Monday, August 25, 2014

Mathematical Recreations : Gravity of a Black Hole

Part of being a good scientist - in any discipline - is learning how to recognize the unconscious assumptions that others are making in their reasoning and then setting about examining those assumptions. I recently came across a video describing the gravity at the center of a black hole as infinite.

This didn't match my intuition, so I spent some time thinking about what might be the assumptions behind it.



My first step was to back away from the extreme case of a black hole and look at something we know more about, Earth. The force of gravity at the surface of the Earth is described by the formula...

\( g = -\frac{GM}{r^2} \).

where M is the mass of the Earth (5.97219x1024 kg), r is the radius of the Earth (6,378.1 km), and G is the gravitational constant (6.67384x10-11 m3kg-1s-2). This calculates to ~9.8 ms-2.

If you start going beneath the surface of the Earth, the equation for calculating the force of gravity becomes more complicated. Any mass below you is pulling you downwards, while any mass above you is pulling you upwards.

Wikipedia has a nice article going into detail about the force of gravity as you go further into the Earth. Because of variations in how density increases at different depths, the force of gravity increases until you reach the transition between the lower mantel and the outer core of the planet. As you go deeper than this transition, the gravity decreases rapidly.

At the very center of the Earth, the calculation is much simpler. The mass pulling you each direction is balanced by an equal amount of mass pulling at you in the the other direction. This cancelation results in the force of gravity at the center being 0.0 ms-2.



What is the gravity at the center of a black hole? The standard answer is that the gravity at the center of a black hole is infinite, because the matter within the object is infinitely compressed.

Nobody knows exactly what goes on within a black hole, because we can't go there and we can't look at it from afar. However, people spend a great deal of time thinking about them because they're interesting cases where our knowledge is limited.

The calculations which show the gravity at the center of a black hole to be infinite rely on the assumption that the mass within a black hole will collapse inwards forever, with all of its mass collected to an infinitesimal point at the center with infinite density.

I think infinite density is something that can't exist in our universe, so a point of infinite gravity also can't exist. If infinite density can't exist, there must be a universal maximum density constant. This would be similar to how there is a maximum speed, the speed of light.

The only certain calculation we can make about the density of mass within a black hole is that all of the matter is compressed to within the Schwarzschild radius of the object. This calculation gives us a lower bound for the maximum density constant. The constant could be much higher, without being anywhere near infinite.

A maximum density constant implies some consequences.
  1. As you pass towards the center of a black hole, eventually you will enter the physical object. Once you are inside the mass at the core of the black hole, the force of gravity will start decreasing until you reach the center. At the center, the force of gravity would equal 0.0 ms-2.
  2. There would be a minimum size for black holes. According to the wikipedia page on the Schwarzschild radius, the average density of a black hole with the mass of Earth would be 2.04x1027gcm-3. If the maximum density constant were lower than this, there could not be a black hole with mass as low as the Earth. For certain values of the constant, this would explicitly rule out the microscopic black holes which some people were worried about being created by the Large Hadron Collider.
  3. Since the Big Bang was an expansion of spacetime itself, a finite maximum density in that spacetime isn't a problem.
  4. As a black hole evaporates (via Hawking Radiation), its Schwarzschid radius will eventually shrink to fall inside the physical surface of its core (because a maximum density implies a minimum size for a black hole). At this point, radiation from the surface of the object will be able to escape. At this time or some point later, the surface itself will begin to decompress (explode). The result is that this form of black hole would end with dramatic incandescence and explosion. The detailed output spectrum from this would probably differ from that produced by the infinite density black hole model decaying away to nothing.
  5. The central singularity of a black hole doesn't exist, so quantum mechanics and general relativity can be unified into a fuller understanding of what gravity is.
There are probably other important consequences, but I haven't worked through the math fully. I'm still working on the thesis for my Genetics PhD, so it will be some time before I can being to examine the math in detail.

The remaining problem is of estimating what the constant of maximum density is. I've conceived of three general approaches:
  1. Analyze of the sizes of black holes (which we have a hard time finding) and note the lower limit to their mass/size range. There is a very large black hole at the center of our galaxy, but this provides for a very limited estimate of the constant because it is at the wrong end of the size range.
  2. Calculate the differences in emission spectra from the different models of black holes and look out into the universe for detonating black holes to test between the models.
  3. Calculate the maximum density from other - more knowable - physical measures. I have no notion of how one would go about doing this, but the inter-connectedness of physics suggests there may be a path.
Can you think of any other approaches to determine what the constant of maximum density is?



So, how does this connect with (or how can I connect it with) biology?

Living things depend upon extracting energy from naturally occurring energy (electrical, chemical, light, etc.) gradients. Around a black hole, there is a profound gradient in gravitational energy. This leads me to the idea of complex self-organizing, self-reproducing structures (life) that feed off of those intense gravitational gradients.

The author Robert L. Forward published the book "Dragon's Egg" in 1980, about intelligent lifeforms (the Cheela) living on the surface of a neutron star.

I'll leave it to you to ponder on what the physical nature of life forms feeding off the surface of the more extreme case of a black hole might be.

Tuesday, August 19, 2014

How I Become a Computational Biologist.

My girlfriend was talking with a friend of hers who is trying to decide what major to go into in college. She is interested in biology and computer science, but is having a hard time deciding which specialty to go into. My girlfriend asked her, "Why not do both? Darren is a 'Computational Biologist'. Maybe I can get him to do a little write up of how he got where is now."

Such a discussion falls under the broad heading of biology which I have as the topic of my blog, so here goes…



I developed a strong interest in biology and computer science from an early age. My mother was a science teacher and my father was an engineer. We had biology textbooks and an early personal computer around when I was young. It wasn't inevitable that I became a computational biologist. I have four siblings who went on to become a computer scientist, a lawyer, an artist, and a public servant.

During high school, I spent much of my free time writing programs exploring different aspects of biology. I wrote programs to let me play with various kinds of cellular automata (CA) systems, which show the complex dynamics that life is known for. I wrote programs to let me play with various kinds of fractal systems, which show the recursive patterns seen in many plant and animal organs. I wrote a simple evolutionary biology system and explored the ecological impacts of changing mutation rates, energy supply, and other characteristics of the system. I was learning the highly ordered thinking needed to program computers, while thinking about biology.

In my first round of college, I majored in Biology. The computer science program at my school didn't allow non-CS majors to take any but the very intro class and the natural sciences program didn't allow students to double-major. I was stuck with biology course-work. This didn't bother me too much, as computers had always been a way for me to study biology. The idea of studying computers directly struck me as odd.

One summer during college at the University of Texas in Austin, I applied for a programming job with at one of the university research labs. I was told that because I wasn't a CS-major, they wouldn't hire me. This seemed unfair. I knew I had more programming experience than most of the CS-major students I knew. I kept programming for my own purposes during the rest of school.

Flickr page for these images.
I began work on a class of CA system which captures some of the mathematics involved in biological pattern formation systems (at right). At this time, I started seeing the underlying mathematics behind the different pattern systems I observed throughout biology. The splotches of color on the back of toads, or the stripes/spots on fish, or the stripes on snakes, all began to appear very simple.

After completing school, I moved to Minnesota and started working for a clinical blood testing lab. Technicians who came in later in the day could always know where I had been working because I would scribble in math on the blotting paper we covered work areas with to help contain any possible biohazardous spills. I continued to play with programming-biology.

Flickr page for these images.
I began implementing a class of CA systems called reaction-diffusion (RD) systems, which model differential equations that capture the physics of how chemicals diffuse and interact. Certain differential equation sets, like the Turing-RD system, spontaneously produce stripes and spots. Calculating the Turing-RD system with diffusional vectors that vary from place to place allowed me to generate the image at left. (It now strongly reminds me of the pattern of trails left by Caenorhabditis elegans worms as they travel over lab growth media.)

Flickr page for these images.
The next step was to work with sets of equations that describe more interesting biological systems. Hans Meinhardt's book, "The Algorithmic Beauty of Seashells" explores numerous biologically inspired RD systems which describe the patterns seen on various seashells.

All of these systems are calculated on rigid square pixel arrays, so I began developing a system for calculating RD systems on a more flexible cell network, where cells could distort, move, and replicate over time. I planned on using the system to explore concepts in development about how differential cell adhesion could drive the formation of tissue layers.

I continued working on these projects until I returned to school at the Genetics graduate program at the University of Minnesota. Until I complete my degree, my personal programming-biology projects have been set aside.



I struggled greatly with deciding how to choose which labs to apply for. I then, and now, am interested in biology in a very wide sense. Limiting myself to one speciality seemed the opposite of how I had approached biology before then. After rotating through a few labs and not finding my place, I spent some time soul-searching to try and determine my next step.

I remembered a professor who lectured one of the core curriculum classes about microarray analysis. Microarrays basically start as glass microscope slides with spots of very precisely defined DNA bonded to them as probes to examine some experimental DNA sample. The experimental DNA is fluorescently labelled and the amount of DNA bound to each spot can then be determined by measuring the amount of fluorescence from each spot on the slide. Typical microarrays contain thousands of spots, so the analysis of the resulting data is heavily reliant on computers and programming. I realized that I like playing with large datasets and approached the professor about joining her lab.

Even though I had no official credentials to suggest I had computational skills, my previous (decades-long) programming experience allowed me to understand and extend the tools used by the lab to analyze previously collected microarray data.

Soon, I began applying my programming skills to other tasks in the lab. I developed a flow cytometry protocol to determine the genome size of the Candida albicans strains we were working with and developed software to process the resulting data down to simple descriptions of biological relevance. When I joined the lab, flow cytometry was an untested technique for this organism and now the lab relies heavily on flow cytometry to track changes during experiments. The technique and computational tools allowed the lab to discover the existence of rare haploid C. albicans cells in the species long thought to be an 'obligate diploid'.

I designed a new microarray to reduce the cost and time needed to analyze C. albicans strains for genomic structural changes. The ~80,000 probes of the microarray were designed to measure the number of DNA copies as well as the alleles present across the C. albicans chromosomes.

Lately I've been designing/implementing a data analysis pipeline to deal with the very large datasets which are produced from whole-genome sequencing experiments. Filtering and compressing the gigabytes of data into relatively simple to interpret figures illustrating changes in DNA copy number and allelic composition requires integrating several tools previously published by others with large chinks of custom code into a single analysis pipeline. We just submitted the paper for this project and are waiting for the review process to complete.

I'm now actively writing my thesis for my PhD, which will be the first official credential illustrating my computational skills. Once I graduate, I won't have to argue with people who think that because I'm a biologist I wouldn't know anything about computers. I still have to work out my next steps, but I'm not worried about my future employment opportunities. Computational biologists are in high demand, in large part because so few people have both the computational and biological intuitions needed for the work.



If you're starting college and are interested in biology and computers, your next steps probably won't include a specialized computational-biology program. In general, there just aren't such programs right now. If you're lucky, your university will have a class or two which covers the topic of computational biology at more than a very basic level.

During college, I studied computers as a side-line to my official biology studies while taking as many math courses for electives as they would let me. I know another computational biologist who was in a computer science program and took as many biology courses as he could for his electives, in addition to his own side studies. Your path will probably look more or less like one of ours, with an official focus on biology of computer science and lots of personal effort dedicated to the other subject in your off-hours. You won't be able to simply rely on college advisors or departments to get you the experience to become a computational biologist. You will have to study long and hard outside of what the college programs are designed for.

You will probably have to continue your schooling with a graduate degree, as the credential you get upon completion of undergrad won't be an indication of your interest and capabilities in this cross-discipline niche. Graduate programs in science pay you to go to school, so you won't have to struggle with another five years of student loans or other outside financing. They don't pay highly, but they do pay plenty enough to get by. When you're done, in addition to the credential of a graduate degree, you will have authorship on one or more papers which highlight your particular style of thinking and skills to potential future employers.

By this time, it will be obvious that you are a Computational Biologist.

Friday, August 1, 2014

The Trouble with Seeds (2/3)

For someone interested in breeding plants, that some plants often don't produce seeds can be a major barrier. A breeder would be limited to looking for selectable mutations, rather than using the more general mixing and segregation of genetics to increase crop diversity.

Many of our common vegetables, fruits, and landscape plants are traditionally propagated by clonal divisions. In some cases (apples, pears, etc.) the complex genetic diversity of the crop means that every seedling would produce a distinct plant. The market demands for consistent production then encourage growers to clone the plants. In other cases, the clonal tradition is enforced by the plants themselves due to their inability to produce seeds. (I'll later discuss a third category of plants which have such long life-cycles that breeding projects become difficult to undertake.)

Researchers have figured out tricks to get seed and allow breeding to be done with plants that might otherwise prefer not to.

The Trouble with Seeds (1/3): Garlic, Horseradish, Potato Onion, Walking Onion, and Banana.
The Trouble with Seeds (2/3): Pineapple, Lily of the Valley, Potato, and Sweet Potato.
The Trouble with Seeds (3/3): Babington's Leek, Crosnes, and Bur Oak.

Most of this post is an accumulation of information from other sources (in colored quotes below) about how to get seeds from these crops. I've also included my thoughts and experiences where I felt something needed to be clarified or extended.


    How to get pineapple (Ananas comosus) to set seed.

    [1] "Pineapple is largely vegetatively propagated. Sexual reproduction is rare in nature because pineapple is self sterile; seeds if produced by self fertilization germinate slowly with low vigour and young seedlings are fragile due to inbreeding depression."

    "Self-sterile" generally doesn't mean there won't be any seeds, but rather that there will be very few seeds.   Plants are somewhat flexible in such things. The examination of one I recently purchased revealed three seeds.

    [2] "Pineapples are usually propagated vegetatively. Seeds are only used by commercial growers for breeding purposes. They are viable for 6 months, are unreliable and difficult to germinate. Pineapples from the store are hybrids and if you do manage to get the seed to germinate and grow into a plant and eventually fruit (2-3 years) the fruit will not resemble what you bought. It still could be a lot of fun. Nick the seeds and plant as deep as it is long in moist (not soggy) potting soil and place the pot inside a plastic baggie. Keep it warm (75-80 F). It will take a long time (up to 6 months to germinate)."

    The best method for germinating the seeds from [3] was to place the seeds in a glass jar with some water, sealed with plastic wrap and stored on a heating-mat.



    If you just want to grow another pineapple like the one you got from the store, simply cut off the leafy top and transfer into soil [4]. Give it plenty of light, keep it moist, and if you're lucky, the new plant will grow you a new fruit in a couple of years. If you're unlucky, the new plant will in time turn into a mold festival.
    1. http://www.ogtr.gov.au/internet/ogtr/publishing.nsf/content/pineapple-3/$FILE/biologypineapple08_2.pdf
    2. http://uk.answers.yahoo.com/question/index?qid=20101129055936AAuEhEl
    3. http://www.youtube.com/watch?v=Z2xK9LigVJQ
    4. http://www.youtube.com/watch?v=vpJHgXaPzFA



    How to get Lilly of the Valley (Convallaria majalis) to set seed.

    Lily of the Valley flowers profusely, but rarely produces fruit. This leaves most propagation of the plant to be via vegetative means. The abundant small rooted bulbs are referred to as 'pips' and readily take to being transplanted.

    I've come across two methods which might trick Lily of the valley to set fruit.

    The first is two have two different clones growing close to each other, so that each will pollinate the other. This is based on the idea that the plant will prefer to cross-pollinate to maintain genetic diversity. I've come across this idea elsewhere, but I am not now able to find any references.

    The second method is to convince the plant that it is dying and thus will have no chance to reproduce except by producing fruit via self-polination. I potted up a single pip from a pink-flowered form, intending only to transfer it to a new garden bed. I placed the pot inside under some fluorescent lights, as I was in the process of moving and didn't have a garden for it, and wasn't that consistent with making sure it had water. Once I finished moving into my new place, I brought over the plants and discovered this one was growing a fruit. The conditions were not consistent with insect based cross-pollination, leading me to the conclusion that the fruit probably derived from a self-polination.

    Both these methods require more experimentation, but are consistent with evolutionary theory and long-term survival of the plant's genes.

    This plant is highly toxic, with reports of fatal poisonings due to eating only a few of its berries, so take care with how you handle and where you store your plants.


      How to get Potatoes (Solanum tuberosum) to set seed.

      Typical garden potatoes are propagated by saving tubers from year to year. A side effect of this process is the gardener is also saving potato viruses from year to year, inside the tubers. Over several years, the amount of viruses can build up and result in stunted plants which don't produce well. There are laboratory techniques to clean viruses from potato tissue clones, allowing the professionals to restore the varieties they grow to high production. Another method is to routinely grow potatoes from true seeds. This will eliminate the viruses which can interfere with production and will let you grow a diverse population of new potato types.

      To do this, you have to get seeds from the potatoes and not all potato varieties are forthcoming about flowering and producing seeds for you. One approach to get potatoes to make seeds for you relies on better living through chemistry.

      [1] "The effect of gibberellic acid containing mixtures, silver thiosulphate and extended photoperiod on flowering induction in 16 non-flowering potato genotypes and on flowering enhancement in 14 normally potato flowering genotypes was studied in sub-tropical plains of India during short-day autumn crop season of 2000-2001 and 2001-2002. Extended photoperiod alone was not successful in induction of flowering. Silver thiosulphate in combination with extended photoperiod effectively induced flowering in 16 potato genotypes studied for flower induction. Induced flowers of some genotypes were male fertile. Normal berry setting was observed on induced flowers and seeds obtained from such berries germinated normally. Gibberellic acid containing treatments were not very effective in flower induction as they induced some flowers only in few genotypes. In the normally flowering genotypes silver thiosulphate enhanced maximum flowering and duration of flowering to a great extent."

      A second approach takes a bit longer and dispenses with the chemistry. If you have varieties which don't produce berries, you eat the tubers and end those genetic lines. Before long, you will only have potato lines which do produce seeds easily and their progeny will also be likely to produce seeds easily. It will only take a few generations to get rid of the gene variations which prevent flowering/seed-production in typical potatoes.

      [2] "The easiest one to deal with is potatoes - mostly commercial varieties - that set a huge load of spuds which act as a photosynthate sink absorbing all the energy and nutrients the plant produces. This prevents berry set because there are not enough resources to go around. Azul Toro is a good example. The way to get these varieties to produce seed is to change the load balance. Plant these potatoes a few weeks later than normal and push them just barely into the surface of the soil. The resulting plants won't have room to make stolons and therefore will have extra resources to devote to producing seed. I accidentally stumbled on this method a few years ago and have used it several times since.

      This probably explains why I found seeds in berries produced by potatoes that were in my garden when I bought the place. They were growing right at the surface, slightly protruding in fact. This may also be why the plants I grew from seed have themselves set seed. They, too, are growing right at the surface."

      This third approach appears to work by playing with the physiology of the potato plant by how they are planted. The mechanism of disrupting the typical source-sink migration of sugar in the plant appears to be a generally useful idea that can apply in various ways to many species.
      1. http://link.springer.com/article/10.1007%2Fs10681-005-9050-y
      2. http://alanbishop.proboards.com/thread/8356/restoring-potato-fertility?page=1#ixzz3Vvu4F6Cb



      How to get Sweet Potatoes (Ipomea batatas) to set seed.

      Sweet potato vines will sometimes flower and set seed, depending on the variety and growing conditions. If you want to encourage flower/seed production, you have to break out the chemical weapons.

      [1] "Most sweet potato cultivers grown in Zimbabwe are poor in agronomic and quality traits and require improvement through breeding. However, most cultivars rarely flower yet the flowers are crucial in genetic improvements. The aim of this study was to determine the effects of different levels of 2,4-dichlorophenoxyacetic acid (2,4-D) on sweet potato flower induction. A 3*4 factorial experiment in a randomized complete block design with three replications was used. The first factor was landrace with three different landraces and the second factor was 2,4-D with four different concentrations (0, 100, 300, and 500 ppm). The 2,4-D was applied 50 days after planting. Sweet potato landraces that were sprayed with 2,4-D showed morphological and physiological disorders that included temporal drooping, petiole pinasty, stem splitting, shoot dieback and root swelling. Extensive morphological and physiological disorders were observed on landraces that were sprayed with the high levels of 2,4-D (300 and 500 ppm). However, within 30 days, all the landraces that were sprayed with 2,4-D managed to initiate buds and set flowers while the plants that were not sprayed did not flower at all. The Friedman's tests showed no significant differences in bud and flower number among the treatment combinations used. Therefore the lowest concentration of 2,4-D (100 ppm) used in this study is probably close to the optimum concentration for flower induction in sweet potato. Although this concentration is not the actual optimum, at the moment this concentration can be used to induce flowering in sweet potato and thus allow sweet potato breeding initiatives to be launched."

      2,4-D is a very commonly used herbicide, readily available in garden centers or from online vendors. Fortunately, there is a method that doesn't rely on toxic chemicals that you might rather not have around you.

      [2] "It was thought that the slight stimulation of flower production in the nonflowering noted in 1951-52 could be due to the limited storage of carbohydrates in the roots of the easy-floweringstock. This might result in a build-up of materials within the aerial portion of the plant and thus induce bud formation. Following this reasoning, grafts were made in 1952-53 on some closely related species commonly found in many home flower gardens. They were selected because they have no storage roots. These species are: Morning Glory (Ipomea purpurea) variety Heavenly Blue, Cardinal-climber-(Quamoclit sloteri), Moonflower (Calonyction aculeatum), and Cypress-vine (Quamoclit pennata). The method of grafting was the cleft graft. After insertion of the scion, the union was tightly wrapped with a strip of rubber made from a cut rubber band. No other treatment was given. A total of 125 such grafts were tried of which 95% were successful. Grafts were successful on all except Cypress-vine which has a stem smaller than that of the sweet potato; however, even these grafts took and grew for a short time before dying.
      On many of the plants flower buds began to appear about one month after grafting or soon after growth of the scion was resumed. These buds continued to appear, develop, and produce normal flowers. In some cases the plants were dwarfed (plant 103) but in others the growth of the scion was normal. The response of the sweet potato scions was the same regardless onto which ornamental they were grafted."

      This method can be performed easily. The best rootstock (I. tricolor) is readily available, grows quickly, and flowers prolifically. The physiological mechanism suggested is interesting, but later research suggests it is incomplete.

      [3] "Grafting has been used to induce flowering and to induce early flowering. For example, sweet potato (Ipomoea batatas) is routinely grafted to other nontuberous root forming Ipomoea species such as I. ruba, I. carnes, and I. tiliaceae to induce flowering. In this case, the presence of leaves on the scion and rootstock have had a profound influence on the flower-inducing response. Flowering was induced only when the rootstocks had expanded leaves, thus suggesting that flower-inducing substances are synthesized in the rootstock leaves and translocated through the graft union to induce flowering in the scion (Kher et al. 1953; Lam and Cordner 1955). Grafting sweet potatoes onto Ipomoea carnea ssp. fistulosa increased flower numbers, percentage of capsule set and number of seeds in all four tested cultivars…"

      Grafting will take some practice, but it appears to be a good way to start a sweet potato breeding project. I would select for lines that didn't require such heroic efforts to get further seed.
      1. http://www.academicjournals.org/article/article1387208292_Mutasa%20et%20al.pdf
      2. https://ucanr.edu/repositoryfiles/ca707p13-71853.pdf
      3. http://alanbishop.proboards.com/thread/7720/sweet-potato-breeding-project?page=6#ixzz3Q345UFcC


      I discuss this topic with respect to a different set of crops in my first post with the shared name. When I wrote the first post, I was thinking of the classic Star Trek episode, "The Trouble with Tribbles". Tribbles are fictional creatures that replicate rapidly by cloning themselves, having been born pregnant. The plants being discussed are generally clonally reproduced, so the name "The Trouble with Seeds" came to mind.

      Most of the lovely plant diagrams in this post were derived from public-domain images hosted at botanicalillustrations.org. Some other diagrams are public-domain images from the same era that I found via google. I chose the original images which depicted the plants under discussion in the way I appreciated and then subtracted out the yellowed background of the page using my favorite image editor (GIMP)