Reference Assisted Nucleic Acid Sequence Reconstruction from

Mass Spectrometry Data

Gabriel Ilie1, Alex Zelikovsky2 and Ion Măndoiu1

1CSE Department, University of Connecticut2CS Department, Georgia State University

MassCLEAVE assay for MS-based nucleic acid sequence analysis

• Signed relative errors assumed to follow a normal distribution with mean 0, standard deviation σ for masses and σ’ for intensities

• Two types of error incurred when matching compomer c to peak of mass m and intensity i(m):– Relative mass error

– Relative intensity error:

Error model

Problem formulation

Given:• Mass spectra MS• Reference sequence r including position of PCR primers• Maximum edit distance D• Standard deviations σ and σ’, tolerance parameter Find: • Target sequence t flanked by PCR primers that

a. is within edit distance D of r, and b. yields a matching of compomers of CS(t) to masses of MS with

minimum total relative error

Naïve Algorithm

• Exhaustive search – Generate all sequences within an edit distance of

D of the reference, and – Compute the minimum total relative error for

matching the compomers of each of these sequences to the masses in MS.

• The number of candidate sequences grows exponentially with D

3-Stage Algorithm

1. Identify regions of the reference sequence that are unambiguously supported by MS data– High probability to be present in the unknown target

sequence

2. Branch-and-bound approach to fill in remaining gaps– Generates set of candidate sequences with compomers

supported by MS data

3. Compute candidate sequences with minimum total relative error – Min-cost flow problem currently solved as linear program– With or without intensities

First stage: finding strongly supported regions of the reference

• Chebyshev’s inequality:

• A detectable compomer c ϵ CSσ(s) is strongly matched to mass m ϵ MSσ(s) if:

where ε = σ / 0.5 is set based on a user specified tolerance

First stage: finding strongly supported regions of the reference

• A strong match between compomer c and mass m is unambiguous if:– c has multiplicity of 1 in reference– c can be strongly matched only to m– m can be strongly matched only to c

• The set M of unambiguous matches can be found efficiently by binary search

First stage: finding strongly supported regions of the reference

which are normally distributed with mean 0 and standard deviation σ /i0.5

• If Chebyshev’s inequality fails for index i, match(ci, mi) is removed from M

• (c1, m1), . . . , (cn, mn) = unambiguous matches for cut base σ, indexed in non-decreasing order of relative errors

• We iteratively apply Chebyshev’s inequality with tolerance to the running means of signed relative errors,

First stage: finding strongly supported regions of the reference

• A position in the reference sequence has strong support if – All detectable compomers overlapping it can be

strongly matched, and – At least one of these matches is in M

(unambiguous + not removed)• Positions in PCR primers automatically marked

as having strong support

Second stage: generating candidate targets by branch-and-bound

• Reference regions with strong support assumed to be present in target

• Gaps filled one base at a time, in left-to-right order, using branch-and-bound– Choice order: reference base, substitutions,

deletion, insertions– Chebyshev test with tolerance applied to running

means of signed relative errors of closest matches• Search pruned when test fails or more than D mutations

Third stage: scoring candidates by linear programming

• Objective:– Minimize total relative error

• Variables:– For each c ϵ CSσ and m ϵ MSσ, xc,m is set to 1 if c is matched to

m, 0 otherwise (integrality follows from total unimodularity)• Constraints:

– No missing peaks: each detectable compomer c ϵ CSσ(t) must be matched to one mass in MSσ

– No extraneous peaks: each mass m ϵ MSσ must be matched to at least one detectable compomer c ϵ CSσ(t)

LP w/o intensities

LP with intensities

Simulation setup

• Reference length: 100-500 bp• Reference sequences/targets– D=1: 10 random references, all sequences at edit

distance 1 used as targets – D=2,3: 100 random reference-target pairs

• Error free MS data: σ = σ’ = 0• Noisy MS data: σ = 0.0001, σ’ =0-1• Tolerance parameter: = 0.01

Precision and Recall

actual target

predicted target(s)

tp(true positive)Prediction is

unique & correct

fp(false positive)

Prediction is unique & incorrect

fn(false negative)

Prediction is not unique

Branch-and-bound vs. Naïve(F-measure for D=1, error free data, w/o intensities)

100 150 200 250 300 350 400 450 50065%

70%

75%

80%

85%

90%

95%

100%

1 substitution Branch-and-Bound1 deletion Branch-and-Bound1 substitution Naïve1 deletion Naïve1 insertion Branch-and-Bound1 insertion Naïve

Branch-and-bound speed-up(D=1, error free data, w/o intensities)

Length 100 150 200 250 300 350 400 450 500

Naïve 18.66 34.95 49.65 65.72 82.60 100.25 120.19 139.90 161.71

Branch-and-Bound 0.06 0.08 0.12 0.16 0.19 0.25 0.33 0.50 0.52

Speed-up 307X 429X 418X 418X 430X 404X 368X 278X 314X

Results on noisy data (F-measure, D=1, σ = 0.0001, w/o intensities)

100 150 200 250 300 350 400 450 50070%

75%

80%

85%

90%

95%

100%

1 substitution, σ=0, τ=01 substitution, σ=0.0001, τ=0.011 deletion, σ=0, τ=01 deletion, σ=0.0001, τ=0.011 insertion, σ=0, τ=01 insertion, σ=0.0001, τ=0.01

Effect of the number of mutations (F-measure, σ = 0.0001, w/o intensities)

100 150 200 250 300 350 400 450 50020%

30%

40%

50%

60%

70%

80%

90%

100%

1 substitution, σ=0.0001, τ=0.011 deletion, σ=0.0001, τ=0.012 substitutions, σ=0.0001, τ=0.011 insertion, σ=0.0001, τ=0.012 deletions, σ=0.0001, τ=0.013 substitutions, σ=0.0001, τ=0.013 deletions, σ=0.0001, τ=0.012 insertions, σ=0.0001, τ=0.013 insertions, σ=0.0001, τ=0.01

Do intensities help?(F-measure, σ = 0.0001, 1 substitution)

100 150 200 250 300 350 400 450 50084%

86%

88%

90%

92%

94%

96%

98%

σ'=0σ'=0.15σ'=0.25σ'=0.35σ'=0.5σ'=1without intensities

Do intensities help?(F-measure, σ = 0.0001)

100 150 200 250 300 350 400 450 50050%

55%

60%

65%

70%

75%

80%

85%

90%

95%

100%

1 substitution σ'=0.352 substitutions σ'=0.351 substitution w/o intensities3 substitutions σ'=0.352 substitutions w/o intensities3 substitutions w/o intensities

Ongoing Work

• Experiments on EPLD clone data– Branch-and-bound relaxation + penalty in LP

objective to handle missing/extraneous peaks– Intensity data normalization: correct for mass and

base composition effects