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arxiv: 2409.06565 · v3 · submitted 2024-09-10 · 🧮 math.PR · math.FA· math.ST· q-bio.QM· stat.ME· stat.TH

Statistical inference for a multiscale stochastic model of enzyme kinetics via propagation of chaos

Arnab Ganguly , Wasiur R. KhudaBukhsh This is my paper

Pith reviewed 2026-05-23 20:46 UTC · model grok-4.3

classification 🧮 math.PR math.FAmath.STq-bio.QMstat.MEstat.TH
keywords stochastic enzyme kineticsstatistical inferencestochastic averaginginteracting particle systemsquasi-steady state approximationproduct formation timesconsistencypropagation of chaos
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The pith

Reaction rates in multiscale enzyme models can be consistently estimated from product formation times alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a stochastic averaging principle that reduces a high-dimensional jump SDE model of multistage enzyme kinetics to a lower-dimensional product-substrate process under a suitable scaling. It introduces an interacting particle system to approximate this reduced dynamics and establishes non-asymptotic bounds and limits for the approximation. These results support an estimator based on an approximate likelihood constructed from product formation time samples, and the consistency of this estimator is rigorously shown without needing full system state observations.

Core claim

Under a scaling regime consistent with the quasi-steady state approximation, the multiscale stochastic model of enzyme kinetics admits a reduced description of the product-substrate dynamics, and an interacting particle system approximation to this reduced process allows construction of a consistent estimator for the reaction rates that uses only a random sample of product formation times.

What carries the argument

The interacting particle system that approximates the product-substrate process at the particle level, enabling the product-form approximate likelihood for the estimator.

If this is right

  • The estimator for reaction rates is consistent as the number of product formation times increases.
  • Non-asymptotic bounds and limiting results hold for the interacting particle system approximation.
  • Inference can be performed without observing the full system states directly.
  • The reduced model captures the essential dynamics of the original multiscale system in the limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This inference method could be adapted to other partially observed multiscale stochastic systems in biology.
  • Computational savings from the reduced model and particle approximation may enable analysis of larger enzyme networks.
  • Further work might explore the rate of convergence of the estimator or its robustness to deviations from the scaling regime.

Load-bearing premise

The scaling regime must be consistent with the quasi-steady state approximation to produce a valid reduced model.

What would settle it

A simulation study in which data generated from the full multiscale model under the scaling regime is used to fit the estimator, and the estimates do not approach the true parameter values as the sample size increases.

Figures

Figures reproduced from arXiv: 2409.06565 by Arnab Ganguly, Wasiur R. KhudaBukhsh.

Figure 1
Figure 1. Figure 1: The accuracy of the sQSSA for the multi-stage MM reaction system in Example 3.1. We compare the deterministic ODE with 100 trajectories of Doob–Gillespie simulations of the original stochastic model. (Left) n = 100. (Right) n = 1000. Other parameters are M = 10, κ1 = 1, κ−1 = 1, κ2 = 1, κ−2 = 1, κP = 0.1. The simulations are performed in Julia programming language v1.9.4 [7]. with the corresponding stochas… view at source ↗
Figure 2
Figure 2. Figure 2: Densities of the MLEs κP and κM in the standard MM kinetic reac￾tion network considered in Example 4.1 obtained by using KDE on 5000 MLEs each obtained from a fresh random sample of size 103 of product formation times. (Left) Density of κP . (Right) Density of κM. True parameter values used in the simulation are n = 106 , M = 10, κ1 = 2, κ−1 = 0.2, κP = 0.1, and T = 2.0 with the true MM constant being κM =… view at source ↗
Figure 3
Figure 3. Figure 3: Posterior densities of the parameters κP and κM in the standard MM kinetic reaction network considered in Example 4.1. (Left) Posterior density of κP . (Right) Poster density of κM. True parameter values used in the simulation are n = 106 , M = 10, κ1 = 2, κ−1 = 0.2, κP = 0.1, and T = 3.0. Therefore, the true MM constant is κM = 0.3. Proof. The proof follows by first writing the integrand as h(ϕ (n) (s), y… view at source ↗
read the original abstract

We study a class of Stochastic Differential Equations (SDEs) with jumps modeling multistage Michaelis--Menten enzyme kinetics, in which a substrate is sequentially transformed into a product via a cascade of intermediate complexes. These networks are typically high-dimensional and exhibit multiscale behavior with a strong coupling between different components, posing substantial analytical and computational challenges. In particular, the problem of statistical inference of reaction rates is significantly difficult and becomes even more intricate when direct observations of system states are unavailable and only a random sample of product formation times is observed. We address this problem in two stages. First, in a suitable scaling regime consistent with the Quasi-Steady State Approximation (QSSA), we rigorously establish a stochastic averaging principle yielding a reduced model for the product-substrate dynamics. Guided by the reduced-order dynamics, we next construct a novel Interacting Particle System (IPS) that approximates the product-substrate process at the particle level. This IPS plays a pivotal role in the inference methodology, and we prove several non-asymptotic bounds and limiting results for this system. These results facilitate the construction of an estimator based on a product-form approximate likelihood requiring only a random sample of product formation times. This approach does not need access to the system states, and we rigorously prove consistency of the estimator.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies a class of jump SDEs modeling multistage Michaelis-Menten enzyme kinetics. In a scaling regime consistent with the Quasi-Steady State Approximation, it establishes a stochastic averaging principle that yields a reduced model for the product-substrate dynamics. It then constructs an interacting particle system (IPS) that approximates the reduced process, proves non-asymptotic bounds and limiting results for the IPS, and uses these to build an estimator from a product-form approximate likelihood that requires only samples of product formation times; consistency of the estimator is proved.

Significance. If the central claims hold, the work supplies a rigorous averaging-plus-IPS route to consistent inference for high-dimensional multiscale biochemical networks under partial observations. The explicit linkage of the scaling regime to QSSA, the propagation-of-chaos analysis, and the derivation of a tractable likelihood from the reduced dynamics are concrete strengths that could influence both theoretical stochastic averaging and applied systems-biology inference.

minor comments (2)
  1. [Introduction] The abstract states that 'several non-asymptotic bounds and limiting results' are proved for the IPS; a short enumerated list of the precise statements (with theorem numbers) in the introduction would help readers locate the key technical contributions.
  2. Notation for the scaling parameters and the QSSA regime is introduced in the abstract and presumably defined later; a single consolidated table or remark collecting all scaling assumptions would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the accurate summary of its contributions and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes a stochastic averaging principle in a QSSA-consistent scaling regime to obtain a reduced product-substrate model, then constructs an IPS approximation whose non-asymptotic bounds and limits are proved directly, and finally builds a product-form approximate likelihood estimator whose consistency is proved from those bounds. No step reduces by definition or self-citation to its own fitted inputs; the averaging, IPS convergence, and consistency results are independent mathematical statements whose assumptions (scaling regime, observation model) are stated separately from the target estimator. The central claim therefore rests on external analytic content rather than tautological renaming or load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the scaling regime and QSSA consistency are invoked as background assumptions without further detail.

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