Original Research ArticleMachine learning provides insight into models of heterogeneous electrical activity in human beta-cells
Introduction
Biological heterogeneity is a fundamental fact, which is well recognized and thought to underlie, e.g., robustness and flexibility of biological systems [1], [2]. Nonetheless, most mathematical modeling of cellular dynamics choose a “typical” set (or a few sets) of parameters representing an “average” cell (cells), see e.g. [3], [4], [5]. The robustness of the model can then be investigated, e.g., by performing bifurcation analyses of the model with respect to a few number of parameters thought to be important [4], [5], [6], [7].
However, many recent models of cellular electrophysiology contain a large number of parameters that preclude a traditional bifurcation analysis, and therefore it has been suggested to vary the parameters randomly by extracting them from an experimentally well-described distribution [8], [9], [10], [11], [12], [13]. Moreover, cells may co-regulate different mechanistic components such as ion channels to obtain or maintain a certain behavior, which mathematically correspond to parameters being correlated within the cell population [14], [15], [16], [17], [18], [19].
If the model and its parameters are realistic, the analysis of its behavior for different combinations of parameters then provide insight into the mechanistic control of cellular dynamics. Such insight can be obtained by direct exploration of parameter space, as has been done for models of neurons [8], [9], cardiac cells [10], pituitary cells [20] and mouse beta-cells [12]. Further understanding can be obtained with statistical analyses of the simulation results. For example, Sobie and colleagues [21], [22] investigated how the duration of the cardiac action potential depends on parameters by performing multivariate linear regression. Montefusco et al. [13] used multinomial logistic regression to investigate which ion currents determine the responses within the very heterogeneous pancreatic alpha-cell population under different conditions. Machine learning provides an alternative toolbox and could provide further insight in addition to the results obtained from the statistical methods described above [11], [23], [24].
Here we explore several machine learning techniques and population modeling based on high-quality single-cell electrophysiological data [25] to investigate how different ion channels influence the dynamics of an updated model of human pancreatic beta-cells. Pancreatic beta-cells release insulin in response to glucose following a cascade of events culminating with electrical activity, calcium influx and exocytosis of insulin containing granules [26], [27]. The heterogeneity of the beta-cell population is well established and believed to contribute to the refined glucose-sensing capabilities of the endocrine pancreas [28], [29], [30], [31], [32], [33]. Mathematical modeling of electrical activity in mouse beta-cells has a 40-year long history [34], [35], [36], and within the last decade electrical activity in human beta-cells has been modeled [37], [38], [39], [40], [41]. Mathematical models of exocytosis and cellular insulin release have also been developed over several decades [42], [43], [44], [45], [46], [47]. However, a systematic investigation of the contributions of the different ion currents and their dynamics to causing and shaping electrical activity in human beta-cells is still lacking.
Section snippets
Human beta-cell mathematical model
We used an updated version of our previous model of human-cells [39] that includes the description of BK-CaV complexes [41]. The model is composed of a metabolic component [48] that drives a model of electrical activity [37], [39], [41], which was developed from electrophysiological data from human beta-cells [49], [50], [51], [52]. In brief, the glycolytic enzyme phosphofructokinase (PFK) can generate metabolic oscillations due to positive feedback by its product fructobisphosphate (FBP). In
Results
We simulated heterogeneous populations of human beta-cells as explained in the Methods and in greater details in the Supplementary Material. The results were analyzed with logistic regression models, classification trees and random forests. A summary of the results are provided in this section ( Table 1). Detailed results are presented for All Cells without the metabolic oscillator in Fig. 1, and for all cases in Figs. S7–S12 in the Supplementary Material. The performances of the classifiers
Discussion
In the present work, we have provided a first analysis of electrical activity in heterogeneous human beta-cells using a combination of mechanistic population modeling and various machine learning methods. Our analysis revealed both expected results as well as novel findings that merit further investigation.
In most of the cases, the dynamics of the gating variables, as modeled via the time constants , did not influence the results much, which to a large degree justifies the typical approach of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was financially supported by MIUR (Italian Minister for Education) under the initiative ‘Departments of Excellence’ (Law 232/2016) and by a grant from the University of Padova, Italy (SEED2020).
References (64)
- et al.
Cellular heterogeneity: do differences make a difference?
Cell
(2010) - et al.
How to address cellular heterogeneity by distribution biology
Curr. Opin. Syst. Biol.
(2017) - et al.
Where to look and how to look: Combining global sensitivity analysis with fast/slow analysis to study multi-timescale oscillations
Math. Biosci.
(2019) - et al.
Heterogeneous alpha-cell population modeling of glucose-induced inhibition of electrical activity
J. Theoret. Biol.
(2020) - et al.
Activity-independent homeostasis in rhythmically active neurons
Neuron
(2003) Parameter sensitivity analysis in electrophysiological models using multivariable regression
Biophys. J.
(2009)- et al.
Patch-seq links single-cell transcriptomes to human islet dysfunction in diabetes
Cell Metab
(2020) - et al.
Pancreatic cell heterogeneity in glucose-induced insulin secretion
J. Biol. Chem.
(1992) - et al.
Cellular communication and heterogeneity in pancreatic islet insulin secretion dynamics
Trends Endocrinol. Metab.
(2014) A biophysical model of electrical activity in human -cells
Biophys. J.
(2010)
Concise whole-cell modeling of BKCa-CaV activity controlled by local coupling and stoichiometry
Biophys. J.
A model of phosphofructokinase and glycolytic oscillations in the pancreatic beta-cell
Biophys. J.
Mathematical modeling and statistical analysis of calcium-regulated insulin granule exocytosis in -cells from mice and humans
Prog. Biophys. Mol. Biol.
A quantitative description of membrane current and its application to conduction and excitation in nerve
J. Physiol.
Mathematical Physiology
Dynamical systems theory in physiology
J. Gen. Physiol.
Dynamical Systems in Neuroscience
Global structure, robustness, and modulation of neuronal models
J. Neurosci.
Alternative to hand-tuning conductance-based models: construction and analysis of databases of model neurons
J. Neurophysiol.
Experimentally calibrated population of models predicts and explains intersubject variability in cardiac cellular electrophysiology
Proc. Natl. Acad. Sci. U S A
Cellular function given parametric variation in the Hodgkin and Huxley model of excitability
Proc. Natl. Acad. Sci. U S A
Activity-dependent regulation of conductances in model neurons
Science
Modeling stability in neuron and network function: the role of activity in homeostasis
Bioessays
Regulation of ion channel expression
Circ. Res.
Robustness of burst firing in dissociated purkinje neurons with acute or long-term reductions in sodium conductance
J. Neurosci.
Ion channel degeneracy, variability, and covariation in neuron and circuit resilience
Annu. Rev. Neurosci.
From global to local: exploring the relationship between parameters and behaviors in models of electrical excitability
J. Comput. Neurosci.
Exploiting mathematical models to illuminate electrophysiological variability between individuals
J. Physiol.
Improved prediction of drug-induced torsades de pointes through simulations of dynamics and machine learning algorithms
Clin. Pharmacol. Ther.
Mechanistic models versus machine learning, a fight worth fighting for the biological community?
Biol. Lett.
Regulation of insulin secretion: a matter of phase control and amplitude modulation
Diabetologia
Cited by (2)
Pancreatic β-cell heterogeneity in adult human islets and stem cell-derived islets
2023, Cellular and Molecular Life Sciences