Compiling and Verifying DNA-Based Chemical Reaction Network Implementations

Amazingly, it has been shown that synthesized DNA molecules can be used to build complex chemical circuits that function without any enzymes or cellular components. First, we present a general-purpose CRN-to-DNA compiler that can apply user-defined compilation schemes for translating formal CRNs into domain-level specifications for DNA molecules.

Molecular Computer

Clearly, the goal of molecular computing is not to compete with electronic computers in speed; after all, a molecule can never move faster than an electric current. Instead, the goal of molecular computing is to learn how to program complex molecular interactions and in doing so establish ways to exercise sensitive control at the molecular level.

Figure 1.1: A sample implementation of the example given in the previous paragraph. For simplic- simplic-ity, the double helix structure of the species C is neglected in this diagram and only the pairing information is given.

Importance of Chemical Reaction Networks

We have seen that CRNs are a good modeling language for DNA systems, as for any other type of chemical system. So if we had a general scheme for building a DNA system that implements the behavior of a given arbitrary CRN, CRNs can effectively function as the “source code” for DNA systems and the CRN formalism becomes a “programming language” [ 39 ].

Implementing Arbitrary Chemical Reaction Networks

Otherwise, spurious reactions will occur that are not explained by the system domain specification. In this case, some of the X1 molecules will be converted into chemomolecules as a result of the first reaction in the operation.

Outline

Finally, actual molecules are synthesized according to the sequence information generated in the second step. Given that data, it will produce domain-specific DNA molecules that implement the given CRN, following the procedures outlined in the given translation scheme.

Figure 1.6: CRN-to-DNA compilation schematics

Features

Here gate is a function that takes one argument of type Domain and returns a Structure object. The name of the function comes from the fact that we call types from the input CRN the formal type.

Examples

Now we can evaluate all the bonds appearing in the ugate function, which is shown in Figure. Now notice that there are tilde (∼) characters in the definition of the two Structure objects that.

Figure 2.5: The values of bindings in the where clause of ugate

Enumeration

Semantics for Enumeration

Elementary Domain Step Semantics

Transient and Resting States

NOTE2: In the current version of the code, 3- and 4-way branch moves are evaluated in the same location (they are very similar), this may cause problems with later extensions. Note that this type of reaction can also be reversible or irreversible, depending on whether the displacement reaches the end of the helices.

Figure 4: 1-2 open reaction. Note that this again assumes that the helix is below the threshold for opening reactions (which can be different from 1-1 open thresholds)

Resting States Semantics

Slower reactions (eg, intermolecular interactions) are assumed to occur only between substrates that have already reached equilibrium with fast reactions. Our work in verifying the CRN implementation uses the Wolfe enumerator with this “rest state semantics”.

Others

Examples

Note that the conversion scheme of [39] is "modular" in the sense that it has a module with which to perform each desired reaction independently. You might even want to use an intrinsic non-modular translation scheme (although no such scheme has been reported so far). From this figure, it can be seen that the translation scheme executes the target reaction correctly.

The Meaning of Correctness

In the false pathway, only one amolecule is produced as opposed to the desired number two, indicating that there is a serious logical error in this translation scheme. The astute reader will also note that this translation scheme will also fail on CRN{X+Y →A+B, Z+W →B+C}, even if each reaction is individually compiled correctly. While the corrected version of this translation scheme can be found in [27], this example shows that it can sometimes be very difficult to detect compilation errors by eye.

Figure 4.2: An incorrect translation scheme applied to X + Y → A + A (intended pathway)

Basic Concepts

The core of our problem is to develop an algorithm that can decide which responses are actually 'implemented' in the implemented CRN. Since the definition ofS⊕r asserts that the reaction rcan occur in state S, the path (r0, ..., rk) can occur if and only if all ofr0, r1,. This route cannot occur in state {A, A, B} because even though the first reaction can occur in that state, the resulting state of the first reaction {C} will not have all the reactants needed for the second reaction .

Preparing the Implemented CRN

Here, {A, A, B} are called reactants and {C} are called products, which is in accordance with the general understanding of the terms. This reaction can also take place in the state {A, A, A, B, B}, but it cannot in the state {A, B, C, C, C, C}, because the latter state does not have all the required reactants. Note that these notations appear different from regular chemistry notations, but they are equivalent.

Equivalence with Respect to a CRN Observer

But why do we distinguish between the set of terminal paths and the set of non-terminal paths. Clearly, these should not be considered equivalent in any case, but if we choose not to distinguish between terminal and non-terminal paths, they will indeed be considered equivalent. However, since we have chosen to make a distinction between terminal and non-terminal pathways, we can now distinguish between these two CRNs because Obs TC1,{A→A+B},{A}(A→A+B ) contains while Obs TC2,{A→A+B},{A}.

Equivalence Based on Pathway Decomposition

Types of Species
Implementing Formal Reactions as Pathways
Pathway Decomposition
Conclusion

For example, waste types can be defined as types that do not affect any other type at all. Since we only see the formal species from the initial state before this breaking point, we can say that we are still in the R state, except that we do not detect some of the molecules in it. In a sense, strong confluence means that we can always get an immediate (formal) interpretation of a chemical state (or an intermediate species in a state).

Figure 4.5: The implemented CRN from Fig. 1.5 before and after the preprocessing one another need not be prohibited as long as such interactions do not produce a species that can interact with non-waste species

History Domains

So we can apply the same procedure as above to obtain a path from C1∪ O that has the same initial state as q0 and has at least one response from O. Then we can simply apply that algorithm to the two CRNs we want to compare and decide if they are equivalent. If necessary, we can also put an artificial limit on the width so that when this limit is exceeded, we will force stop the algorithm and produce a warning.

Figure 4.9: Grouping species using wildcards. Figure taken and modified from [39].

Main Theorems

Ifpi is a formal path, then it is regular if and only if its final state is equal to the regular final state or the regular final state is empty. If its final state is equal to its regular final state, it is trivial that is regular. The quintuple of the start state, the end state, the width, the DFS, and the regular end state of a path is called the signature of the path.

Algorithm

The enumerate subroutine is a function that is designed to enumerate the set of signatures of all semiformal paths of maximum width. It's easy to see from the pseudocode above that enumerate(p, w, ret) generates and returns the signature set of all paths of width at most that contain after a prefix. It is also irrelevant how to derive the formal basis from the set of found signatures.

Testing Confluence and Regularity

Output of the Verifier

Improving Performance

The subroutine enumeratenow has one additional argument which is bound to the size of the initial state (as in line 3). Qian Bug A slightly different version of the above translation scheme shown in Fig. is illustrated. Surprisingly, many of the known translation schemes were shown to be wrong in the four CRNs we tried them on.

Overview

Syntax and Semantics

CRN Language

DOM Language

The next two lines, which hold information about the primary and secondary structure of the complex, use dot-bracket notation (as in NUPACK [47]). In this notation, a plus sign indicates strand breaks, a dot indicates that the corresponding domain is unpaired, and a parenthesis indicates that the corresponding domain is paired with the corresponding bracketed domain. While this notation system cannot express every possible structure that can occur in a DNA molecule (in fact, the class of structures that a dot-bracket can express is exactly those without pseudoknots), it is one of the most convenient notations available. today and is widely used in the field.

TS Language

Function definition
Global declaration
Conditionals
Local bindings
Pattern matching
Function calls, attributes, and list indexing

For example, the following code will generate a foothold domain that can be used universally in the implementation. If one wants to create multiple connections at the same time, one can also use the following syntax. Function calls in the TS language are made in the same way as most other programming languages.

Types

Conventional Data Types

The bindings will be created in the order in which they are listed, so the value of x in this case will be 10. There is no 'true' or 'false' keyword in the TS language, so a boolean value only occurs as a result of of logical operators (such as==or!=).

New Data Types

Species data type
Reaction data type
Domain data type
Structure data type
Solution data type

Unlike the above two data types, aDomainobject can be created using built-in functions like shorter long. The domains used to create a Structure object can be referenced later using the names used at the time of creation. For example, the user can refer to the three domains used in X by writing X.a, X.b and X.c.

Built-In Operators

Arithmetic Operators

The domain data type is typically used to define structure objects that require primary and secondary structure information.

Logical Operators

Built-In Functions

tail function
complement function
infty function
short function and long function
unique function
flip function
rev reactions function and irrev reactions function
sum function
len function
reverse function
unirxn function, birxn function, and rxn degree function
map function

This means that it can be used on either a list of integers, a list of lists, or a list of Solutionobjects because the plus operator is overloaded for these types. Theunirxn function takes an argument which is a list of reaction objects, filters out reactions that do not take exactly one reactant, and returns the resulting list. This function works in the same way as the map function in any other functional language.

Runtime

However, their reactants and products fields contain structure objects obtained by running each species appearing in the given reaction through the formal function. The compiler runs all formal species appearing in the input CRN through the 'formal' function to obtain formal species. The list of reaction objects prepared in the second step is passed to the 'main' function of the given TS code.

Examples

Therefore, the return value of our main function will be sum(map(unimolecular, unirxn(crn))). To calculate the return value of unimolecular(r), we must first evaluate the value of g and t from the where clause. Note that the value of the first argument s is A, which is a Structure object created with the formal function.

Figure A.6: The values of bindings in the where clause of ugate

CRNs