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)

AAGCTGCAACCCTGTA

AAGCTGCAACCCTGTA

AAGCTGCAACCCTGTA

AAGCTGCAACCCTGTA

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 ?

CGTAGCACGAAAACTCGTATCA

CGTAGCACGAAAACTCGTATCA

ACAGCAATGGAGATCGGTACTA

ACAGCAATGGAGATCGGTACTA

>

<

Tm = 3.2°C Tm = 13.8°C

(6,0) match (9,1) match

Definitions

Two strings s1 and s2 have a (c,e)-match if they have substrings t1 and t2 such that:

1. w(t1) = w(t2) ≥ c

2. t1 and t2 differ in ≤ e places

A tag system is an (h,c,e)-code if3. every tag has weight atleast h4. 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=

Two c-tokens differing in at most e symbols

A sphere around CGCA

C G C A

• is a 6-token of weight 7, length 4• how many 4-length codewords at distance 1?

TGCA·GGCA AGCA

CACACCCA·CTCA

CGGACGTACGAA

CGCCCGCTCGCG

How many such spheres pack the whole space ?

Now look at spheres around codewords of optimum code

vol(s) total number of c-tokenss a redsphere

≤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 in ways and options

to change to

REPLACE WEIGHT NUMBER

A→G, A→C, T→G, T→C +1 e1

G→A, C→A, G→T, C→T -1 e2

A→T, T→A, C→G, G→C 0 e3

• One tag of weight h uses (h-c+1) tokens • So size of code ≤

Size of sphere

=

Substitute a = 2l – 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: 13840 ≥ #tags ≥ 12000• 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)

108

64

20

1

100

10000

1000000

100000000

10000000000

1000000000000

12 14 16 18 20 22 24 26 28

uppe

r bou

nd o

nnu

mbe

r of c

odew

ords

e – number of errorsc –

weight of n

ucleation co

mplex

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!