Darwin meets Category Theory in the Tangential Space

For this Darwin Day (February 12), I would like to highlight the relationship between evolution by natural selection and something called category theory. While this post will be rather tangential to Darwin's work itself, it should be good food for thought with respect to evolutionary research. As we will see, category theory also has relevance to many types of functional and temporal systems (including those shaped by natural selection) [1], which is key to understanding how natural selection shapes individual phenotypes and populations more generally.

This isn't the last you'll hear from me in this post!

Category Theory originated in the applied mathematics community, particularly the "General Theory of Natural Equivalence" [2]. In many ways, category theory is familiar to those with conceptual knowledge of set theory. Uniquely, category theory deals with the classification of objects and their transformations between mappings. However, category theory is far more powerful than set theory, and serves as a bridge to formal logic, systems theory, and classification.

A category is defined by two basic components: objects and morphisms. An example of objects are a collection of interrelated variables or discrete states. Morphisms are things that link objects together, either structurally or functionally. This provides us with a network of paths between objects that can be analyzed using categorical logic. This allows us to define a composition (or path) by tracing through the set of objects and morphisms (so-called diagram chasing) to find a solution.

In this example, a pie recipe is represented as a category with objects (action steps) and morphisms (ingredients and results). This monoidal preorder can be added to as the recipe changes. From [3]. Click to enlarge.

Categories can also consist of classes: classes of objects might include all objects in the category, while classes of morphism include all relational information such as pathways and mappings. Groupoids are functional descriptions, and allow us to represent generalizations of group actions and equivalence relations. These modeling-friendly descriptions of a discrete dynamic system is quite similar to object-oriented programming (OOP) [4]. One biologically-oriented application of category theory can be found in the work of Robert Rosen, particularly topics such as relational biology and anticipatory systems.

Animal taxonomy according to category theory. This example focuses on exploring existing classifications, from species to kingdom. The formation of a tree from a single set of objects and morphisms is called a preorder. From [3]. Click to enlarge.

One potential application of this theory to evolution by natural selection is to establish an alternate view of phylogenetic relationships. By combining category theory with feature selection techniques, it may be possible to detect natural classes that correspond to common ancestry. Related to the discovery of evolutionary-salient features is the problem of phylogenetic scale [5], or hard-to-interpret changes occurring over multiple evolutionary timescales. Category theory might allow us to clarify these trends, particularly as they relate to evolving life embedded in ecosystems [6] or shaped by autopoiesis [7]. 

More relevant to physiological systems that are shaped by evolution are gene regulatory networks (GRNs). While GRNs can be characterized without the use of category theory, they also present an opportunity to produce an evolutionarily-relevant heteromorphic mapping [8]. While a single GRN structure can have multiple types of outputs, multiple GRN structures can also give rise to the same or similar output [8, 9]. As with previous examples, category theory might help us characterize these otherwise super-complex phenomena (and "wicked" problems) into well-composed systems-level representations.

NOTES:
[1] Spivak, D.I. (2014). Category theory for the sciences. MIT Press, Cambridge, MA.

[2] Eilenberg, S. and MacLane, S. (1945). General theory of natural equivalences. Transactions of the American Mathematical Society, 58, 231-294. doi:10.1090/S0002-9947-1945-0013131-6 

[3] Fong, B. and Spivak, D.I. (2018). Seven Sketches in Compositionality: an invitation to applied category theory. arXiv, 1803:05316.

[4] Stepanov, A. and McJones, P. (2009). Elements of Programming. Addison-Wesley Professional.

[5] Graham, C.H., Storch, D., and Machac, A. (2018). Phylogenetic scale in ecology and 
evolution. Global Ecology and Biogeography, doi:10.1111/geb.12686.

[6] Kalmykov, V.L. (2012). Generalized Theory of Life. Nature Precedings, 10101/npre.2012.7108.1.

[7] Letelier, J.C., Marin, G., and Mpodozis, J. (2003). Autopoietic and (M,R) systems. Journal of Theoretical Biology, 222(2), 261-272. doi:10.1016/S0022-5193(03)00034-1.

[8] Payne, J.L. and Wagner, A. (2013). Constraint and contingency in multifunctional gene regulatory circuits. PLoS Computational Biology, 9(6), e1003071. doi:10.1371/journal.pcbi.1003071.

[9] Ahnert, S.E. and Fink, T.M.A. (2016). Form and function in gene regulatory networks: the structure of network motifs determines fundamental properties of their dynamical state space. Journal of the Royal Society Interface, 13(120), 20160179. doi:10.1098/rsif.2016.0179.

Fault-tolerant Christmas Trees (not the live kind)

It's an interconnected Christmas scene, but that's not a Christmas Tree! (?) COURTESY: Andrew P. Wheeler.

This year's holiday season post brings a bit of graph-theoretic cheer. That's right, there is a type of network called a Christmas tree [1,2]! It is a class of fault-tolerant Hamiltonian graph [2,3]. So far, Christmas trees have been applied to computer and communications networks, but may be found to have a wider range of applications, particularly as we move into the New Year.

An example of a Christmas Tree directed graph as shown in [2]. The top two graphs are slim trees of order 3 (left) and 4 (right). A Christmas tree (bottom) includes selected long-range connections (longer than the immediate connection to mother, daughter, or sister nodes).

This tree could have used a bit more fault-tolerance!

NOTES:
[1] Hsu, L-H and Lin, C-K (2008). Graph Theory and Interconnection Networks. CRC Press, New York.

[2] Hung, C-N, Hsu, L-H, and Sung, T-Y (1999). Christmas tree: A versatile 1-fault-tolerant design for token rings. Information Processing Letters, 72(1–2), 55-63.

[3] Wang, J-J, Hung, C-N, Tan, J.J-N, Hsu, L-H, and Sung, T-Y (2000). Construction schemes for fault-tolerant Hamiltonian graphs. Networks, 35(3), 233-245.

100 years of Growth and Form!

This year marks the 100th anniversary of "On Growth and Form" [1] by the biologist/ mathematician D'arcy Thompson. "On Growth and Form" has always been an intriguing book from both a historical and technical perspective [2]. This includes the integration of fields such as physics, developmental biology, and geometry. There is an entire website dedicated to the centennial, which demonstrates that his ideas are still useful today [3].

Four bony fish phenotypes related through evolution and transformed through phenotypic deformation. 

D'arcy Thompson provided an account of what we now call evo-devo [4] as a series of mathematical transformations. On the one hand, this provides a mathematical model for the static geometry of the developmental phenotype across species. On the other hand, Thompson provided few if any evolutionary, nor any genetic mechanisms, even in a time when both were becoming ascendant [5]. His physical approach to biological form and morphogenesis has not only been useful in biology, but also as inspiration for computational modeling approaches [6].

NOTES:
[1] Thompson, D.W. (1917). On Growth and Form. Cambridge University Press, Cambridge UK.

[2] Alicea, B. (2011). The Growth and Form of Pasta. Synthetic Daisies blog, October 11.

[3] Much of the contemporary innovation in this area is in the field of architecture. In modern evo-devo, it has taken a back seat to genetic manipulation. Given what we now know about evolution and genetics, there are some potentially interesting biological simulation to be done at the interface of regulatory mechanisms in development and phenotypic fitness based on biomechanical parameters.

[4] Arthur, W. (2006). D'Arcy Thompson and the theory of transformations. Nature Reviews Genetics, 7, 401-406.

[5] Deichmann, U. (2011). Early 20th-century research at the interfaces of genetics, development, and evolution: reflections on progress and dead ends. Developmental Biology, 357(1), 3-12.

[6] Kumar, S. and Bentley, P.J. (2003). On Growth, Form, and Computers. Elsevier, Amsterdam.

Be as Brief as Possible but no Briefer

Nature Highlights article on the Journal of Brief Ideas, which itself is brief.

No, this is not an Einstein quote. But Einstein very well may have submitted to the Journal of Brief Ideas [1], an open access version of Occam's razor. I just submitted a brief paper called "Playing Games with Ideas: when epistemology pays off", which is the equivalent of a fully-indexed abstract [2]. While some people might find 200 words to be too brief, the Journal allows for attachments to be submitted, thus allowing a bit of circumventing with regard to the word limit [3].

According to the Journal FAQ, submitting such brief reports is part of establishing something below the current standard for the minimal publishable unit. It is also important for enforcing good scientific citizenship practices [4]. Very short papers have occasionally been published in regular journals. Mathematics papers by Lander and Parkin [5] and Conway and Soifer [6] accomplished mathematical proofs in less than a paragraph (but with multiple figures). Other than these rather mythical examples, it is quite the challenge to integrate a well-formulated idea into the Journal of Brief Ideas' 200 word limit.


NOTES:
[1] Woolston, C. (2015). Journal publishes 200-word papers. Nature, 518, 277.

[2] Indexing done via document object identification on Zenodo, doi:10.5281/zenodo.167647

[3] If a picture is worth 1000 words, then the Journal of Brief Ideas become less brief than its name implies.

[4] Neisseria (2015). All you need to publish in this journal is an idea. Science Made Easy blog, February 13.

[4] Lander, L.J. and Parkin, T.R. (1966). Counterexample to Euler's Conjecture on sums of like powers. Bulletin of the American Mathematical Society, 72(6), 1079.

[5] Conway, J.H. and Soifer, A. (2004). Can n² + 1 unit equilateral triangles cover an equilateral triangle of side > n, say n + ɛ? American Mathematical Monthly, 1.

Rectified and Ramifying Representations for the Purpose of Theoretical Expediency

One aim of the DevoWorm project is to take a tree structure (in this case a cell lineage tree from an embryo) and extract distributed structural information. This is done to find previously undiscovered patterns in early development (embryogenesis). One way in which this can be accomplished is by building undirected complex networks to represent the relationships between three-dimensional cellular position in a point model of the embryo. Indeed, rather than a branching tree, we are left with a much larger tree with a significant number of cycles. This allows us to examine previously undiscovered interactions between cells based on proximity (such as juxtacrine and paracrine signalling).

A tree with a cycle, indeed. Popular meme or research problem?

Now these ideas have been made concrete in the form of a poster and presentation that describe the methodology and results of representing approximations of cell nuclei in the embryo as a connected network. This work has been featured at the Network Frontiers Workshop (Northwestern University) and the Midwest Regenerative Medicine Meeting (Washington University, St. Louis). Here is the poster in slide form:

Notice how this approach is both geometrically vivid and extensible to different modes of development. The graphs and statistics were rendered in Gephi, and other computation was done in MATLAB and R. Our next steps include developing customized modules in Gephi for drawing differentiation trees, developing hybrid directed acyclic graph (DAG)/undirected network graph structures, and refining the network construction methodology.

We are also working on a methodology called the scalable interactome, which simply involves using graphs to visualize and extract information at multiple spatial and temporal scales. One current example of this is OneZoom explorer, which renders the tree of life in a fractal manner. This can be extended to exploring the fractal and complex geometric nature of the embryo itself.

A slightly different view of human evolution and rejection of human exceptionalism. COURTESY: OneZoom Tree of Life.

"Miscellaneous Polyhedra" by Carol Branch (no pun intended).

With that nod to complexity, I would be remiss if I did not mention the old SimCity dictum? A gratuitous image of fractals and reference to a Wil Wright easter egg is the perfect way to end this post. 

Stardates and Interdigitated Rabbits

Today's Google Doodle animation is in honor of leap year in the Gregorian Calendar. As you can see from the images below, legend has it that rabbit #29 jumps in between rabbits #28 and #1 without disturbing their sleep. Whether any of these cartoon rabbits are related to Inspector #5 is not clear. 

A bit more seriously (but still in the realm of fiction) is the art and science of timekeeping. The leap year, occurring once every four years, is actually a transannual correction on the 365 day year. As it actually takes 365.25 days for the Earth to make a single orbit around the Sun, the Gregorian calendar falls short. In fact, there has yet to be a calendar created that perfectly captures the length of a solar year. This brings us to a potential candidate, the well-known Stardate.

However, despite stardates being the primary mode of timekeeping in a fictional interstellar civilization, they are surprisingly fluid from one part of the galaxy to the next, and from one series to the next. But you can download a more stable version for your own computer, as the concept of a stardate is based on a standard mathematical model.

Regardless of the inconsistencies in  the Stardate system, time travel occurred a number of times in the Star Trek franchise. As this is the 50th anniversary of the first season of Star Trek: TOS, it's a good time to look at instances of time travel in the Trek franchise:

Ex Astris Scientia, Time Travel in the Abramsverse

Memory Alpha Wiki, Temporal mechanics

io9 Blog, Six Theories Of Time Travel In Star Trek

Time travel tech, Trek style. COURTESY: ArsTechnica and Paramount Pictures.

George Boole: 11001000

Today's Google Doodle is a real Boole Doodle, centuries of two. Said three times fast, of course. Here are blogposts by Stephen Wolfram and Mike James at I Programmer with more on its broader significance. The why of the blogpost title can be found here.

Godel's Revenge: All-Encompassing Formalisms vs. Incomplete Formalisms

This content is cross-posted to Tumbld Thoughts. A loosely-formed story in two parts about the pros and cons of predicting the outcome of and otherwise controlling complex sociocultural systems. Kurt Godel is sitting in the afterlife cafe right now, scoffing but also watching with great interest.

I. It's an All-encompassing, Self-regulation, Charlie Brown!

Here is a video [1] by the complexity theorist Dirk Helbing about the possibility of a self-regulating society. Essentially, by combining big data with the principles of complexity would allow us to solve previously intractable problems [2]. This includes more effective management of everything from massively parallel collective behaviors to very-rare events.

But controlling how big data is used can keep us from getting into trouble as well. Writing at Gigaom blog, Derrick Harris argues that the potentially catastrophic effects of AI taking over society (the downside of the singularity) can be avoided by keeping key data away from such systems [3]. In this case, even hyper-complex AI systems based on deep learning can become positively self-regulating.

NOTES:

[1] The World after Big Data: Building the Self-Regulating Society. YouTube, August 14 (2014).

[2] For a cursory review of algorithmic regulation, please see: Morozov, E.   The rise of data and the death of politics. The Guardian, July 19 (2014).

For a discussion as to why governmental regulation is a wicked problem and how algorithmic approaches might be inherently unworkable, please see: McCormick, T.   A brief exchange with Tim O’Reilly about “algorithmic regulation”. Tim McCormick blog, February 15 (2014).

[3] Harris, D.   When data become dangerous: why Elon Musk is right and wrong about AI. Gigaom blog, August 4 (2014).

II. Arguing Past Each Other Using Mathematical Formalisms

Here are a few papers on argumentation, game theory, and culture. My notes are below each set of citations. A good reading list (short but dense) nonetheless.

Brandenburger, A. and Keisler, H.J.   An Impossibility Theorem on Beliefs in Games. Studia Logica, 84(2), 211-240 (2006).

* shows that any two-player game is embedded in a system of reflexive, meta-cognitive beliefs. Players not only model payoffs that maximize their utility, but also model the beliefs of the other player. The resulting "belief model" cannot be completely self-consistent: beliefs about beliefs have holes which serve as sources of logical incompleteness.

What is Russell's Paradox? Scientific American, August 17 (1998).

* introduction to a logical paradox which can be resolved by distinguishing between sets and sets that describe sets using a hierarchical classification method. This paradox is the basis for the Brandenburger and Keisler paper.

Mercier, H. and Sperber, D.   Why do humans reason? Arguments for an argumentative theory. Behavioral and Brain Sciences, 34, 57-111 (2011).

The Argumentative Theory: a conversation with Hugo Mercier. Edge Magazine, April 27 (2011).

Oaksford, M.   Normativity, interpretation, and Bayesian models. Frontiers in Psychology, 5, 332 (2014).

* a new-ish take on culture and cognition called argumentation theory. Rather than reasoning to maximize individual utility, reasoning is done to maximize argumentative context. This includes decision-making that optimizes ideonational consistency. This theory predicts phenomena such as epistemic closure, and might be thought of as a postmodern version of rational agent theory. 

There also seems to be an underlying connection between the "holes" is a culturally-specific argument and the phenomenon of conceptual blending, but that is a topic for a future post.

Starstuff Squared, Rubik's Cubed

Welcome to the 250th Synthetic Daisies post! This post consists of three subthemes cross-posted from Tumbld Thoughts. The first is in honor of the Google Doodle for the 40th Anniversary of the Rubik's Cube, while the latter two are the supplemental readings for the tenth and eleventh episodes of the Cosmos reboot. 

I. Rubik's 3-D CSS Cubes

Today is the 40th anniversary of the Rubik's Cube. Aside from an invention that sold 350 million copies, Rubik's Cube is also an example of a permutation puzzle that contains an interesting problem related to group theory. The doodle itself is unique in that it utilizes a technology called CSS 3-D transforms [1]. Naturally, there is a Google Doodle.

[1] Edidin, R.   How Google Built Its 3-D Interactive Rubik’s Cube Doodle. May 19 (2014). Also check out the Chrome Cube Lab, which uses this technology to render interactive cube-based puzzles beyond Rubik's namesake.

II. All I Want For Christmas is an Electric Charge

Here are the supplemental readings for the tenth episode of the Cosmos reboot ("The Electric Boy"). Readings are organized by theme.

The Electric Boy and his Legacy:

Michael Faraday (1791-1867). BBC History.

History of the Christmas Lectures. The Royal Institution.

Cody, D.   Social Class. The Victorian Web. July 12 (2002).

Burgess, M.P.D.   Semiconductor History: Faraday to Shockley. Transistor History (2008).

Williams, A.T.   Faraday vs. Maxwell and Faraday and the Ether. Consciousness, Physics, and the Holographic Paradigm (2010).

Electromagnetic Spectrum. NASA Goddard Space Flight Center.

Frog Legs and Televisions:

Galvani's animal electricity experiments. Institute of Engineering and Technology.

Luigi Galvani (1737-1798). Center for Integrating Research + Learning, Magnet Lab, Florida State University.

Borgens, R.B., Vanable, J.W., and Jaffe, L.F.   Bioelectricity and regeneration. I. Initiation of frog limb regeneration by minute currents. Journal of Experimental Zoology, 200(3), 403–416 (1977).

Iconoscope. Wikipedia, March 20 (2014).

Philo Farnsworth (1906-1971), Electronic Television. Inventor of the Week Archive, Lemelson-MIT (1999).

Researching Faraday Cages and Electromagnetic Fields on Google is a sad statement on Internet culture:

Chandler, N.   How Faraday Cages Work. How Stuff Works.

Trottier, L.   A Growing Hysteria. Committee for Skeptical Inquiry, CFI. October (2009).

Hruska, J.   New research confirms that our electronics and radio waves disrupt migratory birds. Extremetech, May 8 (2014).

Magnetic Fields of the Earth. HyperPhysics.

Inventions/Discoveries of the Electric Boy:

Faraday's Inventions. Michael Faraday's World.

Homopolar Generator, Wikipedia. April 1 (2014).

Electrolysis, Wikipedia. May 7 (2014).

Faraday Cage, Wikipedia. March 19 (2014).

Electric Motor, Wikipedia. May 10 (2014).

Static Electricity, Wikipedia. May 5 (2014).

III. Leaving Nothing but Footprints, but Still Living On.

Here are the supplemental readings for the eleventh installment of the Cosmos reboot ("The Immortals"). As usual, readings are organized by theme.

Entropy Is Not Immortality, Time Can Be Written Down:

Matson, J.   What Keeps Time Moving Forward? Blame It on the Big Bang. Scientific American, January 7 (2010).

Mlodinow, L. and Brun, T.A.   Relation between the psychological and thermodynamic arrows of time. Physical Review E, 89, 052102 (2014).

Wall, M.   Video Tour of Alien Planets Shows How Time Flies on Strange New Worlds. Space.com, May 07 (2014).

Jones D.L.   Aging and the germ line: where mortality and immortality meet. Stem Cell Reviews, 3(3), 192-200 (2007).

Barksdale, M.   10 Methods of Measuring Time. Discovery TV: Relativity and Time.

Origins of Writing Systems. AncientScripts.com.

Fun With the Origins of DNA:

Akst, J.   RNA World 2.0. The Scientist, March 1 (2014).

Moran, L.A.   Changing Ideas About the Origin of Life. Sandwalk blog, August 7 (2012).

Joshi, S.S.   Origin of Life: the Panspermia Theory. December 2 (2008).

Klyce, B.   Cosmic Ancestry. 

Saenz, A.   Venter creates first synthetic self-replicating bacteria from scratch. SingularityHub, May 20 (2010).

Moving Life (via Dispersal):

Levin, S.A., Muller-Landau, H.C., and Nathan, R.   The Ecology and Evolution of Seed Dispersal: a theoretical perspective. Annual Review of Ecology, Evolution, and Systematics, 34, 575-604 (2003).

Gronstal, A.   Space Rocks Could Reseed Life on Earth. Astrobiology Magazine, May 15 (2008).

Interstellar Dust Could Have Building Blocks Of Life – On Science. RedOrbit, January 28 (2014).

Can our TV signals be picked up on other planets? BBC News, August 6 (2008).

Civilization is (not) Forever:

Chandler, G.   Desertification and Civilization. Saudi Aramco World, 58, 6 (2007).

Arbesman, S.   210 Reasons for the Fall of the Roman Empire. Social Dimension blog, June 26 (2013).

Kunzig, R.   Geoengineering: How to Cool Earth--At a Price. Scientific American, November (2008).

Duncan, R.C.   The Olduvai Theory: sliding towards a post-industrial Stone Age. Institute on Energy and Man, June 27 (1996).

Math Program Cracks Cause of Venus Hell Hole. Space Daily, March 21 (2001).