Biotechnol. Rog. 1904, 10, 45-54
45
Calcium Signaling in Individua€BC3H1 Cells: Speed of Calcium Mobilization and Heterogeneity Patricia A. Mahama and Jennifer J. Linderman* Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigan 48109-2136
Receptor/ligand binding on a cell surface may activate the calcium signal transduction cascade, resulting in the release of calcium from intracellular stores into the cytosol. Changes in intracellular free calcium, [Ca2+li,following ligand stimulation have been linked to a variety of cell responses, from muscle contraction to hormone secretion. We have monitored changes in [Ca2+]iin single smooth muscle-like BC3H1 cells following stimulation by the vasoconstrictor phenylephrine, using the fluorescent calcium probe, fura-2, in a digital fluorescence imaging system. We find that not all cells respond to ligand stimulation with changes in [Ca2+]i. In addition, cells which respond to ligand stimulation exhibit considerable heterogeneity in the speed of calcium mobilization for a given ligand concentration. Both the population-averaged speed for calcium mobilization and the fraction of cells which respond to ligand stimulation are increasing functions of the ligand concentration. In contrast, the magnitude of the ligand-stimulated increase in [Ca2+]ifrom basal to peak levels in responding cells is independent of ligand concentration. We postulate that the heterogeneity seen in the ligand-induced mobilization of calcium among single cells is a function of distinct differences between cells, such as number of receptors, size of the intracellular calcium store, or phospholipase C activity. We have developed a mathematical model, based on the calcium signal transduction cascade, to predict single-cell calcium responses to ligand stimulation. We have systematically incorporated cell-to-cellparameter heterogeneity into the model by randomly selecting single-cellparameter values from a Gaussian distribution. Model simulations predict both single-celland population-averaged trends that we have observed experimentally. The results of this work suggest that increases in a population response may be the result of increased participation in the response as opposed to increases in the magnitudes of individual cell responses.
Introduction Binding of ligandsto receptor moleculeson a cell surface can initiate a remarkable sequence of events. When a
ligand molecule binds to a receptor, a signal transduction pathway may be activated. The external signal of the bound ligand is transduced to an internal signal, which is carried by a second messenger such as calcium. Increases in the free calcium concentration in the cell interior, or cytosol, act as the internal signal. Calcium, one of the body’s most dynamic second messengers, can assist in regulating various components within the cell to produce avariety of cellular responses, such as muscle contraction, cell division, or hormone secretion (1,2).The events that are put into motion following the calcium second messenger response and their relationship to the cellular response are at best unclear. Ideally, we would like to stimulate a cell with a specific ligand and predict its calcium second messenger response and the following cellular response, such as secretion or proliferation. However, because the biochemical events between the calcium second messenger response and the cellular response are still being uncovered, such an undertaking is premature. The logical precursor to predicting the relationship between ligand stimulation and the subsequent cellular response is to predict the relationship between ligand stimulation and the transient calcium second messenger response by modeling the sequence of events leading to the calcium response. This predictive ability would be invaluablein determiningwhich 8756-7938/94/3010-0045$04.50/0
cell or environmental parameters would allow manipulation of the responses of cells in culture or in the body. The events beginning with receptor/ligand binding and leading to the calcium second messenger response are collectively referred to as the calcium signal transduction cascade. The basic features of the calcium signal transduction cascade have been characterized through experimental work in the past several years and are detailed in recent reviews (3, 4). The calcium signaling cascade is illustrated in Figure 1. A receptor/ligand complex (C) on the cell surface can associate with an inactive G-protein, &a-GDP, in the cell membrane, enhancing the exchange of guanine nucleotides by decreasingthe G-protein affinity for GDP and increasing its affinity for GTP. Following the binding of GTP to the G-protein, the receptor and G-protein dissociate, and the trimeric G-protein separates into two parts: the 07 subunit and the a-GTP subunit. a-GTP stimulates the activity of a phospholipase C (PLC), which degrades phosphatidylinositol 4,5-bisphosphate (PIP3 into diacylglycerol (DG) and inositol 1,4,5-trisphosphate (IPS). IPS is soluble in the cytosol and binds to receptors on an internal calcium store,releasing calcium into the cytosol. In some systems, increased calcium influx across the plasma membrane also contributes to elevated levels of calcium in the cytosol. Calcium is removed from the cytosol by ATPases which pump the calcium into the extracellular space or resequester the calcium in the intracellular store. Cytosolic calcium must be carefully regulated to be an effective second messenger. The resting cytosolic calcium
0 1994 American Chemical Society and American Institute of Chemical Engineers
Biotechnol. hog., 1994, Vol. 10, No. 1
48
PIP,
-
DG + IPS-
Q
Release of Ca 2+ A
Cytosol
Calcium Store Figure 1. Signal transduction pathway. The calcium second messenger response is initiated by receptor/ligand binding on the cell surface. The signal is transduced by G-proteinswhich activate phospholipase C, resulting in IP, production and calcium release. The G-protein subunit, a-GTP, has a limited lifetime due to the hydrolysis of GTP to GDP. Details on the reaction cascade are given in the text. concentration is maintained at a 10000-fold lower level than the extracellular environment by several calciumtransporting proteins. Within seconds of receptodligand binding, the cytosolic calcium concentration may increase rapidly from 100 to 1000 nM and then decrease almost as rapidly, returning to its original resting level. Since the introduction of fluorescent calcium-binding dyes, changes in intracellular free calcium, [Ca2+]i,have been monitored in single cells of a variety of cell types. These single-cell [Ca2+limeasurements have shown a broad degree of heterogeneity in the calcium response. Mahoney et al. (5)have observed that the fraction of cells displaying a primary rise in [Ca2+]ifollowing ligand stimulation is ligand concentration dependent. A lag time, or latency, in the onset of the rise in [Ca2+lifollowing ligand stimulation has been observed in many cell types, including fibroblasts (6),HIT insulinoma cells (7),hepatocytes (8, 9), and A10 cells (9). The magnitude of the latency at a single ligand concentration varies from cell to cell (6-10) and shows ligand concentration dependence (8-1O), decreasing with increasing ligand concentration. It may be suggested that the cell-to-cell variability in the latency of onset in [Ca2+lichanges is the result of the stochastic nature of the interactions in the calcium signal transduction pathway or is due to distinct differences between cells, such as differences in receptor number, G-protein number, mobile calcium reserve, or sensitivity of the internal calcium store to the release of calcium by IP3. If the observed variability is due to stochastic effects, then multiple stimulations in the same cell should result in significant differences in latency. However, the [Ca2+li response latency of individual cells stimulated repeatedly with ligand has been shown to be reproducible (6-8). In addition, the shape of the calcium response is also reproducible in some cell types (6-81, suggesting that the calcium signal signature may be unique to each individual cell. Byron et al. (6) have shown that the heterogeneity between cells is not cell cycle dependent and suggest that the cell-to-cell variability in latency may be determined by cell-to-cellvariation in receptor number. Dupont and Goldbeter (11)have also mentioned that heterogeneity in biochemicalparameter values among cells could account for the observed diversity in agonist-stimulated calcium transients.
Several mathematical models have already been proposed to predict the [Ca2+]i second messenger response (12-18). However, these models concentrate on the oscillations of [Ca2+]iobserved in some cell types and not on the primary calcium increase following receptor/ligand binding. For example, the calcium oscillation model proposed by Dupont et al. (151,who define latency as the lag between an assumed instantaneous stimulation by ligand and the first peak in calcium concentration, predicts that the latency decreases with increasing stimulation. Like other published calcium oscillation models (12, 13, 16, 181, the Dupont et al. model ignores the explicit connection between receptodligand binding and calcium signaling, assuming steady-state binding which may not be relevant in many cases. In contrast, Cheyette et al. (19)have developed a simple kinetic model based on the enzymaticallyamplified signal cascade from ligand binding to initial calcium release. The model predicts a prolonged lag phase followed by a rapid increase in [Ca2+]ithat compares well to their experimental data on epidermal growth factor stimulated A431 carcinoma cells. However, the model predicts that stimulation even with very low ligand concentrations will eventually result in an increase in [Ca2+li,a phenomenon not seen experimentally. In this study, we have monitored changes in [Ca2+]iin single BC3H1 smooth muscle-like cells following the stimulation of a1-adrenergic receptors by phenylephrine (PhE) using the calcium indicator, fura-2, in a digital fluorescence imaging system. For this experimental system, receptor/ligand binding is rapid, calculated to reach 95% of equilibrium binding in less than 0.1 s for PhE concentrations as low as 0.1 pM. In comparison to receptodligand binding, calcium mobilization is slow, occurring 3-25 s after PhE stimulation. We find that the fraction of cells that mobilize calcium, as well as the speed of the mobilization, increases with increasing PhE concentration. We see significant cell-to-cell response variability between cells stimulated with the same concentration of PhE. Not only do some cells display no increase in [Ca2+]ibut the response latency of responding cells is also heterogeneous. We believe that cell-to-cell variability in calcium mobilization following PhE stimulation may be due to distinct differencesbetween cells,such as receptor number, PLC activity, and mobile calcium reserve.
Bioted". Rog., 1994, Vol. 10, No. 1
We have developed a mathematical model based on the calcium signal transduction cascade to predict how species evolution and cell-to-cell parameter differences affect the latency and magnitude of the primary calcium second messenger transient. Unknown parameters in our mathematical model were set to produce a stable, resting level of all species prior to PhE stimulation and to give good agreement between model simulations and experimental data.
Materials and Methods Materials. fura-2/AM and fura-2 pentapotassium salt were obtained from Molecular Probes (Eugene, OR). Phenylephrine (PhE), Dulbecco's Modified Eagle's medium (DME), antibiotic/antimycotic solution (penicillin, streptomycin, and amphotericin), and trypsin-EDTA solution were obtained from Sigma (St.Louis, MO). Fetal bovine serum was obtained from Hyclone Laboratories (Logan, UT). All other chemicals used were of reagent grade or better. Tissue Culture. The clone of BC3H1 cells used in this study was obtained from Dr. Linda Slakey (University of Massachusetts). Cells were cultured at 37 OC in a humidified air/COz (955) atmosphere. The cell line was propagated in DME supplemented with 10% fetal calf serum (v/v), 100 units of penicillin/mL, 0.1 mg of streptomycin/mL, and 0.25 pg of amphotericin/mL. Cells were subcultured with trypsin at regular intervals to prevent the cultures from reaching confluency. Experimental cultures were seeded at a density of 3 X lo4cells/mL onto 22 mm2 glass coverslipes in 35-mm six-well plates containing 2 mL of media per well. Seeding was done a t least 24 h prior to fluorescence experiments. Fluorescence Microscopy. BC3H1 cells were visualized using a Nikon Diaphot inverted fluorescence microscope (Nikon Inc., Garden City, NY) equipped with a Hg arc lamp excitation source, a 40X NA 1.3 oil objective, 334- and 365-nm band-pass excitation filters (Oriel Corporation, Stratford CT), a 400-nm dichroic mirror, and a 400-nm long-pass emission filter. A neutral density filter (10% transmission, Newport Corp., Fountain Valley, CA) was used to minimize fura-2 bleaching and cell injury. Images were collected on a charge-coupled device (CCD) camera (camera head CH220, chip Thomson CSF TH7882 CDA,Photometrics Ltd., Tucson, AZ),which was mounted on the side camera port of the microscope. The images were created using a 4 X 4 bin on the CCD chip, resulting in a final image size of 96 X 144 pixels. Exposure times for 334- and 365-nm excitation wavelengths were 0.050 and 0.010 s, respectively. Image pairs were collected every 1.1s. The camera output was collected and analyzed using the ISee graphical programming system (Inovision Corp., Durham, NC) running on a SPARCstation 4/330 computer (Sun Microsystems, Mountain View, CAI. Experimental Procedure. fura-2/AM in DMSO at a final concentration of 5 pM was added to the subconfluent cultures of BC3H1 cells growing on glass coverslips. The cells were incubated with fura-BIAM at 37 OC for 20 min. The glass coverslip was rinsed with physiological buffer (pH 7.41, which contained the following (in mM): NaC1, 140;KC1,10;CaC12,1.8; MgCl2,l.O;Na2HP04,l.Q HEPES, 25; and glucose, 5. The coverslip was then transferred to a 0.8-mL, thermoregulated Teflon flow chamber (maintained a t 37 "C), which fit upon the microscope stage, and 0.25 mL of physiological buffer was added to the flow chamber. Images were collected for 20 s prior to PhE stimulation to establish the [Ca2+]ibase line. For stimulation, 7 mL of PhE solution was rinsed through the flow
47
chamber with a syringe in less than 5 s. The PhE concentration in the flow chamber consistently reached 95% of the PhE stimulant concentration within the first 0.8 s of addition. To ensure that increases in intracellular calcium were the result of PhE stimulation and not a response to fluid shear, control experiments were conducted by sequentially flowing physiological buffer and then PhE solution across the cells. No cells responded to the flow of physiological buffer. Data Analysis. The images collected for each excitation wavelength were corrected for background fluorescence. The [Ca2+liwas calculated from the ratio, R, of the fluorescence emissions collected at the excitation wavelengths of 334 and 365 nm, denoted F334 and Fm, by the following equation (20):
Kd [224 nM (20)] is the equilibrium dissociation constant of fura-2 and Ca2+ at 37 "C. R,, and R- are the maximum and minimum F334/F365 ratios obtained with 1.8 mM Ca2+and no Ca2+plus 0.1 M EDTA, respectively. fl is the ratio of F0,365/F8,365,where F0,365 is the fluorescence emission collected with a 365-nm excitation wavelength in the absence of Ca2+,and Fa= is the fluorescenceemission collectedwith a 365-nmexcitation wavelengthin saturating Rmb, and fl were virtually constant 1.8 mM Ca2+. R,, over the area of each cell because the field was illuminated evenly;therefore, the cell-averaged [Ca2+]iwas calculated using eq 1with R averaged over the entire area of a single cell. The time-dependent cell-averaged value of [Ca2+]iwas plotted for each cell tested. A cell was considered a responder if, following PhE stimulation, its [Ca2+]iincreased by a t least 30% over base line in less than 2 min. A tally of responders and nonresponders was kept for each PhE concentration. Calcium response latency was measured in responding cells by measuring the time between PhE stimulation and the maximum rate of increase of [Ca2+]i. To determine the time of the maximum rate of increase in [Ca2+]i,the calcium data between the time of ligand addition and a maximum in calcium concentration were curve-fit with a third-order polynomial. The second derivative of the curve fit was set equal to zero and solved for the time of the maximum rate of increase in calcium. All curve fits had a correlation value of R 1 0.98. An analysis of variance (ANOVA)was used to determine whether response latencies of individual experiments at the same PhE conditions were statistically different. No statistical difference was seen between experiments at any PhE concentration at a 1% significance level. Thus, calcium response latencies for each PhE concentration were pooled, and the average calcium response latency for the cell sample population was determined. ANOVA was then performed on the pooled latency data for each PhE concentration to determine whether the average response latencies between the PhE concentrations were statistically different. For PhE concentrations ranging from 0.5 to 50 p M , the response latencies were shown to be statistically different at a 0.1% level of significance. However, for high concentrations of PhE stimulation (5-50 pM), no statistical difference a t a 25% level of significance was seen between the response latencies. In addition, ANOVA was performed on responding cells in a similar manner to determine whether the percent increase in [Ca2+lifrom basal to peak level was dependent on ligand concentration. No statistical difference at a 25% level of significance was
rnn
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100 0
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1 1 " 1 1 1 1
10
j
[PhEIt ClM ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
20
30
40
50
60
Time, sec
Figure 2. Calcium second messenger response. The three cells shown are stimulated simultaneously with 10pM PhE at t = 17.9 s. The calcium response latencyis measured as the time between ligand addition and the maximum rate of increase in [Ca2+]i. The calcium response is curve-fit to a third-order polynomial, and the second derivative of the curve fit is set equal to zero to determine the time of the maximum rate of increase in [Ca2+li. The calcium response latencies for cells 1and 3 are 15.5 and 5.1 s, respectively. Cell 2 does not respond to PhE stimulation. seen in the percent increase in [Ca2+]ifor PhE concentrations from 0.5 to 50 pM. Experimental Results Changes in intracellular calcium were studied in single fura-2-loaded BCSH1 cells followingstimulation with PhE. Ligand-induced changes in [Ca2+]ifrom one experiment are shown in Figure 2. Only three cells from a field containing five cells are shown for clarity. The cells are simultaneously stimulated with 10 pM PhE at t = 17.9 s. Cells 1and 3 respond to the PhE stimulation with sharp, distinct increases in [Ca2+]i. The latency of the calcium response, measured from the addition of PhE to the point of the maximum rate of increase in [Ca2+]i,is shorter for cell 3 than for cell 1, illustrating cell-to-cell response variability. In the continued presence of PhE, the [Ca2+]i in both cells reaches a maximum and then returns to a near-basal level. Cell 2 shows very little increase in [Ca2+li following PhE stimulation (((30%); thus, this cell is classified as a nonresponder. To determine the magnitude of the [Ca2+]iincrease in each responding cell following ligand stimulation, the difference between the highest measured calcium concentration and the basal level was calculated. This method assumes that data was collected rapidly enough to obtain an accurate value of the maximum [Ca2+]i. The percent increase in [Ca2+]ifrom base to peak was found to be statistically independent of the ligand concentration, suggesting that the response to ligand stimulation in these cells is essentially an all-or-none phenomenon. The calcium response latency followingPhE stimulation is measured in cells which responded to PhE stimulation. The average calcium response latency for each concentration of PhE stimulation is shown in Figure 3. The average response latency is PhE-dependent, with the highest latency occurring at the lowest PhE concentration. As the concentration of PhE increases, the average response latency decreases, reaching a minimum at 5 pM PhE. No statistical difference is seen between the average response latencies at PhE concentrations greater than or equal to 5 p M .
Figure 3. Calcium response latency of cells stimulated with PhE. Single BCsHl cells are stimulated with one of six concentrations of PhE. Calcium response latencies are calculateci for responding cells. The mean values of latency at each PhE concentrationare plotted with standard deviations. The latencies for 5-50 pM are not statistically different. Standard deviation is calculated for the pooled population of responding cells at all PhE concentrations,assuming logarithmically weighted latencies. Theoreticalestimatesfor calcium response latency are the average of responding cell latenciesfrom a simulation population of lo00 cells at each ligand concentration. In addition to the measurement of calcium response latency, a tally of responders and nonresponders was made for each PhE concentration. The fraction of cells which show an increase in [Ca2+]ifollowing PhE stimulation is shown in Figure 4. At very low ligand concentrations, less than 0.5 pM,cells are unresponsive to PhE stimulation. At higher PhE concentrations, the fraction of responding cells increases with PhE concentration. For PhE concentrations greater than or equal to 10 pM,the fraction of responding cells plateaus to a maximum of approximately 60 5%. The average calcium response latency and the fraction of responding cells are summarized in Table I.
Mathematical Model The calcium signal transduction cascade illustrated in Figure 1 can be modeled mathematically. The goals of this mathematical model are to predict the timing of the initial peak in calcium seen following PhE stimulation and to test the assumption that cell-to-cell response variability may be a function of distinct cell differences such as receptor number. Equations written are for reactions within a single cell. The first step, receptor/ligand binding, is expressed as shown in eq 2, where [Rl is the free receptor concentration, [L] is the ligand concentration, and [Cl is the concentration of receptodligand complexes:
-d[C1 - k,[RI[Ll - k2[Cl
dt In this equation, kl is the association rate constant of ligand and receptor, and 122 is the dissociation rate constant for receptodligand complexes. If we assume the total number of receptors, [Rlt, is constant, the number of free receptors is expressed by [Rl = [RI, - [Cl (3) The G-protein is a heterotrimer consisting of a,0,and y subunits, with GDP bound to the a subunit. The concentration of inactive G-protein, pya-GDP, is represented in the model equations as [GI. G-protein has a low basal activity resulting from dissociation of GDP from the
49
Bbtechnol. Prog., 1994, Vol. 10, No. 1 Table I. Summary of Experimental Rsrultra PhE no.of total fraction of responders responders cella* O1M) 10.3 o/ 106 0.00 0 21/80 0.26 11 0.5 10 1.0 0.35 29/83 19 5.0 26/57 0.46 107/181 0.59 52 10.0 0.60 39/65 21 50.0
0:o
An%. 0.1
,,,,,I
, , ,
1 .o
,.,.., 10.0
1
.
I
. I
100.0
[PhEl, ClM Figure 4. Fractionof cells respondingtoPhEstimulation. Single cells are stimulated with one of six concentrationsof PhE. The fraction of responding cells is shown as a function of PhE concentration. At low PhE concentrations,few cells respond to PhE addition. As the concentration of PhE increases,the fraction of responding cells increases. The maximum level ofresponding cells, 60%, is reached near 10pM PhE. The theoretical fraction of responding cells is determined from a tally of responders and nonresponders from a simulation population of lo00 cells at each ligand concentration.
a subunit and association of GTP. When GTP binds to the a subunit, the G-protein dissociates into fly and a-GTP subunits. Receptodligand complexes act as catalysts in the activation of G-protein. Interaction of a receptor/ ligand complex and a G-protein results in the formation of a ternary receptor/ligand/G-protein complex. The a subunit of the ternary complex has an enhanced affinity for GTP and a diminished affinity for GDP (4). a-GTP is produced following binding of GTP and dissociation of the ternary complex. Taylor (4) states that the transient encounter between an agonist-occupied receptor and a G-protein lasts for only a few milliseconds. In order to compare this time with the likely time between encounters, the time for a receptor/ ligand complex to diffuse to a G-protein can be estimated from W
(4)
where t is the time between encounters between a receptor and any G-protein, D is the sum of the receptor and G-protein diffusion coefficients (estimated as 1 X 10-lo cm2 s-l), and d is one-half the mean distance between G-proteins. If G-proteins are assumed to be uniformly distributed, d can be estimated by 7
d=2/-&
(5)
where A is the cell surface area and N is the number of G-proteins in the cell membrane (21). Using these equations, the time between collisions of a single receptor with any G-protein is calculated to be approximately 150 ms, an order of magnitude greater than the time required for activation of a G-protein in contact with a receptor/ ligand complex. Thus, the time required for a diffusional encounter between receptors and G-proteins appears to be the limiting factor in the activation of G-proteins. The direct in vivo measurements of the effect of receptor diffusion on G-protein-coupled signal transduction have not been made; however, the role of receptor lateral diffusion in signal transduction is supported by experimenta in which the membrane fluidity of the cells is altered (22-241, affecting enzyme activation. Reactions between receptor/ligand complexes and G-protein, as well as all other bimolecular membrane reactions, are therefore
latency (8)
10.6 8.3 6.2 6.2 6.4 a Single cella are stimulated with one of six concentrations of PhE. The average latency time betweenPhEstimulationand the maximum increase in intracellular calcium is calculated for responding celh at each PhE concentration. * Some initial data taken with low time resolution are included in the responderlnonrespondertally but are not included in the latency calculations.
assumed to be diffusion-limited in the formulation of the model equations. Formation of a-GTP can be approximated as shown by eq 6, where k3 is the rate constant of basal exchange of GDP for GTP, k4 is the rate constant for inactivation of a-GTP by hydrolysis of GTP to GDP, ks is the diffusionlimited-encounter rate constant for receptodligand complexes and G-protein, [PLC] is the concentration of inactive phospholipase C, and k6 is the diffusion-limitedencounter rate constant for PLC and a-GTP.
kJPLC1 [a-GTP] (6) a-GTP may interact with PLC and form a binary complex, a-GTP-PLC, shown in eqs 7-12 as PLC*. The inactivation of a-GTP is an intrinsic property of the GTPase (4) and occurs as GTP bound to the a subunit is hydrolyzed to GDP, leaving a-GDP. A species balance on a-GDP is shown in eq 7. The rate constant for GTP hydrolysis, k4, is assumed to be the same for aGTP and a-GTP-PLC. The a-GDP-PLC complex is assumed to dissociate rapidly. a-GDP is removed as a-GDP and By collide,reforming the inactive G-protein;k7 is the diffusionlimited-encounter rate constant for the fly and a-GDP subunits:
(7) The total amount of G-protein, [GIt, is assumed to be constant over the time course of signal transduction. Thus, inactive G-protein, [GI, is expressed as [GI = [GI,- [a-GTP] - [a-GDP] - [PLC*] (8) The quantity of fly subunits produced by activation of the G-protein is equal to the s u m of all a subunit species: [fly] = [a-GTP] + [a-GDPl + [PLC*l (9) Formation of PLC* is assumed to be diffusion-limited and dependent on the concentrations of a-GTP and PLC in the plasma membrane. PLC* is assumed to dissociate rapidly following hydrolysis of GTP to GDP, producing both PLC and a-GDP: d[PLC*l = k6[PLC][a-GTP] - kd[PLC*] (10) dt Total phospholipase C is assumed to be constant over the course of signal transduction and is given by
[PLCI, = [PLCI + [PLC*l (11) IP3 is formed by the action of phospholipase C in degrading PIP2 to DG and IP3. In this model, the activity
Biotechnol. Prog., 1994, Vol. 10, No. 1
50
Table 11. Model Parameter Values.
parameter
physical meaning equilibrium dissociation constant for PhE and al-adrenergic receptor association rate constant for receptodligand binding dissociation rate constant for receptor/ligand complexes basal exchange rate constant of GTP for GDP on a subunit rate constant for hydrolysis of a-GTP to a-GDP encounter rate constant for R/L complex and G-protein encounter rate constant for PLC and a-GTP encounter rate constant for @yand a-GDP rate constant for IP, formation by PLC [Ca2+]ifor half-maximal PLC activation rate constant for degradation of IP3 rate constant for IPS-induced calcium release from store number of IPSmolecules for half-maximal release of stored calcium rate constant for leak of calcium from store to cytosol rate constant for calcium uptake by the SR ATPase [CaZ+]i for half-maximal calcium uptake by SR ATPase number of al-adrenergic receptors in a cell number of G-protein molecules per cell number of PLC molecules per cell ratio of store volume to cytosol volume total mobile calcium in a cell on a cytosol basis
literature range and references 2.25-12.7 pMb
model value varied 5.8 pM C 1 X 107 M-1 s-1 C 58 s-l 0.002-0.01d s-1 0.01 s-1 >0.02-2e s-1 2 s-1 2 x 10-5 f (no./cell)-'s-l 2 x 10-5 (no./cell)-l s-1 not available 2 x 10-5 (no./cell)-l s-1 1 x 10-5 f (no./cell)-l s-1 1 x 10-5 (no./cell)-'s-l 4-1908 s - ~ 1634 s-1 0.07-0.91h pM 0.07 pM 0.001-2' s-1 0.1 s-1 11g-1 0.1 5-1 5 x 106k 1 x 107 0.15'~-~ 0.053 s-l 0.6-60mpM s-l 0.5 pM s-l 0.2-1.0n pM 0.2 pM 14000-250000 19000 loo000 30000op 200004 20000 0.16' 0.16 2s pM 2
Parameters in the model are consistent with literature values where measured. Other parameters are set from initial steady-state conditions. References 49-52. Measurement of kl and kZ have not been reported for the PhE/al-adrenergic receptor system. Values used are similar to those for epinephrine binding to @-adrenergicreceptors (R. Neubig, personal communication). kz was calculated from kz = KD/kl. Reference 4. e References 4 and 53. f Monte Carlo simulation prediction (43). 8 References 26,27,54, and 55. References 26-29. References 12,30,56, and 57. j Reference 58. Reference 31. Reference 17. References 31,58, and 59. References 14,33,60, and 61. References 49,50, and 62. p Reference 63.9 Calculated from the density of PLC in RBC (54) (8.7 PLC/pm2) and surface area of the cell (64) (2200 pm2). ' Reference 64. Reference 17. b
of PLC* is assumed to be much greater than that of PLC; thus, degradation of PIP2 by PLC is neglected. The concentration of PIP2 is approximately constant over the course of signal transduction (25)and is lumped into the rate constant kg. The activity of phospholipase C is moderated by the cytosolic free calcium concentration, [Ca2+]i. Over physiological [Ca2+li ranges, experimentally reported calcium-enhanced phospholipase C activity (2629) can be approximated using Michaelis-Menten kinetics, where KS is the value of [Ca2+liproducing half-maximal phospholipase C activation. In BC3H1 cells, degradation of IP3, due predominantly to the action of a phosphomonoesterase, is first-order in IP3 (30)and is represented by k10[IP3]. The overall production and degradation of IPS are given by
(
d[IP31 = kg[PLC*1 dt
[Ca2+1i
)
K,+ [Ca2+li
convert the calcium released by the internal store to a change in cytosolic calcium concentration. Removal of calcium from the cytosol by SR ATPases is nonlinear; two calcium ions are removed per ATP hydrolyzed (33). KIT is the calcium concentration in the cytosol resulting in half-maximal uptake of calcium into the store by the ATPase. It has been shown previously that the initial increase in [Ca2+liin BC3H1 cells results from the release of calcium from intracellular stores and not from increased influx across the plasma membrane (34). Thus, we neglect the negative feedback of protein kinase C activation on calcium influx that has been seen in other cell types (16, 35,361. Therefore, the total calcium in the cell over the time course of mobilization is assumed to be constant and is expressed as ( 1 7 )
-k1,[1P31 (12)
The concentration of cytosolic free calcium is a function of several opposing forces: cooperative IPS-mediated release of calcium from the calcium store into the cytosol, leak of calcium from the store into the cytosol, and return of calcium from the cytosol to the store by sarcoplasmic reticulum (SR)ATPases. This is represented by the following equation, which is similar to that initially proposed by Keizer and De Young (17):
The initial rate of IPS-mediated calcium release is cooperative (31,321,where Kl2 is the IP3concentration causing half-maximal release of calcium. Both the leak and IP3mediated release of calcium from the store are proportional to the calcium concentration gradient between the cytosol and the calcium store. The ratio VJVi is required to
where [Ca2+18is the concentration of free calcium in the internal store and [Ca2+lbtis the total free calcium in the cell on a cytosol volume basis. The parameters used in the model are consistent with literature values where available and are given in Table 11. Model simulations of stimulation with both low and high ligand concentrations are shown in Figure 5. All species concentrations are dedimensionalized with respect to their initial resting values prior to ligand addition. In Figure 5A, 0.5 pM PhE is added at t = 0 s. Receptor/ ligand binding is rapid and reaches 95% of its steadystate value in less than 0.1 s (not shown). The intracellular concentrations of a-GTP and a-GTP-PLC increase in less than 5 s to an elevated steady state. IP3 increases much more gradually until it plateaus 3.5-fold higher than its value prior to PhE stimulation. This increased level of IP3 is still below the threshhold required for the release of calcium from intracellular stores; thus, [Ca2+liincreases only slightly as a result of ligand addition, remaining near its initial, resting level. When the cell is stimulated with 50 pM PhE as shown in Figure 5B, we see a 2-fold increase
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Biotechnol. hog.., 1994, Vol. 10, No. 1
5, .-
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Figure 6. Model predictionsof the steady-statelevel of [Caz++li following ligand stimulation for one set of parameter values. For ligand greater than -2 pM, the change in [Ca2+liis 230% and the response is classified as responding. The parameters used in this steady-statediagram are the same as those in Table I1 and in Figure 5A,B. For simulation of individual cells with random parameter variation,this figure will vary in the magnitude of the response, the ligand concentration at the nonresponding-toresponding transition,and the steepnessof the calcium increase.
i i
Time, sec Figure 5. Model simulations of PhE stimulation. Model Simulations without Gaussian parameter variation predict an essentiallyall-or-noneresponse to ligand stimulation. The time course of changes in species concentrations following ligand stimulation at t = 0 s is shown. All species concentrations are made dimensionless with respect to the initial species = 376 cell-', a-GTP-PLC = 75 cell-', IPS concentrations: CY-GTP = 7.2 X 106 cell-', and [Caz+]i = 0.1 pM. (A) Simulation of 0.5 KM PhE stimulation shows that, even when the intermediate pathway species increase, [Ca2+limay remain relatively unchanged. (B) However, for 50 pM PhE stimulation, an increase in [CaZ+]iis seen within the first 5 s following stimulation. The [CaZ+]i increases to a new steady state that is approximately 2-fold higher than the resting level. in [Ca2+]i. Again, receptodligand binding is rapid and reaches 95% of ita steady-state value in less than 0.01 s (not shown). Both a-GTP and a-GTP-PLC are produced rapidly, and within the first 5 s of stimulation they actually overshoot their new, elevated steady state. IP3 again increases more gradually than both a-GTP and a-GTPPLC, but it increases 7-fold more than for stimulation with 0.5 pM PhE. As the amount of IP3 produced approaches K12, the half-maximal value for [Ca2+lirelease, the concentration of [CaZ+]i increases quickly. As our model focuses on the earliest events in the signal transduction pathway and neglects the effects of any desensitization processes on the shape and timing of the initial [Ca2+liincrease, it is unable to account for the return of [Ca2+lito basal levels seen experimentally in the continued presence of PhE. In this model, the steady-state level of intracellular calcium reached followingligand stimulation is a smoothly varying function of the ligand concentration. Stimulation with ligand results in a transient increase in intracellular calcium to a new, elevated level. For the parameter values used in Figure 5A,B, the elevated level of intracellular calcium established following ligand stimulation is summarized in Figure 6. For ligand concentrations below 0.2 pM, the intracellular calcium increase is small; the cell is considereda nonresponder. For ligand concentrations over
0.3 pM,the increase in intracellular calcium is more than 30% over the basal level; thus, the cell is considered a responder. For a given set of parameters, the mathematical model will produce either a responder or a nonresponder for any PhE concentration. If a responder is produced following PhE stimulation, only one response latency is possible. Experimentally, for each concentration of PhE, some fraction of cells is found to respond with varying latencies. The experimental heterogeneity of cellular responses suggests that some differences exist between cells. We propose that the variation in the calcium response latency and the concentration dependence of the fraction of responding cells may be attributed to concrete differences between cells, such as differences in receptor number, G-protein number, mobile calcium reserve, and sensitivity of the internal calcium store to the release of calcium by IP3. Thus, we introduce cell-to-cell variability between single cells in our model simulations by using a weighted random selection of model parameters. This is accomplished by allowing parameters in the model to vary according to a Gaussian distribution around the mean values given in Table I1with a standard deviation of 20%. All parameters except Hill exponents are chosen randomly from the distributions for each unique, single cell simulated. The calcium response of each simulated single cell is judged using the same criteria as detailed previously in the data analysis section for responder/nonresponder classification and for the response latency of experimental data. The average calcium response latency and the fraction of responding cells at each ligand concentration are the average of 1000 single-cell simulations. Time-dependent species concentrations are generated from this model by numerically integrating the differential equations using an Adams or backward differentiation formula method (ODEPACK: A. C. Hindmarsh and L. R. Petzold, Lawrence Livermore National Laboratory). For variable parameter input, a Gaussian distribution of parameter values is generated using error functions (RAND.FTN: D. M. Krowitz, Massachusetts Institute of Technology).
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52
The criteria used to judge the quality of the model fit to the experimental data include the following: fraction of responding cells at each PhE concentration, basal and PhE-stimulated levels of [Ca2+li,average PhE-stimulated [Ca2+]i increase, calcium response latency at each PhE concentration, and magnitude of the difference in [Ca2+li increase between responders and nonresponders. Parameters reported in the literature which have a wide range, as well as parameters for which one or no values are reported, are varied according to an experimental design scheme (37) to produce a good fit of the experimental data. Model parameters are set initially to produce stable, resting levels of intracellular species prior to ligand stimulation. A good model fit to the experimental data is shown in Figures 3 and 4. Several parameters are important in determining the calcium response latency and the fraction of responding cells in the model simulations. The calcium response latency is most sensitive to changes in 1210, K12, [Ca2+Ibt, and 1216. The fraction of responding cells is most sensitive to changes in 123, KIZ, [Ca2+Ibt,and [RIt. A thorough discussion of the parameter sensitivity analysis for the mathematical model can be found in ref 38.
Discussion Changes in [Ca2+]i have been linked to a variety of cellular responses in diverse cell types. The ligand-induced changes in [Ca2+]iin cell populations can be significantly different from changes in [Ca2+]iseen on the single-cell level. We have monitored [Ca2+]i in single BC3H1 cells followingstimulation with PhE using the calcium indicator, fura-2. A statistical treatment of the increase in [Ca2+li from the basal level to a peak reveals that, on the singlecell level, the magnitude of the calcium increase in responding cells is not dependent on the ligand concentration. Thus, on the single-cell level, we find that calcium mobilization following PhE stimulation is essentially an all-or-none response. For a wide range of PhE concentrations (0.1-50.0 pM), we find that not all cells respond to ligand stimulation with an increase in [Ca2+li. Combining all single-celldata, we find that the fraction of cells which mobilize calcium in response to PhE stimulation is concentration-dependent. Similar behavior has been observed in single ATPstimulated cultured arterial smooth muscle cells (5). For low PhE concentrations (10.3pM), the cells are unresponsive to PhE stimulation. However, for PhE between 0.5 and 50 pM, the fraction of responders shows a sigmoidal dependence on PhE concentration, plateauing to 60% at 50 pM. Observations of some cell populations show that the magnitude of the ligand-stimulated change in [Ca2+li is an increasing function of ligand concentration (39-42). It usually is not known, however, whether these responses are due to submaximal responses by all cells at low ligand concentrations or due to increasing fractions of cells responding in an all-or-none fashion. In our experiments with BC3H1 cells, we find the latter to be true. For responding cells, we see significant heterogeneity in the speed of calcium mobilization between single cells for the same ligand stimulation. For the summed population of responding cells, we find that as the PhE concentration increases, the average latency of the calcium response decreases from 10 to 6 s. Furthermore, we have developed a mathematical model based on the calcium signal transduction cascade to predict the calcium response latency of individual cells following PhE stimulation. We have used the model to systematically test the hypothesis that cell-to-cellcalcium response
heterogeneity between single cells may result from distinct differences between cells, such as in receptor number or size of the intracellular calcium store. The model consists of six coupled ordinary differential equations which describe the events of receptor/ligand binding, G-protein and PLC activation, IP3production, andchanges in [Ca2+]i. Bimolecular interactions occurringbetween two membrane species such as receptor/ligand complexes and G-protein are all assumed to be diffusion-limited. Heterogeneity in cell parameters is incorporated into our mathematical model by random selection of all parameters for individual cells from a Gaussian distribution about the parameter mean. This model is the first to successfully link receptor/ ligand binding to dynamic increases in [Ca2+liand, in addition, to relate single-cell [Ca2+]iresponses to population behavior. By allowing variability between cells in the model simulations, we are able to produce a heterogeneity in the speed of the calcium response between single cells at each ligand concentration that is similar to that seen experimentally. In addition, the average calcium response latency of our theoretical cell population shows a ligand concentration dependence similar to that seen experimentally. We see both responding and nonresponding cells in our single-cell simulations for PhE concentrations from 0.5 to 50 pM. Our population-averaged fraction of responding cells shows a sigmoidal dependence on the PhE concentration which is similar to that seen experimentally. It is worth pointing out that our model predicts that for a single ligand-stimulated cell the percent increase in [Ca2+]iover the basal level strongly depends on the ligand concentration (Figure 6). This single-cell feature is damped when the responses of a population are considered by allowing parameter variation between cells. The model underestimates the fraction of responding cells at low PhE concentrations and overestimates the fraction of responding cells at high PhE concentrations (Figure 4). The underestimation of responders at low PhE concentrations may be a reflection of the fact that the model is unable to account for movement of the ligand among free receptors. Ligand movement among free receptors may significantly enhance signal transduction at low ligand concentrations (43-45). In model simulations with the addition of a receptor blocker, this limitation is more clearly illustrated. The model predicts that equal numbers of receptor/ligand complexes on the cell surface will produce equivalent responses between cells, regardless of the number of free receptors. However, in experiments with equal numbers of receptor/ligand complexes on the cell surface, with and without blocked receptors, cell responsiveness is significantly reduced when the number of free receptors is reduced with a receptor blocker (43, 45); we have shown using a Monte Carlo approach that this may be a result of reduced ligand movement among free receptors (43). A t higher ligand concentrations, the responsiveness of cells to PhE stimulation may be limited by desensitization mechanisms. Inclusion of desensitization in the model, once these mechanisms are uncovered, may eliminate the overestimation of cell responsiveness at high ligand concentrations. Furthermore, the model formulation only considers the initial increases in intracellular calcium and thus is unable to predict the return of calcium to basal levels in the continued presence of PhE, as is seen experimentally. A more complete model might include later mechanisms for elimination of calcium from the cytosol, returning calcium to prestimulation levels.
Bbtechnol. Rw., 1994, Vol. 10,No. 1
The model predicts that several parameters are key in the timing of the calcium response. Experimental measurements of the single-cell average value of these parameters as well as others such as total PLC and G-protein would be useful in testing the validity of the model structure and parameter choices. The initial model equations may be combined with existing calcium oscillation models (12,13,15,18)to link receptodligand binding to oscillations in [Ca2+liseen in some other experimental systems. For example, recent experimental work on the dynamics of the IPSreceptor ( 4 6 4 8 )may be incorporated into the differential equation for [Ca2+]i, as in the updated model by De Young and Keizer (18).Their expression for the IPS-mediated release of calcium from intracellular stores includes the activation and inhibition of [Ca2+lion calcium release. Use of this more complex expression for calcium release will have little qualitative effect on our model results for the initial timing of the calcium response and the cell-to-cell response variation; however, the use of this expression may improve the later behavior of the model. The model suggests methods for manipulating cell responses that are dependent on calcium signaling. Because the calcium response in single cells is essentially all-or-none, the manipulation of the cell response at a particular ligand dose may be best accomplished by increasing the fraction of responding cells,not by increasing the magnitude of the single-cell response. Such a conclusion is not evident from population studies of cell responses, which are more commonly assessed. Our model predicts that a key parameter in determining the fraction of responding cells in the population is the total number of receptors, a variable which may be altered using modern molecular biology techniques.
Acknowledgment The authors thank Dr. Dave Gross for valuable discussions, Dr. Linda Slakey for the gift of the BC3H1 cell line, and Kimberly Wicklund for experimental assistance. This work was supported by an NSF PYI award, Procter & Gamble Company,and an Institutional Research Grant to the University of Michigan from the American Cancer Society.
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