Chemical Reaction Networks’ Programming for Solving Equations

Round 1

Reviewer 1 Report

In this manuscript, the authors suggested a method to implement chemical reaction networks that can solve equations, where chemical concentrations are used as variables. Using simulations, they demonstrated applications to nonlinear equations such as exponential or triangle equations. Theoretically and mathematically, the method should work as designed.

However, considering actual properties of chemical reactions involving DNA, there are concerns about the reality of the implementation. Particularly, the following points should be clarified, I think.

• First of all, the error evaluation in Section 4 actually evaluated the error observed in the numerical simulation of the chemical reaction network, and I am afraid it just showed the numerical error caused by the simulation algorithm, although Fig. 7 could in some sense show the timescale required for the convergence (i.e. minimum waiting time), and the actual error induced by the nature of chemical reactions is much larger.
• As for the error caused by insufficient waiting time (too short clock cycle for the CMP case), there are some suggestions e.g. Fig. 7; though, more systematic analysis is wanted. The cycle time should be longer than the slowest relaxation.
• There are at least two other sources of errors. One is inverse or incomplete reactions (the yield cannot be 100%). For example, DNA strand separation (Fig. 3) is actually reversible and some by-products may exist. The clock species cannot completely suppress the reaction that should occur in the next step. At least a rough estimation of requirements for the fidelity of reaction is necessary.
• Another source of errors is finite-size effects. Chemicals are actually represented by molecules, and the number of molecules cannot be infinite. The copy number of macromolecules such as DNA is usually limited, and their reactions are stochastic, which may result in fluctuations of concentrations. Even worse, if the number is small, discreteness (integer-ness) of molecules can be significant, particularly in nonlinear reaction cases (so-called "small-number effects" or "discreteness-induced transitions"). The relationship between the system size (number of molecules) and the calculation error should be evaluated (or, at least, it should be shown that the required system size is realistic). Stochastic reaction simulation e.g. by Gillespie's method would be useful.

Minor points

• Lines 271-272 and Fig. 6f: From the result x=0.771566, I guess y=sqrt(0.5), not 0.5.
• Figs. 2 and 4: "The abscissa represents the time unit is seconds". However, in Lines 112-114, it is stated that "We do not consider the change of species concentration in a step, ..., so the simulated concentration will fall or rise vertically". Which is correct?
• Fig. 1: "Moudles" should read "Modules".

 

Author Response

Dear reviewer,

Thank you for your suggestions and the time and effort that reviewers have put into reviewing the previous version of the manuscript. Your suggestions have enabled us to improve our work. Based on the instructions provided in your suggestions, we uploaded the revised manuscript file. Accordingly, we have uploaded a copy of the original manuscript with all the changes highlighted by using the track changes mode in Latex. Appended to this letter is our point-by-point response to the comments raised by the reviewers. The comments are reproduced, and our responses are given directly afterward in a different color (red). We would also like to thank you for allowing us to resubmit a revised copy of the manuscript.

• As for the error caused by insufficient waiting time (too short clock cycle for the CMP case), there are some suggestions e.g. Fig. 7; though, more systematic analysis is wanted. The cycle time should be longer than the slowest relaxation.

For your first comment, we did not use the CMP module in the exponential calculation and trigonometric function simulation in Fig. 7. This module is only used when branch comparison is required. We think it is not clearly introduced in the article, so the description of clock species 133-143 is added in the manuscript. There are chemical oscillators of three clock species in each step, and there are only one clock species in the real reactant. The next step of the reaction requires the output of the previous step. We attach the exponential chemical reaction network.

EulerRsys[a0_] :=
(
  crn = {
    conc[e, 1], conc[element, 1],
    conc[divisor, 1], conc[divisorMultiplier, 1],
    conc[one, 1],conc[xd,a0],
    step[{
      div[element, divisor, elementNext],
      add[divisor, one, divisorNext],
      add[e, elementNextxd, eNext],
      mul[elementNext,xd,elementNextxd]
    }],
    step[{
      ld[elementNextxd, element],
      ld[divisorNext, divisor],
      ld[eNext, e]
    }]
  };
  Return[crn];
)

• There are at least two other sources of errors. One is inverse or incomplete reactions (the yield cannot be 100%). For example, DNA strand separation (Fig. 3) is actually reversible and some by-products may exist. The clock species cannot completely suppress the reaction that should occur in the next step. At least a rough estimation of requirements for the fidelity of reaction is necessary.

In our designed reaction, reactants are equivalent to catalysts. For example, in Figure 3, B → B + X, a multi-step reaction is required. Figure 3 is a method of fitting DNA reaction to a chemical reaction designed by us through a chemical reaction network. In fact, our whole design only considers the calculation of the chemical reaction network, not the actual design of DNA molecules. Allow me to apply the words to the theoretical basis of our design:

“By using strand displacement reactions as a primitive, we construct reaction cascades with effectively unimolecular and bimolecular kinetics. Our construction allows individual reactions to be coupled in arbitrary ways such that reactants can participate in multiple reactions simultaneously, reproducing the desired dynamical properties. Thus arbitrary systems of chemical equations can be compiled into real chemical systems.”Thesis name—DNA as a universal substrate for chemical kinetics, Paper author: David Soloveichika, Georg Seeliga,, and Erik Winfree.

Other papers transform the chemical reaction network into DNA fitting. In our design, we use formal reactions instead of actual species. We first think that the chemical reaction network with DNA as substrate can be fitted to any chemical reaction network. We only care about the design of CRNs, and the specific fitting only depends on the methods designed in other papers. If it does not work, we want to delete this picture.

• Another source of errors is finite-size effects. Chemicals are actually represented by molecules, and the number of molecules cannot be infinite. The copy number of macromolecules such as DNA is usually limited, and their reactions are stochastic, which may result in fluctuations of concentrations. Even worse, if the number is small, discreteness (integer-ness) of molecules can be significant, particularly in nonlinear reaction cases (so-called "small-number effects" or "discreteness-induced transitions"). The relationship between the system size (number of molecules) and the calculation error should be evaluated (or, at least, it should be shown that the required system size is realistic). Stochastic reaction simulation e.g. by Gillespie's method would be useful.

For the third reason you mentioned, fine size effects, the solution we adopt is the reference [35], which is to amplify the concentration in equal proportion. We uniformly amplify the concentration by ten times when calculating a small number. It is added to the manuscript and explained in Discussion361-374. When the number of molecules is small, it will become inaccurate in the deterministic model.

Minor points

1, Lines 271-272 and Fig. 6f: From the result x=0.771566, I guess y=sqrt(0.5), not 0.5.

Thank you for finding this place, although we don't want to ask for it y=sqrt(0.5). The previous result graph is calculated as f(y)=(arcsin1)/2, while the actual calculation result graph f(y)=arcsin (1/2), now corrected.

2, Figs. 2 and 4: "The abscissa represents the time unit is seconds". However, in Lines 112-114, it is stated that "We do not consider the change of species concentration in a step, ..., so the simulated concentration will fall or rise vertically". Which is correct?

Change Initial version{We do not consider the change of species concentration in a step, but only consider the species concentration after reaching the steady-state, so the simulated concentration will fall or rise vertically. }

I think what we're talking about is：

Modified{As can be seen from the results in Fig. 2, in simulating the change of species concentration with time, we did not consider the change of species concentration in each step but only showed the concentration evolution at the end of each step. Each step refers to a calculation step that requires at least three clock species, while only one clock species participates in the calculation. The three clock species will form a chemical oscillator to isolate this calculation from the following calculation. All reactions in the chemical reaction network will have a clock species as reactants in this calculation step. The result chart shows only a straight line in a calculation step; that is, it only shows the time required for this calculation step, and the change of concentration is displayed after the settlement of one step, that is, it needs to reach the steady-state. Therefore, the simulated concentration will change vertically after each calculation step.}

This part introduces 133-143 in the manuscript together with the explanation of clock species

3, Fig. 1: "Moudles" should read "Modules".

Fig. 1 has been modified

Please see the attachment for the revised manuscript.

Author Response File: Author Response.pdf

Reviewer 2 Report

At lines 62-63, the Authors say that their method can "design and solve equations that previous calculation structure could not achieve". I think this point is not very well explained both in the introduction and conclusion, therefore the novelty of the work cannot be grabbed by non-experts, despite working in the area of DNA nanotechnology like this reviewer. It would be good to add such explanatory paragraphs in both sections that clarify in quantitative terms where the novelty is with respect to other schemes. Other than that it is a very solid work, and I think it should be published. 

Author Response

Dear reviewer,

Thank you for your suggestions and the time and effort that reviewers have put into reviewing the previous version of the manuscript. Your suggestions have enabled us to improve our work. Based on the instructions provided in your suggestions, we uploaded the revised manuscript file. Accordingly, we have uploaded a copy of the original manuscript with all the changes highlighted by using the track changes mode in Latex. Appended to this letter is our point-by-point response to the comments raised by the reviewers. The comments are reproduced, and our responses are given directly afterward in a different color (red). We would also like to thank you for allowing us to resubmit a revised copy of the manuscript.

• At lines 62-63, the Authors say that their method can "design and solve equations that previous calculation structure could not achieve". I think this point is not very well explained both in the introduction and conclusion, therefore the novelty of the work cannot be grabbed by non-experts, despite working in the area of DNA nanotechnology like this reviewer. It would be good to add such explanatory paragraphs in both sections that clarify in quantitative terms where the novelty is with respect to other schemes. Other than that it is a very solid work, and I think it should be published. 

We added explanations in the introduction and conclusion, 64-82 in the introduction, and 377-386 in the conclusion.

64-82

The reason why these nonlinear equations could not be solved before was mainly due to the module design of their chemical reaction network. They did not use the basic module more in line with the arithmetic calculation logic, so they could not solve the calculation of nonlinear equations through algorithms in the chemical reaction network. It is also essential that our design uses the chemical oscillator to use the iterative implementation algorithm. We can design and solve this by implementing the discrete and simulation algorithm in the chemical reaction network. This application shows excellent advantages in the simulation algorithm, such as the calculation of polynomial and Newton iteration, and compares with the previous method of using the circuit to construct polynomial. The previous method can only be realized through the cascade of each item but can not be realized by an iterative design algorithm.

Moreover, the previous method of repeatedly constructing the equation instead of directly constructing the analytical solution makes the same kind of solution unnecessary to repeat work. For the discrete algorithm, we realize that the same kind of equation only needs to change the concentration of the coefficient, and there is no need to reconstruct the equation module. In the construction of discrete algorithms, multiple chemical reaction network function modules can be connected, making it possible to connect the multi-step calculation sequence when realizing the equation with an analytical solution.

377-386

This paper simulates and solves the nonlinear equations that cannot be solved before and puts forward the algorithm design for solving three kinds of equations in CRNs. All frameworks are circuit free, which overcomes the limitations of the modular structure used in the previous work and makes it possible to realize the algorithm more in line with mathematical logic rather than circuit logic.

The solution of these three kinds of equations can be attributed to the design of algorithms in CRNs, which can be achieved only when the previous circuit design logic needs to construct very complex modules. In the construction of polynomials, the previous design can undoubtedly approach infinity like the iterative algorithm, which shows the advantage of no circuit structure in the chemical reaction network in solving computational problems.

Please see the attachment for the revised manuscript.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Although I am not convinced that the CRN design by this method is feasible, the manuscript is now acceptable as a pure theory, as its limitations are properly stated.

Just minor points:
- There remains another typo (Moudles) in Figure 1.
- Some parts of the text added in the revised version (e.g. c1 in Page 2) are not very easy to follow; the use of English editing service would be desirable.

Author Response

Dear reviewer,

Thank you again for your suggestions and the time and effort that reviewers have put into reviewing the second version of the manuscript. Your suggestions have played an important role in improving our work. Based on the instructions provided in your suggestions, we uploaded the revised manuscript file. Appended to this letter is our point-by-point response to the comments raised by the reviewers. The comments are reproduced, and our responses are given directly afterward in a different color (red). We would also like to thank you for allowing us to resubmit a revised copy of the manuscript.

• There remains another typo (Moudles) in Figure 1.

Fig.1 has been revised. We apologize for our poor work.

• Some parts of the text added in the revised version (e.g. c1 in Page 2) are not very easy to follow; the use of English editing service would be desirable.

change c1 in Page 2{

The reason why these nonlinear equations could not be solved before was mainly due to the module design of their chemical reaction network. They did not use the basic module more in line with the arithmetic calculation logic, so they could not solve the calculation of nonlinear equations through algorithms in the chemical reaction network. It is also essential that our design uses the chemical oscillator to use the iterative implementation algorithm. We can design and solve this by implementing the discrete and simulation algorithm in the chemical reaction network. This application shows excellent advantages in the simulation algorithm, such as the calculation of polynomial and Newton iteration, and compares with the previous method of using the circuit to construct polynomial. The previous method can only be realized through the cascade of each item but can not be realized by an iterative design algorithm.

}

to  {

Given the limitation of the gate structure of the circuit to realize the calculation, we use a module more in line with the arithmetic logic in the solution process so that we can simulate the algorithm through the combination of basic arithmetic modules in the CRNs. The simulation algorithm in the chemical reaction network realizes the simulation of the polynomial function, which reduces the cost of constructing each term of the gate module, and the simulation of the expansion of the function into a Taylor polynomial also allows us to solve the equation without an analytical solution by the Newton iteration method.

}

change c2 in Page 2{

Moreover, the previous method of repeatedly constructing the equation instead of directly constructing the analytical solution makes the same kind of solution unnecessary to repeat work. For the discrete algorithm, we realize that the same kind of equation only needs to change the concentration of the coefficient, and there is no need to reconstruct the equation module. In the construction of discrete algorithms, multiple chemical reaction network function modules can be connected, making it possible to connect the multi-step calculation sequence when realizing the equation with an analytical solution.}

to {

Furthermore, for previous methods of repeatedly constructing equations rather than directly constructing analytical solutions, we simulate analytical solutions through discrete algorithms so that the same type of equation solving reduces repetitive design. When solving the same type of equation, since the analytical solution is the same, it is only necessary to change the concentration according to the coefficient without rebuilding it. The advantage of the discrete algorithm is that it can connect multiple chemical reaction network function modules, making it possible to effectively connect them according to the order of their calculation when simulating the analytical solution.

}

Author Response File: Author Response.pdf