Benjamin Anderson
  1. Research/

Causal Emergence in Biological Networks

Last updated — October 2022 | What is the role of redundancy in biological networks, and how does it change over time and across scales of spatial organization?

Overview #

To understand that the field determines the behaviour of any local process or constituent, it is necessary fundamentally to modify modern science by revising our theory of first principles in order to justify the unity of Nature as a causal factor. Without this revision of our most elemental concept of Nature, as conceived by science, all field theories, whether in physiology or physics, are mere verbiage.

Harold Saxton Burr1

Causal Emergence is the conversion of redundant information to synergistic information.2 The below effects of course graining existing networks denotes a decrease in the uncertainty of state transitions across the networks. A promising area for continued exploration would be in the seeking of highly causal pathways/Markov chains across these networks that then serve to be extrapolated back down to original scale and perterbed to determine effects of intervention.3 Alternatively, these causal pathways can be enhanced with a literature search to determine corrolary phenotypes tied to involved interactions across the 2 explored scales.

The key idea that I am exploring here is the role of redundancy in networks, and how it changes over time in gene regulatory networks. My hypothesis is that as a byproduct of aging, redundant pathways in the GRNs of our cells are slowly erased through the build-up of epigenetic markers, serving to block certain interactions as a means of adhering to the free energy principal.4

An unaswered question I have in this regard is the direction of causality for promoting this build-up. This is why in addition to GRNs, I choose to run the same network analysis on bioelectricity-integrated gene and reaction(BIGR) networks.5 It has been shown that there is a direct correlation between epigenetic markers and the regulation of intercellular communication through ion channels and gap junctions6,7,8, however it is unknown—at least to me at present—whether the buildup of these markers are themselves a downstream process effected by intercellular signaling dependent transcription. This would imply that they serve as a feedback loop to eachother, wheras if the promotion of buildup was being prompted at the GRN/DNA level alone, then the changes in intercellular communication over time related to epigenetic marker buildup would be exclusively downstream of GRN interactions, and not causal.

For the sake of thinking about where to intervene, I seek to answer the question of which is more causal to the other.


Gene Networks #

Before: #

Image of initial human gene network Human — Aging, Original

After: #

Image of iteration 7 from human gene network Human — Aging, Iteration 7


What I’m doing #

  1. Take a base gene dataset.
  2. Use STRING to create a graph of these genes and their pathways with wieghted edges.
  3. Use this package to determine effective information for this graph9:
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    from ei_net import effective_information
    import networkx as nx
    import numpy as np 
    G = nx.GRAPH()  
    EI_micro = effective_information(G)   
    print("EI micro: %.6f"%EI_micro)
  4. Create a macro graph using the same package. More details on how to do this here.
  5. Return difference between effective information across 2 graphs as well as e:
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    eff_gain = (EI_macro-EI_micro)/np.log2(N)
    print("Effectiveness gain: %.6f"%eff_gain)
    Effectiveness gain: {VALUE}
  6. Repeat until no more macro nodes are found.

Humans — #

Aging #

Initial Effectiveness: 0.335729

RunNodesEdgesEffective InformationEffectiveness Gain
Original30678622.7722478691230394N/A
121567302.84109098489144250.008337
218855742.8504690.009473
318851482.8528020.009755
417746452.8552960.010057
516339132.8601430.010644
614429602.8601430.010644
713626202.8741340.012339

Final Effectiveness: 0.405523737036814

Senescence #

Initial Effectiveness: 0.486386

RunNodesEdgesEffective InformationEffectiveness Gain
Original25925853.899266239854304N/A
118423553.999603838964530.012516
218022624.0024370.012869
317621554.0033780.012987
317420664.0045050.013127

Final Effectiveness: 0.53802705908


If an intervention works for a macronode, with 2 different internal causal pathways, then we may be able to identify more fine grained interventions:

Additionally, what nodes get grouped into a macro-nodes tells us what interventional targets are meaningful within the system. Consider that of the two macro-nodes in the system {μ1, μ2}, each requires the activation of exogenous canWnt II. However, their set of underlying nodes are differentiated solely by the concurrent activation of Foxc1_2, which is not upregulated in μ1 and is upregulated in μ2. This tells us that it is solely Foxc1_2, rather than any other element in the network, that determines which causal path the network takes as long as exogenous canWnt II is activated. The macro-nodes capture which differences are actually relevant to the intrinsic workings of the system itself. cc: https://www.tandfonline.com/doi/full/10.1080/19420889.2020.1802914


Mus Musculus (Mouse) — #

Aging #

Initial Effectiveness: 0.434615

RunNodesEdgesEffective InformationEffectiveness Gain
Original13010563.0520255784285086N/A
1929283.1332830.011571
2868033.1387790.009473
3837673.1390500.012392

Final Effectiveness: 0.48975792415

Physiology #

Initial Effectiveness: 0.458635

RunNodesEdgesEffective InformationEffectiveness Gain
Original49490254.104038136973786N/A
132265084.2404970895748290.015250
231361614.2438130.015620

Final Effectiveness: 0.511918


Drosophila (Fruit Fly) — #

Aging #

Initial Effectiveness: 0.546897

RunNodesEdgesEffective InformationEffectiveness Gain
Original18713034.127372843455749N/A
11289304.2947860.022183
21228534.297222142962440.022506
3837673.1390500.012392

Final Effectiveness: 0.62002380607

Stem Cell Genes #

Initial Effectiveness: 0.511734

RunNodesEdgesEffective InformationEffectiveness Gain
Original28525334.173097660494712N/A
125940214.2022678974293250.003577
225839894.2025620379587080.003613

Final Effectiveness: 0.524584


Caenorhabditis elegans #

Initial Effectiveness: 0.559496

RunNodesEdgesEffective InformationEffectiveness Gain
Original825106555.420533831393049N/A
162094915.5670588247171810.015124
26118535.5687963855191660.015303
360994505.5690211735477270.015327

Final Effectiveness: 0.602037


Saccharomyces cerevisiae (Incomplete) #

Initial Effectiveness: 0.535618

RunNodesEdgesEffective InformationEffectiveness Gain
Original931146895.282608841216231N/A
1761158405.4080624631777760.012720

Final Effectiveness: 0.565002


BIGR Networks #

BIGR Networks = bioelectricity-integrated gene and reaction networks.

The functional properties of BIGR networks generate the first testable, quantitative hypotheses for biophysical mechanisms underlying the stability and adaptive regulation of anatomical bioelectric pattern.9


What I’m doing #

  1. Take a base BETSE config file with a GRN specified.
  2. Run the simulation and output a network of the resulting bioelectric circuit.
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    model.run_pipeline()
    ...(wait for simulation)
    grn = model.phase.sim.grn.core
    grn.init_saving(model.phase.cells, model.p, nested_folder_name='GRN')
    savename = grn.imagePath + 'OptimizedNetworkGraph' + '.svg'
    graph_pydot.write_svg(savename, prog='dot')
    graph_network = networkx.nx_pydot.from_pydot(graph_pydot)
    networkx.write_gml(graph_network, "graphname.gml")
  3. Use this package to determine effective information for this graph8:
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    from ei_net import effective_information
    import networkx as nx
    import numpy as np 
    G=nx.read_gml("graphname.gml")
    EI_micro = effective_information(G)   
    print("EI micro: %.6f"%EI_micro)
  4. Create a macro graph using the same package. More details on how to do this here.
  5. Return difference between effective information across 2 graphs as well as e:
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    eff_gain = (EI_macro-EI_micro)/np.log2(N)
    print("Effectiveness gain: %.6f"%eff_gain)
    Effectiveness gain: {VALUE}
  6. Repeat until no more macro nodes are found.

Tissues that have as little change as possible through time, but with high variance between cells10 #

Initial Effectiveness: 0.664508

RunNodesEdgesEffective InformationEffectiveness Gain
Original26243.1234786725698864N/A
120183.3820720.055015

Final Effectiveness: 0.782538


Tissue that is stable on a specific pattern10 #

“Smiley” Pattern: #

Initial Effectiveness: 0.749480

RunNodesEdgesEffective InformationEffectiveness Gain
Original23193.3903195311147827N/A
120173.4173400.005973

Final Effectiveness: 0.790698

“Bullseye” Pattern: #

Initial Effectiveness: 0.678185

RunNodesEdgesEffective InformationEffectiveness Gain
Original17152.7720552088742005N/A
116142.8230680.012480

Final Effectiveness: 0.705767


To Do #

  1. Run against more complex GRN simulations in BIGR networks.
  2. Test Resilience and Prospective Resilience of all generated networks: https://github.com/jkbren/presilience

Tools #

Greedy Algorithm for identifying subgraphs: https://github.com/jkbren/einet

The greedy algorithm used for finding causal emergence in networks is structured as follows: for each node, vi, in the shuffled node list of the original network, collect a list of neighboring nodes, {vj} ∈ Bi, where Bi is the Markov blanket of vi (in graphical models, the Markov blanket, Bi, of a node, vi , corresponds to the “parents,” the “children,” and the “parents of the children” of vi). *This means that {vj} ∈ Bi consists of nodes with outgoing edges leading into 10 Complexity vi, nodes that the outgoing edges from vi lead into, and nodes that have outgoing edges leading into the out-neighbors of vi. For each node in {vj} , the algorithm calculates the EI of a macroscale network after vi and vj are combined into a macronode, vM. If the resulting network has a higher EI value, the algorithm stores this structural change and, if necessary, supplements the queue of nodes, {vj}, with any new neighboring nodes from vj’s Markov blanket that were not already in {vj}. If a node, vj, has already been combined into a macronode via a grouping with a previous node, vi , then it will not be included in new queues, {vj }′ , of later nodes to check. *The algorithm iteratively combines such pairs of nodes until every node, vj, in every node, vi’s Markov blanket, is tested.

Protien and gene regulatory interaction db: https://string-db.org/


Data Availability #

Aging Genes for Multiple Species: https://genomics.senescence.info/genes/index.html

Senescence Genes for Humans: https://genomics.senescence.info/cells/

Supplementary files for BIGR network simulations: https://www.biorxiv.org/content/10.1101/2022.10.23.513361v1.supplementary-material


Citation #

In academic work, please cite this essay as:

Anderson, Benjamin, “Causal Emergence in Biological Networks”, TheBenjam.in (2022-07-23), available at https://www.thebenjam.in/research/.


Changelog #

August 22, 2022 — Added physiology data for mice and stem cell genes data for drosophilia.

September 14, 2022 — Added quote at the beginning and citation to Free Energy Theory of Aging write-up.

October 25, 2022 — Removed ‘Open questions / Areas to Explore’ section. Added ‘BIGR networks’ section. Edited ‘Overview’ section.


  1. Burr, Harold Saxton. Blueprint for Immortality: The Electric Patterns of Life. ↩︎

  2. Varley Thomas F. and Hoel Erik 2022 Emergence as the conversion of information: a unifying theory Phil. Trans. R. Soc. A. 380: 20210150.20210150 ↩︎

  3. Brennan Klein, Erik Hoel, Anshuman Swain, Ross Griebenow, Michael Levin, Evolution and emergence: higher order information structure in protein interactomes across the tree of life, Integrative Biology, Volume 13, Issue 12, December 2021, Pages 283–294, https://doi.org/10.1093/intbio/zyab020 http://doi.org/10.1098/rsta.2021.0150 ↩︎

  4. Anderson, Benjamin, “The Free Energy Theory of Aging”, TheBenjam.in (2022-09-05), available at https://www.thebenjam.in/research/. ↩︎

  5. Pietak A, Levin M. Bioelectric gene and reaction networks: computational modelling of genetic, biochemical and bioelectrical dynamics in pattern regulation. J R Soc Interface. 2017 Sep;14(134):20170425. doi: 10.1098/rsif.2017.0425. PMID: 28954851; PMCID: PMC5636277. ↩︎

  6. Masahito Oyamada, Kumiko Takebe, Yumiko Oyamada, Regulation of connexin expression by transcription factors and epigenetic mechanisms, Biochimica et Biophysica Acta (BBA) - Biomembranes, Volume 1828, Issue 1, 2013, Pages 118-133, ISSN 0005-2736, https://doi.org/10.1016/j.bbamem.2011.12.031↩︎

  7. Tseng, A.-S. and Levin, M. (2012), Transducing Bioelectric Signals into Epigenetic Pathways During Tadpole Tail Regeneration. Anat Rec, 295: 1541-1551. https://doi.org/10.1002/ar.22495 ↩︎

  8. Pai, V. P., Martyniuk, C. J., Echeverri, K., Sundelacruz, S., Kaplan, D. L., & Levin, M. (2015). Genome-wide analysis reveals conserved transcriptional responses downstream of resting potential change in Xenopus embryos, axolotl regeneration, and human mesenchymal cell differentiation. Regeneration (Oxford, England), 3(1), 3–25. https://doi.org/10.1002/reg2.48 ↩︎ ↩︎

  9. Klein, B., Swain, A., Byrum, T., Scarpino, S. V. & Fagan, W. F. (2022). Exploring noise, degeneracy and determinism in biological networks with the einet package. Methods in Ecology and Evolution, 13, 799– 804. https://doi.org/10.1111/2041-210X.13805 ↩︎ ↩︎

  10. Exploring The Behavior of Bioelectric Circuits using Evolution Heuristic Search. Hananel Hazan, Michael Levin. bioRxiv 2022.10.23.513361; doi: https://doi.org/10.1101/2022.10.23.513361 ↩︎ ↩︎