Source-linked AI summary
Biologically-informed neural networks guide mechanistic modeling from sparse experimental data
John H. Lagergren, John T. Nardini, Ruth E. Baker, Matthew J. Simpson, Kevin B. Flores
TL;DR
The paper addresses the challenge of learning biological dynamics when experimental data are sparse, noisy, and the governing PDE terms are not known in advance. It introduces BINNs, which learn neural-network representations of reaction-diffusion terms while enforcing the PDE, and uses those learned forms to guide mechanistic model selection. Applied to scratch-assay data, the approach identifies a delay term and produces a model reported to outperform established alternatives.
Problem
Biological equation learning is hindered by sparse, noisy data and the need to specify governing equations or candidate-term libraries in advance, despite nonlinear mechanisms in biological systems.
Method
BINNs learn neural-network representations of biologically relevant diffusion, growth, and delay terms while jointly fitting observations and satisfying a generalized reaction-diffusion PDE.
Results
BINNs identified a previously unconsidered delay term and guided a mechanistic model that modeled scratch-assay data more accurately than classical FKPP and Generalized Porous-FKPP models.
Takeaways & Limitations
The results suggest that time-delay terms may be necessary to capture scratch-assay dynamics during the first day after scratching.
Takeaways & Limitations
The model assumes diffusion and growth share one time-dependent delay, which may be inaccurate if migration and proliferation change at different rates.
Abstract
from arXiv · showhide
Biologically-informed neural networks (BINNs), an extension of physics-informed neural networks [1], are introduced and used to discover the underlying dynamics of biological systems from sparse experimental data. In the present work, BINNs are trained in a supervised learning framework to approximate in vitro cell biology assay experiments while respecting a generalized form of the governing reaction-diffusion partial differential equation (PDE). By allowing the diffusion and reaction terms to be multilayer perceptrons (MLPs), the nonlinear forms of these terms can be learned while simultaneously converging to the solution of the governing PDE. Further, the trained MLPs are used to guide the selection of biologically interpretable mechanistic forms of the PDE terms which provides new insights into the biological and physical mechanisms that govern the dynamics of the observed system. The method is evaluated on sparse real-world data from wound healing assays with varying initial cell densities [2].
Introduction
Biological systems are difficult to model from sparse, noisy spatiotemporal data because their governing mechanisms can be nonlinear and are often not known in advance. BINNs address this model-specification problem by learning reaction-diffusion terms from data while enforcing the governing PDE, then using the learned forms to guide interpretable mechanistic modeling.
- Introduction: Scratch assays provide spatiotemporal data for studying collective cell migration, but measurements are sparse and affected by biological observation noise.The experiments quantify cell density during recolonization of an empty region after a scratch.
- Introduction: Reaction-diffusion PDEs model cell-density change through diffusion and growth, but selecting appropriate mechanistic forms for these terms remains an open question.The general equation represents the rate of change of cell density as a combination of diffusion and reaction or growth.
- Introduction: Existing equation-learning approaches face a model-specification problem because governing equations or candidate-term libraries must be chosen before learning.This limitation is especially challenging for biological systems whose underlying mechanics may be nonlinear.
- Introduction: BINNs address this problem by representing biologically relevant PDE terms with function-approximating neural networks while retaining biological and physical constraints.The framework extends physics-informed neural networks and avoids requiring a predefined mechanistic model or candidate library.
- Introduction: In scratch-assay data, BINNs learn nonlinear diffusion and growth forms, and persistent discrepancy motivates a delay term whose learned forms guide an interpretable mechanistic model.The resulting model removes virtually all of the reported model discrepancy in the sparse experimental data.
Results
BINNs learned reaction-diffusion dynamics from sparse scratch-assay data, exposed persistent high-density discrepancies, and motivated a time-delay extension. The resulting delay-reaction-diffusion model closely matched the data and generally outperformed classical alternatives under model-selection criteria.
- Reaction-diffusion BINNs: BINNs learned diffusivity and growth functions from simulated noisy FKPP and Generalized Porous-FKPP data to test whether the networks could recover biologically accurate terms.The networks approximated D and G as functions of cell density u.
- Reaction-diffusion BINNs: Lower-density fits were excellent, but significant model discrepancy remained at higher initial cell densities, with non-i.i.d. GLS residuals.The discrepancy was investigated using the 20,000-cells-per-well dataset.
- Delay-reaction-diffusion BINNs: Early-time discrepancies indicated that diffusion and growth were too rapid, motivating a non-decreasing time-delay network that scales both terms between 0 and 1.The delay was introduced to address persistent discrepancies in high-density datasets.
- Delay-reaction-diffusion BINNs: Including the delay practically eliminated discrepancy across initial densities and produced approximately i.i.d. residuals, while reducing early-time diffusion and growth before convergence to full effect.At t = 0.5 days, the delay model matched the data more accurately than the reaction-diffusion model.
- Guided mechanistic model selection: The learned functions guided a mechanistic model combining generalized porous-FKPP-like diffusivity, logistic growth, and logistic-regression delay.These functional forms were selected from the learned diffusivity, growth, and delay networks.
- Model comparison: The BINN-guided delay-reaction-diffusion model outperformed classical FKPP and Generalized Porous-FKPP models across all datasets except 14,000 cells per well.The delay term was most impactful for the 18,000- and 20,000-cells-per-well datasets, where ΔAIC values were substantially larger.
Discussion
BINNs use learned diffusivity, growth, and delay functions to guide a mechanistic model of sparse scratch-assay data. The resulting delay-reaction-diffusion model matched or exceeded baseline models, while exposing scope limits and future development needs.
- Discussion: BINNs revealed that density-dependent diffusivity and growth alone could not fully capture scratch-assay dynamics.The discrepancy motivated adding a time-delay term.
- Discussion: The learned diffusivity, growth, and delay networks guided a mechanistic model using logistic growth and delay functions.The delay was introduced to represent time-dependent changes in cell dynamics.
- Discussion: The BINN-guided model fit every scratch-assay dataset more accurately than classical FKPP and Generalized Porous-FKPP models by GLS error.The authors attribute this accuracy improvement partly to the BINN-guided model's greater complexity.
- Discussion: AIC scores ranked the BINN-guided model as relatively better across datasets except at 14,000 cells per well, where Generalized Porous-FKPP was slightly better.The delay term contributed most strongly for datasets with initial densities of 18,000 and 20,000 cells per well.
- Discussion: The delay term may reflect longer activation times for density-dependent diffusion and growth at larger initial cell densities.Possible contributors include more damaged cells or more terminally differentiated cells near or within the population.
- Discussion: BINNs were presented as a flexible framework for learning equations from sparse and noisy data and guiding biologically interpretable mechanistic models.The framework was demonstrated on scratch assays and proposed for broader biological and physical applications.
- Discussion: The learned MLP dynamics may not generalize well outside the training domain and can become biologically unrealistic when observations do not approach carrying capacity.The authors suggest replacing unrealistic MLP terms with mechanistic models or adding constraints such as GMLP(u = K) = 0.
- Discussion: Future work includes uncertainty quantification, two-dimensional scratch-assay evaluation, and further development beyond the present one-dimensional setting.The paper identifies these as opportunities rather than completed evaluations.
Methods
The method uses multilayer perceptrons to approximate cell density and learn biologically constrained PDE terms from sparse data. Training combines statistical data fitting, PDE regularization, biological constraints, and numerical forward-solver considerations.
- Preprocessing: Input and output standardization is required because the scratch-assay variables span several orders of magnitude.Without standardization, the neural networks failed to converge on these data.
- Network design: BINNs use a function-approximating MLP for cell density alongside neural networks for diffusivity, growth, and delay terms.The parameter networks are smaller than the cell-density network because their dynamics are assumed simpler.
- Optimization: Training minimizes GLS, PDE-residual, and constraint losses using Adam, with an 80%/20% training-validation split and validation-based model saving.Initial-condition GLS errors are weighted by a factor of 10 to improve the parameter networks' numerical generalization.
- PDE regularization: 10,000 randomly sampled PDE points produce the desired balance: the smallest PDE error and largest GLS error among the tested regularization strategies.This behavior lets the network fit data as accurately as allowed by the governing PDE rather than overfitting observed locations.
- PDE forward solver: The forward solver represents a general quantity u evolving through spatial flux and reaction terms with specified initial and no-flux boundary conditions.The spatial and temporal domains are discretized on equispaced grids before applying a method-of-lines discretization.
Supporting information
The supporting information documents the scratch-assay setup and supplementary simulations, residuals, parameter fits, error-model selection, and PDE sampling validation. These materials cover both classical and generalized reaction-diffusion formulations across varied initial cell densities.
- Experimental setup: Scratch assays seed PC-3 prostate cancer cells at 10,000–20,000 cells per well and estimate one-dimensional density from microscopy columns.The assay uses a uniform vertical scratch and records images over time.
- Simulation validation: Supplementary simulations compare BINN fits for the classical FKPP and Generalized Porous-FKPP equations, including learned diffusivity and growth functions.Solid curves show BINN-based numerical solutions or parameter networks, while dashed curves show reference simulations.
- Experimental fits: Residual and model-solution figures evaluate reaction-diffusion and delay-reaction-diffusion BINNs across experiments initialized with 10,000 to 20,000 cells per well.Additional figures compare classical FKPP and Generalized Porous-FKPP solutions against scratch-assay data.
- Parameter tables: The supplementary parameter tables report fitted classical FKPP and Generalized Porous-FKPP values separately for each initial cell density.Each table column corresponds to one of the seven initial-density conditions.
- Training diagnostics: Supplementary analyses examine unconstrained BINN terms, statistical error-model selection, and PDE regularization through random sampling.The error-model analysis varies γ, while the regularization analysis compares no PDE penalty, data-location penalties, and randomly sampled PDE points.