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Calcium dynamics in cardiac myocytes: a model for drugs effect description
350 Simulation Modelling Practice and Theory 12 (2004) 93–104 www.elsevier.com/locate/simpat
Calcium dynamics in cardiac myocytes: a model for drugs effect description Francßois Rocaries a,*, Yskandar Hamam a,*, Rossany Roche a,b, Marisol Delgado b, Rosalba Lamanna b, Francßoise Pecker c, Catherine Pavoine c, Hubert Lorino d a
b
Laboratoire A2SI Groupe ESIEE, 2 Boulevard Bd Blaise Pascal, B.P. 99, Cite Descartes, F93162 Noisy le Grand Cedex, France Departamiento de Processos y Systemas, Universidad Simon BOLIVAR, Caracas, Venezuela c Unite INSERM 99, CHU Henri MONDOR, F94010 Creteil, France d Unite INSERM 492, CHU Henri MONDOR, F94010 Creteil, France
Received 16 September 2002; received in revised form 26 August 2003; accepted 12 September 2003
Abstract Ca2þ dynamics and handling in cardiomyocytes are critical for both metabolism and contraction of the heart. Action potentials involve membrane and subcellular components. In response to membrane depolarization during an action potential, L-type Ca2þ channels open, allowing the influx of Ca2þ into a restricted subspace where it triggers Ca2þ release from the sarcoplasmic reticulum (SR) via ryanodine-sensitive Ca2þ channels (RyRs). This results in a global increase in the cytosolic Ca2þ concentration which triggers contraction through the contractile proteins. Relaxation follows the reuptake of Ca2þ into the SR by the Ca2þ pump in the SR (SERCA) or its extrusion from the cell essentially ensured by the Na–Ca exchanger. A model of Ca2þ handling model developed in a precedent work give us quite good results but its not sufficient to explain the cell behaviour in presence of some specific substance like arachidonic acid. In order to explain this difference the authors have developed a new model taking into account a modification of some specific components of the SR. 2004 Elsevier B.V. All rights reserved. Keywords: Cardiac cell; Dynamic modeling; Compartmental modeling; Calcium dynamics; Arachidonic acid; Caffeine; Ruthenium red
*
Corresponding authors. Tel.: +33-145926607. E-mail addresses: [email protected] (F. Rocaries), [email protected] (Y. Hamam), [email protected] (R. Roche), [email protected] (M. Delgado), [email protected] (R. Lamanna), [email protected] (F. Pecker), [email protected] (C. Pavoine), [email protected] (H. Lorino). 1569-190X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2003.09.002
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F. Rocaries et al. / Simulation Modelling Practice and Theory 12 (2004) 93–104
1. Introduction In a precedent work Hamam et al. [2] we developed a model of Ca2þ handling based on that developed by Tang and Othmer [6] after determining SERCA activity and RyR conductance. The results obtained with this model are quite good but are not sufficient to explain the cell behaviour in presence of some specific substance like arachidonic acid. In order to explain this difference we have developed a new model taking into account a modification of some specific component of the SR. Our first approach was to identify which component of the Ca2þ handling Ca2þ channels (RyRs) or Ca2þ pump (SERCA) was affected by the presence of this specific substance. This was made by a simple analysis of the experimental data. The main difficulty of the study was to link the results of this analysis to those existing in the literature. Another difficulty was to find what kind of biochemical relationship could produce such dynamical behavior. This paper deals with the various elements of this study. The simulation results presented in this paper are obtained using the package Xmath/SystemBuild. (1) In the first part we present the experimental data obtained in the presence of arachidonic acid or some other drug like caffeine or ruthenium red in the cardiomyocyte and the functional analysis with regards to this results. This analysis leads to the conclusion that the component affected is essentially the Ca2þ pump (SERCA) in the case of arachidonic acid and the RyRs channel for both caffeine and ruthenium red. (2) In the second part a mathematical representation of this particular dynamical behavior is proposed and for each of the cell components, a new extension of the model is proposed in order to improve the simulation results. The results of the simulation model are compared to the experimental data. (3) In the third part an experimental verification using another specific substance (thapsigargin) known as an inhibitor of SERCA is presented and the data of both experiment are compared and discussed.
2. Analysis of the experimental data 2.1. Measurement of Ca2þ concentration Ca2þ imaging, developed by A. Trautmann in collaboration with the IMSTAR Co. (Paris, France) was essentially as described by Sauvadet et al. [5]. Briefly, a Nikon diaphot inverter microscope with epifluorescence was used. The light from a 100W Xenon lamp was filtered alternatively through 350 and 380 nm filters. Mag Fura-2 of Fura-2 fluorescence of 7–10 cells in the microscopic field (Nikon UV-Fluor X40 objective) was filtered at 510 nm and recorded by an intensified CCD Photonic Science camera. In the present study, each fluorescence image was the average of 2 images, in order to improve the signal-to-noise ratio. A Ca2þ ratio image was regis-
F. Rocaries et al. / Simulation Modelling Practice and Theory 12 (2004) 93–104
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tered every 0.3 s. In Mag Fura-2 experiments, Ca2þ in the SR was calculated according to the formula of Grynkiewicz et al. [1]. 2.2. Arachidonic acid action As may be seen in the first row of Fig. 5, after the introduction of arachidonic acid in the cell the amplitude and the average of Ca2þ transient decreased progressively and 38 min after the beginning of the experiment the activity of the electrical stimulated cell is nearly null. Considering the cell elements, arachidonic acid could have an effect on SERCA pump or on RyRs channel. However, as we present in our previous work, Hamam et al. [2], it seems that the essential effect may concern the activity of SERCA pump. In fact, if we look through the model, on the effect of the RyRs channel activity, the opening of these channels produces a notable variation of the dynamic behavior of Ca2þ transients. This effect may be seen on the experimental data, essentially the decrease of both amplitude and average of Ca2þ transients (Fig. 1).
0.14 0.12 0.1 0.08 0.06 10
15
20 (a) Time in s
25
30
0.11 0.105 0.1 0.095 0.09 0.085 0.08 2280
2285
2290 (b) Time in s
2295
2300
2285
2290 (c) Time in s
2295
2300
2205
2210
0.14 0.12 0.1 0.08 0.06 2280
0.1 0.099 0.098 0.097 0.096 2190
2195
2200 (d) Time in s
Fig. 1. (a) Control cell (measured), (b) cell activity (measured) after 10 min, (c) cell control (simulated), (d) cell activity (simulated) after 10 min.
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2.3. Caffeine action Caffeine, despite its various targets (including phosphodiesterase, see [9]), and its complex effects (see [10]), due to the reversibility of its effect in contrast to ryanodine, is the most commonly used RyRs channel activator. Caffeine activates RyRs channels by increasing both the frequency and duration of open events without affecting the unit conductance [11]. As shown in Fig. 2, exposure of electrically stimulated cardiomyocytes to high caffeine concentration, 10 mM caffeine, led to a transient increase in diastolic Ca2þ as previously reported [12] with no change in either the amplitude or the outline of Ca2þ transients. After 20 min, diastolic Ca2þ returned to the initial level, the duration of Ca2þ transients increased and cardiomyocytes progressively lost responsiveness to electrical stimulation. 2.4. Ruthenium red action Ruthenium red when used at 0.5–2 lM is a very specific inhibitor of RyRs [9,13]. As shown in Fig. 3, exposure to 2 lM ruthenium red elicited a rapid disruption of Ca2þ cycling of electrically stimulated cardiomyocytes. This effect was reproduced by tetracaine, another RyR blocker (not shown [14]).
3 Control
Caffeine 10 mM (10 min)
Caffeine 10 mM (20 min)
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2
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1 0
10
0
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20 Time (s)
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620 630 Time (s)
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1250
Simulation results
3
2.5
2
1.5
1 Time (s)
630
Time (s)
1230
Time (s)
Fig. 2. Consequences of RyR activation on cytosolic Ca2þ cycling in electrically stimulated cells. Comparison of experimental data (top part) with simulation (bottom part).
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1.2 2uM Ruthenium Red
1.1
F360/F380
1 0.9 0.8 0.7 0.6 0.5 0
50
100
150
200
Time (s) 1.2 RyR inhibition Simulation Results
1.1 1 0.9 0.8 0.7 0.6 0.5
0
50
100 Time (s)
150
200
Fig. 3. Consequences of RyR inhibition on cytosolic Ca2þ cycling in electrically stimulated cells. Comparison of experimental data (top part) with simulation (bottom part).
3. New mathematical representation Usually the effect of a specific drug should be represented by a diffusion equation but in our specific case we are essentially interested by the effect of this drug both on SERCA pump or RyRs Channels activity. This effect may be represented as a function of the average concentration of arachidonic acid in the cell. To stay consistent with our model the same concept is used (e.g. a compartments model). Two hypothesis have been tested in order to evaluate the most probable one. In the first, it is assumed that it affect the SERCA pump and in the second it is assumed that its influence is on the RyRs channels. 3.1. Action on the SERCA pump In this case we can consider that we have a source of arachidonic acid represented by a concentration csaa and a tank (the cytosol) which will be filled by this source. At the same time we assume that the value of the maximum rate of the SERCA pump p1 is a linear function of the average concentration of arachidonic acid caa ðtÞ. Thus we obtain the following system of equations: c_aa ðtÞ ¼ Kðcaa ðtÞ csaa Þ
ð1Þ
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and
caa ðtÞ p1 ¼ p10 1 caa max where
if caa ðtÞ 6 caa max
else p1 ¼ 0
ð2Þ
• K is a transfer coefficient between the source and the tank. • p10 is the maximum rate value. • caa max is the maximum concentration value of arachidonic acid bearable by the cell. So we just add this system to the following model described in Hamam et al. [2] and schematically represented by Fig. 4. The dynamic differential equations for this model are: x_ 1 ¼ l1 x2 þ l2 ð1 x1 x2 x3 Þ ðl1 þ l2 Þx1 x4
ð3Þ
x_ 2 ¼ l1 x2 þ l2 x3 þ ðl1 x1 l2 x2 Þx4
ð4Þ
x_ 3 ¼ ðl1 þ l2 Þx3 þ ðl2 x2 þ l1 ð1 x2 x3 ÞÞx4
ð5Þ
x_ 4 ¼ ðVr g1 þ g2 þ Vr Chx2 Þx4 þ ðVr g1 þ Vr Chx2 Þx5 ðQðx4 Þ þ Vr P ðx4 ÞÞ þ g2 C0 þ J ðtÞ x_ 5 ¼ ðg1 þ Chx2 Þx4 ðg1 þ Chx2 Þx5 þ P ðx4 Þ
ð6Þ ð7Þ
where P ðx4 Þ ¼
p1 x24 x24 þ p22
ð8Þ
Qðx4 Þ ¼
q1 x24 x24 þ q22
ð9Þ
and
where the variables, constants and parameters are
Fig. 4. Compartment Scheme of the cardiac cell.
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(1) Variables: • x1 : proportion of activatable ryanodine channels (dimensionless); • x2 : proportion of open ryanodine channels (dimensionless); • x3 : proportion of closed ryanodine channels (dimensionless); • x4 : cytosolic calcium concentration ðMÞ ¼ Ccy ; • x5 : calcium concentration inside sarcoplasmic reticulum ðMÞ ¼ Csr ; • caa : arachidonic acid concentration in the cell; • p1 : maximum rate of the sarcoplasmic pump (M s1 ); • J ðtÞ: the excitation signal into the cytosol (represented as a Ca2þ injection); (2) constants: • g1 : leakage conductance between the sarcoplasmic reticulum and the cytosol (s1 ); • g2 : leakage conductance between the cytosol and extracellular medium (s1 ); • Ch: conductance of the ryanodine channels (s1 ); • p2 : threshold concentration of the sarcoplasmic pump (M); • q1 : maximum rate of the sarcolemma pump (M s1 ); • q2 : threshold concentration of the sarcolemma pump (M); • l1 , l2 : RyR channel binding rates; • l1 , l2 : RyR channel dissociation rates; (3) other parameters: • C0 : extracellular calcium concentration; • Vr : cytosol to sarcoplasmic reticulum volume ratio. Eqs. (3)–(5) describe the opening of the RyRs channels, whereas Eqs. (6) and (7) give the dynamics of the Ca2þ concentration in both the cytosol and the sarcoplasmic reticulum. The result obtained are shown in Fig. 5 and it may be noticed that the simulation fits well with the experimental data. 3.2. Extension of the model In certain cases, the external signal of Ca2þ , J ðtÞ, take a major importance in the representation of the cytosolic calcium concentration. In order to avoid this problem we introduce a third compartment corresponding to the so called ‘‘subspace’’ in Jafri et al. [8]. This part of the model is represented by the following equation: x_ 4 ¼ ðVr g1 þ g2 þ Vr Chx2 þ g3 Þx4 þ ðVr g1 þ Vr Chx2 Þx5 ðQðx4 Þ þ Vr P ðx4 ÞÞ þ g2 C 0 þ g3 x 6
ð10Þ
x_ 6 ¼ g3 ðVs x4 x6 Þ þ J ðtÞ
ð11Þ
with the following definition for the new variable and parameters: (1) Variables: • x2 : proportion of open ryanodine channels (dimensionless); • x4 : cytosolic calcium concentration ðMÞ ¼ Ccy ;
100
F. Rocaries et al. / Simulation Modelling Practice and Theory 12 (2004) 93–104 0.8 Arachidonic acid 10uM (1min)
Arachidonic acid 10uM (12 min)
Arachidonic acid 10uM (38 min)
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Simulation results
0.8
0.6
0.4
0.2
0 0
10
20 Time (s)
Time (s)
Time (s)
Fig. 5. Arachidonic Acid 10 lM––first row experimental result––second row simulation result.
• x5 : calcium concentration inside sarcoplasmic reticulum ðMÞ ¼ Csr ; • x6 : calcium concentration inside subspace ðMÞ ¼ Csr ; (2) constants: • g1 : leakage conductance between the sarcoplasmic reticulum and the cytosol (s1 ); • g2 : leakage conductance between the cytosol and extracellular medium (s1 ); • g3 : leakage conductance between the cytosol and subspace (s1 ); • Ch: conductance of the ryanodine channels (s1 ); (3) other parameters: • C0 : extracellular calcium concentration.
3.3. Action on the RyRs channel In this case it is more complicated to describe the effect of arachidonic acid on the proportion of open channels. The system is simulated with a fixed proportion of open ryanodine channels (100%). The only interesting result we can extract from a simulation of this type is the steady state behavior. As may be seen in the Fig. 6 the aspect of the simulation results in term of dynamical behavior is quite different from the experimental data, shown in Fig. 5 (upper row last column).
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0.5
Ca++ Concentration
0.4
0.3
0.2
0.1 2300
2310
2320 time
2330
2340
Fig. 6. Steady state simulation result after 38 min.
3.4. Action on the RyRs channel: a model Another model we can (and we have to!) use for an action on the RyRs channel can be described as follows. Considering that a specific drug a 1 is able to change the value of the conductance of the ryanodine channels, the following model (very similar to the one presented in Eq. (2)) can be used. As described in Section 2.1 the drug a is a source represented by a concentration csa and the cytosol will be filled by this source. At the same time we assume that the value of the RyRs channel conductance Ch is a linear function of the average concentration of a product ca ðtÞ. Thus we obtain the following system of equations: c_a ðtÞ ¼ Kðca ðtÞ csa Þ
ð12Þ
and ca ðtÞ Ch ¼ Ch10 1 ca max
if ca ðtÞ 6 ca max
else Ch ¼ 0
ð13Þ
where • K is a transfer coefficient between the source and the tank. • Ch10 is the maximum conductance. • caa max is the maximum concentration value of a product bearable by the cell.
1
Like caffeine or ruthenium red.
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Thapsigargin 1uM (12 min)
Thapsigargin 1uM (20 min)
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710
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Simulation results
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0
10
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30
40
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710
720 Time (s)
730
740 1200
1210
1220 Time (s)
Fig. 7. Thapsigargin 1 lM––first row experimental data––second row simulation result.
4. Experimental verification with thapsigargin We verify the validity of our hypothesis by using a well known specific inhibitor of the SERCA pump: thapsigargin. The effect of thapsigargin (1 lM) [4,7] was examined on Ca2þ cycling of electrically stimulated cardiomyocytes (Fig. 7, top). The amplitude of Ca2þ transients decreased progressively, as previously described by Kirby et al. [4], and finally cells lost responsiveness to electrical stimulation. This may be attributed to a progressive SR Ca2þ depletion [3]. The mathematical modelling of SERCA inhibition is shown in Fig. 7 (bottom). We use, in this case, the same model as that for arachidonic acid but with different coefficient values in Eq. (2) and with the evolution of thapsigargin concentration instead of arachidonic acid concentration. It is important to note that in this specific case the concentration of thapsigargin probably reaches the maximum concentration value bearable by the cell. Its also important to notice that the experiment was made on another set of cell than for acid arachidonic. Thus is not surprising that the values are quite different.
5. Conclusions In this paper the model previously proposed by the authors is extended to account for the effect of drugs (arachidonic acid or thapsigargin) on the behavior of the heart
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cells. This model is used to explain the effect and relate it to the physiological phenomena in the cell. Some of the constants of the previous model become variables and change in time as a function of the drug concentration. The contribution of this work is twofold: • the extension of the heart cell model in order to account for pathological cases. • the confirmation of the hypothesis concerning the effect of the drug on the various components of the cell. The numerical results presented in this paper are obtained using the package Xmath/SystemBuild. This, however, is easily extended to MatLab/Simulink and SciLab/Scicos software. In fact the model exist for both environment.
Acknowledgements The authors would like to thanks the ECOS-NORD French program and his Venezuelan counterpart CONICIT, Fondacion Gran Mariscal de AYACUCHO and CDCHT for their financial support.
Appendix A. Parameter’s values used for simulation Localisation
Numerical values 1 1
Definition
RyR channels
l1 l1 l2 l2 Ch
0.76 lM s 0.084 s1 1.5 lM1 s1 0.08 s1 0.8 s1
Binding rate Dissociation rate Binding rate Dissociation rate RyR channels conductance
SR pump
p1 p2
0.15 lM s1 0.11 lM
Maximum rate Threshold concentration
Sarcolemmal efflux
q1 q2
0.013 lM s1 0.006 lM
Maximum rate Threshold concentration
Other parameters
C0
1500 lM
Vr g1 g2
0.185 0.04 s1 1.063 103 s1
Extra cellular Ca2þ concentration Volume ratio Leakage conductance Leakge conductance
104
F. Rocaries et al. / Simulation Modelling Practice and Theory 12 (2004) 93–104
References [1] G. Grynkiewicz, M. Poenie, R.Y. Tsien, A new generation of Ca2þ indicators with greatly improved fluorescence properties, J. Biol. Chem. 260 (1985) 3440–3450. [2] Y. Hamam, F. Pecker, H. Lorrino, F. Rocaries, C. Pavoine, R. Natowicz, Identification and modeling of calcium dynamics in cardiac myocites, Simul. Practice Theory J. 8 (2000) 3–15. [3] A.M. Janczewski, E.G. Lakatta, Thapsigargin inhibits Ca2þ uptake, and Ca2þ depletes sarcoplasmic reticulum in intact cardiac myocytes, Am. J. Physiol. 265 (1993) 517–522. [4] M.S. Kirby, Y. Sagara, S. Gaa, G. Inesi, W.J. Lederer, T.B. Rogers, Thapsigargin inhibits contraction and Ca2þ transient in cardiac cells by specific inhibition of the sarcoplasmic reticulum Ca2þ pump, J. Biol. Chem. 267 (1992) 12545–12551. [5] A. Sauvadet, F. Pecker, C. Pavoine, Inhibition of sarcolemmal Ca2þ pump in embryonic chick heart cells by mini-glucagon, Cell Calcium 18 (1995) 76–85. [6] Y. Tang, H.G. Othmer, A model of calcium dynamics in cardiac myocites based on the kinetics of ryanodine-sensitive calcium channels, Biophys. J. 67 (1994) 2223–2235. [7] O. Thastrup, P.J. Cullen, B.K. Drobak, M.R. Hanley, A.P. Dawson, Thapsigargin, a tumor promoter, discharges intracellular Ca2þ stores by specific inhibition of the endoplasmic reticulum Ca2þ -ATPase, Proc. Natl. Acad Sci. 87 (1990) 2466–2470. [8] S. Jafri, J.J. Rice, R.L. Winslow, Cardiac Ca2þ Dynamics: the role of ryanodine receptor adaptation and sarcoplasmic reticulum load, Biophys. J. 74 (1998) 1149–1168. [9] C.W. Taylor, L.M. Broad, Pharmacological analysis of intracellular Ca2þ signalling: problems and pitfalls, Trends Pharmacol. Sci. 19 (1998) 370–375. [10] S.C. O’Neill, D.A. Eisner, A mechanism for the effects of caffeine on Ca2þ release during diastole and systole in isolated rat ventricular myocytes, J. Physiol. (London) 430 (1990) 519–536. [11] E. Rousseau, G. Meissner, Single cardiac sarcoplasmic reticulum Ca2þ -release channel: activation by caffeine, Am. J. Physiol. 256 (1989) H328–H333. [12] G.L. Smith, M. Valdeolmillos, D.A. Eisner, D.G. Allen, Effects of rapid application of caffeine on intracellular calcium concentration in ferret papillary muscles, J. Gen. Physiol. 92 (1988) 351–368. [13] B.E. Ehrlich, E. Kaftan, S. Bezprozvannaya, I. Bezprozvanny, The pharmacology of intracellular Ca2þ -release channels, Trends Pharmacol. Sci. 15 (1994) 145–149. [14] C.L. Overend, S.C. O’Neill, D.A. Eisner, The effect of tetracaine on stimulated contractions, sarcoplasmic reticulum Ca2þ content and membrane current in isolated rat ventricular myocytes, J. Physiol. (London) 507 (1998) 759–769.
Calcium dynamics in cardiac myocytes: a model for drugs effect description Francßois Rocaries a,*, Yskandar Hamam a,*, Rossany Roche a,b, Marisol Delgado b, Rosalba Lamanna b, Francßoise Pecker c, Catherine Pavoine c, Hubert Lorino d a
b
Laboratoire A2SI Groupe ESIEE, 2 Boulevard Bd Blaise Pascal, B.P. 99, Cite Descartes, F93162 Noisy le Grand Cedex, France Departamiento de Processos y Systemas, Universidad Simon BOLIVAR, Caracas, Venezuela c Unite INSERM 99, CHU Henri MONDOR, F94010 Creteil, France d Unite INSERM 492, CHU Henri MONDOR, F94010 Creteil, France
Received 16 September 2002; received in revised form 26 August 2003; accepted 12 September 2003
Abstract Ca2þ dynamics and handling in cardiomyocytes are critical for both metabolism and contraction of the heart. Action potentials involve membrane and subcellular components. In response to membrane depolarization during an action potential, L-type Ca2þ channels open, allowing the influx of Ca2þ into a restricted subspace where it triggers Ca2þ release from the sarcoplasmic reticulum (SR) via ryanodine-sensitive Ca2þ channels (RyRs). This results in a global increase in the cytosolic Ca2þ concentration which triggers contraction through the contractile proteins. Relaxation follows the reuptake of Ca2þ into the SR by the Ca2þ pump in the SR (SERCA) or its extrusion from the cell essentially ensured by the Na–Ca exchanger. A model of Ca2þ handling model developed in a precedent work give us quite good results but its not sufficient to explain the cell behaviour in presence of some specific substance like arachidonic acid. In order to explain this difference the authors have developed a new model taking into account a modification of some specific components of the SR. 2004 Elsevier B.V. All rights reserved. Keywords: Cardiac cell; Dynamic modeling; Compartmental modeling; Calcium dynamics; Arachidonic acid; Caffeine; Ruthenium red
*
Corresponding authors. Tel.: +33-145926607. E-mail addresses: [email protected] (F. Rocaries), [email protected] (Y. Hamam), [email protected] (R. Roche), [email protected] (M. Delgado), [email protected] (R. Lamanna), [email protected] (F. Pecker), [email protected] (C. Pavoine), [email protected] (H. Lorino). 1569-190X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2003.09.002
94
F. Rocaries et al. / Simulation Modelling Practice and Theory 12 (2004) 93–104
1. Introduction In a precedent work Hamam et al. [2] we developed a model of Ca2þ handling based on that developed by Tang and Othmer [6] after determining SERCA activity and RyR conductance. The results obtained with this model are quite good but are not sufficient to explain the cell behaviour in presence of some specific substance like arachidonic acid. In order to explain this difference we have developed a new model taking into account a modification of some specific component of the SR. Our first approach was to identify which component of the Ca2þ handling Ca2þ channels (RyRs) or Ca2þ pump (SERCA) was affected by the presence of this specific substance. This was made by a simple analysis of the experimental data. The main difficulty of the study was to link the results of this analysis to those existing in the literature. Another difficulty was to find what kind of biochemical relationship could produce such dynamical behavior. This paper deals with the various elements of this study. The simulation results presented in this paper are obtained using the package Xmath/SystemBuild. (1) In the first part we present the experimental data obtained in the presence of arachidonic acid or some other drug like caffeine or ruthenium red in the cardiomyocyte and the functional analysis with regards to this results. This analysis leads to the conclusion that the component affected is essentially the Ca2þ pump (SERCA) in the case of arachidonic acid and the RyRs channel for both caffeine and ruthenium red. (2) In the second part a mathematical representation of this particular dynamical behavior is proposed and for each of the cell components, a new extension of the model is proposed in order to improve the simulation results. The results of the simulation model are compared to the experimental data. (3) In the third part an experimental verification using another specific substance (thapsigargin) known as an inhibitor of SERCA is presented and the data of both experiment are compared and discussed.
2. Analysis of the experimental data 2.1. Measurement of Ca2þ concentration Ca2þ imaging, developed by A. Trautmann in collaboration with the IMSTAR Co. (Paris, France) was essentially as described by Sauvadet et al. [5]. Briefly, a Nikon diaphot inverter microscope with epifluorescence was used. The light from a 100W Xenon lamp was filtered alternatively through 350 and 380 nm filters. Mag Fura-2 of Fura-2 fluorescence of 7–10 cells in the microscopic field (Nikon UV-Fluor X40 objective) was filtered at 510 nm and recorded by an intensified CCD Photonic Science camera. In the present study, each fluorescence image was the average of 2 images, in order to improve the signal-to-noise ratio. A Ca2þ ratio image was regis-
F. Rocaries et al. / Simulation Modelling Practice and Theory 12 (2004) 93–104
95
tered every 0.3 s. In Mag Fura-2 experiments, Ca2þ in the SR was calculated according to the formula of Grynkiewicz et al. [1]. 2.2. Arachidonic acid action As may be seen in the first row of Fig. 5, after the introduction of arachidonic acid in the cell the amplitude and the average of Ca2þ transient decreased progressively and 38 min after the beginning of the experiment the activity of the electrical stimulated cell is nearly null. Considering the cell elements, arachidonic acid could have an effect on SERCA pump or on RyRs channel. However, as we present in our previous work, Hamam et al. [2], it seems that the essential effect may concern the activity of SERCA pump. In fact, if we look through the model, on the effect of the RyRs channel activity, the opening of these channels produces a notable variation of the dynamic behavior of Ca2þ transients. This effect may be seen on the experimental data, essentially the decrease of both amplitude and average of Ca2þ transients (Fig. 1).
0.14 0.12 0.1 0.08 0.06 10
15
20 (a) Time in s
25
30
0.11 0.105 0.1 0.095 0.09 0.085 0.08 2280
2285
2290 (b) Time in s
2295
2300
2285
2290 (c) Time in s
2295
2300
2205
2210
0.14 0.12 0.1 0.08 0.06 2280
0.1 0.099 0.098 0.097 0.096 2190
2195
2200 (d) Time in s
Fig. 1. (a) Control cell (measured), (b) cell activity (measured) after 10 min, (c) cell control (simulated), (d) cell activity (simulated) after 10 min.
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2.3. Caffeine action Caffeine, despite its various targets (including phosphodiesterase, see [9]), and its complex effects (see [10]), due to the reversibility of its effect in contrast to ryanodine, is the most commonly used RyRs channel activator. Caffeine activates RyRs channels by increasing both the frequency and duration of open events without affecting the unit conductance [11]. As shown in Fig. 2, exposure of electrically stimulated cardiomyocytes to high caffeine concentration, 10 mM caffeine, led to a transient increase in diastolic Ca2þ as previously reported [12] with no change in either the amplitude or the outline of Ca2þ transients. After 20 min, diastolic Ca2þ returned to the initial level, the duration of Ca2þ transients increased and cardiomyocytes progressively lost responsiveness to electrical stimulation. 2.4. Ruthenium red action Ruthenium red when used at 0.5–2 lM is a very specific inhibitor of RyRs [9,13]. As shown in Fig. 3, exposure to 2 lM ruthenium red elicited a rapid disruption of Ca2þ cycling of electrically stimulated cardiomyocytes. This effect was reproduced by tetracaine, another RyR blocker (not shown [14]).
3 Control
Caffeine 10 mM (10 min)
Caffeine 10 mM (20 min)
F360/F380
2.5
2
1.5
1 0
10
0
10
20 Time (s)
30
40
600
610
620 630 Time (s)
640
650
1200
1210
1220 1230 Time (s)
1240
1250
20
30
40
600
610
620
640
650
1200
1210
1220
1240
1250
Simulation results
3
2.5
2
1.5
1 Time (s)
630
Time (s)
1230
Time (s)
Fig. 2. Consequences of RyR activation on cytosolic Ca2þ cycling in electrically stimulated cells. Comparison of experimental data (top part) with simulation (bottom part).
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1.2 2uM Ruthenium Red
1.1
F360/F380
1 0.9 0.8 0.7 0.6 0.5 0
50
100
150
200
Time (s) 1.2 RyR inhibition Simulation Results
1.1 1 0.9 0.8 0.7 0.6 0.5
0
50
100 Time (s)
150
200
Fig. 3. Consequences of RyR inhibition on cytosolic Ca2þ cycling in electrically stimulated cells. Comparison of experimental data (top part) with simulation (bottom part).
3. New mathematical representation Usually the effect of a specific drug should be represented by a diffusion equation but in our specific case we are essentially interested by the effect of this drug both on SERCA pump or RyRs Channels activity. This effect may be represented as a function of the average concentration of arachidonic acid in the cell. To stay consistent with our model the same concept is used (e.g. a compartments model). Two hypothesis have been tested in order to evaluate the most probable one. In the first, it is assumed that it affect the SERCA pump and in the second it is assumed that its influence is on the RyRs channels. 3.1. Action on the SERCA pump In this case we can consider that we have a source of arachidonic acid represented by a concentration csaa and a tank (the cytosol) which will be filled by this source. At the same time we assume that the value of the maximum rate of the SERCA pump p1 is a linear function of the average concentration of arachidonic acid caa ðtÞ. Thus we obtain the following system of equations: c_aa ðtÞ ¼ Kðcaa ðtÞ csaa Þ
ð1Þ
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and
caa ðtÞ p1 ¼ p10 1 caa max where
if caa ðtÞ 6 caa max
else p1 ¼ 0
ð2Þ
• K is a transfer coefficient between the source and the tank. • p10 is the maximum rate value. • caa max is the maximum concentration value of arachidonic acid bearable by the cell. So we just add this system to the following model described in Hamam et al. [2] and schematically represented by Fig. 4. The dynamic differential equations for this model are: x_ 1 ¼ l1 x2 þ l2 ð1 x1 x2 x3 Þ ðl1 þ l2 Þx1 x4
ð3Þ
x_ 2 ¼ l1 x2 þ l2 x3 þ ðl1 x1 l2 x2 Þx4
ð4Þ
x_ 3 ¼ ðl1 þ l2 Þx3 þ ðl2 x2 þ l1 ð1 x2 x3 ÞÞx4
ð5Þ
x_ 4 ¼ ðVr g1 þ g2 þ Vr Chx2 Þx4 þ ðVr g1 þ Vr Chx2 Þx5 ðQðx4 Þ þ Vr P ðx4 ÞÞ þ g2 C0 þ J ðtÞ x_ 5 ¼ ðg1 þ Chx2 Þx4 ðg1 þ Chx2 Þx5 þ P ðx4 Þ
ð6Þ ð7Þ
where P ðx4 Þ ¼
p1 x24 x24 þ p22
ð8Þ
Qðx4 Þ ¼
q1 x24 x24 þ q22
ð9Þ
and
where the variables, constants and parameters are
Fig. 4. Compartment Scheme of the cardiac cell.
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(1) Variables: • x1 : proportion of activatable ryanodine channels (dimensionless); • x2 : proportion of open ryanodine channels (dimensionless); • x3 : proportion of closed ryanodine channels (dimensionless); • x4 : cytosolic calcium concentration ðMÞ ¼ Ccy ; • x5 : calcium concentration inside sarcoplasmic reticulum ðMÞ ¼ Csr ; • caa : arachidonic acid concentration in the cell; • p1 : maximum rate of the sarcoplasmic pump (M s1 ); • J ðtÞ: the excitation signal into the cytosol (represented as a Ca2þ injection); (2) constants: • g1 : leakage conductance between the sarcoplasmic reticulum and the cytosol (s1 ); • g2 : leakage conductance between the cytosol and extracellular medium (s1 ); • Ch: conductance of the ryanodine channels (s1 ); • p2 : threshold concentration of the sarcoplasmic pump (M); • q1 : maximum rate of the sarcolemma pump (M s1 ); • q2 : threshold concentration of the sarcolemma pump (M); • l1 , l2 : RyR channel binding rates; • l1 , l2 : RyR channel dissociation rates; (3) other parameters: • C0 : extracellular calcium concentration; • Vr : cytosol to sarcoplasmic reticulum volume ratio. Eqs. (3)–(5) describe the opening of the RyRs channels, whereas Eqs. (6) and (7) give the dynamics of the Ca2þ concentration in both the cytosol and the sarcoplasmic reticulum. The result obtained are shown in Fig. 5 and it may be noticed that the simulation fits well with the experimental data. 3.2. Extension of the model In certain cases, the external signal of Ca2þ , J ðtÞ, take a major importance in the representation of the cytosolic calcium concentration. In order to avoid this problem we introduce a third compartment corresponding to the so called ‘‘subspace’’ in Jafri et al. [8]. This part of the model is represented by the following equation: x_ 4 ¼ ðVr g1 þ g2 þ Vr Chx2 þ g3 Þx4 þ ðVr g1 þ Vr Chx2 Þx5 ðQðx4 Þ þ Vr P ðx4 ÞÞ þ g2 C 0 þ g3 x 6
ð10Þ
x_ 6 ¼ g3 ðVs x4 x6 Þ þ J ðtÞ
ð11Þ
with the following definition for the new variable and parameters: (1) Variables: • x2 : proportion of open ryanodine channels (dimensionless); • x4 : cytosolic calcium concentration ðMÞ ¼ Ccy ;
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F. Rocaries et al. / Simulation Modelling Practice and Theory 12 (2004) 93–104 0.8 Arachidonic acid 10uM (1min)
Arachidonic acid 10uM (12 min)
Arachidonic acid 10uM (38 min)
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0.6
0.4
0.2
0 0
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1440
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2330
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2320
2330
2340
Simulation results
0.8
0.6
0.4
0.2
0 0
10
20 Time (s)
Time (s)
Time (s)
Fig. 5. Arachidonic Acid 10 lM––first row experimental result––second row simulation result.
• x5 : calcium concentration inside sarcoplasmic reticulum ðMÞ ¼ Csr ; • x6 : calcium concentration inside subspace ðMÞ ¼ Csr ; (2) constants: • g1 : leakage conductance between the sarcoplasmic reticulum and the cytosol (s1 ); • g2 : leakage conductance between the cytosol and extracellular medium (s1 ); • g3 : leakage conductance between the cytosol and subspace (s1 ); • Ch: conductance of the ryanodine channels (s1 ); (3) other parameters: • C0 : extracellular calcium concentration.
3.3. Action on the RyRs channel In this case it is more complicated to describe the effect of arachidonic acid on the proportion of open channels. The system is simulated with a fixed proportion of open ryanodine channels (100%). The only interesting result we can extract from a simulation of this type is the steady state behavior. As may be seen in the Fig. 6 the aspect of the simulation results in term of dynamical behavior is quite different from the experimental data, shown in Fig. 5 (upper row last column).
F. Rocaries et al. / Simulation Modelling Practice and Theory 12 (2004) 93–104
101
0.5
Ca++ Concentration
0.4
0.3
0.2
0.1 2300
2310
2320 time
2330
2340
Fig. 6. Steady state simulation result after 38 min.
3.4. Action on the RyRs channel: a model Another model we can (and we have to!) use for an action on the RyRs channel can be described as follows. Considering that a specific drug a 1 is able to change the value of the conductance of the ryanodine channels, the following model (very similar to the one presented in Eq. (2)) can be used. As described in Section 2.1 the drug a is a source represented by a concentration csa and the cytosol will be filled by this source. At the same time we assume that the value of the RyRs channel conductance Ch is a linear function of the average concentration of a product ca ðtÞ. Thus we obtain the following system of equations: c_a ðtÞ ¼ Kðca ðtÞ csa Þ
ð12Þ
and ca ðtÞ Ch ¼ Ch10 1 ca max
if ca ðtÞ 6 ca max
else Ch ¼ 0
ð13Þ
where • K is a transfer coefficient between the source and the tank. • Ch10 is the maximum conductance. • caa max is the maximum concentration value of a product bearable by the cell.
1
Like caffeine or ruthenium red.
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Thapsigargin 1uM (12 min)
Thapsigargin 1uM (20 min)
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2.5 2 1.5 1 0.5 0 0
10
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40
700
710
Time (s)
720
730
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1210
Time (s)
1220
1230
1240
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Time (s)
3
Simulation results
2.5 2 1.5 1 0.5 0
0
10
20 Time (s)
30
40
700
710
720 Time (s)
730
740 1200
1210
1220 Time (s)
Fig. 7. Thapsigargin 1 lM––first row experimental data––second row simulation result.
4. Experimental verification with thapsigargin We verify the validity of our hypothesis by using a well known specific inhibitor of the SERCA pump: thapsigargin. The effect of thapsigargin (1 lM) [4,7] was examined on Ca2þ cycling of electrically stimulated cardiomyocytes (Fig. 7, top). The amplitude of Ca2þ transients decreased progressively, as previously described by Kirby et al. [4], and finally cells lost responsiveness to electrical stimulation. This may be attributed to a progressive SR Ca2þ depletion [3]. The mathematical modelling of SERCA inhibition is shown in Fig. 7 (bottom). We use, in this case, the same model as that for arachidonic acid but with different coefficient values in Eq. (2) and with the evolution of thapsigargin concentration instead of arachidonic acid concentration. It is important to note that in this specific case the concentration of thapsigargin probably reaches the maximum concentration value bearable by the cell. Its also important to notice that the experiment was made on another set of cell than for acid arachidonic. Thus is not surprising that the values are quite different.
5. Conclusions In this paper the model previously proposed by the authors is extended to account for the effect of drugs (arachidonic acid or thapsigargin) on the behavior of the heart
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cells. This model is used to explain the effect and relate it to the physiological phenomena in the cell. Some of the constants of the previous model become variables and change in time as a function of the drug concentration. The contribution of this work is twofold: • the extension of the heart cell model in order to account for pathological cases. • the confirmation of the hypothesis concerning the effect of the drug on the various components of the cell. The numerical results presented in this paper are obtained using the package Xmath/SystemBuild. This, however, is easily extended to MatLab/Simulink and SciLab/Scicos software. In fact the model exist for both environment.
Acknowledgements The authors would like to thanks the ECOS-NORD French program and his Venezuelan counterpart CONICIT, Fondacion Gran Mariscal de AYACUCHO and CDCHT for their financial support.
Appendix A. Parameter’s values used for simulation Localisation
Numerical values 1 1
Definition
RyR channels
l1 l1 l2 l2 Ch
0.76 lM s 0.084 s1 1.5 lM1 s1 0.08 s1 0.8 s1
Binding rate Dissociation rate Binding rate Dissociation rate RyR channels conductance
SR pump
p1 p2
0.15 lM s1 0.11 lM
Maximum rate Threshold concentration
Sarcolemmal efflux
q1 q2
0.013 lM s1 0.006 lM
Maximum rate Threshold concentration
Other parameters
C0
1500 lM
Vr g1 g2
0.185 0.04 s1 1.063 103 s1
Extra cellular Ca2þ concentration Volume ratio Leakage conductance Leakge conductance
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References [1] G. Grynkiewicz, M. Poenie, R.Y. Tsien, A new generation of Ca2þ indicators with greatly improved fluorescence properties, J. Biol. Chem. 260 (1985) 3440–3450. [2] Y. Hamam, F. Pecker, H. Lorrino, F. Rocaries, C. Pavoine, R. Natowicz, Identification and modeling of calcium dynamics in cardiac myocites, Simul. Practice Theory J. 8 (2000) 3–15. [3] A.M. Janczewski, E.G. Lakatta, Thapsigargin inhibits Ca2þ uptake, and Ca2þ depletes sarcoplasmic reticulum in intact cardiac myocytes, Am. J. Physiol. 265 (1993) 517–522. [4] M.S. Kirby, Y. Sagara, S. Gaa, G. Inesi, W.J. Lederer, T.B. Rogers, Thapsigargin inhibits contraction and Ca2þ transient in cardiac cells by specific inhibition of the sarcoplasmic reticulum Ca2þ pump, J. Biol. Chem. 267 (1992) 12545–12551. [5] A. Sauvadet, F. Pecker, C. Pavoine, Inhibition of sarcolemmal Ca2þ pump in embryonic chick heart cells by mini-glucagon, Cell Calcium 18 (1995) 76–85. [6] Y. Tang, H.G. Othmer, A model of calcium dynamics in cardiac myocites based on the kinetics of ryanodine-sensitive calcium channels, Biophys. J. 67 (1994) 2223–2235. [7] O. Thastrup, P.J. Cullen, B.K. Drobak, M.R. Hanley, A.P. Dawson, Thapsigargin, a tumor promoter, discharges intracellular Ca2þ stores by specific inhibition of the endoplasmic reticulum Ca2þ -ATPase, Proc. Natl. Acad Sci. 87 (1990) 2466–2470. [8] S. Jafri, J.J. Rice, R.L. Winslow, Cardiac Ca2þ Dynamics: the role of ryanodine receptor adaptation and sarcoplasmic reticulum load, Biophys. J. 74 (1998) 1149–1168. [9] C.W. Taylor, L.M. Broad, Pharmacological analysis of intracellular Ca2þ signalling: problems and pitfalls, Trends Pharmacol. Sci. 19 (1998) 370–375. [10] S.C. O’Neill, D.A. Eisner, A mechanism for the effects of caffeine on Ca2þ release during diastole and systole in isolated rat ventricular myocytes, J. Physiol. (London) 430 (1990) 519–536. [11] E. Rousseau, G. Meissner, Single cardiac sarcoplasmic reticulum Ca2þ -release channel: activation by caffeine, Am. J. Physiol. 256 (1989) H328–H333. [12] G.L. Smith, M. Valdeolmillos, D.A. Eisner, D.G. Allen, Effects of rapid application of caffeine on intracellular calcium concentration in ferret papillary muscles, J. Gen. Physiol. 92 (1988) 351–368. [13] B.E. Ehrlich, E. Kaftan, S. Bezprozvannaya, I. Bezprozvanny, The pharmacology of intracellular Ca2þ -release channels, Trends Pharmacol. Sci. 15 (1994) 145–149. [14] C.L. Overend, S.C. O’Neill, D.A. Eisner, The effect of tetracaine on stimulated contractions, sarcoplasmic reticulum Ca2þ content and membrane current in isolated rat ventricular myocytes, J. Physiol. (London) 507 (1998) 759–769.