Source-linked AI summary
Cellular network entropy as the energy potential in Waddington's differentiation landscape
Christopher R. S. Banerji, Diego Miranda-Saavedra, Simone Severini, Martin Widschwendter, Tariq Enver, Joseph X. Zhou, Andrew E. Teschendorff
TL;DR
The paper addresses the lack of a single quantitative measure for placing a cellular sample in the global differentiation hierarchy. It computes network entropy from expression-linked signaling networks and finds that entropy tracks undifferentiated state across cellular hierarchies, while also distinguishing cancer and normal tissue patterns. The authors conclude that network entropy provides a quantitative measure of a sample’s elevation in Waddington’s landscape.
Problem
No single quantitative measure could place a cellular sample within the global differentiation hierarchy.
Method
The authors compute sample-specific network entropy from gene-expression profiles integrated with protein-interaction signaling networks.
Results
Network entropy recapitulated differentiation hierarchies, correctly classified pluripotent and differentiated samples, and distinguished cancer from normal tissue patterns.
Takeaways & Limitations
Network entropy provides a quantitative measure of a sample’s undifferentiated state and its elevation in Waddington’s landscape.
Takeaways & Limitations
Entropy rates varied between studies profiling the same cell types, despite the measure’s reported robustness.
Abstract
from arXiv · showhide
Differentiation is a key cellular process in normal tissue development that is significantly altered in cancer. Although molecular signatures characterising pluripotency and multipotency exist, there is, as yet, no single quantitative mark of a cellular sample's position in the global differentiation hierarchy. Here we adopt a systems view and consider the sample's network entropy, a measure of signaling pathway promiscuity, computable from a sample's genome-wide expression profile. We demonstrate that network entropy provides a quantitative, in-silico, readout of the average undifferentiated state of the profiled cells, recapitulating the known hierarchy of pluripotent, multipotent and differentiated cell types. Network entropy further exhibits dynamic changes in time course differentiation data, and in line with a sample's differentiation stage. In disease, network entropy predicts a higher level of cellular plasticity in cancer stem cell populations compared to ordinary cancer cells. Importantly, network entropy also allows identification of key differentiation pathways. Our results are consistent with the view that pluripotency is a statistical property defined at the cellular population level, correlating with intra-sample heterogeneity, and driven by the degree of signaling promiscuity in cells. In summary, network entropy provides a quantitative measure of a cell's undifferentiated state, defining its elevation in Waddington's landscape.
Introduction
The paper addresses the absence of a single quantitative measure for placing samples in Waddington’s global differentiation hierarchy. It proposes network entropy as a molecular correlate of undifferentiated state based on signaling promiscuity and cellular heterogeneity.
- Existing molecular signatures distinguish specific differentiation stages, but no single quantitative measure places a sample within the global hierarchy.
- Network entropy is introduced as a sample property that approximates signaling pathway promiscuity from genome-wide expression data.
- Highly undifferentiated cells are expected to have high entropy because they retain options to activate pathways associated with diverse fates.
- Differentiated cells are expected to have low entropy because they maintain activation of fewer fate-specific pathways.
- The authors posit that network entropy directly correlates with a sample’s undifferentiated state and its elevation in Waddington’s landscape.
- Across more than 800 samples, network entropy discriminated pluripotent from non-pluripotent cells, resolved multipotency within lineages, and identified greater heterogeneity in cancer stem cells.
Results
The study constructs network entropy from sample-specific signaling probabilities and tests whether it tracks cellular differentiation across simulations, stem-cell datasets, time courses, and cancer samples. Entropy generally decreases with differentiation and distinguishes pluripotent, multipotent, differentiated, cancer, and normal cellular states.
- Construction and rationale: Network entropy is computed as an entropy rate from local node entropies weighted by a stationary distribution over sample-specific signaling probabilities.
- Simulation: Approximately 70% of single-gene perturbations reduced global entropy, compared with 85% of whole-pathway activations.The corresponding binomial tests were P < 0.001 and P < 10^-10, respectively.
- Pluripotency and multipotency: Network entropy was significantly higher in pluripotent cell lines than non-pluripotent lines and achieved 100% accuracy in an independent pluripotent-versus-differentiated comparison.
- Pluripotency and multipotency: Human embryonic and induced pluripotent stem cells showed higher entropy than differentiated progeny, while committed neural, hematopoietic, and mesenchymal stem cells were intermediate.
- Pluripotency and multipotency: Entropy recapitulated a known blood-cell differentiation hierarchy and was fairly insensitive to normalization or profiling platform, although biological variation was evident among mesenchymal stem cells.
- Dynamic differentiation: During time-course differentiation, entropy decreased over time, whereas de-differentiation increased it before re-differentiation reduced it again.For ATRA-stimulated HL60 differentiation, the time trend had R^2 = 0.96 and P < 10^-8; these changes were independent of proliferation.
- Cancer and normal cells: Cancer tissue had higher entropy than normal tissue, while cancer stem cells had marginally higher entropy than non-stem-like cancer cells.
- Cancer and normal cells: Cancer stem cells and ordinary cancer cells both showed greater heterogeneity than normal differentiated tissue, but cancer stem cells were not higher than normal stem-cell counterparts.
Dynamic changes in local network entropy identifies key differentiation genes and pathways
The paper uses local network-entropy changes to identify signaling pathways associated with differentiation. Notch and JAK-STAT analyses support pathway-specific rather than nonspecific entropy changes.
- The authors test whether dynamic entropy changes target specific signaling pathways rather than reflecting nonspecific network effects.
- Notch signaling: Twelve of thirteen Notch-pathway genes showed the predicted lower entropy in the non-pluripotent state.
- Notch signaling: The Notch-gene pattern exceeded 10,000 random 13-gene selections, with P < 0.0001.
- JAK-STAT signaling: In HL60-to-neutrophil data, ranking genes by entropy changes and applying GSEA identified JAK-STAT signaling as a key pathway.
- JAK-STAT signaling: Randomly permuting expression profiles produced no significantly enriched biological terms, with adjusted P-values > 0.05.
- JAK-STAT signaling: Non-network-based approaches did not identify JAK-STAT signaling in the same analysis.
Discussion
The authors propose network entropy as a computable systems-level measure that correlates with cellular undifferentiation across normal and cancer contexts. It is presented as a robust proxy for differentiation potential, while also reflecting signaling promiscuity and cellular heterogeneity.
- Differentiation hierarchy: Network entropy inversely correlates with a sample’s differentiation status across the global hierarchy.Pluripotent cells show the highest values, followed by multipotent stem cells, whereas terminally differentiated cells show lower entropy.
- Overall interpretation: The authors conclude that network entropy estimates signaling promiscuity and cellular heterogeneity and serves as an in-silico proxy for differentiation potential in Waddington’s landscape.This conclusion frames the measure as a systems property of genome-wide expression profiles rather than a narrowly defined molecular signature.
- Cancer context: Network entropy distinguishes cancer stem cells, cancer cells, and normal tissues, with cancer stem cells showing higher entropy than ordinary cancer cells.Cancer cell lines and cancer tissue also exhibit higher entropy than primary cancers and normal tissue, respectively.
- Cellular heterogeneity: These findings support network entropy as a measure of average cellular heterogeneity and the statistical nature of pluripotency.The proposed interpretation links signaling promiscuity with stochasticity and intra-sample heterogeneity, although genome-wide single-cell expression data were not analyzed.
- Comparison with signatures: Network entropy offers a more refined classification across the differentiation hierarchy than reported pluripotency expression signatures.Those signatures consistently separated pluripotent from non-pluripotent cells but generally failed to distinguish cell types farther down the hierarchy.
- Robustness: The measure is relatively robust across studies and platforms because it is self-calibrating, dimensionless, and independent of feature selection and tunable parameters.Entropy remained associated with differentiation after subsampling at least 40% of network genes, while the authors leave further investigation of the underlying topology-expression interplay for future work.
- Applications: Network entropy could provide a quantitative test of pluripotency or multipotency and help assess iPSC quality or identify mislabeled samples.The proposed measure integrates genome-wide expression profiles with a protein interaction network and may also support identification of oncogenic pathways in cancer stem cells.
Methods
Network entropy is computed from a sample-specific stochastic matrix that integrates gene expression with a protein interaction network, then normalized for network-specific comparison. The resulting entropy rate summarizes local interaction uncertainty across the network, though computation is intensive.
- Sample-specific stochastic matrix: Gene-expression profiles are integrated with a comprehensive protein interaction network to estimate a sample-specific stochastic matrix.Interaction probabilities are approximated from products of normalized expression levels for neighboring genes and normalized across each gene’s neighbors.
- Local entropy: Local network entropy measures uncertainty or promiscuity in interaction probabilities around each gene.Its maximum depends on the gene’s network degree, with max S_i = log k_i, motivating a normalized local entropy.
- Global entropy rate: Global network entropy is derived from the stationary distribution of the stochastic matrix and the unnormalized local entropies.The entropy rate is bounded by a network-dependent maximum determined by the interaction network’s adjacency structure.
- Normalization: Entropy rates are scaled by the maximum attainable value for each network, yielding a quantity bounded between 0 and 1.This normalization facilitates comparisons between networks with different numbers of genes, edges, and topologies.
- Computational cost: Computing entropy rates requires estimating the stationary distribution of a large stochastic matrix and is therefore computationally intensive.For a connected 8,290-node network, one sample’s entropy rate took approximately 10 minutes on the reported workstation.
Conflict of Interest
The authors report no conflict of interest, and funders had no role in the study’s design, analysis, publication decision, or manuscript preparation.
- The authors declare no conflict of interest, and funders had no role in study design, data collection, analysis, publication, or manuscript preparation.