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On dynamic network entropy in cancer
James West, Ginestra Bianconi, Simone Severini, Andrew Teschendorff
TL;DR
The paper asks which network properties distinguish cancer from normal cellular physiology and could help identify therapeutic targets. It integrates gene-expression data with a protein-interaction network to induce phenotype-specific stochastic dynamics, finding increased cancer dynamic entropy and an anticorrelation between expression and local entropy changes. The results associate reduced local entropy with proliferation-related genes and potential drug targets.
Problem
Cancer’s systems-level network principles remain insufficiently understood, limiting explanations of cancer biology and identification of novel drug targets.
Method
The study combines static normal and cancer gene-expression data with a human protein-interaction network, using expression correlations to define phenotype-specific stochastic dynamics and dynamic entropy.
Results
Cancer cells show increased dynamic network entropy, while differential expression is strongly anticorrelated with local differential entropy and overexpressed genes preferentially show entropy reductions.
Takeaways & Limitations
Reduced local entropy in proliferation-related genes and oncogenes may help identify cancer vulnerabilities and promising targeted-intervention candidates.
Abstract
from arXiv · showhide
The cellular phenotype is described by a complex network of molecular interactions. Elucidating network properties that distinguish disease from the healthy cellular state is therefore of critical importance for gaining systems-level insights into disease mechanisms and ultimately for developing improved therapies. By integrating gene expression data with a protein interaction network to induce a stochastic dynamics on the network, we here demonstrate that cancer cells are characterised by an increase in the dynamic network entropy, compared to cells of normal physiology. Using a fundamental relation between the macroscopic resilience of a dynamical system and the uncertainty (entropy) in the underlying microscopic processes, we argue that cancer cells will be more robust to random gene perturbations. In addition, we formally demonstrate that gene expression differences between normal and cancer tissue are anticorrelated with local dynamic entropy changes, thus providing a systemic link between gene expression changes at the nodes and their local network dynamics. In particular, we also find that genes which drive cell-proliferation in cancer cells and which often encode oncogenes are associated with reductions in the dynamic network entropy. In summary, our results support the view that the observed increased robustness of cancer cells to perturbation and therapy may be due to an increase in the dynamic network entropy that allows cells to adapt to the new cellular stresses. Conversely, genes that exhibit local flux entropy decreases in cancer may render cancer cells more susceptible to targeted intervention and may therefore represent promising drug targets.
I. INTRODUCTION
The paper investigates dynamic network entropy as a systems-level property distinguishing cancer from normal physiology and potentially revealing selective therapeutic targets. It combines gene expression with a protein-interaction network to model phenotype-dependent stochastic dynamics.
- Cancer-associated network rewiring remains insufficiently understood, limiting systems-level explanations of cancer biology and discovery of novel drug targets.
- Dynamic network entropy is studied because dynamical-systems theory relates microscopic entropy changes to macroscopic system resilience.The cited framework applies to stochastic dynamics defined on networks.
- Earlier protein-network studies used topology-only stochastic dynamics, leaving phenotype-dependent expression effects unmodeled.
- The method uses static normal and cancer gene-expression data to approximate stochastic dynamics on a human protein-interaction network.The network topology is held constant while phenotype-specific expression correlations modify interaction probabilities.
- The study examines local entropy differences alongside differential expression and discusses implications for future cancer-therapy design.The proposed future direction includes integrating drug-sensitivity data with multidimensional tumour profiles.
- The constructed interaction network contains 10,720 nodes and 152,889 documented interactions after integrating multiple protein-interaction sources.
Normal and cancer tissue gene expression data sets
The study builds phenotype-specific integrated mRNA–protein-interaction networks from curated tissue-expression studies, preserving network topology while expression correlations determine stochastic edge weights.
- Six tissue studies met sample-size, Affymetrix-platform, and data-quality criteria for integrating normal and cancer expression profiles.The tissues were bladder, lung, gastric, pancreas, cervix, and liver.
- For each phenotype, the same protein-interaction network is combined with gene-expression data to construct an integrated mRNA–PIN.
- The stochastic matrix assigns transition probabilities only to neighboring genes, with probabilities normalized across each node’s PIN neighbors.
- Edge weights transform Pearson expression correlations, treating correlations and anti-correlations differently while setting non-edge transitions to zero.The resulting networks approximate positive-correlation signal-transduction flow subject to PIN structure.
- The cancer and normal networks retain identical node degrees; only random-walk weights differ between phenotypes.The study approximates signal-transduction flux using positive expression correlations, acknowledged as a crude limitation because matched molecular data are unavailable genome-wide.
A heat kernel stochastic matrix
The paper converts network transition probabilities into total path-based information flux using weighted walks, then chooses factorial path weights to obtain a heat-kernel-like diffusion process.
- A path of length L carries probability flux from node i to node j equal to (p^L)_ij.
- Total flux E_ij aggregates contributions from paths of different lengths using adjustable weights α_L and a normalization factor.
- Choosing α_L = t^L/L! suppresses long paths while guaranteeing convergence of the infinite path series.
- The parameter t acts as a temperature controlling the resulting modified heat-kernel stochastic matrix.
- For sufficiently large t, the matrix approximates a solution of the heat-diffusion equation and connects to heat-kernel PageRank.
The dynamic network entropy
Dynamic network entropy quantifies uncertainty in phenotype-specific network flux, using normalized transition-derived matrices and local node contributions across paths of varying length.
- The method defines local dynamic entropy S_i from the nonzero flux values K_ij associated with node i’s neighbors.
- The entropy is treated as a non-equilibrium quantity because the stationary distribution of K_ij is excluded.This avoids biasing each node’s entropic contribution toward topological properties such as degree.
- For maximum path length 1, the flux matrix becomes K_ij = p_ij/n, with t set to 1 for convenience.
- The local entropy normalization makes the maximum attainable entropy equal to 1 independently of node degree.
- Dynamic entropies are computed through moment order 5, because the most interesting behavior occurs for h ≤ 3 and higher-order computation is costly.For a typical dataset, h = 5 requires at least approximately 20 hours on a high-performance quad-processor workstation.
Sampling variance using the jackknife
The jackknife estimates entropy variability by repeatedly removing one sample, enabling z-statistics for comparing phenotypes and ranking genes or nodes.
- The jackknife removes one sample at a time and recomputes entropy, producing one estimate for each sample.For n samples, this yields n jackknife estimates.
- Jackknife estimates provide the mean and variance of the entropy estimate across the leave-one-out samples.
- The resulting z-statistic quantifies differential entropy relative to its jackknife variability.
- Applying the procedure per gene or node allows genes to be ranked by the significance of their entropy statistic.
- Because differential entropy and its standard deviation share degree dependence, their ratio is designed to be degree-independent.
III. RESULTS
The study integrated six normal–cancer tissue expression datasets with a human protein interaction network, yielding sparse weighted networks whose topology was checked against expression correlations.
- III. RESULTS: Six datasets covered bladder, lung, stomach, pancreas, cervix, and liver tissues, with sufficient normal and cancer samples.
- III. RESULTS: The integrated mRNA–protein interaction networks contained approximately 7500 nodes and 98500 edges.
- III. RESULTS: Figure 1 compares local dynamic entropy between cancer and normal tissue for approximately 3500 nodes with degree ≥10 across six tissue types.
- III. RESULTS: The analysis verified the assumption that neighboring genes in the integrated networks are more likely to have correlated expression.
Increased local dynamic entropy is a key hallmark of the cancer cell phenotype
Local dynamic entropy distinguishes cancer from normal tissue across tissue types, while degree-corrected differential entropy statistics remain higher in cancer.
- Increased local dynamic entropy is a key hallmark of the cancer cell phenotype: Local dynamic entropy was significantly higher in cancer than normal tissue across the six tissue types.
- Increased local dynamic entropy is a key hallmark of the cancer cell phenotype: Using nodes of degree ≥2 confirmed increased cancer entropy, although its discriminatory power was somewhat reduced.
- Increased local dynamic entropy is a key hallmark of the cancer cell phenotype: The magnitude of differential entropy change was strongly anti-correlated with node degree, motivating a degree-bias correction.
- Increased local dynamic entropy is a key hallmark of the cancer cell phenotype: Degree-corrected differential entropy z-statistics were significantly higher in cancer independently of tissue type.
Non-local dynamic entropy is increased in cancer, albeit weaker than local dynamic entropy
Higher-order dynamic entropy is also increased in cancer, but its discrimination is weaker than local entropy and generally declines beyond nearest-neighbor paths.
- Non-local dynamic entropy is increased in cancer, albeit weaker than local dynamic entropy: The study computed higher-order dynamic entropy over network paths longer than one.
- Non-local dynamic entropy is increased in cancer, albeit weaker than local dynamic entropy: For path length 2, alternative signaling paths are included even between neighboring genes, capturing signaling-path redundancy.
- Non-local dynamic entropy is increased in cancer, albeit weaker than local dynamic entropy: Second-order entropy was higher in cancer across all tissue types, but statistically significant only for the four larger studies.
- Non-local dynamic entropy is increased in cancer, albeit weaker than local dynamic entropy: Higher-order entropies with maximum path lengths of at least 3 generally had reduced discriminatory power, localizing relevant changes to neighbors and nearest neighbors.
Differential dynamic entropy and differential expression are anti-correlated
Differential dynamic entropy was evaluated alongside differential gene expression using per-gene statistics, revealing a strong anti-correlation that remained after accounting for node degree.
- The analysis used a regularized t-statistic for differential expression and a jackknife-derived z-statistic for differential entropy change.
- Differential entropy z-statistics and differential expression t-statistics were strongly anti-correlated across tissue types.The association remained significant after adjustment for node degree.
- Genes overexpressed in cancer preferentially showed reduced dynamic entropy compared with underexpressed genes.The associated odds ratios were statistically significant across all six tissue types.
Cell-cycle/proliferation genes preferentially associate with a lower dynamic entropy in cancer
Cell-cycle genes were enriched among genes with reduced dynamic entropy in cancer, while the entropy–expression anti-correlation persisted in most tissues after removing cell-cycle genes.
- The enrichment analysis compared the top 10% of genes with entropy increases or decreases in cancer relative to normal tissue.
- Cell-cycle genes were strongly enriched among genes exhibiting lower dynamic entropy in cancer, but not among genes exhibiting increases.The enrichment analysis ranked genes separately by increased and reduced entropy.
- The entropy–expression anti-correlation remained in 5 of 6 tissue types after cell-cycle genes were removed.This supports the interpretation that the association is a general systemic feature rather than solely a cell-cycle effect.
- The observed enrichment indicates that entropy changes reflect biology of epithelial tumour cells rather than only tumour–stromal composition changes.The passage links reduced entropy among cell-cycle and proliferation genes to increased tumour-cell proliferation.
DISCUSSION
Dynamic network entropy is increased in cancer, with local entropy changes linking expression patterns to potential therapeutic vulnerabilities. The analysis suggests reduced entropy around overexpressed oncogenic pathways and identifies candidate drug targets, while entropy is not superior to raw expression for classification.
- Cancer-associated entropy changes: Dynamic network entropy increases in cancer compared with normal tissue, with local entropy showing the more significant increase.The authors also report that entropy discriminates normal from cancer tissue, although raw gene expression achieves higher classification accuracy.
- Expression and local entropy: Genes overexpressed in cancer are significantly more likely to exhibit reductions in local dynamic entropy than underexpressed genes.The authors interpret reduced entropy as lower uncertainty in information transfer along an overactivated pathway.
- Candidate drug targets: AURKB exhibited the largest reductions in dynamic entropy in bladder cancer and was proposed as an attractive drug target for bladder cancers that overexpress it.AURKA was also highly ranked, and the authors connect AURKB’s ranking to its reported oncogenic and druggable roles in other cancers.
- Candidate drug targets: Dynamic entropy may help identify neighboring druggable targets within oncogenic pathways when the oncogene itself is not directly druggable.The authors present this as a computational strategy for guiding non-oncogene-addiction therapeutic approaches.
- Implications: The entropy-robustness relation motivates further investigation of cancer gene-network mechanics to rationalize drug-target selection.The authors frame increased dynamic entropy as a key cancer hallmark and connect it to system robustness.