Caring invertebrates

An Oxyopid spider that stayed put for 3 months
L. Shyamal - Creative commons / Wikimedia

We are not brought up on the notion of  invertebrates being caring and that presumably allows us to swat them without pangs of conscience. This spider sat on leaf of a Michelia tree in my garden, without making a movement for nearly three months. The kind of parental care given by invertebrates can be rather subtle. Invertebrates may carry their brood with them, stay over and conceal them, guard them actively, feed larvae or even sacrifice themselves as food for their brood. And it not just females, in some cases the males are involved in care of the brood. From my regular visits to the leaf while watering the plants, I imagine that this mother spider guarding her egg clutch never fed for all of three months unless she hunted by night (but these spiders are said to be diurnal) or found prey that came within reach.

Male water bug carrying eggs
Marshal Hedin - Creative Commons/Wikimedia

The Scientific American article (full text link below) by Douglas Tallamy includes a number of interesting pictures including a praying mantis that positions itself on the lower branch of a plant to intercept any predators that might climb up toward its young on the distal branches past it. 

Lethocerus indicus

Carrion beetles (including the male) regurgitate a fluid to feed their young. Water bugs in the genus Abedus and Belostoma have a peculiar system in which the female lays eggs on the back of the male which he carries until the eggs hatch.  Females  mate with multiple males and encumbered males never mate again until the eggs in their care hatch. The larger water bugs in the genus Lethocerus have the females laying eggs above the water on plant stems, and the male periodically climbs up to moisten the eggs, preventing dessication, and guarding the eggs by night. 

Great eggfly females stand guard near eggs

The social insects (mainly Isoptera, Hymenoptera) are of course well-known for the care of their brood however earwigs, carrion beetles, several bugs, cockroaches, butterflies and a host of others are known to show a range of parental care behaviours.  This extends even into the lower "insects" - the Diplura are known to show parental care in the form of guarding the eggs. Marine amphipods are also known for brood care. Freshwater leeches in the family Glossiphoniidae brood their eggs as do octopuses and other cephalopods. 

In general the idea is that parental care is more likely when the brood is smaller and "more precious" and when the organism is larger and long-lived.

Further reading

• Tallamy, Douglas C (1999) Child care among the Insects. Scientific American. pages 72-77 (PDF)
• Tallamy, DW (2001) Evolution of exclusive paternal care in arthropods. Annu. Rev. Entomol. 46:139–65
• Subsociality - lecture handout
• James D. J. Gilbert and Andrea Manica (2010) Parental Care Trade-Offs and Life-History Relationships in Insects. American Naturalist 176(2):212-226 (Try Google cache if PDF is not downloadable)

Crunching elephants

Apparently the physict Freeman Dyson came up with a complex formula in 1953 and when Enrico Fermi saw it, he just did not like the complexity. Fermi in his quest for simplicity and elegance apparently quoted John von Neumann who had said:

"With four parameters I can fit an elephant, and with five I can make him wiggle his trunk"

I was recently muddling with some techniques for shape quantification which has applications for field biologists, for instance to match or compare biological shapes. Naturally I decided to try out what are termed as elliptic Fourier descriptors - an idea based on the epicyclic movements of planets that goes back to ancient times although the mathematical forms are relatively new. A lovely demonstration of the concept can be found here. The problem of getting the parameters from a given closed curve however is not shown by that applet. The reduced number of terms to describe a closed curve, elliptic Fourier descriptors, have been widely used in analysing shapes for systematics. 

Mathematically reduced elephant

What got my attention was a 2010 paper (I usually stay away from physics) titled "Drawing an elephant with four complex parameters" where the authors try to reconstruct an elephant with four complex numbers and wiggle its trunk with a fifth ! This was attempted earlier in 1975 with least-squares Fourier series fitting but that apparently took 30 parameters. The figures in the new paper show that they managed to do something close.

Program Screenshot: My elephant clearly beats the math

I decided to write up a program and see for myself but have been able to get a passable elephant only with 8 parameters and that is 8 sets of 4 real numbers (a, b, c and d of Kuhl and Giardina can be considered as two complex numbers) and so not as good as the results of the authors of the 2010 paper.  Maybe my elephant is too complex for the math. In case you want to play around with the idea, download (Windows unfortunately, but works in Wine) the small program  that I wrote today from here - you can also find some Matlab code here.

Further reading

• Mayer, J; K Khairy and J Howard (2010) Drawing an elephant with four complex parameters. Am. J. Phys. 78(6):648-649
• Kuhl FP & C R Giardina (1982) Elliptic Fourier features of a closed contour. Computer Graphics and Image processing 18:236-258
• Dyson, Freeman (2004) A meeting with Enrico Fermi. Nature 427:297
• Shape - http://lbm.ab.a.u-tokyo.ac.jp/~iwata/shape/index.html 
• Fitting an Elephant
• R code for the 4 point elephant 
• Morphometrics
  

Math in the wild

If the spots were mobile towers,
the polygons represent the area covered

What is common to mobile towers, giraffes and cracked clay? If you do not already know about it, you might like to play around with the Java Applet here (or this) or see a particularly beautiful visualization of a particular computational method here. You can also look up more about Voronoi tesselations. The connections are well known to most folks with an interest in  "recreational mathematicians"  and although I have been aware of this from my school days, search engine results still surprise me and its great to see lots of new content. This museum/art exhibit seems particulary interesting as a way of introducing such topics  (the exhibit, strangely enough, is named after the thesis of Dr. Theodore Kaczynski - better known as the Unabomber! Incidentally, his manifesto is eminently readable and worthy of examination in our troubled times, quite unlike his thesis which was said to be accessible to only about at best 10 mathematicians when it was published )

Cracking clay

Coming back to giraffes. We usually think that every cell on our body has the same DNA  but this is just the abridged version. There can be major variations and these are large enough sources of error for one to worry about a lot of research. We used to be taught in "genetics 101" about chimerism, but I have recently learnt about the nuanced differences between that and "mosaicism". The patterns by which cells differ, multiply and communicate with each other can be thought of as driving some of the intricate patterns that one can find in nature.  In Siamese cats, for instance, it has been shown that a heat-sensitive mutant of Tyrosinase which leads to a failure in melanin synthesis causing albinism in the warm parts of the body and leaving the cooler nose and extremities black. An increasing number of works are melding ideas from genetics and computer science and some of these demonstrations include the simulation of mammalian coat pattern, such as those of giraffe. The earliest ideas on the mathematical, physical and chemical principles for biological patterns were illustrated  by pioneers like D'Arcy Thompson and Alan Turing. Turing is of course better known for his work in the theory of computability.

Clonal mosaic models to simulate coat patterns (Walter et al. 1998)

Working on the intersections of multiple fields often produces interesting results, and while these are largely ignored by professional biologists, they are becoming increasingly visible, in cartoon animation. Some of the techniques used in computer graphics and to a limited amount, biological studies are becoming more accessible on the Internet. One of these is http://algorithmicbotany.org/ with a PDF version of the book "Algorithmic beauty of plants" on simulating plant forms by Przemyslaw Prusinkiewicz and other related works including the simulation of seashell patterns by Hans Meinhardt (his book "The Algorithmic beauty of Seashells" is into its third edition but is not publicly available online, but a related paper can be found here)

Prum & Williamson (2002)

An interesting 2003 paper on the evolutionary patterns of eyespots in Satyrine butterflies is accompanied by a nice animation that gives a visual way of looking at patterns and correlating them with molecular phylogenies. (Flash animation) Another study has looked at the simulation of feather patterns. Unfortunately, nobody has made an accessible Java Applet to illustrate this idea although for  overall structure there is a nice applet here.

Silhouettes generated using a software toy

Some years ago I considered the idea of simulating the silhouettes of birds. Although rather skeletal and comical, it is something that those with an understanding of bird identification can toy with. You can find the Windows compatible program here. The illustration alongside has a collage of some of the shapes that can be generated using the program. While it is mostly a toy, it might help in sharpening one's idea of how the positions and relative bone lengths alter our perception of shape and how just the relative lengthening or shortening of feathers can give pointed, or rounded wings and graduated, wedge-shaped or forked tails .

Postscript: Unfortunately, nothing that I have written above will change the life of   millions of school children who complain about boring mathematics classes.

Image credits: The Great Wave (public domain); Cracks  in clay (Hannes Grobe, Wikimedia Commons)

Further reading

* Arbesman, S. , Enthoven, L. & Monteiro, A. (2003) Ancient Wings: animating the evolution of butterfly wing patterns. BioSystems, 71:289 - 295  (paper)
* DL Imes, LA Geary, RA Grahn, and LA Lyons (2006) Albinism in the domestic cat (Felis catus) is associated with a tyrosinase (TYR) mutation. Anim Genet. 37(2):175–178.
* Ting-Xin Jiang; R B Widelitz; Wei-Min Shen; P. Will; Da-Yu Wu; Chih-Min Lin; Han-Sung Jung and Cheng-Ming Chuong (2004) Integument pattern formation involves genetic and epigenetic controls: feather arrays simulated by digital hormone models. Int. J. Dev. Biol. 48: 117-136 (2004)
* Shigeru Kondo (2002) The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin. Genes to Cells 7:535–541
* Iwashita M, Watanabe M, Ishii M, Chen T, Johnson SL, et al. (2006) Pigment Pattern in jaguar/obelix Zebrafish Is Caused by a Kir7.1 Mutation: Implications for the Regulation of Melanosome Movement. PLoS Genet 2(11): e197. 

* Plant studio - an interesting software project http://www.kurtz-fernhout.com/summary_plantstudio.html

* Richard O Prum and Scott Williamson (2002) Reaction-diffusion models of within-feather pigmentation patterning. Proc Biol Sci. 269(1493): 781–792.
* Applets - fern leaf -  tree - tree

The Great Wave off Kanagawa by Hokusai (1831)
A fractal wave