On-line version ISSN 1807-0302
Comput. Appl. Math. vol.31 no.2 São Carlos 2012
Shivendra G. Tewari*
Systems Science and Informatics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560059, India E-mail: firstname.lastname@example.org
Calcium plays a significant role in a number of cellular processes, like muscle contraction, gene expression, synaptic plasticity, signal transduction, but the significance of calcium oscillations (CaOs) is not yet completely understood in most of the cell types. It is a widely accepted fact that CaOs are a frequency encoded signal that allows a cell to use calcium as a second messenger while avoiding its toxic effects. These intracellular CaOs are primarily driven by some agonist-dependent pathways or fluctuations in membrane potential. The present mathematical model is of the latter type. The model incorporates expression for all major intracellular ionic species and membrane proteins. Especially, it integrates the coupling effect of sodium pump and Na+ / Ca2+ exchanger over CaOs. By varying sodium pump current, it is found that, sodium pump is a key player in modulating intracellular CaOs. The model predicts that the sodium pump can play a decisive role in regulating intercellular cell signaling process. The present study forms the basis for sodium pump controlled intercellular signaling process and requires further experimental verification.
Mathematical subject classification: 34M10, 92C20.
Key words: Na+/Ca2+ exchanger, sodium pump, calcium oscillations, membrane potential.
But till date the function of cytosolic Calcium oscillations (CaOs) has not been completely understood in most cell types. CaOs are known to play a key role ina number of mechanisms like activation of extracellular signal regulated kinase (ERK) [4, 5], the contraction of smooth muscle , increase in the frequency of synaptic currents  and maturation of Xenopus laevis Oocyte . These CaOs are supposed to contain frequency-encoded signals that help in using Ca2+ as a second messenger while avoiding its high intracellular concentrations . Also, in the process of signal transduction, intracellular Ca2+ behaves like a switch and decides whether a particular signal needs to be further propagated or not. The increase in intracellular concentration is facilitated by the opening of transmembrane Ca2+ channels which lead to the opening of channels at the intracellular stores. There are mainly Rynodine Receptors (RyRs) or Inositol Triphosphate Receptors (IP3Rs) that are located at the membrane of endoplasmic reticulum (in neurons) or sarcoplasmic reticulum (in myocytes) which causes an efflux of Ca2+ from the intracellular stores. The release of Ca2+ through IP3Rs is as a result of some agonist or neurotransmitter binding to its receptor which can cause via G-protein link to phospholipase C (PLC), the cleavage of phosphotidylinositol (4,5)-bisphosphate (PIP2) to inositol triphosphate (IP3) and diacylglycerol (DAG). This released IP3 is free to diffuse through the cytosol and binds with IP3R and leading to the subsequent opening of these receptors and release of Ca2+ from the intracellular stores. CaOs can be classified into mainly two types:
1) that is induced by changing membrane potential as in the case of an action potential; and
2) that occur in the presence of voltage clamp. The latter part can be further categorized based on the fact that the oscillatory Ca2+ flux is from RyRs or IP3Rs but our focus is on the first type.
Jafri et al.  showed CaOs for changing membrane potential of endoplasmic reticulum (ER), Atri et al.  showed CaOs for Ca2+ flux through the IP3Rs in Xenopus laevis Oocyte and determined the intermediate range of IP3 for CaOs. Wagner and Keizer  showed the effect of rapid buffering over CaOs. Later on, Kusters et al.  proposed an integrated model which combines excitable membrane with an IP3 mediated Ca2+ oscillator for normal rat kidney (NRK) fibroblast. Recently, Silva et al.  proposed a mathematical model for endothelial cells which incorporated nearly all the important biophysical parameters but is unable to exhibit CaOs. Thus, we can say that in all the investigations on CaOs carried out by researchers so far, none of the investigators have tried to incorporate the effect of changing cytosolic Na+ and K+ ions over CaOs. Thus, in this article, we have proposed a mathematical model governing CaOs for changing membrane potential in the absence of fluxes from the intracellular stores. Holmgren et al.  determined the three distinct steps of Na+ ion release from the sodium pump with the help of high speed voltage jumps. Here, we have tried to incorporate the impact of these distinct steps of sodium pump over cytosolic CaOs, in case of an action potential. Thus, we have incorporated L-type Ca2+ channel, Na+ channel, K+ channel, Plasma-Membrane (PM) Ca2+ ATPase, Na+ / Ca2+ exchanger (NCX), Na+ / K+ ATPase (Sodium pump), inward rectifier potassium channel (Kir), Ca2+ dependent intermediate potassium channel (IKCa, Ca2+ dependent small potassium channel (SKCa) and dynamic membrane potential. The gating mechanism of the trasmembrane channels emulate the gating mechanism of the famous Hodgkin and Huxley model . Further, the pumps and proteins are modeled to have realistic gating mechanism such that they are in agreement with the biological facts. The proposed mathematical model leads to a system of non-linear ordinary differential equations. We have used Euler's method for the simulation of the proposed model for which a MATLAB script has been written.
2 Mathematical formulation
Our cell model assumes that cell is cylindrical in shape. The diameter of cell is assumed to be 20 µm and its length to be 100 µm (see Fig. 1). Its specific membrane capacitance is taken to be 1 µF/cm2. As mentioned in literature [16, 17, 18], we have taken actual surface area (Acap) of cell to be larger than its geometrical surface area. Further, the channels, proteins and pumps are supposed to be homogenously distributed throughout the membrane. The formulation of the proposed mathematical model comprises of different components which are elaborated in the following subsections:
2.1 The Ca2+ channel
The L-type Ca2+ channel is supposed to have a permeability ratio of 36000:1:18 for Ca2+, Na+ and K+ ions, respectively . The voltage gated Ca2+ channel was modeled using non-linear Goldman-Hodgkin-Katz (GHK) current equation [9, 20], which can be stated as,
where, S is any of the ions, [S]i and [S]o are the intracellular and extracellular concentration ofSion respectively (in mM), PS is permeability (in cm/s) of S ion, zS is its valence, γi, γo are the activity coefficient (a.c.) of the S ion, F is Faradays constant (in Coulombs/moles), Vm is membrane potential (in volts), R is real gas constant (in J/K moles) and T is absolute temperature (in K). The total current through the L-type Ca2+ channel is
Further, equation (1) is converted into fluxes (in mM/second), before being used as an expression for individual ionic concentrations, by using Faraday's constant, Volume of the cytosol (Vcyt) and using the fact that 1 L = 10-3 m3,
where, S is any of the ion and L signifies L-type Ca2+ channel. In equation (2) there is a negative sign against it because by convention inward current is taken to be negative.
2.2 PM Ca2+ ATPase
PM Ca2+ ATPase (PMCA) is a P-type ATPase. The energy required to extrude Ca2+ out of the cytosol is met by ATP. The kinetics of this pump follows the enzyme-substrate formulism and hence using Michaelis Menten type kinetics , one can formulate the net efflux of Ca2+ ions out of the cytosol by,
where, Îpump is the pump current given by the following equation,
where, [Ca2+] is the intracellular Ca2+ concentraton (in mM), H,pump is the Hill's coefficient for PMCA, is the Ca2+ concentration at which the maximum pump current is halved (in mM).
2.3 Na+ / Ca2+ exchanger
This protein is known to play an important role in excitation-contraction coupling in cardiac myocytes . In neurons this protein helps in the extrusion of cytosolic Ca2+ concentration and hence helps in the modulation of neurotransmitter release . It is known that the cardiac type 3Na+ / 1Ca2+ exchanger is dominant in brain . Thus, we have used the same exchange for our model. We know that the amount of energy required to extrude an ion against its concentration gradient is given by [21, 20, 9],
where, S is the extruded ion. Introducing energy barrier, η, and using the fact that
we can write NCX current equation with an allosteric dependence over [Ca2+] as ,
where, [Ca2+]o is the extracellular Ca2+ concentration (in mM), [Na+] is the intracellular Na+ concentration (in mM), [Na+]o is the extracellular Na+ concentration (in mM), is the Ca2+ concentration at which INCX is halved, H, N C X is the Hill's coefficient of NCX, dNCX is constant for saturability of INCX, gNCX is the conductance of NCX (in nS).
2.4 Na+ / K+ ATPase
Na+ / K+ ATPase (NaK) is also a P-type ATPase which is also known as the sodium pump and is a 147 kDa membrane protein . It is known for the extrusion of Na+ ions at an expense of some ATP and inflow of K+ ions. Its formulation is based on the steps given by Holmgren et al. , where heused high speed voltage jumps to determine three distinct steps of Na+ ions deocclusion from the pump. The current through the sodium pump has the following form,
here, INaK is the scaling factor of NaK current (in µA/cm2), kf (in ms) is the forward (deocclusion) rate constant, kb (in ms) is the backward (occlusion) rate constant, K0.5(0) is half activating [Na+]o concentration at 0 mV, H, N a K is the Hill's coefficient for half activating NaK current, λ is the fraction of electrical field dropped along the access channel and τNaK (in ms) is someconstant.
2.5 Cytosolic Ca2+ buffers
It is assumed that a single buffer specie is present inside the cytosol and follows the following bi-molecular reaction
and which can be formulated in terms of the following differential equations,
If we also assume that there are no sources and no sinks present for buffer. Then letting BT represent the total buffer concentration, the above equation can be written in the reduced form as,
where, k+ is the buffer association rate, k- is the buffer dissociation rate, [C a B] represents bound buffer concentration.
2.6 Na+ and K+ channels
To generate action potentials Na+ current and K+ channels are taken as modeled by Hodgkin and Huxley . The transmembrane current due to Na+ and K+ channels has been modeled using the linear current-voltage relationship derived with the help of Ohm's Law,
where, S is either Na+ or K+ ion, gS is conductance of the given ion, VS is the reversal potential of the given ion determined by Nernst Equilibrium Potential equation (or simply Nernst equation),
where, 1000 is used to convert volts into milli-volts. All other symbols have their usual meanings. Here, and in all other instances, individual ionic reversal potential has been determined using Nernst equation at each integration step during runtime.
2.7 Ca2+ activated small and intermediate K+ channel
The current through SKCa is modeled using a linear current voltage relation as follows,
Here, gSKCa is SKCa channel conductance per unit area (in mS/cm2), Po,SKCa is its Ca2+ dependent open probability, VK is the reversal potential of K+ ions. Similarly, Ca2+ activated intermediate K+ current is modeled using a linear current voltage relation as follows,
Here, gIKCa is I KCa channel conductance per unit area (in mS/cm2), Po, I KCa is its Ca2+ dependent open probability, VK is the reversal potential of K+ ions.
2.8 Inward rectifier K+ channel
Inward rectifier K+ current is known to contribute to resting membrane potential. It is also modeled using the linear current voltage relationship,
and gKir is the Kir conductance (in nS), VK is the reversal potential for K+ given by Nernst equation, [K+]o is the extracellular K+ ion concentration. Here, gKir is converted into mS/cm2 using Acap before being used in the equation governing membrane potential.
2.9 Ca2+ and Na+ leak currents
To balance the net effect of INCX and Ipump there is supposed to be a Ca2+ leak current given by,
where, gCa,b is the Ca2+ leak conductance per unit area (in mS/cm2), VCa is the reversal potential (in mV) for Ca2+ given by Nernst equation. Similarly, we can formulate Na+ leak current to balance the net effect of INCX and ÎNaK,
where, gNa,b is the Na+ leak conductance per unit area (in mS/cm2), VNa is the reversal potential (in mV) for Na+ given by Nernst equation. The current due to all other ions is considered as leak and is incorporated as,
where, gL is leak conductance (in mS/cm2) and VL is leak reversal potential assumed to be constant.
2.10 Membrane potential
Like the formulation of Hodgkin and Huxley  we have divided the total membrane current into capacitive current and ionic currents. Thus, for capacitive current we have,
where, Iapp is the applied membrane current density (in µA/cm2), Vm is the membrane potential (in mV), Cm is the specific membrane capacitance (µF/cm2), Ii accounts for all the transmembrane currents discussed earlier and t is time (in ms). The gating mechanism of the transmembrane currents follows Hodgkin and Huxley . Combining equation (1)-(13) we can write the mathematical model governing CaOs with relevance to Na+ and K+ ions, as in the case of an action potential as,
In equation (14),
are rate constants which vary with membrane potential but not with time (ms-1) and m, n,h, mc, hc are dimensionless gating variables with values lying between 0 and 1. In this model we assume that fluxes from IP3Rs are absent. This can be achieved by blocking IP3R channel by using an IP3R antagonist like heparin . This assumption has been taken to exclude the effect of intracellular stores over CaOs. The initial condition of the system is,
All ionic concentrations are in the units of mM. The ordinary differential equations governing the gating variables (m, n, h, mc, hc) are
Here, m and (1 m) are representing on and off state of the variable m, respectively. Variables n, h, mc and hc also follow likewise. The mathematical expressions of the voltage-dependent rate constants in equation (14) are as follows,
For the solution of equations (14)-(16), we have used Euler's method and written a script in MATLAB that has been simulated on an AMD Turion 64 × 2 machine with 1.6 GHz processing speed and 2.5 GB memory. The time taken per simulation is ~9 sec when simulating for 30 ms using 4000 time steps i.e. Δt = 0.0075 ms. The numerical results obtained are used to study the effect of varying transmembrane currents over CaOs which are discussed in the following section.
3 Results and discussion
In all the figures, it is assumed that cytosolic Ca2+ is buffered with 50 µM Ethylene Glycol Tetraacetic Acid (EGTA). The standard biophysical parameters used for simulation of the model are listed in Table 1-4 unless stated along with the figures. Since our main objective is to study CaOs, we have shown results that are pertinent to CaOs only.
In Figure 2 we observe the effect of an impulse of 10 µA/cm2 over membrane potential. Such an effect has been studied in great detail by Hodgkin and Huxley , Luo and Rudy [16, 17, 18] thus, we need not give much emphasis over it here. In Figure 3, we have shown different current densities. All these current densities result in the shown action potential. The rest of the results shown are relevant to CaOs and has been studied in detail in the following figures.
In Figure 4 we have shown Ca2+ oscillation and buffered Ca2+ curves with respect to standard parameters listed in Table 1. Initially it was assumed that 2.3 µM of Ca2+ is buffered. Further it is apparent from Figure 4 that the amplitude of first Ca2+ spike is more than the second spike. It is because of the slow dissociation constant of EGTA which results in higher buffered Ca2+ and lower cytosolic Ca2+ concentration. Comparing Figure 4 with Figure 2, it is clear that both of them are positively correlated. As membrane potential rises the Ca2+ concentration rises and when membrane potential drops Ca2+ concentration drops.
In Figure 5 the effect of increasing total buffer concentration of EGTA isshown. The results are shown for BT = 50 µM (dark line) and BT = 200 µM (broken line). As expected increasing buffer concentration results in lower amplitude of Ca2+ oscillation, which is also evident from Figure 5.
In Figure 6 we observe the effect of increasing NCX conductance. Our simulation is in support of the biological fact that at negative potentials NCX works in reverse direction i.e. outflow of Ca2+ ions and inflow of 3Na+ ions. As increasing NCX conductance results in higher amplitude of Ca2+ oscillation while there is no change resting Ca2+ concentration at more positive membrane potentials as we have used a leak to neutralize the effect of NCX and pump currents.
In Figure 7 we observe the results for which the mathematical model was proposed. It is widely believed that cells encode information in the frequency of Ca2+ oscillations rather than its amplitude. There are a number of authors who have shown different roles of this ubiquitous sodium pump [26, 27]. Matchkov et al.  also experimentally demonstrated that sodium pump plays a significant role in regulating CaOs via regulation of cytosolic Na+ ions. Similar philosophy is suggested by our present simulations. We increased pumping rate, INaK = 1.5 µA/cm2 (dark line), 3 µA/cm2 (broken line), 4.5 µA/cm2 (dotted line), of sodium pump and observed changes in CaOs. It is seen that an increase in INaK results in an increase in the period of Ca2+ oscillations. The changes were quite apparent and are reflected from Figure 7.
In Figures 8-10, the effect of different extracellular concentrations of Ca2+, K+ and Na+ are shown over CaOs. The findings of Figure 8 are quite obvious but should be mentioned to show accordance with the biological facts. The curves are shown for [Ca2+]o = 1.8 mM (dark line) and [Ca2+]o = 1 mM (broken line). It is apparent from Figure 8 that lowering [Ca2+]o results in lower amplitude of Ca2+ oscillation which is because of a corresponding decrease in Ca2+ gradient.
In Figure 9, the findings are worth mentioning as an increase in [K+]o concentration resulted in an increase in frequency of Ca2+ oscillation. Although these findings are also pretty obvious as increasing [K+]o leads to an increase in reversal potential of K+ ions and hence increases the frequency of action potential which in turn increases the frequency of CaOs. In Figure 10, the effect of decreasing [Na+]o concentration is shown. The curves are shown for [Na+]o = 145 mM (dark line) and [Na+]o = 140 mM (broken line), it is apparent from figure that decreasing [Na+]o results in an increase in amplitude of CaOs. As in the previous case changing [Na+]o concentration changes the reversal potential of Na+ ion. But the observed change in VNa is minimal and obviously does not affect amplitude of action potential. The reason behind the increase in amplitude and latency in Ca2+ oscillation is the change in Na+ ion gradient. This gradient regulates the pumping rate of NCX exchanger decreasing the gradient means decreasing the pumping rate of NCX exchanger. Hence, affecting the net extrusion of Ca2+ ions via NCX exchanger; resulting in an increase in amplitude and latency of CaOs.
The results obtained in this paper are new and are subject to CaOs. The intent behind the present study was to investigate the effect of Na+ / K+ ATPase over Ca2+ oscillation influenced by the experimental results obtained by Matchkov et al. . The results obtained by our simulations are quite convincing with biological facts. The obtained results also confirmed the hypothesis of Matchkov et al.  that interaction between NCX and Na+ / K+ ATPase modulates intercellular communication. It was observed that increasing NaK current decreases the frequency of CaOs. The results obtained by previous investigators regarding CaOs have been mainly concerned with membrane potential, inositol triphosphate (IP3) or ryanodine receptor [6, 28, 29, 10, 11, 12, 13]. None of the earlier investigators gave much emphasis over this interaction of NCX and sodium pump which in turn effects CaOs. Thus, in this article, we have looked into and demonstrated a novel mechanism which modulates frequency of Ca2+ oscillation. Here, we have proposed a mathematical model which can be used for problems related to similar cell processes. The results obtained in this paper give new and useful insight for neurologists to look into the paradigm of CaOs at a different perspective. Also, the results obtained are relevant to biomedical scientists for developing protocols for diagnosis and treatment of neurological disorders.
Acknowledgments. The author acknowledges fruitful discussions with Dr. Ronald J. Clarke, School of Chemistry, The University of Sydney, Australia for giving useful insights over the kinetics of Na+ / K+ ATPase.
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*Present address: Biotechnology & Bioengineering Center, and Department of Physiology, Medical College of Wisconsin, 8701 Watertown Plank Road, Milwaukee, WI 53226, USA.