arXiv subjects

Santiago Schnell

Publications and source records attributed to Santiago Schnell.

At least 19 recordsLinked to original sources

arXiv2026

The Stairs of Reconciliation: A Mathematical Tourist in Graz

Inside the Grazer Burg, two late-Gothic stone flights rise about distinct spindles, overlap, share several treads, and separate again. Their plan is governed not by a coaxial double helix but, to first approximation, by two intersecting circles. This elementary geometry yields a model of recurrent meeting and makes explicit the compatibility conditions that meeting requires. It also leads to a second object that geometers call a double spiral staircase - the helicoid - and to a useful distinction between resemblance and identity. The staircase becomes a meditation on how paths, models, and disciplines can meet without becoming the same.

math.HO↗

arXiv2026

Who Wrote This? Turing, Total Variation, and the Mathematics of AI-Text Detection

What can a finished text reveal about the process that produced it? Drawing on Turing's imitation game and statistical decision theory, this article examines the limits of AI-text detection as an inference from a completed object to an unobserved history. For two known source distributions with equal prior probabilities, a standard identity gives the minimum average classification error as half their probability overlap. This overlap is one minus their total variation distance. Perfect detection therefore requires nonoverlapping distributions; useful discrimination does not. In practice, the problem is harder: human and machine writing form changing families of distributions, posterior probabilities depend on base rates, and ``AI-written'' becomes ambiguous when people and software contribute to the same text. Turing's interrogator can ask another question; a detector restricted to finished prose cannot. Distinguishability, source attribution, authentication, and compliance are different inferential tasks. The paper's own documented human-AI provenance shows what a production history can reveal that a binary label cannot.

math.HO↗

arXiv2026

Liquid democracy under vote correlation: On the fallacies of averaging and the excluded middle

Liquid democracy permits voters to vote directly or delegate their votes to others. Existing algorithmic analyses assign each voter a single scalar parameter, interpreted as an independent probability of voting for the ground truth. This representation is inadequate when delegation is fixed before public information changes different voters' reliability in different ways. In this work, we study a minimal common-signal model of this phenomenon. Delegation occurs before a public binary signal is realised, while voting occurs afterwards. Conditional on the signal, sink votes are independent, and each voter has a signal-specific competence; marginally, correctness events are correlated through the common signal. We show that delegation based on average competence is not a safe scalarisation: it can violate do-no-harm, and a beneficial delegation rule may send votes to voters with lower average competence. We present and analyse three novel delegation mechanisms for this setting. The first is a conservative intersection mechanism that delegates only to neighbours whose competence exceeds the delegator's by a prescribed margin in every signal state; the conservative intersection mechanism inherits all the guarantees of delegation in the scalar competence setting. The next mechanism is a confounded-set mechanism that enable the delegation to neighbours who are favoured in one state and worse by at most a prescribed tolerance in the other. For this mechanism, we prove expected-margin bounds, and identify the weight-dispersion and mechanism-concentration conditions needed to obtain majority-correctness guarantees. Finally, on bounded-in-degree graphs, a multi-round certified-path mechanism propagates nonnegative two-dimensional path certificates; it is acyclic, yields statewise terminal competence improvement, and gives uniform bounds on path length and terminal voting weight.

cs.GT↗

arXiv2026

Lumping of reaction networks: Generic and critical parameters

We investigate linear lumping for parameter-dependent mass action reaction networks, distinguishing between generic and critical parameter regimes. For generic parameters -- those ranging in some non-empty open subset of parameter space -- we prove that exact linear lumping yields only "obvious" reductions: elimination of non-reactant species or projections along stoichiometric first integrals. This characterization extends to reaction networks with product-form kinetics, including Michaelis-Menten and Hill-type rate laws. For mass action systems we proceed to develop an algorithmic approach to identify critical parameter sets -- algebraic subvarieties in parameter space where non-trivial lumpings become available. This procedure reduces the determination of lumping maps to a system of finitely many polynomial equations. It also applies to constrained lumping scenarios (which are frequently motivated by chemical considerations). We then review and extend results about proper lumpings. Finally, we discuss lumpings of a self-replicator system, and of a two-pathway enzyme mechanism, to document the viability of our methods in relevant scenarios. Our results clarify the relationship between structural (parameter-independent) and fine-tuned (parameter-dependent) reductions, with implications for approximate lumping when system parameters lie near critical values

math.DS↗

arXiv2026

Strategies of cooperation and defection in five large language models

Large language models (LLMs) are increasingly deployed to support human decision-making. This use of LLMs has concerning implications, especially when their prescriptions affect the welfare of others. To gauge how LLMs make social decisions, we explore whether five leading models produce sensible strategies in the repeated prisoner's dilemma, which is the main metaphor of reciprocal cooperation. First, we measure the propensity of LLMs to cooperate in a neutral setting, without using language reminiscent of how this game is usually presented. We record to what extent LLMs implement Nash equilibria or other well-known strategy classes. Thereafter, we explore how LLMs adapt their strategies to changes in parameter values. We vary the game's continuation probability, the payoff values, and whether the total number of rounds is commonly known. We also study the effect of different framings. In each case, we test whether the adaptations of the LLMs are in line with basic intuition, theoretical predictions of evolutionary game theory, and experimental evidence from human participants. While all LLMs perform well in many of the tasks, none of them exhibit full consistency over all tasks. We also conduct tournaments between the inferred LLM strategies and study direct interaction between LLMs in games over ten rounds with a known or unknown last round. Our experiments shed light on how current LLMs instantiate reciprocal cooperation.

cs.CY↗

arXiv2025

Accuracy versus Predominance: Reassessing the validity of the quasi-steady-state approximation

The application of the standard quasi-steady-state approximation to the Michaelis--Menten reaction mechanism is a textbook example of biochemical model reduction, derived using singular perturbation theory. However, determining the specific biochemical conditions that dictate the validity of the standard quasi-steady-state approximation remains a challenging endeavor. Emerging research suggests that the accuracy of the standard quasi-steady-state approximation improves as the ratio of the initial enzyme concentration, $e_0$, to the Michaelis constant, $K_M$, decreases. In this work, we examine this ratio and its implications for the accuracy and validity of the standard quasi-steady-state approximation as compared to other quasi-steady-state reductions in its proximity. Using standard tools from the analysis of ordinary differential equations, we show that while $e_0/K_M$ provides an indication of the standard quasi-steady-state approximation's asymptotic accuracy, the standard quasi-steady-state approximation's predominance relies on a small ratio of $e_0$ to the Van Slyke-Cullen constant, $K$. Here, we define the predominance of a quasi-steady-state reduction when it offers the highest approximation accuracy among other well-known reductions with overlapping validity conditions. We conclude that the magnitude of $e_0/K$ offers the most accurate measure of the validity of the standard quasi-steady-state approximation.

physics.chem-ph↗

arXiv2024

The Michaelis--Menten reaction at low substrate concentrations: Pseudo-first-order kinetics and conditions for timescale separation

We demonstrate that the Michaelis-Menten reaction mechanism can be accurately approximated by a linear system when the initial substrate concentration is low. This leads to pseudo-first-order kinetics, simplifying mathematical calculations and experimental analysis. Our proof utilizes a monotonicity property of the system and Kamke's comparison theorem. This linear approximation yields a closed-form solution, enabling accurate modeling and estimation of reaction rate constants even without timescale separation. Building on prior work, we establish that the sufficient condition for the validity of this approximation is $s_0 \ll K$, where $K=k_2/k_1$ is the Van Slyke-Cullen constant. This condition is independent of the initial enzyme concentration. Further, we investigate timescale separation within the linear system, identifying necessary and sufficient conditions and deriving the corresponding reduced one-dimensional equations.

math.DS↗

arXiv2024

Rigorous estimates for the quasi-steady state approximation of the Michaelis-Menten reaction mechanism at low enzyme concentrations

There is a vast amount of literature concerning the appropriateness of various perturbation parameters for the standard quasi-steady state approximation in the Michaelis-Menten reaction mechanism, and also concerning the relevance of these parameters for the accuracy of the approximation by the familiar Michaelis-Menten equation. Typically, the arguments in the literature are based on (heuristic) timescale estimates, from which one cannot obtain reliable quantitative estimates for the error of the quasi-steady state approximation. We take a different approach. By combining phase plane analysis with differential inequalities, we derive sharp explicit upper and lower estimates for the duration of the initial transient and substrate depletion during this transitory phase. In addition, we obtain rigorous bounds on the accuracy of the standard quasi-steady state approximation in the slow dynamics regime. Notably, under the assumption that the quasi-steady state approximation is valid over the entire time course of the reaction, our error estimate is of order one in the Segel-Slemrod parameter.

math.DS↗

arXiv2023

Natural parameter conditions for singular perturbations of chemical and biochemical reaction networks

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving "small parameters" (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis--Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.

math.DS↗

arXiv2022

On the anti-quasi-steady-state conditions of enzyme kinetics

Quasi-steady state reductions for the irreversible Michaelis--Menten reaction mechanism are of interest both from a theoretical and an experimental design perspective. A number of publications have been devoted to extending the parameter range where reduction is possible, via improved sufficient conditions. In the present note, we complement these results by exhibiting local conditions that preclude quasi-steady-state reductions (anti-quasi-steady-state), in the classical as well as in a broader sense. To this end, one needs to obtain necessary (as opposed to sufficient) conditions and determine parameter regions where these do not hold. In particular, we explicitly describe parameter regions where no quasi-steady-state reduction (in any sense) is applicable (anti-quasi-steady-state conditions), and we also show that -- in a well defined sense -- these parameter regions are small. From another perspective, we obtain local conditions for the accuracy of standard or total quasi-steady-state. Perhaps surprisingly, our conditions do not involve initial substrate.

physics.chem-ph↗

arXiv2022

Stochastic enzyme kinetics and the quasi-steady-state reductions: Application of the slow scale linear noise approximation à la Fenichel

The linear noise approximation models the random fluctuations from the mean-field model of a chemical reaction that unfolds near the thermodynamic limit. Specifically, the fluctuations obey a linear Langevin equation up to order $Ω^{-1/2}$, where $Ω$ is the size of the chemical system (usually the volume). In the presence of disparate timescales, the linear noise approximation admits a quasi-steady-state reduction referred to as the \textit{slow scale} linear noise approximation (ssLNA). Curiously, the ssLNAs reported in the literature are slightly different. The differences in the reported ssLNAs lie at the mathematical heart of the derivation. In this work, we derive the ssLNA directly from geometric singular perturbation theory and explain the origin of the different ssLNAs in the literature. Moreover, we discuss the loss of normal hyperbolicity and we extend the ssLNA derived from geometric singular perturbation theory to a non-classical singularly perturbed problem. In so doing, we disprove a commonly-accepted qualifier for the validity of stochastic quasi-steady-state approximation of the Michaelis--Menten reaction mechanism.

physics.chem-ph↗

arXiv2021

Hunting $\varepsilon$: The origin and validity of quasi-steady-state reductions in enzyme kinetics

The estimation of the kinetic parameters requires the careful design of experiments under a constrained set of conditions. Many estimates reported in the literature incorporate protocols that leverage simplified mathematical models known as quasi-steady-state reductions. Such reductions often - but not always - emerge as the result of a singular perturbation scenario. However, the utilization of the singular perturbation reduction method requires knowledge of a dimensionless parameter, $\varepsilon$, that is proportional to the ratio of the reaction's fast and slow timescales. Using techniques from differential equations, Fenichel theory, and center manifold theory, we derive the appropriate $\varepsilon$ whose magnitude regulates the validity of the quasi-steady-state reduction employed in the reported experimental procedures for intermolecular autocatalytic zymogen activation reaction. Although the model equations are two-dimensional, the fast/slow dynamics are rich. The phase plane exhibits a dynamic transcritical bifurcation point in a particular singular limit. The existence of such a bifurcation is relevant, because the critical manifold losses normal hyperbolicity and classical Fenichel theory is inapplicable. Furthermore, we show that in some cases chemical reversibility can be interpreted dynamically as an imperfection, since the presence of reversibility can destroy the bifurcation structure present in the singular limit. We show that the reduction method by which QSS reductions are justified can depend on the path taken in parameter space. Specifically, we show that the standard quasi-steady-state reduction for this reaction is justifiable by center manifold theory in one limit, and via Fenichel theory in a different limit.

math.DS↗

arXiv2021

On the validity of the stochastic quasi-steady-state approximation in open enzyme catalyzed reactions: Timescale separation or singular perturbation?

The quasi-steady-state approximation is widely used to develop simplified deterministic or stochastic models of enzyme catalyzed reactions. In deterministic models, the quasi-steady-state approximation can be mathematically justified from singular perturbation theory. For several closed enzymatic reactions, the homologous extension of the quasi-steady-state approximation to the stochastic regime, known as the stochastic quasi-steady-state approximation, has been shown to be accurate under the analogous conditions that permit the quasi-steady-state reduction of the deterministic counterpart. However, it was recently demonstrated that the extension of the stochastic quasi-steady-state approximation to an open Michaelis--Menten reaction mechanism is only valid under a condition that is far more restrictive than the qualifier that ensures the validity of its corresponding deterministic quasi-steady-state approximation. In this paper, we suggest a possible explanation for this discrepancy from the lens of geometric singular perturbation theory. In so doing, we illustrate a misconception in the application of the quasi-steady-state approximation: timescale separation does not imply singular perturbation.

math.DS↗

arXiv2021

On the quasi-steady-state approximation in an open Michaelis--Menten reaction mechanism

The conditions for the validity of the standard quasi-steady-state approximation in the Michaelis--Menten mechanism in a closed reaction vessel have been well studied, but much less so the conditions for the validity of this approximation for the system with substrate inflow. We analyze quasi-steady-state scenarios for the open system attributable to singular perturbations, as well as less restrictive conditions. For both settings we obtain distinguished invariant slow manifolds and time scale estimates, and we highlight the special role of singular perturbation parameters in higher order approximations of slow manifolds. We close the paper with a discussion of distinguished invariant manifolds in the global phase portrait.

math.DS↗

arXiv2020

The Quasi-Steady-State Approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetic

In this work, we revisit the scaling analysis and commonly accepted conditions for the validity of the standard, reverse and total quasi-steady-state approximations through the lens of dimensional Tikhonov-Fenichel parameters and their respective critical manifolds. By combining Tikhonov-Fenichel parameters with scaling analysis and energy methods, we derive improved upper bounds on the approximation error for the standard, reverse and total quasi-steady-state approximations. Furthermore, previous analyses suggest that the reverse quasi-steady-state approximation is only valid when initial enzyme concentrations greatly exceed initial substrate concentrations. However, our results indicate that this approximation can be valid when initial enzyme and substrate concentrations are of equal magnitude. Using energy methods, we find that the condition for the validity of the reverse quasi-steady-state approximation is far less restrictive than was previously assumed, and we derive a new "small" parameter that determines the validity of this approximation. In doing so, we extend the established domain of validity for the reverse quasi-steady-state approximation. Consequently, this opens up the possibility of utilizing the reverse quasi-steady-state approximation to model enzyme catalyzed reactions and estimate kinetic parameters in enzymatic assays at much lower enzyme to substrate ratios than was previously thought. Moreover, we show for the first time that the critical manifold of the reverse quasi-steady-state approximation contains a singular point where normal hyperbolicity is lost. Associated with this singularity is a transcritical bifurcation, and the corresponding normal form of this bifurcation is recovered through scaling analysis.

math.DS↗

arXiv2019

Characteristic, completion or matching timescales? An analysis of temporary boundaries in enzyme kinetics

Scaling analysis exploiting timescale separation has been one of the most important techniques in the quantitative analysis of nonlinear dynamical systems in mathematical and theoretical biology. In the case of enzyme catalyzed reactions, it is often overlooked that the characteristic timescales used for the scaling the rate equations are not ideal for determining when concentrations and reaction rates reach their maximum values. In this work, we first illustrate this point by considering the classic example of the single-enzyme, single-substrate Michaelis--Menten reaction mechanism. We then extend this analysis to a more complicated reaction mechanism, the auxiliary enzyme reaction, in which a substrate is converted to product in two sequential enzyme-catalyzed reactions. In this case, depending on the ordering of the relevant timescales, several dynamic regimes can emerge. In addition to the characteristic timescales for these regimes, we derive matching timescales that determine (approximately) when the transitions from initial fast transient to steady-state kinetics occurs. The approach presented here is applicable to a wide range of singular perturbation problems in nonlinear dynamical systems.

q-bio.QM↗

arXiv2013

Single molecule enzymology a la Michaelis-Menten

In the past one hundred years, deterministic rate equations have been successfully used to infer enzyme-catalysed reaction mechanisms and to estimate rate constants from reaction kinetics experiments conducted in vitro. In recent years, sophisticated experimental techniques have been developed that allow the measurement of enzyme- catalysed and other biopolymer-mediated reactions inside single cells at the single molecule level. Time course data obtained by these methods are considerably noisy because molecule numbers within cells are typically quite small. As a consequence, the interpretation and analysis of single cell data requires stochastic methods, rather than deterministic rate equations. Here we concisely review both experimental and theoretical techniques which enable single molecule analysis with particular emphasis on the major developments in the field of theoretical stochastic enzyme kinetics, from its inception in the mid-twentieth century to its modern day status. We discuss the differences between stochastic and deterministic rate equation models, how these depend on enzyme molecule numbers and substrate inflow into the reaction compartment and how estimation of rate constants from single cell data is possible using recently developed stochastic approaches.

q-bio.SC↗

arXiv2006

A multiscale mathematical model of cancer, and its use in analyzing irradiation therapies

Background: Radiotherapy outcomes are usually predicted using the Linear Quadratic model. However, this model does not integrate complex features of tumor growth, in particular cell cycle regulation. Methods: In this paper, we propose a multiscale model of cancer growth based on the genetic and molecular features of the evolution of colorectal cancer. The model includes key genes, cellular kinetics, tissue dynamics, macroscopic tumor evolution and radiosensitivity dependence on the cell cycle phase. We investigate the role of gene-dependent cell cycle regulation in the response of tumors to therapeutic irradiation protocols. Results: Simulation results emphasize the importance of tumor tissue features and the need to consider regulating factors such as hypoxia, as well as tumor geometry and tissue dynamics, in predicting and improving radiotherapeutic efficacy. Conclusion: This model provides insight into the coupling of complex biological processes, which leads to a better understanding of oncogenesis. This will hopefully lead to improved irradiation therapy.

q-bio.TO↗