Formal design and analysis for DNA implementations of Chemical Reaction Networks

My collaborators, Qing Dong and Erik Winfree, and I defined a specific form of the bisimulating equivalence Chemical Reaction Network, which can compare two such networks and verify that one is (or is not) an exact implementation of the other. . Second, given a programming language and a concept of its compilation, it would be useful to optimize the compilation result. I also note that two of the new schemes we discovered, each satisfying all but one condition of the impossible set, were found in the process of arriving at this proof.

QD provided the initial formulation and proof of the equivalence of the three-notion theorem and the reaction search algorithm, contributed to the initial manuscript describing these parts, and reviewed the final manuscript. RFJ improved existing definitions and proofs, proved transitivity and modularity theorems, designed algorithms for loop search and graph search, proved completeness results, defined and analyzed the extension for implicit catalysts, and was the lead writer of the final manuscript.

LIST OF TABLES

LIST OF ALGORITHMS

INTRODUCTION

While the CRN language is abstract, it turns out that any CRN can be approximated by DNA molecules using DNA Strand Displacement (DSD) mechanisms. In a sense, these schemas can be considered a compiler for CRNs as a programming language; in one more. Because DSD systems (without infinite polymerization) can be described as CRNs, this method is particularly useful for verifying that the DSD system produced by a compiler correctly implements the intended CRN, although this is not the only use case .

We define CRN bisimulation equivalence, prove properties such as transitivity and modularity, show that CRN bisimulation is algorithmically checkable but NP- or PSPACE-complete depending on the assumptions made, and give algorithms that satisfy those lower bounds completed. One such desirable property is the use of only 2-stranded input complexes; in the process of this investigation, Lulu Qian and I found new 2-string mechanisms that can be used for CRN implementations, which I present in Chapter 4.

VERIFYING CHEMICAL REACTION NETWORK IMPLEMENTATIONS: A BISIMULATION APPROACH

Any such pair is a reaction in the language of formal CRN, but it is an informal reaction only if (R, P)∈ R. In Section 2.4 we showed that the translation scheme from [66] is an exact implementation of the single reaction A. +B → C + D according to CRN bisimulation. Apart from cross-correlation, the main reason for the combined implementation to be incorrect under bisimulation is a failure of the enabling condition.

Because paths through a graph can be searched for in space logarithms in the size of the graph [60], we can check the permissive condition in polynomial space when there are no null species. Each of the rules is a valid inference, so any information inferred by the algorithm will be true. We previously referred to members of NS ×NS not necessarily in R as "reactions in the language of the formal CRN".).

Because the implementation reaction could be interpreted as a reaction in the language of the formal CRN that is "invalid", ie.

Figure 2.1: Implementation of A + B → C + D using the scheme described in [66]. Top: DNA complexes and reactions, given as a diagram of the DNA strand displacement circuit

VERIFYING POLYMER REACTION NETWORKS USING BISIMULATION

The reachability problem is, in an informal sense, the CRN equivalent of the Turing machine halting problem;. This PRN bisimulation can be used to verify designs for physical implementations of polymer systems, as we show in Section 3.4 by proving correctly an updated version of the DNA stacking machine of Qian et al. We have previously defined a concept of CRN bisimulation to check whether the implementation CRN is in fact a correct implementation of the formal CRN [40].

The various DNA complexes in the implementation system are modeled as types of the implementation CRN with interpretations as sets of formal types. Reactions are obtained by replacing sets of monomers with wildcards in reaction schemes so that both sides of the reaction observe the compatibility ratio; for example, ∗1 = AB and ∗2 = BC v. Reverse array detection (local model) Σ. Figure 3.4: This example of polymer reaction networks shows different1. features of the PRN model.

For each formal reaction of the formSi+A→Sj+B, we have a module consisting of those formal species and that formal reaction;. Here we show some of the minimal impingement conditions (within the stack 2 module) in which the formal reaction λ2 +Q2 → λ−2 +λf2 should be able to take place. This problem is in the class Π02, the complement of the second level of the arithmetic hierarchy, which is the class of all languages L={x| ∀y∃zφ(x, y, z)}, where φ is a decidable predicate.

We show that this is a correct implementation, according to modular PRN bisimulation up to reachability, of the given formal response scheme. Conversely, given a single-locus amplified PRN (Σ, e,Ψ0), there is an implementation PRN(Σ0, e0,Ψ0) where all schemes inΨ0 are of the types described in Theorem 3.6.1 with PRN bisimulation- interpretation (π, µ ). Recalling that the false catalyst definition of bisimulation removes non-empty µ-interpretations, the interpretation of the implementation reaction so produced will be the formal reaction in question.

For the most part, the main content of this document is orthogonal to this aspect of the model. It also assumes that the model of the implementation system as PRN is accurate, and the model we used in this example does not.

Figure 3.1: Scaling of number of states in well-mixed versus polymer systems.

SIMPLIFYING CHEMICAL REACTION NETWORK IMPLEMENTATIONS WITH TWO-STRANDED DNA

BUILDING BLOCKS

When we discuss CRN implementations, including existing CRN implementations as well as the two new implementation schemes we propose, we describe the implementations mostly in terms of the motifs without needing the details of the low-level DSD reactions. The next step in making CRN programs practical is to "scale up" the size of the CRNs that can be physically built and generally reduce the leakage and failure rates. Finally, we show how, using CRN bisimulation, these schemes can be proven correct, provided that the assumptions of the formal DSD model reflect real DSD systems.

Each of the two products carries only half the information of the original reactants, so the products of different cases of this reaction can interact in the reverse reaction. Since both types of 4-way strand exchange transform complexes of this form into complexes of the same form with different combinations of domains, we find useful an abstract description of this type of molecule. The InitialXandY complexes are combined with a gate that matches their open combinations, producing two 3-stranded complexes each with one of the strands of X and one of the strands of Y.

These schemes can be understood in light of the motifs discussed earlier: the property of pedestal exchange, namely that another pedestal is opened on the gate, allows for merge and fork logic. Because toe hold responses depend on the combination of the long domain and toe hold, this is valid. 4-way Cooperative CRNs The cooperative 4-way strand exchange motif, when its products recombine with products from another instance of the reaction, simultaneously exchanges the support combinations on a long domain X complex and a long domain Y complex.

DNA complexes in columns labeled A, B, C, or D are interpreted as one copy of the corresponding species, while complexes in columns labeled ∅ are fuel. There are indications that 2-chain DSD systems are generally more robust, but it is an open question whether these particular systems are more robust than current state-of-the-art CRN implementations. We have argued that each of the 5 motifs has a certain abstract behavior and that larger systems such as CRN implementations can.

Figure 4.1: Four previously studied reversible 2-stranded DSD motifs, shown through common examples

IMPOSSIBILITY OF SUFFICIENTLY SIMPLE CHEMICAL REACTION NETWORK IMPLEMENTATIONS IN DNA

STRAND DISPLACEMENT

To the extent that it approximates the behavior of DNA strings, these statements can also be applied to DNA strings. I proved that one cannot design a systematic CRN-to-DSD implementation scheme with all of the following properties: correct under modular CRN bisimulation; uses 4-way branch migration, but not 3-way; uses a constant number of footer domains; I dont use. effectively trimolecular DSD mechanisms; uses only reversible DSD reactions; and uses only 1 or 2 filament fuel. Each of these concepts, including the systematic scheme of implementation, has a formal definition in terms of the DSD model.).

Any of the intermediate steps, and in particular this locality theorem, may be an interesting fact in itself and/or may be useful as part of other proofs related to DSD systems. Statements like the locality theorem come with an intuitive understanding of the statement itself and how it was proved. As an example, I show that a physically reversible DSD system without pseudoknots, without efficient trimolecular reactions, and using 4-way but not 3-way branch migration, cannot be a systematic implementation of reactions of the form A B, using a constant number of support domains and not crosstalk occurs when several reactions of this type are combined.

For example, the Nuskell compiler combines all the above tools to answer exactly that question [4]. The semantics of the system will be defined by reaction rules, each of which says that complexes satisfying a certain pattern (and possibly a predicate) will react in a certain way, and define the products of that reaction. Given a given set of complexes, enumerating species and reactions by iteratively applying reaction rules yields a potentially infinite chemical reaction network, which models the behavior of the DSD system.

A DSD system enumerated from an initial set of strings P and a set of rules is the smallest DSD system such that every complex inP is a species in S, every application of one of the rules to reactants in S is a reaction in R, and its products are species in S. This theorem also assumes some sequence of reactions after which P P0, again without specific details, except that the system is reversible and only 4-way, the pathway contains only unimolecular reactions, and the result P0 has the same 4-way junction made of the same strands. If the junction is first broken by u, then apart from the reverse b, the only reaction that can depend on au, ab is with one of the newly opened domains:.