Computational methods for characterizing and learning from heterogeneous cell signaling data

Abstract

Heterogeneity in cell signaling pathways is increasingly appreciated as a fundamental feature of cell biology and a driver of clinically relevant disease phenotypes. Understanding the causes of heterogeneity, the cellular mechanisms used to control heterogeneity, and the downstream effects of heterogeneity in single cells are all key obstacles for manipulating cellular populations and treating disease. Recent advances in genetic engineering, including multiplexed fluorescent reporters, have provided unprecedented measurements of signaling heterogeneity, but these vast data sets are often difficult to interpret, necessitating the use of computational techniques to extract meaning from the data. Here, we review recent advances in computational methods for extracting meaning from these novel data streams. In particular, we evaluate how machine learning methods related to dimensionality reduction and classification can identify structure in complex, dynamic datasets, simplifying interpretation. We also discuss how mechanistic models can be merged with heterogeneous data to understand the underlying differences between cells in a population. These methods are still being developed, but the work reviewed here offers useful applications of specific analysis techniques that could enable the translation of single-cell signaling data to actionable biological understanding.

Introduction

Genetically identical cells, exposed to identical stimuli, exhibit markedly different responses. One key manifestation of this non-genetic heterogeneity is in signaling pathway activation, where individual cells display a range of behaviors in both basal activity and responsiveness to a stimulus. The drivers that control signaling heterogeneity and the behavioral outcomes of signaling heterogeneity are not well characterized, and fundamental questions remain. Heterogeneity in signaling activity is thought to drive cellular behaviors, some of which are binary, such as division or cell death [1–3]. Thus, one area of research focuses on what control points and mechanisms translate continuous variation in some components into binary decisions downstream (Figure 1A). Another focus of research is determining to what extent heterogeneity represents randomness, caused by a cell’s intrinsic inability to faithfully interpret a stimulus. Alternatively, perhaps different responses are caused by pre-existing differences among cells with some poised to strongly activate a signaling pathway (Figure 1B–E). This is an important biological question with deep implications relating to the limitations of signaling pathways and communication in multicellular organisms. Non-genetic heterogeneity also has urgent clinical implications. Heterogeneity drives some of the most disastrous features of cancer, including metastasis [4], invasion [5], and the emergence of drug-resistant cells [6,7]. Therefore, understanding the causes of heterogeneous cell signaling and identifying ways to control it are crucial steps toward effective cancer therapy [8]. To start to answer these questions, researchers have used an array of live-cell microscopy techniques to measure behaviors in single cells. However, interpretation of these data has proven difficult, and an array of modeling techniques have emerged to make meaning out of heterogeneous, single-cell data.

Figure 1:

Sources of heterogeneity. A) Cells can exhibit continuous variation, for instance in signaling response (middle). This continuum may be converted into binary outcomes for phenotypes like cell death. B-E) Comparing the effects of intrinsic and extrinsic noise. Columns represent two cells in different pre-existing states based on the amount of an inhibitory protein in the cell. Rows represent differences in ligand concentration, with low ligand concentration leading to significant differences in response based on stochasticity of receptor-ligand binding. Insets represent signaling responses in 3 separate cells for each condition. F) A kymograph or colormap is commonly used to represent single-cell signaling data. Left: Two individual cell responses, one sustained and one transient. Right: A kymograph summarizing responses of 3000 cells.

In genetically identical cells, heterogeneity is thought to arise from both intrinsic and extrinsic sources [9–11]. Intrinsic heterogeneity occurs due to random fluctuations inherent to individual intermolecular collisions that underlie the chemical reactions that regulate biological processes. These fluctuations are present in any chemical reaction, but they decrease in influence as concentration increases and randomness in intermolecular collisions is smoothed out. However, intrinsic noise can be significant in some biological contexts, typically cellular processes involving low abundance molecules like gene transcripts (below 100 copies in mammalian cells, as a rough estimate) or rare events [12,13]. Intrinsic noise places fundamental limits on the ability of cells to sense their environment and control gene expression [14,15].

Extrinsic noise refers to pre-existing differences in cell state and are thus “extrinsic” from the fundamental fluctuations of chemical reactions. Cells that appear identical may in fact exhibit differences in cell-cycle timing, kinase pathway activation, or the concentration of receptors or other signaling pathway molecules [16,17]. When exposed to a stimulus, these cells will respond in various ways predicated on their pre-existing state. If cells were completely identical, extrinsic noise could be eliminated, but they would still respond differently due to intrinsic noise.

The clinical goals of controlling heterogeneity, e.g. to improve cancer treatment, require that researchers understand the sources of heterogeneity. Therefore, distinguishing between intrinsic and extrinsic noise is a key challenge, which can be addressed with computational and experimental tools. It may be possible to mitigate or alter heterogeneous populations of cells if extrinsic noise is driving their heterogeneity [18,19]. However, if intrinsic stochasticity is the root cause of heterogeneous phenotypes, interventions may not be possible or may need to target a downstream effector that is less impacted by intrinsic noise.

In this mini-review, we outline several methods used to analyze live-cell, time-lapse, fluorescence microscopy data. Given the key role that dynamic processes can play in controlling cellular behaviors, we focus here on live-cell, time-lapse methods, as opposed to endpoint methods including cyclic immunofluorescence, flow cytometry, or single-cell RNA-seq, all of which have been reviewed elsewhere e.g. [20,21]. Dynamic experimental methods can quickly generate hundreds of observations for thousands of cells, necessitating the use of computational methods to answer even simple questions about the data relating to the population level behaviors and variance among cells. Furthermore, watching cells in real time leads to new questions about the dynamics of cellular processes in single cells and their synchronization, which can be answered using dynamic computational models. We discuss data-driven methods that extract patterns from data, which can be quickly and easily applied to quantify and compare heterogeneous signaling behaviors. We then discuss model-driven methods, where researchers construct mechanistic models with varying degrees of complexity to explain heterogeneous behaviors in their data. Throughout this review, we provide examples of specific biological questions that are addressed.

Single-cell Signaling Data Acquisition

Single-cell time domain data can be obtained from any live single-cell imaging modality. Using bright-field microscopy, data about morphology, movement, lineage, and cell division can be measured for individual cells over time [22]. Using fluorescence to monitor biochemical processes including cell signaling can be accomplished with dyes or endogenously expressed fluorescent reporter proteins [23,24]. Ideal fluorescent measurement systems will feature a high quantum yield to minimize light exposure and rapid activation and deactivation kinetics to accurately track the underlying cellular process of interest. Other key factors are the photostability of the reporter protein and maximizing the overlap between reporter emission spectra and detector absorption spectra while minimizing overlap in emission spectra among different reporter proteins. Balancing these considerations is difficult and has been reviewed at length elsewhere [25].

Automated image processing is another key aspect of the experimental pipeline. Although ImageJ [26] can be used to extract intensities and calculate reporter activities, manually processing images will typically not scale effectively to larger datasets. Automated methods such as CellProfiler [27] have been developed to identify individual cells, extract reporter activities, and track cells from frame to frame [28,29]. Ideally, tracking will be consistent even in dense, confluent cultures, and cell division and lineage will be tracked. These programs generate trajectories for each cell tracked based on the activity of the reporters in that cell. These trajectories can be summarized quickly in kymographs, which display individual cell trajectories as rows in a colormap, with the intensity in each column corresponding to the reporter value at that time (Figure 1F).

Data-Driven Inference

Given a heterogeneous distribution of cell trajectories (Figure 2A), the simplest analytical approach is to mitigate the heterogeneity using summary statistics such as averages, medians, and percentiles (Figure 2B). This approach is quick and readily understood, providing a useful start to any analysis of heterogeneous data. However, there are two main problems with applying summary statistics to heterogeneous signaling data. Firstly, they blur the heterogeneity present in the data, leaving the researcher with aggregate behaviors that can be measured with other methods. One solution to this problem is to look not just at means or medians, but instead at various percentiles or outliers, which are only accessible from single cell data [30]. The second issue with mean or median behaviors is that they may not represent any real cells. For instance, if cell responses are binary (either high or low), the average of all cells will be a medium response, behavior that is not exhibited by any actual cells observed (Figure 2B) [31]. It can be difficult to know when aggregate data can provide insight or when aggregation obscures meaningful single-cell behaviors. For instance, Goglia et. al. found meaningful differences in drug responses by aggregating cells and looking at mean population behaviors [32]. However, when deciding what types of aggregation to use, researchers should consider if they are interested in studying collective or individual behaviors and should also balance the potential for greater explanatory power of single-cell data against the increased noise that can come from single cell trajectories. The methods discussed below can provide researchers with more flexibility when considering some of these issues.

Figure 2:

Analysis of dynamic, single-cell data. A) Each line indicates a cellular response from a single-cell experiment, showing two families of responses with variation in each family. B) Mean and percentile measures can capture some dynamics, but no experimental cells behave like the mean. C) Taking the mean of cells assigned to clusters can yield representative dynamics. D, E) Hypothetical data from a dose response experiment. D) The mean increases with dose, but the dynamics are not captured. E) Clustering the data reveals that increasing dose leads to more cells adopting fast, transient responses (blue cluster). F) Functional PCA can identify clusters of behavior while capturing variation within each cluster.

Clustering algorithms (Table 1) can summarize signaling trajectories while ensuring that the aggregation is representative of real behaviors. Clustering finds families of distinct trajectories within the data by first calculating the distance between individual trajectories and then identifying clusters of trajectories that have low within-cluster distances and high between-cluster distances. This allows data aggregation within each cluster (Figure 2C). Euclidean distance is often used as a distance metric to calculate the similarity of each cell with every other cell. However, cells may display dynamics at different times, or for different durations, despite undergoing the same biological processes. For instance, clustering dividing and non-dividing cells would require a distance metric that is sensitive to asynchronous divisions in cells and differences in the duration of cell division. Therefore, distance metrics using dynamic time warping or other algorithms may do a better job of describing similarities in time series data [33]. After defining a distance metric, commonly used clustering algorithms such as hierarchical clustering or k-means are used to label individual observations based on cluster occupancy. Using clustering, researchers can ensure that the aggregate behaviors that compose each cluster are representative of real cell behaviors. Blum et. al. wanted to determine if different growth factors induced distinct ERK activation dynamics in single cells [34,35]. They clustered ERK trajectories from cells activated by various growth factors, allowing them to represent over 1000 signaling trajectories using 6 clusters, drastically simplifying the analysis and presentation of their data. They found that the clustered behaviors were growth-factor dependent. Interestingly, their analysis also showed that for even low doses of growth factors, some cells would respond strongly, and at high doses, the occupancy of this cluster increased (Figure 2D, E). In this work, clustering was useful because it separated cells that had qualitatively different dynamics (e.g. transient vs. sustained activation). However, this may be an inappropriate constraint when cells occupy a continuum of responses.

Table 1:

Computational techniques for dimensionality reduction and mechanistic understanding of single-cell heterogeneous signaling data

Method	Output	Benefits	Shortcomings	Software	Applications
Clustering	Labels for each cell corresponding to cluster belonging.	Aggregates based on real cell behavior, relatively fast, interpretable.	Inappropriate or unhelpful for continuous responses	Time course inspector R package and web application [35]	[22], [34], [47]
FPCA	Low-dimensional projection of each cell along axes of variation in the data.	Enables easy visualization of heterogeneity to identify patterns in data.	Primary axes of variation may not be physiologically meaningful.	FDA package [36,37]	[38]
Information Theory	Numerical quantification of mutual information between input (dose or ligand) and output (reporter activity)	Summarizes heterogeneous outcomes in a single number with a physical meaning.	Mutual information may be of limited interest for particular study.	SLEMI R package [59], EstCC Scala package [60]	[11,39–42,44]
Stochastic Dynamic modeling	A distribution of species dynamics corresponding to each parameter set used by modeler.	Can explicitly model the effects of intrinsic stochasticity and low molecular number.	Requires knowledge of rate constants and species interactions, along with simplifying assumptions to make system tractable.	Simbiology MATLAB package, Hy3S [61], PySB Python package [43]	[43,44,46]
ODE modeling	A set of deterministic species dynamics corresponding to each parameter set used by modeler.	Can vary parameters to test hypotheses about sources of heterogeneity, suggest mechanistic drivers.	Requires knowledge of rate constants and species interactions, along with simplifying assumptions to make system tractable.	Simbiology MATLAB package, PySB Python package [43]	[11,16,43,47,48]

Another method that can summarize an arbitrarily large number of trajectories is functional principal component analysis (fPCA, Table 1) [36,37]. fPCA decomposes a set of trajectories into two different parts, harmonics and principal component scores. The harmonics of fPCA are orthogonal time-dependent trajectories representing times when there is the most variance in the data The key difference between fPCA and PCA is that fPCA captures time-dependence in trajectories, as opposed to principal components that ignore the time-dependence of the data. fPCA will also calculate a numerical score for each individual trajectory provided, based on how much each harmonic represents a specific trajectory. For instance, if some cells exhibited a transient increase in signal, while others had a sustained increase, these would be represented as different functional principal components, with transient cells scoring highly along one axis and sustained cells scoring highly along the other. By representing trajectories in lower-dimensional space, they can be visualized and compared more easily (Figure 2F). Sampattavanich et. al. used fPCA to identify differences in transcription factor FOXO3a dynamics in response to a variety of growth factors [38]. fPCA identified that key variation in trajectories existed through variation in the basal level, the post-stimulus steady-state, and the transient response. Furthermore, the authors were able to cluster responses to ligands based on their dynamics as represented by fPCA. A potential shortcoming of clustering is that it labels each cell discretely, implicitly erasing heterogeneity that may exist within clusters. Since fPCA relaxes this constraint and projects cells along a spectrum of PC scores, this heterogeneity is preserved, which may provide a clearer picture of the data. Furthermore, fPCA can serve as a pre-processing step for later clustering.

Clustering and fPCA focus on revealing structure within heterogeneous data by reducing dimensionality in a discrete or continuous manner. These are related to unsupervised machine learning methods, but supervised learning, where different observations are labeled, can also be applied to heterogeneous cell signaling. For example, by extending experimental acquisition times to several hours or days, cells can be labeled based on cell division, death, or migration. This gives researchers another way to categorize signaling data based on material observations and has the added benefit of reflecting phenotypes that are oftentimes of interest. However, this approach has attendant difficulties, including increased photodamage from long-term microscopy and the increased difficulty of tracking cells over long times. Miura et. al. showed that in cells exposed to UV light, p38 and JNK respond heterogeneously, with p38 suppressing JNK in some cells [2]. They extended imaging to 6–12 hours, and were able to identify JNK-high cells as apoptotic, while p38-high cells survived. This behavioral classification allowed them to associate signaling patterns with functional outputs.

Given that signaling pathways communicate the presence and amount of a ligand in the extracellular space to the cell, key questions are: How effective is this communication, and what tradeoffs do cells make to control effectiveness? Information theory answers these questions by quantifying how severely noise (both extrinsic and intrinsic) corrupts the ability of a signal sender to communicate with a receiver (Table 1). Information theory measures communication in units of bits, which correspond to the amount of information gained about an input from measurement of an output. In a system with high information transfer, measuring an output will give a high-confidence estimate of the input signal, while in a low information transfer system, noise will corrupt the output, making it difficult to determine a specific input based on a measured output. Most commonly, information theory has been used to quantify the ability of cells to differentiate between different doses of a ligand, given that each dose provokes a distribution of responses across different cells. Early work using this approach from Cheong et al. calculated information transfer for experimental measurements of NF- κB activation by TNF and compared them with theoretical results from statistical models incorporating different signaling pathway architectures and noise sources, finding that signaling architecture and the integration of a signal over time by cells can mitigate information loss [39]. In an expansive study, Selimkhanov et. al. quantified information transfer in ERK, calcium, and NF-kB signaling pathways [40]. They showed that information transfer can be much higher if signaling responses at multiple time points are considered, offering one explanation of the previous, static measurements of information transfer, which were surprisingly low. They also used a combination of information theory and experimental perturbation of the ERK pathway to quantify intrinsic and extrinsic sources of noise, finding that extrinsic sources of heterogeneity dominated. More recent work has built on these foundations, measuring information transfer by stimulating the same cells repeatedly [41,42]. These experiments and analyses have revealed that different cells in a population may have very different information capacities. Information theory facilitates the quantification of intrinsic and extrinsic contributions to heterogeneity, and summarizes heterogeneous data in a way that is consistent with signaling pathway function - to transmit information and enable cells to respond accordingly.

Mechanistic Models of Heterogeneity

The data-driven methods described above detail ways of extracting patterns or meaning from within a dataset. However, a separate class of methods involves constructing mechanistic models of cellular behavior, and synthesizing model and experimental data to draw conclusions about observed heterogeneity. Mechanistic modeling requires numerical parameters and information about mechanisms that are oftentimes difficult or impossible to obtain, which can make modeling significantly more difficult and time-consuming than data-driven methods. However, mechanistic models provide unique opportunities to computationally control the relative effects of biological mechanisms and parameters, which can yield deeper insights into the causes and consequences of heterogeneity. Models can be stochastic or deterministic, and can range from simple and qualitative to sprawling and numerically precise. In general, model building should be directed by specific questions that can dictate the specific interactions that need to be included in the model.

To model stochastic sources of intrinsic noise, specific frameworks have been developed that simulate individual molecular events (Table 1) [43]. The framework for these simulations uses the Gillespie algorithm, which calculates the probability of different events occurring based on their pre-defined interaction rate constants and species number. To run stochastic simulations, the researcher must define the species involved, the stoichiometry of each interaction between species, and probabilistic rate constants of each interaction. After these parameters are defined, simulations can be run, which produce individual realization of the random process being modeled. This approach was utilized to study heterogeneity in the ERK response to hormone stimulation by Garner et. al. They constructed a simple model of ERK signaling that incorporated multiple sources of feedback control [44]. They combined this model with calculations of information transfer carried out on their experimental and simulated data, and varied different sources of feedback to study how different regulatory mechanisms could influence information transfer. Their model enabled them to make experimentally testable predictions about the factors that influence information transfer in the presence of intrinsic noise. Notably, stochastic simulations were also used by Iwamoto et. al. to explore variability in EGFR signaling to ERK, as measured by flow cytometry [45]. They found that stochasticity caused by intrinsic noise was insufficient to explain the variability in their data because they measured sufficiently high concentrations of molecules for all components. This highlights the potential complexity of modeling intrinsic and extrinsic factors underlying heterogeneity, and shows how modeling can be used to differentiate between the two. A similar approach was used by Wang et. al. in analyzing p53 dynamics, where they found that in mammalian cells, p53 variability was consistent with extrinsic cell-to-cell differences [46]. These two examples illustrate how computational models can help identify extrinsic, and potentially controllable sources of noise, compared to intrinsic noise.

As consciousness of the role of pre-existing differences in seemingly identical cells grew, other work began to focus on using deterministic mechanistic models based on ordinary differential equations to understand which cellular components were different from cell to cell (Table 1) [11,16,43,47,48]. Yao et. al. observed heterogeneous calcium signaling dynamics (Figure 3A) [48]. They used summary statistics to summarize and cluster their data, but they followed up on those observations by constructing a differential equation model of calcium signaling (Figure 3B). They derived parameter sets that provided good fits to individual cellular responses, and recorded the parameter distributions that were extracted from the population of single-cell fits (Figure 3C, D). They found clusters of parameter distributions that corresponded to clusters of cellular responses. Observed heterogeneity of responses arose from differences in interactions between IP3 import to the endoplasmic reticulum and related regulation of this process by calcium. This approach required the construction of a mechanistic model, which has intrinsic difficulties and complications, but it uniquely facilitated a granular, mechanistic understanding of calcium signaling heterogeneity.

Figure 3:

Mechanistic modeling of heterogeneous signaling data. A) Heterogeneous single cell responses can differ in both magnitude and shape, with some cells exhibiting a sustained peak and others a transient response. B) A simplified schematic of an ODE model which can be used to study the heterogeneity shown in A. Two rate constants, k₁ and k₂, out of many are identified. C) Systemic variation of parameters in the model shows that k₁ increases the magnitude of the response, while changes in k₂ tune the duration of the response. D) Model and experimental outputs can be compared (top) with different cells fitting well with different model outputs. This procedure can be completed for all cell/model combinations (bottom) to extract a parameter set which represents the pre-existing cell state. E) Model-based inference of parameter values – and therefore cell state - enables population of cells to be placed on a spectrum of heterogeneous mechanistic processes, and compared between two different experimental conditions (black distribution vs. red distribution). F) Understanding cell state can provide mechanistic explanations for why some subpopulations of cells are sensitive to a drug and others aren’t. Treating with a drug that only targets some of the cells (green circle) will leave behind an insensitive population of cell. G) Using this knowledge, we can combine drugs temporally, giving one drug (orange) to sensitize the entire population to a second drug (green). H) We can also combine drugs (blue and green) to target two subpopulations simultaneously, eliminating the whole population of cells.

The approach taken by Yao et. al., in varying every model parameter, implies a particular assumption about pre-existing cellular states - that cell state occupancy is pseudo-random, and that variability is equally likely in all components [49]. However, some pathway components are affected by many different inputs, while others are more confined to individual pathways [50]. Furthermore, different components are subject to different regulatory mechanisms which may be better or worse at suppressing variation [51]. Therefore, it is plausible that variability in kinase activation could be due to a small number of highly variable species. Spinosa et. al. explored this hypothesis when studying ERK and Akt signaling in response to CXCL12 (Figure 3B) [16]. They built a differential equation model including both pathways and accounted for the variability observed by varying just three parameters, corresponding to extrinsic noise leading to variable activation in specific pathway components Ras, PI3K, and mTORC1 (Figure 3C,D). This approach allowed them to locate each cell in an experiment to a specific point in 3D space corresponding to pre-existing cell state, and to make measurable predictions about the effects of various inhibitors in moving the population of cells around in state space (Figure 3E). Identifying key nodes of heterogeneity may be useful for inferring treatable therapeutic targets facilitating the control of heterogeneous cell states, or understanding why some cells are resistant to targeted inhibitors (Figure 3F).

Continuing Challenges and Promising Approaches

As experimental methods advance, we will have access to ever-growing forms of data, including multiplexed reporters that can be deployed in new ways and observed at new timescales. Making meaning from all of this data may be a daunting task, but it will yield unprecedented insights into the sources of cellular heterogeneity and means by which it can be controlled. Fortunately, extracting meaning from heterogeneous time-series data is central to the field of signal processing, and many challenges associated with single-cell signaling data are present in other fields. Some techniques, including information theory and functional data analysis, were borrowed from signal processing and creatively applied to single-cell signaling data. Researchers in many fields are studying heterogeneous, dynamic data to answer questions ranging from the geographic differences in bird calls to fault diagnosis in mechanical systems [52,53]. Novel machine learning techniques to extract features from time-series data are an open field of study [54]. All of this suggests that interdisciplinary collaboration inspired by work in many different fields may yield creative methods of analysis for heterogeneous cell signaling data.

There are also key biological challenges to address in order to fully realize the potential of single-cell imaging modalities. Specific challenges, which will require realizing novel computational and experimental methods, include the incorporation of disparate timescales, cellular systems, and streams of data. Meaningful signaling dynamics can occur on the seconds to minutes time scale, as shown by transient ERK pulses that can drive proliferation, on the minutes to hours timescale associated with ligand-receptor signaling, and on the several hour scale associated with cell division [55,56]. Connecting these timescales in a multi-scale model that explains variation in single-cell behaviors represents a major challenge, but also holds the key to understanding how cells integrate information to make decisions. Furthermore, signaling pathways are affected by numerous other cellular processes, including metabolism and cell cycle progression, all of which may help determine a cell’s pre-existing state. Understanding how cell-to-cell variation in these other systems, such as variations in glucose utilization, affect heterogeneity in signaling, and vice versa, will require merging computational models for each individual system [57]. Finally, heterogeneity is understood in a variety of ways, and single-cell sequencing technologies have exploded in recent years. Understanding how observable signaling dynamics shape heterogeneity in RNA transcript number is another major challenge, which again requires merging mechanistic signaling models with data-driven models of heterogeneity in RNA expression [58]. By extending our understanding of the situational balance between various sources of heterogeneity, we will be able to identify targets that will enable control over cellular populations, and possible dead-ends where intrinsic noise dominates over any plausible intervention. Clinically, identifying sources of heterogeneity may enable temporal treatment strategies that first shift heterogeneous cells to a more homogeneous set of states, making the whole population vulnerable to a second drug (Figure 3G) [18]. Another approach might identify treatments that affect specific subpopulations based on an identified axis of variation and combine those treatments to kill all populations simultaneously (Figure 3H). Creative combinations of experimental and computational methods are the key to making progress on all of these fronts.

Highlights:

• Multiplexed fluorescent reporters offer unprecedented measurements of single-cell data.

• Data-driven methods can reveal subpopulations with distinct dynamics or outcomes.

• Mechanistic models enable discovery of drivers of heterogeneity.