Lutherburbanking

Luther Burbank made lots of plant crosses and hybrids. The russet potato? That was him. Shasta daisy? Burbank. And he made many more plants which you probably haven't heard of. He made a white black berry. He made a cactus without needles for use as animal feed. He was a cool guy. He didn't make the first plum-apricot hybrid, but he made lots of good ones and had a big involvement in bringing them to market. There are reportedly 113 named varieties of plums due to his experimentation with Asian plum varieties that he imported to the U.S.

From "Plant Breeding Giants" (Crow, 2001):

His successes depended on judicious employment of several techniques: he selected the best seedlings from large numbers of plants, he imported promising strains from around the world, he made crosses between distantly related varieties and even species, he exploited skillful grafting, and he astutely utilized vegetative propagation of superior recombinants, thereby preserving their genotypes. Probably his greatest contribution to science was discovering nonsegregating, true-breeding hybrids, such as from a cross between a raspberry and blackberry, that were later understood to be amphidiploids. He pioneered in regarding these as another mode of species formation.

What's stopping me or you from Lutherburbanking? Not much that I can see. "Noting but devotion", says my wise friend Feast. He did have financial support from Andrew Carnegie and other rich people, but he didn't start out that way. To start out, you need some related but non-identical plants and the patience to cross them. Maybe it would help to have an idea of what plants to cross? That's what this post is for.

You might have learned in school that "different species can't produce fertile offspring". That's bullshit. Nature doesn't care about our categories, and our categories don't reflect nature very well. Interspecific and intergeneric hybrids are quite common. Interfamilial and even interordinal hybrids are rare but documented.

So, what families of plants might we want to form hybrids from?

Here are some plant families that each include multiple edible species (leaves, seeds, or fruit): 

* Monocots:

Poaceae: bamboo, corn, wheat, rice, barley, millet, oats, rye, sorghum, sugarcane

* Caryophyllales:

Amaranthaceae: amaranth, spinach, beet, quinoa
Cactaceae: prickly pear, saguaro, agave

* Asterids:

Apiaceae: parsley: anise, caraway, carrot, celery, cilantro, cumin, dill
Asteraceae: lettuce, sunflower, artichoke, dandelion, chrysanthemum
Convolvulaceae: sweet potato, water spinach
Ericaceae: blueberry, huckleberry, cranberry
Lamiaceae: basil, heal-all, hyssop, lavender, marjoram, mint, oregano, perilla, rosemary, sage, salvia, savory, thyme

* Rosids:

Fabaceae: string bean, lentil, pea, alfalfa, clover, 
Malvaceae: cacao, kola nut, okra, durian, hibiscus, baobab
Myrtaceae: myrtle, clove, guava, allspice, eucalyptus
Rosaceae: apple, pear, quince, raspberry, strawberry, almond, hawthorn

.

Maybe I should focus on genera with multiple distinct species, but I'm just getting started and documenting my journey as I go.

Some things that have already been done, in the way of fertile edible plant hybrids: Triticale is a hybrid of wheat and rye (different genera, same family). Already mentioned, pluots/plumcots are hybrids of plums and apricots, which are different species in the same genus (Prunus). Basically every citrus you've ever heard of is a hybrid of at least two Citrus species, and in 2013, (Smith, Gultzow, and Newman) established a hybrid between Citrus wakonai and Citropsis gabunensis, previously thought sexually incompatible. Loganberries are a hybrid of the American blackberry with the European raspberry, which are both in the genus Rubus. Rubus is a difficult genus to separate into species to begin with:

Rubus is very complex, particularly within the blackberry/dewberry subgenus, with polyploidy, hybridization, and facultative apomixis apparently all frequently occurring, making species classification of the great variation in the subgenus one of the grand challenges of systematic botany.

I used to think this meant that "Rubus species" and "Rubus hybrids" weren't all that meaningful as categories. And then I learned that Luther Burbank made white blackberries! Who cares if the genetic categories are crisp! You can still get amazing phenotype differentiation.

The most commercially used mint, peppermint, is a hybrid of two Mentha species, spearmint and water mint, but apparently doesn't belong on this list: peppermint is sterile and spreads vegetatively by producing runners. Who knew! 

In 2018, some folks hybridized white rice, Oryza sativa, with a cutgrass from Madagascar, Leersia perrieri. And this will prove to be an illustrative example: they didn't just transfer pollen. They had to do embryo rescue! You take an embryo or an entire ovule from the pollinated plant and put it in a nutrient medium. A solution of Knop's mineral salts (KNO3, Ca(NO3)2, MgSO4•7H2O, KH2PO4) and sucrose is a common old-timey recipe that still works pretty well. The Murashige and Skoog medium, MS0, is a big step up in complexity, but also good; it includes may more trace elements and some vitamins and hormones and stuff. Every modern bio lab that does embryo rescue probably has their own recipes, and they often vary the nutrients with embryo growth phase. Anyway, they did something like that in 2018, and now they've got plants that can be rehybridized with white rice in order to introduce some of the good properties of the Madagascar cutgrass into rice. Genius! So sometimes plant hybridization takes more than devotion. Sometimes a high degree of genetic incompatibility means you have to nurse the first generations of offspring with chemicals. But we could also just skip that. If you're not up to culturing a plant embryo in sugar-mineral water, try making new plants the normal way. It worked for Luther Burbank.

There are lots of reasons why sexually compatible plants might not hybridize in nature: they could be geographically isolated, they could flower at different times, they might not be visited by the same pollinators. Sometimes just bringing plants together in one garden is enough to make new hybrids, although if you want to make an art of it, then manual pollination in a greenhouse is a good way to go.

Cotton (Gossypium hirsutum) occasionally forms a fertile hybrid with hibiscus (Hibiscus panduriformis), and these are in the same family, Malvaceae. One in two-thousand pollinations works and the offspring produces fewer seeds than either parent, and I don't even know if any part of the hybrid is usable as food, as hibiscus is, but it's encouraging! Even if the hybrid is useless, we might be able to re-hybridize it with hibiscus. Or other Malvaceae hybrids might be possible. That same link reports on an attempted hybridization of the cacao tree (Theobroma cacao) and Mountain cocoa (Herrania mariae) (both in Malvaceae), which produced fruit but not viable seeds. But maybe we just need to roll the dice 2,000 more times and then we can get new chocolates. I hear that there's some commercial demand for chocolate.

I've got a suspicion that plants which can double their genomes (auto-polyploids) are good candidates for forming vigorous hybrids containing a complete set of genes from both parents (allo-polyploids). That's something to look into. For example, hexaploid bread wheat (Triticum aestivum) (which you might also know as all-purpose flour) is an inter-specific allo-polyploid hybridization of tetraploid durum wheat (Triticum durum) (which we use for pasta) and the diploid Tausch's goatgrass (Aegilops tauschii). The durum might also be an allo-polyploid? I don't know much about it. Canola (Brassica napus) is an allo-polyploid hybridization of Brassica rapa (turnips, et cetera) and Brassica oleracea (broccoli, et cetera). There are probably others.

What about Lamiaceae? Most of the cooking herbs are in Lamiaceae. It would be cool to get some new ones. We could also just try using some already existing Lamiaceae plants in the kitchen that we haven't used before, but that's not the topic of this post. We've already seen that Mentha species can cross. What about basil with heal-all or perilla with savory? If basil could be hybridized with rosemary, the Mediterraneans probably would have found out hundreds or thousands of years ago, but how about trying some geographically isolated species?

My guess is it still won't work. Basically anyone who has put together an herb garden has collected Lamiaceae plants that were once geographically diverse and we just don't get cool hybrids that way. But if you've already got an herb garden, why not try lutherburbanking it? Spread some pollen around, see what happens.

One kind of hybrid that people do commonly get in their gardens is with squashes/gourds in the genus Cucurbita. Cucurbita maxima and Cucurbita moschata often form seedless hybrids when grown together. Also Cucurbita pepo has a lot of morphological variation between cultivars and you can get some crazy shapes when they cross. I once had a yellow C. pepo in the shape of a trumpet mute with knuckles, which I guess is like halfway between summer squash and pattypan, but the seeds were supposed to be zucchini. That's all within one species - nothing too crazy genetically, but you can get some cool things if you're into knuckled trumpet mute squashes.

Apparently modern sugar cane varieties, genus Saccharum, are often a mix of up to three wild Saccharum species, and there have also been successful hybridizations and back-crosses of Saccharum with corn (Zea mays), with sorghum (Sorghum bicolor), with some wild broomsedge grass (Erianthus), and with some Bamboo (Bambusa), and with cogon grass (Imperata cylindrica). Some of these apparently requires many thousands of attempts to get a single hybrid.

In 1995, (Li, Liu, and Luo) made a hybrid of Brassica napus and Chinese violet cress (both Brassicaceae), and it's crazy? The genomes separate during mitosis but you can keep getting hybrid plants out?:

From the selfed progeny of the hybrid, mainly two kinds of plants, B. napus and the hybrid, were found. The hybrid plants of the selfed progeny again produced two kinds of plants, B. napus and the hybrid.

How about plants in the heather family, Ericaceae? The lingonberry (Vaccinium vitis-idaea) sometimes forms a hybrid in nature with the European blueberry (Vaccinium myrtillus). It was first discovered by Ruthe in the late 1800s and bears the name (Vaccinium intermedium Ruthe). I don't yet know whether it is fertile. The Andean blueberry (Vaccinium meridionale) has been crossed with the lingonberry and the hybrid displayed fertility in backcrosses with both parents. The team of  (Vorsa, Johnson-Cicalese, Polashock)... made a vigorous blueberry x cranberry hybrid, but they weren't sure if it was fertile at the time of publishing. Before we talk about it, let's go over cranberry genetics. The large American cranberry used for juice and sauce is Vaccinium macrocarpon. There are three other species of Cranberry in the U.S. of less commercial importance

1) The southern mountain cranberry (Vaccinium erythrocarpum), which is native to the southeast US and also China+Japan+Korea, weirdly. The other three U.S. species are more closely related to each other and this one is more distant. I don't know much about it

2) The northern cranberry (Vaccinium oxycoccos). More cold tolerant than the American cranberry, smaller fruits, often polyploid.

3) The small cranberry (Vaccinium microcarpum). Micro-carpum versus macro-carpon - It's annoyingly close, I know. V. microcarpum is commonly included as in V. oxycoccos, but I've been convinced - by "Pacbio Sequencing Reveals Identical Organelle Genomes between American Cranberry and a Wild Relative" (Diaz-Garcia et al., 2019) - that it shouldn't be. Microcarpum is diploid, probably branched off from the diploid macrocarpon, and oxycoccos is a polyploid hybrid of microcarpum and macrocarpon. Anyway, this one is also small and also cold tolerant.

Okay, back to the blueberry x cranberry hybrid paper. Vorsa and friends first crossed the Florida evergreen blueberry (Vaccinium darrowii) with "diploid small-fruited cranberry, V. oxycoccos". Diaz-Garcia et al. teach us that diploid oxycoccos is more properly called V. microcarpum. The result of that crossing didn't produce a vigorous hybrid. It was weak, they said. If you tried to make cran-blueberry sauce from it, you'd get weaksauce. Then Vorsa and friends crossed the same Florida blueberry with a cross of V. oxycoccos and V. macrocarpon and they got a plant that lived! They didn't know at the time of publishing whether it was fertile, but they got a vigorous hybrid. But, honestly, the oxycoccos x macrocarpon hybrid is almost as interesting to me as the blueberry x cranberry hybrid. There isn't a lot of literature that I've seen on intentionally crossing cranberry species. I think that's more because I'm doing a poor job searching, but still - if we could get a more cold tolerant large cranberry by crossing, that would be amazing. 

For cold tolerance in the heath family, Ericaceae, I'd previously been looking with hungry eyes at a plant called winter heath (Erica carnea), whose flowers look a lot like cranberry flowers. It springs up in the mountain snow in the Alps, with a interesting pink and black coloration. I was worried we'd have to figure out an Erica x Vaccinium cross to get large cold tolerant cranberries, but maybe we can just cross cranberries with cranberries! That sounds easier. Or cross them with lingonberries or blueberries or whatever. Do it all. Make new crops.

The highbush blueberry (Vaccinium corymbosum) has been crossed with the deerberry (Vaccinium stamineum), and then backcrossed into the blueberry with an aim of making more drought-resistant blueberries.

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My wise friend Feast asked me what plants I actually want to improve in which ways by hybridization and/or selection from natural variation. I have a standing interest in making plants more cold tolerant. This is mostly due to my interest in space colonization and slightly due to living in a moderately cold climate with a moderately short growing season. Dwarf varieties of normally large-growing plants, like grains, are also desirable for use in early space botany.

Another interest of mine is improving the proteinogenic amino acid profile of grains. There are cultivars of corn in use in some countries, so called QPMs or Quality Protein Maizes, that are complete proteins. I'd like to play a role in similarly improving wheat and rice as protein sources. Also, buckwheat already exists and is delicious and are complete protein. The world should use more buckwheat, and if there's any good reason why we're not, then let's Lutherburbank up a solution to that.

You might have heard that Brussels sprouts used to be more bitter and have been made milder over the last ~30 years by the efforts of people including Hans van Doorn. Well, I don't particularly like non-pickled cucumbers; what if we used our ingenuity to make them into palatable food too?

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Integer Relation Algorithms

Last year, I investigated polynomial continued fractions at some length. I found some pretty cool relationships; if you wanted to inscribe them on my grave, I wouldn't be upset - not that I'm planning to die. 

For many continued fractions that seem to converge to real values, I was not able to find separate finite closed-form expressions, but I'm ready to try again. My technique in the last post was to search through coefficients of Möbius transformations of famous mathematical constants for coincidences with convergent values of continued fractions. My technique in this post will be... Integer Relation Algorithms!

One of the simplest and oldest ones is called the Lenstra-Lenstra-Lovász algorithm, or LLL. Another famous one is the PSLQ algorithm, which seems better, but I'm going to start with LLL. Apparently, it's basically a combination of the Euclidean algorithm for finding greatest common divisors with the Gram-Schmidt orthogonalization process. I like both of those things! It seems that you can take a given real-valued constant, and construct from it a matrix that represents some unknown polynomial, perform the LLL algorithm on the matrix, and sometimes you get out integer coefficients for a polynomial which has the given constant as a root! Amazing! An algebraic number explainer. And there are some generalizations for finding expressions for a given constant where the polynomials have non-integer coefficients or even non-algebraic coefficients. That's what this post will be about. Starting with the integer coefficients.

...

Hm. I tried to express six mysterious constants from the PCF post as roots of quadratic polynomials with integer coefficients without luck. Let's do an example. Our constant r will be T3 from the PCF post, 1.1263572396234227708. We set up the following matrix for a quadratic polynomial:

[1, 0, 0, 10000 * (r ** 2)],
[0, 1, 0, 10000 * r],
[0, 0, 1, 10000],

and run LLL reduction on it. The first vector of the resulting basis will have four components. If the fourth component is nearly zero (or at least much smaller in magnitude than the other components?), then the first three components are likely our polynomial coefficients, defined up to a sign change.

But the first vector in the LLL reduced basis with that {r} is

[8, -17, 9, 13],

, and the corresponding quadratic polynomial

y = 8x^2 - 17x + 9

has a root nearby {r} at 

x = 9/8 = 1.125

, but not at {r}. Sad.

Let's try higher degree polynomials!

Let's start with a sanity check. The polynomial x^3 - 2x^2 + x + 2, which I just made up, has a single real-valued root at ~ -0.6956207695598. Let's see if we can recover the coefficients from the root.

Here's the matrix:

[1, 0, 0, 0, 10000 * (r ** 3)],
[0, 1, 0, 0, 10000 * (r ** 2)],
[0, 0, 1, 0, 10000 * r],
[0, 0, 0, 1, 10000],

.  I run LLL reduction on that and the first vector in the new basis that I get out is:

[1, -2, 1, 2, 0]

which has the desired coefficients. Radical. But it didn't work for T3 or any of the other constants I was testing.

I tried fourth degree polynomials and I got a match, but I'm pretty sure it's a false positive. The number {2 / (sqrt(pi) * e * erfc(1))} is pretty close to a root of 

y = 3x^4 - 5x^3 - 4x^2 - 9x - 2

, in particular the erf thing is about

2.6389674131942744

and the root is about

2.6389675142347912

.

The constant T1, {0.4084843294696858}, from the PCF post is moderately close to a root of 

x^4 - 2x^3 + 2x^2 - 3x + 1

and the constant T2, {0.8120409412226914}, is close to a root of 

x^5 - x^4 - 2x^3 - 1x^2 + x + 1

.

I've suddenly lost hope in this project. In the polynomial continued fraction post, the constants were all rational, quadratic, or non-algebraic. There weren't any third roots, for example. My old techniques were good enough to find quadratic constants, and while this technique might find third roots and higher, I doubt that PCFs produce third roots.

Also, I think there's something slightly wrong with my LLL implementation: when I get the coefficeints out, my coefficients match other sources, but the last number in my vector is always an integer and other sources have non-integer entries. So.... I don't think fixing it would change anything substantive above - clearly my code can find polynomials with nearby roots, but I'm a little sad that my code doesn't exactly match my references.

Might come back to this later with a generalization of LLL.

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