[Paper Review] Embodying probabilistic inference in biochemical circuits

Yarden Katz, Michael Springer+1 more|arXiv (Cornell University)|Jun 26, 2018

Gene Regulatory Network Analysis39 references16 citations

TL;DR

This paper proposes a biochemical circuit that implements probabilistic inference in single cells using concentration-based probability encoding and directional counting to estimate Markov transition probabilities in real time. By leveraging conjugate priors (Beta-Binomial) and recursive Bayesian updating, the circuit enables dynamic, stochastic inference without relying on steady states, demonstrating how abstract computation can be embodied in cellular biochemistry.

ABSTRACT

Probabilistic inference provides a language for describing how organisms may learn from and adapt to their environment. The computations needed to implement probabilistic inference often require specific representations, akin to having the suitable data structures for implementing certain algorithms in computer programming. Yet it is unclear how such representations can be instantiated in the stochastic, parallel-running biochemical machinery found in cells (such as single-celled organisms). Here, we show how representations for supporting inference in Markov models can be embodied in cellular circuits, by combining a concentration-dependent scheme for encoding probabilities with a mechanism for directional counting. We show how the logic of protein production and degradation constrains the computation we set out to implement. We argue that this process by which an abstract computation is shaped by its biochemical realization strikes a compromise between "rationalistic" information-processing perspectives and alternative approaches that emphasize embodiment.

Motivation & Objective

• To address how probabilistic inference—essential for learning in uncertain environments—can be implemented in the stochastic, parallel biochemical environment of single cells.
• To develop a mechanism that supports real-time, dynamic inference without relying on steady-state assumptions common in chemical reaction network models.
• To bridge abstract computational frameworks (e.g., Bayesian inference) with concrete biochemical implementations using concentration-dependent representations and directional counting.
• To demonstrate that cellular circuits can embody rational inference processes through biologically plausible mechanisms like protein synthesis and degradation.

Proposed method

• Uses a concentration-dependent scheme to encode probability distributions, where molecular concentrations represent posterior beliefs over transition probabilities in a two-state Markov model.
• Employs directional counting via specialized molecular species to track the number of transitions (e.g., A→B) in real time, enabling sequential Bayesian updating.
• Applies conjugate prior updating via the Beta-Binomial model: posterior parameters are updated recursively using indicator variables for each new transition.
• Formalizes the inference process as a filtering problem, where the posterior at step k+1 is computed from the posterior at step k and the new observation (via indicator variables).
• Models the circuit using Kappa for stochastic simulation and generates ODEs via KaDE for deterministic simulation, enabling validation across dynamic and steady-state conditions.
• Uses the median of the posterior distribution as a point estimate for transition probabilities at each time step, ensuring robustness to noise and variability.

Experimental results

Research questions

• RQ1 How can probabilistic inference be implemented in a stochastic, parallel biochemical system without centralized control?
• RQ2 Can concentration-based molecular species represent probability distributions in a way that supports real-time Bayesian updating?
• RQ3 What biochemical mechanisms can enable directional counting of environmental transitions to support sequential inference?
• RQ4 How can recursive Bayesian updating be encoded in protein production and degradation dynamics?
• RQ5 To what extent can a biochemical circuit approximate rational inference in a dynamic, uncertain environment?

Key findings

• The circuit successfully implements recursive Bayesian inference using only molecular concentrations and transition counts, enabling real-time estimation of Markov transition probabilities.
• The posterior over transition probabilities is updated in real time via a recursive formula: $ b^{k+1}_{\pi_{AB}} = \mathrm{Beta}(\alpha^{k}_{0}+I_{AB}, \alpha^{k}_{1}+I_{AA}) $, where $ I_{AB} $ and $ I_{AA} $ are indicator variables for transitions.
• Stochastic simulations using Kappa and deterministic ODEs via KaDE confirm the circuit's robustness and convergence to expected posterior distributions under varying conditions.
• The use of conjugate priors (Beta) with binomial likelihoods allows exact analytical computation of posteriors, avoiding numerical approximation errors.
• The circuit maintains posterior estimates unchanged when no new transitions occur (e.g., when state remains A), reflecting no update to belief in the absence of new data.
• The median of the posterior distribution is used as a point estimate at each time step, providing a stable and biologically interpretable inference output.