RNA interference and Register Machines (extended abstract)

RNA interference (RNAi) is a mechanism whereby small RNAs (siRNAs) directly control gene expression without assistance from proteins. This mechanism consists of interactions between RNAs and small RNAs both of which may be single or double stranded. The target of the mechanism is mRNA to be degraded or aberrated, while the initiator is double stranded RNA (dsRNA) to be cleaved into siRNAs. Observing the digital nature of RNAi, we represent RNAi as a Minsky register machine such that (i) The two registers hold single and double stranded RNAs respectively, and (ii) Machine’s instructions are interpreted by interactions of enzyme (Dicer), siRNA (with RISC com- plex) and polymerization (RdRp) to the appropriate registers. Interpreting RNAi as a computational structure, we can investigate the computational meaning of RNAi, especially its complexity. Initially, the machine is configured as a Chemical Ground Form (CGF), which generates incorrect jumps. To remedy this problem, the system is remodeled as recursive RNAi, in which siRNA targets not only mRNA but also the machine instructional analogues of Dicer and RISC. Finally, probabilistic termination is investigated in the recursive RNAi system.

💡 Research Summary

The paper presents a novel computational interpretation of RNA interference (RNAi) by modeling it as a Minsky register machine (MRM). Two registers are defined: one holds single‑stranded RNAs (primarily mRNA) and the other holds double‑stranded RNAs (dsRNA). The biochemical actors of RNAi—Dicer, the siRNA‑RISC complex, and RNA‑dependent RNA polymerase (RdRp)—are mapped to the classic MRM instructions of increment, decrement, and conditional jump. In this mapping, Dicer’s cleavage of dsRNA into siRNA corresponds to a decrement of the dsRNA register and an increment of the siRNA counter; the siRNA‑RISC complex’s degradation of target mRNA corresponds to a decrement of the mRNA register; and RdRp’s synthesis of new dsRNA from siRNA templates corresponds to an increment of the dsRNA register.

These operations are initially encoded in the Chemical Ground Form (CGF), a minimal stochastic process algebra that captures reaction rates and probabilistic transitions. The CGF representation, however, suffers from “incorrect jumps”: conditional jump instructions may fire even when the associated register is zero, reflecting the non‑ideal specificity of biochemical reactions. Such behavior violates the deterministic semantics of a Minsky machine and would lead to computational errors.

To eliminate these spurious jumps, the authors introduce a recursive RNAi (recRNAi) extension. In recRNAi, siRNA not only targets mRNA but also down‑regulates the very components that generate it—namely Dicer and the RISC complex. This creates a feedback loop: as siRNA accumulates, production of Dicer and RISC is suppressed, effectively disabling further decrements when the corresponding registers are empty. Formally, the transition system is enriched with additional reactions that set the probability of a jump to zero whenever the relevant register value is zero, thereby restoring the intended Minsky semantics.

The paper then investigates the termination properties of the recursive system. Termination is defined as reaching the configuration where both registers are zero. By modeling the system as a finite‑state Markov chain with the added feedback transitions, the authors prove that, for any finite initial inventory of RNAs and for biologically plausible rate constants, the probability of an infinite execution path is zero. In other words, the expected time to reach the zero‑state is finite, and the system terminates with probability one (almost‑sure termination). This probabilistic termination contrasts with the classic deterministic halting condition of a Minsky machine, highlighting how stochastic biochemical processes can emulate deterministic computation while still guaranteeing eventual cessation.

Overall, the work bridges molecular biology and theoretical computer science. It demonstrates that RNAi, beyond its regulatory role, can be viewed as a limited computational substrate capable of performing register‑machine style operations. By formalizing the process in CGF and then refining it through recursive feedback, the authors provide a rigorous framework for analyzing the computational complexity and reliability of RNA‑based information processing. Their approach opens avenues for designing synthetic biological circuits that exploit RNAi’s intrinsic logic while ensuring predictable termination behavior.