A Bayesian framework for systems model refinement and selection of calcium signaling

Abstract

Calcium (${\text{Ca}}^{2+}$) is a crucial messenger that modulates contractile and electrophysiological processes in eukaryotic cells. Dysregulation of ${\text{Ca}}^{2+}$-signaling influences these processes and is strongly associated with diseases such as cancer, immune disorders, and heart failure. Computational modeling of ${\text{Ca}}^{2+}$ dynamics offers valuable insights into these processes. However, traditional approaches often overlook the inherent heterogeneity within cell populations, including cell-to-cell variability and population-wide differences. To overcome these limitations, we developed and implemented an advanced statistical approach (a Bayesian inference framework using a hierarchical mixture architecture) specifically engineered to capture and model the diverse behaviors seen in fundamental calcium signaling pathways within cells. We applied this framework to myoblasts and to a HEK293 cell line expressing the cardiac proteins SERCA2a and RyR2. Using fluorescence microscopy, we monitored ${\text{Ca}}^{2+}$ dynamics in response to extracellular adenosine triphosphate, as well as spontaneous ${\text{Ca}}^{2+}$ release and uptake between cellular compartments. Our framework leverages the microscopy data to identify the most probable models and parameters that reproduce experimental observations, effectively distinguishing multiple clusters of cells with distinct kinetic behaviors. This approach provides deeper insights into the underlying biological processes and their variability across multiple populations of cells. Our findings demonstrate that this Bayesian method significantly improves our ability to create accurate computational models of ${\text{Ca}}^{2+}$ signaling by explicitly accounting for cellular differences. This, in turn, enhances our capacity to understand the complex regulatory networks that govern how cells use calcium signals.

Significance

Biological data often contain significant heterogeneity, such as cell-to-cell variability and clusters of different cell populations. Although there are numerous tools for fitting theoretical models to such data, most do not model the complexity in a probabilistic and robust way. To address this gap, we developed a Bayesian-based framework that models biological data while capturing both forms of heterogeneity. The tool offers a probabilistic perspective and, by addressing heterogeneity, reveals invaluable insights into the underlying biological processes.

Introduction

${\text{Ca}}^{2+}$ is a critical second messenger that modulates various signaling pathways, influencing cell behaviors as diverse as migration, contraction, gene transcription, and proliferation (1,2,3,4). The modulation by ${\text{Ca}}^{2+}$ is encoded in its waveform (5,6), with the dynamics of intracellular $[{\text{Ca}}^{2+}$] (${\left[{\text{Ca}}^{2+}\right]}_i$) acting as an information-rich signal to regulate ${\text{Ca}}^{2+}$-binding proteins. For this reason, dysregulated ${\left[{\text{Ca}}^{2+}\right]}_i$ dynamics are frequently linked to cardiac dysfunction, neurodegeneration, lysosomal storage disorders, and cancer (7,8,9,10). To study ${\text{Ca}}^{2+}$ waveform generation and regulation, numerous studies have been dedicated to mathematical modeling of ${\text{Ca}}^{2+}$ dynamics using ordinary differential equations (ODEs) (11,12,13,14). ODE-based models can be used to describe the temporal evolution of states, such as channel opening or intracellular ${\text{Ca}}^{2+}$ levels. These models can simulate how intracellular calcium (${\left[{\text{Ca}}^{2+}\right]}_i$) changes in response to stimuli and how these changes regulate downstream pathways, ultimately affecting cellular behavior.

These models are fitted to the experimental data to ensure that ODE-based representations accurately reflect the dynamic processes observed in living cells. However, experimental data frequently exhibit high heterogeneity due to cell-to-cell variability and cell clustering (15), which may stem from differences in gene expression levels and altered Ca²⁺ sensitivity within cell populations (16). Many modeling approaches neglect this heterogeneity, using training data consisting of averaged values across different populations. Consequently, fitted models may not fully capture the underlying heterogeneous states and parameters. Traditional frequentist methods, such as maximum likelihood estimation or nonlinear least squares, can be used for parameter estimation and provide uncertainty quantification through confidence regions. Typically, however, such methods offer point estimates as the primary output, with uncertainty represented through interval estimates that carry a different probabilistic interpretation. These intervals have a different interpretation rooted in long-term error rates from repeated sampling (17), rather than directly reflecting a probability distribution over parameter values.

Given the complexity inherent in heterogeneous biological data, Bayesian inference offers an alternative probabilistic approach based on Bayes’ theorem (18,19). Bayesian inference explicitly models parameters as probability distributions, allowing for direct representation of uncertainty and incorporating prior knowledge. The prior distribution plays a role analogous to regularization in frequentist hierarchical models, offering a natural way to share information across subgroups, such as individual cells. This approach accommodates biological heterogeneity through the probabilistic distributions of the parameters. While frequentist methods also model heterogeneity via hierarchical estimation and regularization, Bayesian inference integrates these aspects through prior distributions and a hierarchical modeling structure. Furthermore, Bayesian inference provides a clear probabilistic basis for model comparison, complementing frequentist criteria like the Akaike Information Criterion through metrics such as leave-one-out cross-validation (LOO) and posterior predictive checks (PPC) that offer probabilistic assessments of model fit and generalizability.

Bayesian inference has been extensively implemented to model a variety of biological problems. To name a few, the BioBayes package was designed to perform Bayesian inference on the parameters of systems biology ODE systems (20). Bayesim is another tool developed to fit biological models using adaptive grid sampling (21). In the same year, Russell-Buckland et al. published a Bayesian framework to analyze brain systems models (22). Recently, a Bayesian approach utilizing the Hamiltonian Monte Carlo sampling algorithm was developed to model cardiac action potentials (23). However, these tools were not designed to address problems that feature heterogeneity often encountered in biological data. To address this, multiple frameworks have been published to model heterogeneity. For example, Thompson et al. developed a hierarchical Bayesian mixture model for the inference of gene expression states (24). Another framework, Brem-SC, was developed to identify cell clusters in transcriptomic and proteomic data (25). However, these tools do not focus on the temporal dynamics of the underlying biological processes.

To address the inherent heterogeneity in dynamic data and provide probabilistic descriptions of dynamic Ca²⁺ signaling processes, we developed a novel Bayesian approach that leverages a hierarchical mixture architecture. This architecture incorporates two key features: 1) a mixture component that accounts for the presence of multiple clusters of cells exhibiting distinct phenotypes and 2) a hierarchy component that models cell-to-cell variability by grouping individual cells into clusters with similar properties. To thoroughly evaluate the performance of our framework, we applied it to two types of time-series microscopy data: single-channel and dual-channel data. We tested the robustness of the approach using both synthetic data and experimental data featuring heterogeneous characteristics. Furthermore, we implemented LOO, a Bayesian cross-validation approach, as the model comparison metric in our framework. LOO yielded expected log pointwise predictive density for LOO (elpd_loo) as the model scores, which could be used to differentiate between models with distinct differences. For models indistinguishable by elpd_loo, PPC, a visual method was used to assess the fit of the models and the uncertainty in the fit as an additional model selection method. Overall, this work provides a novel framework that performs probabilistic model refinement and selection using heterogeneous dynamic data.

Theory

In this work, we develop a model fitting framework based on Bayesian inference for biological microscopy data (Fig. 1). Overall, the framework takes observations from biological experiments as input. It then applies Bayesian inference, given a proposed model and prior distributions of the model parameters. The framework returns posterior distributions of the model parameters, which are estimates of these parameters along with the uncertainty in such estimation. By drawing from the posterior distributions and using them to reconstruct the proposed model, one could generate fits to the experimental observations and obtain the uncertainty of the fits.

Figure 1.

Schematics of the Bayesian framework described in this study. The goal of the framework is to fit hypothesized mathematical models to experimental observations using Bayesian inference. One key challenge of fitting biological data is the heterogeneity in observations, which is addressed in this study using a hierarchical mixture architecture. The fitting process starts with collecting experimental data. Given a hypothesized model and prior knowledge of the underlying system, Bayesian inference is performed. Inference provides posterior distributions of the model parameters, offering both updated knowledge of the parameter range and the uncertainty in estimating the parameters. Finally, by randomly sampling from posterior distributions, the drawn parameters can be used to reconstruct the model and compare it with the observations. Because the parameters used to reconstruct the model are drawn from the posterior distributions, the uncertainty in parameter estimation propagates forward to provide a quantification of the uncertainty of the final fits.

Model definitions

We utilize ODE-based models of ${\text{Ca}}^{2+}$ signaling-associated processes, including ${\text{Ca}}^{2+}$ uptake and ${\text{Ca}}^{2+}$ shuttling between different cellular compartments. These models generally consist of nonlinear equations that represent the time-dependent change of a given state, $S_m\left(t\right)$, such as intracellular ${\text{Ca}}^{2+}$ concentration, as a function of itself and other states such as the activation of a Ca²⁺ exchanger. As an example representation, we have the following:

$$\frac{d{S}_m\left(t\right)}{dt}=\sum _{m}f\left(S_m\left(t\right),\left\{p\right\}\right)$$ (1)

where $\left\{p\right\}\equiv \left\{p_i,\dots \right\}$ are the parameters for determining $S_m\left(t\right)$. Specific equations for the processes modeled in this work are outlined in supporting material, section 1. Numerical simulation of these models with the user-provided parameters yield predictions or observations for each modeled state. These states can be compared against experimental measurements by defining a function, $h_m\left(S_m\left(t\right)\right)$, that maps the value of state $S_m\left(t\right)$ to its corresponding observable $y_m\left(t\right)$. The central challenge is how to determine $\left\{p\right\}$ to 1) minimize the difference between predicted and experimentally measured states and 2) determine the best model for a given set of states.

In this study, we assume $\mathbf{S}\left(t\right)\equiv \left\{S_m\left(t\right),\dots \right\}$ is a set of states that evolve over time subject to a set of $\left\{p\right\}$ of undetermined parameters. We introduced $r\left(t\right)$, an independent process that could serve to represent imposed phenomena such as random oscillations in intracellular Ca²⁺. We also assumed trial models (e.g., functions), $f\left(\left\{S\left(t\right)\right\},\left\{p\right\},r\left(t\right)\right)$. Numerical ODE integrators were used to determine $\mathbf{S}\left(t\right)$ for $t\in \left[0,T\right]$ given suitable initial conditions $\left\{S_m\left(t=0\right),\dots \right\}$ and $\left\{p\right\}$. Here, $y_m\left(t\right)$ could be simulated or measured via experimental means to give $y_{m,obs}\left(t\right)$.

Bayesian inference

We used a Bayesian approach to determine the ODE model and the respective parameter sets that best reproduce the experimentally measured data for each state modeled in Eq. 1. As an example, we assume that our observable $y_{obs}$ corresponds to experimentally measured time-dependent ${\text{Ca}}^{2+}$ concentration via fluorescence indicators. We speculated that the optimal parameter, $p_i$, or more generally the parameter set, $\left\{p\right\}$, could be determined by evaluating the conditional probability of a parameter set $\left(\left\{p\right\}\right)$ given some observable $\left(y_{obs}\right)$, $P\left(\left\{p\right\}|y_{obs}\right)$. Therefore, once this conditional probability is appropriately defined, we can estimate the model parameters (such as reaction rate, expression level), associated with our measurements of $y_{obs}$. According to the Bayes theorem, we define the posterior as

$$P\left(\left\{p\right\}|y_{obs}\right)=\frac{P\left(y_{obs}|\left\{p\right\}\right)P\left(\left\{p\right\}\right)}{P\left(y_{obs}\right)}$$ (2)

The likelihood $P\left(y_{obs}|\left\{p\right\}\right)$ is the conditional probability of an observable, $y_{obs}$, given a trial set of parameters for simulation, $\left\{p\right\}$. The priors, $P\left(\left\{p\right\}\right)$, represent the probabilities for the given parameter set values. It is commonly posited that the parameters’ probabilities follow Gaussian or log-normal distributions with a mean value, $⟨p_i⟩$. For Gaussian-distributed and statistically independent parameter distributions, the priors can be described by

$$P\left(\left\{p\right\}\right)=\prod _i\exp \left({-\left(p_i-⟨p_i⟩\right)}^2/{\sigma }_i^2\right)$$ (3)

The conditional probability $P\left(y_{obs}|\left\{p\right\}\right)$ is determined by simulations of a given system model (Eq. 1) under sets of underlying parameters. Lastly, we evaluate $P\left(y_{obs}\right)$, the probability of observing a given quantity based on a measurement (e.g., how well the measurement reports the true value of an event) via

$$P\left(y_{obs}\right)={\int }_{\left\{p\right\}}P\left(y_{obs}|\left\{p\right\}\right)P\left(\left\{p\right\}\right)d\left\{p\right\}$$ (4)

Hierarchical mixture architecture

Biological data often exhibit heterogeneity, ranging from cell-to-cell variations to cluster-wide differences. To model such heterogeneity, we implemented a hierarchical mixture architecture (26). The hierarchical component accounts for cell-to-cell variation by assuming that all cells share a common set of hyperparameters that govern their observable behavior (e.g., kinetics, expression levels). The hyperparameters define a cluster, and individual cells within that cluster are allowed to vary around these cluster-level trends, reflecting biological noise or intrinsic variability. The behavior of each cell $j$ is represented by its own set of parameters, ${\left\{p\right\}}_j$. The mixture component enables the model to account for the presence of multiple clusters within the data set. Specifically, it assumes that there are K distinct clusters, each cluster $k$ characterized by its own set of hyperparameters. These cluster-specific hyperparameters give rise to different ${y_{obs}}_{,k}$. Together, the hierarchical mixture framework allows for flexible modeling of both within-cluster and between-cluster variability, which is particularly well suited to complex biological systems.

For a given model (e.g., Eq. 1), the weight of cluster k of K total clusters is:

$${\pi }_k\sim \text{Dirichlet}\left(\alpha \right)$$ (5)

where $\alpha$ is the concentration parameter provided by the user as input.

For each cluster k, we draw a trial parameter set ${\mu }_k$ from a lognormal distribution, and ${\sigma }_k$ from a half-normal distribution:

$${\mu }_k\sim \text{Lognormal}\left(\mu ,{\sigma }^2\right)$$ (6)

$${\sigma }_k\sim \text{Half}-\text{normal}\left({{\sigma }_o}^2\right)$$ (7)

where $\mu =⟨{\mu }_k⟩$, ${\sigma }^2=\text{Var}\left({\mu }_k\right)$, ${\sigma }_o^2=\text{Var}\left({\sigma }_k\right)$, and are computed across all clusters.

Then for each cell j, let ${\mu }_{j,k}\equiv {\left\{p\right\}}_{j,k}$. We draw a trial parameter set ${\mu }_{j,k}$ from a lognormal distribution:

$${\mu }_{j,k}\sim \text{Lognormal}\left({\mu }_k,{{\sigma }_k}^2\right)$$ (8)

Given the trial ${\mu }_{j,k}$, we express the likelihood as:

$$P\left({y_{obs}}_{,k}|{\mu }_{j,k}\right)=\exp \left({-\left({y_{obs}}_{,k}-y_{pred,j}\right)}^2/{\sigma }^2\right)$$ (9)

where the predicted observable $y_{pred,j,k}$ is computed using $h_m\left(S_m\left(t\right)\right)$, and $S_m\left(t\right)$ is evaluated using Eq. 1. $y_{pred,j,k}$ is treated as a scalar.

The posterior for ${\mu }_j$ can thus be expressed as:

$$P\left({\mu }_j|{y_{obs}}_{,j}\right)=\sum _{k=1}^K{\pi }_k\frac{P\left({y_{obs}}_{,k}|{\mu }_{j,k}\right)P\left({\mu }_{j,k}\right)}{P\left({y_{obs}}_{,k}\right)}$$ (10)

Model comparison

LOO was used to estimate the out-of-sample predictive accuracy. It iteratively leaves out one observation, $y_d$, to create a new observed data set, $y_{-d}$, to which the model is fitted. The newly generated value for the missing data point is compared with the removed data point to estimate the predictive performance of the model. Although it does not explicitly penalize model complexity, LOO inherently takes into account model complexity, as overfitting is penalized for not being able to reproduce the left-out observations. Overall, LOO is a balanced measure that emphasizes both model interpretability and complexity (27).

Various methodologies are available for model comparison, including cross-validation techniques and a range of information criteria. Here, we choose LOO for its advantages over other popular approaches such as the widely applicable information criterion, the deviance information criterion (DIC), and the Bayesian information criterion (BIC). We base this decision in part on Vehtari et al. who suggested that under circumstances of finite cases with weakly informative priors, LOO is more robust compared with widely applicable information criteria (27). Furthermore, LOO is fully Bayesian and uses the entire posterior distribution for model evaluation, whereas BIC and DIC use point-estimates of parameters (28,29). Additionally, LOO works well for hierarchical models while BIC and DIC are suitable for simple models. Considering the nature of biological data, namely heterogeneity and sometimes poor knowledge of the system, hierarchical models with weak priors are a common occurrence. Hence, LOO serves as a better approach for model comparison in this context.

The main numerical quantity of LOO, elpd_loo, is expressed as

$$elp{d}_{loo}=\sum _{d=1}^D\log P\left(y_d|y_{-d}\right)$$ (11)

where the leave-one-out predictive density for the d-th data point $y_d$, given the remaining observations $y_{-d}$, is expressed as:

$$P\left(y_d|y_{-d}\right)={\int }_{\left\{p\right\}}P\left(y_d|\left\{p\right\}\right)P\left(\left\{p\right\}|y_{-d}\right)d\left\{p\right\}$$ (12)

Materials and methods

Experimental setup and data collection

${\text{Ca}}^{2+}$ transient measurements in C2C12 myoblasts

As previously described (30), undifferentiated C2C12 myoblast cells were grown on a laminin-coated coverslip in 10% FBS DMEM medium. To record ${\left[{\text{Ca}}^{2+}\right]}_i$ transients, a high-affinity ${\text{Ca}}^{2+}$ indicator Fluo-4 AM was used. 24 h after plating, cells were incubated at 37°C with 2.2 $\mu$M Fluo-4 AM for 30 min in Tyrode’s solution: 140 mM NaCl, 4 mM KCl, 1.5 mM${\text{CaCl}}_2$, 1 mM ${\text{MgCl}}_2$, 10 mM glucose, 10 mM HEPES (pH 7.4), followed by a 5 min wash in Tyrode’s solution. Cells were allowed to equilibrate for 20 min and then stimulated with 10 $\mu$M adenosine triphosphate (ATP). Fluo-4 was excited with a 488 nm LED and emitted fluorescence filtered with a band-pass filter (515–530 nm, Chroma). Resultant fluorescent images were acquired at 9 fps on a wide-field fluorescence Zeiss scope, equipped with a sCMOS camera (Orca Flash, Hamamatsu) using Zen Blue software (Zeiss, Germany).

Simultaneous measurements of ER $[{\text{Ca}}^{2+}$] (${\left[{\text{Ca}}^{2+}\right]}_{ER}$) and ${\left[{\text{Ca}}^{2+}\right]}_i$ in HEK293

HEK293 cells stably expressing GFP-hRyR2 and mCer-SERCA2a were transfected with the genetically encoded endoplasmic reticulum (ER)-targeted Ca²⁺ sensor R-CEPIA1er (31). The cells prepared for an experiment were washed with a solution containing: 150 mM K-aspartate, 0.25 mM ${\text{MgCl}}_2$, 0.1 mM EGTA, 10 mM HEPES (pH 7.2). Next, the plasma membrane was permeabilized with 0.005% saponin dissolved in the experimental solution containing: 100 mM K-aspartate, 15 mM KCl, 5 mM ${\text{KH}}_2{\text{PO}}_4$, 5 mM MgATP, 0.35 mM EGTA, 0.22 mM ${\text{CaCl}}_2$, 0.75 mM ${\text{MgCl}}_2$, 10 mM HEPES, dextran (MW 40,000) 2% and pH 7.2. The free [Ca²⁺] and [Mg²⁺] were 200 nM and 1 mM, respectively. After permeabilization, cells were perfused with a saponin-free solution during an experiment. All experiments were carried out using a laser scanning confocal microscope (Radiance 2000 MP, Bio-Rad) equipped with a 40× oil objective lens (N.A. = 1.3). ${\left[{\text{Ca}}^{2+}\right]}_{ER}$ was recorded as changes in the fluorescence intensity of the ER-targeted R-CEPIA1er. R-CEPIA1er was excited with a 514 nm line of the argon laser and the signal was collected at >560 nm. ${\left[{\text{Ca}}^{2+}\right]}_i$ was recorded with the Ca²⁺ sensor Fluo-4 AM. Fluo-4 was excited with a 488 nm line of the argon laser and the signal was collected at $535\pm 10\text{nm}$. The images obtained were analyzed using ImageJ software (NIH).

Computational workflow

Simulation

To test our framework in a robust manner, we first generated simulated data using model 1 (supporting material, sections 1.1 and 1.2) for each data type. Specifically, we created three sets of parameters with initial $\mu$ values that result in distinct kinetics. Each set represents one cluster, or population, of cells. Within each set, samples were randomly drawn from normal distributions of each parameter defined by their respective mean $\mu$ and standard deviation $\sigma$ (where $\sigma =0.2\times \mu$) to allow for cell-to-cell variability. The drawn parameters were used to simulate the model and generate traces for each cell. Simulations were performed with the scipy (v1.10.1) solve_ivp function using LSODA (32) as the integration method to minimize stiffness issues due to the complexity of our models. To mimic the noise in real-life data, we also added synthetic noise that was randomly sampled from a normal distribution with mean $\mu$ of 0 and standard deviation $\sigma$ of 0.01 of the maximum value of each trace. Together, the simulated data contain variability across different populations and different cells. The simulated data were pooled and used for parameter estimation and model selection.

Bayesian inference

In this study, we focus on the kinetic behavior of the biological Ca²⁺ dynamic data. Thus, all data were normalized by their respective minima and maxima. Similarly, the simulation outputs for each iteration of the Markov chain Monte Carlo (MCMC) sampling were normalized by their respective minima and maxima. The experimental data set of the Ca²⁺ shuttling data has an inherent low signal/noise ratio due to the nature of the biological sample (see below). To reduce the noise, we performed smoothed moving averaging on this data set with a window size of 0.15 s that smoothed the data without altering the kinetics of the traces. The simulated and experimental data were used as the observed data in the MCMC sampling for the likelihood calculation.

PyMC (v5.0.0) (33) was used to build the statistical model and perform MCMC sampling. Here, we implemented a hierarchical mixture architecture to account for population-wide and cell-to-cell heterogeneity of our biological data (see hierarchical mixture architecture). Within the hierarchical mixture architecture, the hyperparameters were assumed to follow lognormal distributions with proposed $\mu$ and $\sigma$ to avoid negative values. The distributions were assumed to have multiple components to represent the mixture property. Dirichlet distribution, a continuous multivariate probability distribution, was used to model the weight of each component. As the prior distribution, each component was assigned the same weight. The model parameters follow lognormal distributions defined by the $\mu$ and $\sigma$ drawn from the hyperparameter distributions. sunode (v0.4.0) was used as the ODE solver to speed up the simulations. nut-py (v0.5.1) was used for MCMC sampling. Due to the complexity of our models, the automatic differentiation variational inference (34) was used to initialize the sampling. The MCMC sampling was performed with 5 chains, 500,000 automatic differentiation variational inference steps, 1000 tune steps, 1000 sampling steps, target_accept = 0.99. Table 1 compiles the computation cost of the inferences performed in this study. To compare different models, inference data from each model were used as input for the PyMC function pm.compare, and LOO (27) was used as the information criterion. The final elpd_loo scores computed were used to evaluate the models. In essence, models with higher elpd_loo scores are considered models that perform better.

Table 1.

Simulations performed in this study

System	Model	No. of parametersa	No. of cells	Computation time (h)
Ca2+ uptake	1	15	7	60
			12	120
	2	9	7	33
			12	116
	3	21	7	82
			12	124
Cytoplasm/ER Ca2+ shuttling	1	9	7	48
			12	92
	2	6	7	33
			12	50
	3	12	7	56
			12	90

All inferences were performed using 5 MCMC chains with 1000 steps/chains.

^aThe number of parameters include that of both hyperparameters and model parameters.

Data and software availability

All code written in support of this publication is publicly available at https://github.com/pkhlab/bayes.git.

Results

Data types

In this work, our goal is to build a general framework for the refinement and selection of models that are suitable for various types of microscopy data. To this end, we tested our framework on the following types of fluorescence microscopy measurements: 1) ${\text{Ca}}^{2+}$ uptake data that consist of a single fluorescent channel and 2) cyto/ER ${\text{Ca}}^{2+}$ shuttling data that consist of dual fluorescent channels. Each data type will be described and discussed in detail in the following sections.

Model parameter estimation

We first assessed the performance of the framework in parameter estimation. To evaluate the effectiveness and robustness of the framework, we tested our framework on both simulated data and experimental data. The simulated data were generated with randomized parameter values as the ground truth.

Fitting to single channel: ${\text{Ca}}^{2+}$ uptake

In this experiment, ATP was applied to C2C12 cells, which activated the purinergic receptors on the plasma membrane to conduct Ca²⁺ from the extracellular space (35). To capture this phenomenon, we propose a model (Fig. 2 A) where ATP is consumed $\left(\frac{dATP}{dt}\right)$ at rate $k_{decay}$ to open the purinergic receptor. The open probability of the receptor $\left(\frac{d{P}_{open}}{dt}\right)$ is dependent on the ATP concentration and its ATP binding affinity $\left(K_{d,ATP}\right)$. Counteracting this, the receptor closes at rate $k_{close}$. Finally, the opening of the receptor allows for Ca²⁺ entry $\left(\frac{dC{a}_{in}}{dt}\right)$ at rate $k_{flux}$ with a countering Ca²⁺ leaking rate $\left(k_{leak}\right)$. This model is represented mathematically by a set of ODEs (supporting material, section 1.1, model 1). Prior distributions of the model hyperparameters are listed in Table 2 and are assumed log-normal to ensure positive values for the ODE simulations. The priors were selected to deviate substantially from the parameter values used to generate the simulated data. This design choice was made to evaluate the robustness of our framework in converging to the correct solution even when the prior information is weak or uninformative, a situation often encountered in real-world biological applications where prior knowledge is often limited or qualitative.

Figure 2.

Fitting of model 1 to the simulated Ca²⁺ uptake data set. (A) Proposed model 1 of Ca²⁺ uptake. The binding of extracellular ATP opens the purinergic receptor on the plasma membrane, allowing for Ca²⁺ entry. The receptor closes as ATP decays, returning the cytosolic Ca²⁺ to the prestimulation level by extruding Ca²⁺ from the cell. (B) Observed data generated by simulation are classified into three populations, based on their kinetic differences (e.g., time to peak, time from peak back to baseline). (C) PPC fits to the observed data showing how well the model is fitted to the data, as well as the uncertainty in the fits. (D) Posterior distributions of the model hyperparameters (blue) relative to the prior distribution (gray). (E) Posterior distributions of the model parameters for each cell are color-coded based on the cell indexes.

Table 2.

Hyperparameter prior distributions of the Ca²⁺ uptake data set

Hyperparameter	Observable	Model 1	Model 2	Model 3
${\mu }_{K_{d,ATP}}$	sim	Lognormal(0.5, $0.3^2$)		Lognormal(0.5, $0.3^2$)
	expt	Lognormal(0.5, $0.1^2$)		Lognormal(0.5, $0.1^2$)
${\mu }_{k_{decay}}$	sim	Lognormal(0.4, $0.3^2$)	Lognormal(0.4, $0.3^2$)	Lognormal(0.4, $0.3^2$)
	expt	Lognormal(0.04, $0.1^2$)	Lognormal(0.04, $0.1^2$)	Lognormal(0.04, $0.1^2$)
${\mu }_{k_{close}}$	sim	Lognormal(0.25, $0.3^2$)
	expt	Lognormal(0.5, $0.1^2$)
${\mu }_{k_{flux}}$	sim	Lognormal(0.1, $0.1^2$)	Lognormal(0.1, $0.1^2$)	Lognormal(0.1, $0.1^2$)
	expt	Lognormal(0.05, $0.1^2$)	Lognormal(0.05, $0.1^2$)	Lognormal(0.05, $0.1^2$)
${\mu }_{k_{leak}}$	sim	Lognormal(0.55, $0.3^2$)	Lognormal(0.55, $0.3^2$)	Lognormal(0.2, $0.3^2$)
	expt	Lognormal(0.3, $0.1^2$)	Lognormal(0.3, $0.1^2$)	Lognormal(0.3, $0.1^2$)
${\mu }_{V_{\max ,open}}$	sim			Lognormal(0.7, $0.3^2$)
	expt			Lognormal(0.7, $0.1^2$)
${\mu }_{K_{d,open}}$	sim			Lognormal(0.2, $0.3^2$)
	expt			Lognormal(0.2, $0.1^2$)
${\mu }_{V_{\max ,close}}$	sim			Lognormal(0.3, $0.3^2$)
	expt			Lognormal(0.6, $0.1^2$)

Prior distribution selected for hyperparameters of the models for the Ca²⁺ uptake data set.

All $\sigma$ hyperparameters follow a half-normal distribution with a scale of 0.1.

We first tested our framework on simulated Ca²⁺ uptake data. Simulated data were generated by randomizing model 1 parameters and adding synthetic noise (noise magnitude equals 1% of the trace maximum). To generate data mimicking real-life biological data that exhibit high variability (cell-to-cell and population-wide variations), we created three sets of parameters with drastically different values and randomized values, each set representing one population. Specifically, with the first set as the reference, the second set has a substantially lower $k_{leak}$, while the third set has a substantially higher $k_{decay}$. Furthermore, within each set, the parameters for each cell were randomly drawn from a normal distribution defined by the parameter values of that set as $\mu$, and 20% of the $\mu$ values as $\sigma$ to allow for variations between cells. This resulted in three populations of cells that displayed different Ca²⁺ uptake kinetics (e.g., time to peak, time from peak back to baseline), within which individual cells also show variable kinetics (Fig. 2 B).

Model 1 was first fitted to estimate the model parameters with respect to the simulated data. As diagnostics of the sampling, trace plots and autocorrelation plots (Figs. S6 and S8) indicated good convergence and mixing of the sampling. Namely, the sampling traces explored sufficient parameter space (oscillation) rather than being stuck at certain values. Traces from individual chains also converged to the same area. Furthermore, the autocorrelation quickly decayed to 0, indicating that the samples drawn are independent and uncorrelated.

We next examined the hyperparameters that characterize the kinetic behavior of a population. Posterior distributions of the hyperparameters (Fig. 2 D) showed that the three components of the mixture model progressed away from the prior distribution (gray), indicating that the framework successfully updated the prior beliefs based on the observed data to form posterior distributions. More importantly, the framework was able to identify three distinct populations in the data. The extent of separation among components varied across different hyperparameters. For instance, the mixture components of ${\mu }_{k_{decay}}$ and ${\mu }_{k_{close}}$ showed clear separation, while those of ${\mu }_{k_{flux}}$ and ${\mu }_{k_{leak}}$ partially overlapped with each other. In the case of ${\mu }_{K_{d,ATP}}$, components 2 and 3 were overlapped, with component 1 being well separated. This suggests that the dominant contributors to the population-level kinetic differences are ${\mu }_{k_{decay}}$ and ${\mu }_{k_{close}}$, with ${\mu }_{K_{d,ATP}}$ also contributing to the variability. Within each mixture component, the posterior distributions of individual cells (Fig. 2 E) were well aligned with the corresponding hyperparameter distributions, supporting the hierarchical structure of the model. To further assess the relationships between hyperparameters, we examined the pairwise posterior distributions (Fig. S10). The resulting plots showed circular or narrow ellipsoidal shapes aligned with one axis, indicating a low or no correlation between hyperparameters within each component. These results suggest that the model parameters are largely independent of one another.

To assess how well the estimated parameters reproduce the observed Ca²⁺ uptake kinetics, PPC was used (Fig. 2 C). In the PPC process, parameter values are randomly sampled from their posterior distributions, and these sampled parameters are then used to simulate the behavior of the system. This process was repeated 1000 times to generate a distribution of predicted trajectories. PPC thus serves as a visual tool to evaluate both the quality of model fit and the associated uncertainty. To complement the visual assessment, we also quantified the goodness of fit using mean-squared error (Fig. S5). The PPC results indicate that model 1 fits the observed data well overall. However, for cells 5, 6, and 7, the credible intervals are relatively wider, and their corresponding mean-squared errors are around 0.01, higher than the 0.002 observed for other cells. This suggests a higher uncertainty in parameter estimation for these specific cells.

We also evaluated how the assumed number of components (K) influences the ability of the framework to differentiate the underlying populations. To this end, we conducted additional inference runs using K = 2 and K = 4. With K = 3 (as discussed above), the inferred means of the parameters ${\mu }_{k_{decay}}$ and ${\mu }_{k_{close}}$ showed clear separation into three distinct components, while ${\mu }_{k_{flux}}$ and ${\mu }_{k_{leak}}$ exhibited overlapping distributions across the components. For ${\mu }_{K_{d,ATP}}$, components 2 and 3 were overlapped, whereas component 1 remained distinct, effectively suggesting two populations for this parameter. In the case of K = 2, ${\mu }_{K_{d,ATP}}$ maintained a similar two-cluster structure as K = 3 (Fig. S12), and ${\mu }_{k_{flux}}$ and ${\mu }_{k_{leak}}$ remained overlapped. However, the framework failed to identify a third distinct component in ${\mu }_{k_{decay}}$ and ${\mu }_{k_{close}}$, effectively merging components 2 and 3. This likely contributed to the broader credible intervals observed in the PPC, indicating higher uncertainty in parameter estimation (Fig. S13). For K = 4, ${\mu }_{k_{flux}}$ and ${\mu }_{k_{leak}}$ again showed minimal separation, with overlapping clusters. In contrast, ${\mu }_{k_{close}}$ displayed further refinement as the original component 2 split into two subclusters (now components 2 and 3). However, ${\mu }_{k_{decay}}$ collapsed back to two main clusters, similar to what was observed in the K = 2 case. Additionally, ${\mu }_{K_{d,ATP}}$ revealed a new cluster, while components 2 and 3 remained centering around a value of 1. The PPC fits for K = 4 exhibited credible intervals comparable with those of K = 3, suggesting similar levels of predictive uncertainty in both cases. Overall, this suggests that K = 3 is the simplest mixture that minimizes uncertainty with the fewest components.

With the validation of the simulated results, we next considered real data (Fig. 3 A). These data recorded the Ca²⁺ uptake of C2C12 cells induced by 10 $\mu$M ATP in the presence of 1.5 mM extracellular Ca²⁺. Similar to the simulated data, the sampling of the parameters were well mixed and converged (Figs. S7 and S9). Visually, the data can be roughly classified into three populations, i.e., population 1 (cells 1, 2), population 2 (cell 3, 4, 5, 6), population 3 (cell 7).

Figure 3.

Fitting of model 1 to the experimental Ca²⁺ uptake data set. (A) Observed data were obtained from the Ca²⁺ imaging experiments. The data can be roughly classified into three populations: i.e., population 1: cells 1, 2; population 2: cell 3, 4, 5, 6; population 3: cell 7. (B) PPC fits to the observed data showing how well the model is fitted to the data, as well as the uncertainty in the fits. (C) Posterior distributions of the model hyperparameters (blue) relative to the prior distribution (gray). (D) Posterior distributions of the model parameters for each cell are color-coded based on the cell indexes.

We verified that the hyperparameters are not well correlated and are therefore independent (Fig. S11). The posterior distributions of the hyperparameters diverged from the prior distributions (Fig. 3 C), signifying that the incorporation of observed data modified the parameter estimates. The hyperparameter distributions captured different populations that represent clusters of cells. Interestingly, due to the smaller kinetic variations among different cells, the hyperparameter ${\mu }_{k_{close}}$ was sufficient to account for the major difference. Furthermore, while we enforced a three-component mixture model for the hyperparameters, components 2 and 3 were partially overlapped (${\mu }_{K_{d,ATP}}$, ${\mu }_{k_{close}}$, ${\mu }_{k_{leak}}$), indicating that there are effectively two populations. This was further confirmed by the additional inference run assuming two components (Fig. S14). Namely, K = 2 produced results comparable with K = 3. The resulting PPC fits of K = 2 also have comparable credible intervals, with a slightly narrower interval for cell 3 (Fig. S15). Lastly, the PPC analyses indicate that the framework estimated parameter values that captured experimental traces, albeit with a higher uncertainty for cell 3, 6, and 7 (Figs. 3 B and S5). When performing the inference with four components, K = 4, we identified a new cluster for ${\mu }_{k_{close}}$ (component 4) (Fig. S14). However, the identification of the new cluster did not lead to PPC fits with reduced uncertainty (Fig. S15). Rather, it yielded fits with broader credible intervals. This indicates that overparameterization of K risks overfitting, and therefore poorer generalizations.

Fitting to multiple channels: Cytoplasm/ER ${\text{Ca}}^{2+}$ shuttling

We next assessed whether the framework would perform well with multichannel microscopy data, which is essential for probing Ca²⁺ signaling between various cellular compartments such as the cytoplasm, ER, and mitochondria, using different organelle-localized fluorophores. This requires simultaneous measurement of ${\text{Ca}}^{2+}$ levels within different regions of the cell, and can be challenging.

In this experiment, we monitored the spontaneous shuttling of ${\text{Ca}}^{2+}$ between the cytoplasm and ER in HEK cells. The complete data set captures both the release of ${\text{Ca}}^{2+}$ from the ER and the uptake of ${\text{Ca}}^{2+}$ back to the ER. We were particularly interested in the ER Ca²⁺ uptake, as during this phase we noted that, while the decrease of ${\left[{\text{Ca}}^{2+}\right]}_i$ occurred rapidly, the Ca²⁺ uptake to the ER lagged behind (Fig. 5 B). This implies that the Ca²⁺ may be shuttled to another unobserved compartment. That is, we postulate that the lagging kinetics of ER Ca²⁺ uptake is due to the compartmentalization of the ER (Fig. 4 A). Specifically, we hypothesize that within the ER there are two compartments: a hidden compartment (ER1) that actively uptakes ${\text{Ca}}^{2+}$ from the cytoplasm via sarcoplasmic/ER calcium ATPase (SERCA) (36) with rate $k_{SERCA}$ and transfers ${\text{Ca}}^{2+}$ to the other compartment with rate $k_{shuttle}$, and a second compartment that releases ${\text{Ca}}^{2+}$ to the cytoplasm via ryanodine receptor (RyR) (37) with $k_{Ryr}$. Furthermore, we postulate that the second compartment (ER2) is where the ER ${\text{Ca}}^{2+}$ sensor is located, resulting in the observed lagged ER ${\text{Ca}}^{2+}$ kinetics. This model is mathematically represented by a set of ODEs (supporting material, section 1.2, model 1). Prior distributions of the model hyperparameters are listed in Table 3 and are assumed log-normal to ensure positive values for the ODE simulations.

Figure 5.

Fitting of model 1 to the experimental cytoplasm/ER Ca²⁺ shuttling data set. (A) Observed data were obtained from the Ca²⁺ imaging experiments. (B) PPC fits to the observed data showing how well the model is fitted to the data, as well as the uncertainty in the fits. (C) Posterior distributions of the model hyperparameters (blue) relative to the prior distribution (gray). (D) Posterior distributions of the model parameters for each cell are color-coded based on the cell indexes.

Figure 4.

Fitting of model 1 to the simulated cytoplasm/ER Ca²⁺ shuttling data set. (A) Proposed model 1 of Ca²⁺ shuttling between cytoplasm and ER. We posited that there are two compartments of ER. ER1 is the site where Ca²⁺ uptake by SERCA from the cytoplasm occurs, whereas ER2 is where the Ca²⁺ fluorophore is situated and RyR Ca²⁺ release takes place. (B) Observed data were generated by simulations, and are classified into three populations, based on their kinetic differences. (C) PPC fits to the observed data show that the model is fitted to the data well, as well as the uncertainty of the fits. (D) Posterior distributions of the model hyperparameters (blue) relative to the prior distribution (gray). (E) Posterior distributions of the model parameters for each cell are color coded based on the cell indexes.

Table 3.

Hyperparameter prior distributions of the cytoplasm/ER Ca²⁺ shuttling data set

parameter	Observable		Model 1	Model 2	Model 3
${\mu }_{k_{SERCA}}$	sim		Lognormal(2.0, $1.0^2$)	Lognormal(1.0, $0.5^2$)
	expt		Lognormal(3.0, $1.0^2$)	Lognormal(3.0, $1.0^2$)
${\mu }_{k_{Ryr}}$	sim		Lognormal(0.02, $0.5^2$)	Lognormal(0.02, $0.5^2$)	Lognormal(0.02, $0.5^2$)
	expt		Lognormal(0.02, $0.5^2$)	Lognormal(0.02, $0.5^2$)	Lognormal(0.02, $0.5^2$)
${\mu }_{k_{shuttle}}$	sim		Lognormal(1.0, $0.5^2$)		Lognormal(1.0, $0.5^2$)
	expt		Lognormal(0.2, $0.5^2$)		Lognormal(0.2, $0.5^2$)
${\mu }_{V_{\max ,SERCA}}$	sim				Lognormal(1.5, $0.5^2$)
	expt				Lognormal(1.5, $0.5^2$)
${\mu }_{K_{d,SERCA}}$	sim				Lognormal(0.5, $0.5^2$)
	expt				Lognormal(0.5, $0.5^2$)

Prior distribution selected for hyperparameters of the models for the cytoplasm/ER Ca²⁺ shuttling data set. All $\sigma$ hyperparameters follow a half-normal distribution with a scale of 0.1.

We first assessed our framework on simulated data. Similar to the Ca²⁺ uptake data, these simulated data were generated by randomizing model 1 parameters and adding synthetic noise. Additionally, we created three sets of parameter values representing three populations of cells that exhibit different kinetics (Fig. 4 A).

To estimate the parameter values, model 1 was fit to the simulated data. Trace plots and autocorrelation analyses showed good convergence and mixing of the sampling (Figs. S22 and S24). Pairwise posterior distributions showed no or low correlation between hyperparameters (Fig. S26).

The posterior distribution of the hyperparameter ${\mu }_{k_{shuttle}}$ showed a distinct separation of the three components with means around 0.4, 1.2, and 2.6, respectively (Fig. 4 D). In contrast, the posteriors of ${\mu }_{k_{Ryr}}$ are moderately separated and those of ${\mu }_{k_{SERCA}}$ are clustered. The posterior results indicate that the difference in kinetics among the three populations of cells is largely attributed to ${\mu }_{k_{shuttle}}$. Within each component, posterior distributions of the parameters for each cell are aligned with the respective hyperparameter distributions best for $k_{shuttle}$ (Fig. 4 E).

PPC was used to assess the model fits to the observed data (Fig. 4 C). While fits to most cells captured the observed data well, in cells 6, 9, 10, 11, and 12, credible intervals of the PPC fits were less precise. Consequently, there was a higher uncertainty in estimating parameter values for these cells.

We also examined different numbers of components (K = 2 or K = 4). With K = 3, ${\mu }_{k_{shuttle}}$ showed three clearly separated components, which account for the kinetic differences between different cells. In contrast, ${\mu }_{k_{SERCA}}$ has three overlapped components and ${\mu }_{k_{Ryr}}$ showed two effective components (component 1 and 2 overlapped) (Fig. S28). With K = 2, ${\mu }_{k_{SERCA}}$, ${\mu }_{k_{Ryr}}$, and ${\mu }_{k_{shuttle}}$ exhibited different patterns from K = 3. However, the PPC fits of K = 2 have comparable credible intervals to K = 3 (Fig. S29). On the other hand, posterior distributions of K = 4 exhibited similar patterns to K = 3, with component 2 of ${\mu }_{k_{shuttle}}$ split into two components, component 2 and 3. The PPC fits of K = 4 did not show substantial improvement relative to K = 3. Hence, our framework recovered the correct number of clusters from the simulation conditions.

We then challenged our framework with the experimental data (Fig. 5 A), which are noisy in comparison. We smoothed the raw data using the smoothed moving average method with a window size of 0.15 s. The window size was carefully chosen so that the noise level was reduced without changing the overall trend kinetics. Using the smoothed data exhibited good convergence and mixing (Figs. S23 and S25), as observed with the simulated data. The hyperparameters were confirmed to be uncorrelated (Fig. S27). The framework identified three components of ${\mu }_{k_{shuttle}}$ that account for the kinetic differences among the cell populations (Fig. 5 C). The hyperparameter distributions were able to approximate their underlying parameter distributions of each cell (Fig. 5 D). Finally, the estimated parameters produced good fits to the ${\left[{\text{Ca}}^{2+}\right]}_i$ trace and moderate fits to the ${\left[{\text{Ca}}^{2+}\right]}_{ER}$ trace due to the signal/noise ratio even after smoothing, especially for cells 3 and 6 (Figs. 5 B and S21). Overall, challenged with the noisy experimental data, our framework was able to identify multiple components to explain the population-wide difference and provide an assessment of the uncertainty in parameter estimation.

When we assessed the effect of varying the number of components, K = 2 poorly estimated ${\mu }_{k_{shuttle}}$, and identified a new component of ${\mu }_{k_{SERCA}}$ around 2 (Fig. S30) On the other hand, K = 4 found a new component of ${\mu }_{k_{SERCA}}$ around 6 and showed effectively the same clusters of ${\mu }_{k_{Ryr}}$. As for ${\mu }_{k_{shuttle}}$, component 2 was separated into components 2 and 3. Overall, K = 3 and K = 4 yielded fits with comparable credible intervals, while K = 2 exhibited higher uncertainty (Fig. S31).

Bayesian versus differential evolution

We compared and contrasted our Bayesian framework with a widely used frequentist method, differential evolution (DE). DE is an evolutionary algorithm that optimizes model parameters by iteratively refining a population of candidate solutions based on the principles of biological evolution (38). The process begins with an initial population of parameter sets, whose fitness is evaluated using a likelihood function. At each iteration, new candidate solutions are generated through mutation, crossover, and selection, ensuring that better-performing solutions are retained. This iterative process continues until convergence, typically defined as a minimal change in fitness over successive iterations. Given the stochastic nature of DE, we performed 500 independent runs per data set to assess its general performance and variability.

When applied to the Ca²⁺ uptake data set, all 500 DE runs successfully produced fits that closely followed both the simulated and experimental data (Fig. S16, A and C). To examine the variability in the estimated parameters, we binned the results of the 500 runs to approximate their empirical distributions (Fig. S16, B and D). These distributions spanned a wide range (approximately 0–1), and notably, in the simulated case, DE failed to reveal distinct clusters in parameters such as ${\mu }_{K_{d,ATP}}$, ${\mu }_{k_{decay}}$, and ${\mu }_{k_{close}}$, which were clearly identified by our Bayesian framework (Fig. 2 C). When addressing the experimental data, DE yielded distributions for parameters like $K_{d,ATP}$ and $k_{leak}$ that exhibited multimodal distributions, indicating the presence of multiple local minima. The multimodality could potentially be addressed through regularization by penalizing parameter similarity. However, these outcomes reflect the variability of DE and the high dimensionality of the parameter space, which increase the likelihood of becoming trapped in local optima. In contrast, our use of Bayesian inference leverages prior knowledge and systematically explores the parameter space via sampling techniques such as NUTS, leading to more stable convergence and well-defined posterior distributions.

For the cytoplasm/ER Ca²⁺ shuttling data set, DE produced satisfactory fits to the simulated data (Fig. S32 A). Several DE runs struggled to fit cell 1 in the experimental data set (Fig. S32 C). However, the distribution of $k_{shuttle}$ exhibited three clusters (ranging from 0 to 3; Fig. S32 B), closely resembling the clusters identified by the Bayesian framework (Fig. 4 D). Similarly, the distributions for $k_{SERCA}$ and $k_{Ryr}$ showed patterns consistent with the Bayesian results, which is likely due to the relative simplicity of the model in this case.

In summary, DE offers a computationally efficient solution (e.g., hours versus days for the Bayesian approach). However, for complex problems, in our opinion the Bayesian approach is more intuitive as it systematically explores the parameter space while incorporating prior information, leading to more robust and well-constrained solutions.

Model comparison

${\text{Ca}}^{2+}$ uptake

To assess the ability of our framework to select among different theoretical models, we challenged the framework with a variety of models with varied interpretability and complexity (supporting material, section 1.1). Model 1 describes the ATP-dependent opening of the purinergic receptor, which allows for the influx of Ca²⁺. In this model, the opening rate of the purinergic receptors, $\frac{1}{1+\frac{K_{d,ATP}}{ATP}}$, was dependent on the ATP concentration and the receptors’ affinity for ATP. In model 2, we relate the Ca²⁺ influx directly to the ATP concentration, thus simplifying the model by effectively omitting the receptor. In model 3, we added complexity to the closing rate of the receptor by formulating the rate to be dependent on the open probability and to be scaled by a maximum rate $V_{m,close}$. The closing rate is thus expressed as $\frac{V_{m,close}}{1+\frac{K}{P_{open}}}$. Parameters of the models and their respective priors are summarized in Table 2.

Using elpd_loo as the model comparison metric, models 1 and 3 both demonstrated better predictive performance over model 2 across both simulated and experimental data sets (Fig. 6). PPC supported these findings as models 1 and 3 closely followed the observed trajectories in the simulated data, whereas model 2 failed to capture the dynamics in cells 6 and 8 (Figs. S1 and S5 A). Despite their comparable fits, model 3 exhibited broader credible intervals than model 1, indicating greater uncertainty in fitting to the observed data. In the experimental data set, all three models captured the overall trends, but model 1 maintained substantially narrower credible intervals, while models 2 and 3 showed much higher uncertainty and struggled to generalize across all cells (Figs. S2 and S5 B). Taking into account both elpd_loo scores and PPC results, model 1 consistently outperformed the others, and therefore is the most robust choice for both simulated and experimental data.

Figure 6.

Comparison of models fit to the Ca²⁺ uptake data set. The elpd_loo metric was used to quantify model performance. The higher the elpd_loo, the more suitable the model was to describe the observed data.

Cytoplasm/ER ${\text{Ca}}^{2+}$ shuttling

For the cytoplasm/ER Ca²⁺ shuttling data set, we propose three models. Model 1 assumes a hidden compartment that shuttles Ca²⁺ from the cytoplasm to the ER as described in fitting to multiple channels: cytoplasm/ER ${\text{Ca}}^{2+}$ shuttling. Model 2 does not assume a hidden compartment and instead models direct Ca²⁺ shuttling between the cytoplasm and the ER, and hence was a simpler model. Model 3 uses a more detailed expression of SERCA activity that models the regulation of SERCA function by Ca²⁺ $\left(\frac{V_{m,SERCA}}{1+{\left(\frac{K_d}{Cyto}\right)}^2}\right)$. Parameters of the models and their respective priors are summarized in Table 3.

elpd_loo scores indicate that for the simulated ${\left[{\text{Ca}}^{2+}\right]}_i$ data sets, models 1 and 3 performed slightly better than model 2, and the performance differences between the three models were greater for the ${\left[{\text{Ca}}^{2+}\right]}_{ER}$ data (Fig. 7). Notably, the elpd_loo scores for the experimental ${\left[{\text{Ca}}^{2+}\right]}_{ER}$ data set were negative, primarily due to high data noise. In effect, this indicates poorer predictive accuracy across all models compared with their performance on the ${\left[{\text{Ca}}^{2+}\right]}_i$ data. In terms of PPC, model 2 produced narrow credible intervals for the simulated data (${\left[{\text{Ca}}^{2+}\right]}_i$ and ${\left[{\text{Ca}}^{2+}\right]}_{ER}$) but did not align well with the observed trajectories in cells 1 to 8. Models 1 and 3 fit well to the ${\left[{\text{Ca}}^{2+}\right]}_{ER}$ traces with narrow credible intervals. In the case of ${\left[{\text{Ca}}^{2+}\right]}_i$ data, models 1 and 3 yielded larger credible intervals for cells 6 to 12 but provided better fits. For the experimental data set, model 2 again struggled to capture the observed dynamics. Taking into account both the simulated and experimental data, model 2 did not capture the lagging kinetics in ${\left[{\text{Ca}}^{2+}\right]}_{ER}$ relative to ${\left[{\text{Ca}}^{2+}\right]}_i$, implying that the hidden compartment modeled in models 1 and 3 could be attributed to the kinetic differences. Models 1 and 3 resulted in comparable fits with the ${\left[{\text{Ca}}^{2+}\right]}_{ER}$ data. Regarding the ${\left[{\text{Ca}}^{2+}\right]}_i$ traces, while both models 1 and 3 adequately reflected the kinetics in the experimental ${\left[{\text{Ca}}^{2+}\right]}_i$ data, model 1 stood out by producing narrower credible intervals, suggesting lower uncertainty in its predictions. Overall, model 1 was the best performing model of the three for both the simulated and experimental data.

Figure 7.

Comparison of models fit to the cyto/ER Ca²⁺ shuttling data set. The elpd_loo metric was used to quantify model performance. The higher the elpd_loo, the more suitable the model was to describe the observed data.

Discussion

Our Bayesian approach inherently quantifies uncertainty (arising from various sources, including data collection, parameter estimation, and model prediction) through posterior probability distributions, allowing intuitive probabilistic interpretations of variability and uncertainty, while effectively capturing the inherent heterogeneity through hierarchical mixture modeling. Bayesian inference inherently integrates these components, providing a cohesive and intuitive probabilistic understanding of complex biological systems.

We acknowledge that traditional frequentist methods like maximum likelihood estimation or least-squares fitting offer robust frameworks for parameter estimation, including quantification of uncertainty via confidence intervals. Nevertheless, these methods frequently provide point estimates with interval estimates interpreted as long-term frequency statements, which may not always naturally translate into direct probabilistic interpretations of model parameters and uncertainties. To offer a complementary perspective, we developed a Bayesian-based framework of model optimization to represent Ca²⁺ signaling processes in myoblasts and a cell line expressing cardiac proteins.

Within our Bayesian framework, we designed a hierarchical mixture architecture that effectively accounts for the heterogeneity present in biological samples when fitting to systems biology models. The framework was tested using both simulated and experimental data sets with varied levels of complexity, showcasing its ability to provide excellent fits to data that feature significant cell-to-cell variability. Moreover, it successfully identified multiple subpopulations of cells within these data sets, each characterized by a distinct parameter set unique to their behavior. Our framework currently accepts a finite number of components (K), thus choosing such a number is critical, as we have shown by comparing the performance with different K’s. A systematic search for the appropriate K typically requires heavy computation. As such, we suggest here a mixed method between empirical and systematic. Namely, the initial K values can be determined by estimation based on the observed data. One could then perform inferences with K, along with additional inferences with numbers around K (e.g., K−1, K+1). One should also be aware that overparameterization of K could lead to under- or overfitting, as our results have shown.

In addition to providing accurate fits, the framework offers a probabilistic interpretation for parameter estimation and fitting. Instead of relying on point estimates, each parameter was inferred as a posterior distribution, from which we can derive the estimated value and its associated uncertainty. To leverage this probabilistic information, we drew parameters from the posterior distributions to produce fits that not only accurately capture the data but also provide the uncertainty in these fits. We also applied our framework to the challenging task of model selection. Our findings suggest that elpd_loo serves as an effective measure to select among models that have distinctly different performance. Based on the elpd_loo scores, if some models are indistinguishable, PPC is useful for assessing the goodness of fit for further selection among similar models. Overall, our analyses reveal that diverse parameter sets are often necessary to accurately fit the proposed models to the experimental data. Furthermore, we observed that experimental data can be well captured by multiple mathematical descriptions, suggesting that model uniqueness was not guaranteed.

Our Bayesian framework helps reveal novel insights into the intricate workings of cellular function by systematically identifying parameter clusters and linking them to biological mechanisms. When applied to the Ca²⁺ uptake data set (Fig. 3), we uncovered three distinct clusters of the parameter $k_{close}$, the closing rate of purinergic receptors. These clusters likely reflect varied levels of receptor desensitization, possibly due to differential expression or modulation by regulatory proteins such as G-proteins or kinases (39,40). Such differences may correspond to distinct cellular states, where some cells exhibit rapid receptor closure in response to repeated stimulation, while others maintain prolonged receptor activity, resulting in sustained Ca²⁺ influx.

Similarly, in the cytoplasm/ER Ca²⁺ shuttling data set (Fig. 5), we identified clusters in the parameter $k_{shuttle}$, which describes the effective diffusion rate of Ca²⁺ within the ER. The variation in $k_{shuttle}$ values may be attributed to differences in ER morphology, Ca²⁺-buffering protein expression (e.g., SERCA pump activity), or cytoskeletal interactions that influence ER structure and Ca²⁺ handling (41). These clusters were associated with distinct ${\left[{\text{Ca}}^{2+}\right]}_{ER}$ recovery kinetics, implying that the intracellular redistribution of Ca²⁺ is not uniform across the population and is regulated by cellular substructures and organelle interactions. Together, these insights underscore the biological relevance of capturing cell-to-cell variability. This practice reveals how heterogeneity in biophysical parameters contributes to functional diversity in the signaling dynamics of Ca²⁺.

Our approach has several areas for improvement that can be explored in future studies. One limitation is the computational complexity introduced by handling ODE-based models, which significantly slows down our inference procedure. For instance, even simple ODEs with 10–15 parameters require hours of simulation to obtain reasonable fits, partly due to the large number of cells and inherent heterogeneity in the data sets. Additionally, preselecting the number of clusters using data reduction techniques may still be necessary to achieve reasonable runtime. Furthermore, populating the normalization coefficient in our Bayesian inference equation is a computationally costly process that requires sampling over many degrees of freedom. To address these challenges, we are exploring machine learning approaches that can efficiently extrapolate and interpolate estimates (42), which could help reduce the simulation time. By developing more efficient methods for Bayesian inference using machine learning, we aim to accelerate our analysis and uncover new insights into complex biological systems. This computational complexity challenge is related to another limitation of our approach: because of the complexity, we could not fit a fairly sophisticated model that encompasses multiple pathways and a vast number of parameters (usually in the range of dozens to hundreds). One solution is to divide the large model into small submodels and fit each submodel independently. The trained submodels subsequently could be connected to form a large model. This is analogous to the common practice in experiments. A complicated system is broken down into small pieces that are studied individually. The collective data from these small pieces could eventually be used to obtain a complete picture of the entire system. In addition, simplification of large and complex models can also help reduce computational cost. For example, Smith et al. has used a lumping scheme to reduce a 15-state model to a 4-state representation (43). Lastly, our current model does not consider the interdependence of model parameters. For example, many processes are coregulated at a transcriptional level, which adds more complexity to the systems biology models.

Conclusions

As experimental techniques for characterizing system-level cellular function continue to advance, we are confronted with the challenge of analyzing exponentially growing data sets. This increasing data complexity highlights the imperative need for more sophisticated analytical approaches capable of capturing the inherent heterogeneity in cellular behavior and gene expression profiles. Our Bayesian-based framework marks a significant milestone in this endeavor, providing a flexible and adaptive toolkit for modeling complex biological systems. By identifying the most suitable models for our data sets, we can uncover valuable insights into molecular mechanisms and formulate novel mechanistic hypotheses that can be experimentally tested and refined. Ultimately, our overarching goal is to develop a profound understanding of the intricate dynamics that underlie biological processes, and this framework represents a significant step toward achieving that objective.

Acknowledgments

We thank Dr. Ivana Kuo for the insightful discussion and data sharing. Research reported in this publication was supported by the Maximizing Investigators’ Research Award (MIRA) (R35) from the National Institute of General Medical Sciences (NIGMS) of the National Institutes of Health (NIH) under grant number GM148284 to P.M.K.-H. This work was also supported by NIH grant R01HL151990 and R01HL176568 to A.V.Z. This work used Expanse at San Diego Supercomputer Center (SDSC) through allocation CHE140116 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants nos. 2138259, 2138286, 2138307, 2137603, and 2138296.

Author contributions

X.F. and P.M.K.-H. designed the research. X.F. and P.V. performed all simulations and analyzed the data. S.O.-V. and A.V.Z. conducted the experiments. X.F., P.V., and P.M.K.-H. wrote the manuscript.

Declaration of interests

The authors declare no competing interests.

Editor: Jeremiah Zartman.