A Two-Pulse Cellular Stimulation Test Elucidates Variability and Mechanisms in Signaling Pathways

Abstract

Mammalian cells respond in a variable manner when provided with physiological pulses of ligand, such as low concentrations of acetylcholine present for just tens of seconds or TNFα for just tens of minutes. For a two-pulse stimulation, some cells respond to both pulses, some do not respond, and yet others respond to only one or the other pulse. Are these different response patterns the result of the small number of ligands being able to only stochastically activate the pathway at random times or an output pattern from a deterministic algorithm responding differently to different stimulation intervals? If the response is deterministic in nature, what parameters determine whether a response is generated or skipped? To answer these questions, we developed a two-pulse test that utilizes different rest periods between stimulation pulses. This “rest-period test” revealed that cells skip responses predictably as the rest period is shortened. By combining these experimental results with a mathematical model of the pathway, we further obtained mechanistic insight into potential sources of response variability. Our analysis indicates that in both intracellular calcium and NFκB signaling, response variability is consistent with extrinsic noise (cell-to-cell variability in protein levels), a short-term memory of stimulation, and high Hill coefficient processes. Furthermore, these results support recent works that have emphasized the role of deterministic processes for explaining apparently stochastic cellular response variability and indicate that even weak stimulations likely guide mammalian cells to appropriate fates rather than leaving outcomes to chance. We envision that the rest-period test can be applied to other signaling pathways to extract mechanistic insight.

Introduction

Cellular response variability is evident when a population of cells is exposed to two pulses of ligand: some cells respond only to the first or to the second pulse, some respond to both pulses, and some cells respond to neither pulse. Such variability is important to consider given the multiple circumstances under which cells are exposed to pulsatile or changing ligand concentrations in physiology (1, 2, 3, 4, 5). As described in (5), all the examples show frequency- and rest-period-dependent outcome changes. Different biological processes such as neuronal growth or myocyte differentiation have been shown to be impacted by the frequency of ligand stimulation for the corresponding signaling pathways. For example, pulsed stimulation of muscarinic receptors elicits nuclear translocation of a transcription factor, nuclear factor of activated T-lymphocytes (NFAT) (1). The frequency of stimulation determines whether NFAT4 or NFAT1 is translocated. Similarly, pulsed stimulation of neurites with epidermal or nerve growth factors impacts growth and differentiation via extracellular-regulated kinase signaling based on the frequency of stimulation (2). In another work, the frequency of transforming growth factor β pulses has been shown to impact the ability of myocytes to differentiate or not (3).

Noise in the signaling system is thought to influence the fidelity of the signaling under pulsatile stimulation (6). This noise can be either due to stochastic reactions after exposure (intrinsic noise) or due to the preexisting heterogeneity in cell state (extrinsic noise), and both can lead to cell-to-cell heterogeneity in the response. For intrinsic noise, if the concentration of components is held constant, there would still have different outcomes from cell to cell. In contrast, with extrinsic noise, variation in component concentrations are the source of the variability from cell to cell. Small microorganisms, with volumes and protein numbers roughly three orders of magnitude smaller than that for the mammalian cells, display stochasticity in response to weak stimuli. Signaling variability in mammalian cells had also been generally considered to be due to stochastic processes. However, this has been debated more recently because there are typically significantly larger volumes and numbers of protein molecules in mammalian cells as well as experimental observations at single-cell levels with deterministic explanations. For examples, cellular events such as CDK2-dependent mitotic exit or TNFα-induced NFκB activation, which were earlier described as stochastic processes, have recently been shown to have deterministic origins (7, 8, 9). Tools are needed to understand the sources of and mechanisms behind this variability, which can lead to highly disparate cell fates.

Here, we describe a tractable, two-pulse experimental test and accompanying mathematical analysis. The “rest-period test” entails stimulating cells with two pulses of ligand, increasing the rest period between pulses, and observing the effect on cellular responses. Because the rest period between stimulation pulses is associated with signaling pathway recovery properties, our test provides a characteristic signature for a signaling process without measuring or inferring individual cell parameters. We used the rest-period test to gain mechanistic insight into cellular response.

Single HEK 293 cells stably expressing muscarinic M₃ receptors, a G-protein-coupled receptor (GPCR), were analyzed using the rest-period test by exposing these cells to two consecutive low-concentration pulses of an M₃ agonist within a microfluidic device and monitoring the resulting intracellular calcium responses (10). Calcium responses were quantified using live-cell imaging as described previously (1). Building upon our previous mathematical model of GPCR-induced calcium signaling (1, 11, 12), we additionally incorporated cell population heterogeneity (sometimes referred to as extrinsic noise or biochemical variability). Using this approach, we found that cellular response variability is in large part dictated by a high Hill coefficient step, a short-term memory of stimulation, and variability in protein levels between cells. Additionally, to illustrate broad applicability of our findings beyond the M₃ signaling pathway, we also analyzed published data on the TNFα-NFκB pathway (6, 13, 14), which operates at an entirely different timescale (several minutes to hours) and found that highly variable cellular responses here are also consistent with similar deterministic processes and a heterogeneous cell population.

In both these signaling systems, we observed that as the rest period between stimulation pulses was increased, cells either responded to both pulses or responded to neither pulse. This key observation demonstrates that even at low levels of stimulation, cells can be guided to their appropriate fate based on stimulation parameters, such as concentration, duration of stimulation, and rest period between successive stimulations, rather than through stochastic processes that leave outcomes to chance. Our findings are supported by a number of recent works that have used microfluidics or analysis of single cells in a population (15, 16, 17, 18, 19, 20, 21, 22). Our rest-period test and accompanying mathematical analysis is a complementary approach, which provides unique insight into signaling states and can be broadly applied for analysis of other signaling systems.

Materials and Methods

Materials

For microfluidic experiments on single cells, high-glucose Dulbecco’s modified Eagle’s medium (DMEM) with phenol red, 0.05% trypsin, 10× HBSS (Hank’s Balanced Salt Solution; with Ca²⁺, Mg²⁺), HEPES buffer (1 M), geneticin, and OPTIMEM were obtained from Gibco (Life Technologies, Grand Island, NY), fetal bovine serum (FBS) was obtained from Gemini Bioproduct (West Sacramento, CA), Lipofectamine 3000 was obtained from Invitrogen (Life Technologies, Carlsbad, CA), D-glucose 10% (w/v) from Sigma (St. Louis, MO), carbachol (CCh) from Calbiochem (EMD Biosciences, La Jolla, CA), and polydimethylsiloxane (PDMS) and curing agent from Dow Corning (Midland, MI). Imaging media (1× HHBSS (pH 7.4)) was prepared as described by Palmer and Tsien (23). CMV-R-GECO1 was a gift from Robert Campbell (plasmid no. 32444; Addgene, Watertown, MA) (24).

Methods

Cell culture

HEK 293 cells with stable transfection of human muscarinic acetylcholine receptor M₃ as described previously (25) were cultured in DMEM with 10% FBS and geneticin (400 mg/mL) in T-25 flasks. Transient transfection with R-GECO1 plasmid was carried out using Lipofectamine 3000 following prescribed protocol.

Microfluidic device

PDMS microfluidic devices for cell culture and pulsatile stimulation were fabricated based on the computerized microfluidic cell culture system using Braille display and is described in (10). The PDMS chips were filled with laminin (100 mg/mL) and incubated overnight. Subsequently, the chips were washed and incubated with DMEM/10% FBS under sterilized conditions. HEK 293 cells transiently transfected with R-GECO1 were seeded in the outlet channel of the device as described in Sumit et al. (1). The microfluidic device was mounted on a custom-built setup and controlled by a custom-written software to deliver periodic stimulation of CCh as described in (10).

Time-lapse imaging and analysis

Cells in the microfluidic device were simultaneously stimulated with ligand and imaged with a TE-2000 U Nikon microscope (Nikon, Tokyo, Japan) using a 20× Fluor objective illuminated by a 100-W Hg lamp. Sequential acquisition of R-GECO1 fluorescence was carried out every 3 s using ET572/35 excitation filter and ET630/50 emission filter, respectively (Chroma Technology, Rockingham, VT). The excitation and emission filters were equipped in filter wheels controlled by a Lambda 10-3 shutter controller (Sutter Instruments, Novato, CA). Regions of interest (ROIs) within the cytoplasmic region were selected for single cells using MetaFluor Software (Molecular Devices, Downingtown, PA) to determine the area averaged intensity I(t) in cytoplasmic region (I(t)_cyto). Calcium response was measured by calculating the ratio I(t)_cyto/I₀. Here, I₀ corresponds to the basal R-GECO1 intensity in the cytoplasmic ROI. Background signal was subtracted before the ROI intensity analysis in all the experiments by choosing a region with no cells and assigning it as the background.

Calculating signal/noise ratio and intrinsic/extrinsic noise ratio

We adapted the method for experimental noise analysis provided in the supporting material of a recent article (26). We first calculate the signal magnitude σ²_r (variance of average responses of n cells over all m input concentrations) as follows:

$${\sigma }_r^2=\frac{1}{m}\sum _{i=1}^m{\left(\left(\frac{1}{m}\sum _{w=1}^m\frac{1}{n_w}\sum _{j=1}^{n_w}{r}_{wj}\right)-\frac{1}{n_i}\sum _{j=1}^{n_i}{r}_{ij}\right))}^2.$$

Here, r represents calcium peak responses from single cells; i (indexed from 1 to m) represents input signals such as various concentrations of CCh; j (indexed from 1 to n_w) represents all the calcium peak response signals from a given input i; and j (indexed from 1 to n_i) represents all the calcium peak response signals from all the m inputs. Different input concentrations are indexed as w ranging from 1 to m. Noise magnitude σ²_n is calculated as the average of the variances of n_i responses to a single input level as follows:

$${\sigma }_n^2=\frac{1}{m}\sum _{i=1}^m{\left(\frac{1}{n_i}\sum _{j=1}^{n_i}\frac{1}{n_i}\sum _{w=1}^{n_i}{r}_{iw}-r_{ij}\right)}^2.$$

The signal/noise ratio (SNR) is then defined as the ratio ${\sigma }_r^2/{\sigma }_n^2$.

To calculate intrinsic/extrinsic noise ratio (IER), we used the portion of calcium traces from step changes for which the intensity did not change significantly. Intrinsic noise $\left({\sigma }_{\xi }^2\right)$ was estimated as the variance of the differences in R-GECO intensities between successive time points. Total noise $\left({\sigma }_t^2\right)$ was calculated similar to the noise magnitude, ${\sigma }_n^2$, but taking only the static region of the calcium traces. Extrinsic noise is then calculated as ${\sigma }_e^2={\sigma }_t^2-{\sigma }_{\xi }^2$. The ratio ${\sigma }_i^2/{\sigma }_e^2$ is IER.

Based on the noise analysis, we incorporated cell-to-cell variability in the form of a uniform distribution of ±20% in the model parameters, namely the receptor, total G-protein, and the phospholipase-C (PLC) concentration. The intrinsic noise (±5%, normally distributed throughout the dynamic data) was minimal and did not affect the calculation for “0” (nonresponding) and “1” (responding) cells.

Mathematical model and computational analysis

GPCR-calcium model with extrinsic noise. Previously, we developed a mathematical model that links step/pulsatile ligand stimulation, receptor/ligand binding, and calcium signaling and captures the characteristic features of the microfluidic pulsatile experiments (1, 11, 12). The model description, reactions, and parameter values are as detailed in our previous work (1). Briefly, the ligand CCh binds to M₃ muscarinic receptors, promoting G-protein coupling. After the exchange of guanosine diphosphate (GDP) for guanosine triphosphate on the α-subunit (Gα), Gα dissociates from the receptor and binds to PLC, initiating downstream signaling. Guanosine triphosphate on activated Gα is rapidly hydrolyzed to GDP, forming inactive Gα-GDP. If Gα is bound to PLC, then this hydrolysis reforms inactive PLC as well as inactive Gα. The ligand-receptor complex is reversibly phosphorylated to form inactive L-R-P state. The inactive complex can be dephosphorylated to reform the free receptor or can be internalized and either degraded or recycled back to the surface via endosomal sorting. Gα binding to PLC increases inositol triphosphate (IP₃) production; IP₃ binds to the IP₃ receptor on the endoplasmic reticulum (ER), triggering the release of Ca²⁺ from ER into the cytosol. Cytosolic Ca²⁺ acts both to stimulate and to inhibit its release from the ER through multiple pathways. The oscillatory release of Ca²⁺ from the ER is achieved by the sarcoendoplasmic reticulum calcium transport ATPase pump, which pumps cytosolic Ca²⁺ back into the ER. Ca²⁺ can also enter or leave the cell through the plasma membrane. To incorporate extrinsic noise, the values for three major nodes in the pathway, the receptor, G-protein, and PLC concentrations, were chosen from a uniform distribution around the mean value so that the SNR matches with the experimentally calculated value.

Computational analysis of response subpopulations

A system of ordinary differential equations was generated for the model and solved in MATLAB (The MathWorks, Natick, MA) with the ode15s stiff solver. Experimental data for more than 1000 cells under pulsatile stimulation conditions for various concentrations (C = 10, 20, 40, and 80 nM) were analyzed to determine SNR and IER following a method described in Selimkhanov et al. (26). These quantities provided an estimate of plausible range of extrinsic noise in our signaling system. Based on the noise analysis of experimental data, the initial values for core signaling components (i.e., GPCR, G-protein, and PLC) were chosen from a 20% uniform distribution around the mean value using Latin hypercube sampling (27). Thus, in silico cell-to-cell variability was generated for three independent runs each consisting 250 Latin hypercube sampling parameter sets.

Assignment of “0” and “1” binary responses

Analysis of the individual traces for peak finding and determining fraction of subpopulations of (0,0), (0,1), (1,0), and (1,1) was done by writing script files in MATLAB. Peak calcium responses from single cells for each of the stimulation pulses were normalized with the basal cytoplasmic R-GECO1 intensity to calculate I_peak/I₀. Accounting for the variability in the peak intensities from cell to cell, the following conditions were imposed to define the binary responses: for a two-pulse stimulation, if both peaks were such that I_peak/I₀ < 1.5 (defined as subthreshold peaks), it was counted as a (0,0) response. For the rest of the cases, if the first peak was 0.3 times the second peak, it was counted as (0,1) response and vice versa. The remaining peaks were assigned as (1,1) responses. For the simulation results, the calcium concentration was normalized to match the R-GECO1 normalized intensity (multiplied by 10), and the same algorithm was used to determine the binary responses.

TNFα-NFκB deterministic model with extrinsic noise

We used the deterministic as well as the stochastic versions of the TNFα-NFκB model from (6). Although both uniform and lognormal distributions were applied to the model variables, results shown in the manuscript are based on the uniform distribution. The lognormal distribution showed similar trends (data not shown). No changes were made in the model equations. The parameter values used for the model analysis were taken from (6). We performed sensitivity analysis of all the parameters used in the model to determine the parameters sensitive to both first and second peak in the in silico two-pulse experiment (Table S1). Subsequently, extrinsic noise was incorporated in the deterministic model using a similar method as described in the previous section by generating a distribution in sensitive parameters limited within the same log scale. Similar to the GPCR-calcium model, the model parameter values were drawn from a uniform distribution for all the signaling components with nonzero initial concentrations. The rest of the analysis was performed similar to that for the GPCR model. Computational codes and data can be made available upon request.

Results

Microfluidic two-pulse experiments reveal cellular calcium response heterogeneity upon stimulation in a physiologically relevant concentration regime

We designed a microfluidic experiment entailing cell stimulation with two pulses of CCh, which binds and activates the M₃ muscarinic receptor, eliciting an intracellular calcium signal (Fig. 1 a). The stimulation pulse can be characterized by its concentration (C), pulse duration (D), and the rest period between two consecutive pulses (R). The concentration regime chosen (≈5–10 nM CCh) was low enough to be insignificant for receptor desensitization and sufficient to generate cell populations with a wide range of activation probability (11, 12, 28). We measured cytoplasmic calcium as our readout for hundreds of HEK 293 cells transiently transfected with the R-GECO1 calcium sensor (1). These cells were stimulated within a microfluidic device that can deliver temporal rectangular pulses of ligand as described previously (1, 10, 11, 12). For each pulse of ligand, a cell may or may not respond with a calcium peak. A full or near-full peak response is designated as output “1” in the scheme, whereas no response or a subthreshold peak is designated as output “0” (see Materials and Methods). Thus, for a two-pulse scheme, the four combinatorial responses possible for any cell are no response to both pulses (0,0), a calcium peak response only to the second stimulation pulse (0,1), a calcium peak response only to the first stimulation pulse (1,0), or calcium peak responses to both stimulation pulses (1,1).

Figure 1.

Microfluidic two-pulse experiments reveal cellular calcium response heterogeneity upon stimulation in a physiologically relevant concentration regime. (a) Shown is the design of pulsatile stimulation experiments, in which two outcomes are possible for each stimulation (0 and 1), leading to four possibilities for two-pulse experiments. (b) The top panel shows time-montage images for two representative cells depicting a (0,1) response (top arrow) and a (1,0) response (bottom arrow) under the two-pulse test with C = 7.5 nM, D = 16 s, and R = 120 s. The bottom panel shows a heat map of normalized R-GECO intensity for single cells. Representative data for 20 single-cell responses out of >100 cells per condition are shown. (c) Representative single-cell data demonstrate all four possible outcomes. (d) The subpopulation varies with stimulation parameters D and R; here R = 24 s is compared to R = 120 s for the indicated D values. Data represents three sets of two-pulse microfluidic experiments with ≈30 cells per set, with a total of >100 cells per condition. Error bars are standard errors of the mean. Asterisks represent the p-value for unpaired t-test. ^∗0.01 < p < 0.05, ^∗∗0.01 < p < 0.001, and ^∗∗∗p < 0.001.

Although the experiment was conducted for a wide range of rest periods, results are shown only for the rest-period regime for which the cells exhibited most of their response variability. Choosing an intermediate concentration in the test regime (C = 7.5 nM) and pulse duration that produces nonsaturating calcium responses (D = 16 s), we observed population heterogeneity in the calcium responses (Fig. 1 b). The responses could be separated into the four aforementioned outcomes (Fig. 1 c). Stimulation of cells with different duration and rest period of pulses suggests that the subpopulation varies with varying stimulation parameters (Fig. 1 d). Increasing rest period, R, from 24 to 120 s results in increased (0,0) or (1,1) subpopulation. Increasing pulse duration increases the (1,1) subpopulation and decreases (0,0) subpopulation for the two rest-period regimes tested. The existence of (1,1) and (0,0) responses is relatively easy to explain within a deterministic framework. However, the explanation for the simultaneous existence of (0,1) and (1,0) responses using a deterministic mathematical model has not been shown and often leads to the conclusion that the response outcomes have stochastic origins.

A mathematical model with deterministic reaction kinetics and extrinsic noise recreates cellular response variability

We previously reported a deterministic mathematical model that captures population-scale calcium response characteristics under both step increases and pulsatile stimulation with ligand (1, 11, 12). To account for heterogeneity in responses at the single-cell level, we additionally incorporated extrinsic noise into our mathematical model. Note that we and others (29, 30) use the term extrinsic noise to refer to cell population heterogeneity in the levels of various molecules (e.g., molecules in signaling or metabolic pathways) that may have an influence on cell behavior. The origin of this heterogeneity is generally unknown but may be related to cell cycle, microenvironmental factors, previous signaling events, etc. Introducing cell-to-cell variability in signaling response in a computational model has been approached in different ways. Ideally, the parameter values should be drawn from a concentration distribution established experimentally using techniques such as flow cytometry (20, 31, 32, 33, 34, 35). However, most often, it is not practical to quantify distribution for each signaling component in the pathway. An alternative approach developed by Selimkhanov et al. (26) uses an information theoretic approach to quantify intrinsic and extrinsic noise in the output signaling response for several input levels (see supporting material of (26)), and this is the approach we used. Using this approach, extrinsic noise can be quantified and used to draw parameter values from a distribution. Thus, population heterogeneity was introduced in the model by incorporating a uniform distribution in the concentration of the signaling proteins in the model (Fig. 2 a). Briefly, hundreds of single-cell calcium time traces were recorded for a step change in ligand concentration from C = 0 nM to C = 10, 20, 40, or 80 nM using our device. We then calculated the SNR and IER for the CCh-induced calcium response, as described in Methods. Subsequently, the distributions of the concentration of signaling components were set such that simulations matched the experimentally observed SNR. Using this updated model, we then generated hundreds of in silico single-cell calcium traces under the two-pulse stimulation scheme (Fig. 2 b). These simulated traces show features similar to those observed in our microfluidic pulsatile stimulation experiments, and we were able to obtain all four possible calcium response outcomes (Fig. 2 c). Similar to the experimental results, the in silico subpopulation also varies with varying stimulation parameters (Fig. 2 d). Increasing R from 24 to 120 s results in increased (0,0) and (1,1) subpopulations. Increasing pulse duration increases the (1,1) subpopulation and decreases the (0,0) subpopulation for the two rest-period regimes tested. Similar results were obtained with a lognormal distribution of the signaling components (Fig. S1). We also determined how individual components in the signaling pathway influence the fate of cells in terms of one of the four response outcomes. Although protein expression variability in each of the signaling components affected the responses, an overall cumulative parameter Q (calculated by multiplying together the concentrations of the protein components in the pathway that vary from cell to cell) clearly distinguishes the four outcomes, indicating that the biochemical variability cumulatively may affect the downstream cell fates (Fig. S2). Taken together, our deterministic mathematical model with added extrinsic noise captures cellular response variability, which includes all four possible response outcomes under the two-pulse scheme.

Figure 2.

Deterministic mathematical model with extrinsic noise in key components exhibits all four possible outcomes. (a) In the model schematic, the ligand (L) binds to the receptor (R) to form a ligand-receptor (LR) complex, which initiates G-protein-mediated PLC activation and IP₃ generation, ultimately leading to the observed calcium responses. The shaded nodes indicate where extrinsic noise is introduced in the model. Phosphatidylinositol 4,5-bisphosphate to IP₃ pathway activated by PLC denotes a high Hill coefficient pathway. Model equations can be found in (1). (b) Shown is an in silico two-pulse stimulation test with ligand (CCh) to generate representative single-cell traces by introducing cell-to-cell variability (biochemical noise) in the model (C = 7.5 nM, D = 16 s, and R = 120 s). (c) Intracellular calcium concentrations for simulation data showing all the four possible outcomes similar to the experimental data are shown. (d) Similar to the experimental results, the in silico subpopulation varies with varying stimulation parameters. Data represent three sets of two-pulse in silico experiments with 250 cells per set. Error bars are standard errors of the mean.

Rest-period test reveals a deterministic basis for cellular response variability

The existence of all four possible calcium response outcomes for the two-pulse test (both in the model and in experiments) raises an interesting question: is the fraction of cells responding in each way related to the rest period R between the pulses? If deterministic sources are primarily responsible for cellular response variability, increasing R should allow the two responses of the two-pulse test to uncouple from each other (i.e., increased R should result in less influence of the first pulse on the second calcium response). In other words, as R is increased, the fractions of cells falling into different subpopulations should change, with cells moving into the (0,0) or the (1,1) subpopulations. However, if stochastic sources are primarily responsible for cellular response variability, the responses to two stimulation pulses are expected to be uncorrelated, and hence, the rest-period changes should not affect the response outcomes.

We tested this hypothesis with two widely separated rest periods (R = 24 s and R = 120 s), both in silico and with microfluidic experiments (Fig. 3). The concentration (C = 7.5 nM) and durations of pulsed stimulation (D = 8, 16, and 24 s) were carefully chosen based on previous data (1, 28) so that a wide range of response variability could be observed. Furthermore, under these stimulation conditions, GPCR desensitization and gene- expression-induced feedback were minimal (1). Applying these stimulation conditions, we found that the summation of (0,1) and (1,0) responses decreases with increasing R (Fig. 3 a) whereas the repeated response, denoted by the summation of (0,0) and (1,1) responses, increases with increasing R (Fig. 3 b).

Figure 3.

Rest period determines subpopulation composition. (a and b) Microfluidic two-pulse test with widely separated rest periods (R = 24 s and R = 120 s) shows that subpopulation composition changes with R in a predictable fashion for different pulse durations tested (D = 8, 16, and 24 s; C = 7.5 nM). (a) The response denoted by the summation of (0,1) and (1,0) subpopulations decreases with increasing R for all the pulse durations tested. (b) The response denoted by the summation of (0,0) and (1,1) subpopulations increases with increasing R. The scatter points show the corresponding model result for 250 simulation conditions from the distribution drawn on model parameters. (c) Response variability for extremely large rest periods (R = 640 s) is shown. (d) Response variability for extremely large stimulation concentrations (C = 100 nM) is shown. (e) A simulation study showing the effect of rest period on subpopulation composition in a two-pulse in silico experiment with GPCR-calcium model (C = 10 nM; D = 10 s) is shown. Experimental data (bars) are presented as mean ± SDs for three sets of experiments, with ≈30 cells in each set, with a total of >90 cells for each condition. Modeling results are shown for 250 in silico simulations with distribution drawn from model parameters. Asterisks represent the p-value for unpaired t-test. ^∗0.01 < p < 0.05, ^∗∗0.01 < p < 0.001, and ^∗∗∗p < 0.001.

For extremely large rest periods, most of the cells responded either as (0,0) or as (1,1) (Fig. 3 c). Similarly, for extremely large stimulation concentrations, most of the cells responded either as (0,0) or as (1,1) (Fig. 3 d). Our deterministic model also suggests that the effect of R on each subpopulation in GPCR-calcium signaling is monotonic and predictable for various pulse concentrations and durations in the tested regime (Fig. 3 e; Fig. S3). When the rest-period test was extended to three pulses in the model, we observed a similar effect of rest-period (i.e., with increasing R, the system tends to either respond completely (1,1,1) or not at all (0,0,0)) (Fig. S3). Taken together, we find that longer rest periods reduce cellular response variability.

Rest-period test reveals a deterministic basis for cellular response variability in another signaling system: TNFα -NFκB signaling

Our model and experiments support a deterministic origin for the cellular response variability we observed in the GPCR-calcium signaling pathway. Next, we asked whether the same might be true for a signaling pathway that signals at a much longer timescale as compared with the intracellular calcium signaling described above. NFκB oscillations are observed upon TNFα stimulation. The timescale of TNFα-induced NFκB signaling is from several minutes to hours, unlike the GPCR-calcium system for which signaling is on the order of seconds to minutes. Briefly, nuclear localization and transcriptional activity of NFκB upon stimulation of the TNF-receptor is regulated through negative feedbacks by A20 and IκBα at different levels of signaling. The activation of IKK-α is modeled as a high Hill coefficient pathway (Fig. 4 a). Model equations and parameters can be found in (6). Using microfluidic two-pulse stimulation of TNFα in 3T3 mouse fibroblasts and measuring nuclear localization of NFκB, Tay et al. (6) showed the existence of cells with all four possible signaling outcomes (i.e., (0,0), (0,1), (1,0), and (1,1) responses). They suggested that the single-peak responses (i.e., (0,1) and (1,0)) were a combination of stochastic and deterministic effects and developed a hybrid model incorporating stochasticity at certain nodes based on a previously existing deterministic model (4). We hypothesized that their deterministic model alone may be able to generate all four possible signaling response outcomes upon incorporation of extrinsic noise and performed a sensitivity analysis of the entire parameter set of their model to identify parameters that significantly influence NFκB peak height (27). We limited exploration of the parameter space such that the parameters were of the same order of magnitude as reported in Tay et al. (6). Table S1 lists parameters significantly correlated with first- and second-peak responses (Fig. 4 a; Table S1). We generated extrinsic noise by sampling from a uniform distribution of these parameters. Under the same stimulation conditions used by Tay et al. (6), we found that this modification of their deterministic model that incorporates cell-to-cell variability in the sensitive parameters exhibits all four response outcomes of the two-pulse experiment with subpopulation compositions similar to the reported experimental values (Fig. 4, b and c). The effect of increasing rest period in the NFκB model is similar to our GPCR model. With increasing R, the subpopulation (1,1) increases whereas (1,0) decreases, similar to the trends observed in the GPCR model. In contrast, the (0,0) response increases whereas (0,1) decreases with increasing R, similar to the trend we observe in GPCR signaling, although these trends are nonmonotonic around R ∼200 min (Fig. S4). The effect of increasing concentration and pulse duration in the NFκB model is also similar to our GPCR model (Fig. S5, a and b). We also applied the rest-period test to a recently reported experimental data for pulsatile stimulation with low-concentration TNFα (0.1 ng/mL) (13) (Fig. 4 d). We found that the “response variability,” denoted by the summation of (0,1) and (1,0) responses, is reduced at a longer rest period (R = 150 min) as compared to a shorter rest period (R = 45 min), whereas the repeated responses (summation of (0,0) and (1,1) fractions) increase at longer R, similar to what we observe for the GPCR-calcium experiments. These computational and experimental findings suggest that the cellular response variability in NFκB signaling is consistent with a deterministic process.

Figure 4.

Deterministic NFκB signaling model also shows cellular response variability under low-concentration pulsatile stimulation, similar to the GPCR-calcium signaling. (a) A schematic of the TNFα-NFκB signaling model is shown. Nuclear localization and transcriptional activity of NFκB upon stimulation of TNF-receptor is regulated through negative feedbacks by A20 and IκBα at different levels of signaling. (b) Using sensitivity analysis to determine parameters affecting the first and second peaks differentially and adding extrinsic noise to the parameters, the deterministic NFκB model can produce the four subpopulation outcomes. Representative esponses from 20 out of n = 250 simulated cells are shown. (c) Representative in silico single-cell traces of nuclear NFκB from (b), showing all the four possible outcomes. (d) Rest-period test of NFκB signaling is shown. Experimental data from (13) was examined by calculating the percent of (0,1) and (1,0) responses and the rest of the subpopulation (0,0) and (1,1) responses at low TNFα stimulation (0.1 ng/mL). A shorter rest period (R = 45 min; D = 45 min) and a longer rest period (R = 150 min; D = 30 min) were compared to find that the response variability decreases with longer rest periods, similar to our experimental data for calcium response. The modified deterministic NFκB model also predicted similar results.

Rest-period test can identify deterministic and stochastic origins

Analysis of our GPCR-calcium model showed that cellular response variability, such as quantified by the summation of the fractions constituted by (0,1) and (1,0) responses, decreases with increase in D or R (Fig. 5 a). A similar response was observed when we decreased the concentration of stimulation “C” while keeping C × D the same (Fig. S6). Although it was known that the fractions of cells responding to a pair of ligand pulses are governed by concentration C and stimulation duration D, we show that the role of rest period R is equally important in determining the distribution of possible signaling response outcomes. We found similar trends for an NFκB deterministic model, wherein a low D and R regime resulted in maximal cellular response variability (Fig. 5 b). Importantly, the stochastic model reported in Tay et al. (6) did not exhibit any significant changes in response variability with changes in R (Fig. 5 c). This difference in subpopulation shifts between the stochastic and deterministic mechanisms identifies our tractable, two-pulse rest-period test as a means to distinguish between the two mechanisms. The basis of this test is that increasing R results in a reduction of response variability for deterministic mechanisms because cellular responses tend toward (0,0) or (1,1) responses. In contrary, stochastic responses appear to remain unchanged.

Figure 5.

Rest-period test can identify characteristic difference between cellular response variability of deterministic and stochastic origins. (a) Cellular response variability in the deterministic GPCR-calcium model with extrinsic noise decreases with D and R. (b) Cellular response variability in the TNFα-NFκB deterministic model with extrinsic noise decreases with D and R. (c) The TNFα-NFκB stochastic model with no extrinsic noise shows no significant effect of D or R on response variability. To see this figure in color, go online.

A high Hill coefficient process and pathway recovery properties govern cellular response variability

The results above raise interesting questions about the internal mechanism of the signaling pathway that leads to cellular response variability. We analyzed each signaling pathway node in our GPCR-calcium model for the two response subpopulations (0,0) and (1,1) to understand the potential differences leading to the disparate signaling outcomes. Although the two subpopulations differed slightly in the concentrations of activated receptor complex, total G-protein, and activated PLC, a much greater difference was seen with IP₃ levels, leading to a seemingly bifurcated downstream response (Fig. 6 a). This difference is due to the high Hill coefficient (H≈2) at this point in the signaling pathway (Fig. S7).

Figure 6.

Component-wise analysis of the GPCR model. (a) The two subpopulations (0,0) vs (1,1) suggest a high Hill coefficient reaction kinetics as the prime step that leads to amplification of the difference in responses. Basal PLC activity brings about the major difference leading to either (0,0) or (1,1) response (a node of high Hill coefficient (H≈2)). (b) A model analysis of why (0,1) calcium response translates into a (0,0) response upon increasing R in the GPCR model is shown. The first light-gray pulsatile shape around t = 0 is an overlap for all the three conditions, with the gray line on the top, along with dotted and black lines underneath. All line plots are a mean of n = 250 in silico simulations per condition.

We also tested the deterministic NFκB model by analyzing the individual pathway nodes. Similar to our GPCR model, we found that the bifurcation in signaling responses is a result of a high Hill coefficient (H≈2) in the signaling pathway, in this case in the generation of active IKK, a major component in TNFα-NFκB signaling (Document S1. Supporting Text, Figs. S1–S9, and Table S1, Document S2. Article plus Supporting Material).

We investigated how the (1,0) and (0,1) subpopulation transition to (0,0) and (1,1) responses with increasing R. Although transition from (1,0) to (1,1) responses with increasing R is reported for both the signaling pathways and is attributed to the pathway recovery properties (4, 12), the transition from (0,1) to (0,0) is less intuitive. So, using the model, we analyzed the transition from (0,1) to (0,0) responses upon increase in R. Our analysis of this transition entailed performing a component-wise analysis of the GPCR model using the rest-period test for three different R values (120, 200, and 300 s) (Fig. 6 b). Our simulation data show that for cases in which the cell does not respond to the first stimulation pulse ((0,0) and (0,1) cases), a short-term memory of the stimulation in the calcium concentration in the ER is created. This calcium gradually leaks out of the ER reservoir. An early second stimulation helps the ER reach its threshold to release the calcium, and, hence, the cell responds, resulting in a (0,1) response; a delayed second stimulation does not have this advantage (28), resulting in a (0,0) response as R is increased. Similarly, we find that in the TNFα-NFκB model, receptor activation creates a short memory of the stimulation through IKK kinase signaling. For shorter R, the active IKK kinase level increases more upon a second stimulation compared to larger R, despite a lack of an NFκB response to the first stimulation. This increase in IKK kinase is amplified via a high Hill coefficient pathway downstream (Fig. S8 b), thus rendering cellular response variability sensitive to R. Although it is challenging to test experimentally, a high Hill coefficient pathway seems to impart similar effects on signaling responses based on the two models analyzed in this work. Taken together, cellular response variability is altered by the length of the rest period (R) because of slow recovery of intermediate signaling processes.

Discussion

Physiologically relevant weak stimulations, often low-concentration pulsatile bursts of ligand, can lead to downstream signaling responses that are highly variable. A better understanding of how stimulation parameters govern response patterns and overall response fidelity in a signaling system is fundamental to biology (1, 5, 6, 11, 36). It will also be useful in the analysis of pharmacological, in vitro culture, and synthetic biology applications.

There are three main conclusions from our work. First, within the stimulation regime in which cells exhibit significant response variability, a two-pulse rest-period test is able to distinguish whether the variability is likely driven by deterministic or stochastic mechanisms. Second, response variability in GPCR-mediated calcium signaling and in TNFα-mediated NFκB nuclear localization is consistent with a deterministic mechanism. The models suggest that the basis of such response variability is a short-term memory of stimulation along with a high Hill coefficient pathway and cell-to-cell variability in signaling protein expression levels. Third, although often ignored, looking at and accounting for the nonresponding population of cells, such as the (0,0) response here, is essential to the understanding and analysis of single-cell data.

Our rest-period test provides a simple method for revealing effects of stimulation parameters on cellular responses. When applied to GPCR-induced calcium signaling, the test revealed that as the time between stimulation pulses increased, cells tended either to respond to every stimulation or did not respond at all. We envision that pulsatile stimulation parameters could thus be used as a means to precisely control the percentage of fully responding and nonresponding cells. This fascinating observation has implications on various facets of biology, providing a potential explanation for how organisms are able to selectively control genetically homogeneous groups of cells and yet guide them to particular cell fates (15, 16, 17).

The combined experimental and computational approach described in this article enabled extraction of mechanistic insight for the intracellular calcium and NFκB pathway studied. In particular, our analysis revealed that the bifurcation in cellular responses described above could be attributed to extrinsic noise, a short-term memory of stimulation, and a high Hill coefficient process operating within a deterministic process. Although the models studied in this work suggest bifurcation properties based on high Hill coefficient motifs in the pathways, in general, a motif or mechanism that provides a strong bottleneck will likely provide similar bifurcation properties and influences over cell-to-cell variability. These results are in accord with previous studies that have indicated that mammalian cellular signaling pathways, such as intracellular calcium and NFκB, are largely dictated by deterministic mechanisms (4, 11, 13, 19, 37). Similar to our approach, these studies have applied pulsatile microfluidics and computational analysis to reveal deterministic components of signaling responses. Other studies have combined computational analysis with analysis of daughter cells (20) or contextual data making up a single cell’s microenvironment (8, 9) and have shown that deterministic processes largely dictate phenotypic cell-to-cell variability.

Most cell studies focus on high-ligand-concentration stimulation to determine the minimal “reset” time a signaling system requires to recover between stimulation events (4). Experimental metrics like peak amplitude of cellular signals are often used to assess this reset time by monitoring the amount of time needed between stimulation events to produce responses of equal peak amplitudes. It is often assumed that reset times that are assessed at high-ligand-concentration stimulation are sufficient for resetting the signaling system upon exposure to low concentrations (6). However, our work provides new, to our knowledge, insights into how this assumption could be problematic and suggests that low-concentration regimes, in which receptor desensitization does not dominate recovery time, should be treated differently (i.e., the role of signal strength and pathway recovery properties must be considered). Rather, the positive and negative feedback mechanisms and high Hill coefficient processes of the pathway play crucial roles along with cell-to-cell variability in determining the signaling response. Recently, Yao et al. (38) have shown that the variability in intracellular calcium signaling may be a consequence of structured heterogeneity between cells that gives rise to distinct cellular states. Our results provide a possible explanation of how these cellular states may be achieved; it is not only based on cell-to-cell variability but also depends on signaling pathway properties.

Many analyses of single-cell responses discard data from nonresponding cells (the (0,0) case we describe above) and use data only from responsive cells (12, 39, 40). In other studies, cells are stimulated at a very high concentration of ligand to avoid this nonresponsive regime (4). Such analyses can miss important aspects of signaling architecture and discount the role of cell-to-cell variability in governing the population response. Our results suggest that nonresponding cells represent a legitimate cell subpopulation that need to be factored into cellular response analyses.

An interesting feature of nonlinear dynamic systems is that stochasticity can lead to enhanced signaling, such as increased transcription relative to a linear system, in a process referred to as stochastic resonance (41). Mice fibroblasts have been reported to exhibit stochastic resonance in NFκB expression upon weak, sawtooth-shaped TNFα stimulation (41). We wondered whether the deterministic mechanism we find consistent with observed signaling shown above with the GPCR-calcium and TNFα-NFκB systems could also produce “apparent” stochastic resonance. To test this, we performed simulations with sawtooth stimulation, similar to the stimulation strategy used in (41). Geometrical features can potentially influence the stimulation parameters (C, D, and R). In case of square wave pulses, the rest period does not change with an increase or decrease in the ligand concentration (Document S1. Supporting Text, Figs. S1–S9, and Table S1, Document S2. Article plus Supporting Material). In contrast, in case of sawtooth stimulation, a decrease in C is accompanied by a shortening in D and an increase in R. Thus, under conditions of weak stimulation, or lower C, there is a simultaneous increase in R, as in our rest-period test. Our work above suggests that it will be important to analyze all cells, including (0,0) or nonresponding cells, in these conditions. Indeed, we can produce an “apparent” stochastic resonance effect using our deterministic model in a system with behavior previously attributed to stochastic resonance (41) (Fig. S9 b).

Conclusions

In summary, our results suggest that the rest-period test can be used to determine whether observed cellular response variability is consistent with a deterministic origin, as we found to be the case for several mammalian signaling pathways. Physiological, low-concentration pulsing of signals can represent a deterministic strategy that allows the body to guide distinct subsets of cells to their appropriate fate.

Author Contributions

M.S., A.J., J.J.L., and S.T. conceived and designed experiments. M.S. carried out experiments, analyzed data, and wrote the article. A.J., R.R.N., J.J.L., and S.T. consulted on data analysis and edited the article.

Editor: Kevin Janes.