Abstract

Background

   The rewiring of molecular interactions in various conditions leads to
   distinct phenotypic outcomes. Differential network analysis (DINA) is
   dedicated to exploring these rewirings within gene and protein
   networks. Leveraging statistical learning and graph theory, DINA
   algorithms scrutinize alterations in interaction patterns derived from
   experimental data.

Results

   Introducing a novel approach to differential network analysis, we
   incorporate differential gene expression based on sex and gender
   attributes. We hypothesize that gene expression can be accurately
   represented through non-Gaussian processes. Our methodology involves
   quantifying changes in non-parametric correlations among gene pairs and
   expression levels of individual genes.

Conclusions

   Applying our method to public expression datasets concerning diabetes
   mellitus and atherosclerosis in liver tissue, we identify
   gender-specific differential networks. Results underscore the
   biological relevance of our approach in uncovering meaningful molecular
   distinctions.

   Keywords: Differential network, DINA, Differential network analysis,
   Atherosclerosis, Diabetes

Introduction

   The emergence of high-throughput technologies in genomics, proteomics,
   and non-coding RNA studies has revolutionized our understanding of how
   variations in the abundance of these biological molecules correlate
   with diseases [[30]1]. This deluge of data has spurred the creation of
   innovative analytical techniques that adopt a system-level approach
   through the lens of network science. These techniques employ networks
   to depict the complex interactions between biological molecules under
   specific conditions, as inferred from empirical data [[31]2]. A
   particularly compelling use of network theory is comparing biological
   networks under disparate conditions, such as contrasting a disease
   state with a healthy one.

   Differential network analysis (DINA) has gained recognition for its
   ability to delineate the differences between two states by
   encapsulating them within a singular differential network that
   highlights their variances [[32]3–[33]6]. DINA has found application in
   contrasting various experimental conditions or phenotypes, with recent
   studies underscoring the significance of factors like the age and sex
   of patients on drug response, disease progression, and comorbidity
   prevalence in chronic diseases [[34]7–[35]9]. Empirical findings have
   demonstrated notable disparities in disease incidence and progression
   based on sex and age, such as the heightened susceptibility of older
   diabetic patients to comorbidities and the observed sex-based
   differences in COVID-19 mortality rates [[36]10–[37]12]. This
   underscores the need for advanced algorithms to unravel the molecular
   mechanisms driving these age- and sex-dependent disparities.

   DINA algorithms are designed to pinpoint changes in network structures
   by identifying association measures that differ between two biological
   states,
   [MATH: <mrow><msub><mi
   mathvariant="script">C</mi><mn>1</mn></msub><mo>,</mo><msub><mi
   mathvariant="script">C</mi><mn>2</mn></msub></mrow> :MATH]
   . When presented with two disparate biological conditions, represented
   by two networks of molecular interactions, DINA algorithms aim to
   uncover the network rewiring that underpins the mechanistic differences
   between these states. DINA algorithms construct networks
   [MATH: <mrow><msub><mi
   mathvariant="script">N</mi><mn>1</mn></msub><mo>,</mo><msub><mi
   mathvariant="script">N</mi><mn>2</mn></msub></mrow> :MATH]
   for each condition, starting with gene expression datasets from two
   conditions. These networks feature nodes for each gene and weighted
   edges that denote the strength and nature of associations or causal
   relationships between genes. A differential network
   [MATH: <msub><mi mathvariant="script">N</mi><mi>d</mi></msub> :MATH]
   is then derived to represent the variation in associations across
   conditions, a technique previously applied to investigate
   disease-related alterations [[38]8, [39]13].

   Traditional methods often assume that gene expression data adhere to
   specific parametric distributions, such as Gaussian or Poisson
   distributions [[40]1, [41]14–[42]16]. However, the count-based nature
   of Next-Generation Sequencing (NGS) data challenges these assumptions,
   prompting the necessity for non-parametric DINA analysis approaches.

   We introduce a novel DINA methodology that identifies differential
   edges between networks and integrates differential gene expressions,
   taking into account sex-based differences [[43]12]. This approach
   employs multivariate count data for predicting gene expression levels
   and constructs a conditional dependence graph using pairwise Markov
   random fields [[44]17]. This departure from traditional methods, which
   often presuppose parametric distributions for gene expression data,
   highlights the imperative for non-parametric techniques in DINA
   analysis.

   Our proposed DINA algorithm is initiated by constructing two
   condition-specific graphs, from which a final differential graph is
   derived. This graph is then pruned to emphasize edges related to genes
   exhibiting differential expression. Our DINA method facilitates the
   identification of differential networks while incorporating
   considerations of gender differences, thereby advancing our
   understanding of the molecular basis of disease in the context of sex
   and age.

Relevance of the DNA

   Differential networks are used to compare two or more networks based on
   changes in connectivity or interactions between nodes under different
   conditions or across different datasets. They are commonly applied in
   biological networks, such as gene expression or protein-protein
   interaction networks, to identify how interactions change in response
   to disease, treatment, or other stimuli [[45]18–[46]26].

   For instance, DNA may be applied on Expression Networks in Healthy vs.
   Diseased States. Let imagine that Network 1 represents Healthy State.
   In the healthy state, a set of genes (nodes) are connected based on
   their co-expression (edges) For example, if Gene A and Gene B are
   highly co-expressed, there would be an edge between them. This network
   shows how genes typically interact in a healthy individual. Let imagine
   a Network 2 who models a diseased state (such as cancer) in which the
   expression levels of genes may change, and therefore the interactions
   between them may be altered. Some gene-gene interactions may be lost,
   and new interactions may emerge. The differential network is created by
   taking the difference between the two networks (Healthy vs. Diseased).
   This network will highlight both lost and novel connections.
   Interactions present in the healthy network but missing in the diseased
   network, and interactions that appear in the diseased state but were
   absent in the healthy state.

   For example: In the healthy state: Gene A interacts with Gene B and
   Gene C. In the diseased state: Gene A no longer interacts with Gene B,
   but interacts with Gene D instead. The differential network would show
   a lost edge between Gene A and Gene B and a new edge between Gene A and
   Gene D. This type of analysis helps in identifying key genes and
   pathways that may be responsible for the diseased condition or that
   could be potential therapeutic targets see Fig. [47]1.

Fig. 1.

   Fig. 1
   [48]Open in a new tab

   Toy example of differential network

   Examples of the relevance of DNA are reported in many works. For
   instance, by Ha et al [[49]27], who describes differential networks
   between two different subtypes of glioblastoma estimated from genomic
   data. Basha et al,. [[50]28] introduces an extensive differential
   network analysis of multiple human tissue interactomes. As a results
   they are able to evidence differences of processes between tissues.

Related work

   DINA’s application in distinguishing differentially expressed genes
   among various sample groups is invaluable, especially in contrasting
   individuals with specific diseases against healthy controls. This
   methodological approach is crucial in molecular biology and
   bioinformatics for pinpointing genes with variable expression levels
   between diseased and healthy sample groups.

   Central to DINA-based research are algorithms designed to detect
   alterations in network structures under varying conditions [[51]29].
   These algorithms have been pivotal in biology for mapping the
   transition from healthy to diseased states within the same biological
   framework [[52]2]. Our focus narrows to networks that maintain constant
   node sets yet exhibit variable edge sets. Specifically, in the presence
   of two distinct conditions
   [MATH: <msub><mi mathvariant="script">C</mi><mn>1</mn></msub> :MATH]
   and
   [MATH: <msub><mi mathvariant="script">C</mi><mn>2</mn></msub> :MATH]
   , represented by graphs
   [MATH: <mrow><msub><mi>G</mi><mn>1</mn></msub><mrow><mo
   stretchy="false">(</mo><mi>V</mi><mo>,</mo><msub><mi>E</mi><mn>1</mn></
   msub><mo stretchy="false">)</mo></mrow></mrow> :MATH]
   and
   [MATH: <mrow><msub><mi>G</mi><mn>1</mn></msub><mrow><mo
   stretchy="false">(</mo><mi>V</mi><mo>,</mo><msub><mi>E</mi><mn>2</mn></
   msub><mo stretchy="false">)</mo></mrow></mrow> :MATH]
   , the objective of DINA analysis is to pinpoint the modifications of
   the network.

   In biological systems analysis, it is pertinent to note that while
   nodes represent directly quantifiable entities, the derivation of edges
   necessitates observing a sequence of temporal data. For instance, gene
   networks originating from microarray experiments necessitate the
   inference of edges from data through statistical graphical models
   [[53]30–[54]32]. In these models, each node within the graph
   [MATH: <mrow><mi>G</mi><mo>=</mo><mo
   stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   is one of these measurable random variables
   [MATH:
   <mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><m
   sub><mi>X</mi><mi>M</mi></msub></mrow> :MATH]
   , and the edges quantify a pre-specified notion of associations between
   the pairs of these variables. In this setting, the focus is
   predominantly on undirected graphs where the directions or the
   causality of these associations are not of interest. Among different
   metrics of associations, partial correlation is one of the most common
   ones as it measures conditional dependencies. Probabilistic graphical
   models allow conditional dependency-based graph estimation.
   Differential associations within these models are scrutinized by
   evaluating the variance in partial correlations across experimental
   conditions, utilizing specific statistical tests to measure the
   alterations in correlations among entities. Additionally, changes in
   gene expression levels are assessed using the classical Student’s
   t-test [[55]33]. Subsequently, these statistical evaluations are
   amalgamated into a singular optimization model aiming to elucidate the
   hierarchical network structures. However, certain assumptions inherent
   to previous models, such as Gaussian data distribution, may not hold
   across all experimental conditions, necessitating non-parametric
   methods. While computationally efficient and simpler to implement,
   these methods demand adherence to specific distributional
   prerequisites, failing which could skew or invalidate the results.

   Several studies have opted for a nonparanormal data distribution (or
   Gaussian copula) approach [[56]34], employing rank-based correlation
   matrices like Spearman correlation or Kendall’s
   [MATH: <mi>τ</mi> :MATH]
   . There are other variations available too [[57]29]. However, the
   nonparanormal models are primarily only suitable for continuous data,
   which limits its applicability in other settings.

   These models have found applications in analyzing brain data and
   sequencing counts, circumventing the temporal limitations of
   non-parametric methods. Efficient Bayesian models have emerged
   [[58]29], calculating edge probabilities by inferring their likelihood.
   Some methods adopt diverse heuristics for probability inference,
   challenging direct data derivation, as highlighted in [[59]17], with
   this method surpassing other contemporary techniques.

   Non-parametric methods, recognized for their minimal assumptions
   regarding data distribution, leverage data-driven approaches to
   evaluate network connectivity differences between conditions. Their
   flexibility and robustness are advantageous in handling complex,
   non-linear node relationships within networks, albeit at the cost of
   computational intensity and reduced interpretability.

   The decision between parametric and non-parametric approaches for
   differential network analysis hinges on data characteristics,
   foundational assumptions, and the investigative query. Researchers
   frequently use sensitivity analysis and result cross-validation to
   ensure their findings’ robustness and reliability. Integrating insights
   from both methodological spectrums can yield a more detailed
   comprehension of the differential network architecture.

Materials and methods

Non parametric differential network analysis algorithm

   Let us consider two different expression datasets encoded in two
   matrices
   [MATH:
   <mrow><msub><mi>N</mi><mi>j</mi></msub><mo>×</mo><mi>M</mi></mrow>
   :MATH]
   (
   [MATH: <msub><mi>N</mi><mi>j</mi></msub> :MATH]
   samples, M genes) for
   [MATH: <mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow>
   :MATH]
   denoted as
   [MATH: <msup><mi>X</mi><mn>1</mn></msup> :MATH]
   ,
   [MATH: <msup><mi>X</mi><mn>2</mn></msup> :MATH]
   representing two biological conditions
   [MATH: <msub><mi mathvariant="script">C</mi><mn>1</mn></msub> :MATH]
   [MATH: <msub><mi mathvariant="script">C</mi><mn>2</mn></msub> :MATH]
   . Each row of
   [MATH: <msup><mi>X</mi><mi>j</mi></msup> :MATH]
   stores the expression values of M genes of different samples.
   Therefore,
   [MATH:
   <msubsup><mi>X</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mi>c</mi
   ></msubsup> :MATH]
   [MATH: <mrow><mo
   stretchy="false">(</mo><mi>c</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</m
   n><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo
   ><msub><mi>N</mi><mi>c</mi></msub><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</
   mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>M</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   denotes expression of j-th gene in i-th sample under condition c. Note
   that the sample sizes under the two conditions may be different. We
   model this data under a Bayesian non-parametric framework.

   Each column representing a gene may be encoded as a network node and
   compute conditional independence-based graphical relation. Let
   [MATH: <mrow><mi>M</mi><mo>×</mo><mi>M</mi></mrow> :MATH]
   dimensional matrices
   [MATH: <msub><mi mathvariant="script">P</mi><mn>1</mn></msub> :MATH]
   and
   [MATH: <msub><mi mathvariant="script">P</mi><mn>2</mn></msub> :MATH]
   represents the conditional independence relation among the M genes
   [[60]17]. We then define the differential relation between two
   conditions based on the posterior samples of
   [MATH: <msub><mi mathvariant="script">P</mi><mn>1</mn></msub> :MATH]
   and
   [MATH: <msub><mi mathvariant="script">P</mi><mn>2</mn></msub> :MATH]
   .

   Following the pairwise Markov random field (MRF) model for counts from
   [[61]17], we consider the following joint probability mass function for
   M-dimensional count-valued data X,
   [MATH: <mrow><mtext>Pr</mtext><mrow><mo
   stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mo>…
   </mo><mo>,</mo><msub><mi>X</mi><mi>M</mi></msub><mo
   stretchy="false">)</mo></mrow><mo>∝</mo><mo>exp</mo><mfenced close=")"
   open="("><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mr
   ow><mi>M</mi></munderover><mrow><mo
   stretchy="false">[</mo><msub><mi>α</mi><mi>j</mi></msub><msub><mi>X</mi
   ><mi>j</mi></msub><mo>-</mo><mo>log</mo><mrow><mo
   stretchy="false">(</mo><msub><mi>X</mi><mi>j</mi></msub><mo>!</mo><mo
   stretchy="false">)</mo></mrow><mo
   stretchy="false">]</mo></mrow><mo>-</mo><munderover><mo>∑</mo><mrow><mi
   >ℓ</mi><mo>=</mo><mn>2</mn></mrow><mi>M</mi></munderover><munder><mo>∑<
   /mo><mrow><mi>j</mi><mo><</mo><mi>ℓ</mi></mrow></munder><msub><mi>β</mi
   ><mrow><mi
   mathvariant="italic">jl</mi></mrow></msub><mi>F</mi><mrow><mo
   stretchy="false">(</mo><msub><mi>X</mi><mi>j</mi></msub><mo
   stretchy="false">)</mo></mrow><mi>F</mi><mrow><mo
   stretchy="false">(</mo><msub><mi>X</mi><mi>ℓ</mi></msub><mo
   stretchy="false">)</mo></mrow></mfenced><mo>,</mo></mrow> :MATH]

   where
   [MATH: <mrow><mi>F</mi><mo stretchy="false">(</mo><mo>·</mo><mo
   stretchy="false">)</mo></mrow> :MATH]
   is a monotone increasing bounded function with support
   [MATH: <mrow><mo
   stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   . As in [[62]17], we let
   [MATH: <mrow><mi>F</mi><mrow><mo stretchy="false">(</mo><mo>·</mo><mo
   stretchy="false">)</mo></mrow><mo>=</mo><msup><mrow><mo
   stretchy="false">(</mo><msup><mo>tan</mo><mrow><mo>-</mo><mn>1</mn></mr
   ow></msup><mrow><mo stretchy="false">(</mo><mo>·</mo><mo
   stretchy="false">)</mo></mrow><mo
   stretchy="false">)</mo></mrow><mi>θ</mi></msup></mrow> :MATH]
   for some positive
   [MATH: <mrow><mi>θ</mi><mo>∈</mo><msup><mrow><mi
   mathvariant="double-struck">R</mi></mrow><mo>+</mo></msup></mrow>
   :MATH]
   . Since the data is positive-valued, the range of F(X) is
   [MATH: <mrow><mrow><mo maxsize="1.2em" minsize="1.2em"
   stretchy="true">(</mo></mrow><mn>0</mn><mo>,</mo><msup><mrow><mo
   stretchy="false">(</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo
   stretchy="false">)</mo></mrow><mi>θ</mi></msup><mrow><mo
   maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo></mrow></mrow>
   :MATH]
   and the exponent
   [MATH: <mi>θ</mi> :MATH]
   is specified as a minimizer of the loss, quantifying the difference in
   covariance between F(X) and X following [[63]17]. For detailed
   descriptions of the method, readers are encouraged to check [[64]17].

   Under this model, if
   [MATH:
   <mrow><msub><mi>β</mi><mrow><mi>j</mi><mi>ℓ</mi></mrow></msub><mo>=</mo
   ><mn>0</mn></mrow> :MATH]
   , we have
   [MATH: <msub><mi>X</mi><mi>j</mi></msub> :MATH]
   and
   [MATH: <msub><mi>X</mi><mi>ℓ</mi></msub> :MATH]
   to be conditionally independent, i.e.
   [MATH: <mrow><mi>P</mi><mrow><mo
   stretchy="false">(</mo><msub><mi>X</mi><mi>j</mi></msub><mo>,</mo><msub
   ><mi>X</mi><mi>ℓ</mi></msub><mo>∣</mo><msub><mi>X</mi><mrow><mo>-</mo><
   mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>ℓ</mi><mo
   stretchy="false">)</mo></mrow></msub><mo
   stretchy="false">)</mo></mrow><mo>=</mo><mi>P</mi><mrow><mo
   stretchy="false">(</mo><msub><mi>X</mi><mi>j</mi></msub><mo>∣</mo><msub
   ><mi>X</mi><mrow><mo>-</mo><mo
   stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>ℓ</mi><mo
   stretchy="false">)</mo></mrow></msub><mo
   stretchy="false">)</mo></mrow><mi>P</mi><mrow><mo
   stretchy="false">(</mo><msub><mi>X</mi><mi>ℓ</mi></msub><mo>∣</mo><msub
   ><mi>X</mi><mrow><mo>-</mo><mo
   stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>ℓ</mi><mo
   stretchy="false">)</mo></mrow></msub><mo
   stretchy="false">)</mo></mrow></mrow> :MATH]
   , where
   [MATH: <msub><mi>X</mi><mrow><mo>-</mo><mo
   stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>ℓ</mi><mo
   stretchy="false">)</mo></mrow></msub> :MATH]
   stands for all the variables excluding
   [MATH: <msub><mi>X</mi><mi>j</mi></msub> :MATH]
   and
   [MATH: <msub><mi>X</mi><mi>ℓ</mi></msub> :MATH]
   . Our estimated graphical relation thus would rely on
   [MATH: <msub><mi>β</mi><mrow><mi>j</mi><mi>ℓ</mi></mrow></msub> :MATH]
   ’s.

   We take a Bayesian route for inference and put the same priors as in
   [[65]17]. Specifically, for the
   [MATH: <msub><mi>β</mi><mrow><mi
   mathvariant="italic">jl</mi></mrow></msub> :MATH]
   ’s, we set simple independent and identically distributed mean zero
   Gaussian priors. The parameter
   [MATH: <msub><mi>λ</mi><mi>j</mi></msub> :MATH]
   ’s are treated as random effects and given distributions
   [MATH: <msub><mi>D</mi><mi>j</mi></msub> :MATH]
   . The distribution
   [MATH: <msub><mi>D</mi><mi>j</mi></msub> :MATH]
   governs the over-dispersion and the shape of the marginal count
   distribution for the
   [MATH: <msup><mi>j</mi><mtext>th</mtext></msup> :MATH]
   node. To allow these marginals to be flexibly determined by the data,
   we take a Bayesian nonparametric approach using Dirichlet process (DP)
   priors, where
   [MATH:
   <mrow><msub><mi>D</mi><mi>j</mi></msub><mo>∼</mo><mtext>DP</mtext><mrow
   ><mo
   stretchy="false">(</mo><msub><mi>M</mi><mi>j</mi></msub><msub><mi>D</mi
   ><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mrow> :MATH]
   , with
   [MATH: <msub><mi>D</mi><mn>0</mn></msub> :MATH]
   as a Gamma base measure and
   [MATH: <msub><mi>M</mi><mi>j</mi></msub> :MATH]
   as a precision parameter. The precision parameter
   [MATH: <msub><mi>M</mi><mi>j</mi></msub> :MATH]
   follows a Gamma distribution,
   [MATH:
   <mrow><msub><mi>M</mi><mi>j</mi></msub><mo>∼</mo><mtext>Ga</mtext><mrow
   ><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo
   stretchy="false">)</mo></mrow></mrow> :MATH]
   , allowing for greater adaptivity to the data. We run MCMC to
   approximate the posterior and generate posterior samples of
   [MATH: <msub><mi>β</mi><mrow><mi
   mathvariant="italic">jl</mi></mrow></msub> :MATH]
   ’s. The model follows a structure, similar to the Poisson auto-model
   [[66]35]. When
   [MATH: <msub><mi>β</mi><mrow><mi
   mathvariant="italic">jl</mi></mrow></msub> :MATH]
   ’s are zero, the marginals lead to Poisson-type marginals with
   sample-specific means as
   [MATH: <msub><mi>λ</mi><mi>j</mi></msub> :MATH]
   ’s are modeled as random effects. Thus, they also need to be sampled
   for all the samples. This puts a moderately high computational cost.
   The general code to fit this model is in the second author’s GitHub
   page [67]https://github.com/royarkaprava/CONGA.

   After running the Markov chain Monte Carlo (MCMC) sampling for the
   above model under two conditions, we get the matrices
   [MATH: <msub><mi mathvariant="script">P</mi><mn>1</mn></msub> :MATH]
   and
   [MATH: <msub><mi mathvariant="script">P</mi><mn>2</mn></msub> :MATH]
   with (j, k)-th entries as
   [MATH:
   <msubsup><mi>β</mi><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mrow><mo
   stretchy="false">(</mo><mn>1</mn><mo
   stretchy="false">)</mo></mrow></msubsup> :MATH]
   and
   [MATH:
   <msubsup><mi>β</mi><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mrow><mo
   stretchy="false">(</mo><mn>2</mn><mo
   stretchy="false">)</mo></mrow></msubsup> :MATH]
   respectively. Consequently, a differential network is defined as the
   difference
   [MATH:
   <mrow><msubsup><mi>β</mi><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mr
   ow><mo stretchy="false">(</mo><mn>1</mn><mo
   stretchy="false">)</mo></mrow></msubsup><mo>-</mo><msubsup><mi>β</mi><m
   row><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mrow><mo
   stretchy="false">(</mo><mn>2</mn><mo
   stretchy="false">)</mo></mrow></msubsup></mrow> :MATH]
   for each edge (j, k) where
   [MATH:
   <msubsup><mi>β</mi><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mrow><mo
   stretchy="false">(</mo><mn>1</mn><mo
   stretchy="false">)</mo></mrow></msubsup> :MATH]
   and
   [MATH:
   <msubsup><mi>β</mi><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mrow><mo
   stretchy="false">(</mo><mn>2</mn><mo
   stretchy="false">)</mo></mrow></msubsup> :MATH]
   are the coefficients under two conditions 1 and 2. From the MCMC
   samples, we can get the posterior mean of these differences as
   [MATH: <mrow><msubsup><mover accent="true"><mi>β</mi><mo
   stretchy="false">^</mo></mover><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mr
   ow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo
   stretchy="false">)</mo></mrow></msubsup><mo>-</mo><msubsup><mover
   accent="true"><mi>β</mi><mo
   stretchy="false">^</mo></mover><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mr
   ow><mrow><mo stretchy="false">(</mo><mn>2</mn><mo
   stretchy="false">)</mo></mrow></msubsup></mrow> :MATH]
   using the individual posterior means. Alternatively, we can compute
   other posterior summaries such as
   [MATH: <mrow><mi>P</mi><mo stretchy="false">(</mo><mo
   stretchy="false">|</mo><msubsup><mi>β</mi><mrow><mi>j</mi><mo>,</mo><mi
   >k</mi></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo
   stretchy="false">)</mo></mrow></msubsup><mo>-</mo><msubsup><mi>β</mi><m
   row><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mrow><mo
   stretchy="false">(</mo><mn>2</mn><mo
   stretchy="false">)</mo></mrow></msubsup><mo
   stretchy="false">|</mo><mo>></mo><mi>c</mi><mo>∣</mo><msub><mi>D</mi><m
   n>1</mn></msub><mo>,</mo><msub><mi>D</mi><mn>2</mn></msub><mo
   stretchy="false">)</mo></mrow> :MATH]
   , which is the posterior probability that
   [MATH: <mrow><mrow><mo
   stretchy="false">|</mo></mrow><msubsup><mi>β</mi><mrow><mi>j</mi><mo>,<
   /mo><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo
   stretchy="false">)</mo></mrow></msubsup><mo>-</mo><msubsup><mi>β</mi><m
   row><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mrow><mo
   stretchy="false">(</mo><mn>2</mn><mo
   stretchy="false">)</mo></mrow></msubsup><mrow><mo
   stretchy="false">|</mo></mrow></mrow> :MATH]
   is greater than some pre-specified cutoff c given the two datasets,
   denoted as
   [MATH: <msub><mi>D</mi><mn>1</mn></msub> :MATH]
   and
   [MATH: <msub><mi>D</mi><mn>2</mn></msub> :MATH]
   . We take the second approach to define our differential networks. To
   choose c adequately, we run a sensitivity test. Specifically, we vary c
   over a range, and compute
   [MATH: <mrow><msub><mi>f</mi><mrow><mi
   mathvariant="italic">ij</mi></mrow></msub><mrow><mo
   stretchy="false">(</mo><mi>c</mi><mo
   stretchy="false">)</mo></mrow><mo>=</mo><mrow><mi>P</mi><mo
   stretchy="false">(</mo><mo
   stretchy="false">|</mo></mrow><msubsup><mi>β</mi><mrow><mi>j</mi><mo>,<
   /mo><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo
   stretchy="false">)</mo></mrow></msubsup><mo>-</mo><msubsup><mi>β</mi><m
   row><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mrow><mo
   stretchy="false">(</mo><mn>2</mn><mo
   stretchy="false">)</mo></mrow></msubsup><mrow><mo
   stretchy="false">|</mo><mo>></mo><mi>c</mi><mo>∣</mo><msub><mi>D</mi><m
   n>1</mn></msub><mo>,</mo><msub><mi>D</mi><mn>2</mn></msub><mo
   stretchy="false">)</mo></mrow></mrow> :MATH]
   for each choice. Then, we monitor
   [MATH:
   <mrow><msub><mo>∑</mo><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub
   ><msup><mrow><mo stretchy="false">(</mo><msub><mi>f</mi><mrow><mi
   mathvariant="italic">ij</mi></mrow></msub><mrow><mo
   stretchy="false">(</mo><msub><mi>c</mi><mn>1</mn></msub><mo
   stretchy="false">)</mo></mrow><mo>-</mo><msub><mi>f</mi><mrow><mi
   mathvariant="italic">ij</mi></mrow></msub><mrow><mo
   stretchy="false">(</mo><msub><mi>c</mi><mn>2</mn></msub><mo
   stretchy="false">)</mo></mrow><mo
   stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow> :MATH]
   and find the smallest c showing stability around its neighborhood to
   assess its sensitivity on the estimate.

Databases

   The T2DiACoD database, as described by Rani et al. (2017) [[68]36], was
   employed to collate a comprehensive list of genes linked to comorbid
   conditions associated with Type 2 Diabetes Mellitus (T2DM). Gene
   expression datasets were also sourced from the GTEx database [[69]37].

   T2DiACoD is a meticulously compiled database, the result of rigorous
   research and systematic literature review. It catalogues genes and
   noncoding RNAs that are crucial to understanding T2DM and its frequent
   comorbidities, including atherosclerosis, nephropathy, diabetic
   retinopathy, and cardiovascular disorders. This repository, enriched
   through meticulous data integration from existing databases,
   encapsulates 650 genes and 34 microRNAs related to these conditions,
   providing a reliable resource for your research.

   The genotype-tissue expression (GTEx) project is a vast open-access
   platform that empowers researchers like us with the distribution of
   genomic data collected from various individuals. This repository
   encompasses a broad spectrum of genomic data, from sequencing to
   methylation analyses, providing a wealth of information at your
   fingertips.

   GTEx offers detailed metadata for each sample, covering aspects such as
   tissue type, sex, and age, categorized into six distinct groups. This
   makes GTEx an invaluable asset for investigating the interplay between
   age and tissue-specific gene expression. As of February 1st, the GTEx
   database boasts a collection of 17,382 samples across 54 tissue types
   from 948 donors, all accessible via the GTEx web portal. This portal
   enables users to efficiently search for and visualize data [[70]12,
   [71]38, [72]39]. Furthermore, the data can be downloaded for in-depth
   analysis with custom scripts. In our research, we leveraged data from
   various tissues, dividing samples into two categories based on sex. Our
   study focused on the genes that play a pivotal role in developing
   T2DM-related complications, examining nine tissues (including blood,
   brain, adipose, amygdala, aorta, colon, coronary, liver, and lung
   tissues).

   To ensure the validity of our research, we meticulously selected an
   equal number of samples from each tissue type, maintaining a balanced
   representation of age groups and uniformity in sample sizes across
   tissues. This approach facilitated an equitable distribution of age
   groups within each sex-based category for each tissue analyzed,
   ensuring the precision and reliability of our findings.

   Focusing on atherosclerosis, we obtained a list of 115 genes related to
   this disease in T2DiACoD database, while for diabetes we obtained a
   list of 650 genes. We retrieved expression data by employing
   GTExVisualizer [[73]8, [74]40], and metadata related to tissue, sex and
   age of the sample are extracted using genes identified in the T2DiACoD
   database in the previous step. Expression data are measured as
   Transcript per Million (TPM). This data integration and gene enrichment
   process was performed using an ad-hoc realised script that has been
   integrated into GTExVisualizer. We performed the analysis at tissue
   level, thus for each considered tissue, we split the data into male and
   female samples and randomly selected the same number of samples. We
   first generated DN by using non parametric methods and we evaluated the
   biological significance by means of enrichment methods.

Results

   To show the effectiveness of our method we present two case studies on
   two chronic diseases to show differential mechanisms related to sex
   differences. We evaluated differential networks between men and women
   focusing on genes related to diabetes and atherosclerosis as reported
   in the T2DiaCoD database [[75]36]. The characteristics of all the DNs
   are reported in Tables [76]1 and [77]2. Then, to evaluate the
   biological significance of the resulted DNs, we performed a pathway
   enrichment analysis. Functional enrichment analysis, also known as
   pathway enrichment analysis (PEA), is a bioinformatics technique used
   to identify biological pathways that are significantly over-represented
   in a given list of genes compared to what would be expected by chance.
   These biological functions are stored in bioinformatics databases such
   KEGG, and statistical methods like Fisher’s exact test, that computes
   the p-value of the enrichment are used to determine the most enriched
   pathways. All PEA tools require a gene list as input. However, some
   tools accept genomic regions instead of gene lists and first map these
   regions to their associated genes. This process, known as genomic
   regions enrichment analysis, is helpful for uncovering biological
   pathways related to specific chromosome regions.

Table 1.

   Characteristics of the differential networks related to diabetes
   DN    Nodes Edges
   Liver 128   3340
   Aorta 237   2116
   Heart 238   4316
   [78]Open in a new tab

Table 2.

   Characteristics of the differential networks related to atherosclerosis
   DN               Nodes Edges
   Adipose Visceral 12    32
   Artery Coronary  11    30
   Artery Tibial    11    12
   Blood            2     1
   [79]Open in a new tab

   Thus, for each network, we performed a pathway based on the KEGG
   pathway database [[80]41] available on STRING enrichmennt app [[81]42]
   of the Cytoscape software [[82]43].

Diabetes related differential networks

   Liver Tissue For liver tissue in diabetes, we obtained a DN with 128
   nodes and 3340 edges, see Fig. [83]2. The enrichment analysis
   highlighted the presence of some enriched pathways between sex, see
   Table [84]3. Figure [85]3 depicts the subnetworks of the differential
   network for Liver tissue in Diabetes, where, we highlighted the genes
   involved in the resulted pathways in different colours. In order to
   evaluate

Fig. 2.

   [86]Fig. 2
   [87]Open in a new tab

   The figure shows the differential network between men and women
   focusing on genes in liver tissue related to diabetes disease

Table 3.

   Top Enriched Pathways of liver tissue in Diabetes
   Description Term name P-value with Fisher
   Cytokine-cytokine receptor interaction hsa04060
   [MATH:
   <mrow><mn>5.15</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>8</mn></mrow></m
   sup></mrow> :MATH]
   Viral protein interaction with cytokine and cytokine receptor hsa04061
   [MATH:
   <mrow><mn>3.48</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>7</mn></mrow></m
   sup></mrow> :MATH]
   Chemokine signaling pathway hsa04062
   [MATH:
   <mrow><mn>3.47</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>6</mn></mrow></m
   sup></mrow> :MATH]
   Inflammatory bowel disease hsa05321
   [MATH:
   <mrow><mn>3.47</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>6</mn></mrow></m
   sup></mrow> :MATH]
   Toll-like receptor signaling pathway hsa04620
   [MATH:
   <mrow><mn>5.97</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>5</mn></mrow></m
   sup></mrow> :MATH]
   Malaria hsa05144
   [MATH:
   <mrow><mn>2.6</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>4</mn></mrow></ms
   up></mrow> :MATH]
   Rheumatoid arthritis hsa05323
   [MATH:
   <mrow><mn>2.6</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>4</mn></mrow></ms
   up></mrow> :MATH]
   Renin-angiotensin system hsa04614
   [MATH:
   <mrow><mn>3.5</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>4</mn></mrow></ms
   up></mrow> :MATH]
   Th17 cell differentiation hsa04659
   [MATH:
   <mrow><mn>4.1</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>4</mn></mrow></ms
   up></mrow> :MATH]
   Chagas disease hsa05142
   [MATH:
   <mrow><mn>4.1</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>4</mn></mrow></ms
   up></mrow> :MATH]
   [88]Open in a new tab

Fig. 3.

   [89]Fig. 3
   [90]Open in a new tab

   The figure shows selected subnetworks of the differential network for
   Liver tissue in Diabetes. In detail, starting from the differential
   network, we highlighted the genes involved in the pathways resulted
   from enriched analysis in different colours, whereas the genes not
   involved in pathways are identified in grey. In particular, the genes
   involved in Cytokine-cytokine receptor interaction are coloured in
   purple, the genes involved in Inflammatory bowel disease are coloured
   in green, the genes involved in Viral protein interaction with cytokine
   and cytokine receptor are coloured in pink and the genes involved in
   Chemokine signaling pathway are coloured in blue

   Aorta Tissue For Aorta tissue in Diabetes we obtained a DN with 237
   nodes and 2116 edges, see Fig. [91]4. The enrichment analysis
   highlighted the presence of some enriched pathways between sex, see
   Table [92]4. Figure [93]5 depicts the subnetworks of the differential
   network for Liver tissue in Diabetes, where, we highlighted the genes
   involved in the resulted pathways in different colours.

Fig. 4.

   Fig. 4
   [94]Open in a new tab

   The figure shows the differential network between men and women
   focusing on genes in aorta tissue related to diabetes disease

Table 4.

   Top Enriched Pathways of aorta tissue in Diabetes
   Description Term name P-value with Fisher
   Toll-like receptor signaling pathway hsa04620
   [MATH:
   <mrow><mn>7.05</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>19</mn></mrow></
   msup></mrow> :MATH]
   NOD-like receptor signaling pathway hsa04621
   [MATH:
   <mrow><mn>7.57</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>16</mn></mrow></
   msup></mrow> :MATH]
   Shigellosis hsa05131
   [MATH:
   <mrow><mn>7.57</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>16</mn></mrow></
   msup></mrow> :MATH]
   Hepatitis B hsa05161
   [MATH:
   <mrow><mn>1.08</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>15</mn></mrow></
   msup></mrow> :MATH]
   Salmonella infection hsa05132
   [MATH:
   <mrow><mn>1.5</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>15</mn></mrow></m
   sup></mrow> :MATH]
   TNF signaling pathway hsa04668
   [MATH:
   <mrow><mn>4.27</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>15</mn></mrow></
   msup></mrow> :MATH]
   NF-kappa B signaling pathway hsa04064
   [MATH:
   <mrow><mn>1.31</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>14</mn></mrow></
   msup></mrow> :MATH]
   Toxoplasmosis hsa05145
   [MATH:
   <mrow><mn>1.55</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>14</mn></mrow></
   msup></mrow> :MATH]
   PD-L1 expression and PD-1 checkpoint pathway in cancer hsa05235
   [MATH:
   <mrow><mn>1.82</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>14</mn></mrow></
   msup></mrow> :MATH]
   PI3K-Akt signaling pathway hsa04151
   [MATH:
   <mrow><mn>6.36</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>14</mn></mrow></
   msup></mrow> :MATH]
   [95]Open in a new tab

Fig. 5.

   [96]Fig. 5
   [97]Open in a new tab

   The figure shows Selected subnetwork of the differential network for
   Aorta tissue in Diabetes. In detail, starting from the differential
   network, we highlighted the genes involved in the pathways resulted
   from enriched analysis in different colours, whereas the genes not
   involved in pathways are identified in grey. In particular, the genes
   involved Toll-like receptor signaling pathway are coloured in blue, the
   genes involved in NOD-like receptor signaling pathway are coloured in
   yellow, the genes involved in Hepatitis B are coloured in pink, and the
   genes involved in Shigellosis are coloured in purple

   Heart Tissue For Hearth tissue in Diabetes we obtained a DN with 238
   nodes and 4316 edges, see Fig. [98]6. The enrichment analysis
   highlighted the presence of some enriched pathways between sex, see
   Table [99]5. Figure [100]7 depicts the subnetworks of the differential
   network for Liver tissue in Diabetes, where, we highlighted the genes
   involved in the resulted pathways in different colours.

Fig. 6.

   Fig. 6
   [101]Open in a new tab

   The figure shows the differential network between men and women
   focusing on genes in heart tissue related to diabetes disease

Table 5.

   Top Enriched Pathways of heart tissue in Diabetes
   Description Term name P-value with Fisher
   NOD-like receptor signaling pathway hsa04621
   [MATH:
   <mrow><mn>2.33</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>18</mn></mrow></
   msup></mrow> :MATH]
   Toll-like receptor signaling pathway hsa04620
   [MATH:
   <mrow><mn>8.48</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>18</mn></mrow></
   msup></mrow> :MATH]
   TNF signaling pathway hsa04668
   [MATH:
   <mrow><mn>6.15</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>16</mn></mrow></
   msup></mrow> :MATH]
   Cytokine-cytokine receptor interaction hsa04060
   [MATH:
   <mrow><mn>4.69</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>14</mn></mrow></
   msup></mrow> :MATH]
   NF-kappa B signaling pathway hsa04064
   [MATH:
   <mrow><mn>2.34</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>13</mn></mrow></
   msup></mrow> :MATH]
   Salmonella infection hsa05132
   [MATH:
   <mrow><mn>3.34</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>13</mn></mrow></
   msup></mrow> :MATH]
   Inflammatory bowel disease hsa05321
   [MATH:
   <mrow><mn>3.79</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>13</mn></mrow></
   msup></mrow> :MATH]
   Shigellosis hsa05131
   [MATH:
   <mrow><mn>5.73</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>13</mn></mrow></
   msup></mrow> :MATH]
   PD-L1 expression and PD-1 checkpoint pathway in cancer hsa05235
   [MATH:
   <mrow><mn>3.01</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>12</mn></mrow></
   msup></mrow> :MATH]
   Viral protein interaction with cytokine and cytokine receptor hsa04061
   [MATH:
   <mrow><mn>1.04</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>11</mn></mrow></
   msup></mrow> :MATH]
   [102]Open in a new tab

Fig. 7.

   [103]Fig. 7
   [104]Open in a new tab

   The figure shows selected subnetwork of the differential network for
   Heart tissue in Diabetes. In detail, starting from the differential
   network, we highlighted the genes involved in the pathways resulted
   from enriched analysis in different colours, whereas the genes not
   involved in pathways are identified in grey. In particular, the genes
   involved Toll-like receptor signaling pathway are coloured in blue, the
   genes involved in NOD-like receptor signaling pathway are coloured in
   red, the genes involved in Cytokine-cytokine receptor interaction are
   coloured in pink, and the genes involved in Inflammatory bowel disease
   are coloured in yellow

   Figure [105]8 shows a selected subnetwork of the differential network
   in Liver, Hearth and Aorta Tissue.

Fig. 8.

   Fig. 8
   [106]Open in a new tab

   Differential network related to diabetes expressed in Liver, Heart,
   Aorta tissues. The genes expressed in different tissues are reported in
   green, whereas the genes of specific tissue are reported in blue

   Finally, we applied the Markov Clustering Algorithm to perform a
   cluster analysis on differential networks. Figures [107]9, [108]10,
   [109]11 depict the clustered differential networks for Liver, Aorta,
   Heart tissues.

Fig. 9.

   Fig. 9
   [110]Open in a new tab

   Clustered Differential subnetwork for Liver tissue in Diabetes

Fig. 10.

   Fig. 10
   [111]Open in a new tab

   Clustered Differential subnetwork for Aorta tissue in Diabetes

Fig. 11.

   Fig. 11
   [112]Open in a new tab

   Clustered Differential subnetwork for Heart tissue in Diabetes

Atherosclerosis related differential networks

   Adipose Visceral Tissue For Adipose Visceral tissue in Atherosclerosis
   we obtained a DN with 12 nodes and 32 edges, see Fig. [113]12. The
   enrichment analysis did not report the presence of some enriched
   pathways between sex.

Fig. 12.

   Fig. 12
   [114]Open in a new tab

   Differential network for Adipose Visceral tissue in Atherosclerosis

   Artery Coronary Tissue For Artery Coronary tissue in Atherosclerosis we
   identified a network of 11 nodes and 30 edges, see Fig. [115]13. The
   enrichment analysis highlighted the presence of some enriched pathways
   between sex, see Table [116]6. Figure [117]14 depicts the subnetworks
   of the differential network for Liver tissue in Diabetes, where, we
   highlighted the genes involved in the resulted pathways in different
   colours.

Fig. 13.

   Fig. 13
   [118]Open in a new tab

   The figure shows the differential network between men and women
   focusing on genes in artery coronary tissue related to atherosclerosis
   disease

Table 6.

   Enriched Pathway of Artery Coronary and Artery Tibial tissue in
   Atherosclerosis
   Description Term name P-value with Fisher
   PPAR signaling pathway of Artery Coronary hsa03320
   [MATH:
   <mrow><mn>1.48</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>7</mn></mrow></m
   sup></mrow> :MATH]
   PPAR signaling pathway of Artery Tibial hsa03320
   [MATH:
   <mrow><mn>2.59</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>5</mn></mrow></m
   sup></mrow> :MATH]
   [119]Open in a new tab

Fig. 14.

   [120]Fig. 14
   [121]Open in a new tab

   The figure shows selected subnetwork of the differential network for
   Artery Coronary tissue in Atherosclerosis. In detail, starting from the
   differential network, we highlighted the genes involved in the pathways
   resulted from enriched analysis in yellow, whereas the genes not
   involved in pathways are identified in grey. In particular, the genes
   involved in PPAR signaling pathway are reported in yellow

   Artery Tibial Tissue For Artery Tibial tissue in Atherosclerosis we
   identified a network of 11 nodes and 12 edges, see Fig. [122]15. The
   enrichment analysis highlighted the presence of some enriched pathways
   between sex, see Table [123]4. Figure [124]16 depicts the subnetwork of
   the differential network for Liver tissue in Diabetes, where, we
   highlighted the genes involved in the resulted pathways in different
   colours.

Fig. 15.

   Fig. 15
   [125]Open in a new tab

   The figure shows the differential network between men and women
   focusing on genes in artery tibial tissue related to atherosclerosis
   disease

Fig. 16.

   [126]Fig. 16
   [127]Open in a new tab

   A Selected subnetwork of the differential network for Artery Tibial
   tissue in Atherosclerosis. In detail, starting from the differential
   network, we highlighted the genes involved in the pathways resulted
   from enriched analysis in yellow, whereas the genes not involved in
   pathways are identified in grey. In particular, the genes involved in
   PPAR-
   [MATH: <mi>γ</mi> :MATH]
   signaling pathway are reported in yellow

   Blood Tissue For Blood tissue in Atherosclerosis we identified a
   network of 2 nodes and 1 edges, see Fig. [128]17. The enrichment
   analysis did not report the presence of some enriched pathways between
   sex.

Fig. 17.

   Fig. 17
   [129]Open in a new tab

   Differential network for Blood tissue in Atherosclerosis

   We summarize in Fig. [130]18 a selected subnetwork of the differential
   network in Adipose Visceral, Artery Coronary, and Aorta Tissue, Artery
   Tibial and Blood tissues.

Fig. 18.

   Fig. 18
   [131]Open in a new tab

   Differential network related to atherosclerosis expressed in Adipose
   Visceral, Artery Coronary, Artery Tibial tissues. The genes expressed
   in different tissues are reported in green, whereas the genes of
   specific tissue are reported in blue

   Finally, we applied the Markov Clustering Algorithm to perform a
   cluster analysis on differential networks. Figures [132]19, [133]20,
   [134]21 depict the clustered differential networks for Adipose
   Visceral, Artery Coronary, Artery Tibial tissues.

Fig. 19.

   Fig. 19
   [135]Open in a new tab

   Clustered Differential network for Adipose Visceral tissue in
   Atherosclerosis

Fig. 20.

   Fig. 20
   [136]Open in a new tab

   Clustered Differential network for Artery Coronary tissue in
   Atherosclerosis

Fig. 21.

   Fig. 21
   [137]Open in a new tab

   Clustered Differential network for Artery Tibial tissue in
   Atherosclerosis

Comparison with baseline methods

   In order to evaluate the effectiveness of our method with respect to
   other parametric ones, we compared Conga with a R package iDINGO
   [[138]44]. iDINGO is able to infer group-specific dependencies and
   build differential networks. We built the differential network for
   diabetes dataset and atherosclerosis dataset. We run iDINGO using
   default parameters. However, iDINGO built the differential networks
   according to NP-complete approach by including in the building of the
   network the whole set of edges. We reported the characteristics of the
   differential networks related to diabetes and atherosclerosis in Tables
   [139]7 and [140]8.

Table 7.

   Characteristics of the differential networks related to diabetes built
   with iDINGO
   DN    Nodes Edges
   Liver 128   8127
   Aorta 238   28679
   Heart 238   28679
   [141]Open in a new tab

Table 8.

   Characteristics of the differential networks related to atherosclerosis
   built with iDINGO
   DN               Nodes Edges
   Adipose Visceral 12    45
   Artery Coronary  11    45
   Artery Tibial    11    45
   Blood            2     1
   [142]Open in a new tab

   We subsequently performed the path enrichment analysis on the DN, and
   found that the pathways detected for each tissue for diabetes were
   fewer than those built with our method, whereas none pathway was
   detected for atherosclerosis. This demonstrates that our methodology is
   able to build networks that have a greater biological information
   content than classical methods. We reported the results of pathway
   enrichment analysis in Tables [143]9, [144]10 and [145]11.

Table 9.

   Top Enriched Pathways of liver tissue in Diabetes related to DNs buitl
   with iDINGO
   Description Term name P-value with Fisher
   Cytokine-cytokine receptor interaction hsa04060
   [MATH:
   <mrow><mn>2.31</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>6</mn></mrow></m
   sup></mrow> :MATH]
   Viral protein interaction with cytokine and cytokine receptor hsa04061
   [MATH:
   <mrow><mn>2.7</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>6</mn></mrow></ms
   up></mrow> :MATH]
   Chemokine signaling pathway hsa04062
   [MATH:
   <mrow><mn>4.57</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>5</mn></mrow></m
   sup></mrow> :MATH]
   Inflammatory bowel disease hsa05321
   [MATH:
   <mrow><mn>2.31</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>6</mn></mrow></m
   sup></mrow> :MATH]
   Toll-like receptor signaling pathway hsa04620
   [MATH:
   <mrow><mn>3.46</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>4</mn></mrow></m
   sup></mrow> :MATH]
   Malaria hsa05144
   [MATH:
   <mrow><mn>7</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>4</mn></mrow></msup
   ></mrow> :MATH]
   [146]Open in a new tab

Table 10.

   Top Enriched Pathways of aorta tissue in Diabetes related to DNs buitl
   with iDINGO
   Description Term name P-value with Fisher
   Salmonella infection hsa05132
   [MATH:
   <mrow><mn>6.12</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>14</mn></mrow></
   msup></mrow> :MATH]
   TNF signaling pathway hsa04668
   [MATH:
   <mrow><mn>8.55</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>14</mn></mrow></
   msup></mrow> :MATH]
   NF-kappa B signaling pathway hsa04064
   [MATH:
   <mrow><mn>2.24</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>13</mn></mrow></
   msup></mrow> :MATH]
   Toxoplasmosis hsa05145
   [MATH:
   <mrow><mn>2.65</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>13</mn></mrow></
   msup></mrow> :MATH]
   PD-L1 expression and PD-1 checkpoint pathway in cancer hsa05235
   [MATH:
   <mrow><mn>2.68</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>13</mn></mrow></
   msup></mrow> :MATH]
   PI3K-Akt signaling pathway hsa04151
   [MATH:
   <mrow><mn>3.5</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>12</mn></mrow></m
   sup></mrow> :MATH]
   [147]Open in a new tab

Table 11.

   Top Enriched Pathways of heart tissue in Diabetes related to DNs buitl
   with iDINGO
   Description Term name P-value with Fisher
   NOD-like receptor signaling pathway hsa04621
   [MATH:
   <mrow><mn>2.25</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>18</mn></mrow></
   msup></mrow> :MATH]
   Toll-like receptor signaling pathway hsa04620
   [MATH:
   <mrow><mn>1.17</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>17</mn></mrow></
   msup></mrow> :MATH]
   TNF signaling pathway hsa04668
   [MATH:
   <mrow><mn>7.96</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>16</mn></mrow></
   msup></mrow> :MATH]
   Cytokine-cytokine receptor interaction hsa04060
   [MATH:
   <mrow><mn>6.43</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>14</mn></mrow></
   msup></mrow> :MATH]
   NF-kappa B signaling pathway hsa04064
   [MATH:
   <mrow><mn>2.91</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>13</mn></mrow></
   msup></mrow> :MATH]
   [148]Open in a new tab

Discussion

   In our research study, we introduced a method called differential
   network analysis. This approach is particularly beneficial for datasets
   that contain count data or non-parametric data, which are prevalent in
   biological research but pose challenges due to their non-normal
   distribution and discrete nature. By employing differential network
   analysis, we can delve into the underlying network structures between
   different conditions, providing a more comprehensive understanding of
   biological variations and interactions that are not discernible through
   traditional methods. Given the absence of an established standard in
   this area, the rigorous evaluation of the biological relevance of the
   networks derived from our differential network analysis was paramount.
   We analysed the structure of the networks, ensuring their biological
   significance and consistency with known biological pathways and
   mechanisms. This comprehensive analysis not only validated the
   practical utility of our methodology but also provided novel insights
   into the biological systems under study. By correlating our findings
   with existing biological knowledge and experimental data, we improved
   the biological relevance of our results, further underscoring the value
   of differential network analysis in uncovering biological patterns and
   interactions. In the context of diabetes, we found some differential
   networks in Liver, Aorta, and Heart tissues are enriched for the
   Toll-like receptor signaling pathway, as indicated in Tables [149]3,
   [150]4, [151]7. The toll-like receptor 4 (TLR-4) pathway has been
   associated with various pathophysiological conditions, including
   cardiovascular diseases (CVDs) and Rheumatoid Arthritis (RA),
   underscoring the relevance of our findings in the broader context of
   human health. Different studies demonstrated that TLR4 activates the
   expression of several pro-inflammatory cytokine genes that play pivotal
   roles in myocardial inflammation, particularly myocarditis, myocardial
   infarction, ischemia-reperfusion injury, and heart failure
   [[152]45–[153]48].

   Also, we found in Aorta and Heart tissues the NOD-like receptor
   signalling pathway; see Tables [154]6 and [155]7. The NOD-like Receptor
   (NLR) family of proteins is a group of pattern recognition receptors
   (PRRs) known to mediate the initial innate immune response to cellular
   injury and stress. Different studies reveal the role of the activation
   of the Nod-like receptor protein 3 (NLRP3) inflammasome in the
   pathogenesis of many metabolic diseases, including diabetes and its
   complications [[156]49].

   Furthermore, the differential networks in the Aorta and Heart tissues
   were enriched for the TNF signalling pathway; see Tables [157]6 and
   [158]7. Lamki et al. [[159]50] reported that Tumor necrosis factor
   (TNF) represent a central mediator of a broad range of biological
   activities from cell proliferation, cell death and differentiation to
   induction of inflammation and immune modulation. TNF mediates the
   inflammatory response and regulates immune function. Inappropriate
   production of TNF or sustained activation of TNF signalling has been
   implicated in the pathogenesis of a broad spectrum of human diseases,
   including diabetes, cancer, osteoporosis, allograft rejection, and
   autoimmune diseases such as multiple sclerosis, rheumatoid arthritis,
   and inflammatory bowel diseases [[160]50, [161]51].

   Instead, for the differential networks in Liver tissue were reported
   the chemokine signalling pathway that promotes changes in cellular
   morphology [[162]52] and insulin signalling pathway that accounts for
   selective insulin resistance [[163]53], see Table [164]6.

   Also, the differential network in the Aorta tissue was enriched for the
   PI3K-Akt signalling pathway, see Table [165]7. Phosphatidylinositol
   3-kinases (PI3Ks) are crucial coordinators of intracellular signalling
   in response to extracellular stimulators. The hyperactivation of PI3K
   signalling cascades is one of the most common events in human cancers.
   The high recurrence of phosphoinositide 3-kinase (PI3K) pathway
   adjustments in cancer has led to a surge in the progression of PI3K
   inhibitors [[166]54]. For diabetes, we found that differential networks
   in Artery Coronary and Artery Tibial tissue were enriched for the PPAR
   signalling pathway, which activation is linked to a correlation between
   metabolic syndromes and cancer, see Table [167]6.

Conclusion

   Differential network analysis (DINA) may help the understanding the
   intricate interactions within biological systems, especially regarding
   specific conditions or phenotypes. This study utilizes DINA to explore
   the differential molecular interactions related to diabetes mellitus,
   taking into account age and gender-specific variations. Differential
   networks are critical since they enable the detection and visualisation
   of variations in gene and protein interactions between males and
   females. This approach helps to identify subtle biological
   discrepancies that conventional analytical methods could miss.
   Non-parametric methods for DINA are emphasized in the study since they
   enhance the robustness and applicability of the findings, given the
   complex nature of biological data, rather than assuming normal
   distributions of gene expression data.

   The study applies this method to liver tissue gene expression data
   related to diabetes, identifying distinct networks that may be involved
   in sex-specific disease mechanisms. These findings can provide insight
   into why certain diseases exhibit different manifestations in men and
   women, ultimately leading to more targeted approaches to treatment and
   management. Identifying gender-specific differential networks helps to
   understand the molecular basis of diabetes and its variation with sex,
   providing potential pathways for therapeutic intervention and a deeper
   understanding of the etiology of the disease.

   This study demonstrates the effectiveness of the differential network
   approach in discriminating between male and female biological samples.
   The method effectively identifies and visualizes differences in
   molecular interactions related to diabetes in liver tissue between the
   two sexes. Such discrimination is vital for personalized medicine,
   leading to more precise diagnostic and therapeutic strategies. By
   effectively highlighting the differences in gene expression and
   interactions, the study supports the potential of DINA in improving our
   understanding of sex-specific traits in diseases, which is crucial for
   advancing gender-specific medicine.

   The article discusses the use of differential networks in biological
   research, especially in complex diseases such as diabetes. Integrating
   non-parametric methods improves the analysis by accounting for the
   inherent complexities and variations in biological data often
   overlooked in parametric approaches. This research contributes
   significantly to our understanding of diabetes and sets a precedent for
   future studies exploring other complex diseases with potential
   variations in expression and interaction patterns across different
   groups.

Acknowledgements