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

Life is built upon amazingly sophisticated molecular machines whose behavior combines mechanical and chemical action. Engineering of similarly complex nanoscale devices from first principles remains an as yet unrealized goal of bioengineering. In this paper we formalize a simple model of mechanical motion (mechanical linkages) combined with chemical bonding. The model has a natural implementation using DNA with double-stranded rigid links, and single-stranded flexible joints and binding sites. Surprisingly, we show that much of the complex behavior is preserved in an idealized topological model which considers solely the graph connectivity of the linkages. We show a number of artifacts including Boolean logic, catalysts, a fueled motor, and chemo-mechanical coupling, all of which can be understood and reasoned about in the topological model. The variety of achieved behaviors supports the use of topological chemical linkages in understanding and engineering complex molecular behaviors.

Keenan Breik, Austin Luchsinger, and David Soloveichik. Molecular Machines from Topological Linkages. In 27th International Conference on DNA Computing and Molecular Programming (DNA 27). Leibniz International Proceedings in Informatics (LIPIcs), Volume 205, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{breik_et_al:LIPIcs.DNA.27.7, author = {Breik, Keenan and Luchsinger, Austin and Soloveichik, David}, title = {{Molecular Machines from Topological Linkages}}, booktitle = {27th International Conference on DNA Computing and Molecular Programming (DNA 27)}, pages = {7:1--7:20}, 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.27.7}, URN = {urn:nbn:de:0030-drops-146749}, doi = {10.4230/LIPIcs.DNA.27.7}, annote = {Keywords: chemical computation, mechanical computation, bioengineering, models of biochemistry, molecular machines, mechanical linkages, generic rigidity} }

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

Formal methods have enabled breakthroughs in many fields, such as in hardware verification, machine learning and biological systems. The key object of interest in systems biology, synthetic biology, and molecular programming is chemical reaction networks (CRNs) which formalizes coupled chemical reactions in a well-mixed solution. CRNs are pivotal for our understanding of biological regulatory and metabolic networks, as well as for programming engineered molecular behavior. Although it is clear that small CRNs are capable of complex dynamics and computational behavior, it remains difficult to explore the space of CRNs in search for desired functionality. We use Alloy, a tool for expressing structural constraints and behavior in software systems, to enumerate CRNs with declaratively specified properties. We show how this framework can enumerate CRNs with a variety of structural constraints including biologically motivated catalytic networks and metabolic networks, and seesaw networks motivated by DNA nanotechnology. We also use the framework to explore analog function computation in rate-independent CRNs. By computing the desired output value with stoichiometry rather than with reaction rates (in the sense that X → Y+Y computes multiplication by 2), such CRNs are completely robust to the choice of reaction rates or rate law. We find the smallest CRNs computing the max, minmax, abs and ReLU (rectified linear unit) functions in a natural subclass of rate-independent CRNs where rate-independence follows from structural network properties.

Marko Vasic, David Soloveichik, and Sarfraz Khurshid. CRNs Exposed: A Method for the Systematic Exploration of Chemical Reaction Networks. In 26th International Conference on DNA Computing and Molecular Programming (DNA 26). Leibniz International Proceedings in Informatics (LIPIcs), Volume 174, pp. 4:1-4:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{vasic_et_al:LIPIcs.DNA.2020.4, author = {Vasic, Marko and Soloveichik, David and Khurshid, Sarfraz}, title = {{CRNs Exposed: A Method for the Systematic Exploration of Chemical Reaction Networks}}, booktitle = {26th International Conference on DNA Computing and Molecular Programming (DNA 26)}, pages = {4:1--4:25}, 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.2020.4}, URN = {urn:nbn:de:0030-drops-129574}, doi = {10.4230/LIPIcs.DNA.2020.4}, annote = {Keywords: molecular programming, formal methods} }

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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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