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Efficient Synthesis for Two-Dimensional Strand Arrays with Row Constraints

Boaz Moav, Ryan Gabrys, Eitan Yaakobi

TL;DR

This work analyzes row-constrained two-strand synthesis in a dense $m\times k$ DNA-array, modeling progress as a Markov chain under a fixed periodic synthesis sequence. It delivers tight asymptotic bounds on the expected completion time: for general alphabets, a laggard-first policy achieves ${\mathbb E}[T] \approx \frac{L(q+3)}{2}$, and no online no-lookahead policy can beat this leading term; in the binary case a one-symbol look-ahead improves the bound to ${\mathbb E}[T] \approx \frac{7}{3}L$. It further ties the problem to Chvátal-Sankoff constants via an interleaving/LCS interpretation, offering bounds based on known constants. Finally, the paper provides a dynamic programming algorithm that computes the exact offline optimal schedule for any fixed pair of sequences in ${O}(L^2 q)$ time and space, laying groundwork for optimal policy design under spatial synthesis constraints.

Abstract

We study the theoretical problem of synthesizing multiple DNA strands under spatial constraints, motivated by large-scale DNA synthesis technologies. In this setting, strands are arranged in an array and synthesized according to a fixed global synthesis sequence, with the restriction that at most one strand per row may be synthesized in any synthesis cycle. We focus on the basic case of two strands in a single row and analyze the expected completion time under this row-constrained model. By decomposing the process into a Markov chain, we derive analytical upper and lower bounds on the expected synthesis time. We show that a simple laggard-first policy achieves an asymptotic expected completion time of (q+3)L/2 for any alphabet of size q, and that no online policy without look-ahead can asymptotically outperform this bound. For the binary case, we show that allowing a single-symbol look-ahead strictly improves performance, yielding an asymptotic expected completion time of 7L/3. Finally, we present a dynamic programming algorithm that computes the optimal offline schedule for any fixed pair of sequences. Together, these results provide the first analytical bounds for synthesis under spatial constraints and lay the groundwork for future studies of optimal synthesis policies in such settings.

Efficient Synthesis for Two-Dimensional Strand Arrays with Row Constraints

TL;DR

This work analyzes row-constrained two-strand synthesis in a dense DNA-array, modeling progress as a Markov chain under a fixed periodic synthesis sequence. It delivers tight asymptotic bounds on the expected completion time: for general alphabets, a laggard-first policy achieves , and no online no-lookahead policy can beat this leading term; in the binary case a one-symbol look-ahead improves the bound to . It further ties the problem to Chvátal-Sankoff constants via an interleaving/LCS interpretation, offering bounds based on known constants. Finally, the paper provides a dynamic programming algorithm that computes the exact offline optimal schedule for any fixed pair of sequences in time and space, laying groundwork for optimal policy design under spatial synthesis constraints.

Abstract

We study the theoretical problem of synthesizing multiple DNA strands under spatial constraints, motivated by large-scale DNA synthesis technologies. In this setting, strands are arranged in an array and synthesized according to a fixed global synthesis sequence, with the restriction that at most one strand per row may be synthesized in any synthesis cycle. We focus on the basic case of two strands in a single row and analyze the expected completion time under this row-constrained model. By decomposing the process into a Markov chain, we derive analytical upper and lower bounds on the expected synthesis time. We show that a simple laggard-first policy achieves an asymptotic expected completion time of (q+3)L/2 for any alphabet of size q, and that no online policy without look-ahead can asymptotically outperform this bound. For the binary case, we show that allowing a single-symbol look-ahead strictly improves performance, yielding an asymptotic expected completion time of 7L/3. Finally, we present a dynamic programming algorithm that computes the optimal offline schedule for any fixed pair of sequences. Together, these results provide the first analytical bounds for synthesis under spatial constraints and lay the groundwork for future studies of optimal synthesis policies in such settings.
Paper Structure (17 sections, 12 theorems, 88 equations, 3 figures, 1 algorithm)

This paper contains 17 sections, 12 theorems, 88 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definitions and Problem Statement
  3. Definitions and Preliminaries
  4. Problem Statement
  5. Previous Work
  6. Expected Synthesis Time
  7. Upper Bound for the General Case
  8. Improved Upper Bound for the Binary Case
  9. Connection to the Chvátal-Sankoff Constants
  10. Lower Bounds
  11. Optimal Synthesis Order
  12. Optimality of the Greedy Approach
  13. The x-first Policy
  14. No Look-ahead Policies
  15. One Symbol Look-ahead
  16. ...and 2 more sections

Key Result

Lemma 1

There exists an optimal schedule that never remains idle when progress is possible.

Figures (3)

  • Figure 1: Illustration of the array model in light-directed DNA synthesis under neighborhood-wise constraints, e.g., using a separate mirror per row.
  • Figure 2: Illustration of \ref{['example:ordering-matters']}, demonstrating the impact of scheduling under row constraints. The two schedules shown for ${\boldsymbol x} = (1, 3, 2, 2)$ and ${\boldsymbol y} = (0, 1, 3, 0)$ differ in their second time slot.
  • Figure 3: $\mathbf{P}$ and $\pi$ for \ref{['claim:stationary-state-one-look-ahead']}.

Theorems & Definitions (35)

  • Definition 1
  • Example 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Example 2
  • Claim 1
  • Claim 2
  • Claim 3
  • Claim 4
  • ...and 25 more