"The first law of thermodynamics," he said, "is you do not talk about thermodynamics."
"No!" I corrected. "The laws are: you can't win, you can't break even, you can't even quit the game!"
Either way I should have expected that his reaction would be to throw me to the pavement and pin me.
"Ouch!" I writhed struggling to get out from under him. "I'm sorry! Uncle — UNCLE!"
"Sorry," he said, "there's no getting out of this one."
This explains a lot. Be sure to check out the full illustration of the process. Of course, such a process would only be a boon to math texts, which are seemingly written by calculator manufacturers. Hence, one of my students today needed a calculator to find 3 - 1.
Every Thursday the department holds a general coloquium to hear speakers on various mathematical subjects. I started attending regularly when my advisor berated a seminar full of grad students for missing an apparently fascinating talk on Erdös. Last week Dr. Cunningham, my field theory professor, gave a lecture on some arcane conjecture in higher algebra. Although I have no idea about what the conjecture was about, nor did I follow the general arguments presented, I at least did learn about some new, interesting things. There's apparently an object called the "Adels" which contain the reals and all p-adics as subfields, so you can work in all the completions of the rationals at once. Cool. Anyway, I was still recovering from this talk, so I was happy enough that yesterday's would be given by a non-mathematician.
I didn't know who Stuart Kauffman was, but the crowd seemed pretty energetic before the lecture. Partly this was because there was going to be a meeting about the deanship afterwards, and so they had moved the free donut time up to before the meeting (though I learned this too late). A doctoral student gave a testimonial about how he converted from driving schoolbuses to doing graduate work after reading one of Kaufmann's books.
This is pretty understandable, given his talk. Kaufmann gave us a low-level overview of molecular biology with the stated goal of recruiting mathematicians to beat the Americans in deciphering gene regulation in humans.
Anyway, here's the deal. The activity of certain genes can act to promote or repress the action of other genes through several mechanisms. A significant amount of non-coding DNA is actually used for proteins to attach to in order to regulate expression. An example of this is a gene responsible for digesting lactose. There are 4 areas nearby which act as promoters. If all of these have the proper proteins attached, then the gene becomes active. However, there is also a repressor site. If a repressor is attached, then the gene is inactive regardless of what promoters are present. So, one can (perhaps over-simply) treat the expression of this gene as a boolean function of 5 inputs. The first 4 must be on, and the last off, for the gene to be active.
In general certain genes may both be regulated and regulate other genes. For example, one could imagine a system with two genes: A and B. Suppose A acts as a repressor for B, and likewise B acts as a repressor for A. Then if A is on, its activity will tend to shut off B, and vice-versa, so we should expect to see only two states: A on, B off; or A off, B on. In general these networks could be extremely complicated, involving periodic activity: A then B, then back to A again and so on. You know your computer is constructed of binary logic gates which act like the boolean functions in this model of gene activity. And just like your computer can get stuck in an endless loop of activity, forcing a reboot, barring external input cells must get caught in an endless loop of some sort. These states which cells must eventually end up in are called attractors of the dynamical system.
The canonical explanation of attractors is as follows. Suppose you let loose a marble on some kind of warped surface. The marble will roll around for awhile, but eventually end up in a valley or divot of some kind. These divots are called the attractors of the systems, because they "attract" the marble from less stable areas of the surface. As another example, there is a point in Glacier National Park in which mere inches determine whether a drop of water will end up in the Pacific Ocean, the Gulf of Mexico, or Hudson Bay. These bodies of water can be considered attractors of water in North America.
Now here's the point. When the embryo begins to divide, its cells are undifferentiated, but after some 50 divisions the full organism has developed into roughly 260 distinct cell types. Kaufmann's idea was maybe these different cell types represent different attractors for the genetic system. If so, he figured they would probably be relatively stable and simple, with low periodic behavior. This hypothesis would make the most sense if random boolean networks had these desireable traits, so he decided to test this using the means available to him: renting a computer to run the program he had typed up on punch cards. Of course, he wanted to test a random network so just prior to handing over his program to be run, he riffle-shuffled the lower half of the deck. The computer's operator was amazed.
It was actually something of a bet with himself, since if the periodicities were fairly long he could easily have gone broke considering the cost of computation at the time. Fortunately ten minutes later the computer spat out the answer, giving periodicities of lengths no greater than four.
So then we flash forward to today, when the mathematics behind such systems are somewhat better understood, and we have a wealth of biological knowledge including the human genome. The full regulatory patterns of cells are not well known, but we can see that the number of inputs per gene follows a power law distribution, and we can state given such a random distribution about how many attractors are expected. Using the model developed here and a couple parameters one comes up with an estimated 150 attractors, which is not too far off (in the engineering sense) from the number of distinct cell types, 260.
Of course, it also indicates that there is much work to do to discover all the regulatory mechanisms at play and the exact patterns of gene expression in every cell type, but Kauffman reminds us it is our responsibility to help Canada beat the United States to such discoveries.
Great talk. I leave and head to the department lounge. There are still three plain donuts left. I take one and head up to field theory.
Calgary has yet to send me a 100% official acceptance letter, but it now appears that they are preparing to send me a letter saying that I am accepted as long as one of my references decides to send page two of the his form. I hope this will be good enough to submit with my requires:
On the other hand, I need to decide on housing - whether on university
or off campus. There is a nonrefundable campus application fee of $50, so I should decide on that soon.
