Responses of Ca2+-binding Proteins to Localized, Transient Changes in Intracellular [Ca2+]

J. theor. Biol. (2003) 221, 245–258 doi:10.1006/jtbi.2003.3187, available online at http://www.idealibrary.com on

Responses of Ca2+-binding Proteins to Localized, Transient Changes in Intracellular [Ca2+] Gary J. Kargacinnw wDepartment of Physiology and Biophysics, University of Calgary, 3330 Hospital Drive NW, Calgary, Alberta, Canada T2N 4N1 (Received on 19 March 2002, Accepted in revised form on 4 October 2002)

In smooth muscle cells, various transient, localized [Ca2+] changes have been observed that are thought to regulate cell function without necessarily inducing contraction. Although a great deal of effort has been put into detecting these transients and elucidating the mechanisms involved in their generation, the extent to which these transient Ca2+ signals interact with intracellular Ca2+-binding molecules remains relatively unknown. To understand how the spatial and temporal characteristics of an intracellular Ca2+ signal influence its interaction with Ca2+-binding proteins, mathematical models of Ca2+ diffusion and regulation in smooth muscle cells were used to study Ca2+ binding to prototypical proteins with one or two Ca2+-binding sites. Simulations with the models: (1) demonstrate the extent to which the rate constants for Ca2+-binding to proteins and the spatial and temporal characteristics of different Ca2+ transients influence the magnitude and time course of the responses of these proteins to the transients; (2) predict significant differences in the responses of proteins with one or two Ca2+-binding sites to individual Ca2+ transients and to trains of transients; (3) demonstrate how the kinetic characteristics determine the fidelity with which the responses of Ca2+-sensitive molecules reflect the magnitude and time course of transient Ca2+ signals. Overall, this work demonstrates the clear need for complete information about the kinetics of Ca2+ binding for determining how well Ca2+-binding molecules respond to different types of Ca2+ signals. These results have important implications when considering the possible modulation of Ca2+- and Ca2+/calmodulin-dependent proteins by localized intracellular Ca2+ transients in smooth muscle cells and, more generally, in other cell types. r 2003 Elsevier Science Ltd. All rights reserved.

1. Introduction It is becoming increasingly clear that, in addition to global changes in cytoplasmic Ca2+ concentration ([Ca2+]i), Ca2+ gradients and localized, transient changes in [Ca2+] may also be important in determining cell function. The detection of Ca2+ transients (including Ca2+-waves, Ca2+-oscillations or fluctuations and discrete n

Corresponding author. Tel.: +1-403-220-3873; fax: +1-403-270-2211. E-mail address: [email protected] (G. J. Kargacin). 0022-5193/03/$35.00

Ca2+ release or influx events such as Ca2+sparks) in smooth muscle and other cell types and the elucidation of the mechanisms involved in their generation have been the subject of intense experimental investigation (see Asada et al.,1999; Bolton & Gordienko, 1998; Janiak et al., 2001; Kirber et al., 2001; Miriel et al., 1999; Nelson et al., 1995; Ruehlmann et al., 2000; ZhuGe et al., 2000; Zou et al., 1999). These studies have provided information about the diversity in the temporal and spatial characteristics of these transients and the differences in the r 2003 Elsevier Science Ltd. All rights reserved.

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mechanisms by which they are initiated. The effects of transient Ca2+ signals on cell function, however, are much less well understood. There are numerous Ca2+-binding proteins in muscle and other cell types that could potentially respond to these Ca2+ signals. The magnitude of the Ca2+ change that can occur near a plasma membrane or a sarcoplasmic/endoplasmic reticulum (SR/ER) Ca2+ release channel in response to a single channel opening has not been precisely measured; however, Ca2+concentrations in the 1 – 10 mM range appear to be possible. Since the Kd’s of many Ca2+-binding proteins are in this same range, one might expect that even Ca2+ signals due to the opening of a single Ca2+ channel would alter the activity of these proteins. On the other hand, the transient nature of such rapid Ca2+ signaling events could limit their ability to interact with some proteins. Single channel openings and closings, for example, are virtually instantaneous and, in smooth muscle, reported open times of L-type Ca2+ channels range from 2 to 10 ms (see Knot et al., 1996). Thus, although the [Ca2+] near an open channel can reach a high level, the speed with which single-channel Ca2+ signals develop and their short duration suggests that knowledge of a protein’s Kd and the assumption of equilibrium binding may not provide a true picture of interaction of the protein with a single-channel signal. The purpose of this study was to examine the potential interaction of Ca2+-binding proteins with transient Ca2+ signals in smooth muscle cells and to determine how the Ca2+ binding characteristics of different proteins influence this interaction. The mathematical models of Ca2+ diffusion and regulation that we and others have developed have proven to be very useful for studying the mechanisms involved in modulating transient Ca2+ signals. As discussed above, in addition to understanding how Ca2+ signals themselves are generated and modified, it is also important to know how the spatial and temporal characteristics of a Ca2+ signal, in turn, influence the activity of Ca2+-binding proteins or other target molecules. We have shown, for example, that the high Ca2+ concentrations that are likely to develop near sites of Ca2+ influx or SR release in smooth muscle cells during

contractile activation could result in significant changes in the concentration of free ATP and the ratio of CaATP to MgATP in these regions (Kargacin & Kargacin, 1997). In general, however, the way in which transient Ca2+ and other second messenger signals interact with effector molecules and regulate cell function has received relatively little attention. In this study, the mathematical models of Ca2+ diffusion and regulation in smooth muscle cells that were described previously (Kargacin, 1994; Kargacin & Fay, 1991; Kargacin & Kargacin, 1997) were modified and used to study the possible interactions of Ca2+ with proteins having one or two Ca2+-binding sites and different binding kinetics. The characteristics of these interactions and their implications for cell signaling were determined and compared in cellular regions where free diffusion was possible and where diffusion was restricted by the presence of physical barriers to diffusion. Responses of Ca2+-binding proteins to Ca2+ transients with different temporal characteristics and to trains of transient input signals were also examined. The results of this study are particularly relevant for understanding how near-membrane Ca2+- or Ca2+/calmodulin-dependent pathways are affected by localized, transient changes in [Ca2+]. 2. Methods The two-dimensional model of Ca2+ diffusion and regulation in smooth muscle cells was described in previous publications (Kargacin, 1994; Kargacin & Fay, 1991; Kargacin & Kargacin, 1997) and will only be described briefly here. The central longitudinal plane of a cylindrical cell segment with a radius of 1mm and of variable length was modeled (see Fig. 1A). The equation describing diffusion and regulation of Ca2+ in 2-dimensions (radial and longitudinal) is
 @½Ca 1 @ @½Ca ¼ rD @t r @r @r
 @ @½Ca D þ Gð½Ca; t; r; lÞ; þ @l @l

ð1Þ

where l is the position along the length axis, r is the radial position, D is the diffusion coefficient

Fig. 1. Two-dimensional diffusion model. (A) The segment of the smooth muscle cell modeled in the simulations was assumed to be a cylinder with a radius of 1 mm and of variable length. For some simulations, a barrier to free diffusion (black bar) was imposed between the plasma membrane and the central cytoplasm of the modeled smooth muscle cell segment. The diffusion coefficient through the barrier was 0.1 of the diffusion coefficient through the rest of the cell. The diffusion coefficients in the radial direction and the longitudinal direction were assumed to be the same throughout the segment. Additional details of the model are given in Section 2. (B) The Ca2+ regulatory processes included in the model at the plasma membrane and the sarcoplasmic reticulum included Ca2+ influx pathways and Ca2+ extrusion sites on the plasma membrane and Ca2+ pumps and Ca2+ leak channels in the SR membrane. Ca2+ leak from the SR and into the cytoplasm through the plasma membrane leak channels balanced Ca2+ uptake by the SR Ca2+ pump and Ca2+ extrusion through the plasma membrane in the resting cell (when [Ca2+]i was 150 nM) so that net Ca2+ movement across the SR and plasma membranes was 0. SERCA, SR Ca2+ pump; PMCA, plasma membrane Ca2+ pump; Ca2+ leak, plasma membrane and SR leak channels [see text and Kargacin and Fay, 1991; Kargacin (1994) for additional details]. (C): Ca2+ transients. The trace on the left shows the time course (in ms) of the integrated Ca2+ transient used in some of the simulations; the trace on the right shows the square-wave [Ca2+] increase used to model influx through a single open Ca2+ channel.

doi: 10:1006 jtbi.2003.3187 G. J. KARGACIN

CALCIUM-DEPENDENT SIGNAL TRANSDUCTION

for Ca2+ (assumed to be the same in both the radial and longitudinal directions) and the function G([Ca],t,r,l) includes the [Ca2+], time and position-dependent terms that represent Ca2+ regulatory processes in the model cell [described by eqns (2)–(6) below]. This equation was solved using the explicit finite-differences equations for diffusion into a cylinder described by Crank (1975; also see Kargacin, 1994; Kargacin & Fay, 1991). For these computations, the modeled cell segment was divided into 40 (25 nm) radial elements and up to 140 (25 nm) length elements. The diffusion coefficient for Ca2+ diffusion in the bulk of the cytoplasm in the model cell was taken as 2.2  106 cm2 s1 (the same diffusion coefficient was used for both radial and longitudinal diffusion) based on the estimates of this value in the literature (Allbritton et al., 1992; Kushmerick & Podolsky, 1969; also see Kargacin, 1994). For some simulations, an area near the plasma membrane with a reduced diffusion coefficient [represented by the black bar in Fig. 1(A)] was included in the model. This was done to simulate diffusion in areas where there are physical barriers present that inhibit the free movement of Ca2+ from the near-membrane space into the bulk of the cytoplasmic space. Such regions are thought to be present in smooth muscle cells where intracellular organelles such as the SR come into close apposition to the plasma membrane. It has been postulated that the restricted diffusion spaces that are formed in this way between a barrier and the plasma membrane create local environments in which signal transduction is facilitated (Sturek et al., 1992; van Breemen & Saida, 1989). The barrier in the model cells extended 50 nm in the radial direction and was 1.75 mm long; it was located 25 nm from the plasma membrane. The dimensions of the barrier and its distance from the plasma membrane of the model cell are in keeping with the size and location of near-membrane SR elements that have been observed in electron micrographs of smooth muscle cells (Devine et al., 1972; Gabella, 1983; Somlyo, 1980; Somlyo & Franzini-Armstrong, 1985). The diffusion coefficient through the barrier region was taken to be 0.22  106 cm2 s1 (see Kargacin, 1994). Choosing a diffusion coefficient

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greater than zero for the barrier region allowed some diffusion to occur through the barrier region and was done to account for the fact that, in a real cell, movement of Ca2+ through fenestrations in the barrier and around the barrier in the angular dimension that was not included in the model would also allow some Ca2+ to move into the cytoplasm behind the barrier (also see discussion in Kargacin, 1994). The equations from Crank (1975) describing Ca2+ diffusion at a boundary between composite media were used to determine the diffusion of Ca2+ across the interface between the barrier and the near-membrane space and the barrier and the central cytoplasm of the modeled cell segment. Diffusion at the center of the cell was treated as described by Crank (1975). For the diffusion calculation in the longitudinal dimension, it was necessary to extend the calculation into one additional length element at each radial position at the longitudinal borders of the modeled segment. This was done by setting the concentrations in each of these additional elements beyond the boundary equal to the concentration on the other side of the boundary (i.e. when the model included 40 radial elements and 140 length elements, Ca(r,0) ¼ Ca(r,1) and Ca(r,141) ¼ Ca(r,140) for 0prp40) so that no diffusion occurred across the longitudinal boundaries of the modeled segment. These boundary conditions do not affect the results of the simulations reported here because the localized changes in [Ca2+] used in the model did not reach the longitudinal borders of the modeled cell segment (also see discussions in Kargacin, 1994; Zou et al., 1999). The Ca2+ regulatory processes included in the model are shown in Fig. 1B. Uptake of Ca2+ by the SR Ca2+ pump was described by the equation: D½Ca Vmax ½Can ; ¼ n Dt Km þ ½Can

ð2Þ

where Vmax (maximum velocity ¼ 0.35 m M s1), Km ([Ca2+]free at half-maximal velocity ¼ 219 nM) and n (Hill coefficient ¼ 2) were based on values described in the literature (see Kargacin, 1994; Kargacin & Fay, 1991;

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Kargacin & Kargacin, 1997). To balance SR Ca2+ influx under resting conditions so that there was no net movement of Ca2+ across the SR membrane, a leak term described by the equation: D½Ca ¼ Kleak ð½CaSR  ½Cacytoplasm Þ Dt

ð3Þ

was included on the SR membrane with Kleak adjusted to balance Ca2+ uptake at rest ([Ca2+]i ¼ 150 nM). For the simulations, active SR Ca2+ uptake occurred at sites 25 nm away from the plasma membrane along the entire length of the cell segment being modeled. Similar equations were used to model Ca2+ extrusion from the modeled cell segment through the plasma membrane. The parameters for eqn (2) for Ca2+ extrusion used in the model were: Vmax ¼ 0.032 mM s1; Km ¼ 200 nM; and n ¼ 1 (Carafoli, 1987; Lucchesi et al., 1988). As discussed previously (Kargacin & Fay, 1991), the magnitude of Vmax was increased to 0.064 mM s1 in the simulations to account for the possible contribution of Na+/Ca2+ exchange to Ca2+ extrusion (see Cooney et al., 1991). The binding of Ca2+ to intracellular Ca2+ buffers and Ca2+-binding proteins was described by equations of the form: D½Ca ¼ kon ð½bufferfree ½CaÞ Dt þkoff ½Ca buffer;

ð4Þ

where [Ca buffer] is the concentration of the Ca2+ bound form of the buffer. The total Ca2+ buffer concentration in the cell was 230 mM (in keeping with estimates in the literature; see for example Bond et al., 1984; Fink et al., 2000; Jafri & Keizer, 1995; Kargacin, 1994; Kargacin & Fay, 1991) and included contributions from the main Ca2+ buffer in the model ([buffer]total ¼ 195.5 mM) and three prototypical Ca2+-binding proteins (each [protein]total ¼ 11.5 mM) that were fixed and distributed uniformly throughout the modeled cell segment. For the reasons discussed below, for most of the simulations, the Kd values for the buffer and binding proteins were all set at 1 mM giving free concentrations of 170 mM for the main buffer and 10 mM for each of the other Ca2+-binding proteins at the start of the

simulations when [Ca2+]i was 150 nM. The onrate (kon) for the main buffer was 107 M1 s1 and the off-rate (koff) was 10 s1. These values are in keeping with the approximate mean of values reported in the literature for different cytoplasmic Ca2+-binding molecules (see Kargacin, 1994; Kargacin & Fay, 1991; Robertson et al., 1981). The on- and off-rates for the three prototypical Ca2+-binding proteins were set at different values for the simulations described in Section 3. For some simulations, a two-site Ca2+-binding protein was included in the model and was described by a sequential binding process: k1on

½Ca þ ½protein 2 ½Ca protein þ ½Ca k1off

k2on

2 ½Ca2 protein;

k2off

ð5Þ

where [Ca protein] and [Ca2 protein] are the singly and doubly bound forms of the protein respectively. The on- and off-rates for the two binding steps were set as described in Section 3. Two types of Ca2+ influx were used for the present simulations. The first, which was used in previous papers (Kargacin, 1994; Kargacin & Fay, 1991; also see Backx et al., 1989; Cannell & Allen, 1984), modeled Ca2+ influx by an exponential function according to the following equation: D½Ca ¼ P0 ð1  expt=ton Þðexpt=toff Þ Dt

ð6Þ

with the time constants ton and toff equal to 3 and 50 ms, respectively, so that the time course of the simulated Ca2+ influx [shown by the trace on the left in Fig. 1(C)] matched the time course of the Ca2+ current measured by Becker et al. (1989) from voltage-clamped smooth muscle cells. The constant P0 was adjusted to set the maximum amplitude of the Ca2+ influx. A Ca2+ influx of the type described by eqn 6 represents the influx pattern that might be expected from an ensemble of Ca2+ channels along an extended region of the membrane and will be referred to here as an integrated input signal. It has also been proposed that some Ca2+ channels in cardiac and smooth muscle cells are localized into clusters (see DeFelice, 1993; Gathercole et al., 2000;

CALCIUM-DEPENDENT SIGNAL TRANSDUCTION

Risso & DeFelice, 1993; and discussions in Janiak et al., 2001; ZhuGe et al., 2000; Zou et al., 1999). Influx through such a localized cluster of channels would also be expected to be mimicked by the integrated input signal. The second type of influx used in the present simulations was represented by a square-wave Ca2+ pulse lasting for 1 or 4 ms [shown by the trace on the right in Fig. 1(C)]. This influx form more accurately depicts the type of Ca2+ influx one would expect near the mouth of a single ion channel and is described further in Section 3. Influx in the model cell occurred at one or two points on the plasma membrane (see Section 3) 3. Results 3.1. RESPONSES OF CA2+-BINDING PROTEINS WITH DIFFERENT RATE CONSTANTS TO LOCALIZED CA2+ TRANSIENTS

To understand how the interaction of Ca2+ with Ca2+-binding proteins might influence subsequent localized Ca2+-dependent signaling, simulations were carried out in which the kinetics of Ca2+ binding to the modeled proteins were varied. Reported forward-rate constants for Ca2+-binding proteins range over at least three orders of magnitude from approximately 106 M1 s1 (e.g. calmodulin, C-terminal) to greater than 108 M1 s1[e.g. calmodulin, Nterminal; also see Davis et al. (1999), Robertson et al. (1981) and Smith et al. (1996) for information about other proteins]. For the simulation shown in Fig. 2, three prototypical Ca2+-binding proteins (total concentration of each protein ¼ 11.5 mM) with on-rates of 106, 107 and 108 M1 s1 were included simultaneously in the model. Off-rates were adjusted to keep the Kd’s of the proteins the same in all cases (Kd ¼ 1 mM). The concentrations of the three Ca2+-binding proteins and their Kd’s were all chosen to be the same so that the effects of the rate constants on Ca2+ binding could be examined and compared directly. The Kd’s used in the model are of the same order of magnitude as the Kd’s of a number of muscle proteins (e.g. calmodulin C-terminal, Kd ¼ 1 mM; calmodulin N-terminal, Kd ¼ 2.5 mM; striated muscle troponin C, Kd ¼ 1.8 mM; see Davis et al., 1999;

249

Johnson et al., 1996; Robertson et al., 1981). Figure 2(A)–(C) are images of the Ca2+-bound form of each of the proteins in the model at several time points in a simulation. For the simulation, influx occurred into an unrestricted diffusion space in the modeled cell segment from a single point on the plasma membrane [Ca2+ influx was described by eqn (6)]. The solid lines in Fig. 2(D) show the concentrations of the Ca2+-bound form of the proteins as a function of time at the site of Ca2+ influx. These results demonstrate the extent that the rate constants for Ca2+-binding influences the time course and the spatial distribution of Ca2+ binding to the proteins. The two faster proteins bound, and then started to release Ca2+ before the third, slower protein, bound Ca2+ to its maximum extent. The dashed line in Fig. 2(D) shows the amount of Ca2+ binding and the time course of binding to the proteins that would be predicted if one calculated Ca2+ binding assuming that equilibrium was reached at each time point in the simulation (i.e. using the Kd of the binding-proteins for the calculation). Of the three proteins modeled in the simulation, the fastest protein (on-rate ¼ 108 M1 s1; Fig. 2(D), black line] followed the equilibrium-based binding prediction most successfully at times greater than 20 ms, although the simulation using the Kd inaccurately predicted faster binding at earlier time points. The results in Fig. 2(D) clearly show that the equilibrium determination cannot be used to predict the early responses to the Ca2+ transient especially for the proteins with on-rates of 107 and 106 M1 s1 (e.g. the C-terminal Ca2+-binding sites of calmodulin; see Davis et al., 1999; Johnson et al., 1996). At longer times (1 – 4 s), the equilibrium-based determination of Ca2+ binding and the results obtained from the simulations using rate equations produced results that were in better agreement (discussed below). The results clearly demonstrate that knowing the Kd for Ca2+-binding is not enough information for determining if a given Ca2+-binding protein can respond to a localized, transient Ca2+ signal; the nature of the Ca2+ transient (time course, localization) and the rate constants for Ca2+ binding to the protein must also be taken into account.

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G. J. KARGACIN 3.2. LOCALIZED SIGNAL TRANSDUCTION IN A RESTRICTED DIFFUSION SPACE

To determine how the presence of a barrier to free diffusion is likely to affect the responses of Ca2+-binding proteins to a localized, transient Ca2+ signal, the simulations in Fig. 2 were repeated when a barrier impeded the free movement of Ca2+ into the central cytoplasmic space in the modeled cell segment. The simulation was done with the input parameters used for the simulation in Fig. 2 except that a diffusion barrier (D ¼ 0.22  106 cm2 s1 through the barrier; barrier thickness ¼ 50 nm; distance between plasma membrane and barrier ¼ 25 nm; see Section 2) was included in the model. Figure 3(A) shows the [Ca2+] (after 10 ms) in the modeled cell segment in a region with the diffusion barrier and in a region where Ca2+ was allowed to freely diffuse into the central cytoplasm in response to two localized Ca2+ input signals. Figure 3(B) shows the concentration of the Ca-bound form of three Ca2+binding proteins in the space adjacent to the plasma membrane of the modeled cell segment at the 10 ms time point for the same simulation. In addition to increasing the maximum [Ca2+ ]free and [Ca protein] near the site of Ca2+ influx, the barrier also caused a greater spread of Ca2+ and increased Ca2+ binding to the proteins lateral to the site of influx. At the concentration shown by the dashed line in Fig. 3(B) (when the [Ca protein] for the protein with an on-rate of 108 M1 s1 was approximately half-maximal, 5.5 mM), the lateral spread of the Ca protein signal in the region between the barrier and the plasma membrane was roughly twice that in the space without a barrier. 3.3. CA2+ BINDING FOLLOWING SINGLE-CHANNEL INFLUX EVENTS

The simulations described thus far assumed that localized Ca2+ influx into the model cell could be described by an exponential function [see Section 2 and Fig. 1(C)]. Although one would expect the Ca2+ influx arising from the opening of an array of activated Ca2+ channels to resemble this waveform, a square-wave function more accurately represents Ca2+ influx into the cytoplasmic space directly in front of a

plasma membrane or an SR Ca2+ channel. The results shown in Figs 1 and 2 indicate that, even with the relatively long time course of the integrated Ca2+ influx signal [see Fig. 1(C)], one would only expect the Ca2+-binding proteins with on-rate constants of 108 or 107 M1s1 to be affected significantly by prolonged transients of this type. The types of proteins able to respond to a single-channel like influx event should be even more restricted. Smooth muscle L-type Ca2+ channels have reported mean open times ranging from 2 to 10 ms and single-channel currents in the range of 0.1–0.5 pA (reviewed in Knot et al., 1996). From the computer models developed by Smith (1996) and Smith et al. (1996), in which values for [buffer]total and the cytoplasmic Ca2+ diffusion coefficient were used that are similar to those used in the present simulations, it can be inferred that the [Ca2+]free 30 nm in front of a Ca2+ channel passing a 0.1 pA current would reach a magnitude between 1 and 10 mM. Figure 4 shows the results of simulations in which the opening of a single plasma membrane Ca2+ channel allowed [Ca2+]free to increase by 5 mM for 1 ms [Fig. 4(A) and (C)] or 4 ms [Fig. 4(B) and (D)] at a nearmembrane site in the free diffusion space of the model cell. The three prototypical Ca2+-binding proteins (i.e. [protein]total’s ¼ 11.5 mM; on-rates of 108, 107 and 106 M1 s1; Kd’s ¼ 1 mM) were also included in the models. As can be seen in Fig. 4(C) and (D), the Ca2+-binding protein at the influx site with the highest on-rate (108 M1 s1) was best able to respond to the nearmembrane Ca2+ signal and did so in a phasic manner that was partially able to follow the driving Ca2+ transient. The [Ca protein] did not return to baseline, however, over the 100 ms time interval shown. Ca2+ binding to the protein with an on-rate of 107 M1 s1 increased when the transient pulses occurred; however, the magnitude of the response remained relatively constant over the entire time course of the simulation and the phasic nature of the driving Ca2+ signal was not clearly perceptible in the response. The maximum [Ca protein]’s in the near-membrane space for the proteins with on-rates of 108 or 107 M1 s1 were considerably higher for the 4 ms Ca2+ pulse than those for the 1 ms pulse. The Ca2+-binding protein with an on-rate of

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Fig. 2. Binding of Ca2+ to three prototypical Ca2+-binding proteins with the same Kd’s but different rate constants for Ca2+ binding. (A) Spatial distribution of the Ca2+-bound form of a protein with an on-rate of 108 M1 s1 and an off-rate of 102 s1 following influx of Ca2+ at a single site on the plasma membrane of the model cell. The images show the concentration and distribution of the Ca2+-bound form of the protein at 7 time points in the simulation; total protein concentration was 11.5 mM. (B) As in (A) except that the on- and off-rates for the Ca2+-binding protein were 107 M1 s1 and 10 s1, respectively. (C): As in (A) and (B), except that the on- and off-rates for the Ca2+-binding protein were 106 M1 s1 and 1 s1, respectively. The numbers under the images in (C) are the time points (in ms) for the images in (A)–(C) the color scale bar under image (C) shows the relationship between [Ca protein] and color in the images. The images shown in (A)–(C) show 1 mm  1 mm cellular regions composed of 40  40 25 nm square pixels; the plasma membrane of the modeled cell segment is on the right and the center of the cell on the left. (D) Near-membrane concentration of the Ca2+-bound forms of the proteins plotted as a function of time. The results shown give [Ca protein] in the cellular element into which Ca2+ influx occurred. The left axis shows the concentration of the Ca2+-bound form of the protein and the right axis the percent to which the Ca2+-binding site on the protein was saturated. The solid lines in D are the predicted concentrations of the Ca2+-bound form of the proteins [on-rate of 108 M1 s1 (black line), 107 M1 s1 (red line) and 106 M1 s1 (blue line)]; the dashed line is the concentration of the Ca2+-bound form of the proteins that was predicted when the equilibrium equation (i.e. using only the Kd for the calculation) was used to determine the extent of Ca2+ binding at the influx site to a protein with a total concentration of 11.5 mM and a KdCa2þ of 1 mM.

doi: 10:1006 jtbi.2003.3187 G. J. KARGACIN

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Fig. 3. Effect of a barrier to free diffusion on Ca2+ binding to proteins in the modeled smooth muscle cell segment. (A) [Ca2+]free in the modeled cell segment 10 ms after Ca2+ influx occurred at two points on the plasma membrane. The plasma membrane is on the bottom of the image, the cell center on the top. The cellular region modeled was divided into 40 (25 nm) radial elements (total radial distance ¼ 1 mm) and 140 (25 nm) length elements (total length ¼ 3.5 mm); the black bar indicates the barrier to free diffusion in the modeled cell segment (D in this space was 0.22  106 cm2 s1; D throughout the rest of the cell was 2.2  106 cm2 s1). The color scale bar at the bottom of A shows the relationship between [Ca2+]free and color in the image. (B) Near-membrane [Ca protein] (10 ms after Ca2+ influx started) as a function of position along the length of the modeled cell segment for the simulation in (A). Ca2+ influx sites were located at length element 35 (into a restricted diffusion apace) and at length element 105 in the free diffusion space. The diffusion barrier extended from element 0 to element 70. The traces (10 ms time point) show the concentration (left vertical axis) and the % saturation (right vertical axis) of the Ca2+-bound form of proteins with an on-rate for Ca2+ binding of 108 M1 s1 (black line), 107 M1 s1 (red line) or 106 M1 s1 (green line) as a function of position in the radial elements immediately adjacent to the plasma membrane of the model cell. The lateral spread of the Ca protein species (defined as signal width at half-maximal [protein Ca], i.e. 5.5 mM) for the protein with the on-rate of 108 M1 s1) is shown by the dashed line in the figure and was 325 nm in the region between the barrier and the plasma membrane and 175 nm in the free-diffusion space (i.e. no barrier present).

doi: 10:1006 jtbi.2003.3187 G. J. KARGACIN

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Fig. 4. Response of Ca2+-binding proteins to a single-channel-like Ca2+ influx. (A) and (B) Distribution and concentration of the Ca2+-bound form of a protein with an on-rate of 108 M1 s1 and an off-rate of 102 s1 following a 1 ms (A) or 4 ms (B) square-wave influx (that increased [Ca2+]free to 5 mM) into a single near-membrane site (25 nm square). The numbers between A and (B) give the time points (in ms) in the simulation; influx started at the 6 ms time point in the simulation. The color bar at the bottom of (B) shows the relationship between [Ca protein] and color in the images. The images show a cellular region composed of 40  40 (A) or 40  100 (B) square 25 nm pixels; the plasma membrane of the modeled segment is on the right and the center of the cell on the left of each image. (C) and (D) Time course of Ca2+ binding to three prototypical Ca2+-binding proteins following a 1 ms (C) or 4 ms (D) square-wave Ca2+ influx that increased nearmembrane [Ca2+]free at a single site by 5 mM. The traces in the upper portion of each graph show the concentration (left vertical axis) and the % saturation (right vertical axis) of the Ca2+-bound form of the protein as a function of time for a binding protein with an on-rate of 108 M1 s1 (solid lines), 107 M1 s1 (dashed lines) or 106 M1 s1 (dotted lines). The lowest traces in each graph show the time course of the square-wave Ca2+ influx. The [protein]total was 11.5 mM and the Kd was 1 mM for all of the Ca2+-binding proteins in (A)–(D).

doi: 10:1006 jtbi.2003.3187 G. J. KARGACIN

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CALCIUM-DEPENDENT SIGNAL TRANSDUCTION

WITH DIFFERENT KD’S BUT THE SAME ON-RATES

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% saturation

[protein-Ca] uM

It is also of interest to compare the responses of proteins with the same on-rates but different Kd’s for Ca2+ binding to transient Ca2+ signals. For this, Ca2+-binding proteins with on-rates of 108 M1 s1 but different off-rates were included in the model cell. Figure 5 shows the responses of proteins with off-rates of 102, 103 and 104 s1 (i.e. Kd’s of 1, 10 and 100 mM) to the integrated Ca2+ influx signal and to influx due to the opening of a single Ca2+channel. It is clear from the results shown that the high off-rates of the proteins with Kd’s of 10 and 100 mM limited the extent to which these proteins bound Ca2+ during the Ca2+ transients, even though their on-rates for Ca2+ binding were 108 M1 s1. On the other hand, the responses of these proteins more accurately reflected the time course of the square-wave Ca2+ input signal than did the protein with a Kd ¼ 1 mM. Cellular proteins with kinetics represented by those in the model with Kd’s 10 and 100 mM could, therefore, maintain very localized, temporally accurate responses to single channel events.

ING SITES TO LOCALIZED, TRANSIENT CA2+ SIGNALS

The results described above indicate that the kinetics of the interaction of various Ca2+binding proteins with Ca2+ play a major role in restricting the proteins that are able to interact with, or respond to, a given transient Ca2+ signal. For the simulations described below a second stage in a Ca2+-dependent signaling process was also included in the models by modifying them so that the Ca2+-bound form of a protein then bound a second Ca2+ ion. The responses to square-wave Ca2+ input signals were then determined when the kinetics of the second Ca2+ binding step were varied (the first binding step was the same in all cases and had on- and off-rates of 108 M1 s1 and 102 s1, respectively). Figure 6(A)–(C) show the results of simulations for single channel-like Ca2+ transients (open time of 4 ms; [Ca2+]free ¼ 5 mM in the cell element immediately next to the membrane at the influx site). In these simulations, the doubly bound protein species showed a significant increase in concentration when the second Ca2+ bound with an on-rate of 108 M1 s1 [Fig. 6(A)]. There were only slight increases in the doubly bound form of the other two proteins [Fig. 6(B) and (C)] because of their lower on-rates for binding the second Ca2+ ion. Although the responses of the one- and two-site proteins with the highest on-rates (shown in Figs 4(D) and 6(A) respectively) to the 4 ms squarewave Ca2+ influx event appear quite similar, there are some differences. In Fig. 7(A), the results of the simulations in Figs 4(D) and 6(A) were normalized to the maximum change in

(B)

100 80 60 40

10 8 6 4 2 0

20

% saturation

3.4. RESPONSES TO CA2+ TRANSIENTS OF PROTEINS

3.5. RESPONSES OF PROTEINS WITH TWO CA2+-BIND-

[protein-Ca] uM

106 M1 s1 was not able to interact significantly with the Ca2+ transients in either of the simulations. Comparison of these results with those shown in Fig. 2 indicates that the time courses of the responses of fast proteins can reflect, at least qualitatively, the time course of the single channel-like Ca2+ transients. Slower proteins on the other hand, would respond to these types of transients with low-amplitude sustained responses.

0 0 20 40 60 80 100 time ms

Fig. 5. Responses of Ca2+-binding proteins with forward rate constants of 108 M1 s1 but with different Kd’s to integrated (A) and single channel-like (B) Ca2+ influx signals. [protein]free at the beginning of the simulations was 10 mM for all of the proteins. The three traces starting from the top are for proteins with Kd’s of 1, 10 and 100 mM, respectively. The bottom trace in (B) shows the time course of the square-wave Ca2+ input for this simulation.

252

25 0 0 25 50 75 100

50 25 0 0 25 50 75 100

time ms [protein-Ca] uM

(A)

75

% saturation

50

100

10 8 6 4 2 0

time ms

(B)

100

10 8 6 4 2 0

75 50 25 0

% saturation

75

[protein-Ca] uM

100

10 8 6 4 2 0

% saturation

[protein-Ca] uM

G. J. KARGACIN

0 25 50 75 100 (C)

time ms

Fig. 6. Responses of Ca2+-binding proteins with two Ca2+ binding sites to localized, single channel-like [Ca2+] transients. (A)–(C) Predicted responses to a square-wave Ca2+ influx that increased [Ca2+] at a single near-membrane site in the model cell to 5 mM for 4 ms. Ca2+ binding to the first Ca2+-binding site of each protein had an on-rate of 108 M1 s1; on rates for binding to the second Ca2+-binding sites were 108 M1 s1 (A), 107 M1 s1 (B) or 106 M1 s1 (C). KdCa2+ values for binding to the two sites were 1 mM in all cases. In (A)–(C) the open circles show the concentration of the protein with 1 Ca2+ bound; the filled squares the concentration of the protein with 2 Ca2+’s bound; the lower traces show the time course of the square-wave Ca2+ influx for these simulations.

concentration of the Ca2+-bound form of the protein with one binding site. The maximum change in concentration of the doubly bound form of the protein with two Ca2+-binding sites was less than that of the protein with a single binding site and, as expected, the rate at which the concentration of the fully bound form of the two-site protein increased was also less than that for the single site protein. The response of the protein with one Ca2+-binding site to the square-wave Ca2+ influx consisted of an initial rise in Ca2+ binding followed by a decline to a level of Ca2+ binding that remained higher than that at rest during the remainder of the time period modeled. In contrast, the concentration of the doubly bound form of the two-site protein returned to its approximate baseline level more rapidly during the course of the 100 ms simulation. In this sense, formation of the doubly bound protein species more accurately reflected the transient nature of the Ca2+ input to the cell. An important difference between the responses of the one- and two-site binding proteins is the way in which they would be expected to

respond to a second Ca2+ channel opening or to a series of openings. As shown in Fig. 7(B), the responses of the one-site protein summed when a series of such Ca2+ influx events occurred at 80 ms intervals whereas the individual responses of the two-site protein remained essentially the same. Summed responses were evident for the two-site protein when the frequency of the influx train increased; however, there were still differences between the two proteins [Fig. 7(C)]. These results indicate that the Ca2+ binding responses of a single-site protein like the one modeled would more readily saturate during a train of channel openings than would those of a protein similar to the two-site protein. 3.6. SOME GENERALIZATIONS FOR SINGLE CHANNEL-LIKE CA2+ TRANSIENTS

Although the Ca2+ regulatory system considered in the models is complex, it is possible to make some generalizations about the responses of the system to transient Ca2+ signals by considering individual regulatory steps or the diffusion process in isolation. As noted above,

relative increase

CALCIUM-DEPENDENT SIGNAL TRANSDUCTION

1.0 0.8 0.6 0.4 0.2 0.0 0 20 40 60 80 100 time ms

(A)

75 50 25

% saturation

[protein-Ca] uM

100 10 8 6 4 2 0

0 0

(B)

50 100 150 200 250 300 350 400 time ms

(C)

75 50 25

% saturation

[protein-Ca] uM

100 10 8 6 4 2 0

0 0 20 40 60 80 100 time ms

Fig. 7. Comparison of the responses of a Ca2+-binding protein with a single Ca2+ binding site to those of a protein with two Ca2+-binding sites to an increase in [Ca2+]free described by a square-wave. (A) The solid line [taken from Fig. 4(D)] shows the relative increase in concentration of the Ca2+-bound form of a protein with a single Ca2+binding site to a square-wave Ca2+ influx; the dotted line [taken from Fig. 6(A)] shows the relative increase in concentration of the doubly bound form of a protein with two binding sites to the same Ca2+ signal. The responses were normalized to the maximum change in concentration of the Ca2+-bound form of the single-site Ca2+-binding protein. The time course of the square-wave Ca2+ influx is shown at the bottom of the figure. (B) and (C) Comparison of Ca2+ binding to a Ca2+-binding protein with one Ca2+ binding site with the binding of a protein with two Ca2+binding sites to a series of 4 ms square-wave input signals spaced 80 ms (B) or 20 ms (C) apart in time. The responses of the single-site protein are shown by the solid line; the responses of the two-site protein are shown by the dotted lines; the solid line at the bottom of the figures shows the square-wave Ca2+ input signals. The single-site protein in (A)–(C) had an on-rate of 108 M1 s1 and a Kd of 1 mM; the two-site protein had on-rates of 108 M1 s1 and Kd’s of 1 mM for both binding sites. The [protein]free at the beginning of the simulation was 10 mM for both the one and the two-site proteins.

253

the [Ca2+] near a site of localized influx can reach very high levels; however, once influx stops [Ca2+] declines due to diffusion away from the site and because of the regulatory processes described in the models. The simulations indicate, however, that some fast Ca2+-binding proteins could equilibrate with very rapid shortduration Ca2+ signals. Longer signals would be expected to allow slower Ca2+-binding proteins to also equilibrate with Ca2+. By considering solutions to eqn 4 alone in response to step changes in [Ca2+] it is possible to make some generalizations about the time required for various proteins to equilibrate with such changes in [Ca2+]. Let c ¼ [Ca2+], b ¼ [protein]free and a ¼ [Ca protein] in a cell segment prior to the onset of the Ca2+ change. Because the system is in equilibrium before the step change, d[Ca]/ dt ¼ 0 and eqn (4) can be solved to yield: kon cb ¼ koff a:

ð7Þ

After a step change in [Ca2+] occurs, this equilibrium is disturbed and eqn (4) must be integrated to determine a solution. If Dc is the amplitude of the step change in [Ca2+], then, when this occurs: ½Ca2þ  ¼ c þ Dc:

ð8Þ

As the Ca2+ binding protein responds to this change in [Ca2+], the Ca2+-bound form of the protein increases by an amount x, and [Ca2+] and [protein]free decrease by this same amount. Equation 4 can therefore be rewritten as d½Ca dða þ xÞ ¼ ¼ kon ðc þ Dc  xÞ dt dt ð9Þ ðb  xÞ þ koff ða þ xÞ: Expanding this equation and incorporating eqn (7) into it yields: dx ¼ kon Dcb  ðkon ðc þ Dc þ bÞ þ koff Þx þ koff x2 ; dt ð10Þ For the simulations in this study, kon ranged from 106 to 109 M1 s1, DcE106 M and b E 105 M; therefore, konbDc ranges from 105 to 102 M s1 making this constant smaller that the other two constants in the equation. Therefore,

254

G. J. KARGACIN

Table 1 Approximate times required for equilibrium to be reached for Ca2+ binding to proteins with different rate constants kon (M1s1)

koff (s1)

t (s)n

Time to equilibrium calculated (s)w

Time to equilibrium simulations (s)z

109 108 107 106

103 102 103 10

B104 B103 B102 B101

B103 B102 B101 B1

B103 B1.6  102 B0.6  101 B0.9

Computed from eqn (11) assuming a starting [protein]free (b) ofB105 M and a [Ca2+] increase (Dc) of B 106M. w The system was assumed to approach equilibrium when t ¼ 10 t. z Determined when the value of [Ca protein] calculated assuming equilibrium binding was within 5% of that predicted by simulations with the model. n

neglecting the first term on the right in eqn (10) and also making the assumption that c+DcEDc, eqn (10) can be integrated to yield:
 1 t=t x ¼ kon x  e ; t where t¼

1 : kon ðb þ DcÞ þ koff

ð11Þ

When t b t, x approaches zero (i.e. there is no net change in [Ca2+], [Ca protein] or [protein]free) and the system approaches equilibrium. Table 1 gives approximate values for the time required for Ca2+-binding proteins with the rate constants used in the simulations to approach equilibrium following a step increase in [Ca2+] to 5 mM from a starting value of 150 nM. The values for t shown in Table 1 were influenced primarily by kon and the concentration of unbound protein (b) at the start of the simulations (i.e. konb4konDcXkoff). As one approaches the mouth of a channel more closely, or for Ca2+-binding proteins with concentrations much lower than those considered in the simulations, the magnitude of the term konDc could approach or exceed that of konb and the magnitude of the Ca2+ change would, consequently, play a greater role in determining conditions under which one could use equilibrium constants to approximate Ca2+ binding to a given protein. The results in Table 1 predict that a Ca2+-binding protein would require a forward rate constant of B109 M1 s1 to equilibrate with a Ca2+ transient during a time

period of 1–5 ms. The values given in right column of Table 1 are the times that were required for various Ca2+-binding proteins to equilibrate with Ca2+ in simulations with the model following a square-wave influx event (D[Ca2+] ¼ 5 mM, open time ¼ 4 ms). The values predicted by eqn (11) are in good general agreement with those predicted by the model if one assumes that the simplified system described by eqn 11 approaches equilibrium when tX10t. Thus, computing t can provide a rough guideline for determining when equilibrium equations can be used to predict Ca2+-protein interactions following a transient Ca2+ increase. It is also possible to use exact solutions to the diffusion equation alone to estimate the distance over which a localized Ca2+ transient might be expected to significantly influence nearby Ca2+binding proteins. The [Ca2+] at time t at a given distance from a point source that produces an increase in Ca2+ at a constant rate is given by the equation (see Crank, 1975): ½Ca ¼

q r erfc pffiffiffiffiffiffi; 4pDr 2 Dt

ð12Þ

where q is the rate of increase of Ca2+, r the distance from the point source of Ca2+ and D the diffusion coefficient. If Co is the Ca2+ concentration at distance ro from the source at time t, then the concentration of Ca2+ at a second position a distance r from the source is pffiffiffiffiffiffi ro erfcðr=2 DtÞ pffiffiffiffiffiffiffi: ½Car ¼ Co r erfcðro =2 DtÞ

ð13Þ

CALCIUM-DEPENDENT SIGNAL TRANSDUCTION

pffiffiffiffiffiffi If ro is very close to the source, erfcðro =2 DtÞ ) 1 and eqn (13) becomes ½Car ¼ Co

ro r erfc pffiffiffiffiffiffi: r 2 Dt

ð14Þ

When this equation was used to predict the [Ca2+] profile along the length of the cell in the near-membrane space for the simulations shown in Fig. 4(B) and (D), [Ca2+] was predicted to be half-maximal approximately 100 nm away from the site of influx at the 2 ms time point. In the simulations with the model, [Ca2+] was halfmaximal approximately 200 nm away from the influx site. Equation 14 gave a better approximation of the near-membrane [Ca2+] profile for the simulation with the integrated influx shown in the right half of the model cell in Fig. 3. After 2 ms [Ca2+] was predicted by eqn (14) to be halfmaximal at a distance B 100 nm from the site of influx and predicted by the model to be halfmaximal B130 nm from the influx site. The presence of the plasma membrane, which restricts diffusion in one direction in the model cell, would allow more Ca2+ to diffuse along the cell length in simulation with the model and is the reason that the diffusion distances estimated by eqn 14 were less than those predicted by the model (when a single channel-like Ca2+ increase was introduced into the central cytoplasm of the modeled cell segment, both the model and the calculation using eqn (14) predicted half-maximal [Ca2+] B100 nm from the site of the Ca2+ increase after 2 ms). In general, therefore, eqn (14) can be used to obtain rough estimates of the distance over which localized Ca2+ transients are likely to significantly influence Ca2+-binding proteins within a cell. 4. Discussion The model described here was based on the dimensions and the Ca2+ regulatory mechanisms of smooth muscle cells; however, the results are generally applicable to Ca2+ regulation in other cell types as well as to other second messenger systems. From the results of the simulations, it is clear that knowing the equilibrium constant for the interaction of Ca2+ or another second messenger with an intracellular protein is not enough information to determine

255

if the protein is likely to respond to a transient increase in the second messenger or to determine the magnitude of the response. Both the rate constants for binding and the temporal and spatial characteristics of the second messenger signal itself are necessary to make these determinations. This has implications whether one considers the interaction of second messengers with regulatory proteins clustered into microdomains or with proteins that are more widely distributed throughout the cytoplasm. Recent advances in our ability to detect transient Ca2+ signals have led to an increased appreciation for the variety of potential signaling mechanisms that are possible in cells. The results of this study point out the clear need for additional information about the kinetics of Ca2+ binding to intracellular molecules to determine how well these molecules can respond to these different types of signals. The clustering of some regulatory proteins into microdomains as a result of their association with scaffolding proteins such as the caveolins is thought to play an important role in facilitating interactions among these proteins and provides a mechanism by which cellular responses can be localized (see Lockwich et al., 2000 and recent reviews by Okamoto et al., 1998; Shaul & Anderson, 1998; Taggart, 2001). The results of the simulations reported here indicate that more uniformly distributed proteins could also bind Ca2+ or other second messengers to different extents in different parts of a cell in response to localized transient signals and thus give rise to localized cellular responses. In general, the modeled proteins with Kd’s for Ca2+ binding of 1 mM and on-rates of 108 or 107 M1 s1 and a single Ca2+-binding site were able to bind a significant amount of Ca2+ following an integrated Ca2+ input signal (Figs 2 and 3); however, only the proteins with an onrate of 108 M1 s1 were able to do so following square-wave Ca2+ input signals (Fig. 4). Thus, only a limited subset of all cellular Ca2+-binding proteins are likely to be able to respond such transient Ca2+ signals. These results are particularly relevant for predicting the regulatory pathways that may be influenced by single channel openings of plasma membrane or SR/ ER Ca2+ channels. The results further indicate

256

G. J. KARGACIN

that additional signal discrimination can take place in cells depending upon the number of binding sites present on a protein. All of these factors will determine the responses of both clustered proteins and more uniformly distributed proteins to a given transient input signal. The simulations in which Ca2+ bound to oneand two-site binding proteins produced a number of interesting results, especially when one considers the multitude of Ca2+-dependent processes that are mediated by Ca2+ binding to the multiple binding sites on calmodulin. Although the Ca2+-binding proteins with one and two sites and high on-rates responded in a qualitatively similar manner to single squarewave input signals, there were important differences that were revealed when responses to trains of square-wave transients were considered. During the declining phase of the Ca2+ transients, the single-site protein distributes into two pools (free-protein and Ca2+-bound protein), whereas the two-site protein distributes into three pools. Thus, the concentration of the saturated form of the one-site protein remains elevated for a longer period of time than does the fully bound form of the two-site protein and the responses of singlesite proteins are additive over a broader range of input frequencies as shown by the results in Fig. 7. When the Ca2+-bound form of a protein is required for its subsequent interaction with a second signal transduction molecule, the cumulative nature of the Ca2+ binding to single-site proteins would be expected to allow them to interact to a greater extent with a second protein or molecule. The Ca2+ binding to the two-site proteins, on the other hand, was better able to reflect the pulsatile nature of the input signal train. The two N-terminal Ca2+-binding sites of calmodulin have on- and off-rates of 1.6  108 M1 s1 and 405 s1, respectively, and the two C-terminal binding sites have on- and off-rates of 2.3  106 M1 s1 and 2.4 s1, respectively (Davis et al., 1999; Johnson et al., 1996). Thus, if the activation or inactivation of a Ca2+/calmodulin-dependent protein requires saturation of the 4 Ca2+-binding sites on calmodulin, this protein would be unlikely to respond to a singlechannel-like Ca2+ transient even if the protein was localized near a Ca2+ influx site. A protein that directly bound Ca2+ with a Kd E1 mM and

an on-rate E108 M1 s1, on the other hand, would be expected to be more fully affected. These considerations are important for predicting the relative responses of Ca2+- and Ca2+/ calmodulin-dependent ion channels localized near plasma membrane Ca2+ channels or plasma membrane channels separated from SR Ca2+ release channels by a restricted diffusion space. Both Ca2+-dependent and Ca2+/calmodulin-dependent channels would be expected to respond to prolonged, global changes in [Ca2+]i; however, only the channels modulated by direct Ca2+ binding might be expected to respond to localized, brief transient [Ca2+] changes. The Ca2+-binding proteins in the modeled cell segment were assumed to be fixed and uniformly distributed throughout the cytoplasm. Many cytoplasmic proteins in living cells are associated with the cytoskeleton and would be expected to behave in a manner similar to that of the fixed proteins included in the model. Although other Ca2+-binding proteins such as calmodulin are thought to be mobile, the cytoplasmic diffusion coefficients for these proteins are likely to be at least an order of magnitude less than the Ca2+ diffusion coefficient (see Luby-Phelps et al., 1986; Smith et al., 1996; Zhou & Neher, 1993). Thus, it is unlikely that, for the time intervals considered here, a significant movement of the Ca2+-bound form of mobile proteins would occur in a cell. Small cytoplasmic Ca2+-binding molecules could be considerably more mobile, however. A number of investigators have modeled the effects of such small molecules (particularly those added exogenously) on intracellular Ca2+ transients and the effects of mobile ion-sensitive fluorescent probes on the outcome of experiments designed to study intracellular Ca2+ signaling. The results of these studies indicate that the effective diffusion coefficient for Ca2+ in cells can be increased significantly (2 – 3 fold; see Wagner & Keizer, 1994; Zhou & Neher, 1993) by high concentrations of mobile buffers. Thus, a high mobility for a Ca2+-binding molecule could enhance the spread of its regulatory signal away from its site of origin. It is also important to consider cell size when evaluating the potential for significant interaction between Ca2+-sensitive proteins and a localized, transient change in [Ca2+]. The cell

CALCIUM-DEPENDENT SIGNAL TRANSDUCTION

modeled here had a radius of 1 mm. For the Ca2+ transient shown in Fig. 4(B), [Ca2+] was half-maximal 200 nm from the membrane (10% of the cell diameter) and 1/10 maximal 750 nm from the membrane after 2 ms. In this case, therefore, one would not expect the transient to significantly influence central cytoplasmic proteins as it would in a much smaller cell. The results of this study point out a number of important considerations relevant to the role that transient, localized changes in second messenger concentration can play in cell regulation. The results also point to the value of using computational models to examine regulatory functions in cells. Mathematical models of the type used in these simulations are conceptually straightforward and are very versatile in their application (e.g. any regulatory process can be located anywhere within a cell and turned on or off at any time point during a simulation; buffer or binding protein concentrations can be varied; diffusion coefficients can be varied). The models are important tools for answering questions about cell regulation in a context that allows one to manipulate various regulatory processes and to predict the results of these manipulations on cell function. The models also provide an excellent framework for posing experimental questions and for interpreting experimental results. The author thanks Dr Margaret E. Kargacin, Dr. Michael P. Walsh and Ms Marnie L. Dodds for their critical comments on this manuscript. This work was supported by the American Heart Association, the Canadian Institutes of Health Research and the Heart and Stroke Foundation of Alberta. The author is an Alberta Heritage Foundation Senior Scholar. REFERENCES Allbritton, N. L., Meyer, T. & Stryer L. (1992). Range of messenger action of calcium ion and inositol 1,4,5trisphosphate. Science (Wash DC) 258, 1812–1815. Asada Y., Yamazawa, T., Hirose, K., Takasaka, T. & Iino, M. (1999). Dynamic Ca2+ signalling in rat arterial smooth muscle cells under the control of local renin– angiotensin system. J. Physiol. (Lond) 521.2, 497–505. Backx P. H., de Tombe, P. P., Van Deen, J. H. K., Mulder B. J. M. & ter Keurs H. E. D. J. (1989). A model of propagating calcium-induced calcium release mediated by calcium diffusion. J. Gen. Physiol. 93, 963–977.

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