MATHEMATICAL APPROACH on the FREE Ca+2 KINETICS in the MUSCLE CELL

MATHEMATICAL APPROACH ON THE FREE Ca+2 KINETICS IN THE MUSCLE CELL

MATHEMATICAL APPROACH ON THE FREE Ca+2 KINETICS IN THE MUSCLE CELL

D.E.Creanga

Univ. Al. I. Cuza, Fac. of Physics, 11 A Bd. Carol I, Iasi

The theoretical model presented inhere was proposed to describe the in vivo circulation of Ca+2 ions through the plasma membrane as well as through the sarcoplasmic reticulum (SR) storage membranes, based on in vitro experimental findings. A set of differential equations was written and solutions were identified according to the “standard” bi-compartmental model. The time constants k12, k21 and k01 describing ion transport through the system compartments can be calculated.

Theoretical background.The two-compartment physiological model is able to describe the kinetics of a labeled substance s, which is passing between two physiological compartments C1 and C2, so that C1 communicates with C2 as well as with the physiological pool C0 while C2 is a closed compartment (communicating only with C1) [1]. According to the phenomenological model, s1(t) and s2(t) represent the quantities of the substance s that can be found at the time t in the compartment C1, respectively, C2; k12 and k21 are the time constants characterizing the circulation of the substance s between the two compartments; k01 and are the time constant for the passing of s into the physiological pool and respectively the turnover rate characterizing the physiological pool ability of replacing the substance s from the compartment C1.In our case the calcium ions represent the substance s, labeled, studied during the movement from the SR stores (the compartment C2) toward the cytoplasm (the compartment C1) and reversibly, as well as during the calcium exchanges between cytoplasm and the extra-cellular medium (C0). The differential equation system, describing the communication between C1, C2 and C0 by means of the calcium ions is [1]:

(1)

(2)

with the initial conditions:

(3)

written in the hypothesis that a certain amount of calcium ions that we follow enters the cell at the time t=0, throw the plasma membrane calcium channels and, therefor they are not arrived yet in the second compartment at this initial moment of time. We have to mention that the turnover process (corresponding to the rate constant) does not explicitly appear in the equation system solutions, since the labeled ions are not yielded naturally. The solutions of the differential equation system written for the description of the labeled substance kinetics are:

(4)

(5)

where: (6) (7) (8)

that is: (9)

(10)

where: (11)

S1 and S2 being the equilibrium values of the substance s mass in the two compartments C1 and C2, and s1(0) is the initial amount of s in C1, (s2(0) =0).

The premises for the model application

The exponential form of the above system solutions suggested a correlation with the next experimental evidences reported in literature.

(i)First, the results reported in [2] where a fluorescent dye was used to evaluate the Ca+2 taken up by smooth muscle skinned cells, that revealed the rapid decreasing of calcium concentration in the cells suspension. The experimental curve describing the calcium decay in the skinned cell suspension is similar to the graphic representation of s1(t) (equivalent to the calcium kinetics in cytoplasm, namely the compartment C1).

(ii) Further, it was found that the release of Ca+2 ions from isolated triad vesicles (equivalent with the circulation from C2 to C1), induced by a calcium ionophore was fit to a single exponential with a known time constant (corresponding to k12)[3].

(iii)Then, the depolarization-induced calcium release in isolated muscle triads (equivalent also to the circulation from C2 to C1) is reported as being fitted by a sum of exponential: a fast one and a slow one, both having negative time constants[3]:

(12)

where the fast one is considered as the dominant, i.e.;

(13)

In the above equations the symbol y is designing the calcium level, x is the time while T1 and T2 are the time constants.

(iv) In [2] is mentioned that in [4], the study of calcium extrusion from cytoplasm, led to the idea that: “…the plasma membrane calcium extrusion processes account for 20-40% of the calcium removal seen experimentally”[4]. Thus the SR pumps cover about 80-60% of the calcium removal from cytoplasm.

Discussion. In Figure 1 one can see a comparison between the experimental curve representing only calcium decay in the cell suspension, obtained in [2] and the solution s1(t) representing the calcium kinetics in cytoplasm (compartment C1) provided by the theoretical model. The latest may be taken as corresponding to the calcium decay within the cytoplasm due to the Ca2+ pumps from plasma membrane and from the SR and also to the calcium release through SR (compartment C2) ion channels and to the calcium uptake from the extra-cellular medium (physiological pool, C0)). In Figure 2 are given the experimental curve obtained in [3] for the calcium released in cytoplasm from SR stores upon depolarization together with the theoretical curve of the whole calcium kinetics in the cytoplasm.The analysis of Figures 1 and 2 reveals the following limitations of the model:

- if we take into account that calcium intrusion in the cytoplasm and calcium storage in SR vesicles are not in phase processes, then the theoretical curve from Figure 1 (left) can be taken as describing only the calcium kinetics during the time interval when the SR pumps occurs diminishing calcium level in cytoplasm;

-the theoretical curve from Figure 2 (left) corresponds to the overlapping of calcium intrusion in the cytoplasm from extra-cellular medium and calcium extrusion from SR storages followed by the action of calcium pumps from SR membranes, while the experimental curve (right) corresponds only to Ca2+ extrusion from the SR storages through ion channels. So, in order to evaluate in vivo time constants of calcium ion transport, in the first stage k12is to be determined from in vitro experiments designed for calcium release by ionophore action from isolated muscle triades. In the second stage T2 and c may be determined from in vitro experiments focused on depolarization-induced calcium release from isolated muscle triades. Then, if we approximate the two percentage values mentioned above [4] with 30% and respectively with 70%, and considering that the ratio may be applied to the time constants, we may obtain:

(14)

In the hypothesis that the value of could be considerable, so that the second term in (10) is negligible, we can obtain an approximation of y from (13) with s2(t) from (10) and, following this:

(15)

and, from (11):

(16)

From (ii), (14) and (16) we could determine all the three rate constants while from (15) it results the value of the ratio: (17)

The rate S1/S2 gives the ratio of the calcium quantities in C1 and C2 when the dynamic equilibrium is established and could be related to the correspondent concentrations of calcium ions in the cytoplasm and respectively in the triad vesicles. The s1(0) parameter corresponds to the arbitrarily established initial moment when hypothetical labeling of a certain amount of calcium ions is carried out.

Conclusion. The model offers the possibility to evaluate the time constant k01 describing the in vivo calcium transport from cytoplasm to the extra-cellular medium, on the basis of the invitro approximated time constants k12 and k21 (corresponding to Ca2+ circulation between cytoplasm and sarcoplasmic reticulum). It gives also an evaluation of the ratio between the calcium quantities (and then between the correspondent concentrations) in cytoplasm and sarcoplasmic reticulum at the dynamic equilibrium, characterizing calcium kinetics.The model should be improved by taken into account the timing of the intrusion and extrusion processes that are not simultaneous while new experimental evidences are expected to give experimental curves describing in vitro behavior of calcium ions within the muscle cell.

References

(1) Gremy, F., Perrin, J., Elements de biophyisique, Ed.Flammarion, Paris, 1970

(2) Kargacin, M.E., Kargacin, G.J., Direct measurement of Ca 2+ uptake and release by the sarcoplasmic reticulum of saponin permeabilized isolated smooth muscle cells, Journal of General Physiology, 1995, 106, 467-484

(3) Kramer, J.W., Corbett, A.M., The voltage dependence of depolarization-induced calcium release in isolated skeletal muscle triads, Membrane Biology, 1995, 144: 217-230

(4) Conney, R.A.,Honeyman,T.W., Scheid, R.C., Contribution of Na+ dependent and ATP-dependent Ca 2+ transport to smooth muscle calcium homeostasis, Sodium exchange: Proceedings of the Second International Conference, 1991, 639:558-560