The helix chooses the lattice

Two ceilings computes what stops a routed shape being buildable, and it computes both ceilings on a square lattice: every helix has four neighbours a quarter of a turn apart, a route steps between neighbours, and a block of ten by ten is the largest square shape one viral strand can thread. It ends by naming the obvious next measurement. Honeycomb lattices are used as well, their adjacency graph is different, and a different graph would change the routing results.

Before the routing, there is a question the square lattice skipped. Why would anybody draw a helix bundle on a honeycomb? A square grid is simpler, every neighbour is equivalent, and the routing figures so far were drawn on one. If the honeycomb is an equal alternative, the routing results are results about one convention; if it is not, they are results about a lattice the molecule does not want.

The answer is not a matter of taste. It is arithmetic on the helix’s own pitch, and it comes out before any shape is chosen.

Where a strand can cross #

A sheet that routes itself describes the object: one long strand held into a shape by several hundred short ones, with the long strand running along one helix, crossing to a neighbour, running back along that one, and so on until it has visited every helix in the shape. A crossover is where the strand leaves one helix for the next, and a staple strand that holds two helices together crosses between them in the same way.

A crossover is not available everywhere along a helix. The strand’s backbone spirals round the helix’s axis, and the strand can only step across to a neighbour at a base whose backbone points at that neighbour. Everywhere else the neighbour is on the wrong side.

The spiral has a pitch. The double helix makes one turn in about ten and a half bases, so each base turns the backbone by $360^\circ/10.5 = 34.29^\circ$. A neighbour whose direction from the helix’s axis is $\psi$ can be crossed to at base $b$ only when $b \times 34.29^\circ$ lands on $\psi$, give or take a few degrees the molecule will absorb.

So the question of which lattice a bundle can be drawn on is a question about which sets of directions a sequence of multiples of 34.29° keeps landing on.

Three neighbours, every seven bases #

On a honeycomb lattice every helix has three neighbours, a third of a turn apart: at 0°, 120° and 240°.

Seven bases turn the backbone by $7 \times 360^\circ/10.5 = 240^\circ$ exactly. So a strand starting at a crossover to the neighbour at 0° faces the neighbour at 240° after seven bases, the neighbour at 120° after fourteen, the neighbour at 0° again after twenty-one, and so on for as long as the helix lasts. Every seventh base is a crossover to some neighbour, and not one of them misses.

That is not a coincidence of rounding. Ten and a half is twenty-one halves, a third of a turn is a third of twenty-one halves’ worth of bases — three and a half — and twice that is seven. The helix’s own pitch divides evenly by the honeycomb’s angle, which is why the first row of the figure is a row of exact marks.

Four neighbours, never exactly #

On a square lattice every helix has four neighbours, a quarter of a turn apart: 0°, 90°, 180° and 270°.

A quarter of a turn is $10.5/4 = 2.625$ bases, which is not a whole number, and neither is any odd multiple of it. The question is whether some larger number of bases lands on a quarter turn anyway, and it can be settled in one line. A crossover to a side neighbour needs $b \times 360^\circ/10.5$ to be $90^\circ$ more than a whole number of turns, which is $8b \equiv 21 \pmod{84}$. The left side is even for every b and the right side is odd, so no whole number of bases ever faces a side neighbour exactly.

The nearest bases miss by 4.3°: eight bases turn the backbone 274.3°, thirteen turn it 445.7°, which is 85.7°, and the pattern repeats with the same miss at twenty-nine and thirty-four. The neighbours in line — at 0° and 180° — are faced exactly at twenty-one and forty-two, because twenty-one bases is exactly two turns, but a lattice that could only cross to two of its four neighbours is not a square lattice in any useful sense.

The second row of the first figure is that argument drawn: exact marks only every twenty-one bases, and 4.3° misses at every other mark.

Each lattice has one pitch #

Turn the question round and ask what helix each lattice would be exact for.

The honeycomb’s designed crossover is seven bases to a two-thirds-turn neighbour, which is exact when a turn is 10.5 bases. The square lattice’s natural crossover is eight bases to a three-quarter-turn neighbour, which is exact when a turn is $32/3 = 10.667$ bases. Each lattice has one pitch at which it is exact and misses by an amount growing linearly with the distance from that pitch.

The second figure draws both lines. They cross zero at 10.5 and at 10.667, and the double helix sits at the first. A designer who uses the square lattice is designing for a helix that turns once every 10.667 bases — which is to say, for a double helix wound about one and a half per cent more loosely than the molecule is.

How far the pitch can move before the verdict turns #

The double helix does not turn at exactly 10.5 bases in every molecule. Sequence, the ions in solution and temperature move it by a few tenths of a base per turn, so the claim that the honeycomb is exact deserves a range as well as a point.

The two lines cross. At 10.4 bases a turn the honeycomb’s seven-base crossover lands at 242.3°, a miss of 2.3°, and the square lattice’s eight-base crossover lands at 276.9°, a miss of 6.9°. At 10.6 the honeycomb misses by 2.3° the other way and the square lattice misses by only 1.7°. Past about 10.58 bases a turn the square lattice would be the nearer of the two, because its exact point, 10.667, is closer.

So the arithmetic’s verdict is not unconditional. It is that the molecule’s usual pitch sits on the honeycomb’s exact point and eighth of a turn away from the square lattice’s, and that the honeycomb stays the nearer lattice across the range of pitches ordinarily met. A molecule wound more loosely — a different helix, a different chemistry — could reverse the preference, and the argument would reverse with it for exactly the reason it holds here.

That is a better form for the claim than “the honeycomb is right”. The lattice a design should be drawn on is the one whose exact pitch is nearest the helix it will actually be built from, and for the double helix in ordinary conditions that is the lattice with three neighbours.

Both lattices are used, and the arithmetic says what each is used as. The honeycomb is the lattice on which the molecule’s natural geometry places crossovers at whole bases. The square lattice is a lattice on which it does not, and a design drawn on it asks the helix to be something slightly other than itself.

A small miss, repeated #

A miss of 4.3° at one crossover looks like something a flexible molecule would take up without noticing. The trouble is that it is not one miss.

A square-lattice design places a crossover every eight bases along every helix, and every one of those crossovers asks the backbone to be at 270° when it is at 274.3°. The misses are all in the same direction. They do not scatter around zero and average out; they add. Over thirty-two bases — four crossovers — the helix is 17.1° short of where the design assumes it is. Over sixty-four bases, 34.3°. Over two hundred and fifty-six, 137°, which is more than a third of a turn.

That is the shape of error the tolerance argument distinguishes as the dangerous one, and the same shape a tolerance spent at the end of a motion has when every step of a folded mechanism inherits the one before. An error that is different every time grows as the square root of the count; an error that is the same every time grows with the count itself. A helix whose crossovers all miss by 4.3° in the same sense has a systematic error, and a bundle of such helices responds by twisting as a whole: the strain has to go somewhere, and a global twist is where a long, thin bundle can put it.

Square-lattice bundles published in 2009 were reported to twist in exactly this way, and the remedy that came with them is the one the arithmetic suggests: take bases out, or put them in, at intervals along each helix, so that the design’s assumed pitch and the molecule’s actual one agree on average. Nothing here models the molecule’s response. What the arithmetic supplies is the size of the mismatch those corrections had to cancel, and the reason it accumulates rather than averaging.

A leap base #

The correction has a familiar shape, and seeing it makes the lattice choice easier to hold.

A calendar year is 365 days and the solar year is about 365.2422. Every year the calendar falls a quarter of a day behind, in the same direction, so the error adds: after four years it is a whole day, and without a correction the seasons would walk round the calendar. The correction is a leap day every fourth year, a further correction every hundredth, and another every four hundredth — a whole unit inserted wherever the accumulated error has reached a whole unit.

A square-lattice helix is a calendar with the wrong year length. Its crossovers assume 10.667 bases a turn and the molecule turns every 10.5, so the design falls behind by a fixed angle every eight bases, and an inserted or deleted base is a leap day. The honeycomb is the calendar whose year happens to be a whole number of days: seven bases, two thirds of a turn, exactly, and nothing ever needs inserting.

What the grid costs, and who chose it #

The question of what a grid costs has come up before, from the design side.

Spelling a tree on a grid rounds a subject’s limbs onto the lattice box pleating uses, and pays for the rounding in paper; the designer’s grid is the dearest single decision in that kind of design. There the grid is chosen by the designer, for what it makes easy, and the cost is the price of that convenience.

What the grid settles makes the same point from inside box pleating: a grid decides a great deal before any design is drawn on it, and designing on a grid is a decision about which designs are cheap.

Here nobody chooses. The grid that costs nothing is decided by a molecule’s pitch, and the grid that costs a twist is the one a designer would pick for simplicity. The medium picks the lattice, and choosing another one is a debt paid in strain. That reverses the usual relation between a design grid and the thing being designed, and it is the reason the honeycomb is not merely one convention among two.

Which results were computed on the wrong lattice #

The earlier routing results were computed on the square lattice, and it is worth being exact about what the arithmetic above does and does not do to them.

It does not make them wrong. A route on a graph is a route on a graph, and the square-lattice verdicts are correct verdicts about the square-lattice graph. Designs on that lattice exist and are built, with the pitch mismatch corrected along each helix.

What it does is make them results about the less natural of the two lattices, and it adds a qualification the account of two things called folding would want: the graph a strand is routed through is itself a consequence of the molecule, so even the combinatorial half of the analogy has chemistry under it. The honeycomb’s graph is different — three neighbours, not four, and a row that reaches the row above only at every second helix — and a shape that routes easily on one need not route at all on the other. The ceiling on routing and the parity obstruction both deserve to be recomputed on the lattice the helix chose.

What an idealised helix cannot show #

The first figure draws an idealised helix and cannot show the molecule’s slack.

A real double helix is not a rigid spiral. Its twist varies with sequence, with the ions around it and with temperature, by amounts comparable to the 4.3° the square lattice misses by. A crossover is not a point but a pair of backbones bending toward one another over a base or two, and it will tolerate some misalignment. The figures treat a 4.3° miss as a miss and a zero miss as a hit, and the molecule does not draw that line so sharply.

What survives the slack is the accumulation. A random variation in local twist averages out along a helix; the square lattice’s miss does not, because it is the same miss at every crossover in the same direction. That is why the figure of accumulated drift is the one to believe even where the single-crossover figure overstates how exact the honeycomb needs to be.

Nor can the figures show a single-layer design. A flat sheet of parallel helices has each helix’s two neighbours at opposite sides, half a turn apart, and 5.25 bases is half a turn — another number no whole base reaches, met in practice by alternating crossovers a turn and a half apart and absorbing the rounding the same way.

The helix the arithmetic assumes #

The pitch is 10.5 bases a turn and constant. It is the standard figure for the double helix in solution, and it is a mean. A different pitch moves the exact points of both lattices proportionally and changes none of the argument about which lattice has an exact point near the molecule’s.

A crossover needs the backbone to face its neighbour. That is the geometric core of the design rule, stated without the molecule’s detail — the major and minor grooves, the positions of the two backbones, the exact angular window.

Neighbours sit exactly at the lattice’s angles. A real bundle’s helices are not at exact positions either, and they spread apart where crossovers are sparse.

And nothing responds. The drift is computed as if the helix held its natural pitch against the design; the molecule’s actual answer is some mixture of twisting the bundle, bending helices and straining crossovers, and which mixture is not modelled.

How the arithmetic was checked #

The exact points are checked as exact. Seven bases at 10.5 a turn must be two thirds of a turn and eight bases at $32/3$ a turn three quarters, each to a part in a billion, before anything is drawn.

Every honeycomb crossover in forty-two bases is required to land exactly, and every square-lattice crossover to a side neighbour is required to miss by more than a degree, which is the whole-number argument above observed on the bases rather than taken on trust. None of this makes a claim about a real molecule’s response, which is where the model always stops in this field.

The drift is required to grow with the helix’s length and the honeycomb’s to be nothing at every length, so a figure in which the misses averaged away would refuse to draw.

Still open: which shapes the honeycomb can route #

The routing question the square lattice answered has to be asked again on the lattice the helix picks, and the change of graph is not small.

On the honeycomb a helix has three neighbours rather than four, and in a block drawn as rows every helix reaches the rows beside it at only every second position. A row of such helices can trap a route: if enough of its members have no neighbour off the row, a route that enters the row may be forced to run its whole length and have nowhere left to go. Whether the blocks that route trivially on squares route at all on the honeycomb is a finite search, and it is the obvious first computation on the new lattice.

The second is the ceiling. The strand’s length still divides by the helix length to give a hundred and thirteen helices, and which honeycomb shapes reach that before the routing stops them is the honeycomb’s version of the first result.

The habit worth carrying is a question to ask of any grid a design is drawn on. Is the grid the medium’s, or the designer’s? A grid the medium supplies costs nothing to use; a grid the designer imposes has a residual somewhere, and when the same residual recurs at every step it adds up to something the finished object cannot hide.