Traveling Pulse Phase Diagrams

Working on understanding the behavior of my amorphous traveling pulse, "Mexican Wave". On the right is a marked up phase diagram of the two states "standing" on the x axis and "tired" on the y axis. The mark ups show the regions where different parts of the circuit are operational. This has helped me get my head around what has to be adjusted to make the system more predictable. One lesson is that the mystery of why the pulse is traveling at different speeds has something to do with the fact that the system does not usually get all the way back down into the same steady-state. The bottom steady-state point "not standing and not tired" should be determined by the relationship of the pull down gate 1 & 2 and the grounding resistors. So, next thing I'm going to do is try to adjust things so that I give the system enough time always settle down into that same point. Then I can tackle understanding how the other gates reshape this phase chart.

An observation. The one directional traveling pulse on the left is making a pattern that looks like the branching pattern on a plant stem. This reminds me of a plant branching model Wolfram talked about in NKS.

More fun with Traveling Pulse

I started messing around today with the amorphous traveling pulse from yesterday. First thing I did was try creating an asymmetric starting condition by "pipetteing" in both a spot of "standing" as yesterday and also a spot of "tired" adjacent to that so that the pulse could travel only in one direction. As before, the x axis is cyclical space which is why the pulse travels off to the left and then reappears on the right.

Inexplicably, the pulse does not always travel at the same velocity. I have no idea why, maybe its an artifact of the integration but it seems periodic -- like its accelerating and decelerating at some predictable way.

I then start exploring parameter space of the circuit, repeated here for reference.


(Drawing revised 19 May)


I started with 3 vs 5. All things being equal, it should be the case that the concentration of gate 5 needs to be greater than the concentration of gate 3 so that it can overpower "standing" when "tired". As the following phase chart of 3 vs 5 illustrates, this is true. Also, as 3 grows so does the pulse width. This is intuitive because the harder p3 works to pull up "standing", the longer it will take for the discharge circuit to overpower it. Graph of P3 vs P5:

Then I started on P3 vs P4. P4 determines how fast it gets "tired" so more P4 should create a narrower pulse width, which is indeed the case. As you would expect, there's a limit, P4 can make the system tired so quickly that the pulse disappears (it becomes tired the instant it stands). However, there's a relationship between P3, the charging circuit and P4 the "getting tired" drive. As the standing driver is increased, you have to compensate with fast you become "tired". Makes sense. Ratios in the kind of 5-7 ball park seem to work well given the arbitrary other settings I have. Graph of P3 vs P4:



Crazy things happen when you change the two stabilizing gates p1 and p2. When pull down resistors are set to 0.01 and diffusion to 0.3 in this simulation. As p1 increases the pulse travels slower which makes sense as it is harder to charge standing. (Thanks to Xi for pointing out that I had previously stated this backwards.) At some critical value, it the charge circuit can't keep up with the diffusion and pull down sides and the pulse evaporates. Really weird things start happening around p1=0.01 and p2=0.07, looks like it becomes unstable and pattern forming, which is cool.


Some close ups of instability patterns. They look a like Sierpinski triangles which makes some vague sense because the standing and tired are in opposition to each other and can act as some kind of binary counter where diffusion permits the next space over to act as the carry bit. (I say this with while waving my hands furiously :-)

Traveling Pulse Amorphous Computer

After a few meetings with John, Nam, Xi, Edward, and Andy in the last few weeks I think I have a plausible molecular gate model that can make some interesting amorphous computations. Specifically, I've been trying to make the "Mexican Wave" -- an amorphous pulse wave.




A variable "A" is encoded by the log ratio of the concentration of two RNA species: a sense strand called "A+" and its anti-sense strand called "A-".




(Image updated 21 May -- Thanks for Erik for pointing out I left off the promoter completion domain)

Gates are molecular beacons that use promoter disruption to squelch the generation of some output strand. For now, all gates are unary operators. The RNAs can be displaced off the beacons by toe-hold mediated strand displacement. This design is basically Winfree lab's transcriptional circuits but where the gate is a hairpin DNA molecular beacon and where variables are encoded by log ratio of sense and anti-sense instead of as a proportionality to concentration of an ssRNA.



(Note I updated this diagram to change the naming convention on this 17 May 2009. Again on 21 May thanks for Erik for noticing I left off the promoter completion domain.)

Gates are modeled as having hyperbolic production curves and can be built according to one of four choices of sense and anti-sense sequence on the inputs and outputs. As a matter of convention, the sense strand is labeled "+" relative to the ssRNAs, not relative to the DNA because the concentration of the RNAs is the variable of interest in these systems.

To explore the model, I created a circuit that I hoped would make an amorphous pulse propagating wave. Below, I switch into electrical analogy which I do for my own sanity. The charge across capacitors represent the two variables which I call "standing" and "tired" by analogy with the Mexican Wave. The gates are labeled like "i+o+" meaning "when input is + the output will be -". (I've changed around the naming convention several times, this update is as of 17 May) The gates without inputs are under constitutive promotion and are labeld only by what they output. All nodes are pulled down by the same RNAases represented here as resistors to ground from each capacitor. The two variables are assumed to diffuse at equal rates. The only changeable parameter is assumed to be the concentrations of the gates.


(Thanks to Xi and John for help reworking this diagram. I updated it on 19 May.)

This circuit can be thought of like this. "Standing" and "tired" are constantly being pulled low by the gates 1 & 2 against the action of the resistors. If the rest of the gates weren't there, this would ensure the system will be "not standing" and "not tired". Gate 3 puts feedback on "standing" thus a small threshold level of "standing" will generate more until it saturates in steady-state against the resistor. Gate 4 increases "tired" if "standing". Gate 5 is in high concentration relative to the other gates and can thus overpower the "standing" variable when "tired".



Here are the 1D amorphous results. The two plots are "standing" (left) and "tired" (right). The X axis of each is space (cyclical coordinates). The Y axis from bottom to top is increasing time. Blue represents a high ratio of - to + strands. Red represents a high ratio of + to - strands. Black represents an even ratio. At time zero, a pulse of + is added to the "standing" variable representing a manual pipetteing operation at some point in space. As time passes (bottom to top) the pulse propagates in both directions at a constant rate until the two pulses hit each other and then stop.

Understanding Principal Component Analysis via cool Gapminder graphs

Gapminder.org is a wonderful site full of "statistical porn". This chart in particular is a fascinating graph that demonstrates the correlation between income and child mortality rates. It is also a great example to teach about a cool statistical tool: "Principal Component Analysis".

In this graph of regions there is an obvious negative correlation between infant mortality and income illustrated by the fact that the data points scatter along a line from upper left to lower right. In other words, if you knew only the infant mortality rate or the income of a region you could make a reasonable guess at the other.

Principal Component Analysis (PCA) is a statistical tool that’s very useful in situations like this. PCA delivers a new set of axes that are well aligned to correlated data like this -- I've illustrated them here with black and red lines. For each axis, it also returns a “variance strength” which I’ve represented as the length of the black and red axes. (Actually I just hand approximated these axes by eye for the purposes of illustration).

The strongest new axis returned by PCA (the black one) aligns well with the primary axis of the data. In other words, if one were forced to summarize a region with a single number it would be best to do so with the position along this black axis. The zero point on the axis is arbitrary but is usually positioned in the center of the data (the mean). Positive valued points along this black axis would be those regions further toward the lower right and negative valued regions would be those further toward the upper left. Let’s call this new axis “wealth” to separate it in our minds from “income” which is the horizontal axis of the original data set. Increases in “wealth” represent an increase in income and drop in infant mortality simultaneously.

The second axis returned by PCA is shown as the red axis. Countries that lie far off the main diagonal trend-line (black axis) have particularly unique infant mortality rates given their wealth which we’ll assume is because of something unique about their health care systems. Points well below the black axis are regions that have very good health care given their wealth and those above it have particularly poor health care given their wealth.

Because PCA gives us convenient axes that are well aligned to the data, it makes senses to just rotate the graph to align to these new axes as illustrated here. Nothing has changed here, we've simply made the graph easier to read.



Before you even look at specific regions on these new axes, one could guess that socialist countries would score more negatively along this red axis and those whose economy is heavily biased towards mineral extraction -- where income tends to be very unevenly distributed -- would score more positively. Indeed, this is confirmed. The most obvious outliers below the black axis are Cuba and Vietnam where communist governments have directed the economy to spend disproportionately on health care and the outliers on the other side are: Saudi Arabia, South Africa, and Botswana -- all regions heavily dependent on resource extraction where the mean income statistics hide the reality that few are doing very well while the vast majority are in extreme relative poverty.

One particularly interesting outlier is Washington DC which is located as far along the red axis as is Botswana! In other words, based on this realigned graph, you might guess that the wealth in DC is as unevenly distributed as it is in Botswana. Fascinating! (The observation is probably at least partially explained by the fact that it is the only all urban "state" and urban areas will tend to have wider income distributions than rural/suburban areas.) Also note that all of the points in the United States (orange) are well into positive territory on the red axis -- our health care system is as messed up relative to our wealth as is Chad, Bhutan, and Kazakhstan -- countries with completely screwed-up governmental agendas. Think of it this way: the degree to which our infant mortality rates are "good" owes everything to our wealth and is despite the variables independent of wealth! In other words, countries that provide average health-care relative to their wealth like El Salvador, Ukraine, Australia and the UK fall right on the black axis but we fall significantly above that line -- roughly the same place as countries that are, independent of their wealth, really messed up like Chad and Kazakhstan. (A caveat: the chart is on a log scale so the comparative analysis is more subtle than I'm making it out here.)

PCA returns not only the direction of the new axes but also the variance of the data along those axes. To understand this, imagine for a moment that all the regions of the world had exactly the same health care given their income; in this case all the points would align perfectly along the main trend line (the black axis) and the variance of the red axis would be zero. In this imaginary case, the data would be “one dimensional”, that is income and infant mortality would be one in the same statement; if you knew one, you'd know the other exactly. Now imagine the opposite scenario. Imagine that there was no relationship at all between income and infant mortality; in that case we would see a scattering of points all over the place and there wouldn’t any obvious trend lines. Neither of these imaginary scenarios are what we see in the actual data. It isn’t quite a line along the black axis but neither is it a buckshot scattering of points, so we can say the data is somewhere between 1 dimensional and 2 dimensional. If both variances are large and equal to each other, then the system is 2 dimensional while if one of the variances is large while the other is near zero, then we know the system is nearly 1 dimensional. In other words, PCA permits you to summarize complicated data by finding axes of low variance and simply eliminate them. This technique is called “dimensional reduction” and is a very powerful tool for summarizing complicated data sets such as would arise if we looked at more than two variables. For example, we might include: car ownership, water accessibility, education, average adult height, etc to the analysis at which point performing a dimensional reduction would help to get our heads around any simplifications we might wish to make.

External link: My Manhattan Project

This is an excellent article in New York Magazine about a software engineer on Wall Street.

Some quotes and thoughts.

> "Over time, the users of any software are inured to the intricate nature of what they are doing."

Well put. This is the heart of all software successes and failures. Software is the perfect tool to lie to others and lie to ourselves with. It is the ultimate obfuscatory tool if you let it be.

Thomas Jefferson fought against a Hamilton-supported economy based on industry and banking. Hamilton was right, of course, but Jefferson had a good point. A detail of that eighteenth century debate that has intrigued me is: If financial instruments were already obfuscated to Jefferson in the 18th century, imagine what it must be like now? This article confirmed my intuition for what must have been going on: software sold and maintained by an external company helped to obfuscate the transactions to everyone involved. Of course the technology should have allowed it to be understood too but sounds like some monopolies of thought took hold because it was short-term profitable for them to. It was Jefferson's worries manifest in twenty-first century technology. Maybe there is some law lurking like "The Law of Constant Obfuscation" -- at any given time technology will permit obfuscation to a constant level.

> “Mike,” he told me when denying my request, “can you really look for people dumber than you and then take advantage of them? That’s what trading is all about.”

Ha!

> "I was very good at programming a computer. And that computer, with my software, touched billions of dollars of the firm’s money. Every week. That justified [my salary]. When you’re close to the money, you get the first cut. Oyster farmers eat lots of oysters, don’t they?"

Rationalization is such a powerful force! As is momentum. Feynman has an excellent point in one of his books where he's talking about forgetting why he worked on the Manhattan Project. He joined because, like his collaborators, the idea of Hitler having unilateral nuclear power was unimaginably scary. But, after Hitler was defeated, he forgot why it was he was working on this project and the momentum of the technical challenge remained. He regretted that he didn't re-evaluate his thinking after the VE day.

External Link: Energy Flux Graph from Lawrence Livermore Nat. Labs

I love this graph from Lawrence Livermore National Laboratory illustrating the flux of energy through the US economy. Some things that surprised me:

1) The amount of energy wasted in the transport of electricity is staggering, slightly more than the total amount of oil imported (in energy equivilent units); technological improvement in that sector would make an enormous contribution.

2) Transportation, as I expected, is woefully inefficient. What I didn't appreciate was the magnitude, the energy wasted by transport is approximately equal to all the coal burned!

3) The residential / commercial waste is surprisingly low. One assumes that some fraction of that waste is insulation and so-forth, but even if you took a big bite out of that with building improvements, you wouldn't make a dent in the big picture. It boils down to this: if one's goal is to reduce waste (which is a very different goal than reducing consumption) then electrical and transport are the obvious primary targets.

Vaccinate your child or gramps gets it in the stomach!

There seems to be a growing ignorance about vaccination. From my informal queries of friends and acquaintances who have chosen not to immunize their children or who do not get flu vaccines, I have found that few people understand that vaccination is part of a greater social compact, not merely a personal cost/benefit analysis. The effect is called Herd Immunity. When we vaccinate ourselves, and especially our children, we are adding to the communal common defenses. Obviously, everyone would like to have a defensive wall built to protect a community yet everyone would prefer to not contribute. But that's not the way a good society works; we share the costs of doing things that benefit the common good.

Immunization of children is particularly important for two reasons: 1) Children's immune systems respond to vaccination much more effectively than do others, especially the elderly who are the most likely to die of viral diseases such as influenza. 2) Children are responsible for much of the transport viruses throughout a community owing to their mobility and lack of hygiene.

For example, a controlled experiment conducted in 1968 by the University of Michigan demonstrated that large-scale vaccination of children conferred a 2/3 reduction in influenza illnesses to all age groups. For a nice article on the subject, here's a Slate article from 2008.

I propose an old-fashioned poster campaign to inform about the social benefits of vaccination. Here's a couple of prototype posters I photoshopped up this afternoon. (Apologies to Norman Rockwell!)






(Original photo Adam Quartarolo via WikiCommons)