A statistical mechanics investigation of Unfolded Protein Response across organisms

doi:10.21203/rs.3.rs-4908180/v1

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A statistical mechanics investigation of Unfolded Protein Response across organisms

https://doi.org/10.21203/rs.3.rs-4908180/v1

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published 12 Nov, 2024

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Living systems rely on coordinated molecular interactions, especially those related to gene expression and protein activity. The Unfolded Protein Response is a crucial mechanism in eukaryotic cells, activated when unfolded proteins exceed a critical threshold. It maintains cell homeostasis by enhancing protein folding, initiating quality control, and activating degradation pathways when damage is irreversible. This response functions as a dynamic signaling network, with proteins as nodes and their interactions as edges. We analyze these protein-protein networks across different organisms to understand their intricate intra-cellular interactions and behaviors.

In this work, analyzing twelve organisms, we assess how fundamental measures in network theory can individuate seed-proteins and specific pathways across organisms. We employ network robustness to evaluate and compare the strength of the investigated PPI networks, and the structural controllability of complex networks to find and compare the sets of driver nodes necessary to control the overall networks. We find that network measures are related to phylogenetics, and advanced network methods can identify main pathways of significance in the complete Unfolded Protein Response mechanism.

endoplasmic reticulum stress

complex networks

protein–protein interactions

network analysis

The Unfolded Protein Response (UPR) [1] is a mechanism adopted by cells to maintain homeostasis within the endoplasmic reticulum (ER) compartment in response to an accumulation of unfolded or improperly folded proteins (Fig. 1) [2–4]. When protein concentration exceeds physiological levels, pro-survival mechanisms are activated to restore the balance between folded and unfolded proteins [5–7]. The heat shock protein family A member 5 (HSPA5), also known as binding immunoglobulin protein (BiP) [8], is a key promotor of the UPR, activating three stress sensors in the ER: the activating transcription factor 6 (ATF6), the endoplasmic reticulum to nucleus signaling 1 (ERN1), and the eukaryotic translation initiation factor 2 alpha kinase 3 (EIF2AK3), respectively [9–11]. If the adaptive UPR response fails, other pathways are activated, leading to apoptosis and autophagy [6,12]. This mechanism is essential for cell survival in mammals [1,13,14] and is strongly preserved across various organisms, from mammals to yeasts and worms [15–17], as well as in fungi [18] and plants [19,20].

The advancement of network theory has significantly contributed to the study of biological networks, particularly protein-protein interaction (PPI) networks [21–24]. Indeed, by conducting network modeling and topological analysis, researchers can gain insights into genes and proteins involved in various biological functions and disease mechanisms [25–27] The UPR pathway can be described as a PPI network [28–30] and analyzed using complex network tools [31–36]. Information of protein interactions are stored in public databases [37–40], obtained via direct and indirect information (i.e., obtained from experimental Y2H test and homology). Classic measures in network theory, whose definition is briefly reported in Table 1, provide valuable insights into network structure and function, but they do not adequately address the dynamic aspects of network behavior and vulnerability to disruptions. Therefore, structural controllability [41,42] and network robustness [43–45] can be used to identify driver nodes and exploit whether the network withstands failures or attacks.

In this study, we study the properties of the ER stress response network in twelve different organisms, to determine if network analysis methods can provide insight into characteristics of PPI networks. Specifically, we want to identify and analyze the factors that impact the "strength" of various UPR networks and their resistance to potential alterations. This includes looking at random-attack strategies and various metric-based attack strategies and identifying similarities between different organism models. Our findings indicate that the various methods we adopt can uncover different network characteristics, such as phylogenetic similarities [1], distinguishing mammals and their animal models, and the identification of relevant molecular pathways within the UPR mechanism across organisms. Thus, we hypothesize that these network methods can be applied more widely to characterize unknown PPI networks in silico.

Table 1

Definition of quantities used to describe PPI networks.
Term / metric	Definition
Barycenter	The node with the lowest value of eccentricity (for us, the absolute center of the networks)
Betweenness centrality [46]	Measure of how often a node occur on the shortest paths between other nodes
Closeness centrality	C(x)=(∑_y d(x,y))^-1 is a measure of how close a node is to all other nodes in the network
Clustering coefficient	Proportion of edges between the nodes within the i^th neighborhood divided by the number of links that could possibly exist between them
Average clustering coefficient	CC = (∑_i CC_i/n) is the arithmetic mean of clustering coefficient of all the nodes
Density [47]	D = 2M/[N(N-1)] where M is the total number of connections in an N nodes network
Degree	Number of edges of one node
Average Degree	Arithmetic mean of degrees of all network nodes
Diameter	It is defined as the eccentricity of a node with the maximum distance to the other nodes
Edges	Physical or functional connections between pairs of proteins
Modules or Communities	Sub networks that include a high number of inside-sub network edges and a low number of between-sub network edges
Modularity	A measure of network tendencies to divide in communities.
Nodes	Proteins composing the network
Shortest path length	Number of edges needed to connect every pair of nodes through their shortest path

The results are presented in three subsections, each elucidating the potential descriptive and predictive power of the methods employed. These methods allow the association of network properties with phylogenetic analogies and assist in identifying biological weaknesses through advanced network descriptors. These subsections correspond to the methods tested in this study: i) standard network descriptors [48], as well as topological analysis [49], ii) robustness [43–45], and iii) structural controllability [41,42]. The methodological pipeline starts by establishing a native UPR model network for each organism, as detailed in Table 2, utilizing PPI data sourced from public databases. We then create configuration models by randomizing connections while preserving the same number of connections per protein. Well-established network theory measures and advanced network methods are then applied to evaluate the networks for each organism and model.

Table 2

Average values of common network features for all native models.
	Barycenter	#nodes	#edges	Density	Diameter	Degree	Closeness	Betweenness	Cust. coeff.	Modularity	Communities
Homo sapiens [2,52]	HSPA5	216	5286	0.114	7	25	0.0020	145.5	0.522	0.311	5
Rattus norvegicus [53,54]	Hspa5	139	2224	0.116	5	16	0.0030	95.3	0.533	0.368	5
Mus musculus [55–57]	Hspa5	172	3200	0.109	6	19	0.0025	119.1	0.548	0.352	6
Macaca fascicularis [58]	HSPA5	64	578	0.143	6	9	0.0067	45.8	0.544	0.419	5
Bos taurus [59]	HSPA5	93	1102	0.129	5	12	0.0047	64.2	0.528	0.398	4
Oryctologus cuniculis [60,61]	HSPA5	54	528	0.184	5	10	0.0086	33.8	0.597	0.334	4
Gallus gallus [62,63]	HSPA5	29	252	0.310	4	9	0.0199	12.0	0.646	0.273	2
Danio rerio [64–66]	Hspa5	97	1058	0.114	7	11	0.0042	76.9	0.465	0.371	5
c [67–69]	Hsc70-3	57	760	0.238	5	14	0.0089	29.9	0.560	0.324	3
Caenorhabditis elegans [70–72]	hsp-90	109	1212	0.103	7	11	0.0034	98.9	0.594	0.450	6
Saccharomyces cerevisiae [16,73,74]	KAR2	150	2834	0.127	6	19	0.0028	108.0	0.607	0.352	6
Arabidopsis thaliana [75–78]	BIP2; BIP3	62	1188	0.314	5	19	0.0092	25.6	0.680	0.245	3

a. Network and topological analyses

Standard Network Characteristics

Barycenter. We find that the barycenter of all network models corresponds to the Binding Immunoglobulin Protein (KAR2 and BIP2/BIP3 proteins in Saccharomyces cerevisiae and Arabidopsis thaliana are homologous to mammal HSPA5), apart from Caenorhabditis elegans, for which the Heat shock protein 90 (hsp-90) results as the key protein (Table 2). Indeed, literature shows that in Caenorhabditis elegans, hsp-90 plays a crucial role in the chemotaxis to non-volatile and volatile attractants detected by AWC sensory neurons [50,51].

Density. Arabidopsis thaliana, Gallus gallus and Drosophila melanogaster are the most densely connected networks, with the values of 0.314, 0.310 and 0.238, respectively (Table 2). The less densely connected network is found for Caenorhabditis elegans (0.103) and Mus musculus (0.109). Interestingly, we note a threshold value in the number of nodes (about 100 nodes) above which we have a linear trend of both average and normalized average degrees. This may be since increasing the size of the network can provide a decrease in the available public information for some organisms rather than others.

Average Degree. The highest average degree value of the native models is observed for Homo sapiens (25) followed by Mus musculus, Arabidopsis thaliana and Saccharomyces cerevisiae with a value of average degree of 19. The lowest average degree value is found for Macaca fascicularis and Gallus gallus, with a numeric value of 9, followed by Oryctologus cuniculis with a value of 10.

Closeness centrality. Gallus gallus has the highest closeness, with a value of 19.9$\:\cdot\:$10^-3, followed by Arabidopsis thaliana (9.2$\:\cdot\:$10^-3) and Drosophila melanogaster (8.9$\:\cdot\:$10^-3). Homo sapiens shows the smallest value of closeness, with a numeric value of 2.0$\:\cdot\:$10^-3.

Betweenness centrality. As the average degree, the highest betweenness is observed for Homo sapiens (145.5) followed by Mus musculus (119.1) and Saccharomyces cerevisiae (108.0). On the contrary, Gallus gallus (12.0), Arabidopsis thaliana (25.6) and Drosophila melanogaster (29.9) show the lowest betweenness values.

Clustering coefficient. The highest clustering coefficient value is observed for Arabidopsis thaliana (0.680) followed by Gallus gallus (0.646) and Saccharomyces cerevisiae (0.607). All the remaining organisms show clustering coefficient < 0.600 suggesting that, in the smaller networks, proteins tend to be directly connected with their neighbors.

Communities and Modularity. The modularity and number of communities provide a different description of alteration in a network, since it is an evaluation based on its configuration models. In Table 2 we report the number of communities and values of modularity calculated with the Louvain algorithm. Mus musculus and Saccharomyces cerevisiae have the same number of communities (6) and the same modularity (0.352). Caenorhabditis elegans has the highest modularity (0.450), together with Macaca fascicularis (0.419) although the number of communities is different (6 and 5 respectively). Gallus gallus provides the smallest number of communities (2), followed by Drosophila melanogaster and Arabidopsis thaliana (3), probably because the sizes of the networks are quite smaller and comparable among them, concerning the other organisms (29, 57 and 62 nodes). A graph representation of main communities for all the native networks is shown in Fig. 3.

Correlation Analysis. Pairwise comparisons for degree, closeness and betweenness are evaluated via Pearson correlation coefficient. This analysis is performed only across these quantities because they are the more common centrality measures. Betweenness centrality increases with the degree (r = 0.65; p = 0.02) but it shows a strong negative correlation with the closeness (r = − 0.78; p = 0.003). Degree and closeness centrality show a non-significant moderate negative correlation (r = − 0.43; p = 0.16), indicating opposite behavior for these measures.

Normalized Network Characteristics

The native UPR models for the different organisms are characterized by different sizes that may affect the standard measures reported in the previous paragraph. Thus, when analyzing normalized metrics (based on the network size), analogies and differences between organisms becomes clearer (Table 3).

Table 3

Normalized values and statistical analyses of common network features for all native models.
	Degree	Closeness			Betweenness			Clust. coeff
	Norm. value	Norm. value	z-score	p-value	Norm. value	z-score	p-value	z-score
Homo sapiens	0.116	0.440	-3.0	$\:\ll\:$ 0.05	0.063	1.9	$\:<$ 0.05	3.0
Rattus norvegicus	0.115	0.434	-3.0	$\:\ll\:$ 0.05	0.043	-1.1	0.50	3.0
Mus musculus	0.110	0.432	-3.0	$\:\ll\:$ 0.05	0.042	-1.1	0.50	3.0
Macaca fascicularis	0.141	0.424	-3.0	$\:\ll\:$ 0.05	0.080	-0.5	0.10	3.0
Bos taurus	0.129	0.433	-3.0	$\:\ll\:$ 0.05	0.053	-0.9	$\:>$ 0.10	3.0
Oryctologus cuniculis	0.185	0.458	-2.9	$\:\ll\:$ 0.05	0.012	1.1	$\:>$ 0.50	3.0
Gallus gallus	0.310	0.556	-2.3	$\:>$ 0.10	0.130	-1.1	0.50	2.9
Danio rerio	0.113	0.401	-2.9	$\:\ll\:$ 0.05	0.083	0.2	$\:>$ 0.10	3.0
Drosophila melanogaster	0.246	0.501	-2.0	$\:<\:$0.05	0.070	-1.6	$\:>$ 0.50	3.0
Caenorhabditis elegans	0.101	0.368	-3.0	$\:\ll\:$ 0.05	0.048	-2.7	$\:<$ 0.05	3.0
Saccharomyces cerevisiae	0.127	0.422	-3.0	$\:\ll\:$ 0.05	0.092	-1.3	$\:<$ 0.05	3.0
Arabidopsis thaliana	0.306	0.563	-3.0	$\:\ll\:$ 0.05	0.135	0.7	$\:>$ 0.50	3.0

Average Degree. Three mammals (i.e. Homo sapiens, Rattus norvegicus and Mus musculus) share similar values of degree (0.116, 0.115 and 0.110), together with the Danio rerio (0.113). The Caenorhabditis elegans model provides the lowest values of normalized degree (0.101) while Gallus gallus and Arabidopsis thaliana have the highest degree values (0.310 and 0.306). Yet, the Oryctologus cuniculis shows a value of normalized degree (0.185) between the variation range.

Closeness centrality. Homo sapiens, Rattus norvegicus, Mus musculus and Bos taurus share also similar values of closeness centrality (0.440, 0.434, 0.432 and 0.433). Caenorhabditis elegans model provides the lowest values of closeness centrality (0.368) while Gallus gallus, Arabidopsis thaliana and Drosophila melanogaster have the highest closeness values (0.556, 0.563 and 0.501).

Betweenness centrality. Rattus norvegicus, Mus musculus and Caenorhabditis elegans share comparable values of normalized betweenness (0.043, 0.042 and 0.048). Gallus gallus and Arabidopsis thaliana show the highest betweenness values > 0.1 (0.130 and 0.135, respectively).

Configuration models and z-score distributions. To assess concrete differences between organisms models, we resort to the use of configuration models. For each real-world UPR model we reconstruct 10 configuration models, to compute the z-scores of closeness centrality, betweenness centrality and clustering coefficient, reported in Table 3. Mus musculus and Rattus norvegicus show same values of z-scores for all the quantities, while similarities with Homo sapiens model are observed only for closeness centrality and clustering coefficient. Drosophila melanogaster and Saccharomyces cerevisiae have comparable z-score for closeness centrality while Caenorhabditis elegans and Saccharomyces cerevisiae show similar z-score for betweenness centrality. Overall, z-scores of closeness and clustering coefficient are included in small variation ranges ([-3.0, -2.3] and [2.9, 3.0] respectively). Moreover, z-score for betweenness centrality shows significant differences among organisms (from − 2.7 for Caenorhabditis elegans to 1.9 for Homo sapiens), suggesting that betweenness centrality provides a potential useful tool for identifying similarity or differences between organisms regarding this specific mechanism.

Configuration models and statistical analysis. In Table 3, we also report results from nonparametric signed rank test for closeness and betweenness centralities. Regarding closeness centrality, we obtain that the native and the configuration models are significantly different for all species, except for Gallus gallus (p $\:>$ 0.10). In fact, all the p-values for closeness are smaller (or much less) than 0.05, so the test rejects the null hypothesis of zero median at the 5% significance level. On the other hand, for the betweenness centrality distribution, the test provides a significant difference only for half of the sample (Homo sapiens, Caenorhabditis elegans, and Saccharomyces cerevisiae), but distributions cannot be distinguished in the rest of the organisms. Overall, the configuration models alter the network features regarding closeness and only partially regarding the betweenness centrality.

Highest metrics nodes. To more precisely relate the evaluated network metrics with the biological content, we also analyze the role of protein within the UPR pathways. Few specific genes or their homologues appear in all the sets across organisms, associated with high values of degree, closeness and betweenness centralities. In Table 4 we report the nodes with the highest values of the three principal metrics.

A percentage varying between the 5% and 30% of the total number of proteins in each native network is represented by heat shock cognate proteins (HSC), that are members of the heat shock protein family (HSP), one of the most ubiquitous and conserved protein families across organisms [79–82]. They are fundamental in the correct functioning of cells, maintaining the cellular proteostasis and protecting cells from induced stresses [83,84]. Their genes are associated to the highest values of the main three network features, across all the twelve species. They are found as relevant nodes and hubs also in various diseases, like cancers and strokes [85–88]. HSP/HSC proteins for each organism belong to the same community, the one associated with the largest node size (Fig. 2). Other proteins related to the highest values of the three metrics for the various networks are: i) X-box binding protein 1 (XBP1), which is an important initiator and modulator factor of ER stress response [89,90], ii) ATF6 and iii) ERN1, both of which are ER membrane receptors (together with EIF2AK3) [9,91–93], in charge of initiating and regulating the stress response after the activation promoted by HSPA5/BiP [94–96]. Moreover, focusing on the proteins that can be associated to the main subnetwork of UPR [97] (Table 4), another main result is that the most relevant pathway in the UPR mechanism for Saccharomyces cerevisiae and Arabidopsis thaliana is related to the IRE1/ERN1 signaling cascade [98–103], also notable from the fact that we cannot find homologues for the other two ER stress sensors.

Table 4

*Nodes sets with highest values of network metrics, for native models. Proteins’ homologues are found in the Homologous Gene Database (HGD*, https://ngdc.cncb.ac.cn/hgd) [104] *and UniProt database* [105–107]. *Not all proteins have a corresponding homologous in our reference minimal network model* [97].
	Degree	Closeness	Betweenness	Homologous of UPR proteins from Ref. [97]
Homo sapiens	HSPA5, HSPA4, HSP90AB1, HSP90AA1, HSPA8	HSPA5, HSPA4, HSP90AB1, HSP90AA1, HSPA8	HSPA8, HSP90B1, XBP1, HSP90AA1, HSP90AB1, ATF6	ATF4, ATF6, DDIT3, EIF2AK3, EIF2S1, ERN1, HSPA5, NFE2L2, PPP1R15A, XBP1
Rattus norvegicus	HSPA5, HSP90AB1, HSP90AA1, HSPA1B, XBP1	HSPA5, HSP90B1, XBP1, HSPA1B, HSP90AA1, ATF6, DNAJC3	HSP90B1, XBP1, ATF6, ERN1, HSP90AA1, HSPA1B	Atf4, Atf6b, Ddit3, Eif2ak3, Ern1, Hspa5, Nfe2l2, Ppp1r15a, Xbp1
Mus musculus	HSP90B1, HSPA1B, HSP90AB1, HSP90AA1, XBP1	HSPA5, HSP90B1, HSPA1B, XBP1, HSP90AB1, HSP90AA1, ATF6	HSPA1B, CCND1, HSP90B1, ERN1, HSP90AB1	Atf4, Atf6, Ddit3, Eif2ak3, Ern1, Hspa5, Nfe2l2, Ppp1r15a, Xbp1
Macaca fascicularis	HSPA5, HSP90B1, XBP1, DNAJC3, ATF6, HSPA1L	HSPA5, XBP1, HSP90B1, DNAJC3, ATF6, CALR, ATF6B	HSPA5, XBP1, CALR, HSP90B1, DNJC3, ATF6	ATF4, ATF6, DDIT3, EIF2AK3, EIF2S1, ERN1, HSPA5, NFE2L2, PPP1R15A, XBP1
Bos taurus	HSPA5, HSP90B1, DNAJC3, ATF6, XBP1, HSP90AA1, HSPA9	HSPA5, HSP90B1, DNAJC3, ATF6, XBP1, CALR, DERL1	HSPA5, DNAJC3, HSP90B1, ATF6, DERL1, TMED2	ATF4, ATF6, DDIT3, EIF2AK3, EIF2S1, ERN1, HSPA5, NFE2L2, PPP1R15A, XBP1
Oryctologus cuniculus	HSPA5, HSP90B1, XBP1, DNAJC3, ATF6, CALR	HSPA5, HSP90B1, XBP1, DNAJC3, ATF6, EIF2S1, CALR	HSPA5, HSP90B1, XBP1, RNASEL, ATF6, ERN1	ATF4, ATF6, DDIT3, EIF2AK3, EIF2S1, ERN1, HSPA5, NFE2L2, PPP1R15A, XBP1
Gallus gallus	HSPA5, HSP90B1, HSP90AB1, HSP90AA1, DNAJB1	HSPA5, HSP90(B1, AB1, AA1), DNAJ(B1, C3, A2)	HSPA5, HSP90B1, DNAJB1, HSP90AB1, CRYAB, HSP90AA1, DNAJC3	ATF4, ATF6, DDIT3, EIF2AK3, EIF2S1, ERN1, HSPA5, NFE2L2, PPP1R15A, XBP1
Danio rerio	hspa5, hsp90b1, hspa9, sec63, hsp90aa1.1, xbp1, hsp90ab1, hsp90aa1.2, dnajc3a	hspa5, hsp90b1, sec63, hspa9, dnajc3a, canx, xbp1, hsp90aa1.1, edem1, dnajc10	hspa5, hsp90b1, sec63, txnl1, xbp1, eprs1, edem1, hspa9	atf4, atf6, ddit3, eif2ak3, eif2s1b, ern1, hspa5, nfe2l2a, ppp1r15a, xbp1
Drosophila melanogaster	HSC70-3, GP93, HSP83, HSP70AB, HSP70BB, HSC70-5, DROJ2	HSC70-3, GP93, HSP83, DROJ2, CG2918, HSC70-5	HSC70-3, XBP1, GP93, HSP83	crc, Atf6, -, EIF2AK3, Ire1, Hsc70-3, cnc, PPP1R15, Xbp1
Caenorhabditis elegans	HSP-70, HSP-90, HSP-4, HSP-3, HSP-6, ENPL-1	HSP-90, HSP-70, HSP-4, HSP-3, HSP-6, ENPL-1, XBP-1	HSP-90, HSP-70, PQN-91, DAF-16, HSP-4, HSP-4 ENPL-1, HSP-3, HSP-6	Atf-4, atf-6, -, pek-1, ire-1, hsp-4, sknr-1, -, xbp-1
Saccharomyces cerevisiae	HSC82, HSP82, KAR2, LHS1, SSA2, SSA1	KAR2, HSC82, HSP82, SSA1, SSB1, SSA2, HSP104, SSB2	ATG8, HSC82, SSA1, SSB1, SIL1, IRE1, HAC1, CDC48, SSB2	MET4/MET28, -, -, ISR1, IRE1, KAR2, -, -, Xbp1
Arabidopsis thaliana	BIP(3, 2), HSP70-4, HSP90-(1–4, 7)	BIP(3, 2), HSP70-4, HSP90-(1–4, 7)	BIP(3, 2), HSP70-4, HSP90-7, CPN60, CCT6A, BZIP60	-, -, -, -, -, IRE1(A, B), BIP(2, 3), -, -, BZIP(18, 34, 60, 61)

Multi-comparison test. In Fig. 3 we report the correlation matrices related to multiple comparison test applied to normalized metrics distributions (degree, closeness centrality and betweenness centrality) cross-species. Representation of metrics distributions are shown in Figure S5 of Supplementary material. We observe that degree and closeness centrality provide similar results. On the contrary, correlation matrix of betweenness centrality shows that most of organisms have very similar normalized distributions (most of matrix elements are closer to 1), so this metrics does not discriminate among all organisms.

We can conclude that statistical and correlation analyses show that closeness centrality, together with the degree, results in a better network feature to discriminate among the same organism and across different organisms.

Topological analysis. A different type of information comes from the topological analysis of adjacency matrices, evaluated with the Generalized Hamming Distance (GHD, Table 5). It provides a degree of difference between two NxN matrices, by comparing paired matrix elements [49]. In Table 5 we report the average value of GHD calculated for the native network with respect to each of the 10 associated configuration models. The most different models are provided by Arabidopsis thaliana (0.2559), Gallus gallus (0.2414) and Drosophila melanogaster (0.2143); this can be rationalized because the size of the network is small compared to the other organisms, and we have poor biological information about the nodes, so the null models generate quite different related networks. The most similar matrices are obtained for Mus musculus, and it provides a similar result to Rattus norvegicus (0.1359 and 0.1376). Other comparable GHD values are for Homo sapiens, Saccharomyces cerevisiae and Danio rerio (0.1439, 0.1457 and 0.1422).

Homo sapiens	Rattus norvegicus	Mus musculus	Macaca fascicularis	Bos taurus	Oryctologus cuniculis	Gallus gallus	Danio rerio	Drosophila melanogaster	Caenorhabditis elegans	Saccharomyces cerevisiae	Arabidopsis thaliana
0.1439	0.1376	0.1359	0.1617	0.1519	0.1873	0.2414	0.1422	0.2143	0.1397	0.1457	0.2559

We can conclude, from statistical and correlation analyses, that closeness centrality, together with the degree, are convenient network features, able to discriminate across different organisms, and to identify native or null networks within the same organism.

Summarizing the results for all the above quantities, the mammals (Homo sapiens, Rattus norvegicus, Mus musculus and Bos taurus) and their biological models (Danio rerio, Caenorhabditis elegans and Saccharomyces cerevisiae) share similar values, in particular of closeness, and have similar connections, as shown by the correlation matrices and the GHD values.

Overall, the description of UPR networks via standard network quantities and GHD allows to identify phylogenetic similarities, and to characterize the networks of mammals with respect to other phyla.

b. Network robustness

The networks robustness (Fig. 4) has been tested both with random- and metrics-based target attacks to the native networks. In Table 6 we report normalized values of the average path length and efficiency for all investigated network models. Average shortest path length is observed for two of the smallest network models (1.840 and 1.860 for chicken and plant, respectively), with Caenorhabditis elegans providing the highest value (2.832). As expected, the opposite situation is reflected in the normalized global efficiency of the networks, since higher values are related to more robust networks. Comparable values of robustness quantifiers are obtained for the five mammals (~ 2.3–2.4 and ~ 0.48 for human, murine, monkey and the bull), together with the yeast. Another comparable couple is provided by the rabbit and the fruit fly (~ 2.1–2.2 and ~ 0.5).

Table 6

Normalized values of average path length and efficiency of all native models.
	Average path length	Efficiency
Homo sapiens	2.354	0.488
Rattus norvegicus	2.382	0.483
Mus musculus	2.393	0.479
Macaca fascicularis	2.453	0.486
Bos taurus	2.396	0.487
Oryctologus cuniculis	2.277	0.523
Gallus gallus	1.860	0.629
Danio rerio	2.601	0.456
Drosophila melanogaster	2.068	0.571
Caenorhabditis elegans	2.832	0.429
Saccharomyces cerevisiae	2.450	0.478
Arabidopsis thaliana	1.840	0.633

The evolution of the largest connected component (LCC) in each network is evaluated by removing at each step one node from the network based on ascending index (random) [108], degree [109] and centralities [110] attacks, i.e. the least “important” nodes are removed first. The LCC identifies a connected component of a given graph that contains a significant fraction of the entire graph's vertices. The removal based on random node choice provides a linear trend for all organisms (Fig. 4). An attack strategy relying on random removal of nodes requires the removal of many nodes for significantly decreasing the potency of the attack, so targeted attacks result more efficient in degrading the network [111,112]. In most cases, the betweenness-based attack strongly affects the behavior of the network, except for Caenorhabditis elegans, for which the opposite is observed. In general, degree-based and closeness-based attacks show the same behavior of LCC degradation, except for Macaca fascicularis, Bos taurus and Arabidopsis thaliana which provide similar trends for the two centralities attacks. Gallus gallus provides a situation in which all the three metrics-based attacks have comparable effects on the robustness of the network model.

Overall, degree and closeness are quite correlated features in this type of analysis. This finding is also supported by sets of nodes with highest values of metrics (see Table 4), where it is shown that sets for degree and closeness are more similar compared to sets obtained for betweenness centrality, as confirmed also by the computed correlation coefficients between network metrics (see above). Interestingly, Macaca fascicularis, Bos taurus, Danio rerio and Arabidopsis thaliana show a degree-based degradation in the robustness quite similar to the random removal, so much slower compared to the two centrality-based attacks.

By removing nodes with increasing value of metrics, we expect a slower degradation of the networks integrity (notable especially for the human, murine, bull, zebrafish, yeast and plant models); the results could be explained considering a larger number of “external” and less important nodes within the networks (very low values of network metrics). Attacking the networks based on decreasing values of metrics provides a random-like trend in network robustness.

The sudden jumps in the robustness can be associated not only to a precise role of a specific protein removal, because their removal at the beginning of the process could not produce the same results, like in a purely star network. Instead, it can also be the result of a cumulative effect, because the overall removal of all the previous proteins builds a star-like network, that collapses upon removal of a specific protein.

However, going through the principal functions and pathways that are involved in the UPR and arising from the network robustness, across all, or most of, the organisms, we highlight that those proteins with the role of chaperones, i.e. initiating the signaling pathways, are present in all considered organisms (apart Caenorhabditis elegans) as the ones that induce a sudden collapse of metrics robustness. Also, proteins involved in pro-apoptosis and ERAD mechanisms emerge from the robustness analysis as present in most organisms. Other mechanisms, with corresponding proteins as obtained from robustness stress analysis, are listed in Table 7.

Table 7

UPR key mechanisms, pathways and functions sustaining the network robustness.
	Chaperone	TM receptor and transfer	Kinase	Regulator and messenger	Pro-apoptotic and ERAD	Pro-survival	Initiation and transcription factor	Protein folding	Nonspecific in UPR
Homo sapiens	HSPD1	–	–	–	UBQLN1, TMBIM6	–	–	–	–
Rattus norvegicus	Hspb2	–	–	–	Tomm20	–	–	Erp44	–
Mus musculus	Hspb6, Dnajb12	–	–	–	Vapb, Os9	–	Creb3, Creb3l1	–	Cops5
Macaca fascicularis	CRYAA	–	–	–	BAG6, HERPUD2	–	CREBRF	–	–
Bos taurus	HSPB1, DNAJA2, DNAJC10	TMTC4	ERN2	–	ERN2, DNAJC10, HERPUD2	–	–	–	–
Oryctologus cuniculis	HSPA5	–	–	–	RNASEL	–	–	–	–
Gallus gallus	DNAJA4, DNAJB1, DNAJA3	ERN1	–	–	–	–	–	–	–
Danio rerio	hsp90aa1.2, dnajc10	PEK	–	–	dnajc10, heprud2	ddrkg1	–	–	–
Drosophila melanogaster	CG4461, CG5504, Hsp67Ba	–	–	–	Edem2, Der-1	Ficd, wfs1	–	–	–
Caenorhabditis elegans	–	–	–	–	–	–	pqn-2, nhr-80, pqn-79, pqn-90	–	set-6
Saccharomyces cerevisiae	APJ1, GIM3	–	–	–	–	–	BTT1	–	–
Arabidopsis thaliana	PFD6, DJA5, HSP70-6, HSP90-3, HSP70-4	F26K24.16	–	F23H11.4	–	–	BZIP60	–	–

c. Structural controllability

The structural controllability allows to determine the number and identity of the minimum driver nodes (Table 8). For all organisms, the study of the native model may produce different number of driver nodes sets, and of driver nodes.

For Homo sapiens, Gallus gallus, and Drosophila melanogaster the algorithm finds 1 single set of driver nodes (namely, of RPA2P, HSP70AB/I(2)EFL, and STT3B).

For all other organisms, the algorithm finds different possible driver nodes sets, moreover, for the same organisms, few nodes occur in all the possible allowed driver nodes sets. Recurrent proteins in the driver nodes sets are: UFL1 for Rattus norvegicus, Ficd for Mus musculus, CRYAB/NCK1/VAPB for Macaca fascicularis, abu-12 for Caenorhabditis elegans, and NAC062/PFD1 for Arabidopsis thaliana. The algorithm cannot find driver nodes sets for Bos taurus, Oryctologus cuniculis and Danio rerio. This is not surprising, because it is possible that the description of these networks lacks connections necessary to define the scheme of structural controllability.

Gene analysis of driver nodes. We provide some details on genes identified as driver nodes, to identify general trends and behaviors in the different networks.

Table 8

Structural controllability results for all species models (# DNs and nodes sets). Details on genes identified as DNs in the case of multiple DNs sets are in the text.
	DNs sets	# DNs sets
Homo sapiens	1 (RPAP2)	1
Rattus norvegicus	2 (UFL1, CREB3L4 or CREB3L2)	2
Mus musculus	4	8
Macaca fascicularis	5	4
Bos taurus	–	–
Oryctologus cuniculis	–	–
Gallus gallus	1 (STT3B)	1
Danio rerio	–	–
Drosophila melanogaster	2 (HSP70AB, l(2)EFL)	1
Caenorhabditis elegans	3	3
Saccharomyces cerevisiae	4	16
Arabidopsis thaliana	9	8

The RPAP2 gene, associated with transcription and RNA processing, connects to RUBVL2 and has minimal network metrics [113]. In metrics terms, this node is characterized by 1 degree (connected to RUBVL2), 1.48·10^− 3 closeness and 0 betweenness (we can define it an “external node”).

In Rattus norvegicus, the two alternate driver nodes (CREB3L family of transcription factors), in combination with UFL1, are homologues and they possess distinctive transcriptional activities through their binding to cAMP response elements [114].

The FIC domain protein adenylyltransferase (FICD) is an enzyme belonging to the Fic (filamentation induced by cAMP) domain family. FICD is associated with various cellular pathways, particularly the ATF6 and EIF2AK3 branches of the UPR pathway, which regulate ER homeostasis. In humans, FICD is typically present at very low basal levels in most cell types, and its expression is tightly regulated [115,116].

In mammals, STT3B facilitates the degradation of misfolded proteins via the ERAD pathway [117]. The HSP70AB gene in fruit flies, part of the Hsp70 family, stabilizes non-native protein conformations [118–122]. The I(2)EFLS gene in Drosophila[123] induces phosphorylation of eIF2α, playing a role in aging [124–127]. The Abu-12 gene in C. elegans encodes UPR proteins that function alongside the canonical UPR pathway in xbp-1 mutants under ER stress [128–130].

In addition, we also analyze the possible altered behavior of the native models by excluding from the networks the DNs obtained in the first round of analysis. The algorithm is not able to find other combinations of driver nodes if the original identified nodes are excluded from the networks.

Control theory applied to UPR networks identifies driver nodes related to common molecular pathways. In rat, two driver nodes are transcription factors; in fruit fly, two are chaperones. The gene Ficd is present as jump node in Drosophila melanogaster robustness and as driver node in mouse and pertains to two of the main pathways of signaling. Macaca fascicularis provides the chaperone CRYAB as recurrent driver node, together with VAPB that also results as jump node for Mus musculus, and NCK1 that is an ERN1 regulator within its signaling pathway. PFD1 protein of Arabidopsis thaliana is another molecular chaperone, like PFD6 which arises from the network robustness.

Overall, it is not possible to identify one single mechanism or protein that is always present across all or most organisms, and this could be due to missing information and/or intrinsic network differences, but some genes and/or mechanisms are recurrent as in the robustness analysis.

Unfolded Protein Response stands as one of the highly conserved fundamental biological mechanisms in organisms [15,16,131,132], occurring primarily in the endoplasmic reticulum (ER). Its main function is to restore cell homeostasis after a pathological accumulation of non-properly folded proteins [6,7,10,11,133,134]. Any inefficiency in the adaptive response to ER stress can lead to the accumulation of unfolded or misfolded proteins at different levels. These proteins tend to aggregate, posing a threat to cellular and tissue integrity and serving as a primary driver for the onset of amyloidosis and neurodegenerative diseases [25–27,135].

A key signaling pathway governing the UPR, originally discovered in Saccharomyces cerevisiae during the 1970s [52,136–138], is characterized by a single transmembrane protein, ERN1, responsible for detection of ER stress provoked by over-accumulation of unfolded/misfolded proteins [98–100]. The major ER chaperone BiP triggers the dimerization of ERN1, which leads to its subsequent autophosphorylation and the activation of its signaling cascade [9]. This pathway reinforces the ER function, and it is conserved across eukaryotes [139]. Indeed, the basic features of the UPR mechanism result to be highly preserved throughout metazoans; most species have homologues of the three main stress sensors, ERN1, ATF6 and EIF2AK3. Signaling pathways of the stress sensors cooperate to restore and/or bolster ER function, primarily through upregulation of many components of the protein folding machinery (as the action of XBP1 on regulation of BiP chaperone [89,90] within the ERN1 pathway) and the quality control machinery within the ER. Additionally, these signaling pathways work to limit ER stress by dampening the attenuation of translation and potentially engaging the regulated IRE1-dependent decay (known as RIDD) [140–142].

A network-based description of cell mechanisms, using protein-protein interactions (PPI) networks, offers a valuable tool for comprehending the behavior of complex systems. Some points must be considered before discussing the analysis results. PPI network models are built utilizing biological data sourced from publicly available PPI databases. These databases collect various types of data, from thousands of experimental works. Despite their current large size, the databases are not exhaustive, because only part of the molecular pathways have been completely understood and characterized. The absence or bias in information stored in interaction databases must be considered when creating and analyzing biological networks, and PPI databases must contain sufficient information for a specific pathway to yield qualitative accurate results from network analysis techniques.

With these premises, we apply a classical network metrics investigation to assess the degree of similarity and differences existing among UPR models from twelve different organisms, considering the existing phylogenetic pathway in all the analyzed models. As model organisms, we choose for our investigation Homo sapiens, Rattus norvegicus and Mus musculus due to the quite complete genomic accordance with human [143–148], and other model organisms, as Macaca fascicularis [58], Bos taurus [57,59], Oryctologus cuniculis [60,61], Gallus gallus [62,63], Danio rerio [65,66], Drosophila melanogaster [149,150], Caenorhabditis elegans [151], Saccharomyces cerevisiae [152], and Arabidopsis thaliana [76,77,103].

Comparisons made for common metrics provide that network measures are very sensitive to the network features (Table 1, and Figures S2–S4 from Supplementary); however, from a biological point of view, the network analysis shows that there are strong similarities among the model organisms (Table 3), in particular among mammals and their model organisms. Moreover, there are highly conserved genes and pathways across species (Table 4). Chaperones belonging to the heat shock protein (HSP) family are among the most central proteins in terms of network metrics, as the densest connected, and they also play a fundamental role in the correct functioning of cells [79–83]. This result can be interpreted as a support to significance of these proteins, as it implies a correlation between the structure and the biological properties of protein networks. In summary, network theory and statistical mechanics confirm that it is possible to identify similarities among organisms phylogenetically related via their PPI networks. In addition, we find that there are some network metrics that better discriminate among organisms, in the sense that are useful for identifying similarities and differences, as closeness centrality and node degree.

Additionally, in this study we employ a network scientific methodology to investigate the levels of tolerance of multiple systems when subjected to external perturbations. We achieve this by adapting a measure of network robustness [153–156], to characterize the potential resilience of several PPI networks. The levels of robustness of the twelve models are evaluated by removing nodes, and consequently altering the network integrity, adopting different network-based metrics target strategies. Obviously, in these cases, network robustness is influenced by network features and degree of accuracy of biological information accessible online, but similarities in resilience behavior arise for organisms that are phylogenetically closer, for example Homo sapiens and murine species, despite the difference between network features. This result can be supported by the closeness between these organisms in terms of phylogeny, since 99% of the genome is conserved between human and murine species [143–148]. Similar behaviors can be also found based on network-size – i.e., bovine, plant and machaca, or zebrafish and fruit fly. A detailed analysis of genes that are related to jumps in the network robustness highlights that, despite different proteins are involved in different organisms, there is a recurrence of pathways across most species. In particular, the sensing role of the chaperones is fundamental, together with the apoptosis and endoplasmic-reticulum-associated protein degradation (ERAD) functions.

Lastly, the minimum drive nodes methodology, based on the structural controllability analysis [41,42] and employing Kalman's rank condition [157], yields significant biological insights into the proteins involved in specific mechanisms. Through the application of this theory to network models, it becomes feasible to pinpoint the key nodes that exert control over the entire network [41,42,158]. We apply structural controllability to all organism models investigated in this work. Driver nodes sets are related to the topological structure of the network; as for the other network methods applied, results are sensitive to the biological information available online (also resulted in the high variability of the possible DNs sets obtained – several sets for the same organism in Table 8). Attributing a biological significance to the set of driver nodes is not straightforward. When analyzing networks with a high number of nodes and edges, there is a possibility of losing biological information in the models due to the consequent increase of missing information stored in the databases. However, some specific mechanisms and genes, already identified via robustness and network statistics, arise also in the list of driver nodes.

This study represents a comprehensive and innovative analysis of the biological behavior and characteristics of a fundamental cellular control mechanism, the Unfolded Protein Response (UPR). By modeling the UPR as a protein-protein interaction (PPI) network, it uses both standard and advanced techniques to extract meaningful information from the network. The study establishes a direct correlation between specific network features and biological components. It finds that the three classes of methods employed – standard network metrics and topology, network robustness, and network control theory – offer complementary and non-conflicting characterizations of the systems studied.

The first class of methods offers a comprehensive portrayal of the PPI networks and clearly discerns phylogenetic similarities, distinguishing mammals and their animal models (Danio rerio, Caenorhabditis elegans, and Saccharomyces cerevisiae) from organisms of other phyla. Additionally, it identifies key genes (network nodes) that significantly influence the overall UPR mechanism. Network robustness utilizes network metrics to assess the network's ability to withstand node removal, identifying pivotal proteins and sub-pathways, and simulating disease onset. This robustness analysis also highlights phylogenetic similarities among organisms.

Among the various UPR sub-mechanisms, chaperone sensing, apoptosis, and ERAD are identified as the most relevant. The control theory, though less distinctly, also pinpoints proteins with central roles, particularly chaperones, transcription factors, and ERAD proteins. Using network models for molecular and cellular pathways is a powerful yet underutilized approach. This study's combination of diverse network descriptors and methods provides profound insights into complex mechanisms and emphasizes the need to revise and enrich PPI databases to create increasingly accurate biological models.

In this work, we present a study of the unfolded protein response mechanism among twelve organisms.

Below we report general information of the network models investigated in this study, and the description of the performed analyses. We perform calculation of i) network descriptors: degree, betweenness and closeness centralities, ii) modularity and communities, iii) topological distance, iv) network robustness, and v) structural controllability. We perform comparisons among native models, with the help of the configuration models (degree-based reconstruction). Native models are built using Python v. 3.11 programming language and NetworkX Python library [159], based on information on paired connection between couples of nodes, and configuration models are built using MATLAB v. R2023a programming language. Modularity and communities, and robustness analyses are performed using NetworkX library, while network descriptors analysis and structural controllability [160,161] have been implemented in MATLAB.

UPR network models

Here we present the UPR network models proposed in this work. Investigated organisms are: Homo sapiens, Rattus norvegicus, Mus musculus, Macaca fascicularis, Bos taurus, Oryctolagus cuniculis, Gallus gallus, Danio rerio, Drosophila melanogaster, Caenorhabditis elegans, Saccharomyces cerevisiae, and Arabidopsis thaliana. We combine information stored in UniProt database [105–107,162] and String protein-protein interaction database [163–166] to identify proteins involved in the mechanism for all investigated organisms and build undirected network models. We impose a minimum required interaction score of medium confidence, and we consider experimental evidence, curated databases, text-mining and co-expression associations to construct the set of connections among nodes. In this framework, original models, also referred as native, are built directly from biological information obtained by the consulted databases, while connections in configuration models are semi-randomly built based on total degree of each node in the original models. Matrix representation of native UPR models is shown in Figure S1 of Supplementary material.

Analysis of network models using network metrics

We first characterize the models using the usual metrics of network descriptors, (i) to highlight differences and similarities between models related to different organisms and (ii) to evaluate the influence of constructing a network with random connections starting from the degree of nodes of original models.

For each model, we calculate (i) the total degree of the nodes, (ii) the betweenness and closeness centralities, which are measures of how often each graph node appears on the shortest path between two nodes in the graph, (iii) local clustering coefficient, and (iv) the modularity and number of communities, as measures of the structure of networks to evaluate the strength of division into different modules, calculated using the Louvain community detection Algorithm. As definition, networks with high modularity have dense connections within communities.

In addition, we also include normalized values of degree ($\:\mathbf{n}\mathbf{o}\mathbf{r}\mathbf{m}\left(\mathbf{d}\mathbf{e}\mathbf{g}\right)\:=2\mathbf{M}/\mathbf{N}$, where 2M is the total degree of a node in a N nodes network) [167], betweenness ($\:\mathbf{n}\mathbf{o}\mathbf{r}\mathbf{m}\left(\mathbf{b}\mathbf{e}\mathbf{t}\right)=(\mathbf{b}\mathbf{e}\mathbf{t}-\mathbf{m}\mathbf{i}\mathbf{n}(\mathbf{b}\mathbf{e}\mathbf{t})/(\mathbf{m}\mathbf{a}\mathbf{x}\left(\mathbf{b}\mathbf{e}\mathbf{t}\right)-\mathbf{m}\mathbf{i}\mathbf{n}\left(\mathbf{b}\mathbf{e}\mathbf{t}\right))$) and closeness centrality ($\:\mathbf{n}\mathbf{o}\mathbf{r}\mathbf{m}\left(\mathbf{c}\mathbf{l}\mathbf{o}\mathbf{s}\right)=\left(\mathbf{N}-1\right)\cdot\:\mathbf{c}\mathbf{l}\mathbf{o}\mathbf{s}$) [168] (Table 3). All measures are normalized by removing the dependence from the network dimensionality (normalized distributions shown in Figures S2-S4 of Supplementary material). We compute z-scores of averaged normalized metrics distributions, defined as $\:\varvec{x}-\varvec{\mu\:})/\varvec{\sigma\:}$, where $\:\varvec{x}$ is the value of the variable, $\:\varvec{\mu\:}$ and $\:\varvec{\sigma\:}$ are the mean value and standard deviation of the population, respectively.

Analysis of topological differences between original and configuration models

As well explained in our previous work [97], we use the algorithm relies on the Generalized Hamming Distance (GHD) [169], which can be used for assigning a “weight” to the topological difference between networks, and evaluating its statistical significance, based on comparison between matrix elements. We apply this theory to evaluate the degree of difference between original models (built from the databases) and the configuration models (re-created from original models).

If we consider two distinct networks, labeled X and Y, with the same number of nodes (N), we can calculate the distance dGHD between the two networks as follows:

$$\:\mathbf{d}\mathbf{G}\mathbf{H}\mathbf{D}\left(\mathbf{X},\mathbf{Y}\right)=\:\frac{1}{\mathbf{N}(\mathbf{N}-1)}\sum\:_{\mathbf{i}\ne\:\mathbf{j}}({{\mathbf{x}}^{\mathbf{{\prime\:}}}}_{\mathbf{i}\mathbf{j}}-{\mathbf{y}}_{\mathbf{i}\mathbf{j}}^{\mathbf{{\prime\:}}})$$

where $\:{\mathbf{x}}_{\mathbf{i}\mathbf{j}}^{\mathbf{{\prime\:}}}$ and $\:{\mathbf{y}}_{\mathbf{i}\mathbf{j}}^{\mathbf{{\prime\:}}}$ are mean centered edge-weights, and depend on the topology of the network, providing a measure of connectivity between every pair of i^th and j^th nodes in X and Y, respectively.

Statistical analysis of network models

The graph analysis, together with the network robustness, is performed using Python 3.11 [170]. The statistical analysis, the implementation of structural controllability and the GHD algorithm are performed using MATLAB R2023a [171]. We apply non-parametric tests because our variables are not normally distributed. For paired comparisons between the centralities distributions of native and null models considering one specific organism we use the Wilcoxon signed-rank test, and for non-paired comparisons across different organisms we use a multi comparison test computing values with the Bonferroni method [172–178], on the results of a one-way Kruskal Wallis analysis of variance (shown in Figure S4 of Supplementary material). If p < 0.05, the results are regarded statistically significant.

Network robustness exploitation

Considering a network X, composed of N nodes set denoted as $\:\mathbf{V}\hspace{0.17em}=\hspace{0.17em}\{{\mathbf{v}}_{1},\hspace{0.17em}{\mathbf{v}}_{2},\hspace{0.17em}\dots\:,\hspace{0.17em}{\mathbf{v}}_{\mathbf{N}}\}$, interconnected by M links represented by $\:\mathbf{E}\hspace{0.17em}=\hspace{0.17em}\left\{\right({\mathbf{v}}_{\mathbf{i}},\hspace{0.17em}{\mathbf{v}}_{\mathbf{j}}):\hspace{0.17em}{\mathbf{v}}_{\mathbf{i}},\hspace{0.17em}{\mathbf{v}}_{\mathbf{j}}\hspace{0.17em}\in\:\hspace{0.17em}\mathbf{V}\}$, the robustness $\:\mathbf{R}$ of the X network is defined by the ratio [179,180]

$$\:\mathbf{R}=\:\frac{1}{\mathbf{N}}\sum\:_{\mathbf{i}=1}^{\mathbf{N}}{\mathbf{G}}_{\mathbf{i}}$$

where $\:{\mathbf{G}}_{\mathbf{i}}={\mathbf{n}}_{\mathbf{i}}/\mathbf{N}$ is the size of the large connected component after the removal of i^th node. The normalization factor $\:{\mathbf{N}}^{-1}$ is useful for the comparison of networks of different sizes. To numerically quantify the robustness of a network, some metrics can be used. The average path length $\:\varvec{l}$ can provide a quantification of network robustness, since large values of $\:\varvec{l}$ mean that nodes are farther apart from each other, and removal of a node can significantly increase the average paths between many other nodes, decreasing the robustness of the network [181]:

$$\:\varvec{l}=\frac{1}{\mathbf{N}\cdot\:(\mathbf{N}-1)}{\sum\:}_{\mathbf{i}\ne\:\mathbf{j}}{\mathbf{d}}_{\mathbf{i}\mathbf{j}}$$

where the sum of all possible paired-nodes distances is normalized over all the possible couples of N nodes.

Another useful metrics to quantify network robustness is the variation of the efficiency $\:\varvec{\Delta\:}\mathbf{E}$ depending on increasing number of removed nodes [179]. A robust network would have a small drop in the network efficiency with the removal of a node.

$$\:\varvec{\Delta\:}{\mathbf{E}}_{\mathbf{i}}=\frac{\mathbf{E}-{\mathbf{E}}_{\mathbf{i}}}{\mathbf{E}}$$

where the efficiency is defined as $\:\mathbf{E}={\left(\mathbf{N}\cdot\:\left(\mathbf{N}-1\right)\right)}^{-1}{\sum\:}_{\mathbf{i}\ne\:\mathbf{j}}{\mathbf{d}}_{\mathbf{i}\mathbf{j}}^{-1}$.

In the investigation presented here, we employ the theory of network robustness to assess the capability of networks to deliver and maintain an acceptable level of service in the presence of faults, as outlined by our models. Our analysis involves subjecting each network to both random and targeted attacks, following specific strategies (degree-, closeness-, and betweenness-based).

Structural controllability and minimum driver nodes identification

Lastly, we employ the structural controllability theory to assess the nodes ranking using the Kalman’s rank condition for continuous linear time-invariant systems [157]. In addition, we also implement the Minimum Driver Nodes (MDN) algorithm [160,161], proposed by Liu et al., which is based on the minimal set of input signals required to fully control the network, and the MDN selection algorithm used by Zhang et al. [158], which can be used to identify the driver nodes — the nodes on which an input signal must be injected to obtain full control of the network.

Generally, the time-evolution of a network system consisting of N nodes and M input signals, with M ≤ N, can be described with the following linear differential equation

$$\:\frac{\mathbf{d}\mathbf{x}\left(\mathbf{t}\right)}{\mathbf{d}\mathbf{t}}=\mathbf{A}\mathbf{x}\left(\mathbf{t}\right)+\mathbf{B}\mathbf{u}\left(\mathbf{t}\right)$$

where $\:\mathbf{x}={\left({\mathbf{x}}_{1},\:{\mathbf{x}}_{2},{\mathbf{x}}_{3},\:\dots\:,\:{\mathbf{x}}_{\mathbf{N}}\right)}^{\mathbf{T}}$ is the state vector for the N nodes system and $\:\mathbf{u}={\left({\mathbf{u}}_{1},\:{\mathbf{u}}_{2},{\mathbf{u}}_{3},\:\dots\:,\:{\mathbf{u}}_{\mathbf{M}}\right)}^{\mathbf{T}}$ is the control vector. A is the N$\:\times\:$N state matrix, in which each element $\:{\mathbf{a}}_{\mathbf{i}\mathbf{j}}$ identifies the connection between the i^th and j^th nodes, while B is the M$\:\times\:$N control matrix, whose dimension M depends on the number of input signals

$$\:\mathbf{B}=({\mathbf{e}}_{1}^{\mathbf{T}}\:{\mathbf{e}}_{2}^{\mathbf{T}}\:{\mathbf{e}}_{3}^{\mathbf{T}}\dots\:{\mathbf{e}}_{\mathbf{M}}^{\mathbf{T}})$$

where $\:\{{\mathbf{e}}_{1},\:{\mathbf{e}}_{2},\:{\mathbf{e}}_{3},\:.\:.\:.\:,\:{\mathbf{e}}_{\mathbf{M}}\}$ are the vectors of the canonical base.

Given A and B, it is possible to assemble the controllability matrix C:

$$\:\mathbf{C}=(\mathbf{B},\:\:\mathbf{A}\mathbf{B},\:\:{\mathbf{A}}^{2}\mathbf{B},\:\:{\mathbf{A}}^{3}\mathbf{B},\:\:\dots\:,\:\:{\mathbf{A}}^{\mathbf{N}-1}\mathbf{B})$$

If the controllability matrix has a full rank, i.e., rank(C)$\:\:=\:$N, the network is fully controllable. Theory and algorithms are well-explained in Ref. [97].

Acknowledgements

A.L., S.F. and L.C. acknowledge the support of the International Center for Relativistic Astrophysics Network

(ICRANet), Italy, and the support of the Italian National Group for Mathematical Physics (GNFM-INdAM).

Author contributions

Conceptualization: N.L., K.M.S., S.F. and L.C.

Methodology: N.L., K.M.S., M.A.G.M., A.L. and L.C.

Formal analysis: N.L.

Investigation: N.L.

Writing – Original draft: N.L., M.A.G.M. and L.C.

Writing – Review and editing: all authors

Visualization: N.L. and M.A.G.M.

Supervision: K.M.S., S.F. and L.C.

Competing interests

The authors declare no competing interests.

Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

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A statistical mechanics investigation of Unfolded Protein Response across organisms

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Journal Publication

Version 1

Abstract

Figures

Introduction

Results

a. Network and topological analyses

Standard Network Characteristics

Normalized Network Characteristics

b. Network robustness

c. Structural controllability

Discussion

Materials and Methods

UPR network models

Analysis of network models using network metrics

Analysis of topological differences between original and configuration models

Statistical analysis of network models

Network robustness exploitation

Structural controllability and minimum driver nodes identification

Declarations

References

Additional Declarations

Supplementary Files

Status:

Journal Publication

Version 1