Characterizing the Asymmetry in Hardness between Synthesis and Destruction of Heteropolymers

We present a simple model describing the assembly and disassembly of heteropolymers consisting of two types of monomers $A$ and $B$. We prove that no matter how we manipulate the concentrations of $A$ and $B$, it takes longer than the exponential function of $d$ to synthesize a fixed amount of the desired heteropolymer, where $d$ is the number of $A\text{−}B$ connections. We also prove the decomposition time is linear for chain length $n$. When $d$ is proportional to $n$, synthesis and destruction have an exponential asymmetry. Our findings may facilitate research on the more general asymmetry of operational hardness.

Figure 1

Schematic illustration of the synthesis and destruction of a one-dimensional molecular chain $XABA$. The green triangle, blue circle, and red square represent the reaction nucleus $X$, molecule $A$, and molecule $B$, respectively. We assume that the only possible reactions are the sequential binding or detachment of the molecules.

Figure 2

Schematic illustration of the setup. (a) Setup for synthesis process. The reaction tank is connected to particle reservoirs that supply reaction nucleus $X$ and material molecules $A$ and $B$. Molecule $X$ is supplied at a constant rate $J_X$. The operator’s task is to synthesize the desired molecule ($XABA$ in this example) as efficiently as possible by controlling the concentrations $c_A(t)$ and $c_B(t)$. (b) Setup for destruction process. The concentration of the molecular chain to be decomposed ($XABA$ in this example) is kept constant and $X$ is recovered to the particle bath at rate ${\tau }_X^{-1}{c}_X(t)$. The total amount of $X$ recovered is ${\mathrm{\Sigma }}_X^{\prime }[0,T]$.

Figure 3

Reaction network in our model. (a) Schematic illustration of the chain synthesis. The color intensity represents the number of molecules. The number of molecules decreases exponentially with the chain length because of the inevitable formation of byproducts. (b) Schematic illustration of chain decomposition. In this case, there is no by-product, and the decomposition proceeds along a single path (red solid line).