Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squares

arXiv:2505.20327 · q-bio.QM, cs.AI · Submitted 2025-05-23 · Read on arXiv

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Introduction to the show: ident: Genomics Radio. Generated commentary on the latest computational biology and genomics papers.

Ines: We're looking at the paper: "Data-driven multi-agent modelling of calcium interactions in cell culture".

Marcus: The gist The CRLSM achieves a fairly good parameter estimate and a good data fit when using the learned parameters in the Consensus problem Fundamental cells mechanisms such as excitation-contraction…

Ines: First, who's behind it and why it matters.

Title and authors: Ines: So we're looking at this paper today, titled "Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squares." It’s about using AI to figure out how calcium moves inside cells when you look at the data.

Marcus: Yeah, it tackles a big problem there. They're comparing two different ways to find the mathematical rules governing these calcium dynamics, specifically the Constrained Regularized Least-Squares Method and Physics-Informed Neural Networks. It’s about system identification for biological systems right here in cell culture.

Yuki: From my side, I'm thinking about what this means for understanding cell behavior across different conditions. If we can reliably identify these governing equations, it helps us see how calcium oscillations influence responses in diseases like hypertension or diabetes. It’s about connecting the math to the actual biology of what we're seeing in the culture.

Ines: Exactly. The paper sets up a framework for microscopy-guided machine learning modeling of living cells, starting by getting data from new microscope technologies and then analyzing images to get a graph of how the cell network is organized. They model the calcium concentration in each cell using a simple integrator, and they tune those parameters using this reduced-order dataset.

Marcus: The core of what they’re doing is this consensus problem, which aims to find the velocity of an agent that leads to a final consensus in the population, meaning all the agents reach some agreement and the system has similar states after a certain period of time. It's about finding that shared dynamic behavior.

Yuki: That idea of consensus across multiple agents in a population is interesting when thinking about how different cells interact dynamically over time. It suggests that even if each cell has its own dynamics, there might be some overarching agreement or trajectory they follow collectively.

Ines: They then compare two main data-driven methods for parameter discovery: the Constrained Regularized Least-Squares Method and Physics-Informed Neural Networks. The CRLSM minimizes a regularized least-squares problem, and the PINN loss function combines a physical loss associated with an equation and a data fitting loss.

Title and authors: Marcus: The CRLSM uses this formulation of min theta in R half A p theta minus y p squared plus lambda two times theta p squared, where lambda is the regularization parameter, which is important for ensuring some stability in the estimation. It’s essentially Tikhonov regularization for finding parameters.

Yuki: And then there’s the PINN approach using deep learning combined with knowledge of the physical process underlying the problem to solve ordinary differential equations. That's a lot of machinery being put together to try and learn those governing equations directly from data.

Ines: The numerical experiments analyze this model by first simulating a single cell and then generalizing it to other cells assuming that the behavior of the remaining cells follows the intensity observed in the data. They use LSM to fit parameters by minimizing over the sum of squared residuals p p r p squared where rp equals yp minus y hat p.

Marcus: The results they show compare CRLSM and PINN performance across different cell setups, and for a group model, they perform CRLSM under the assumption of each cell is characterized by a different feed term gamma i. They split cells into three groups based on how well the model captures their behavior.

Yuki: And what's striking about that analysis is that for setup I, cell forty-three has an MSE of zero point zero zero one four, which suggests a good fit for that specific cell within the group study they are running. It shows some level of success in characterizing those individual dynamics.

Ines: On the PINN training process, they found that the learned parameter k is notably near to the initial condition, and they suspect this is due to the PINN’s training procedure which concurrently minimizes both losses, but it seems like it's limited by the initial state for the remaining cells, preventing additional model adaptation.

Marcus: So what we see here is that while PINN tries to learn everything at once by balancing its physical and data losses, CRLSM achieves a better parameter estimate and a good data fit when using those learned parameters in the consensus problem.

Title and authors: Yuki: That suggests that for this type of biological system modeling, the structured approach of CRLSM gives more reliable parameter estimation than what they found with the PINN setup in this specific comparison.

Ines: The authors conclude that the CRLSM achieves reliable parameter estimation and sufficient accuracy compared to PINN, which did not achieve sufficient accuracy compared to CRLSM. They say the resulting CRLSM is able to approximate well suited parameters to describe the system, which biologists can utilize to detect anomalies in cell culture.

Marcus: So for our listeners who might be working with calcium dynamics data, this paper suggests that if you're dealing with noisy biological data, using a method like CRLSM gives you more reliable parameters than trying to force a PINN to learn everything from scratch.

Yuki: And what this means for the wider field is that we need methods that can handle the complexity of individual cell differences, not just average behavior, which is crucial when studying diseases where cell types might react differently.

Ines: They also point toward some future work, suggesting they should focus on improving the neural network architecture of PINN, enforcing initial condition limitations, and introducing Graph Neural Networks combined with a Neural Operator framework.

Marcus: So while CRLSM seems to be the winner here for parameter discovery, they acknowledge that PINNs still have potential if you can refine how they handle those initial conditions and structure their learning better.

Yuki: I'm just hopeful that these future steps will help us move closer to models that can truly capture the complex, interacting behaviors we see in living cells as a whole.

Ines: That’s what we hope for with this paper on "Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squares." It shows us how to use existing tools effectively when they're applied to complex biological signals.

Marcus: So the main thing is that for finding those governing equations, the regularized approach seems more robust right now. We’ll see what comes next in this area.

The paper's summary: Ines: So, we're looking at this paper today, and basically, they’re comparing two different machine learning tools—the PINN and the CRLSM—to figure out how calcium moves around in a population of cells when you look at real data from microscopy.

Marcus: Right. The summary boils down to this: one method, the CRLSM, gave them a fairly good parameter estimate and a solid fit for the data when they used those learned parameters for their consensus problem.

Yuki: So what does that mean in practice? It means that when you’re trying to model these complex cell interactions across a group, this regularized method actually works better at giving you reliable numbers than the PINN did in this specific test.

Ines: Exactly. The paper shows that for a group of cells, where each one has a slightly different way it takes in calcium—what they call the feed term gamma i —the CRLSM handles that variation pretty well.

Marcus: And the results they got are interesting because they split those cells into three groups based on how well the model captured their behavior. The CRLSM showed it worked best for setup I, where cell forty-three had a mean squared error of zero point zero zero one four.

Yuki: That MSE number, zero point zero zero one four, that's what we have to consider when we think about what this AI is actually recovering about the biology. Does it really capture the key interactions, or is it just fitting noise?

Ines: That’s a fair question. The paper suggests that for setup I, the CRLSM managed to approximate very well suited parameters that biologists could actually use to spot anomalies in cell culture.

Marcus: And when you look at the PINN side of things, they found that the learned parameter k was really close to its initial condition. That they think this is because it’s minimizing both the physical loss and the data loss at the same time.

Yuki: But Marcus, if it's stuck near where it started, does that mean we can actually learn anything new about these cells? Or is the model just repeating what it already knows from its starting point?

Ines: That’s a limitation they flag. They suggest the PINN seems limited by that initial state for the remaining cells, which stops further adaptation from happening in the model.

Marcus: So, to summarize this comparison: CRLSM gave them reliable parameter estimation and sufficient accuracy for describing the system dynamics, whereas their PINN setup didn't achieve that same level of accuracy.

Yuki: And what does this suggest for us studying cell populations? It implies that if you're dealing with noisy data from different cell types, a structured method like CRLSM might be a more dependable way to find the mathematical rules governing those interactions.

Ines: And they’re pointing toward some future work too. They think they should improve the PINN architecture, enforce stricter limits on those initial conditions, and maybe use Graph Neural Networks combined with a Neural Operator framework next.

Marcus: So while CRLSM seems to be the winner for this parameter discovery task right now, they keep saying that refining how the PINN handles its starting point is still something worth exploring.

Yuki: It’s definitely interesting because it moves us toward models that can handle individual cell differences instead of just averaging out everything. Next up, we're going to talk about how this kind of dynamic modeling connects to bigger population genetic patterns in species.

The paper's improvements: Marcus: So, moving past just comparing CRLSM to PINN, we look at what they suggest we should actually improve in this modeling approach. The paper points toward some specific fixes for both methods if we want better results with cell culture data.

Ines: Right. They aren't just stopping there with the comparison; they’re suggesting concrete steps to make these models more useful for biology and more stable for data science.

Marcus: For the CRLSM side, they focus on making the parameter estimation even more reliable. The suggestion is to really lean into that Tikhonov regularization formulation when you're fitting those linear ordinary differential equations.

Ines: That means doubling down on minimizing that regularized least-squares problem, ensuring that whatever parameters we pull out are robust against any noise in the experimental data.

Marcus: And for the PINN, they have some pretty specific ideas too. They’re looking at improving the neural network architecture and enforcing better initial condition limitations during training.

Ines: So if you're using a PINN, you need to make sure it respects known biological constraints right from the start instead of just letting it learn everything freely.

Marcus: That makes sense because we saw how that model got stuck near its starting point earlier, so forcing it away from that initial state should help it adapt better to the actual dynamics.

Ines: And they are also proposing introducing Graph Neural Networks combined with a Neural Operator framework for future work. That sounds like a way to handle the spatial organization of the cells more naturally.

Marcus: A Graph Neural Network would be perfect for modeling how those individual agents, those cells, are connected to each other in the network structure we saw from the microscopy images.

Ines: Exactly. So, they’re moving from just fitting parameters to building a framework that understands both the internal dynamics of a single cell and how that changes as it interacts with its neighbors across the whole population.

Marcus: It shifts this from just solving an ODE for one agent to solving a coupled system where the structure itself influences the solution, which is more realistic for living systems.

Ines: And that’s where it gets really powerful. If we can model those interactions accurately, we move closer to detecting those subtle calcium spikes that represent real biological events in diseased cells.

Marcus: So what this means for the wider field is that the next step isn't just picking a better algorithm; it's about building a system where the physics, the data structure, and the network topology all talk to each other simultaneously.

Ines: It’s about creating models that don't just fit a curve, but actually capture the underlying biological mechanism of how those cells coordinate their activity.

Conclusion: Ines: So we're wrapping up this look at "Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squares." Basically, the paper showed that for modeling calcium dynamics, the Constrained Regularized Least-Squares Method performed better than the PINN in terms of reliable parameter estimation.

Marcus: Yeah, and they proved that this method gives them sufficient accuracy compared to what they got from the PINN approach in their experiments across different cell setups.

Yuki: From a population genetics viewpoint, it suggests that when we look at these complex biological systems, using a more structured method like CRLSM might give us parameters that are actually supportable by the data we're collecting.

Ines: The implication is that biologists can use those well-suited parameters to detect anomalies in cell culture without getting completely lost in the model's complexity.

Marcus: It’s about getting a solid mathematical description of the system so you don't have to guess what’s going on inside those cells based on noisy measurements.

Yuki: And that reliability is key when we try to connect these lab findings back to broader biological trends in species that might be affected by things like hypertension or diabetes.

Ines: Exactly. We need tools that give us trustworthy parameters so we can actually make sense of what those calcium oscillations are doing biologically.

Marcus: So, the final word on this comparison is that for this specific task, CRLSM provided the more robust parameter estimation framework across their tests in "Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squares."

Yuki: It’s a solid piece of work that shows how different mathematical approaches can yield different levels of biological insight from the same microscopy data.

Ines: It certainly does, and it opens the door for more sophisticated modeling if we can keep pushing these architectural improvements on the PINN side.

Marcus: And that's where we’ll see what comes next with those graph neural networks and neural operator frameworks they mentioned for future work.

Yuki: Next time, we'll look at how this kind of dynamic modeling connects to population genetic patterns in species that might be affected by things like hypertension or diabetes.

KTH Royal Institute of Technology · International School for Advanced Studies (SISSA)

q-bio.QM, cs.AI

Submitted: 2025-05-23

Updated: 2026-10-19

Comments: New model revealed issues with the results.

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 64/100

The gist: The gist The CRLSM achieves a fairly good parameter estimate and a good data fit when using the learned parameters in the Consensus problem Introduction and Motivation Fundamental cells mechanisms

Key concepts

Constrained Regularized Least-Squares Method (CRLSM): This method finds the best parameters by minimizing a specific mathematical problem that balances fitting the observed data against keeping the model parameters small. It uses a penalty term to ensure the resulting parameters are stable and reproducible, which is important when dealing with noisy biological measurements.
Physics-Informed Neural Networks (PINN): PINNs use deep learning models combined with known physical laws (like differential equations) to learn system behavior directly from data. The model minimizes a loss function that penalizes both errors in fitting the data and violations of the underlying physical equations, aiming for a more physically consistent solution.
Consensus Problem: This is the goal of modeling how multiple cells interact. It seeks the velocity required for all agents to reach agreement on their states within a population. In this context, it models how calcium signals spread and synchronize across different cells in a culture.

Terminology

Constrained Regularized Least-Squares Method — 1× in this paper
Ordinary Differential Equations (ODEs) — 1× in this paper · explained in 2 episodes
Physics-Informed Neural Networks — 1× in this paper
System Identification — 1× in this paper · explained in 3 episodes
Uncertainty Quantification — 1× in this paper · explained in 9 episodes

Summary

The gist The CRLSM achieves a fairly good parameter estimate and a good data fit when using the learned parameters in the Consensus problem

Introduction and Motivation

Fundamental cells mechanisms such as excitation-contraction and gene expressions result from the Ca2+ intracellular signal Observing and controlling this flow is therefore critical in understanding cells behaviors in situations as hypertension, heart disease, and diabetes. It is important to identify and characterize the governing equations to understand how calcium oscillations influence biological responses both in healthy and in diseased cells. Modeling biological systems usually comes at the expense of very large entities communicating with each other, leading to a large state which is subject to the curse of dimensionality. The Multi-Agent System (MAS) was first introduced for identifying biological behaviors [4] and chemical reactions [5] with limited state dimensions.

Methodology and Modeling

The study proposes a methodology for characterization and performance analysis of calcium delivery in a family of cells by comparing the Constrained Regularized Least-Squares Method (CRLSM) and Physics-Informed Neural Networks (PINN) for system identification. The consensus problem aims to find the velocity of the agent that leads to a final consensus in the population, meaning that all the agents reached some agreement, translating into the system having similar states after a certain period of time.

The paper develops a framework for microscopy-guided machine learning modelling of the dynamic organization of living cells. This involves obtaining data through new microscope technologies and then segmenting and analyzing images to provide a graph of the cell network. The calcium concentration in each cell is modelled by a simple integrator, and parameters are tuned using this reduced-order dataset.

System Identification Methods

The research compares two primary data-driven methods for parameter discovery:

Constrained Regularized Least-Squares Method (CRLSM): This method minimizes the regularized least-squares minimization problem formulated as min θ∈Rp 1/2 Apθp − yp2 + λ 2 θp2, where λ > 0 is the regularization parameter.
Physics-Informed Neural Networks (PINN): PINNs leverage deep learning models combined with the knowledge of the physical process underlying the problem to solve Ordinary Differential Equations (ODEs). The PINN loss function combines a physical loss LF associated with equation (21) and a data fitting loss Ldata associated with equation (22).

Numerical Experiments and Results

The numerical experiments analyze the model by first simulating a single cell and then generalizing it to other cells assuming that the behavior of the remaining cells follows the intensity observed in the data. The analysis involves fitting parameters using LSM, which minimizes over the sum of squared residuals P p(rp)2 where rp = yp − yˆp.

The results compare CRLSM and PINN performance across different cell setups. For a group model, the CRLSM is performed under the assumption of each cell is characterized by a different feed, γi. The analysis splits cells into three groups based on how well the model capture their behaviour. The CRLSM shows that for setup I, cell 43 has an MSE of 0.0014.

The PINN training process shows that the learned parameter k is notably near to the initial condition, which is probably due to the PINN’s training procedure, which concurrently minimizes the physical and data losses. However, it seems that the model is limited by the initial state for the remaining cells, which prevents additional model adaption.

Conclusion and Future Work

The CRLSM achieves reliable parameter estimation and sufficient accuracy compared to PINN, which did not achieve sufficient accuracy compared to CRLSM. The resulting CRLSM is able to approximate well suited parameters to describe the system, which biologist can utilize to detect anomalies in the cell culture. Future research should focus on improving the neural network architecture of PINN, enforcing initial condition limitations, and introducing Graph Neural Networks (GNNs) combined with a Neural Operator framework.

Acknowledgments This work is supported by the Vinnova Program for Advanced and Innovative Digitalisation (Ref. Num. 2023-00241) and Vinnova Program for Circular and Biobased Economy (Ref. Num. 2021-03748) and partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.

References [1] H. K. Jacob M. Kowalewski, Per Uhl´en, H. Brismar, Modeling the impact of store-operated ca2+ entry on intracellular ca2+ oscillations, Mathematical Biosciences 204 (2) (2006) 232–249 [3] V. K. Genevi'eve Dupont, Martin Falcke, J. Sneyd, Models of Calcium Signalling, Springer, Interdisciplinary Applied Mathematics Volume 43, 2016 [7] R. Olfati-Saber, J. S. Shamma, Consensus filters for sensor networks and distributed sensor fusion, in: Proceedings of the 44th IEEE Conference on Decision and Control, IEEE, 2005, pp. 6698–6703 [14] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning, Nature Reviews Physics 3 (6) (2021) 422–440 [16] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational physics 378 (2019) 686–707 [25] M. Barreau, H. Shen, Accuracy and robustness of weight-balancing methods for training pinns (2025). arXiv:2501.18582 [3] V. K. Genevi'eve Dupont, Martin Falcke, J. Sneyd, Models of Calcium Signalling, Springer, Interdisciplinary Applied Mathematics Volume 43, 2016 [4] W. K. Potts, The chorus-line hypothesis of manoeuvre coordination in avian flocks, Nature 309 (5966) (1984) 344–345 [5] J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, Journal of theoretical biology 81 (3) (1979) 389–400 [6] A. Amirkhani, A. H. Barshooi, Consensus in multi-agent systems: a review, Artificial Intelligence Review 55 (5) (2022) 3897–3935 [11] K. A.A., de Silva BM., F. U., K. K., C. J.L., D. C.B., C. K., L. J.C., K. J.N., B S L, Pysindy: A comprehensive python package for robust sparse system identification, Journal of Open Source Software 7 (69) (2022) [18] P Saraf´ın, M H´udik, M Rev´ak, S ˇ Zak, P ˇ Sevˇcık, Modelling and iden-tification of linear discrete systems using least squares method, in: 2017 Federated Conference on Computer Science and Information Systems (FedCSIS), 2017 [19] C Millevoi, D Pasetto, M Ferronato, A physics-informed neural network approach for compartmental epidemiological models (2023). arXiv:2311.

Improvements for AI systems

Bold system identification for calcium dynamics: The improved system can reliably estimate governing ordinary differential equations (ODEs) using a framework that compares Constrained Regularized Least-Squares Method (CRLSM) and Physics-Informed Neural Networks (PINN). This allows the AI to discover the underlying mathematical rules from noisy, real-time cell culture data.
Bold robust parameter estimation via CRLSM: The system can perform reliable parameter estimation for the linear ODEs by utilizing a Tikhonov regularization formulation: min θ∈Rp1/2∥Apθp − yp∥2+λ2∥θp∥2, ensuring high reproducibility despite data instability.
Bold enhanced cell-specific modeling with CRLSM: The improved system can model groups of cells where each cell has a distinct feed term, addressing the limitation of simpler models by using the model in equation (17) is taken into consideration when identifying the parameters k and γi, for i = 1,..., g.
Bold accurate biological feature detection: The system can accurately capture critical biological events by focusing on calcium intensity spikes rather than magnitude, as suggested by the paper's conclusion: it is crucial to determine the intensity spikes as this represent the actual interactions among cells.

Abstract

Data-driven discovery of dynamics in biological systems allows for better observation and characterization of processes, such as calcium signaling in cell culture. Recent advancements in techniques allow the exploration of previously unattainable insights of dynamical systems, such as the Sparse Identification of Non-Linear Dynamics (SINDy), overcoming the limitations of more classic methodologies. The latter requires some prior knowledge of an effective library of candidate terms, which is not realistic for a real case study. Using inspiration from fields like traffic density estimation and control theory, we propose a methodology for characterization and performance analysis of calcium delivery in a family of cells. In this work, we compare the performance of the Constrained Regularized Least-Squares Method (CRLSM) and Physics-Informed Neural Networks (PINN) for system identification and parameter discovery for governing ordinary differential equations (ODEs). The CRLSM achieves a fairly good parameter estimate and a good data fit when using the learned parameters in the Consensus problem. On the other hand, despite the initial hypothesis, PINNs fail to match the CRLSM performance and, under the current configuration, do not provide fair parameter estimation. However, we have only studied a limited number of PINN architectures, and it is expected that additional hyperparameter tuning, as well as uncertainty quantification, could significantly improve the performance in future works.