Coding for Optimized Writing Rate in DNA Storage Siddharth Jain, - - PowerPoint PPT Presentation

Coding for Optimized Writing Rate in DNA Storage

Siddharth Jain, Farzad Farnoud, Moshe Schwartz, Shuki Bruck

IEEE ISIT 2020

DN DNA Stor

• rage

DNA Synthesis (Writing) DNA Sequencing (Reading) Reconstruction Storage Medium (Multiple Strands of DNA)

Information

In this talk

Current DNA Synthesis Systems

• Slow
• Expensive

Terminator Free DNA Synthesis

(H. H. Lee, R. Kalhor, N. Goela, J. Bolot, and G. M. Church, “Terminator-free template-independent enzymatic DNA synthesis for digital information storage,” Nature Communications, vol. 10, no. 2383, pp. 1–12, 2019. )

• Faster
• Cheaper
• Noisy

Terminator Free DNA Synthesis Channel

(H. H. Lee, R. Kalhor, N. Goela, J. Bolot, and G. M. Church, “Terminator-free template-independent enzymatic DNA synthesis for digital information storage,” Nature Communications, vol. 10, no. 2383, pp. 1–12, 2019. )

C C C C C A

Sequence Write

Noise

Previous Symbol Current Symbol

Distribution 𝐸

Write Time 𝑢

(Sticky Insertion)

Terminator Free DNA Synthesis Channel

(H. H. Lee, R. Kalhor, N. Goela, J. Bolot, and G. M. Church, “Terminator-free template-independent enzymatic DNA synthesis for digital information storage,” Nature Communications, vol. 10, no. 2383, pp. 1–12, 2019. )

C C C C C A

Sequence Write

Noise

Previous Symbol Current Symbol

Distribution 𝐸!→#

Write Time 𝑢

(Sticky Insertion)

Terminator Free DNA Synthesis Channel

(H. H. Lee, R. Kalhor, N. Goela, J. Bolot, and G. M. Church, “Terminator-free template-independent enzymatic DNA synthesis for digital information storage,” Nature Communications, vol. 10, no. 2383, pp. 1–12, 2019. )

C C C C C A

Sequence Write

Noise

Previous Symbol Current Symbol

Distribution 𝐸!→#(𝑢)

Write Time 𝑢

(Sticky Insertion)

Terminator Free DNA Synthesis Channel

(H. H. Lee, R. Kalhor, N. Goela, J. Bolot, and G. M. Church, “Terminator-free template-independent enzymatic DNA synthesis for digital information storage,” Nature Communications, vol. 10, no. 2383, pp. 1–12, 2019. )

Sequence to be synthesized: ACTAG

A ACCC ACCCTT ACCCTTA ACCCTTAGGGGG Round 1 Round 2 Round 3 Round 4

𝐸

!→#(𝑢$)

𝐸#→%(𝑢&) 𝐸%→!(𝑢') 𝐸

!→((𝑢))

Length of run in each round is given by a distribution 𝐸 which depends on

• previous symbol - current symbol
• time of synthesis

Approach

(H. H. Lee, R. Kalhor, N. Goela, J. Bolot, and G. M. Church, “Terminator-free template-independent enzymatic DNA synthesis for digital information storage,” Nature Communications, vol. 10, no. 2383, pp. 1–12, 2019. )

ACCCTTAGGGGG ACTAG

Forget Runs and Encode Information in Transitions

Rate: 𝐦𝐩𝐡𝟑 𝟒 Can we do better?

Precision Resolution (PR) Framework

(M. Schwartz and J. Bruck, “On the capacity of the precision-resolution system,” IEEE Trans. Inform. Theory, vol. 56, no. 3, pp. 1028–1037, 2010. )

0100010010010101

• Information encoded in length of runs of 0’s.
• Clock frequency mismatch at Tx and Rx can result in

erroneous measurement of run lengths.

• PR framework provides an optimal set of run lengths that

can be recovered without any error.

Precision Resolution (PR) Framework

• Assumptions:
• 1. Run Length noise is independent of the location of the run.
• 2. Noisy Run Lengths have a finite support.

PR framework cannot be directly applied for the Terminator Free DNA Synthesis Channel.

Why?

Memory

ACCC ACCCTT ACCCTTA ACCCTTAGGGGG Round 1 Round 2 Round 3 Round 4

𝐸

!→#(𝑢$)

𝐸#→%(𝑢&) 𝐸%→!(𝑢') 𝐸

!→((𝑢))

Distribution 𝑬 depends on the previous symbol

Quantization Error

• Distribution 𝑬 doesn’t have a finite support.
• The quantizer may have an error in detecting the

round duration.

• We assume this error to be ≤ 𝜺.

Multiple Copies

• Multiple DNA strings can be synthesized for the

same user information.

• They can be used to improve the overall scheme.

T A C G 1 2 1 3 2 1 1 2 1 3 1 2 1 4 1 3 1 2 1 3 1 2 1 4

𝑢!→# = {1, 2} 𝑢!→$ = {1, 3} 𝑢!→% = {1, 3} 𝑢#→! = {1, 2} 𝑢#→$ = {1, 2} 𝑢#→% = {1, 3} 𝑢$→! = {1, 2} 𝑢$→# = {1, 3} 𝑢$→% = {1, 2} 𝑢%→! = {1, 4} 𝑢%→# = {1, 2} 𝑢%→$ = {1, 4}

𝑇(𝐻)

Encode Information in round times

Convert 𝐻 to a simple graph 𝐻’

A d1 d2 T 1 1 1

Perron-Frobenius Theory 𝑑𝑏𝑞 𝑇 𝐻 = log% 𝜇 𝐵&!

𝐵$!: 𝐵𝑒𝑘𝑏𝑑𝑓𝑜𝑑𝑧 𝑁𝑏𝑢𝑠𝑗𝑦 𝑝𝑔 𝐻&, 𝜇 𝐵$! : 𝑁𝑏𝑦𝑗𝑛𝑣𝑛 𝐹𝑗𝑕𝑓𝑜 𝑊𝑏𝑚𝑣𝑓 𝑝𝑔 𝐵$!

Add auxiliary vertices A T 3

Framework Description

• Maximal round time decoding error (𝜺 > 𝟏)
• Maximal round time (𝑵)
• Allowable round times for a given transition 𝑐 → 𝑏

𝟐 ≤ 𝒖𝒄→𝒃

(𝟐) < 𝒖𝒄→𝒃 𝟑

< ⋯ < 𝒖𝒄→𝒃

(ℓ)

≤ 𝑵

Example

T A C G 1 2 1 3 2 1 1 2 1 3 1 2 1 4 1 3 1 2 1 3 1 2 1 4

𝑢!→# = {1, 2} 𝑢!→$ = {1, 3} 𝑢!→% = {1, 3} 𝑢#→! = {1, 2} 𝑢#→$ = {1, 2} 𝑢#→% = {1, 3} 𝑢$→! = {1, 2} 𝑢$→# = {1, 3} 𝑢$→% = {1, 2} 𝑢%→! = {1, 4} 𝑢%→# = {1, 2} 𝑢%→$ = {1, 4}

𝑇(𝐻)

𝑚 = 2, 𝑁 = 4

Framework Description

• Maximal round time decoding error (𝜺 > 𝟏)
• Maximal round time (𝑵)
• Allowable round times for a given transition 𝑐 → 𝑏

𝟐 ≤ 𝒖𝒄→𝒃

(𝟐) < 𝒖𝒄→𝒃 𝟑

< ⋯ < 𝒖𝒄→𝒃

(ℓ)

≤ 𝑵

• Number of copies (𝑶)
• Quantizing Function ℚ𝒄→𝒃: ℕ𝑶 → [ℓ]

Receiver: 𝒔𝟐, 𝒔𝟑, … , 𝒔𝑶 𝒕. 𝒖 . 𝒔𝒌 ~ 𝑬𝒄→𝒃(𝒖𝒄→𝒃

(𝒋) )

ℚ 𝒔𝟐, 𝒔𝟑, … , 𝒔𝑶 = [ 𝒋 . 𝒕. 𝒖. 𝐐𝐬 [ 𝒋 = 𝒋 𝒋 ≥ 𝟐 − 𝜺.

𝐸"→$(1) 𝐸"→$(2)

CGGG CGGGGG CGGGG CGGGGGG CGGGGG

Receiver

Quantizing Function

𝑢,→. = {1, 2}

State Splitting encoder

𝑛 ∈ {0,1}! 𝑒 ∈ [ℓ]"

Error Correction Coding Multiple Sequence Alignment + Quantizing Function ℚ Terminator Free DNA Synthesis Channel

Decoding for ECC

, 𝑒 ∈ [ℓ]"

State Splitting decoder

• 𝑛 ∈ {0,1}!

𝑶 𝒅𝒑𝒒𝒋𝒇𝒕 𝑡 rounds where for each round there are ℓ possible round times 𝑐𝑓𝑑𝑏𝑣𝑡𝑓 𝑝𝑔 𝑢ℎ𝑓 𝜀 𝑓𝑠𝑠𝑝𝑠 𝑗𝑜𝑢𝑠𝑝𝑒𝑣𝑑𝑓𝑒 𝑐𝑧 𝑢ℎ𝑓 𝑑ℎ𝑏𝑜𝑜𝑓𝑚

Theorem

Let Gʹ be the ordinary version of G. Further assume the k user informaMon bits are i.i.d. uniform random bits. Then for all large enough k, the user informaMon bits may be encoded into a sequence using at most Here α is the sum of probabilities of non-auxiliary vertices in the stationary distribution of the max-entropic Markov chain and 𝐷1,ℓ ≜ 1 + 𝜀𝑚𝑝𝑕ℓ 𝜀 ℓ − 1 + (1 − 𝜀)𝑚𝑝𝑕ℓ(1 − 𝜀)

𝑙 𝑑𝑏𝑞 𝑇 𝐻 (1 + 𝛽 1 𝐷*,ℓ − 1 𝑚𝑝𝑕-.$(ℓ)) synthesis time, and be decoded correctly with high probability.

State Splitting encoder

𝑛 ∈ {0,1}! 𝑒 ∈ [ℓ]"

Error Correction Coding Multiple Sequence Alignment + Quantizing Function ℚ Terminator Free DNA Synthesis Channel

Decoding for ECC

, 𝑒 ∈ [ℓ]"

State Splitting decoder

• 𝑛 ∈ {0,1}!

𝑶 𝒅𝒑𝒒𝒋𝒇𝒕

𝑙 𝑑𝑏𝑞 𝑇 𝐻

(1 + 𝛽 1 𝐷1,ℓ − 1 𝑚𝑝𝑕345(ℓ))

Experimental Results – Binomial Run lengths

𝑂 = 5

𝜀 = 0.02

Poisson Run Lengths

𝜇! for different values of 𝑂

Achievable Rates for different values of 𝜀

Conclusion

• Method for encoding information in DNA sequences based on PR

framework and terminator free DNA synthesis method.

• Rate above 𝐦𝐩𝐡𝟑 𝟒 can be achieved.
• Method accounts for quantizer error 𝜺.
• Provided method for designing quantizers for Binomial and Poisson

run lengths.

• As we have multiple copies, alignment can be used to account for

deletion of runs.

Thank You. Questions

sidjain@caltech.edu