Sub-GRNN Search Algorithm

• Sub-GRNN search algorithm is a specialized method that extracts functional subnetworks from biological gene regulatory networks for both mathematical calculation and classification tasks.
• It employs a two-stage search with binarization and margin ranking to match gene expression profiles against predefined patterns, enabling tasks such as Fibonacci recognition and prime identification.
• Robustness is ensured through individual and collective perturbation analyses along with Lyapunov stability checks, demonstrating potential for scalable and parallel biocomputing.

The Sub-GRNN Search Algorithm is a specialized neural-network-based methodology designed to identify and extract functional subnetworks (“sub-GRNNs”) from complex biological gene regulatory neural networks (GRNNs), with the objective of constructing a biocomputing library of mathematical solvers. Operating on experimentally measured gene expression patterns in bacteria exposed to combinatorially encoded input conditions, the algorithm addresses both calculation and classification tasks by evaluating output profiles and stability properties under perturbation (Ratwatte et al., 25 Sep 2025).

1. Algorithm Structure and Workflow

The Sub-GRNN search algorithm uses a two-stage search paradigm tailored to the intended computational task. For mathematical calculation tasks, it systematically scans the GRNN for output genes whose expression profiles, across all considered input codes, closely match a predefined target pattern representing the solution sequence:

• For each gene $p$ and timepoint $t$, the fold-change under input code $i$ and replicate $r$ is computed as

$$d_{i,p,t}^{(r)} = \frac{x_{i,p,t}^{(r)}}{x_{i_0,p,t}^{(r)}}$$

where $x_{i,p,t}^{(r)}$ denotes the measured expression and $i_0$ is the baseline input code.

• The algorithm retains a gene $p$ at time $t$ if, for all relevant input codes $i = 2,\dots,7$ and replicates, $|d_{i,p,t}^{(r)} - f_i| \leq \varepsilon$ for prespecified target values $f_i$ and tolerance $\varepsilon$.

For classification tasks, the procedure is bifurcated:

1. Binarization: A threshold is established (mean expression across inputs) for each gene. Expression values above the threshold are “on” (1), below are “off” (0). Only genes whose binary profile matches the expected class pattern are kept.
2. Margin Ranking: Candidate genes are ranked by the separation between above/below-threshold expression values, optimizing distinctiveness for class assignments.

The final output is therefore a set of functional gene–timepoint pairs $(p^*, t^*)$ satisfying stringent matching criteria for the mathematical or classification task at hand.

2. Functional Applications in Biocomputing

Demonstrated task repertoire achieved through Sub-GRNN search includes:

• Recognition of the $i$th Fibonacci number
• Prime number identification
• Multiplicative factor reconstruction (e.g., $2i$, $3i$, $4i$, $5i$)
• Collatz step count determination
• Classification of “lucky” numbers
• Detection of repeating cycle lengths in decimal expansions (e.g., $1/i$)

These tasks are encoded via chemical perturbations—the biological system is exposed to different input media, each corresponding to an integer code. The regulatory response is measured via gene expression, exploiting the inherent parallelism and adaptability of the transcriptional machinery.

For calculation tasks involving discrete outputs (e.g., Collatz steps), gene expression levels are thresholded using the largest gap between adjacent values:

$\hat{\theta}_p^{(r)} = \frac{x_+ + x_-}{2}$

where $x_+$ and $x_-$ flank the gap. The resulting binary vector is then pattern-matched to the solution.

3. Stability and Robustness Evaluation

To ensure computational reliability, Sub-GRNNs are subjected to rigorous perturbational and stability analyses:

• Gene-wise Perturbation: Each hidden gene is perturbed individually by adding Gaussian noise ($\alpha$, $\sigma^2$). Propagation through the network uses the correlation-weighted adjacency matrix $W$ ($W_{ij}$: regulator effect from $i$ to $j$). Output deviations (in $d_{i,p,t}^{(r)}$ or Hamming distance) yield an $R^2$ score for calculation tasks—high impact genes are thus ranked by sensitivity.
• Collective Perturbation: Sensitivity-ranked critical genes are simultaneously perturbed, with robustness quantified by aggregate $R^2$ or Hamming distance metrics.
• Lyapunov-Based Stability: System robustness is formally characterized by Lyapunov functions:

$$V(t;\alpha,\sigma) = \sum_\text{outputs} (\text{deviation})^2$$

with the stability condition $\frac{dV}{ds} < 0$ (where $s$ is the perturbation scale). For calculation tasks, stability under perturbation is analyzed via the root of

$$2\ell - \frac{k\sigma(s)}{\alpha(s)^3\|\Delta_q\|} = 0$$

setting an upper bound (positive root $s_1$) on permissible disturbance for reliable operation.

4. Architecture Features and Biological Significance

By exploiting the native transcriptional network as a biocomputing substrate, Sub-GRNN search capitalizes on several architectural advantages:

• Parallelism: All genes operate and process information simultaneously, enhancing computational efficiency.
• Stable Cores: The identification of regulatory edges with consistent sign and correlation across input codes yields stable subnetworks resistant to noise.
• Reusability: The same GRNN can be repurposed for multiple distinct computational tasks by changing input encoding and pattern-matching parameters, demonstrating adaptability.

Challenges relate to noise susceptibility and scalability. Perturbing critical genes impacts reliability; cross-talk and metabolic burden present obstacles to large-scale in vivo implementation.

5. Formulas and Quantitative Criteria

Key equations and quantitative measures from the algorithm and evaluation pipeline include:

Concept	Formula	Description
Fold-change (calculation)	$$d_{i,p,t}^{(r)} = x_{i,p,t}^{(r)} / x_{i_0,p,t}^{(r)}$$	Relative gene expression for matching solution values
Binary threshold (classif)	$\hat{\theta}_p^{(r)} = \frac{x_+ + x_-}{2}$	Threshold for discrete on/off classification
Edge consistency score	$$(\text{Same Sign Proportion}) / [1 + \text{Std. Dev. of Correlations}]$$	Quantifies regulatory edge stability across conditions
Lyapunov stability	$$2\ell - [k\sigma(s)] / [\alpha(s)^3 \|\Delta_q\|] = 0$$	Sets perturbation limit for computational stability

6. Implications for Biocomputing

The Sub-GRNN search algorithm demonstrates that native biological networks can be systematically mined for computational functionality. By matching chemical input–encoded gene expression outputs to mathematical patterns, bacterial transcriptional machinery is repurposed for analog computation without hardware. This paradigm introduces new opportunities for robust, adaptable, and parallel biocomputing, while also highlighting intrinsic limitations related to biological noise and stability under perturbation.

A plausible implication is the feasibility of building scalable biological libraries of mathematical solvers, contingent on further advances in minimizing off-target effects and maximizing perturbation tolerance. Moreover, the clear computational mapping from chemical inputs to distinct mathematical tasks showcases the versatility and programmability of biological GRNNs under the Sub-GRNN search framework.