Tuesday, December 30, 2014

Mathematical Recreations : Ramanujan's Nested Radical

Srinivasa Ramanujan posed the following puzzle to the Journal of Indian Mathematical Society in April 1911.

What is the value of the following?
\( \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)

He waited over six months, and when nobody replied he gave his solution. The result he provided was 3.



The algebraic solution provided by Ramunujan:

Consider the identity 
\( (x+n)^2 = n^2 + 2nx + x^2 = n^2 + x[(x+n)+n] \)
Take square roots to get
\( [x+n] = \sqrt{n^2+x[(x+n)+n]} \)
Now inside the brackets you have (something + n), so you can
substitute in the same equation with (x+n) replacing (x) to get
\( x+n = \sqrt{n^2+x\sqrt{n^2+(x+n)[(x+2n)+n]}} \)
and repeat the process to get
\( x+n = \sqrt{n^2 + x\sqrt{n^2+(x+n)\sqrt{n^2+(x+2n)\sqrt{\cdots}}}} \)
If you now set n = 1, x = 2 you get
\( 3 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)



An alternate way of calculating the equation reveals some interesting behavior.

\( 3 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)
\( 3^2 = 1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}} \)
\( 3^2-1= 2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}} \)
\( \frac{3^2-1}{2}= \sqrt{1+3\sqrt{1+4\sqrt{\cdots}}} \)
\( 4 = \sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{\cdots}}}} \)
\( \vdots \)
\( n = \sqrt{1+(n-1)\sqrt{1+(n)\sqrt{1+(n+1)\sqrt{\cdots}}}} \)

The value of each sequential nested radical forms a series...

\( 3, 4, 5, 6, \cdots, \infty \)

...which is perfectly valid. If 3 was not a valid answer to the equation, then this calculation would have revealed a contradiction. Proof by contradiction is one of my favorite methods of proving.


So, what happens if we start with a different starting assumption?

\( 4 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)
\( \frac{3\cdot5}{2} = \sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{\cdots}}}} \)
\( \frac{13\cdot17}{2^{2}3} = \sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{\cdots}}}} \)
\( \frac{11\cdot19\cdot223}{2^{6}3^{2}} = \sqrt{1+5\sqrt{1+6\sqrt{1+7\sqrt{\cdots}}}} \)
\( \frac{17\cdot27902701}{2^{12}3^{4}} = \sqrt{1+6\sqrt{1+7\sqrt{1+8\sqrt{\cdots}}}} \) 
\( \vdots \)

The value of each sequential radical forms another series.

\( 4, 7.5, 18.41\bar6, 84.543402\bar7, 1429.71739065, \cdots, \infty \)

It is harder to predict subsequent entries of the series than for the original series, but no less valid. I've put some effort into constructing an algebraic function that also produces this series, but have set aside the challenge for now.



So, let's try another starting assumption.

\( 2 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)

The value of each sequential radical forms a third series...

\( 2, 1.5, 0.41\bar6, -0.206597\bar2, -0.191463517554, \cdots, -\infty \)

...which is not valid under the convention that a radical can't have a negative value.



Generalizing this computation method, Ramanujan's Nested Radical has an infinite number of valid answers...

\( 3 \ge \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \)

Every starting value greater than 3 results in a series of radical values that increase at more than 1 per step and increase to positive infinity. Every starting value less than 3 results in a series of radical values that increases at less than 1 per step and eventually turns negative. I don't yet know how to prove this general statement, but the pattern holds. The system behaves very simply when the starting value is 3, more than 3, or less than 3.



Ramanujan's solution to his puzzle is correct, but it is also incomplete.

The algebraic method is unable to find the full solution to the equation. Algebra doesn't effectively deal with infinites, so it shouldn't entirely be a surprise that an algebraic intuition might make some faulty (or at least incomplete) inferences when infinity is involved. This is interesting because it hints at the limits of algebra as a way of knowing truth. The Gรถdel Incompleteness Theorems discuss something similar to this and are worth reading more about.

How does this connect with biology?

One detail of my training to be a scientist is learning to look for untested assumptions in arguments that are presented to me. This was never explicitly stated in any of my graduate coursework, but it has been important to do well in the academic environment I experienced in getting my PhD. When I find such an assumption, I see it as a puzzle to be explored. Am I seeing something real? What inferences is the presenter drawing from the assumption? Why is what I'm seeing not already addressed?

What assumptions have you made today?

References:
  1. Ramanujan: www.thefamouspeople.com/profiles/srinivasa-ramanujan-503.php
  2. Algebraic solve: mathforum.org/library/drmath/view/52674.html
  3. Numerical solve: www.johndcook.com/blog/2013/09/13/ramanujans-nested-radical/ 
  4. Proof by contradiction: en.wikipedia.org/wiki/Proof_by_contradiction
  5. Incompleteness theorems: en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Saturday, December 27, 2014

Mystery Plant ID Found!

1. omnilexica.com/?q=silphium
A little over a year ago, I was involved in a wedding party and we were walking through a park to a site for some of the arranged photographs. Along the walk, we crossed a stream and there I came across an interesting plant.

The plant was done flowering, but the seed heads retained a nice green color and had the lovely geometry seen in image #1. I picked a sample of the plant, using the pocket-knife I almost always carry (largely for such purposes as this).

Neither myself, my mother, nor my aunt were able to identify the plant. Now this was getting interesting, as my mother and aunt have a long history of gardening and interest in wildflowers. My aunt noted the material had the scent of sunflowers, but this only helped identify it to the family level, which the structure itself had already provided. I tucked the sample into my pocket, hoping to keep it intact until I could identify it at some later time. On returning home, I consulted the field guides I owned. I had no luck, since they all seem to focus on the most charismatic aspect of the wildflowers…  usually, the flowers themselves.

As the plant material dried, I realized that I had collected very mature looking seeds of this unknown plant. I cleaned the seeds and stored them in a vial, hoping to identify them when I later had the chance to grow plants from them. Part of the motivation was also because I thought that the green seed heads would make a nice display in a vase, so I could use the plant as part of my home flower cutting garden.
2. diggingdog.com/pages2/plantpages.php/P-1403

…a year passes…

I am in the final stages of completing my PhD thesis and found my mind wandering. I was reading about the efforts to breed perennial versions of annual crops and to domesticate new perennial crops from wild plants. While looking through a document on the efforts to generate perennial seed crops, I saw a dried plant specimen of Silphium integrifolium (Deam's Rosinweed) and it struck a memory. A quick web-search later and I found an image of exactly what I discovered on the creekside. However, there are several potential species in the genus Silphium that live in the area and have the memorable seed heads, so it will still take growing the seeds I saved to identify the plant down to the species level.

The plant was being studied for its domestication potential as a new seed crop. I still think it would be a worthwhile project to domesticate it as green material for florist use. Domestication of a single plant species can easily go in multiple directions at once, as the plethora of cabbage/broccoli/etc. plants indicate.

Now that I know what the plant looks like and that it will produce some really nice flowers (image #2), I can better plan for where the plant will go in my garden next year.

References:

Tuesday, December 23, 2014

GMO labeling : The right to know what is in our food.

Sometimes biology intersects with political-charged topics that engage large-numbers of people. In the USA, there has been recent activity around the labeling of foods made from genetically modified organisms (GMOs). The following video epitomizes one viewpoint on the subject.

http://www.upworthy.com/a-14-year-old-explains-food-labeling-in-language-even-condescending-tv-hosts-should-get-3?c=reccon3



One of the major arguments presented in the video is that people have the right to know what is in our food. Because of this, GMO foods should be labeled.

I think this is a wonderful idea!

However… there's a big problem. People think they know what is in regular crops, while they think they don't know what is in GMO crops. Can you list the compounds found in the last organic heirloom tomato that you ate? Do you know what is in "natural" corn? How about the poisons found in the various types of beans that people eat? How about the poisons sprayed on any crops (even organic) that you might find in the store? The reality is that people, in general, have no idea what is in any of the food they eat.

A second problem I have is that the GMO-labeling ideas being pushed will do nothing to assist people in knowing what is in their food. If you're eating a tomato which included a gene from a fish and the label only says the tomato has been altered with a certain form of technology, then the label does nothing at all to inform you about what you're eating.

We do already have an established method for indicating the presence of small amounts of diverse substances in our food that can be utilized for labeling GMOs: the ingredients list. If your tomato product also includes some fish genes, then the ingredients list for the tomato should include the fish (and any bacterial) genes that are found in it. Finding an ingredients label on a single tomato will discourage those who aren't interested in purchasing GMOs, but will allow those who are interested in learning what is in their food to be able to easily educate themselves.

So far, no organization is pushing for such an educational labeling idea. The people pushing for GMO-labeling don't want honest and educational labeling. They're pushing for labels to inspire fear, uncertainty, and doubt (FUD) about GMOs.

I think such a labeling system should also be extended to indicate what chemicals were sprayed on the fruit and thus are likely to remain on the fruit. This issue doesn't involve GMOs, but these chemicals are something we do know with certainty are poisons...  and yet there is no push for this information on labels.



I am a biologist, so I realize my perceptions of this subject are likely to be distinct from those of most people.

I agree with the reasons stated for GMO labeling, but I am totally against every GMO-labeling proposal I have so far heard being discussed because they don't help attain the goal of the stated reasons.

Monday, December 15, 2014

Hybrid sunflower roots.

1. Plant 3, roots & tuber.
A few weeks ago, we had a solid freeze and the sunflower season came to an end. I dropped by the old place and dug up my sunflowers to see if any of the F1s had developed tubers. The largest hybrid (plant 3) produced one skinny tuber, while the smaller two plants appeared to produce no tubers at all. This was a disappointment, but a more detailed examination of the remaining plant material led to some positive surprises.

2. Plant 1, seeds.
The first plant looked very much like the Helianthus tuberosus mother, but with more red pigment on the stems. I had assumed this was the result of some recombination of maternal alleles and was at best a control to compare the hybrid plants against. When I looked at the remaining two dried flowers on this plant and found them to be full of seeds (image #2), I realized this plant is likely to also be a hybrid.
3. Plant 3, root bud.

The plant that grew the tuber in image #1 also produced a second type of perennial structure, a new bud growing from the old root crown (image #3). This is a structure that isn't seen in either parent species. This feature has been observed before in crosses of this type, however, so I should have expected the possibility.

4. Plant 2, roots.
The other two plants had mildly-swollen roots (images #4 & #5) that have appeared alive when I checked on them. I don't know if they will show new growth in the spring, or if the roots will at some point finally start to rot.
5. Plant 1, roots.

I'm storing the tuber and all the root structures in a dedicated small cube-fridge over the winter. This should keep them safe from the mice that wander through my basement and allow me to grow at least one of them again next year alongside additional F1s from the ~70 seeds that remain from the first cross.

References:
  1. H. annuss x H. tuberosus : bulbnrose.x10.mx/Heredity/sunflowerXchoke/sunflowerXchoke.html

Wednesday, December 10, 2014

Black Nightshade

I've recently written a new post on the topic of toxicity in this plant that you should also read if you've found this page via search or other methods.



1. Solanum nigrum.
Myths of edibility are much shorter-lived than myths of toxicity. If something is poisonous and you keep eating it, you (or your surviving friends and relatives) will soon learn your error. If something is perfectly edible, but you never eat it for fear of poison, then you will never learn what you're missing.

The common weedy plant Solanum nigrum (Black Nightshade) is a premiere example of this. The berries are routinely considered to be poison, even though there are no recorded fatal poisonings unambiguously associated with the plant.

The berries of every S. nigrum plant I've come across have been very edible, tasting like a somewhat floral and mildly sweet tomato. The ripe berries and green leaves are used over much of the world, with the leaves being used as a pot-herb comparable to spinach. Research has suggested that it might not be a good idea to eat the unripe berries, as they sometimes contain a limited amount of solanine.

Plant poisons tend to be polite, in that they have the trait of tasting poisonous. The putative toxin in S. nigrum is solanine, which has a bitter taste. There are reports of some S. nigrum plants having bitter leaves and unpleasant berries, while others have bland leaves and mildly sweet berries. I suppose I should advise you to not eat the unpleasant tasting plants.

The myth of toxicity of S. nigrum seems to have been spread by the European diaspora. European-derived cultures everywhere seem to think it is deadly poison, even while the natives living in the same places continue eating it routinely. Why would Europeans think this plant is poison?

2. Atropa belladonna; UK range map.
In the UK and much of western Europe, there grows another plant with black berries. This one, Atropa belladonna (Deadly Nightshade), is deadly poisonous, with a long recorded history of deaths… but only in Europe where it grows. In Europe, if you taught children that one black berry (S. nigrum) was edible, but another (A. belladonna) was poison, there would be the risk of them making a deadly mistake in identification. In this context, it is perfectly reasonable for European parents to teach their children that black berries are poison.

A. belladonna has spread to a few other places in the world, but isn't something you will generally run into. If you don't know plants well enough to tell the difference between S. nigrum and A. belladonna, then you really shouldn't be eating anything you find outside. The plants are as distinct as a dog is from a cat. You almost assuredly have experience with identifying those animals, so there is absolutely no way you would mistake one for the other. It still is a good idea to teach children not to eat things you can't identify, but you shouldn't be claiming poison is the reason.

3. Diospyros texana.
The aversion to black berries has even carried over to entirely unrelated plants, that just happen to have black, round fruit.

Diospyros texana (Texas Persimmon) is a tree that produces perfectly edible black fruit that many (of European cultural extraction) consider to be poisonous, even though there are no toxic relatives or mimics. It has a long history of utilization as a food source by American natives of the arid Southwest, but has in recent times been marginalized to a landscaping plant because of the peculiar attitudes of the now-dominant culture.
4. S. nigrum.



Forms of S. nigrum have been partly domesticated under the name "Garden Huckleberry". These plants have slightly larger berries and a more upright growth form than most of the wild plants. There are red ("Makoi") and orange ("Otricoli") varieties that people might be more likely to believe are edible.

I've collected numerous seeds from a local (Minnesota) form of S. nigrum, with the goal of using them in a mutation breeding experiment. The basic idea is to expose a batch of seeds to some mutagen, like X-rays or some chemicals, and then grow out the resulting plants to look for variations which might be more useful. Larger or different colored fruit are the most obvious things to look for, but other interesting traits may also appear. I would like to use ultraviolet light as a source of mutations, as UV-light is easy to control and keep contained, but I still need to determine if it will work for these seeds.



There are still occasional reports of people eating S. nigrum and experiencing gastrointestinal distress. They could have had a specific allergic reaction to the new food source. For this reason, people should be conservative about eating plants they don't have experience with.

5. Solanine-rich S. dulcamara.
The putative poison found in S. nigrum is the bitter-tasting solanine. It is not entirely clear if everyone can taste this compound. You can experimentally determine your ability to taste Solanine by tasting the very common S. dulcamara (image #5), which has elongated orange/red berries and purple flowers. S. dulcamara is definitely toxic due to the high levels of solanine found in its leaves and berries. For several years, I have been occasionally tasting the berries (looking for a 'sweet' version), but have found very little variation in the amount of poison. If the fruit of this plant tastes sweet to you, then you should have someone else taste it before you really eat any and you might want to avoid tasting wild things like this as a rule.



If you eat some S. nigrum (or S. dulcamara) berries and get sick, you really can't blame me for it. "Some guy on the internet told me it was ok!" won't hold up in court.



References:
  1. Solanum nigrum
  2. Atropa belladonna
  3. Diospyros texana
  4. Solanum dulcamara

Tuesday, December 2, 2014

Nicotine

Nicotine is highly addictive, and not just for us humans.

I observed this black vulture (Coragyps atratus) apparently looking for and eating discarded cigarettes at a roadside park in central Florida. I also noted the same behavior with a different group of vultures at another park during the same trip.

If we were to eat tobacco like this, it would quickly make us sick. Vultures are adapted to eat decaying meat and all the nasty toxins that go with it. Because of this, they have what would be described as an, "iron stomach". I wonder how common the behavior is and if addicted vultures get angry when they haven't had a fix in too long.

Cigarette butts have reportedly been found in the intestines of whales (presumably having been washed out to the ocean and ingested along with their normal food). There are photos of other birds investigating cigarette butts they come across, but I haven't found any reports of any animals actually eating them.

(While trying to find other reports of this behavior, I did find a report of people smoking dried vulture brains. People are strange things.)

Thursday, November 27, 2014

Genetic Assimilation.

1. [source]
Tomatoes sporadically produce fruit with horns, fleshy extensions adjacent to the calyx. Do a web search for, "Devil Tomato" and you will find several like the one in image #1. Generally, there is no evidence for these being the result of a genetic mutation. Rather, they represent the sort of thing that can happen when the normal development program of the fruit is disrupted in some way. Seeds taken from such a horned fruit will be no more likely to produce a plant that has similar fruit than seeds taken from any other fruit on the plant.

2. [source]
 There is a related species, Solanum mammosum, that has multiple such horns (image #2). (Though, there are example plants without horns.) The fruit of S. mammosum are rather toxic, so it wouldn't be a great idea to try and make a hybrid between the species and domesticated tomatoes.

3. [source]
Because there is the developmental potential for horns to be generated in tomatoes, there is the potential for a mutation to emphasize the trait. In the Tomato-TILING project, a few such mutations turned up (image #3). I'm not a professional plant developmental biologist, so I don't expect to get access to these interesting mutant seed lines any time soon.

I like the idea of looking for something that everyone else is trying to avoid. Every tomato breeder I've come across has been trying to breed away from a horned tomato, to produce a more "perfect" fruit shape, so I instead want a tomato that is all horns. I have the mental image of a tomato covered in fleshy projections featuring on a counter in some new science fiction movie.

As the previous examples have certain difficulties as a source for this trait, I've been looking for tomato lines which show a higher rate of these "deformations" to use as starting material in a project to breed a tomato that has the trait more consistently.

A rarely studied evolutionary model called "Genetic Assimilation" describes the process where an aberrant trait produced as the result of some stress is selected for and eventually becomes genetically fixed even without the presence of the stress. This mechanism sounds like Lamarckian evolution, except that it relies on the natural selection and the developmental plasticity of organisms…  rather than the personal experiences and intention of the organism that was favored by Lamark. It works because every trait is impacted by the genetic background, the combination of many subtle influences from other genes throughout the genome.

I frequent the Tomatoville forums, including the "Crosstalk: Tomatoville Research and Development™" forum. I started doing so because people there have a tendency to post lovely photos of the interestingly colored and patterned tomatoes they have been growing. Recently, a user was posted images from the results of a complex cross (["Pink Furry Boar" x "Ananas Noir"] x "Bosque Green Cherry") that they were working with. One of the diverse progeny they grew (image #4) had horns on 4 of the 20 fruit. 20% is a far higher rate than I'd otherwise come across, so I asked for a few seeds.

In a few years, I'll have a better idea of where this project is going. The good thing is that I can eat all the rejects along the way.



References:
  1. Genetic Assimilation:
    1. http://jeb.biologists.org/content/209/12/2362.full
    2. http://en.wikipedia.org/wiki/Genetic_assimilation
    3. http://eebweb.arizona.edu/faculty/badyaev/ecol596e/assimilation.pdf
    4. In tiger snakes: http://blogs.discovermagazine.com/notrocketscience/2009/10/30/big-headed-tiger-snakes-support-long-neglected-theory-of-genetic-assimilation/
    5. In fruit flies: Waddington, C. H. (1942) Canalization of development and the inheritance of acquired characters. Nature 150:563-565.
  2. Horned Tomatoes:
    1. http://www.tomatoville.com/showthread.php?t=34162
    2. https://www.flickr.com/photos/farflung/6462879911/
  3. Solanum mammosum
    1. https://www.flickr.com/photos/30372914@N03/3895429577/
    2. https://www.flickr.com/photos/22012266@N02/7164709249/
  4. Tomato Tiling project
    1. http://tilling.ucdavis.edu/index.php/Tomato_Tilling
  5. Tomato Varieties:
    1. Pink Furry Boar
    2. Ananas Noir
    3. Bosque Green Cherry

Wednesday, November 26, 2014

A Requiem.

Jonathan Abbey, my brother.
I've never really fit in with those around me. I accept this and don't need those around me to think the way I do. All I need is for them to accept me for who I am. I have had the good fortune to find someone to share my life with who does this. Barring some unexpected misfortune, by this time next year, she and I will be married.

The way I think about the world is very rarely linear. This has caused conflict between me and my academic advisor, as she wants me to construct lists of what I am working on and how I will set about completing then. I generally think in images, patterns, and relationships. When I am working hard on a puzzle, I tend to see my thought processes as some form of abstract math, even though I don't always have the vocabulary to convey that math to those around me. There are conceptual problems that I've thought about for a while and came to solutions that I'm absolutely certain are true, but I don't yet know how to show them to anyone else. Sometimes, I don't even have a glimpse of how to explain.

There have only ever been a few people that I looked to as role models, for inspiration. Athletes, artists, politicians, and other people who arguably have large positive (or negative) impacts on the people of the world have never felt like role models to me. The people I have ever felt this sort of connection with, that remind me of how I see and want to see the world around me, I can count (in no particular order) on one hand.
  1. Albert Einstein.
  2. Richard Feynman.
  3. Stephen Hawking.
  4. Jonathan Abbey.
None of them were biologists. Perhaps this shouldn't be a surprise, as I often don't fit the standard model of a biologist all that well. They all shared a clarity and depth of thought that I aspired to.

The first three are names you are probably familiar with. Well, you're probably familiar with them if you've had a long-running interest in science and how the universe works. Einstein and Feynman died before I became aware of them and I don't expect to ever meet Stephen Hawking. (I wouldn't know what to do or say if I did.) It was only when I started learning about how they came to the discoveries they're known for that I started looking to them as role models.

Jonathan Abbey was the older of my two older brothers, my parents' first child. A few weeks ago, he died unexpectedly. The proximate cause of his death was cardiac disease, atherosclerosis. This is what is colloquially referred to as "hardening of the arteries". The ultimate cause of his death was his inability or refusal to keep to the schedule for his medication. He had type-1 diabetes and ankylosing spondylitis, two auto-immune diseases which amplify the effect of high blood-pressure on the damage to cardiac arteries which causes atherosclerosis. He went to the emergency room in the week before with chest pain. They gave his heart a clean bill of health and sent him home.

He spent a great deal of time thinking about thinking (meta-cognition). He encouraged me to pursue a PhD and was very proud of the work I have been doing when I last visited with him. He lamented his own choice of not pursuing a higher academic degree for himself. His professional work involved designing and managing very complex systems. He liked video games, music, and poetry as hobbies. He spent time thinking very deeply about people and how the world works. He pursued knowledge and argued vehemently against "belief". He strongly felt that what was real, what was verifiable, was most important. He was a good father, but maybe not so good of a husband or boyfriend. Many people who knew him thought he was a genius. He was my brother and I'm having a hard time dealing with his passing.

I've gotten past the shock. I've gotten past the sporadic moments of denial. I've even gotten past the moments of anger. I never really went through a bargaining stage. Now, I mostly just feel old. I think this is a mix of depression and acceptance.

I don't believe in a soul or an afterlife and neither did he. Attempts to comfort me by saying, "he's in a better place", in any form or variation are misplaced. Such efforts will anger me, even if not obviously so. If I know you, they will discourage me from interacting with you in the future. If I don't know you, I'll just delete your comment and maybe ban you.

I'm in the very final stages of completing my PhD in the department of Genetics at the University of Minnesota. By the time I post this, I'll have handed off my written thesis to my committee for review. In another two weeks, I'll defend my thesis and be done with it.

I'm sad that my brother won't get to know.

Monday, November 17, 2014

What is a chicken?


We refer to them by the species name Gallus gallus domesticus, but there was a time before they had any connection to us. The wild species is Gallus gallus, also known as the Red Jungle Fowl, and it can still be found running around the wilds of south-east Asia.

There is genetic evidence that modern chickens arose from multiple independent domestication events. The diversity of alleles found in domestic chickens encompasses those found in wild populations of G. gallus spread through India (G. g. murghi), Burma (G. g. spadiceus), and Tailand (G. g. gallus). This is best explained by the early incorporation of Red Jungle Fowl from different regions into the common pool of chickens being cared for by people.

It turns out that there are three other related species of jungle fowl (grey, Ceylon, and green) roaming the area of south-east Asia. A trait found in domesticated chickens that causes yellow skin on the legs and feet is due to an allele which shows most similarity to an allele found in the Grey Jungle Fowl.

A. Green stars indicate putative domestications.
B. Domesticated chicken.
C. Red Jungle Fowl. (Range in red in A.)
D. Grey Jungle Fowl. (Range in grey in A.)
At least four different populations across two (of what we consider) separate species contributed to modern domesticated chickens.

How could the process of domestication start in multiple places at the same time? Well... it can't, but it can happen close enough in time to be indistinguishable to modern researchers.

It is a common pattern in domestication for the idea of domesticating an animal or plant to spread faster than the newly domesticated organism can spread. This results in multiple independent domestication of a single species, or of similar species, found across a wide area.

Cattle appear to have been domesticated two or three times (from Bos tauros, B. indicus, and possibly B. africanus). Sheep and goats appear quite distinct to us now, but when they were domesticated, they were very similar creatures.

Chile peppers have been domesticated at least five times (Capsicum annum, C. chinense, C. frutsecens, C. bacatum, C. pubescens). Squash were domesticated at least five times (Curcurbita pepo, C. moschata, C. maxima, C. mixta, C. ficifolia). Carrots (Daucus carota), parsnips (Pastinaca sativa), celery (Apium graveolens), parsley (Petroselinum crispum), Dill (Anethum graveolens), and chervil (Anthriscus cerefolium) all belong to the family Apiaceae and look very similar in their wild state.

So.  What is a chicken?

It is an example of how the rapid spread of ideas through human culture impacts the process of wild things becoming integral to our civilization.



References
  1. http://en.wikipedia.org/wiki/Red_junglefowl
  2. Multiple domestication : http://www.biomedcentral.com/1471-2148/8/174
  3. Hybrid between red and grey jungle fowl : http://www.plosgenetics.org/article/info%3Adoi%2F10.1371%2Fjournal.pgen.1000010 
  4. Cattle : http://archaeology.about.com/od/domestications/qt/cattle.htm
  5. Chile peppers : http://archaeology.about.com/od/cbthroughch/qt/Chili-Peppers.htm
  6. Squash : http://en.wikipedia.org/wiki/List_of_gourds_and_squashes
  7. Apiaceae : http://science.jrank.org/pages/1240/Carrot-Family-Apiaceae-Edible-species-in-carrot-family.html

Tuesday, November 11, 2014

Evolution

While thinking about the evolvability of different artificial life simulations, as discussed some in my last posting, I realized that it would be helpful to talk about what is required for a system to evolve. It comes down to four basic traits.

1. Reproduction: Some unit in the system has to reproduce. This unit could be bacterial cells in your gut, or it could be numerical representations in a computer. (Even fire can be described as reproducing when it spreads through a house or forest.)

2. Inheritance: During reproduction, each new unit in the system has to gain traits from its parent(s). The traits could be hidden, as in recessive alleles, or it could be obvious, as in dominant alleles. The number of parents can be one or more than one. (We have two, but maybe some aliens have three or more.)

3. Mutation: At some point in the reproductive cycle, there has to be the potential for changes in the traits (mutations) that are inherited.

4. Death: Death is generally required to remove individuals from a population, thus freeing up room for the next generation. However, there are scenarios where death isn't required. If the population is continuously expanding into new territory, the front-line sub-population can evolve over time without individual death. In this case, the older organisms being left behind fills the same role of actual death.



It is relatively easy to prove mathematically that a system with these four traits will experience evolution.

Lets give it a go in a simulation that has a maximum population of four organisms represented by letters and driven by the following rules.
  1. Reproduction with inheritance: A -> AA; B -> BB
    • A or B can duplicate.
  2. Mutation: A -> B.
    • A can mutate into B.
  3. Death: A -> A 
    • Only A can die.
We start the simulation with "A" .

"A" -> "AA" -> "AAAA" -> "AAAB" -> "AAAB" -> "AB" -> "AABB" -> "AABB" -> "ABB" -> "ABBB" -> "ABBB" -> "BBBB"

This may not look like the sort of math you are familiar with, but it is math nonetheless. Math is the manipulation of abstract symbols that represent precise concepts with the extremely rigid rules of logic. 2+2 always equals 4. A system with the described traits will always experience evolution.

Now, this little toy system I've described has an extremely low evolvability. The starting state of the system ("A") does meet the four requirements and thus evolves. However, once the system has reached the final state ("BBBB"), it no longer meets the four requirements and thus cannot evolve further.



If you argue that life doesn't evolve, then you are logically arguing that life does not meet one of the four requirements discussed above. Unequivocally, life meets the four requirements.

Life evolves. The math doesn't provide any other possibility.

Wednesday, November 5, 2014

Artificial Life

Way back in 1989, I read an article in Scientific American ("Mathematical Recreations") which described a simulation which showed evolution of a very simple "bug".

The genome of the bugs consisted of six numbers that weighted a random selection of which direction each bug would go in the next time step (forward, left, hard-left, backward, hard-right, and right). The bugs would eat "bacteria" which rained down around them, reproducing if they ate enough, but starving to death if they didn't.

Along the way, bugs would evolve different strategies depending on the environment they found themselves in. If they found themselves in a highly food-rich environment, it was more advantageous for them to stay in the same place. If they found themselves in food-poor environments, it was more advantageous for them to keep moving whatever direction they were going.

I thought this was a really cool idea and set about writing my own version of the program. My bugs had a genome of eight numbers, but otherwise they were identical to the original. I worked on the project periodically over several years, eventually producing the version seen at left in 2006. Thousands of bugs (green) cruising around a screen full of bacteria/food (blue) that rains evenly over the screen (at a higher rate in gardens) can be seen at left.

In the later versions of the program, I took some effort at visualizing how the bugs were evolving. At first, I simply plotted the population size over time and compared the resulting curves across different runs of the program. I figured out that the system would consistently support a higher number of bugs if they were allowed to mutate instead of just replicate. The bugs reached population densities about 10% higher than in the no-mutation control.

I then realized I could convert the genome into a 2D coordinate, describing the propensity of each bug to go in each direction. This would let me observe the behavior of the population at a glance.


The video at right starts with the population of genomes clustered around the center. They have no initial tendency to go any where and walk around randomly.

The distribution soon widens to the right and left, as many of the bugs replicating in the gardens are turning in tight circles.

By about 0:02, the population has split into three groups. The left and right have moved downwards as the garden-bugs are increasingly turning around during each time step, going nowhere at all. The center region has started to move upward, showing the specialization of the wide-open-bugs for moving forward at increased rates.

 The three populations continue moving through the remainder of the video. The wide-open-bugs experience large periodic population cycles as they deplete their food source and die back, while the garden-bugs with their much more rich food source show a more constant population over time.



The video above shows a version where mutations are introduced every time a bug divides. If mutations are introduced only when a bug is starving, which potentially allows a single bug to mutate into a better strategy and so keep on living, a much tighter population distribution results. In the figure at left, three separate runs have been overlaid (in red, green, and blue, respectively). In these simulations, six gardens were available, but only a limited number were colonized. Each colony is represented by a single cluster of genomes in the lower half of the image.



Being able to experiment with evolutionary concepts on my computer helped me learn about biology and represented a stepping stone on my way to my current approaches to understanding biology. This system is limited, it can only evolve to a few end points.

Other systems can evolve in a much more complicated way. In Tierra, the evolution of short computer programs develops into a rich ecosystem of interacting organisms. Parasites evolve, followed by hosts that are resistant to those parasites.

The term "evolvability" is used to describe these differences in how different systems can evolve. Our biology has a very high evolvability, while my little simulation has a very low evolvability. What allows a system to have a higher evolvability seems to be related to how complicated the interaction of an "organism" is to its environment, as well as how complicated its genome is.

My bugs can only interact with the density of food and their genome is eight numbers. The bacteria living in my gut can interact with me, my food, other bacteria, radiation from the sun, etc. and their genome can grow or shrink as needed. Simulations with higher evolvability invariably show more of the features that we see in living things and so are more useful/interesting for studying real living things

Systems that show any evolvability at all are interesting and included in the subject of "artificial life". Hypothetically, you could be a "biologist" and never look at a messy living thing. I like studying the messy living things too.  ;-)



Links:
  1. http://martin-gardner.org/MGSAindex.html
  2. http://lifesciassoc.home.pipeline.com/instruct/evolution/
  3. Tierra :
  4. Critters :
  5. https://www.youtube.com/watch?v=ZpW_ojpmTWk
  6. Polyworld : https://www.youtube.com/watch?v=_m97_kL4ox0

Tuesday, October 28, 2014

Sunflower crosses.

Last year I crossed the perennial (tuber-forming) sunflower Helianthus tuberosus (image #1) to an annual sunflower H. annuus "Russian Mammoth". I used the much larger, 1ft wide, flower of "Russian Mammoth" (image #2) to pollinate as many of the tiny H. tuberosus flowers as I could.

2. H. annuus "Russian Mammoth".
1. H. tuberosus & seeds.
At the end of the season, I collected ~70 seeds from the H. tuberosus seed heads. Many seed heads had already been destroyed by the local birds which resulted in some scattered seed.

Squirrels got to the seed heads of the "Russian Mammoth" (image #2), because I later put them out in the sun to dry. As a result, I don't have the many more seeds of this variety I was expecting to have. Fortunately, the variety has been available since roughly 1870 and should be easy to find more seeds for.

3. Giant hybrid.
I didn't plant any of the seeds I collected from the H. tuberosus plant, since I would be moving before the plants had matured. However, three of the seeds that the birds had scattered managed to grow up out of the weed patch. Of these, two were obviously hybrids (they had traits found separately in both parents). One of the hybrid plants has a thin stem and flopped over (yellow flowers at lower-left in image #3), even with my efforts to keep it upright. The second hybrid has a robust stem that has let it withstand all the wind and rain of this season. So far, this plant is pretty much exactly what I was hoping the F1 hybrid plant would be.

I finally got a picture of me (6'4") standing next to the hybrids yesterday. Both hybrid plants are still green and thriving, even though the H. tuberosus plants have all shut down for winter. Once we get a killing freeze, I'll cut down the plants and dig up any tubers they've produced.

With luck I'll be able to collect some F2 seed off these plants, but since I no longer live where the plants are, I'm expecting the birds to get to them before I do. As the F1s are supposed to produce tubers generally, I should be able to regrow these plants next year from the tubers they are now likely to be producing.



4. en.wikipedia.org/wiki/Perennial_sunflower
The sunflower genus (Helianthus) contains a wide range of species. Some species are difficult to cross, while others will cross readily. Image #4 illustrates the use of hexaploid species to break down reproductive barriers between annual and perennial diploid species (at left and right). Crossing the tetraploid hybrids to either parent type results in uneven chromosome sets and high rates of infertility due to aneuploidy. The tetraploids can easily cross, however, allowing genes from diverse sources to be recombined in their progeny.



5. www.edenbrothers.com
The common sunflower (H. annuus) has been bred to produce a range of colors in addition to the yellow of wild sunflowers (such as those in image #5). The genes for these color changes could be added to a perennial sunflower using the same method I'm using to add traits for giant growth. (Someone else has this project under way.)

Because of the differences in ploidy between the annual sunflowers from the commonly available perennial (H. tuberosus), it would likely be in the F3 generation or later before such rich colors could be regained. This is discussed in the link below.

Wednesday, October 15, 2014

Making a new "Blue" tomato


1. Tomato "Indigo Rose".
"Blue" is the color label applied to the new breed of anthocyanin rich tomatoes. "Indigo Rose" (image #1 at left) is the first officially available variety with the trait. The variety was bred at OSU, using two genes from wild relatives of tomatoes. The atroviolaceum ('atv') gene was introgressed from Solanum cheesemanii. The anthocyanin fruit ('Aft') gene is a transcription factor introgressed from S. chilense. The two genes combine to result in a tomato with dark purple anthocyanin pigment production when exposed to sunlight.

The high-anthocyanin traits managed to escape from the OSU breeding program before the official release, under the names "OSU Blue" or "P20". This variety was not yet stable and didn't taste very good to most people, but it did successfully introduce tomato breeders to the interesting traits a few years early. Breeders quickly took to trying to incorporate anthocyanin expression into better tasting types of tomatoes.



2. F2 tomatoes, showing pigment on fruit and calyx.
I've been growing a miniature tomato variety called "Tiny Tim" for the last several years. I saved a batch of open-pollinated seeds two years ago, as my previous batch was running out. Last year, one of the seedlings turned out to grow much faster and larger than all the others. It was the result of a cross to one of the other tomatoes growing the previous year. I grew several F2s this year, allowing me to identify the other parent as a "Roma" tomato.

Among the F2s, I noted a range of anthocyanin phenotypes in the fruit and leaves/stems. The anthocyanin pigment produced on the fruit when sun-exposed came in three levels (none, middle, and high in image #2.)

3. F2s (top two) & "Indigo Rose" (bottom).
The anthocyanin pigment produced in the calyxes also came in three levels (high, none, and middle in image #2), but independent from the fruit pigment. The pigment produced in the rest of the plant wasn't as obvious. The no-pigment plants were entirely green. the medium-pigment plants had the anthocyanin highlights on the calyxes and leaf edges. The high-pigment plants showed increased pigment over the entire plant where sunlight hit, at a level about half of that seen in "Indigo Rose" tomato plants (image #3).

4. Original; color-enhanced; postureized.
The high-pigment plants also appeared more of a red/brown color rather than the purple of "Indigo Rose" plants. Image #4 shows a section of the image #3 after using the color-enhance filter in GIMP (center) and then the posterize filter in GIMP (right). The enhanced images more clearly convey the difference in color which is visually seen on examining the plants. This either indicates a different mix of anthocyanin pigments, or is a visual artifact caused by the blending of green chlorophyll with the anthocyanin purple. I would need to do some chromatography or micro-dissection experiments to discriminate between these possibilities.



5. Anthocyanins on unripe "Tiny Tim".
The level of pigment in the F2s was a surprise as I hadn't noted any anthocyanin expression in the parent varieties. After seeing the F2s, I re-examined some "Tiny Tim" plants grown this year and found they did have anthocyanin expression. The fruit show a level of anthocyanin production comparable to the high-fruit-pigment F2s (image #5), but the small size of the fruit made it hard to notice. The F1 showed a pigment level like the medium-fruit-pigment F2s, suggesting it is a single trait with partial dominance. The color of the calyx and leaf/stem was comparable to the F2 plants with the middle level of pigment for each feature. Looking into the lineage of "Tiny Tim" suggests the middle-pigment trait was contributed from S. pimpinellifolium, used in the breeding to contribute small fruit size to "Tiny Tim". As anthocyanin pigments on the shoulder is common in many wild tomato relatives, I suspect the fruit trait also came from S. pimpinellifolium.

"Tiny Tim" is an open-pollenated variety, so it should be homozygous for any alleles impacting pigment production. The increased calyx/leaf/stem pigment intensity in the F2s over what is seen in "Tiny Tim" suggests the involvement of a second gene from the "Roma" parent that enhances the expression of the first gene. This second gene would have been hidden in "Roma" because that variety doesn't have any anthocyanin pigment production.



What are the expected genetics for this cross?

The fruit pigment appears driven by one gene. Under the model of partial dominance, the cross ...

1tt1tt x 1R1R

… produces an F1 …

1tt1R

… that shows a low level of anthocyanins in the fruit. Low amounts of anthocyanin pigment was noted in the fruit of the real F1. Selfing the F1 produces F2s …

1tt
1R
1tt 
1tt1tt
1tt1R
1R
1tt1R
1R1R

… where 1/4 have high-pigment on the fruit (1tt1tt) and another 1/2 have low pigment on the fruit (1tt1R). I only grew 10 F2s this year, so it is hard to estimate real ratios, but all three color classes were observed.


The calyx/leaf/stem pigment appears to involve two genes. If we assume both involved alleles are recessive, the cross …

2tt2tt3TT3TT x 2R2R3r3r

… produces an F1 …

2tt2R3TT3r

… that shows no anthocyanins in the calyx/leaf/stem. No anthocyanin pigments were observed in the calyx/leaf/stem of the real F1. Selfing the F1 produces F2s …

2tt3TT
2tt3r
2R3TT
2R3r
2tt3TT 
2tt2tt3TT3TT2tt2tt3TT3r2tt2R3TT3TT2tt2R3TT3r
2tt3r
2tt2tt3TT3r2tt2tt3r3r2tt2R3TT3r2tt2R3r3r
2R3TT
2tt2R3TT3TT2tt2R3TT3r2R2R3TT3TT2R2R3TT3r
2R3r
2tt2R3TT3r2tt2R3r3r2R2R3TT3r2R2R3r3r

… where 1/16 are expected to express the recessive alleles from both parents and thus show the high-pigment trait. Another 3/16 are expected to express the recessive allele from "Tiny Tim" and show the medium-pigment trait. The remaining 12/16 should only have green chlorophyll evident in the unripe fruit. This year I grew 10 F2s and only one shows the high-pigment trait. 1/10 approximates 1/16 reasonably well for the numbers I grew. Some, but not all showed the middle-pigment trait. I didn't note exactly how many F2s showed the middle-pigment trait and they've begun dying back from the cold, so I will have to screen more F2s next year at an earlier stage to better estimate the true ratios of the different color classes.



The dark pigment of "Indigo Rose" fruit is due to the interaction of two traits, the anthocyanin fruit ('Aft') trait combined with the atroviolaceum ('atv') trait. The 'Aft' trait by itself only produces a small amount of pigment on the fruit shoulder. The 'atv' trait by itself only produces dark pigment on the calyx/leaf/stem of the plant.

If the fruit pigment in the F2s is driven by a single gene, as it appears, and two genes are responsible for the calyx/leaf/stem pigment, then 1/64 of the F2s will contain both high-anthocyanin traits.



6. Derived from S. hirsutum.

There are several anthocyanin traits floating around that have been introgressed from different wild tomato relatives.
  1. S. cheesemanii
    • "atv" gene: pigment throughout plant. Seen in variety "Indigo Rose" (image #1).
  2. S. chilense
    • "Aft" gene: pigment on fruit shoulder. Seen in variety "Indigo Rose". (image #1)
  3. S. hirsutum
    • [unnamed] gene: pigment on fruit shoulder, similar to "Aft". Described at maprc.blogspot.com. (image #6)
  4. S. peruvianum

  5. 7. Derived from S. peruvianum.
    • [unnamed] gene: pigment on fruit shoulder, similar to "Aft". Seen in variety "Purple Smudge". (image #7)
  6. S. pimpinellifolium
    • gene #1: pigment on fruit shoulder, similar to "Aft". Described here.
    • gene #2: pigment throughout plant, similar to "atv". Described here.
  7. Conventional tomatoes
The four fruit pigment traits and the two plant pigment traits seem to behave similar to the others in each category. Because the species are so closely related, the traits may represent different alleles of the same genes. If so, combinations of a trait from each category (like the "atv" and "Aft" in "Indigo Rose") should result in a strong increase in the total pigment produced relative to either trait alone, especially when the modifier trait (gene #3) is also present.

I've isolated a line that appears homozygous for gene #1 and one that appears homozygous for both genes #2 and #3. Unfortunately, crossing these two lines would simply recreate the F1 (heterozygous for all three traits) rather than help me generate a triple-homozyous line.

Comparing the lightly-pigmented fruit in images #6 and #7 to my pigmented F2s suggests they are showing a different mix of anthocyanins from the other "blue" lineages. I look forward to finding one of the rare segregants which contains all three genes, so I can find out!



References:
  1. "Indigo Rose" tomato: http://extension.oregonstate.edu/gardening/purple-tomato-debuts-indigo-rose
  2. 'Aft' gene: http://www.esalq.usp.br/tomato/Aft.pdf
  3. 'atv' gene: http://www.esalq.usp.br/tomato/atv.pdf
  4. Escape of "P20" tomato : http://www.tomatoville.com/showthread.php?t=16989
  5. "Tiny Tim" tomato: http://tatianastomatobase.com/wiki/Tiny_Tim
  6. "Roma" tomato: http://tatianastomatobase.com/wiki/Roma
  7. "Orange Smudge" tomato: http://tatianastomatobase.com/wiki/Purple_Smudge