invertible, this will prove the proposition. Let

On the other hand,

Since

□

The second proposition relates the surjective map μ^{e}(^{V}.

Proposition 8.3. Let ^{V}(

Since λ^{V} is an involution, this will prove the proposition.

Let ^{e}(_{w}(_{w}(

Now consider the

where the last line follows from

□

Finally, we prove that projections commute and that successive projections are equivalent to projecting every edge simultaneously.

Proposition 8.4. The projection map Φ satisfies the following properties.

Φ(·;∅, {_{2}}) ∘ Φ(·;{_{1}}, ∅) = Φ(·;{_{1}}, {_{2}}) = Φ(·;{_{1}}, ∅) ∘ Φ(·;∅, {_{2}}).

Φ(·;{_{1}}, ∅) ∘ Φ(·;{_{2}}, ∅) = Φ(·;{_{1}, _{2}}, ∅) = Φ(·;{_{2}}, ∅) ∘ Φ(·;{_{1}}, ∅).

Φ(·;∅, {_{1}}) ∘ Φ(·;∅, {_{2}}) = Φ(·;∅, {_{1}, _{2}}) = Φ(·;∅, {_{2}}) ∘ Φ(·;∅, {_{1}}).

The projection map Φ^{in} satisfies

Finally,

Let _{1}}, {_{2}}), and

We will show that

First, we show that

where we now explicitly include the dependencies on the parameter

To prove

Applying _{u}, we have

Since _{u} below

We therefore conclude

Next, we show that

Since we have shown

To show the second equality, we apply Theorem 8.1 and the fact that ^{V} is an involution to the first equality:

Both equalities in (2) follow from a similar argument for the first equality in (1). Statement (3) follows from applying Theorem 8.1 to (2). Statement (4) follows from statement (3) and a realization that the only difference between Φ and Φ^{in} is independence of the latter on division of cut edges into

□

We describe the embeddings of the parameter graph of negative feedback loop

We first consider the negative loop which requires that we cut the edge ^{e}(

There are two embeddings: If

Checking the

Therefore, the embedding Ψ_{1}(

The embedding Ψ_{2}(

Note that the fact that these embeddings intersect does not contradict Theorem 7.7, since both of these embeddings have a single anchor logic.

We also note that the oscillatory behavior in the

We conclude that these two nodes in

The embedding of _{yx}:_{xz}:_{yx}, _{xz}. Evaluating the MBF at node

We note that the bistable behavior in the

We conclude that these four nodes in

So far, we only considered cutting of output inessential edges which results in the embedding of the entire parameter graph. We now consider parameters at which the edges that are being cut are input inessential.

We now describe input inessential parameters (

support essential edges forming the negative feedback loop NL; and

the edge

To satisfy the first condition, the only choice in

Therefore at 4 parameters

Theorem 5.4 applies. There is a semi-conjugacy of the dynamics on the state transition graph, and therefore, these parameters support oscillations generated by the negative feedback loop.

Similar arguments show that all parameters where at least one of the edges _{yx} and _{xz} is input inessential, but the bistable loop is essential, will exhibit bistable dynamics.

We first consider the first condition that _{yx} is input inessential. This does not affect the choice of parameter in _{yx} is input inessential, _{yz} is input essential, and where the edge _{zy} is output essential is the function IV. In _{yz} is output essential are {2, 4, 9, 11}. Therefore for parameter combinations

there is a semi-conjugacy of state transition graph dynamics onto dynamics of positive loop PL, which exhibits bistability.

The second condition that _{xz} is input inessential is satisfied in _{yz} input essential, while choices {2, 4, 9, 11} in _{yz} output essential. Therefore for parameters

Theorem 5.4 also implies existence of bistability. Both edges _{yx} and _{xz} are input in-essential on parameters where two collections (

It is now possible to investigate relative positions of parameters that support bistability and those that support oscillations. For instance, parameter (

In this article, we study a question of the relationship between network structure and its dynamics. In the DSGRN approach, the parameter space of switching ODE systems compatible with the network structure is first decomposed into a finite number of domains that correspond to multi-level Boolean systems [

DSGRN analysis is in some sense complementary to the analysis of equations of mass action kinetics. Enzymatic chemical reactions in cellular biology are often modeled by saturating Hill functions. While DSGRN follows a switching system approximation of Hill functions by piecewise constant functions, thus emphasizing the saturating behavior of the Hill functions, the mass action polynomial approximation emphasizes the non-saturated part of the Hill function. General lack of experimentally determined values of parameters requires the development of combinatorial, often graph based, approaches for both types of approximations [

The main result of the study describes how dynamics of a subnetwork manifests itself in the dynamics of a larger network. In systems biology, the concept of a network is central to understanding the function of a cell and its responses to the environment. Unfortunately, the networks studied by biologists are always incomplete, uncertain, and the parameters mostly unmeasurable. Yet the need for predictive models is very high. The results presented here suggest that the DSGRN approach can provide precise understanding of whether or not a network can produce distinct phenotypes (i.e., normal vs. cancer) and how robust are such predictions under network embedding, provided that the expectations imposed on predictive modeling are properly adjusted, that is, there is no expectation that our approach can reproduce precise time series trajectories of multiple genes.

A popular link between structure and function has been proposed in the theory of

Our present study provides some answer to these questions. We show that the dynamics of the subnetwork can be always found in the dynamics of the larger network, provided that it is “uncovered” by setting parameters of the additional edges of the larger network to be constitutively ON or OFF. As such, the larger network is always capable of reproducing dynamics of each of its subnetworks for an appropriate choice of parameters.

Another important conjecture in systems biology asserts that the reason why we observe large networks exhibiting dynamics that smaller networks can produce on their own (say oscillations) is that redundancy enhances robustness of the phenotype. Our observation that the parameter graph

We believe that the proposed framework of DSGRN to study the relationship between network structure and its function (i.e., dynamics) will provide many important insights in systems biology.

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

WD: Writing – original draft, Visualization, Investigation, Formal analysis. BC: Writing – review & editing. TG: Writing – review & editing, Supervision, Conceptualization.

The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This research was partially supported by NIH 5R01GM126555-01.

WD was employed by Immunetrics.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.

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