The Computational Universe

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   Note 1. Liping Zhu, Song-Ju Kim, Masahiko Hara, and Masashi Aono, “Remarkable problem-solving ability of unicellular amoeboid organism and its mechanism,” Royal Society Open Science 5 (2018): 180396. Both listed affiliations (the Faculty of Environment and Information Studies and the Graduate School of Media and Governance) are at Keio University’s Shonan Fujisawa Campus in Fujisawa, Kanagawa, roughly 50 km southwest of central Tokyo. Earlier work in the same programme: Aono et al., “Amoeba-based computing for traveling salesman problem,” BioSystems 112 (2013): 83-89.

Three details from the paper bear on how the result should be read. Solution quality is stated against the mean tour length rather than the optimum: the paper reports Lexp/Lmean ratios in the range 0.90 to 0.93, concluding that “the average length of the tour found by the plasmodium remained significantly shorter than the mean value for all n,” with no measurement of the gap to the shortest route. Problem sizes ran from four to eight cities, and the authors give the reason for the ceiling: “The reason why in this study the number of cities n was limited to eight was that we were not able to fabricate the chip with more than 64 lanes, owing to the size limit of the photolithography equipment.” The lane encoding is n x n, so hardware cost grows quadratically even as solution time grows linearly.

1. 1a

   Note 1a. Aono, quoted in Lisa Zyga, “Amoeba finds approximate solutions to NP-hard problem in linear time,” Phys.org, 20 December 2018. The remark is from correspondence with the reporter, not from the paper, and the terms are glossed loosely there: n is the body area needed to express a finished solution, which the paper’s encoding ties to the number of cities, and x is described only as a constant rate of gelatinous supply, with no units or measured value given in any source.

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   Note 4. Mirna Kramar and Karen Alim, “Encoding memory in tube diameter hierarchy of living flow network,” PNAS 118 (2021): e2007815118. The tubes that persist longest “directly bear the memory of the nutrient stimulus.” Note ¹⁶ below gives the mechanism from this same paper: the two notes cite one study, not two.

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   Note 5. Larson, B.T., Garbus, J., Pollack, J.B., and Marshall, W.F., “A unicellular walker controlled by a microtubule-based finite-state machine,” Current Biology 32 (2022). Source for the Euplotes microtubule mechanical computer.

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   Note 6. Sharon Glotzer, interview in Quanta Magazine (2017): “Digital Alchemist Sharon Glotzer Seeks Rules of Emergence.” See Damasceno et al., “Predictive Self-Assembly of Polyhedra into Complex Structures,” Science 337 (2012): 453-457.

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   Note 9. Susanne Still et al., “Thermodynamics of prediction,” Physical Review Letters 109 (2012): 120604.

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   Note 10. Jeremy England, “Statistical physics of self-replication,” Journal of Chemical Physics 139 (2013): 121923.

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   Note 11. Andrew Adamatzky, Physarum Machines: Computers from Slime Mould (World Scientific, 2010). See also Adamatzky et al., arXiv:1108.4956 for maze-solving dynamics.

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   Note 13. Alan Turing, “The Chemical Basis of Morphogenesis,” Philosophical Transactions of the Royal Society B 237 (1952): 37-72. For the shark denticle research confirming conservation across 450 million years: Cooper, R.L. et al., Science Advances 4 (2018): eaau5484 (from Gareth Fraser’s lab).

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   Note 14. Division of labor, kin structure, and competition in biofilms: Nadell et al., Nature Reviews Microbiology 14 (2016): 589-600.

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    Note 16. Slime mold tube restructuring mechanism: Mirna Kramar and Karen Alim, PNAS 118 (2021): e2007815118. A nutrient source “locally releases a softening agent that gets transported by the cytoplasmic flows within the tubular network. Tubes receiving a lot of softening agent grow in diameter at the expense of other tubes shrinking. Thereby, the tubes’ capacities for flow-based transport get permanently upgraded toward the nutrient location, redirecting future decisions and migration.” The paper does not name the agent chemically. The paper’s own term is “permanently upgraded,” which supports durability but not irreversibility, since the same mechanism grows and shrinks tubes as new events arrive.

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    Note 18. Leslie Valiant, Probably Approximately Correct: Nature’s Algorithms for Learning and Prospering in a Complex World (Basic Books, 2013).

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    Note 19. Hector Zenil et al., “Evolution of algorithmic complexity,” PLOS Computational Biology (2018).

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    Note 20. Electronic amoeba: Hokkaido University (2020), “Electronic amoeba finds approximate solution to traveling salesman problem in linear time.” Ising machines: Stanford/NTT implementations described in Physics World (2019).

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    Note 21. Joseph Burchett et al., “Revealing the Dark Threads of the Cosmic Web,” Astrophysical Journal Letters 891 (2020): L35. The paper describes the Monte Carlo Physarum Machine algorithm in detail and reports an “almost perfect fit” to the density fields of the Bolshoi-Planck cosmological simulation.

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    Note 22. Atsushi Tero, Toshiyuki Nakagaki, et al., “Rules for biologically inspired adaptive network design,” Science 327 (2010): 439–442. The Tokyo railway experiment won the 2010 Ig Nobel Prize in Transportation Planning.

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    Note 24. Krishna Palem et al., “Inexact computing improves answers,” Rice University/Argonne/UIUC (2016). Sacrificing intermediate precision improved final solution quality by 1000x at same energy cost.

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    Note 27. Sangram Bagh et al., “Distributed computing with engineered bacteria and its application in solving chemically generated 2x2 maze problems,” ACS Synthetic Biology (2021). Six types of engineered E. coli collectively solve maze problems with 100% accuracy.

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    Note 28. Programmable DNA gate arrays: Lv, H. et al., Nature 622 (2023): 292-300. Over 100 billion distinct circuits implementable via DNA origami architecture. (The often-quoted ~1 exabyte per cubic millimeter is a theoretical ceiling for DNA storage in general, not a property of these arrays; see note ⁴⁸.)

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    Note 29. Michael Levin, “Bioelectric signaling: Reprogrammable circuits underlying embryogenesis, regeneration, and cancer,” Cell 184 (2021): 1971-1989. See also Levin, “The bioelectric code,” Biosystems 164 (2018): 76-93.

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    Note 30. John Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” PNAS 79 (1982): 2554-2558. Nobel Prize in Physics 2024 (shared with Geoffrey Hinton). Simulated annealing was introduced separately by Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., “Optimization by simulated annealing,” Science 220 (1983): 671-680. For memristor-based annealing: Cai et al., Nature Electronics 3 (2020): 409-418.

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    Note 33. Daniel Wolpert, “The real reason for brains,” TED Talk (2011). Wolpert uses the sea squirt, which resorbs much of its larval nervous system once it settles and stops moving, to argue that brains evolved to produce adaptable movement.

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    Note 34. Andrew Adamatzky, “Slime mould logical gates: exploring ballistic approach,” arXiv:1005.2301 (2010). Source of the size-discriminated gate described in the text: “In input scenarios (0,1) and (1,0) size of propagating wave-fragment was not enough for the fragment to branch into output channel p… When two wave-fragments are initiated… they merge into a single larger-wave fragment. This new fragment propagates towards output q and also expands into output channel p.” Two caveats the prose compresses. The discriminator is fragment size and front curvature, not depletion of the medium; and the gate is a two-output device, so a lone input still leaves by the straight-ahead channel rather than vanishing. The “no strict rules on repelling and merging” observation about laboratory plasmodia is from the same paper. On Turing completeness in this substrate see also Adamatzky, “Physarum machine: implementation of a Kolmogorov-Uspensky machine on a biological substrate,” Parallel Processing Letters 17 (2007): 455-467.

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    Note 35. Andrew Adamatzky, Ben De Lacy Costello, and Tomohiro Shirakawa, “Universal Computation with Limited Resources: Belousov-Zhabotinsky and Physarum Computers,” International Journal of Bifurcation and Chaos 18(8) (2008): 2373-2389; preprint arXiv:0711.2711. The quoted sentence opens the abstract, which continues: “In situations of limited resources the systems studied develop travelling localizations. The localizations are elementary units of dynamical logical circuits in collision-based computing architectures.” The claim is about universal computation in an unconstrained medium. Nakagaki’s maze-solving and network-optimization results do not contradict it, since there the organism was fed only at chosen points on plain agar, itself a limit on resources. What abundance destroys is localization, not pattern. The nutrient-rich regime produces uniform circular growth, the organism’s analogue of the target waves seen in an excitable chemical medium.

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    Note 37. “Cellular supremacy”: The term emerged in 2024 biocomputing literature. See Communications of the ACM (2024): “Biocomputation: Moving Beyond Turing with Living Cellular Computers.”

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    Note 38. Brain organoid computing: See arXiv (2025): “Brain Organoid Computing—an Overview.”

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    Note 38a. Organoid criticality: Itatani, N. and Zavaglia, M., “Criticality emerges within coherent functional organization in human forebrain organoids,” Research Square preprint (2026), DOI: 10.21203/rs.3.rs-8640242/v1. Not yet peer-reviewed. Chapter 8b returns to this result and its caveats.

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    Note 39. Slime mold as memristor: Whiting et al., Frontiers in Soft Matter (2025).

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    Note 40. Andrew Adamatzky, “We are chemical computers,” interview published for the launch of npj Unconventional Computing (2024).

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    Note 41. George Dyson, Turing’s Cathedral: The Origins of the Digital Universe (Pantheon, 2012). Primary source for the Barricelli, Bigelow, and IAS computer history in this chapter.

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    Note 42. Nils Aall Barricelli, “Symbiogenetic Evolution Processes Realized by Artificial Methods,” Methodos 9 (1957): 143-182.

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    Note 43. Julian Bigelow’s distinction between sequence and time appears in unpublished notes and interviews conducted by Dyson. See Turing’s Cathedral, chapters 11-12.

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    Note 45. Wolpert, D.H., “The stochastic thermodynamics of computation,” Journal of Physics A 52 (2019): 193001.

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    Note 47. Sender, R., Fuchs, S., and Milo, R., “Revised Estimates for the Number of Human and Bacteria Cells in the Body,” Cell 164 (2016): 337-340.

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    Note 48. The exabyte-per-cubic-millimeter figure is a theoretical ceiling, from Zhirnov, V., Zadegan, R.M., Sandhu, G.S., Church, G.M., and Hughes, W.L., “Nucleic acid memory,” Nature Materials 15 (2016): 366–370. DOI: 10.1038/nmat4594. It applies to data written directly into the base sequence, and the paper is explicit that densities attained in practice are far lower, since approaches relying on DNA secondary structure need on the order of a hundred base pairs per bit. For a demonstrated write-and-read at a density orders of magnitude below the ceiling, see Church, G.M., Gao, Y., and Kosuri, S., “Next-Generation Digital Information Storage in DNA,” Science 337 (2012): 1628, which encoded a digital book in DNA and recovered it. The distinction matters: the ceiling is what the chemistry permits, not what anyone has built.

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    Note 49. Campbell, John O., “Universal Darwinism as a process of Bayesian inference,” Frontiers in Systems Neuroscience 10 (2016): 49.