Improved Models and Algorithms for Universal DNA Tag Systems continued … a.k.a. what did we do?

Nucleation Model When do two tags form a match? 1.sum of score of matches ≥ c ? (not stable complex!) 2.score of heaviest match ≥ c ? (as in [BKSY]) 3.score of heaviest match with e errors ≥ c ! (we propose) AAGCTGCA ACCCTGTA AAGCTGCA ACCCTGTA AAGCTGCA ACCCTGTA AAGCTGCA ACCCTGTA

Score of a single match (recap) May be computed via either of 2-4 Rule – easy approximation: A-T = 2, G-C = 4 – sum gives melting temperature Nearest Neighbor Rule – sum energies due to contiguous A-T & C-G pairs – A-T different from T-A different from A-G etc.. 

It’s an improvement.[BKSY] would predict We predict mfold predicts Is this a realistic model ? CGTAGCACGAA AACTCGTATCA CGTAGCACGAA AACTCGTATCA ACAGCAATGGA GATCGGTACTA ACAGCAATGGA GATCGGTACTA > < T m = 3.2°C T m = 13.8°C (6,0) match(9,1) match

Definitions Two strings s 1 and s 2 have a (c,e)-match if they have substrings t 1 and t 2 such that: 1.w(t 1 ) = w(t 2 ) ≥ c 2.t 1 and t 2 differ in ≤ e places A tag system is an (h,c,e)-code if 1.every tag has weight atleast h 2.no two tags have a (c,e)-match

Design of (h,c,e)-code with large size Outline of Upper Bound on size How? Via upper bound on number of c-tokens (the substrings t that have weight ≈ c) Choosing one c-token in a tag knocks out a sphere of nearby c-tokens from further use in any other tag. Similar to sphere packing bound in coding theory. Algorithms for generating optimal codes Modify alphabetic tree-search algorithm of [MPT]

c-tokens (recap) strings with weight ≥ c no proper suffix of weight ≥ c have weight either c or c+1 length ranges from c (all C/G) to 2c (all A/T) can’t use tailweight method of [BKSY]  nucleation complexes nucleation complexes = Two c-tokens differing in at most e symbols Two c-tokens differing in at most e symbols

A sphere around CGCA C G CA is a 6-token of weight 7, length 4 how many 4-length codewords at distance 1? TGCA·GGCA AGCA CACA CCCA·CTCA CGGA CGTA CGAA CGCC CGCT CGCG

How many such spheres pack the whole space ? Now look at spheres around codewords of optimum code vol(s) total number of c-tokens s a red sphere ≤ must be disjoint ! size of code × vol(sphere) total number of c-tokens ≤

Size of a sphere Suppose string s has a A/T and b C/G symbols weight = a + 2b, length = a + b Introduce e errors into s to get t weight of t same as weight of s, so e1 = e2 for errors of type 1, pick inways and options to change to REPLACEWEIGHTNUMBER A → G, A → C, T → G, T → C +1e1 G → A, C → A, G → T, C → T e2 A → T, T → A, C → G, G → C 0e3

One tag of weight h uses (h-c+1) tokens So size of code ≤ Size of sphere = Substitute a = 2 l – c and b = c - l l varies from c/2 to c, c-tokens of weight c or c+1 = number of strings of length l =

Can tighten the bound further our sphere knocked out only c-tokens of the same length we should also remove similar c-tokens of other lengths.. reduce bound by factor e ? In comparison to [BKSY] bound h = 30, c = 12, e = 0: ≥ #tags ≥ h = 30, c = 12, e = 2: #tags ≤ 1268 if nucleation does occur with errors then we can’t assume so many tags

Plot of upper bound vs. c,e (h = 50) upper bound on number of codewords e – number of errors c – weight of nucleation complex

Open Problems & Remarks design, analyze efficient algorithms for model can we use random deBruijn sequences to generate codewords ? analyze using mixing techniques on Markov chain of [KMUW] ? exciting new question for coding theory: alphabets with weighted Hamming distances!