Paper List
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An AI Implementation Science Study to Improve Trustworthy Data in a Large Healthcare System
This paper addresses the critical gap between theoretical AI research and real-world clinical implementation by providing a practical framework for as...
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The BEAT-CF Causal Model: A model for guiding the design of trials and observational analyses of cystic fibrosis exacerbations
This paper addresses the critical gap in cystic fibrosis exacerbation management by providing a formal causal framework that integrates expert knowled...
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Hierarchical Molecular Language Models (HMLMs)
This paper addresses the core challenge of accurately modeling context-dependent signaling, pathway cross-talk, and temporal dynamics across multiple ...
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Stability analysis of action potential generation using Markov models of voltage‑gated sodium channel isoforms
This work addresses the challenge of systematically characterizing how the high-dimensional parameter space of Markov models for different sodium chan...
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Approximate Bayesian Inference on Mechanisms of Network Growth and Evolution
This paper addresses the core challenge of inferring the relative contributions of multiple, simultaneous generative mechanisms in network formation w...
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EnzyCLIP: A Cross-Attention Dual Encoder Framework with Contrastive Learning for Predicting Enzyme Kinetic Constants
This paper addresses the core challenge of jointly predicting enzyme kinetic parameters (Kcat and Km) by modeling dynamic enzyme-substrate interaction...
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Tissue stress measurements with Bayesian Inversion Stress Microscopy
This paper addresses the core challenge of measuring absolute, tissue-scale mechanical stress without making assumptions about tissue rheology, which ...
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DeepFRI Demystified: Interpretability vs. Accuracy in AI Protein Function Prediction
This study addresses the critical gap between high predictive accuracy and biological interpretability in DeepFRI, revealing that the model often prio...
A multiscale discrete-to-continuum framework for structured population models
Mathematical Institute, University of Oxford, OX2 6GG Oxford, UK | Ludwig Institute for Cancer Research, University of Oxford, OX3 7DQ Oxford, UK
30秒速读
IN SHORT: This paper addresses the core challenge of systematically deriving uniformly valid continuum approximations from discrete structured population models, overcoming ambiguities in truncation order and boundary conditions inherent in traditional Taylor expansion methods.
核心创新
- Methodology Introduces a discrete multiscale framework combining the method of multiple scales with matched asymptotic expansions to systematically derive continuum approximations, identifying regions where continuum representation is appropriate versus fundamentally discrete.
- Methodology Provides asymptotically consistent boundary conditions through discrete boundary layer analysis, resolving the ambiguity in boundary condition selection that plagues traditional Taylor expansion approaches.
- Methodology Demonstrates the framework on a lipid-structured model for early atherosclerosis, showing consistency between discrete and continuum descriptions and validating the method's practical applicability.
主要结论
- The method identifies distinct asymptotic regions: outer regions (e.g., O1-O4) describable by continuum PDEs (nonlinear advection equations) and inner boundary layers (e.g., IN1-IN5, B1-B4) that remain fundamentally discrete and require separate analysis.
- For the paradigm problem (Eq. 1), the framework yields a composite solution (Eq. 16) asymptotically consistent with the exact discrete steady state (Eq. 10), unlike the truncated PDE solution (Eq. 9) which predicts an incorrect decay rate (a/(εb) vs. log((2b+a)/(2b-a))/ε).
- The framework successfully derives a continuum approximation for a lipid-structured atherosclerosis model, verifying consistency and demonstrating transferability to biological systems with discrete internal states (e.g., lipid accumulation in macrophages).
摘要: Mathematical models of biological populations commonly use discrete structure classes to capture trait variation among individuals (e.g. age, size, phenotype, intracellular state). Upscaling these discrete models into continuum descriptions can improve analytical tractability and scalability of numerical solutions. Common upscaling approaches based solely on Taylor expansions may, however, introduce ambiguities in truncation order, uniform validity and boundary conditions. To address this, here we introduce a discrete multiscale framework to systematically derive continuum approximations of structured population models. Using the method of multiple scales and matched asymptotic expansions applied to discrete systems, we identify regions of structure space for which a continuum representation is appropriate and derive the corresponding partial differential equations. The leading-order dynamics are given by a nonlinear advection equation in the bulk domain and advection-diffusion processes in small inner layers about the leading wavefronts and stagnation point. We further derive discrete boundary layer descriptions for regions where a continuum representation is fundamentally inappropriate. Finally, we demonstrate the method on a simple lipid-structured model for early atherosclerosis and verify consistency between the discrete and continuum descriptions. The multiscale framework we present can be applied to other heterogeneous systems with discrete structure in order to obtain appropriate upscaled dynamics with asymptotically consistent boundary conditions.