One strand, through every helix, once
scaffold-route is one function. Everything below came out of it during this
build, at arguments taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and when the generator changes, this
page changes with it.
At its defaults
view: "sequence", show: "thresholds"
view: "sequence", show: "uniqueness"
view: "sequence", show: "measured"
What it checked while it drew
Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.
- 24 helices of 64 bases need 1536 of the 7249-base scaffold — 21% of it ×7
- the shape's 24 helices are one connected piece ×7
- the route visits all 24 helices exactly once, every step between lattice neighbours ×5
- along the top row 3 of the 5 helices have no neighbour off the row, and the two corners have one neighbour each ×4
- so a route must start and end at the two corners and pass along the whole row between them — 5 helices of 25 — and the exhaustive search agrees there is none, although the colours balance 13 to 12 ×4
- the route visits all 30 helices once, and every step is between honeycomb neighbours ×3
- every one of the 12 placements is drawn, not a sample of them ×2
- every shape any cheap test refuses is searched anyway, and none of them has a route — 691,268 shapes on the square lattice ×2
- every shape any cheap test refuses is searched anyway, and none of them has a route — 691,268 shapes on the square lattice, 3,898,510 shapes on the honeycomb lattice ×2
- the exhaustive search refuses all 12 of them, in 748 to 920 partial routes each ×2
- this 11-helix shape passes the colour of every forced end, the exhaustive search finds no route ×2
- 1 shape is balanced and still unroutable — the count is necessary and not sufficient ×1
- 1 shape is refused by one pass over the cells, before any search ×1
- 20 of the 80 honeycomb blocks do not route, against none on the square lattice ×1
- a helix built to the square lattice at 10.5 bases a turn is out by 17.1° over 32 bases and by 137° over 256, and one built to the honeycomb by nothing ×1
- at 10.5 bases a turn every crossover the honeycomb offers in 42 bases lands exactly — 7, 14, 21, 28, 35, 42 ×1
- at one crossover period a design expects 201 of its domains to have somewhere else to go, and at 11 bases it expects fewer than one ×1
- each added test leaves the smallest unrouted shape the same size or larger — 9, 11, 11 helices on the square lattice; 12, 15, 16 helices on the honeycomb lattice — and the honeycomb's is never the smaller ×1
- each added test leaves the smallest unrouted shape the same size or larger — 9, 11, 11, 12 helices on the square lattice; 12, 15, 16, 16 helices on the honeycomb lattice — and the honeycomb's is never the smaller ×1
- each family is grown one step at a time and both ceilings are tested at every step, so which one stops it is found rather than predicted ×1
- every block from one by one to 10 by 8 routes on the square lattice ×1
- every shape any cheap test refuses is searched anyway, and none of them has a route — 3,898,510 shapes on the honeycomb lattice ×1
- every shape is put through the colour count and through the exhaustive search, and the verdict is what the two of them said together ×1
- every test has a smallest survivor on both lattices inside the sizes listed, so every row of the table is a measurement rather than a blank ×1
- no route exists although the colours are balanced 5 to 4 — the parity test is necessary and not sufficient ×1
- no route exists, and the colour counts 1 to 4 said so before the search ran ×1
- on both lattices every set of cheap tests has a smallest shape it passes and no route reaches, within the sizes listed ×1
- on the honeycomb every block odd in both directions, from three by three up, passes the colour count and has no route ×1
- one helix cuts off a piece with no end in it, so the route has to finish there, and the stretch it must cross first has its colours the wrong way round ×1
- over 8 shapes the colour count never refuses one the search can route — the necessary condition holds ×1
- seven bases at ten and a half to a turn is exactly two thirds of a turn, and eight bases at thirty-two thirds is exactly three quarters ×1
- staying bound is the harder of the two demands — 17 bases against 11 for being unique ×1
- the hand-drawn raster is a valid route too, so the abstraction agrees with what a designer draws ×1
- the shortest domain the honeycomb admits that clears the mean leaves 15 domains below the hold on the sequences tried, and the shortest the square lattice admits leaves none ×1
- the shortest lengths really do repeat in one sequence and the longest really do not, so the expectation is checked against a sequence rather than trusted ×1
- the strand's 7249 bases thread 113 helices, and a shape that cannot be routed fails at whatever size it first cannot be ×1
- the two lattices are exact at 10.500 and 10.667 bases a turn, and nowhere else in the range ×1
- the weakest domain of this sequence melts 14.6 degrees below the mean, so the mean is not the number an anneal is held above ×1
- this 11-helix shape passes the steps the ends force, the exhaustive search finds no route, and the colour of every forced end refuses it ×1
- this 9-helix shape passes the colour count, the ends and the cuts, the exhaustive search finds no route, and the steps the ends force refuses it ×1
- two crossover periods is not enough on either lattice — 37 and 43 degrees against a hold at 45 — and three is enough on both ×1
- while the square lattice's neighbours a quarter turn round are never faced exactly — the nearest bases miss by 4.3° ×1
Where it is called
Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.
A domain too short to be unique
A staple holds the scaffold by pairing with a stretch of it, and two arguments decide how long that stretch has to be. One is combinatorics — a stretch of seven bases has about four hundred other places in a 7,249-base strand it would also match. The other is thermodynamics, and it is the one that binds: a duplex that is unique at eleven base pairs still comes apart at the temperature the design is held at, and staying paired takes seventeen. Rounded up to the crossover period, that is three periods on both lattices, and the lattice the helix prefers is the one whose three-period domain leaves some of its staples unattached.
A row the route cannot leave
Every rectangular block of helices up to ten by eight routes on the square lattice. On the honeycomb, the lattice a double helix's pitch prefers, twenty of the eighty do not — and every block odd in both directions fails for a reason visible along one row: every second helix on the top row has no neighbour off it, the two corners have one neighbour each, and a route forced through them runs the length of the row and ends. The colour count passes all of them, and a degree count along a single row refuses them.
A sheet that routes itself
DNA origami folds one long strand into a shape by holding it against itself with a few hundred short ones. There is no sheet and no crease — what has to be designed is a route — and the first thing that can go wrong is a counting argument crease patterns already know under another name.
A test that only knows one lattice
The cheapest argument that refuses the smallest shape no cheap test could refuse was read off that shape: cut at one helix, find the piece with no end in it, and count the colours of the stretch the route is then forced to cross. Added to the census it refuses every one of the square lattice's thirty-two eleven-helix survivors and pushes the smallest survivor to twelve, where twelve placements of two shapes survive out of half a million. On the honeycomb it refuses none of the six at sixteen. A test inherits the lattice of the witness it was read off, and the staircase is two staircases.
Every cheap test misses a shape
A strand routed through a bundle of helices has to visit each once, and whether a shape allows that is hard to decide — so the cheap tests that refuse shapes are necessary and never sufficient, and for every set of them there is a smallest shape they pass and no route reaches. Listing every connected shape and searching the ones the tests let through finds it: nine helices on the square lattice for the colour count, the ends and the cuts, eleven once the steps a route's ends force are added, and still eleven once the ends' colours are checked. On the honeycomb the same three stages give twelve, fifteen and sixteen. Each test pushes the smallest unroutable shape out or leaves it where it is; none removes it.
The helix chooses the lattice
A strand can cross to the next helix only where its backbone faces that helix, and a double helix turns about 34.3° a base. Three neighbours a third of a turn apart are faced exactly every seven bases. Four neighbours a quarter of a turn apart are never faced exactly by any whole number of bases — the nearest miss by 4.3°, and the misses do not average out, they add: 17° over thirty-two bases, 137° over two hundred and fifty-six. The lattice a design is drawn on is decided by the molecule before any shape is chosen.
Two ceilings
A DNA origami is limited by the length of one viral strand and by whether its helices can be visited once each in a single pass. Grown one step at a time, a square block runs into the first at a hundred helices and a plus runs into the second at five — so which limit a shape meets is decided by the shape and not by the chemistry.
Two things called folding
A protein folds and a sheet folds, and the word is the same word by accident. Both have exponentially many states and that is not the difference. The difference is that one of them can be filtered by four conditions checked at a single point, and the other cannot be filtered by anything local at all.
Every generator · The folding nobody designed field · The patterns a reader can fold