# A*B/C^2 Design For Both DEC and INC

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A while back I was asked how I designed my A*B/C^2 puzzles, and at the time I couldnâ€™t remember. A few days working on round 4 seems to have refreshed my memory and I think I can answer now.

This time around my paper designs did serve as a good starting point, and this one in particular resulted in a INC solution for me and a DEC solution for Astromon.

A little disclaimer here, when putting together these â€śpaperâ€ť designs, I donâ€™t use exact math. I write the oligos down on paper and see where they could share lanes. And I donâ€™t want them sharing to big or to small of a lane. And by â€ślaneâ€ť I mean reverse complements in the design sequence. And of course the shared lanes are not perfect matches, this is where I have to ballpark, or estimate. If I come to a point where there is no complementary sequence for both oligos sharing a lane, I know one or the other states is going to have a bulge or an internal loop. And Iâ€™ve found these bulges and loops come in very handy later on when itâ€™s time to â€śbalanceâ€ť the oligos. So if in a shared lane the [C] oligo will be missing a bond, further down the line I will chose to match [C] over matching [A] or [B], maintaining a rough balance.

Another thing I am shooting for in my paper design is a scheme for part of the reverse complements of each [A] and [B] to be able to form a static stem, or crash together in states 1-3. In the diagram I labeled them Magnets.

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At this point I am not yet really considering a reporter. What I am going for is a balanced design which will bind [CC] in state 1, [CC] in state 2, [CC] in state 3, and [AB] in state 4. This is what Iâ€™ve started refering to as a neutral design. Once this is achieved the #neutral design can become either a DEC or a INC solution.

For both the DEC and INC puzzles, the presence and concentrations of the oligos are the same.

State 1 has 2 [C] oligos present at low (100nM) concentration and only an [A] oligo at high (100nM) concentration. [B] not present.
State 2 has 2 [C] oligos present at low (100nM) concentration and only a [B] oligo at high (100nM) concentration. [A] not present.
State 3 has 2 [C] oligos present at high (300nM) concentration and both [A] and [B] oligos at low (50nM) concentration.
State 4 has 2 [C] oligos present at low (100nM) concentration and both [A] and [B] oligos at low (50nM) concentration.

Reporter oligos are present in all states at the same concentration, but can be ignored for now.

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In a nutshell:

1.) The true balance for this type of design comes by the ability of the [A] and [B] reverse complements to crash and form a â€śmagnetâ€ť stem.
2.) All while having [A] and [C]1 share a partial lane.
3.) And having [B] and [C]2 share a partial lane.
4.) The â€śmagnetâ€ť stem that forms in states 1, 2, and 3, the one that is prohibiting the bonding of [A] and [B], is only overcome in state 4 where [A] and [B] are both present and the 2 [C]s are at low concentration.
5.) State 1 is missing the [B] oligo, so the â€śmagnetâ€ť stem forms and [C]1 and [C]2 bind.
6.) State 2 is missing the [A] oligo, so the â€śmagnetâ€ť stem forms and [C]1 and [C]2 bind.
7.) State 3 has both [A] and [B] oligos, almost the right condition for [A] and [B] to bind, but here the [C] oligos are in high concentration and win the battle, and the â€śmagnetâ€ť stem still forms.
8.) State 4 is where the perfect conditions are present for [A] and [B] to bind. Both [A] and [B] oligos are present, and the [C] oligos are at the low concentration, allowing for the â€śmagnetâ€ť stem to be pulled apart and for [A] and [B] to finally bind.

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At this point I have something to work with, but the puzzle is far from being solved and will require some balancing to get it to the desired #neutral state. Last year this balancing took me 3 and a half days with plenty of blind alleys and dead ends. This round I spent 2 days floundering with my first design, but these last ones have taken about 2 hours. I would suggest that if you get stuck for more than a few hours maybe check your logic and try a different approach.

This is the initial sequence I derived from my paper design.

AAAAAAAAACCACUUGUCUCAGUCUGUACCUGCAUUUGUAAAAACAAAUCAUCCAUCACCGUGCUCUCUGUGUGCAAAAAAAAAA

And what it looks like in the INC lab.

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At this point my static or magnet stem is not forming as I wanted, and by looking it appears the [B] oligo is binding too strongly, so I mutate base 50 from a Â© to an (A).

AAAAAAAAACCACUUGUCUCAGUCUGUACCUGCAUUUGUGAAAACAAAUAAUCCAUCACCGUGCUCUCUGUGUGCAAAAAAAAAA

And partial success. My â€śmagnetâ€ť or static stem now forms in state 2.

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Now looking at the design above, it looks to me that the [A] oligo might be binding to strongly. So I mutate base 33 from a Â© to a (U).

AAAAAAAAACCACUUGUCUCAGUCUGUACCUGUAUUUGUGAAAACAAAUAAUCCAUCACCGUGCUCUCUGUGUGCAAAAAAAAAA

And another partial success. The magnet stem now forms in state 1 and 2.

I donâ€™t want to give the wrong impression, I am only showing the â€śrightâ€ť moves I made. The wrong ones took a couple of hours. Saving your sequences in notepad and annotating what you did and where you are at helps a lot. Flash will eventually crash and you will lose your undo stack. It happens.

But back to the design.

Iâ€™m missing some steps at this point, but with some tinkering with the [A] and [B] bonds and the resulting bonds in the â€śmagnetâ€ť stem I came to a point where it was a close energy balance between two designs, one with [A] and [B] bonding in state 3 and 4 like above, and a design where all 4 states had only 2 [C]s bonding identically like in state 1 and 2 above. At that point I just started trimming of bonds at the beginning of the first [C], and at the end of the second [C]. Without too much trouble I came up with this. (Below)

AAAAAAAAAAAUCUUGUCUCAGUCUGUACCUUUAUUUGUAAAAACAGAUAAUCCGCCACCGUGCUCUCAGUGAGAAAAAAAAAAA

That design worked, CC in state 1-3 and AB in 4, but I didnâ€™t like the pairing so I played with it some more and got this. (Below)

AAAAAAAAAAAUCUUGUCUCAGUAUGUACCUGUAUCUGGAAAAGCAGAUAAUCCGCCACCGUGCUCUCAGUGAGAAAAAAAAAAA

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So I have it where I wanted it. At this point Astromon started on a DEC design and I continued on with the INC design. Looking at what I had, I saw that it was possible to match the reporter to the unpaired bases at the beginning of the design in state 4.

After grafting some extra nts from the end of the design back to the beginning, and completing the complementary reporter sequence I got this. (Below)

AAAAAAAAGUCAGAACUUAGUCUUGUCUCAGUAUGUACCUGUAUCUGGAAAAGCAGAUAAUCCGCCACCGUGCUCUCAGUGAGAA

This would appear to have broken the design, but it was expected. The reporter is now bonding strongly to what are the first nts of the [C] oligo complementary sequence. Nothing else has changed. All I need to do is balance the reporter sequence so it only forms with the unpaired end of [C]s complement in state 4.

An easy way to do this is to add a â€śtailâ€ť magnet. The tail magnet will fold back on the reporter sequence and form a hairpin in states 1-3.

AUGACAAAGUCAGAACUUAGUCUUGUCUCAGUAUGUACCUGUAUCUGGAAAAGCAGAUAAUCCGCCACCGUGCUCUCAGUGAGAA

This gives a lot of added flexibility when working with the reporter. The reporter is shorter and therefore more sensitive than the other oligos.

Some more mutations resulted in this. (Below)

AUGACAAAGUCAGUAUUUAGUCUUGUCUCAGUAUGUACCUGUAUCUGGAAAAGCAGAUAAUCCGCCACCGUGCUCUCAGUGAGAA

Almost there. Now the tail magnet is too strong, and pairing in an unexpected way with part of the unpaired [C] complement. (Above) A single mutation, base 7 from an (A) to a (U) results in this, a solution.

AUGACAUAGUCAGUAUUUAGUCUUGUCUCAGUAUGUACCUGUAUCUGGAAAAGCAGAUAAUCCGCCACCGUGCUCUCAGUGAGAA

And shortly after Astromon solved the DEC version starting with the same #neutral design. (Below)

http://www.eternagame.org/game/browse/8489874/?filter1=Id&filter1_arg2=8526453&filter1_arg1=â€¦

I thought I might have to put the reporter in between [A] and [B], but he made it work at the beginning.

This is the second example this week of starting from neutral and getting a DEC and INC solution from the same design.

These two solutions came from the same base or #neutral design.

http://www.eternagame.org/game/browse/8489874/?filter1=Id&filter1_arg2=8523202&filter1_arg1=â€¦

http://www.eternagame.org/game/browse/8489875/?filter1=Id&filter1_arg2=8524687&filter1_arg1=â€¦

I spent considerable time on the â€śdiagramâ€ť at the beginning of this thread, so that it would be readable for others. But paper designs donâ€™t have to be for anyone but yourself. Here is the actual paper design that I used for this post. It was drawn on the back of some old homework.

For me, making these paper designs has led to a better understanding of the parts of this problem. All of my successful designs have started this way. I would encourage players who are struggling with these puzzles to give it a try.

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Andrew, thank you so much for taking the time to write this up!

Just to drive home how much these designs are the shared sequence, I made the above picture.

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Hereâ€™s another #neutral design I created today, designed first without a reporter, then it worked out for 2 DEC solutions and an INC solution. It seems to work on the same principle as the first design in this thread, except itâ€™s an opposite. The â€śmagnetâ€ť or static stem only forms in the 4th state when the conditions are right for [B] and [A] to kick out the 2 [C]s.

From my diagram, I got this sequence.
AAAAAAAAAACAGACCGUUCCACCACCUUGGUCUUAUGUACAAAAAGUACUUGUCACAGUAUGUACCUGUAUUUGUAAAAAAAAA

And after a couple hours, this one.

AK99 #neutral
AAAAAAAAAAAAAAAAAAACGGACCGUUCCACCACCUUGGUCUUAUGUACAAUCAGUACUUGUCUCAGUAUGUACCUGUAUUUGA
http://www.eternagame.org/game/browse/8489874/?filter1=Id&filter1_arg2=8533383&filter1_arg1=â€¦

State 1-3 (below)

State 4 (below)

Then my first DEC solution. (below)
AK99.1 R4
http://www.eternagame.org/game/browse/8489874/?filter1=Id&filter1_arg2=8532146&filter1_arg1=â€¦

States 1-3 (below)

State 4 (below)

The one above had the reporter in between the 2 C oligos, so I used the same base design and created another solution with the reporter at the end after the [A].

state 1-3 (below)

state 4 (below)

And finally an INC solution from the same neutral design. (below)
AK99.1 R4
http://www.eternagame.org/game/browse/8489875/?filter1=Id&filter1_arg2=8533127&filter1_arg1=â€¦

state 1-3 (below)

state 4 (below)

I think I need to invest in a whiteboard, GIMP is a pain.

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I came up with another INC design today, again starting on paper and from a neutral state. My goal was a more compact version in the style of AK2.2, except being an INC design.

state 1-3 neutral

state 4 neutral

solved state 1-3

solved state 4

This one took a bit longer, it gave me a little extra trouble because in making it more compact I had to increase the amount of lane sharing between A - C1 and B - C2.

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