Cancer Metastatic Processes Contextualized by Networks: Role of Entropy, Symmetry and Controllability.Networks are pervasive in biomedicine, especially in the realm

of experimental studies and computational applications. Their

relevance reflects the ability to model systemic complexity. While

entropy has been central to many studies focused on equilibrium,

more recently other concepts like symmetry, controllability and

synchronization have emerged. In light of such properties,

networks’ impacts are still partially unknown in domains like

cancer research, for which a thorough assessment is here provided.

Depending on the question we ask, the use of networks can be

different. In general, the analysis of complex processes such

carcinogenesis involve the consideration of the so-called cancer

hallmarks. In particular, it would be important to establish how

the metastatic processes take place and progress through the

disease stages. One way to tackle the problem is to analyze how

network states can reflect these metastatic processes, possibly

identifying dysregulation paths whose characteristic features

might guide inference. Our main goal is to cast into network

cancer domain information such that it can be translated into

well-defined and interpretable features. Therefore we consider a

spectrum of entropies, at both local and global network scale,

and we look at both symmetry and controllability. A few examples

are provided to demonstrate the role that typical network

constraints may play when measures or indicators of metastasis

are available.

by Enrico Capobianco

Introduction

1.1. Complexity through the lens of Networks

Cancer research is a paradigmatic complex

context representing the natural setting for

network science studies and applications.

It is well-known that multiple factors must

be considered to explain the carcinogenetic

process, from endogenous ones like DNA

replication or cellular interactions, to

exogenous ones, like for instance lifestyle,

diet and environmental influences. Given the

redundancy (number, variety, etc.) of these

influencing factors, the impact exerted on

cancer heterogeneity is clear. Heterogeneity

is the most inherently complex characteristic

of cancer, the one that makes hard to cure

it because the one that eventually fuels the

resistance mechanisms that negatively affecting

treatment response1.

Redundancy in cancer is typically recognized

because of the multiple bioentities to be

considered and the various types of associ-

ations. Especially through the recourse to

genomics studies, redundancy has become

even more an issue. Genomes can reveal

differentiated somatic alterations of normally

functional cells between and within

cancet types2. Most importantly, functional

redundancy is fundamental to establish a sort

of biological robustness, and the complexity

of the relationship between redundancy and

robustness is often described by quantitative

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66

approaches3. Finally, only a fraction of all

possible genomic measurements can be

detected through profiling techniques: it is

especially important at the translational level

the fact that relatively few examples are really

useful to either biological or clinical validation

purposes. In fact, most experimental validation

techniques, including clinical trials, are not

designed for high-throughput redundant

analysis, thus reducing the list of candidate

genes or proteins playing the role of putative

diagnostic biomarkers or therapeutic targets.

Another property of absolute relevance for

cancer treatment purposes is synergy.

Synergistic interactions are typically found

in cancer contexts, and they involve genes,

proteins, drugs etc.4. However, synergies

are quite hard to explain5. In principle,

exploiting synergy in a complex system means

to be able to limit the overall redundancy

effects. Equivalently, its informative contents

can be assigned to a reduced number of

components able to combine together and

amplify their individual contributions. Gene

regulation, pathway activation or clinical

markers imply the role of synergy, especially

because with such ensembles the use of

pairwise associations cannot ensure a global

understanding of complex dynamics. Despite

the clear need of identifying synergistic

regulation or dysregulation for inference

scopes, most of the times the identifications

that one can get are computationally

intractable, such that one turns again to

pairwise relationships. This is clearly a

coarse approximate solution.

In general, networks support the co-existence

of redundancy and synergy, but offer also

means to deal with them. Specifically in the

cancer context, as we aim to show. First of all,

consider modularity. A modular architecture

is known to add information content. This

is the case when the identified modules and

sub-networks are associated with functional

aspects and even complex functions,

possibly part of processes for which a

sequence of states can be defined. Cell types

can indeed be subjects to transitions between

the different states, depending on their

reference conditions. Intuitively, cellular

states could be studied by discrete-time

Markov models. Knowing the transition rates

for healthy and diseased cells could be

important for understanding disease

mechanisms, for estimating the temporal

effect of a certain perturbation or stressor,

and also for predicting the composition of

the cell state at a given time.

Other concepts addressing states and disease

phenotypes of interest involve the recourse

to so-called combinatorially dysregulated

subnetworks, which are obtained by genomic

systems decompositions aimed to identify

states that are most informative for pheno-

types 6,7. Synergistic dysregulation can leverage

differential expression, in particular when

synergy acts towards those genes for which

significant measure is lacking. The presumed

functional similarity in biological modules

legitimates the importance assigned to

coordinated actions that may exert effects

over phenotypes, including cancer ones.

1.2 Cancer Networks Taxonomy

Networks are powerful tools to infer the

dynamics of cancer systems because their

structural and functional characteristics

combine with the ability of visualizing and

abstracting complex features of experimental,

computational, clinical conditions. To this end,

networks can effectively synthesize different

types of data (qualitative and quantitative),

as well as inputs from various data generating

processes, such as sources of genomic,

metabolomics, proteomic, and imaging

data. Moreover, they provide visibility into

cross-disciplinary problems and areas, not

just for the aims of reframing and envisioning.

Networks guide discovery, deliver solutions,

and aid in managing the transition between

phases of complex processes.

A taxonomy of network types and models has

emerged in the recent years across disciplines.

Notably, this taxonomy remains still partial

in biomedicine, but progresses diffusively

in cancer research due to the advances in

epigenetics, non-coding RNA, imaging, just to

mention only a few examples. Multiple steps

are involved in the definition of ‘Next

Generation Networks’, following what has

been already seen in the Information

Technology and in the Telecommunication

fields, and providing ubiquitous connectivity

with pervasive accessibility to service,

application, content and information. Both

algorithms and methods are destined to

adapt to emerging applications covering

topics at the intersection of disciplines,

such as optimization of network design and

functionality, measuring influences of physical

network structural characteristics in

characterizing robust solutions, cross-layer-

ing algorithms for in-depth understanding of

cellular networks, considering the interactions

between multitype networks while balancing

their coexistence at the infrastructural level,

processing information packets instead of

single entities, surveying anomaly detection

approaches and monitoring early warnings,

facing ever increasing heterogeneity across the

application contexts.

When considering the exposome, it is clear

that this is becoming a crucial source of data

contributing significantly to cancer prevention

by prospectively reducing the global burden

of disease while remaining a prominent public

health issue8. The translation of environmental

epigenetics research to environmental policy

and public health solutions will enhance

chemical risk assessment and enable clinicians

to identify at-risk populations prior to disease

onset. For instance, environmental toxicants

exert their impact on health and disease

in part by modifying the epigenome, DNA

modifications that do not affect the underlying

sequence but can result in altered gene

expression and downstream phenotypic

changes. Research addressing perturbation

mechanisms at the epigenome level, from the

biological and functional significance of small

epigenetic changes (e.g. DNA methylation,

histone modifications, and non-coding RNA

expression), to the impact of epigenetic change

longitudinally on health outcomes are key

aspects.

The network taxonomy that is here considered

specific to the cancer context but is general-

izable to other diseases, is described in view

of challenges centered on three main network

properties: Entropy, Symmetry, Controllabil-

ity. In particular, entropy can be associated

with quantification of uncertainty and this

affects the discovery process; symmetry can

be associated with system robustness and from

here with resilience, but also generalization;

controllability concerns the ability to perform

feasible and accurate selection and to improve

prioritization. It is in particular the multifaceted

metastatic process that will be used in the

presented examples and use cases.

2 Results

2.1 Entropy to decipher Complexity.

There are many possible ways to introduce

entropy in complex biological systems,

including cancer ones; it is perhaps useful

to look at concepts whose potential is not

yet completely exploited. For example, a

prominent principled concept is set

complexity9-11. In biomedicine, this might

help the interpretation of network constraints

of functional relevance from a biological

viewpoint. The associated measure is called

differential interaction information, and the

relevance at biological levels is conveyed by

the global dependence idea and its collective

flavor, typical say of regulatory dynamics

between transcription factors and genes, or

between microRNAs and genes, to mention

two examples among other possible causal

associations revealed in multiple types of

networks. Biological networks are clearly

neither random not regular, instead they are

interesting mixtures of the two properties.

In particular, the network modular architecture

is able to maximize the set-complexity of a

graph. Set complexity is defined below for

two nodes i,j of a network as:

1where S is the Shannon entropy and MI

is the mutual information between nodes i,j.

Modularity is basically enforcing the

connectivity between nodes and represents

a characterizing structural feature, which is

quantitatively reflected in the MI measure.

As said, the gain comes from functional

significance of the dependence that goes

beyond simple correlation. In genomics,

this might be relevant to capture nonlinear

or high-order interactions due to highly

complex regulation signatures.

A second relevant concept is basin entropy,

used to parametrize the dynamical uncertainty

of a network from the connectivity of its

components, thus probing its behavior with

varying parameters12. Seen in relation with

basins of attraction that link initial conditions

to final states, this measure is useful to quantify

the possible unpredictability of the dynamical

system underlying the network13. For gene

regulatory networks, basin entropy can be

defined by considering the basins as the state

space components, as follows14:

S = - Σi si log(si)

2where si indicates the size of the

basin I, and S is minimal when one basin has

all the states and maximal when the states are

distributed into different attractors.

Since the state of a network defines its activity

through the component nodes, a state is clearly

a dynamic entity destined to fluctuate according

to trajectories determined by the various

transitions occurring from state to state. In

general, trajectories are purely random in a

stochastic system, and in noisy systems these

trajectories can exhibit multiple dynamical

regimes, unlike in deterministic systems. It may

also be possible that states tend to recur, in

such case there are attractor states, or simply

attractors. Other states may tend to flow into

attractors forming their basin, or basin of

attraction. The important aspect is thus that

the size of an attractor is contributed by the

number of basin states gravitating around it.

Entropy can be interestingly computed

for both these types of states, and being

state-space partitions not unique for a given

network, the same holds for the related state

entropy. This offers a rationale for relying on

average entropy measures, computed on

network ensembles.

Speaking of average entropies, a reference

goes to the centrality of fluctuation theorems 15.

They describe systems’ non-equilibrium

states analytically, through the statistical

fluctuations in time-averaged properties. This

can be rephrased as stating that the resilience

(or robustness) R in a system is correlated to

its level of uncertainty, hence its entropy S,

according to:

dR dS > 0

Ψ = c ∑ i ∑ j max{ Si Sj} MIij (1-MIij)

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32.2 Metastasis: Stochastic Dynamics.

An entropy increase in the cancer context

can depend on the robustness of cancer cells

to perturbing factors or environmental

stressors, including the tumor microenviron-

ment16. However, it was observed that

differential expression, which leads to cancer

identification, and differential network

entropy are indeed two anticorrelated

entities17. An example is offered with

overexpressed proliferation genes (carrying

oncogenes) that were found associated with

reduced network entropy across several

cancers. Notably, the impact in cancer research

is relevant because this relationship suggests

that the metastatic potential could be

established at early stages of disease progres-

sion through proper entropy measurements.

It has also been established that from the

analysis of transcriptional signatures associated

with perturbed cancer-related genes, metastatic

cancer seems to fall into two main subtypes:

epithelial to mesenchymal transition or ETM-

like associated to an inflammation state, and

proliferative state (increased metabolism and

systemic stress signaling)18. Typically, ETM

transitions occur in cell morphology with

a corresponding reduction of cell function

synchronization and increased motility levels.

Another possible finer subtype derived from

the adaptive stress signaling could be referred

to the so-called phenotype switching, which

occurs after the inhibition of receptor

(non-receptor) kinases, or in response to

(chemo-) therapy. Most importantly, signaling

events occur not only dynamically but rapidly

in cancer cell that respond to therapy, and

then can turn to survival or not depending on

resistance. Adaptive stress signaling has been

indicated as the reference emerging research

area to such developments19.

Metastasis evolves over time and in response to

therapy. Therefore, cancer might in principle

be modelled according to a stochastic sampling

process, and the genotypes would be the result

of random drawing from mutational pooling,

this latter acting as the process underlying

the data generating mechanism20. A mutation

ensemble would thus be composed of various

states to which perturbed networks can be

associated depending on the probability of

state activation or repression following a

certain mutation. Under conditions of

equilibrium, or stationarity, the observed

experimental data endowed with the

phenotypic response (metastatic or non)

can be considered the outcome of ensemble

averages operating over many states. These

states have differentiated entropy degrees,

and because they interact between them

and with the environment, the influences at

systems level fluctuate depending on entangled

versus decoherent conditions.

A stochastic matrix can be formed by the

transition probabilities between states Xi

and Xj, assuming they are constant over time

and depend only on the reference state. This

way, the transition matrix would define a

discrete-time Markov process explaining state

evolution dynamics of the population of cells

under consideration21,22. At a network scale, a

dynamic rewiring occurs through node

interactions that determine the evolving

network state. When no change of state is

observed over time, an attractor can be

observed, or an attractor landscape if all

possible state trajectories are considered.

Both genetic and epigenetic alterations can

fuel such network evolution process, and

genomic instability or loss of proteostasis are

among the possible observed causes of

uncontrolled cell proliferation23.

A steady-state uncertainty measure of the

information flow is provided by the network

entropy computed from the entropy rate ∆S24:

∆S = ∑iπi Si

4considering the stationary distribution

of the transition probabilities π and with the

entropies Si computed locally (at each state).

A change in entropy rate would imply that

the effects of signaling are likely present in

response to some type of perturbation.

While an interesting interpretation has been

offered in a context of primary and metastatic

versus non-metastatic cancer, showing that

in the former case the entropy increases and

thus becoming an indicator of cancer progres-

sion, a finer formulation would be required to

consider network nodes instead of the states.

The local entropy for a given node would be a

special case:

Si = -[log( k)]-1 ∑i pi log pi

5with k as the degree of the reference

node i, and pi the probability linking the

reference node with other nodes. An

important consequence is that cancer

phenotypes can be assessed by the use of

their local entropies from node-wise degrees.

The distribution obtained this way can allow

the assessment of network heterogeneity, and

in particular how randomness characterizes

metastatic vs non-metastatic networks.

A principled elucidation of the dynamical

responses of cancer networks naturally

involves therapy, and the starting point is

measuring the differential gene expression

that determines phenotype-driven network

configurations. Under the influence of external

factors, a phase transition may occur once

certain critical levels are reached. A normal

cell can undergo a transition and thus become

not apoptotic due to cumulated mutations

too. Epigenetics play a central role in such

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regards25. However, relevance goes to genome

stability too, in parallel and in light of genomic

rearrangement studies elucidating concepts like

chromothripsis and chromoplexy that underlie

catastrophic events. In general, some types of

compensatory perturbations might be designed

and used to direct a network to a desirable

state, thus deviating from undesirable ones.

This topic is re-considered next, within the

control arguments.

While a phase space represents all possible

states in a dynamical system, it is important

to note that biological systems tend to be

sparse, with a high number of degrees of

freedom reflected over the big set of parameters

characterizing the system. Similarly, generators

of complexity are the interactions among

bio-entities like proteins, many of them being

false negatives or missing values, or the profiles

of genes, only in part significant due to noisy

signals and presenting multiple transcripts.

Depending on the possibility to limit the

overall uncertainty, this complexity can be

mitigated by dimensionality reductions aiming

at preserving the structural information, Due

to sparsity, the local structures are those that

usually orient the phase space while the system

evolves. For example, multiple evolutionary

trajectories are observed in cancers even

before achieving a transformed state, such as a

metastatic one, thus establishing the role of the

temporal dimension of cancer heterogeneity.

2.3. Symmetry: General concepts.

Biological signals that cause a shift in the

normal balance of symmetric and asymmetric

cell division can trigger differentiation arrest,

and for instance cancer progression26.

Studies on symmetric stem cell divisions

have considered their role in homeostasis27. While asymmetric division is very

important in generating diversity during

development, its dysregulation is relevant

for promoting oncogenesis. It is well-known

that a small fraction of cancer stem cells or

cancer-initiating cells exist in the affected

cell populations exhibit resistance to many

anticancer drugs and produce many heteroge-

neous tumor cell populations, resulting in

a high frequency of tumor recurrence

and metastasis Cancer stem cells undergo

asymmetric cell division, which is a physio-

logical event of normal stem cells. It occurs

during development and tissue homeostasis

and maintains a balance between self-renewal

stem cells and differentiated cells in a single

division, therefore between the stem cell

pool and the progenitor cell pool28,29.

Symmetry is also a key factor in brain network

functionality, to which neural activity must

be coordinated according to mechanisms that

remain largely unknown. Recently, behavioral

hypotheses have emerged, and they suggest

that symmetry breaking of network connectivity

shapes the timescale hierarchy that eventually

ends in some target functional subspace. In

turn, the behavior emerges when the appropriate

conditions imposed upon the couplings are

satisfied, justifying the conductance property

of synaptic couplings30,31.

The structure of complex networks is increas-

ingly chosen as the context of application

of symmetry concepts. This depends quite

naturally on the presence of connectivity

patterns that are rarely unique in the same

network. Therefore, of interest here is how

these patterns that may determine symmetry

can influence structurally and functionally

biological networks.

A few open questions remain:

l Do symmetric patterns represent localized

network dynamics, such as modules, clusters,

motifs? If yes, how specifically vs redundantly

so?

l Are such patterns robust or fragile structures

with regard to symmetry breaking?

When considering the context of biomedicine,

the following questions emerge in relation

with cancer progression and therapy:

l What influences do symmetry patterns exert

at a global network disease scale? Are they

diffusive, heterogeneous, redundant?

l Is some form of control possible over these

patterns?

Detecting symmetries facilitates the achieve-

ment of reliable inference, particularly in

presence of multimodal distributions or

spurious correlations. For instance, likelihood

functions can have symmetries. Consider

sample data x ϵ X and the choice of model

M(x,θ), given the parameters θ ϵ θ*. These

parameters may be non-identifiable due to the

underlying presence of symmetry. Namely, a

symmetry s: θ θ is a measurable function

that makes θ no longer identifiable from the

data x. In such cases, the recourse to symmetry

breaking solutions can be needed to augment

inference performance. The natural way to

obviate to the effects of symmetry is to change

the model parameterization. A re-parametriza-

tion means to enable a transform θ ψ, with

ψ ϵ ψ*, or as an alternative strategy to

constraint the given parameters, say θc for

given c. There are several types of symmetries

and some are local (when likelihood-preserv-

ing) and some are not. Common symmetries

are permutation ones, but there are also

scaling and translation ones.

When the reference parametric context is a

network N, it has been shown that mesoscale

symmetries imply the existence of localized

dynamical modes. These might be unstable no

matter what the structure of the network is, or

might be stable, suggesting that the symmetry

allows to nodes reduction without influencing

the overall dynamics32. In general, defining

symmetries is something inherent to nodes and

their permutations. Two nodes are permuted

when their links are rewired and thus a switch

occurs. Collecting all nodes into the adjacency

matrix A, in which Aij = 1 (if node i and node

j are adjacent) or 0 (otherwise), a symmetry is

present when a permutation P is applied to A,

leaving it unchanged, i.e. PA = A. This estab-

lishes a so-called automorphism, indicating

that nodes are topologically equivalent if

their permutation does not affect the network

structure. This is useful to collapse redundancy

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into network quotients or skeletons33-35, in

which no structural repetition is present but

other structural network properties remain,

together with the inherent complexity. Note

that symmetries may also be combined, and

all the participating nodes form a symmetric

structure. Then, an orbit is present with a set

of structurally equivalent nodes, i.e. all vertices

in the same orbit have the exact same degree.

Formally, given a set of nodes in a network N

and a set of automorphisms A(N) (forming a

group), the orbit of a node n ϵ n(N) is the set

(see36):

O(n) = {λn ϵ n(N) : λ ϵ A(N)}

7as the associated probabilities (ratios

between AP cell nodes and network nodes).

The normalized entropy is then defined simply

as:

(SAP)* = SAP/log N

8Of a certain utility is also to know

that the eigenvalues of a symmetric structure

are decomposable into two types: redundant

and non-redundant eigenvalues. The former

correspond to the eigenvectors localized

on the symmetric structure, while the latter

refers to the eigenvectors whose eigenvalues

are the same in relation with the nodes of a

given orbit. This is relevant for considering the

network based on orbits instead of nodes, thus

eliminating redundancy. Another important

aspect is the consideration of a system at the

steady state versus a possible departure from it.

Such dynamics are regulated by the previously

addressed eigenvalues, regardless the network

structure.

With a probability measure inserted in the

network, a probability distribution P is defined

on its nodes, thus leaving with the network

entropy as a measure of the randomness degree

in the network:

SN(P) = min ∑i pi log pi

9and this implies the importance of the

stochasticity of symmetries (see fuzzy

symmetries38) based on network ensembles.

Furthermore, network communities, which are

cohesively connected sets of likely functionally

similar nodes, exert also an impact on another

important aspect, synchronization39. The

relationship between network topology and

synchronization are not completely clear yet.

The synchronization depends on the spec-

tral properties of networks, as the topology

is summarized by the Laplacian, a symmetric

matrix in general. The second eigenvalue of

this matrix may tend to 0, implying that a state

of synchronization cannot be reached by the

network. An influence comes also from the

highest of the eigenvalues. Besides determining

or not the property, no information is available

from these two eigenvalues on the relation-

ships with network topology, of interest here.

For instance, betweenness is defined as the

fraction of shortest paths between node pairs

that cross the given node, and indicates the

influence that the node exerts in terms of

information flow at network scale. It clearly

refers to synchronization capability. This latter

is also enhanced with high clustering in the

network, measuring the number of neighbors

of adjacent nodes, especially because the

higher is clustering value the closer are the

nodes. Quite intuitively, synchronization is

also enhanced by degree heterogeneity, as this

reduces the average distance, and it has also

been shown to be affected by the ratio be-

tween the highest and lowest degree nodes40.

5Therefore, an orbit is the set whose

nodes can be obtained from one another by

simple permutations in A(N), and a symmetric

network partitioning of nodes into orbits

establishes disjoint equivalence classes for

each node, i.e. an automorphism partition (AP) 37.

Of interest is the corresponding entropy,

a measure of the network structural

heterogeneity, such that for n=1,N:

SAP = - ∑n pn log pn

6Pn = |ni| / nN

72

There are cases in which the functional

similarity refers to symmetry. In such cases,

node similarity could not be due to the

community effect. A typical example is

provided by brain areas. Note that the role

of symmetries with reference to network

synchronization is investigated in41, while the

computational methods to break such

symmetries (isolated desynchronization)

are discussed in42. The construction of

functional networks depends on the relation-

ships between their coupling components,

which makes synchronization motifs central

features. Functional networks are in general

heterogeneous, thus non-symmetrical

structures. This reflects the fact that disruption

of the couplings generates symmetry-breaking

in the network, and also loosens the inherent

synchronization motifs distribution. However,

if the couplings are able to sustain synchroni-

zation, then symmetry will be characterizing

the functional network43.

2.4. Controllability: Relevance for Cancer

In general, networks evolve through a series

of transitions between states, but also by

changing structure. The typical hierarchical

structure presents clear directionality, while

in the symmetric structure more fluctuations

can be observed, inducing alternating

order-disorder transitions. In cancer, different

sets of control parameters may be present

to characterize the cancer type and to drive

the transition from normal to cancer state.

Using signaling as a reference, a human cancer

signaling network allowed the identification of

driver nodes and classification of their role in

the structural controllability of the network.

The underlying hypothesis is that nodes that

are critical for achieving centralized control

are therapeutically important. In particular,

anti-cancer drugs primarily act through such

genes that serve as their targets, thus revealing

utility for analysis of disease networks with

therapeutic implications.

Cancer forms a robust system that maintains

stable functioning (cell sustenance and

proliferation) despite perturbations. Cancer

progression occurs stage-wise over time

with increasing aggressiveness and worsening

prognosis. The characterization of these

stages and identification of the genes driving

the transitions are critical steps for an

understanding of cancer progression and the

development of effective anti-cancer therapies.

Valuable insights into cancer progression

was recently revealed by a few factors:

l The presence of interactions involved in core

cell-cycle and DNA-damage repair pathways

that are significantly rewired in tumors,

indicating significant impact to key

genome-stabilizing mechanisms;

l Several flipped genes that are serine/

threonine kinases which act as biological

switches, reflecting cellular switching

mechanisms between stages;

l Different sets of genes flipped during the

initial and final stages, indicating a

progressive pattern.

Based on these results, the robustness of

cancer can be derived from exchange

mechanism between genes at different stages

and from different biological processes and/

or cellular components. These are all involved

in progression stages and thereby allow

cancerous cells to evade targeted therapy,

therefore suggesting that a possible effective

therapy should target a “cover set” of these

genes. The control problem can in general be

concentrated on the search for a master

regulator of perturbations affecting target

bioentities. The principle of structural

controllability can thus be used to identify

a minimal set of driver nodes that exert

control the network states.

Since nonlinearity induces dynamics hard to

control, the attractor principle and associated

landscape analysis can be also be usefully

considered for differentiating between normal

and cancer states and optimize possible control

strategies44-47. Co-existence of multiple

attractors allows stepping to an attractor

network, or equivalently a coarse representa-

tion of the system phase space, would then be

possible by identifying all possible attractors

in the system, and once nodes are assigned to

them the gain for control purposes appears

from the fact that driving the network to a

desired state in a finite time would be more

efficiently done48. A control kernel would be

identified as the minimal set of nodes that

once regulated would drive the network from

any initial state to any target state49. Clearly

enough, such kernel inspires the possibility

of identifying its components with cellular

phenotypes, and possibly with drug targets,

and such kernel structure leverage the inherent

property of state coherency (distribution over

the attractors).

Because noise is always a possible inducer of

transitions between states and also network

components, a complementary aspects of

such developments refers to the role that

compensatory perturbations used to direct

the network to a desired state, by taking

advantage of its basin of attraction and thus

exerting a so-called network reprogramming50.

This concept refers to seminal work on survival

signaling in lymphocyte leukemia51,52 and on

cancer subtypes53.

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74

2.5 Controllability: Working Definitions

Controllability, namely the ability of a

dynamical system to step between states, say

from an initial to a final state in finite time,

can be verified quite straightforwardly in

linear time-invariant systems by the so-called

Kalman’s rank condition54. In particular, given

a canonical system:

dz = Az + Bv

10 with state vector z, and A

and B as state and control parameter matrices,

respectively, the condition to obey is:

rank{B,AB,….., AP-1 B} = P

11Naturally enough, things

complicate when the system’s parameters are

unknown. Furthermore, a common strategy

to establish network controllability turns to

the identification of a minimum set of driver

nodes. This implies that controlling the set is

equivalent to controlling the entire system55-57.

Referring back to symmetry, it is of interest

to consider the impact of individual rather

than global dynamics, departing therefore

from the analysis of network topology and

instead considering the diversity of individual

dynamics. This condition is helpful particularly

with real-world networks, as it can be

expected that some assumptions, for instance

the independence of parameters, must be

relaxed. Notably, a global symmetry aspect

seems to emerge, one at which the highest

controllability or lowest number of key nodes

is achieved. Thus, the specific intrinsic

dynamics become less relevant for controlla-

bility purposes than the densities with which

they present themselves58.

This seems coherent with what anticipated

before, namely that the real networks

complexity may be reduced to a context

considering node-specific hidden variables

that once transformed, may reveal latent

symmetries. If at each network N is assigned

a probability P(N), then statistical ensembles

of graphs can be designed by considering fam-

ilies of such stochastic networks59. These will

be stochastically symmetric under a transfor-

mation if each member network has the same

property under the same transformation. An

entropy optimization problem is then present-

ed when searching for the maximum entropy

probability. This problem usually implicates the

existence of a topologically constrained net-

work, particularly when the ensemble network

functions as a null model.

2.6 Applications

A series of examples on cancer systems is

now presented. References are recent

studies in which metastatic cancers have

been analyzed through various methodologies,

and differentially expressed gene profiles have

been obtained.

Use Case 1: Metastatic Breast Cancer

We built networks in Figure 1 and in Figure

2 with data from the study in60 on 123 paired

primary and metastatic tissues from breast

cancer patients. Here, 47 genes (16 significantly

over-expressed and 31 significantly under-

expressed) were found differentially expressed

in metastatic ones. The whole protein-protein

interaction network (source STRING db,

confidence level = 0.7) is shown for the

over-expressed protein coding genes (counting

for 375 nodes and 476 edges) and for the

under-expressed protein coding genes

(counting for 772 nodes and 1113 edges). As

these two networks are big, their minimum

connected configuration is also computed (28

nodes and 52 edges for over-expressed protein

coding genes, and 53 nodes and 141 edges for

under-expressed protein coding genes).

The modules from the reduced networks are

also computed using the algorithm WalkTrap 61

(Label Propagation62 and InfoMap63

were applied too). The second type of

constraint we examine for these data, is gene

regulation according to transcription factors

(TF) (we use ENCODE Chip-seq data as the

source db). The whole network we found for

the overexpressed genes accounts for 217

nodes and 413 edges (reduced to 51 nodes

and 136 edges when the minimum connected

network is considered), and is visibly dense in

communities (each of the employed algorithms

find several modules, while in the reduced

one only WalkTrap finds 6 modules). With

the under-expressed genes the whole TF-gene

network accounts for 313 nodes and 1208

edges, while the reduced one accounts for 73

nodes and 354 edges, and here the WalkTrap

algorithm finds 4 modules.

The metastatic BC data state space has been

thus sequentially constrained, first by the

sign of the gene differential expression, then

by both the retrieved interactions between

DEGs and the TF regulation over the target

genes. Modularity is an additional topological

constraint applied to both the gene network

and the TF-gene network. Modularity remains

present also after reducing the original

network to a minimally connected one.

Therefore, this fraction of the metastatic

profile is highly structured and organized in

modules which imply specific functional

activity. This holds for both over-expressed

and under-expressed genes in the metastatic

BC profile, with a richer modularity map found

for the under-expressed fraction of the gene

profile, possibly induced by the bigger size

more than by inherent structure.

A claim in the original paper on BC subtyping

was that large absolute expression changes

at the RNA level between primary and

76

metastatic disease were not identified, and

indeed this fact weakens the constraints that

can be applied to the state space from gene

profiling. A certain pivotal role (top overex-

pressed gene) was assigned to FGFR4 or

Fibroblast Growth Factor Receptor 4

(well-known player in both ovarian and BC

tumorigenesis, and its overexpression in BC is

characteristic of the HER2-E intrinsic subtype),

and in Figure 1 it appears that it plays a role of

modular driver at whole network scale, but

less at reduced network scale (not in the blue

group). Therefore FGFR4 exerts an influence

but this is specific to a subtype, and therefore

explains why it is not present when the

minimally connected configuration is used.

Being the subtype a specific state of cells, a

lower entropy would be expected due to a

decreased number of degrees of freedom

in the system. Being metastatic both the

configurations we displayed, this local entropy

decrease could also represent a more general

signature.

Overall, the presence of constraints in a

network tends to naturally decrease the value

of its entropies. The topology of the network

establishes several types of constraints. It is

surely more complicated to interpret the

entropy changes in light of the context in

which they occur, say cancer. As a general rule,

higher entropies are associated with increased

network plasticity, as when a re-modulation

takes place, this might be observed from

DEG profiles or activated pathways or wired

modules. It was reported that over-expressed

gene profiles show reduction of entropy

compared to under-expressed gene profiles,

and according to the fluctuation theorem the

latter could indicate superior robustness, for

instance by adapting to the selective pressure

of the tumor microenvironment. Conversely,

lower entropies are typically induced by

constraints, and these limit the state activities

and therefore, at systems scale, the degrees of

freedom.

Figure 1. PIN from 16 metastatic over-expressed genes in BC. Top panel: Left, whole network; Right, modular map

of minimum connected sub-network. Bottom panel: left, whole TF-gene network, Right, modular map of minimum

connected sub-network.

Figure 2. PIN from 31 metastatic under-expressed genes in BC. Top panel: Left, whole network; Right, modular

map of minimum connected sub-network. Bottom panel: left, whole TF-gene network, Right, modular map of

minimum connected sub-network.

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Use Case 2 – Metastatic Stomach Cancer

In a proposed study with 21 metastatic

samples and 11 non-metastatic samples of stom-

ach adenocarcinoma, 2979 (of which

1251 overexpressed, 1728 under-expressed,

and 2593 specific) DEGs were obtained in the

first group and 491 in the second group64. Other

386 DEGs appeared in both groups, therefore

showed similar expression patterns. We took

the top-50 DEGs in this group and built a

network from them (Figure 3, top panel). We

can see that a quite organized and modular

structure appears, even after suitable reduction

(42 nodes and 58 edges).

When we consider specific metastatic

sub-groups (top-ten overexpressed and

under-expressed DEGs), we see that the

small induced networks are not modular,

but centered on only two hubs, HSPD1

(over-expressed) and ESRRG (under-

expressed). Therefore, the topological

constraint of modularity is not due to

metastatic characteristics, and very likely a

substantial degree of entropy in the system

remains undirected to any specific phenotype.

The subtypes that are well-known and usually

investigated are mainly a few, say 4, thus a

relatively low number compared with other

cancers. HSPD1 and ESRRG are considered

promising metastasis biomarkers for this

cancer, the former explaining how the cancerous

cells escape from apoptosis, the latter is one of

the 5 gastric cancer signature genes

Use Case 3 – Melanoma state transitions

In a recently published paper on melanoma 65, the metastatic process is investigated by

the analysis of transcriptional reprogramming

occurring during transition from proliferative

to invasive states. Many genomic regulatory

regions were revealed for the melanoma states

from joint transcriptome and methylome

profiling, and master regulators were identified

in SOX10/MITF (SOX-10 is a marker for

melanocytic differentiation, and its dysfunction

Figure 3. PIN from 21 metastatic and 11 non-metastatic stomach adenocarcinoma genes. Top panel: whole (left)

and reduced networks induced by mixed gene group; Bottom panel: over-expressed (left) and under-expressed

hubsFigure 3. PIN from 21 metastatic and 11 non-metastatic stomach adenocarcinoma genes. Top panel: whole (left)

and reduced networks induced by mixed gene group; Bottom panel: over-expressed (left) and under-expressed hubs.

Figure 4. PIN from proliferative DEG signature in melanoma. Top: whole network. Bottom: reduced networks,

with top150 over-expressed values (left) and the rest (right).

GeneticProfiling

Disease ManagementProtocol

Patient Informatio

78

impairs MITF, the Microphthalmia transcription

factor expression, as well as melanocytic

development and survival), and AP-1/TEAD

(together, they critically regulate transcriptional

and functional mechanisms in cancer cells,

particularly invasion and resistance66).

We present in Figure 4 and in Figure 5 the

protein-protein interactions networks obtained

from two signatures, proliferative (over-

expressed) and invasive (under-expressed)

genes, together with their reductions when

only the top-150 over-expressed (proliferative

DEGs) and top-150 under-expressed (invasive

DEGs) are taken into consideration. It is

interesting to note that while the two global

signature networks are densely built around

key hubs (PDGFRB, ITGB1, EGFR, NFKBIA,

JUN in invasive, and CDK2, STAT5A, TP53,

MYC, BCL2, PTEN, MAP3K1, KDR, KIT in

proliferative), lots of communities appear to

be part of their structures (data not shown).

We thus considered the top-150 DEGs in

both signatures, and the rest of the DEGs,

and plotted their networks. In both signatures,

particularly in the proliferative one, the

contribution to modularity of the top DEGs

is strong compared to the rest of the DEGs,

indicating that the strong differentially

expressed values are driving the interaction

dynamics and establishing through modularity

a more constrained network configuration

with invasive and especially proliferative genes.

Since the transition between the two states,

from proliferative to invasive cells, is a sign of

disease progression and metastasis, it might be

justified to a certain extent that communities

appear characterized as shown.

Figure 5. PIN from invasive DEG signature in melanoma. Top: whole network. Bottom: reduced networks, with

top 150 down-expressed values (left) and the rest (right).

80

Further evidence is obtained when other sets

of DEGs are considered. For instance, following

TEAD knockdown experiments, we notice in

Figure 6 that the under-expressed DEGs are

densely bulky while the over-expressed DEGs

are sparsely modular, while the set of the TEAD

target DEGs with log2fc > |1| expression is highly

modular, indicating a strong influence on

modularity induced by TEAD through its target

genes, i.e. a subset of the previous two DEG

sets, and therefore a prevalent sparsification of

this feature (and targets neutralization as well)

after knockdown of the regulator.

Figure 6. PIN from TEAD knockdown experiments. Top-left under-expressed DEGs; Top-right: over-expressed

DEGs.. Bottom: TEAD target DEGs.

3. Discussion

Due to the importance of state transitions in

cancer networks, many dynamic properties

that are currently analyzed through entropy,

symmetry and controllability can become

interpretable from a biological and hopefully,

in the future, clinical standpoint. Among the

most important network states exerting

influences in such regards, the attractor states

are stable states and can be considered

differentiated cell types, thus also phenotypes

such a subtypes or tumor cell characteristics.

We know that cell state transitions have

associated patterns of gene expression, and

from such patterns we can investigate the

network configurations associated to them

A system likely spends most of the time within

the basins of attraction, and relatively little

time moving between them. This depends

on the fact that stationarity tends to recur

and non-stationarity transiently breaks it.

Underlying both states there are differenti-

ation processes whose stochastic dynamics

can be tracked by monitoring the inter-state

trajectories covering different regions of the

state-space. The activation of the states is

the element inducing the observed changes,

therefore the measurable trajectories. The

signaling activity patterns usually measured by

gene profiles and by the related pathways once

cast within network representations, allow

more dimensions and more constraints to be

investigated through the topological properties.

The control of a large dynamical complex

network may be a very hard task in biological

networks67-70. It might be intuitively

efficient to control the state of subsets of

nodes instead of single nodes, but identifying

functional sets as markers or targets is a

highly context-specific problem that requires

complex algorithms rather than simplified

solution paths. The identification of

symmetries in a complex system can be

very important in order to decipher its

organizational principles and rules. The key

question is thus to understand the role of

symmetries in reconstructing or controlling

network dynamics.

Naturally enough, it is important to compute

an optimal network decomposition into

observable/controllable and unobservable/

uncontrollable sub-networks likewise into

symmetry-driven versus non-symmetry-driven

sub-networks to study how they synchronize

or desynchronize. The problems remain

much harder in nonlinear networks with

symmetries, due to the fact that observability

and controllability present more complicate

dependence relationships.

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4. Materials and Methods

The datasets used for the application

section and the described examples have been

retrieved from published sources reported

in each use case. Standard network software

(http://igraph.org/, http://www.cytoscape.

org/) was used to build the graphical evidences

from annotation sources such as

https://string-db.org.

Conflicts of Interest. The author declares

no conflicts of interest.

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Enrico Capobianco, core expertise is in quantitative

methods and computational modeling for conducting

inference on complex data.

In particular, network science and stochastic methods

from statistics and machine learning have been

instrumental to his wide spectrum work in biomedicine.

While participating in worldwide academic and scientific

activities at several international institutes, Enrico

collaborates with major research agencies, and organizes

scientific programs especially designed to advance

Systems and Precision Medicine, and Big Data in Health.

Enrico holds a Doctorate in Statistical Sciences from the

University of Padua (IT) and after conducting postdoc

research in computational fields at Stanford University

(1994-98), he received in 1999 a NATO-CNR grant in

Denmark at the Niels Bohr Institute and at the Danish

Technical University, before becoming in 2001-02 an

ERCIM (European Research Consortium for Informatics

and Mathematics) fellow at the Center for Mathematics

and Computer Science in Amsterdam, the Netherlands.

After returning to the USA, he worked as a Staff Scientist

at the Mathematical Sciences Research Institute in

Berkeley (2003), and as a Senior Scientist at Boston

University, Biomedical Engineering (2004-05). In 2005,

Enrico was appointed Head of Methods at Serono,

Evry, France, and in 2006 he joined the Center for

Advanced Studies, Research and Development in

Sardinia at the Polaris Science and Technology Park,

leading a quantitative systems biology group till 2011.

Enrico received a Professorship from the Chinese

Academy of Sciences, Shanghai University, in 2011,

was a Visiting Professor at the Fiocruz Foundation in

Brazil (2008-10), and was Visiting Scientist at the

Institut des Hautes Études Scientifiques in France (2011).

Enrico leads since 2011 the Computational Biology and

Bioinformatics program at the Center of Computational

Science, University of Miami, FL (USA). He has been

associate scientist with the National Research Institute

in Italy, at the Institute of Clinical Physiology in Pisa,

where he founded LISM, the Laboratory of Integrative

Systems Medicine (2012–15) – and where he has

coordinated Big Data in Health activities, in Milan,

Siena and Pisa (2015-17).

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