Parameter space of "standing" circuit

I've been working on decomposing the traveling pulse circuit in order to understanding the parameter space. Today I've worked on the isolated "standing" circuit.



There's two parts. The "pull down" gate that is constantly trying to pull the system to a negative value against the action of the resistor which is trying to pull it to zero. The ratio of the pull down gate (1) to the resistor (RNAase) determines the steady-state level when the feedback gate 3 is not active. The RNAase resistor must be common to all nodes so I treat it as a fixed parameter; I picked the value 0.01 out of thin air for it.

For the following graphs, I pick different starting conditions for "standing" and let this circuit evolve. Each colored trace in the chart is one run of the circuit. Note that there are two steady states. One is about 28 and the other is about -1. If the "standing" value falls below about -0.5 then it goes to the low steady-state and above that it goes high. I like this chart in comparison to typical transform function plots because it lets you see both the kinetics and the steady-states in one place.


Here's the same chart but zoomed in around the origin so you can see that the critical point is about -0.5 which is determined by the gate model.

I varied the two parameters over a range and plotted the parameter space result (best viewed on large monitor).


From top to bottom p1 is increasing. From left to right p2 is increasing. Increasing p2 shifts the steady-state of the "standing" state upwards and thereby separates the two states more dramatically. As p1 is increased -- moving from top to bottom -- both the top and bottom steady-states shift downwards but the bottom one seems to move faster. In the lower left, the two states blur into each other and are poorly defined. So, in general you'd like to push p2 and p1 fairly high but this comes at the cost of slowing down the approach to steady-state as they are pushed further away. When the other half of the circuit is added, p2 value will have to be smaller than p5, so that will determine the upper bound of p2.

My conversation with Alpha

I tried out Wolfram's Alpha this morning. First, something technical and mathematical as it suggests:

Where are the tidal phase singularities?
> Wolfram|Alpha isn't sure what to do with your input. ...

The same search on Google not only brings up links to maps but also brings up the scanned and OCR pages from Winfree's book -- via Google books -- where I got the phrase! Google is amazing.


Why should I use wolfram alpha?
> Wolfram|Alpha isn't sure what to do with your input.

The same search on Google came up with the pages on Wolfram's own site and many more reviews.


Why is Stephen Wolfram so cocky?
> Wolfram|Alpha isn't sure what to do with your input. ... person: Stephen Wolfram ... chemical element: element Wolfram

Tungsten (according to Wiki) is also called "Wolfram" which is why it has the the chemical symbol "W", but nowhere on Wolfram's summary page about Tungsten does it mention this. If you do this same search on Google the first hit is a Slashdot article about the outrageous TOS on alpha that's only *16 hours* old! Google continues to amaze.


How big of an ego does Stephen Wolfram have?
> Wolfram|Alpha isn't sure what to do with your input. ...

The same search on Google returns all kinds of hits from book reviews and whatnot complaining about his inflated ego.


All joking aside, I did like its stock summary page (one of its suggested searches). When you do ask it about something it knows does present a very well formatted result with lots of good technical information. But the TOS is absurd.

Complementary logic ideas

Talking with John this morning about the equivalence between the gates we're proposing and electrical analogs. John points out that our gates are like "half of a tri-state gate". We started thinking about higher-order logic cells using the proposed gates and realized that you can be logically complete assuming that you can mix gates with complementary inputs and only lose some fraction of them to a bi-molecular cancellation. If this is not the case -- if you lose everything -- then there might still be a way to do it with extra translation stages, but I haven't thought that through yet.


(Image update 21 May. Thanks to Erik for pointing out that I forgot the promoter completion domain.)

Assuming that the above gate cancellation reaction is not favorable (or that tethering them reduces the favorability) then you could combine the gates to make buffers, inverters, and a biased-and-gate that doesn't produce a very clean output, but which would have the property that when inputs A & B are + then output would be + and all other input combination would give output slightly - to very -.

Traveling pulse - a stable orbit

I started hunting around in parameter space trying to get my head around what makes the traveling pulse stable and predictable. I don't yet have a set of exact rules, but what I've learned is that the reactions need to be slow compared to the diffusion. This is achieved by simply lowering the concentration of the gates and resistors appropriately. Next, the pull down gates 1 & 2 are very small compared to the feedback and shutdown gates. Also, the "tired" charging gate is very small so that you can delay the onset of the shutdown.

The biggest point is obvious when you look at the phase diagram: you have to let the system get back into steady-state before another pulse hits it. Also interesting is how perfectly straight are the edges of the phase diagram. I think that this means that the gates are run way out of their linear regions and are running in steady-state most of the time. I'm going to try to make a graph to make sense of that.

I also found that it is easy to make complex patterns form when you push the system really hard as in the following class-3-like cellular automata. Note that the system was started with symmetric initial conditions and has full symmetric rules yet is symmetric only until it starts to interact with itself; once it reaches the boundaries, it becomes asymmetric. Fascinating. I suppose this is because the "periodicity" of the pattern is not related to the size of the container so the two periods start to alias in some weird sense.

Idea: Cut healthcare costs? Reduce the patent duration.

Brooks has a good essay today about the proposed underwhelming health care cost-cutting measures. I agree that none of the proposed changes sound like enough to take a reasonable bite out of our growing health care costs; and I doubt that for such a big problem there exists many easy fixes. But, there is one very easy fix that would have an huge impact -- cut patent duration times from 20 years to, say, 10. Of course, innovating companies will hate the idea of reducing their patents and boring-old manufacturers will love it but I guarantee that 10 years from now there will be an incredible drop in drug prices.

We have a fundamental problem that no one wants to admit: until some revolution in drug development takes place (e.g: if it turns out that siRNAs are a magic bullet) then we simply can not have guns, butter, and bandages -- at least we can't have every newfangled "bandage" being made at such an incredible pace.

We have an impossible expectation for our health care that we don't have for any other sector of our economy. We simultaneously want the free market to invent new treatments on a for-profit motive and then we want everyone to have access to the result. In contrast, we don't expect every driver in the country to have access to a Lamborghini just because they exist. We don't expect everyone to have access to the latest iPhone gadget just because they exist. But we do expect -- for good ethical and moral reasons -- that everyone should have access to whatever the latest, best treatments are. While this expectation is understandable, it's nevertheless schizophrenic: "Pharma: go be innovative, invest a lot of money to make amazing drugs! Oh my god, why are they so expensive?" We don't say: "Apple: go be innovative, invest a lot of money to make amazing phone! Oh my god, why are they so expensive?" (Actually some people do, but most just recognize that if the phone is too expensive they'll just do without.)

Health care is always going to involve an insurance middle man be it private, public, or all-messed-up-in-between as it is now. So, health care will always be a collective venture. It is simply irrational to expect that we can collectively afford every possible innovation, just as it would be irrational to expect that we could all collectively own the latest iPhone gadgets. Thus, the systemic way to change the collective system is to simply lower the profit bar. And this can be done by changing one simple variable: the duration of patents. Make patents last 10 years and drug companies won't build as many expensive drugs and, yes, more people will die of things that could have been prevented. But, recognize that this is already the case! The 20 year limit is totally arbitrary. Had it been set at, say, 30 years then there would exist, right now, more amazing but even more expensive drugs and therefore because the number is set at 20 and not 30 we are "heartlessly" letting people go untreated because of an arbitrary number. The number has changed before (upwards) and we can change it again, downwards -- at least for drugs -- if we collectively choose to. It's the only "easy" fix.

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 :-)