Elsevier

Mathematical Biosciences

Volume 354, December 2022, 108927
Mathematical Biosciences

Original Research Article
Machine learning provides insight into models of heterogeneous electrical activity in human beta-cells

https://doi.org/10.1016/j.mbs.2022.108927Get rights and content

Highlights

  • Cellular heterogeneity of pancreatic beta-cells is of physiological importance.

  • We exploit a dataset of electrophysiological parameters in single human beta-cells.

  • We perform population modeling to understand human beta-cell dynamics.

  • Machine learning is used to relate ion channels to different patterns of activity.

  • Our approach provides insight and generates new hypotheses to be investigated.

Abstract

Understanding how heterogeneous cellular responses emerge from cell-to-cell variations in expression and function of subcellular components is of general interest. Here, we focus on human insulin-secreting beta-cells, which are believed to constitute a population in which heterogeneity is of physiological importance. We exploit recent single-cell electrophysiological data that allow biologically realistic population modeling of human beta-cells that accounts for cellular heterogeneity and correlation between ion channel parameters. To investigate how ion channels influence the dynamics of our updated mathematical model of human pancreatic beta-cells, we explore several machine learning techniques to determine which model parameters are important for determining the qualitative patterns of electrical activity of the model cells. As expected, K channels promote absence of activity, but once a cell is active, they increase the likelihood of having action potential firing. HERG channels were of great importance for determining cell behavior in most of the investigated scenarios. Fast bursting is influenced by the time scales of ion channel activation and, interestingly, by the type of Ca2+ channels coupled to BK channels in BK-CaV complexes. Slow, metabolically driven oscillations are promoted mostly by K(ATP) channels. In summary, combining population modeling with machine learning analysis provides insight into the model and generates new hypotheses to be investigated both experimentally, via simulations and through mathematical analysis.

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 τX, 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).

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