Bed

I haven't been posting house progress for a while. My night-time project for the last week has been construction of a bed with a little bit of a floating cloud theme. I laminated four 4x8 sheets of maple plywood together and then cut out circles of various radii.


After Alex's sanding for many hours...


Here's the rough-cut end table before I cut out the circles...

Molecular model transfer function

Today I got around to trying out a simplified molecular version of the gate model that will replace my hyperbolic function.



The kinetics are all arbitrary for the model, but the shape of the transfer function looks even better than the made-up model from before. There's an almost perfectly linear section in the middle -- it looks more made-up than my made-up model! This is assuming that all three reactions have the same strength. Next, I need reasonable terms for the three reaction rates.

More parameter space of "standing" circuit

Using the parameter space maps made last time, I've set the "standing" circuit into a place where it has a nearly symmetric bi-stable steady-state at p1 =0.25 and p2=0.50.



The following is the derivative at a given concentration of standing. This dy/dt vs y plot (I don't know if there is a correct name for this find of plot) shows that there are two stable steady states at the zero crossings -5, and +5. There's also the unstable point near zero. It is not exactly at zero because the gate model functions do not cross at zero as seen below.






Now I continue the analysis with the "tired" half of the circuit. I'm interested in the response of "tired" when the "standing" input reaches 0, the point at which the tired circuit will charge fully.



Charging of the tired circuit when standing is 0 and tired starts at its steady-state value of -5


So, "tired" reaches 0 (the point at which the gate 5 is going to be fully on) within about 20 time units when standing = 0.

The following is a sampling of the parameter space for p1 and p2 given "standing" = 0. The steady-state value of tired changes as a function of p1, so for each graph I've started "tired" off at the appropriate steady-state and then watch the evolution when "standing" = 0. This demonstrates that I can delay both the onset of tired (when it hits zero) and how high tired gets at steady-state by adjusting these two parameters.


Next up, I put the circuit back together again...

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.