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**Published in:** LIPIcs, Volume 276, 29th International Conference on DNA Computing and Molecular Programming (DNA 29) (2023)

We present an abstract model of self-assembly of systems composed of "crisscross slats", which have been experimentally implemented as a single-stranded piece of DNA [Minev et al., 2021] or as a complete DNA origami structure [Wintersinger et al., 2022]. We then introduce a more physically realistic "kinetic" model and show how important constants in the model were derived and tuned, and compare simulation-based results to experimental results [Minev et al., 2021; Wintersinger et al., 2022]. Using these models, we show how we can apply optimizations to designs of slat systems in order to lower the numbers of unique slat types required to build target structures. In general, we apply two types of techniques to achieve greatly reduced numbers of slat types. Similar to the experimental work implementing DNA origami-based slats, in our designs the slats oriented in horizontal and vertical directions are each restricted to their own plane and sets of them overlap each other in square regions which we refer to as macrotiles. Our first technique extends their previous work of reusing slat types within macrotiles and requires analyses of binding domain patterns to determine the potential for errors consisting of incorrect slat types attaching at undesired translations and reflections. The second technique leverages the power of algorithmic self-assembly to efficiently reuse entire macrotiles which self-assemble in patterns following designed algorithms that dictate the dimensions and patterns of growth.
Using these designs, we demonstrate that in kinetic simulations the systems with reduced numbers of slat types self-assemble more quickly than those with greater numbers. This provides evidence that such optimizations will also result in greater assembly speeds in experimental systems. Furthermore, the reduced numbers of slat types required have the potential to vastly reduce the cost and number of lab steps for crisscross assembly experiments.

David Doty, Hunter Fleming, Daniel Hader, Matthew J. Patitz, and Lukas A. Vaughan. Accelerating Self-Assembly of Crisscross Slat Systems. In 29th International Conference on DNA Computing and Molecular Programming (DNA 29). Leibniz International Proceedings in Informatics (LIPIcs), Volume 276, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{doty_et_al:LIPIcs.DNA.29.7, author = {Doty, David and Fleming, Hunter and Hader, Daniel and Patitz, Matthew J. and Vaughan, Lukas A.}, title = {{Accelerating Self-Assembly of Crisscross Slat Systems}}, booktitle = {29th International Conference on DNA Computing and Molecular Programming (DNA 29)}, pages = {7:1--7:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-297-6}, ISSN = {1868-8969}, year = {2023}, volume = {276}, editor = {Chen, Ho-Lin and Evans, Constantine G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.29.7}, URN = {urn:nbn:de:0030-drops-187908}, doi = {10.4230/LIPIcs.DNA.29.7}, annote = {Keywords: DNA origami, self-assembly, kinetic modeling, computational modeling} }

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**Published in:** LIPIcs, Volume 276, 29th International Conference on DNA Computing and Molecular Programming (DNA 29) (2023)

The field of chemical computation attempts to model computational behavior that arises when molecules, typically nucleic acids, are mixed together. By modeling this physical phenomenon at different levels of specificity, different operative computational behavior is observed. Thermodynamic binding networks (TBNs) is a highly abstracted model that focuses on which molecules are bound to each other in a "thermodynamically stable" sense. Stability is measured based only on how many bonds are formed and how many total complexes are in a configuration, without focusing on how molecules are binding or how they became bound. By defocusing on kinetic processes, TBNs attempt to naturally model the long-term behavior of a mixture (i.e., its thermodynamic equilibrium).
We study the problem of signal amplification: detecting a small quantity of some molecule and amplifying its signal to something more easily detectable. This problem has natural applications such as disease diagnosis. By focusing on thermodynamically favored outcomes, we seek to design chemical systems that perform the task of signal amplification robustly without relying on kinetic pathways that can be error prone and require highly controlled conditions (e.g., PCR amplification).
It might appear that a small change in concentrations can result in only small changes to the thermodynamic equilibrium of a molecular system. However, we show that it is possible to design a TBN that can "exponentially amplify" a signal represented by a single copy of a monomer called the analyte: this TBN has exactly one stable state before adding the analyte and exactly one stable state afterward, and those two states "look very different" from each other. In particular, their difference is exponential in the number of types of molecules and their sizes. The system can be programmed to any desired level of resilience to false positives and false negatives. To prove these results, we introduce new concepts to the TBN model, particularly the notions of a TBN’s entropy gap to describe how unlikely it is to be observed in an undesirable state, and feed-forward TBNs that have a strong upper bound on the number of polymers in a stable configuration.
We also show a corresponding negative result: a doubly exponential upper bound, meaning that there is no TBN that can amplify a signal by an amount more than doubly exponential in the number and sizes of different molecules that comprise it. We leave as an open question to close this gap by either proving an exponential upper bound, or giving a construction with a doubly-exponential difference between the stable configurations before and after the analyte is added.
Our work informs the fundamental question of how a thermodynamic equilibrium can change as a result of a small change to the system (adding a single molecule copy). While exponential amplification is traditionally viewed as inherently a non-equilibrium phenomenon, we find that in a strong sense exponential amplification can occur at thermodynamic equilibrium as well - where the "effect" (e.g., fluorescence) is exponential in types and complexity of the chemical components.

Joshua Petrack, David Soloveichik, and David Doty. Thermodynamically Driven Signal Amplification. In 29th International Conference on DNA Computing and Molecular Programming (DNA 29). Leibniz International Proceedings in Informatics (LIPIcs), Volume 276, pp. 8:1-8:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{petrack_et_al:LIPIcs.DNA.29.8, author = {Petrack, Joshua and Soloveichik, David and Doty, David}, title = {{Thermodynamically Driven Signal Amplification}}, booktitle = {29th International Conference on DNA Computing and Molecular Programming (DNA 29)}, pages = {8:1--8:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-297-6}, ISSN = {1868-8969}, year = {2023}, volume = {276}, editor = {Chen, Ho-Lin and Evans, Constantine G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.29.8}, URN = {urn:nbn:de:0030-drops-187917}, doi = {10.4230/LIPIcs.DNA.29.8}, annote = {Keywords: Thermodynamic binding networks, signal amplification, integer programming} }

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**Published in:** LIPIcs, Volume 276, 29th International Conference on DNA Computing and Molecular Programming (DNA 29) (2023)

Discrete chemical reaction networks formalize the interactions of molecular species in a well-mixed solution as stochastic events. Given their basic mathematical and physical role, the computational power of chemical reaction networks has been widely studied in the molecular programming and distributed computing communities. While for Turing-universal systems there is a universal measure of optimal information encoding based on Kolmogorov complexity, chemical reaction networks are not Turing universal unless error and unbounded molecular counts are permitted. Nonetheless, here we show that the optimal number of reactions to generate a specific count x ∈ ℕ with probability 1 is asymptotically equal to a "space-aware" version of the Kolmogorov complexity of x, defined as K̃s(x) = min_p {|p|/log|p| + log(space(𝒰(p))) : 𝒰(p) = x}, where p is a program for universal Turing machine 𝒰. This version of Kolmogorov complexity incorporates not just the length of the shortest program for generating x, but also the space usage of that program. Probability 1 computation is captured by the standard notion of stable computation from distributed computing, but we limit our consideration to chemical reaction networks obeying a stronger constraint: they "know when they are done" in the sense that they produce a special species to indicate completion. As part of our results, we develop a module for encoding and unpacking any b bits of information via O(b/log{b}) reactions, which is information-theoretically optimal for incompressible information. Our work provides one answer to the question of how succinctly chemical self-organization can be encoded - in the sense of generating precise molecular counts of species as the desired state.

Austin Luchsinger, David Doty, and David Soloveichik. Optimal Information Encoding in Chemical Reaction Networks. In 29th International Conference on DNA Computing and Molecular Programming (DNA 29). Leibniz International Proceedings in Informatics (LIPIcs), Volume 276, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{luchsinger_et_al:LIPIcs.DNA.29.9, author = {Luchsinger, Austin and Doty, David and Soloveichik, David}, title = {{Optimal Information Encoding in Chemical Reaction Networks}}, booktitle = {29th International Conference on DNA Computing and Molecular Programming (DNA 29)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-297-6}, ISSN = {1868-8969}, year = {2023}, volume = {276}, editor = {Chen, Ho-Lin and Evans, Constantine G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.29.9}, URN = {urn:nbn:de:0030-drops-187920}, doi = {10.4230/LIPIcs.DNA.29.9}, annote = {Keywords: chemical reaction networks, Kolmogorov complexity, stable computation} }

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Complete Volume

**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

LIPIcs, Volume 257, SAND 2023, Complete Volume

2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 1-278, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{doty_et_al:LIPIcs.SAND.2023, title = {{LIPIcs, Volume 257, SAND 2023, Complete Volume}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {1--278}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023}, URN = {urn:nbn:de:0030-drops-179357}, doi = {10.4230/LIPIcs.SAND.2023}, annote = {Keywords: LIPIcs, Volume 257, SAND 2023, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

Front Matter, Table of Contents, Preface, Conference Organization

2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{doty_et_al:LIPIcs.SAND.2023.0, author = {Doty, David and Spirakis, Paul}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {0:i--0:x}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.0}, URN = {urn:nbn:de:0030-drops-179367}, doi = {10.4230/LIPIcs.SAND.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

The population protocol model describes a network of anonymous agents that interact asynchronously in pairs chosen at random. Each agent starts in the same initial state s. We introduce the dynamic size counting problem: approximately counting the number of agents in the presence of an adversary who at any time can remove any number of agents or add any number of new agents in state s. A valid solution requires that after each addition/removal event, resulting in population size n, with high probability each agent "quickly" computes the same constant-factor estimate of the value log₂(n) (how quickly is called the convergence time), which remains the output of every agent for as long as possible (the holding time). Since the adversary can remove agents, the holding time is necessarily finite: even after the adversary stops altering the population, it is impossible to stabilize to an output that never again changes.
We first show that a protocol solves the dynamic size counting problem if and only if it solves the loosely-stabilizing counting problem: that of estimating log n in a fixed-size population, but where the adversary can initialize each agent in an arbitrary state, with the same convergence time and holding time. We then show a protocol solving the loosely-stabilizing counting problem with the following guarantees: if the population size is n, M is the largest initial estimate of log n, and s is the maximum integer initially stored in any field of the agents' memory, we have expected convergence time O(log n + log M), expected polynomial holding time, and expected memory usage of O(log²(s) + (log log n)²) bits. Interpreted as a dynamic size counting protocol, when changing from population size n_prev to n_next, the convergence time is O(log n_next + log log n_prev).

David Doty and Mahsa Eftekhari. Dynamic Size Counting in Population Protocols. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{doty_et_al:LIPIcs.SAND.2022.13, author = {Doty, David and Eftekhari, Mahsa}, title = {{Dynamic Size Counting in Population Protocols}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.13}, URN = {urn:nbn:de:0030-drops-159558}, doi = {10.4230/LIPIcs.SAND.2022.13}, annote = {Keywords: Loosely-stabilizing, population protocols, size counting} }

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**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

SIMD||DNA [Wang et al., 2019] is a model of DNA strand displacement allowing parallel in-memory computation on DNA storage. We show how to simulate an arbitrary 3-symbol space-bounded Turing machine with a SIMD||DNA program, giving a more direct and efficient route to general-purpose information manipulation on DNA storage than the Rule 110 simulation of Wang, Chalk, and Soloveichik [Wang et al., 2019]. We also develop software (https://github.com/UC-Davis-molecular-computing/simd-dna) that can simulate SIMD||DNA programs and produce SVG figures.

David Doty and Aaron Ong. Simulating 3-Symbol Turing Machines with SIMD||DNA. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{doty_et_al:LIPIcs.SAND.2022.14, author = {Doty, David and Ong, Aaron}, title = {{Simulating 3-Symbol Turing Machines with SIMD||DNA}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {14:1--14:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.14}, URN = {urn:nbn:de:0030-drops-159568}, doi = {10.4230/LIPIcs.SAND.2022.14}, annote = {Keywords: DNA storage, strand displacement, parallel computation} }

Document

**Published in:** LIPIcs, Volume 205, 27th International Conference on DNA Computing and Molecular Programming (DNA 27) (2021)

The thermodynamic binding networks (TBN) model [Breik et al., 2021] is a tool for studying engineered molecular systems. The TBN model allows one to reason about their behavior through a simplified abstraction that ignores details about molecular composition, focusing on two key determinants of a system’s energetics common to any chemical substrate: how many molecular bonds are formed, and how many separate complexes exist in the system. We formulate as an integer program the NP-hard problem of computing stable (a.k.a., minimum energy) configurations of a TBN: those configurations that maximize the number of bonds and complexes. We provide open-source software solving this integer program. We give empirical evidence that this approach enables dramatically faster computation of TBN stable configurations than previous approaches based on SAT solvers [Breik et al., 2019]. Furthermore, unlike SAT-based approaches, our integer programming formulation can reason about TBNs in which some molecules have unbounded counts. These improvements in turn allow us to efficiently automate verification of desired properties of practical TBNs. Finally, we show that the TBN has a natural representation with a unique Hilbert basis describing the "fundamental components" out of which locally minimal energy configurations are composed. This characterization helps verify correctness of not only stable configurations, but entire "kinetic pathways" in a TBN.

David Haley and David Doty. Computing Properties of Thermodynamic Binding Networks: An Integer Programming Approach. In 27th International Conference on DNA Computing and Molecular Programming (DNA 27). Leibniz International Proceedings in Informatics (LIPIcs), Volume 205, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{haley_et_al:LIPIcs.DNA.27.2, author = {Haley, David and Doty, David}, title = {{Computing Properties of Thermodynamic Binding Networks: An Integer Programming Approach}}, booktitle = {27th International Conference on DNA Computing and Molecular Programming (DNA 27)}, pages = {2:1--2:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-205-1}, ISSN = {1868-8969}, year = {2021}, volume = {205}, editor = {Lakin, Matthew R. and \v{S}ulc, Petr}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.27.2}, URN = {urn:nbn:de:0030-drops-146694}, doi = {10.4230/LIPIcs.DNA.27.2}, annote = {Keywords: thermodynamic binding networks, integer programming, constraint programming} }

Document

**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

The standard population protocol model assumes that when two agents interact, each observes the entire state of the other. We initiate the study of message complexity for population protocols, where an agent’s state is divided into an externally-visible message and externally-hidden local state.
We consider the case of O(1) message complexity. When time is unrestricted, we obtain an exact characterization of the stably computable predicates based on the number of internal states s(n): If s(n) = o(n) then the protocol computes semilinear predicates (unlike the original model, which can compute non-semilinear predicates with s(n) = O(log n)), and otherwise it computes a predicate decidable by a nondeterministic O(n log s(n))-space-bounded Turing machine. We then introduce novel O(polylog(n)) expected time protocols for junta/leader election and general purpose broadcast correct with high probability, and approximate and exact population size counting correct with probability 1. Finally, we show that the main constraint on the power of bounded-message-size protocols is the size of the internal states: with unbounded internal states, any computable function can be computed with probability 1 in the limit by a protocol that uses only 1-bit messages.

Talley Amir, James Aspnes, David Doty, Mahsa Eftekhari, and Eric Severson. Message Complexity of Population Protocols. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{amir_et_al:LIPIcs.DISC.2020.6, author = {Amir, Talley and Aspnes, James and Doty, David and Eftekhari, Mahsa and Severson, Eric}, title = {{Message Complexity of Population Protocols}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.6}, URN = {urn:nbn:de:0030-drops-130848}, doi = {10.4230/LIPIcs.DISC.2020.6}, annote = {Keywords: population protocol, message complexity, space-optimal} }

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**Published in:** LIPIcs, Volume 174, 26th International Conference on DNA Computing and Molecular Programming (DNA 26) (2020)

We introduce scadnano (short for "scriptable cadnano"), a computational tool for designing synthetic DNA structures. Its design is based heavily on cadnano [Douglas et al., 2009], the most widely-used software for designing DNA origami [Paul W. K. Rothemund, 2006], with three main differences:
1) scadnano runs entirely in the browser, with no software installation required.
2) scadnano designs, while they can be edited manually, can also be created and edited by a well-documented Python scripting library, to help automate tedious tasks.
3) The scadnano file format is easily human-readable. This goal is closely aligned with the scripting library, intended to be helpful when debugging scripts or interfacing with other software. The format is also somewhat more expressive than that of cadnano, able to describe a broader range of DNA structures than just DNA origami.

David Doty, Benjamin L Lee, and Tristan Stérin. scadnano: A Browser-Based, Scriptable Tool for Designing DNA Nanostructures. In 26th International Conference on DNA Computing and Molecular Programming (DNA 26). Leibniz International Proceedings in Informatics (LIPIcs), Volume 174, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{doty_et_al:LIPIcs.DNA.2020.9, author = {Doty, David and Lee, Benjamin L and St\'{e}rin, Tristan}, title = {{scadnano: A Browser-Based, Scriptable Tool for Designing DNA Nanostructures}}, booktitle = {26th International Conference on DNA Computing and Molecular Programming (DNA 26)}, pages = {9:1--9:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-163-4}, ISSN = {1868-8969}, year = {2020}, volume = {174}, editor = {Geary, Cody and Patitz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.2020.9}, URN = {urn:nbn:de:0030-drops-129624}, doi = {10.4230/LIPIcs.DNA.2020.9}, annote = {Keywords: computer-aided design, structural DNA nanotechnology, DNA origami} }

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Brief Announcement

**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

We study population protocols: networks of anonymous agents whose pairwise interactions are chosen uniformly at random. The size counting problem is that of calculating the exact number n of agents in the population, assuming no leader (each agent starts in the same state). We give the first protocol that solves this problem in sublinear time.
The protocol converges in O(log n log log n) time and uses O(n^60) states (O(1) + 60 log n bits of memory per agent) with probability 1-O((log log n)/n). The time to converge is also O(log n log log n) in expectation. Crucially, unlike most published protocols with omega(1) states, our protocol is uniform: it uses the same transition algorithm for any population size, so does not need an estimate of the population size to be embedded into the algorithm.

David Doty, Mahsa Eftekhari, Othon Michail, Paul G. Spirakis, and Michail Theofilatos. Brief Announcement: Exact Size Counting in Uniform Population Protocols in Nearly Logarithmic Time. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 46:1-46:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{doty_et_al:LIPIcs.DISC.2018.46, author = {Doty, David and Eftekhari, Mahsa and Michail, Othon and Spirakis, Paul G. and Theofilatos, Michail}, title = {{Brief Announcement: Exact Size Counting in Uniform Population Protocols in Nearly Logarithmic Time}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {46:1--46:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.46}, URN = {urn:nbn:de:0030-drops-98359}, doi = {10.4230/LIPIcs.DISC.2018.46}, annote = {Keywords: population protocol, counting, leader election, polylogarithmic time} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Population protocols are a distributed computing model appropriate for describing massive numbers of agents with very limited computational power (finite automata in this paper), such as sensor networks or programmable chemical reaction networks in synthetic biology. A population protocol is said to require a leader if every valid initial configuration contains a single agent in a special "leader" state that helps to coordinate the computation. Although the class of predicates and functions computable with probability 1 (stable computation) is the same whether a leader is required or not (semilinear functions and predicates), it is not known whether a leader is necessary for fast computation. Due to the large number of agents n (synthetic molecular systems routinely have trillions of molecules), efficient population protocols are generally defined as those computing in polylogarithmic in n (parallel) time. We consider population protocols that start in leaderless initial configurations, and the computation is regarded finished when the population protocol reaches a configuration from which a different output is no longer reachable.
In this setting we show that a wide class of functions and predicates computable by population protocols are not efficiently computable (they require at least linear time), nor are some linear functions even efficiently approximable. It requires at least linear time for a population protocol even to approximate division by a constant or subtraction (or any linear function with a coefficient outside of N), in the sense that for sufficiently small gamma > 0, the output of a sublinear time protocol can stabilize outside the interval f(m) (1 +/- gamma) on infinitely many inputs m. In a complementary positive result, we show that with a sufficiently large value of gamma, a population protocol can approximate any linear f with nonnegative rational coefficients, within approximation factor gamma, in O(log n) time. We also show that it requires linear time to exactly compute a wide range of semilinear functions (e.g., f(m)=m if m is even and 2m if m is odd) and predicates (e.g., parity, equality).

Amanda Belleville, David Doty, and David Soloveichik. Hardness of Computing and Approximating Predicates and Functions with Leaderless Population Protocols. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 141:1-141:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{belleville_et_al:LIPIcs.ICALP.2017.141, author = {Belleville, Amanda and Doty, David and Soloveichik, David}, title = {{Hardness of Computing and Approximating Predicates and Functions with Leaderless Population Protocols}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {141:1--141:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.141}, URN = {urn:nbn:de:0030-drops-75044}, doi = {10.4230/LIPIcs.ICALP.2017.141}, annote = {Keywords: population protocol, time lower bound, stable computation} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap.
This answers an open question of Chen and Doty (SODA 2012), who showed that so-called "partial-order" systems producing a unique finite assembly and avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem.

Ho-Lin Chen, David Doty, Ján Manuch, Arash Rafiey, and Ladislav Stacho. Pattern Overlap Implies Runaway Growth in Hierarchical Tile Systems. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 360-373, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chen_et_al:LIPIcs.SOCG.2015.360, author = {Chen, Ho-Lin and Doty, David and Manuch, J\'{a}n and Rafiey, Arash and Stacho, Ladislav}, title = {{Pattern Overlap Implies Runaway Growth in Hierarchical Tile Systems}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {360--373}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.360}, URN = {urn:nbn:de:0030-drops-50935}, doi = {10.4230/LIPIcs.SOCG.2015.360}, annote = {Keywords: self-assembly, hierarchical, pumping} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

We show that the Tile Assembly Model exhibits a strong notion of universality where the goal is to give a single tile assembly system that simulates the behavior of any other tile assembly system. We give a tile assembly system that is capable of simulating a very wide class of tile systems, including itself. Specifically, we give a tile set that simulates the assembly of any tile assembly system in a class of systems that we call \emph{locally consistent}: each tile binds with exactly the strength needed to stay attached, and that there are no glue mismatches between tiles in any produced assembly.
Our construction is reminiscent of the studies of \emph{intrinsic universality} of cellular automata by Ollinger and others, in the sense that our simulation of a tile system $T$ by a tile system $U$ represents each tile in an assembly produced by $T$ by a $c \times c$ block of tiles in $U$, where $c$ is a constant depending on $T$ but not on the size of the assembly $T$ produces (which may in fact be infinite). Also, our construction improves on earlier simulations of tile assembly systems by other tile assembly systems (in particular, those of Soloveichik and Winfree, and of Demaine et al.) in that we simulate the actual process of self-assembly, not just the end result, as in Soloveichik and Winfree's construction, and we do not discriminate against infinite structures. Both previous results simulate only temperature 1 systems, whereas our construction simulates tile assembly systems operating at temperature 2.

David Doty, Jack H. Lutz, Matthew J. Patitz, Scott M. Summers, and Damien Woods. Intrinsic Universality in Self-Assembly. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 275-286, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{doty_et_al:LIPIcs.STACS.2010.2461, author = {Doty, David and Lutz, Jack H. and Patitz, Matthew J. and Summers, Scott M. and Woods, Damien}, title = {{Intrinsic Universality in Self-Assembly}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {275--286}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2461}, URN = {urn:nbn:de:0030-drops-24619}, doi = {10.4230/LIPIcs.STACS.2010.2461}, annote = {Keywords: Biological computing, Molecular computation, intrinsic universality, self-assembly} }