Erratum: A Sign Error in the Proof of Theorem 4 (DNA25)

2026-04-01

The Error

In the paper:

X. Huang, T.H. Klinge, and J.I. Lathrop. “Real-Time Equivalence of Chemical Reaction Networks and Analog Computers.” International Conference on DNA Computing and Molecular Programming (DNA 25), 2019.

Theorem 4 proves that $e \in R_{RTCRN}$ using the CRN:

X \xrightarrow{1} \varnothing, \qquad X + Y \xrightarrow{1} X,

with initial conditions $x(0) = y(0) = 1$. The paper states that this induces the ODE system

x'(t) = -x(t), \qquad y'(t) = -x(t)\,y(t),

and claims the solution is

$$x(t) = e^{-t}, \qquad y(t) = e^{1 - e^{-t}}.$$

The closed-form solution for $y(t)$ has a sign error in the exponent. The correct solution is:

$$y(t) = e^{e^{-t} - 1}.$$

Verification

From $y' = -xy$ with $x = e^{-t}$:

\frac{dy}{y} = -e^{-t}\,dt \implies \ln y = e^{-t} + C.

With $y(0) = 1$: $\ln 1 = 1 + C$, so $C = -1$. Therefore:

$$\ln y(t) = e^{-t} - 1, \qquad y(t) = e^{e^{-t} - 1}.$$

As $t \to \infty$: $y(t) \to e^{0-1} = e^{-1} = 1/e$, not $e$.

The paper’s claimed solution $y(t) = e^{1-e^{-t}}$ would satisfy $y' = +xy$ (positive sign), which does not match the CRN’s mass-action kinetics.

Impact on the Theorem

Theorem 4 ($e \in R_{RTCRN}$) remains correct. The CRN computes $1/e$ (not $e$ directly), but since $R_{RTCRN}$ is a field (Theorem 2 in the same paper), $e = 1/(1/e) \in R_{RTCRN}$.

Alternatively, one can fix the CRN by replacing the reaction $X + Y \to X$ (which removes $Y$) with a reaction that produces $Y$, such as $X \to X + Y$ (giving $y' = +x$ rather than $y' = -xy$). However, the simplest fix is to note that the original CRN computes $1/e$, and the field closure gives $e$.

Corrected Statement

The original CRN computes $1/e$, not $e$. To compute $e$ directly, flip the sign in the second reaction:

Corrected CRN:

X \xrightarrow{1} \varnothing, \qquad X \xrightarrow{1} X + Y,

with $x(0) = 1$, $y(0) = 0$. This gives $x'=-x$, $y'=x$, so $x(t) = e^{-t}$ and $y(t) = 1 - e^{-t}$. By Theorem 3 (shifting initial values), this computes $1$, not $e$. Instead, use:

X \xrightarrow{1} \varnothing, \qquad Y' = X \cdot Y, \quad x(0)=1,\; y(0)=1.

This gives $y'(t) = +x(t)\,y(t)$, with solution $y(t) = e^{1-e^{-t}} \to e$. The corrected ODE is the one the paper intended; the CRN reaction notation should be read as $X$ catalyzing $Y$ production (e.g., $X \to X + Y$ with rate $xy$ via suitable bimolecular encoding).

Theorem 4 (corrected). With $y' = xy$, $x' = -x$, $x(0) = y(0) = 1$: $y(t) = e^{1-e^{-t}}$, so $|y(t) - e| = O(e^{-t})$, and $e \in R_{RTCRN}$. $\square$