A Model to Study Effect of Rapid Buffers and Na on Ca2+ Oscillations in Neuron Cell

Ca2+ plays a vital role in muscle mechanics, cardiac electrophysiology, secretion, hair cells, and adaptation in photoreceptors. It is a vital second messenger used in signal transduction. Calcium controls cell movement, cell differentiation, ciliary beating. Many cells exhibit oscillations in intracellular [Ca2+] in response to agonist such as hormones and neurotransmitters. Many cells use oscillations in calcium concentration to transmit messages (Sneyd J. et al, 2006, p. 151-163). In this paper, an attempt has been made to develop a model to study calcium oscillations in neuron cells. This model incorporates the effect of variable Na influx, sodium-calcium exchange (NCX) protein, Sarcolemmal Calcium ATPase (SL) pump, Sarco-Endoplasmic Reticulum CaATPase (SERCA) pump, sodium and calcium channels, and IP3R channel. The proposed mathematical model leads to a system of partial differential equations which has been solved numerically using Forward Time Centered Space (FTCS) approach. The numerical results have been used to study the relationships among different types of parameters such as buffer concentration, disassociation rate, calcium permeability, etc.


Introduction
Calcium oscillations are known to play a key role in a number of mechanisms like the secretion in the pituitary and parotid glands, the contraction of smooth muscle, and cardiac inotrophy and induction of arrhythmias (Jafri M.S. et al, 1992, p. 235-246).These oscillations are supposed to contain frequency encoded signals that help in using Calcium as a second messenger while avoiding its high intracellular concentrations (Keener J. et al, 1998, p. 53-56).A number of investigators have reported the oscillatory behaviour of calcium due to intracellular concentration of inositol 1, 4, 5-trisphosphate (IP 3 ).In the process of signal transduction, intracellular calcium acts like a switch and decides whether a particular signal needs to be further propagated or not.There are mainly two types of receptors, Rynodine Receptors (RyRs) or Inositol Triphosphate Receptors (IP 3 Rs) that are located at the membrane of endoplasmic reticulum (in neurons) or sarcoplasmic reticulum (in myocytes) which cause an efflux of Ca 2+ in the cytosol.The release of Ca 2+ through IP 3 Rs 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 (PIP 2 ) to inositol triphosphate (IP 3 ) and diacylglycerol (DAG).This released IP 3 is free to diffuse through the cytosol and binds with IP 3 R leading to the subsequent opening of these February, 2010 receptors and release of Ca 2+ from the intracellular stores.Calcium oscillation can be classified into mainly two types: 1) that is induced by changing membrane potential as in the case of an action potential and the associated periodic entry of Ca 2+ through voltage-gated Ca 2+ channels,2) that occur in the presence of voltage clamp.The latter part can be further categorized based on the fact that the oscillatory Ca 2+ flux is from RyRs or IP 3 Rs.The period of IP 3 -dependent oscillations ranges from a few seconds to a few minutes.There is a great deal of evidence that in many cell types, these oscillations occur at constant [IP 3 ] and are therefore not driven by oscillations in [IP 3 ].It is also observed that as [IP 3 ] increases, the steady state [Ca 2+ ] increases, the oscillation frequency increases, and the amplitude of the oscillations remains approximately constant.Calcium oscillations usually occur only when [IP 3 ] is greater than some critical value and disappear again when [IP 3 ] gets too large.Thus, there is an intermediate range of IP 3 concentrations that generate Ca 2+ oscillations (Keener J. et al, 1998, p. 53-56).In this paper, we have also studied oscillations induced in calcium due to change in IP 3 concentration.Here a mathematical model is proposed which incorporates nearly all important and necessary biophysical parameters like buffers, L-type calcium channel, calcium pump, sodium calcium exchanger (NCX), and calcium leak.Further, we assume that buffers exhibit rapid buffering.
Another factor which might have significant effect on calcium oscillation and which has not been given much importance is the sodium ion concentration.We have incorporated it through the NCX protein.We have considered an exchange ratio of 4:1 (Fujioka Y., 2000, p.611-623) with respect to sodium and calcium ions respectively.To make the model more realistic we have included the ER.The proposed mathematical model leads to a system of partial differential equations.We have used finite difference approach for the simulation of the proposed model for which a program has been developed in MATLAB and run on a Pentium IV Dual Core 1.00 GB RAM, 1.73GHz processor to obtain the numerical result.The time taken per simulation is 240 seconds for time, t = 30 seconds.

Mathematical Model
Our mathematical model assumes the following reaction-diffusion kinetics (Wagner J. et al, 1994, p.447-456) (Smith G.D., 1996, p.3064-3072), where, [B m ] and [CaB m ] are free and bound buffers respectively.Using Ficks law of diffusion and law of mass action and assuming rapid buffering approximation, we have the following partial differential equation (Smith G.D., 1996, p.3064-3072), where, D Ca , and D CaB m are the diffusion coefficients of free calcium, and calcium bound buffer respectively, and K m is disassociation rate constant.For stationary buffers, D CaBm = 0.
Our proposed mathematical model also contains the following parameters, to study the effect of rapid buffer, Na + ions and ER over Ca 2+ oscillations,

Ion channels
The Ca 2+ and Na + channels have been modeled using the Goldman-Hodgkin-Katz (GHK) current equation (Keener J. et al, 1998, p. 53-56): where [S ] i ,[S ] o , are the intracellular and extracellular ion concentration (Molar),respectively .P S is the permeability (m/s) of S ion, z s is valence of S ion.F is Faradays constant (C/moles).V m is membrane potential (Volts).R is Real gas constant (J/K moles) and T is Absolute temperature (Kelvin).Equation ( 3) is converted into molar/second by using the following equation ¢ www.ccsenet.org/jmrISSN: 1916-9795 The negative sign in equation ( 4) is taken because by convention inward current is taken to be negative.The GHK equation is derived from the constant field approximation which assumes that the electric field in the membrane is constant, and thus decoupled from the effects of charges moving through the membrane.

Na
The NCX protein is essential for excitation-contraction coupling in cardiac myocytes (Fujioka Y., 2000, p.611-623).It helps in the extrusion of cytosolic calcium in neurons and hence regulates neurotransmitter release, (Blaustein M.P., 1999, p. 763-854).The pump is assumed to be electrogenic in nature as one calcium leaves the cytosol for intake of four sodium ions.In our model we have taken an exchange ratio of 4:1 with respect to sodium and calcium ions respectively (Fujioka Y., 2000, p.611-623).The amount of energy required to extrude an ion against its concentration gradient is given by: So using Ca 2+ = 4 Na + , we have,

Sarcolemmal Calcium ATPase pump (SL CaATPase pump)
It is a P-type ATPase which is also known as Plasma Membrane Calcium ATPase pump (PMCA).Energy obtained from ATP is used to extrude calcium ions out of the cytosol.The kinetics of the pump follows MichaelisMenten kinetics (Nelson D.L., 2005) (Blackwell K.T., 2005, p.1-27).So the net efflux of calcium ions out of the cytosol is given by: where, V S Lpump is the maximum pump capacity, K S Lpump is half of the maximum pump capacity at steady state and H is the Hills coefficient.

Sarco Endoplasmic Reticulum CaATPase (SERCA) pump
The SERCA pump uses the chemical energy produced from the conversion of adenosine triphosphate (ATP) into adenosine diphosphate (ADP) to transport calcium ions across the membrane from the cytosol to the ER, against its concentration gradient, (Sneyd J. et al, 2006, p. 151-163).It binds calcium on the cytosolic side and releases it on the ER side.The SERCA pump has been modelled by a simple Hills equation as follows: where, v S ERCA is the maximal pump rate, k S ERCA is half of the maximum pump capacity at steady state and H is the Hills coefficient.

Endoplasmic Reticulum leak and IP 3 R Flux
There is a certain amount of leak from the ER to the cytosol along the concentration gradient and an IP 3 flux given by the following term (Jafri S. et al, 1995(Jafri S. et al, , p.2139(Jafri S. et al, -2153)): Where, c 1 is the ratio of volume of ER to cytosol, leakcons is the calcium leak rate constant, ipflux is the maximum IP 3 receptor flux, X 110 fraction of open channels; u er and u are the calcium concentrations in the ER and cytosol respectively.
Variation of channel states, that is, whether closed or opened is given as follows

Journal of Mathematics Research
February, 2010 Where, X 100 and X 110 represent the fraction of closed channels and open channels respectively.a 2 is inhibitory receptor binding constant,a 5 is activation receptor binding constant, b 2 is inhibitory receptor disassociation constant, and b 5 is activation receptor disassociation constant.

ER Calcium Concentration
Calcium in the ER is assumed to be under rapid buffering approximation (Jafri S. et al, 1995(Jafri S. et al, , p.2139(Jafri S. et al, -2153)), and is as given below, where, The IP 3 concentration varies with time according to the equation given below, (Keizer J. et al, 1992, p. 649-660) where, f(t)=0 or 1 (defines pulses of IP 3 ), v 3 is external IP 3 input rate, v 6 is maximum calcium dependent IP 3 input rate, v 7 is IP 3 decay rate constant and k 6 is activation constant, calcium dependent IP 3 input.The first term in above equation denotes IP 3 pulse, second term denotes calcium induced IP 3 production and last term denotes IP 3 degradation.
Combining equations (1-14) we get the proposed mathematical model as given below, Along with the initial-boundary conditions, A. Initial condition: B. Boundary conditions: Our problem is to solve equation ( 15) and ( 16) coupled with equation (17-19).For our convenience we are writing 'u' in lieu of [Ca 2+ ] and 'v' in lieu of [Na + ].Applying finite difference method (Forward Time Centered Space) on equation (15 16), we get ) And where, ε = FV m /RT is a dimensionless quantity, 'h' represents spatial step and 'k' represents time step, 'i' and 'j' represents the index of space and time respectively.Since, the above expression is not valid at the mouth of the channel; therefore the approximation at the mouth of the channel is given by And

Result and Discussion
In this section the results and discussion of the obtained numerical solution is given.The values of all the biophysical parameters used are as given in table 1. Figures (1-6) are the plots obtained for [IP 3 ] = 0.6µM whereas figures (7-12) are obtained for variable IP 3 concentration.

Mobile Buffers (Simulation Time =48s)
Figure 1 shows the oscillations in calcium concentration against time with variable mobile buffer concentration for a constant [IP 3 ] concentration of 0.6µM.This figure shows that it is not necessary to have oscillations in [IP 3 ] to observe oscillations in calcium concentration, (Keizer J. et al, 1992, p. 649-660).As the buffer concentration increases the free calcium concentration decreases but in the time interval t= [8-15]s and t= [27-30]s it is vice-versa.The reason for this is that as the calcium concentration increases the NCX protein activates which causes larger extrusion of calcium ions from the cytosol.Hence the calcium concentration inside the cytosol decreases.
Figure 2 shows the variation with time and buffer disassociation rate for two mobile endogenous buffers, calmodulin (K m =2µM) and (K m =3µM).As the disassociation rate increases more calcium ions become free and so the calcium concentration increases.However, in the time duration [7][8][9][10][11][12][13][14][15]s and [25-30]s it is vice-versa, i.e., lower disassociation rate has higher calcium concentration.This is because as the calcium ions increases the NCX protein activates which extrudes 1 calcium ion for four sodium ions entering the cytosol.Also the frequency has increased on increasing the buffer disassociation rate.This is due to the faster kinetics involved with higher disassociation rate.Also there is latency in the oscillation, that is, delay in time required to attain maximum value.When the disassociation rate is increased the calcium achieves the peak value faster.

Stationary buffers (Simulation Time =48s)
Figure 3 shows that as the stationary buffer concentration increases the calcium concentration decreases as more number of ions get bound to the buffer.However, in the intervals t= [7][8][9][10][11][12][13][14][15]s and [26-30]s the trend reverses.This is due to the fact that initially only the Sarcolemmal pump is working but as the calcium concentration increases the NCX protein gets activated which extrudes larger amount of calcium ions from the cytosol and hence the concentration decreases.The stationary buffers are immobile and hence they affect only the temporal variation of calcium and do not affect the spatial profile.The Calcium achieves the peak value faster when the buffer concentration is lower.This explains the latency observed.
Figure 4 shows variations in calcium concentration with time and buffer disassociation rate.Initially calcium concentration increases with increase in disassociation rate.This is so because with increases in disassociation rate more calcium ions ¢ www.ccsenet.org

Journal of Mathematics Research
February, 2010 become free.However in the time intervals [8][9][10][11][12][13][14][15]s and [26-30]s calcium decreases with increase in disassociation rate.This is because as the calcium ions reach higher concentrations the NCX protein gets activated which extrudes more calcium ions and hence the calcium ion concentration inside the cytosol decreases.As the buffer disassociation rate increases the frequency of oscillation also increases.This is because higher disassociation rate implies faster Calcium kinetics, that is, the binding and disassociation increases.We also observe latency in the oscillations.As the disassociation rate is increased the peak value is achieved faster.
Figure 5 shows the variations of calcium concentration with time and the leak rate constant of calcium from the ER into the cytosol are as shown in the above figure.As the calcium leak from the ER into the cytosol is increased the amount of free calcium in the cytosol increases.However, in intervals [5][6][7][8][9][10][11][12]s and [24-30]s the opposite is observed because with increase in calcium ions the NCX protein extrudes larger amount of calcium ions and hence the net calcium ion concentration in the cytosol decreases.
Figure 6 shows the sodium concentration variation with time.For the sodium concentration we have incorporated (1) sigmoidal deactivation of the NCX protein (2) calcium dependent inactivation of the L-type sodium channel and (3) a leak term which balances the sodium entering the cytosol via sodium channel and NCX protein.The sodium decreases up to 4s and thereafter increases.It decreases first due to deactivation of the NCX protein and calcium dependent inactivation of the sodium channel.After that it increases as the NCX protein is no longer deactivated and the sodium channel is also no longer inactivated.

Variable IP 3
Figure 7 shows a comparison of calcium concentration and IP 3 concentration on time scale.It is observed that the peaks of calcium and IP 3 occur at about the same time.This is because IP 3 oscillations are believed to induce calcium oscillations.By incorporating variable IP 3 it is observed that the frequency of calcium oscillation increases.Since earlier we observed only two peaks in 30 s time interval but now we observe 3 peaks.This has also been proved by Kusters et al, (Kusters et al, 2005, p. 3741-3756).Calcium is released from the ER when the IP 3 binds to the IP 3 R and is refilled back into the ER by the SERCA pump.This increases and decreases the calcium in cytosol more rapidly when we consider a varying IP 3 which is dependent on the fraction of open channels.Also the calcium maximum concentration increases from 0.37 µM to 0.7 µM.This is because of increased released of calcium from the ER.In the earlier case we had taken the IP 3 value to be 0.6 µM whereas the maximum IP 3 value here is 1.15 µM.IP 3 is regulated by the fraction of open channels in the ER which regulates the amount of calcium released from the ER.
Figure 8 compares cytoplasmic and ER calcium concentration on time scale.Both show oscillations with respect to time.Calcium concentration is high in the ER compared to the cytoplasm.There is a leak of calcium from the ER to the cytoplasm and calcium uptake by ER through the SERCA pump.It is thus observed that when the calcium concentration in the ER is minimum it has a maximum in the cytoplasm.This has also been proved by Kusters et al, (Kusters et al, 2005, p. 3741-3756).It is also observed that the calcium concentration in the ER is continuously decreasing.This is so because the calcium extrusion from the cytosol via the NCX protein and Sarcolemmal pump is large and hence the amount of calcium being refilled into the ER by the SERCA pump is decreasing.
Figure 9(a) depicts variation of cytosolic calcium variation with ER buffer concentration and time.The effects of changing buffer concentration are negligible near the source due to rapid buffering.As we move away from the source the effect of other parameters becomes visible.So the effects are more visible at the second peak compared to the first peak.It is observed that as the ER buffer concentration increases the cytosolic calcium increases.This is because as ER buffer concentration increases more calcium inside the ER gets bound to the buffers so concentration gradient decreases and hence lesser calcium ions leak from the ER to cytosol.The pumping by the SERCA pump depends on the cytosolic calcium concentration which pumps lesser amount of calcium from the cytosol and hence the cytosolic calcium concentration increases.Also ER buffer concentration has little effect on cytoplasmic calcium oscillations.This is in agreement with the findings of Kusters et al (Kusters et al, 2005, p. 3741-3756).
Figure 9(b) depicts variation of cytosolic calcium variation with cytosolic buffer concentration and time.The effects of changing buffer concentration are less near the source due to rapid buffering.So the effects are more visible at the second peak compared to the first peak.It is observed that as the cytosolic buffer concentration increases the cytosolic calcium decreases.This is because as buffer concentration increases more calcium in the cytosol gets bound to the buffers.Hence the cytosolic calcium concentration decreases.
Figure 10(a) depicts variation of ER calcium concentration with ER buffer concentration and time.The effects of changing buffer concentration are more evident at the second peak compared to the first peak due to rapid buffering in the ER.
It is observed that as the ER buffer concentration increases the ER calcium increases.This is because as ER buffer concentration increases more calcium inside the ER gets bound to the buffers so there is lesser concentration gradient so lesser calcium ions leak from the ER.Hence the ER calcium concentration increases.ISSN: 1916-9795 to the cytosolic calcium so lesser cytosolic calcium implies that there is lesser refilling of the ER and hence the ER calcium concentration decreases.Also as calcium concentration in the cytoplasm decreasing there is more concentration gradient and hence more calcium leaks from the ER and hence the ER calcium concentration decreases.
Figure 11(a) depicts variation of cytosolic calcium variation with ER buffer disassociation rate and time.The effects of changing buffer disassociation rate are negligible at first spike due to rapid buffering.As we move away from the source the effect of disassociation rate becomes more evident.It is observed that as the ER buffer disassociation rate increases the cytosolic calcium increases.This is because as ER buffer disassociation rate increases more calcium inside the ER becomes free so there is a larger leak of calcium ions from the ER into the cytosol.Thus calcium ion concentration inside the cytosol increases.ER buffer disassociation rate has negligible effect on cytosolic calcium oscillations as also shown by Kusters et al (Kusters et al, 2005, p. 3741-3756).
Figure 11(b) depicts variation of cytosolic calcium variation with cytoplasmic buffer disassociation rate and time.The results are evident, that is, as the disassociation rate increases the calcium concentration in the cytosol also increases.
Figure 12(a) depicts variation of ER calcium concentration with ER buffer disassociation rate and time.The effects of changing buffer disassociation rate are more evident at second spike due to rapid buffering.It is observed that as the ER buffer disassociation rate increases the ER calcium increases, this is in agreement with the physiological facts.As ER buffer disassociation rate increases more calcium inside the ER gets free.This is because disassociation rate is the ratio of disassociation to association rate, so larger disassociation rate implies that disassociation is greater than association so more calcium ions are released from the buffer.
Figure 12(b) depicts variation of ER calcium concentration with cytosolic buffer disassociation rate and time.As the cytosolic buffer disassociation rate increases the calcium concentration inside the cytosol increases and hence a greater amount of refilling of calcium into the ER by the SERCA pump.Hence the calcium concentration inside the ER increases.Also the concentration gradient across the ER decreases and hence there is less amount of calcium leaking from the ER into the cytosol so the ER calcium concentration increases.

Conclusion
Here we have studied the effects of sodium ions on calcium oscillations which have not been given much importance till date.We have used an exchange ratio of 4:1 with respect to sodium and calcium ions.We have tried to make the model more realistic by incorporating the different parameters as mentioned above.Some of the results obtained above are in agreement with the work done by previous researchers.The new results obtained are also in agreement with the physiological facts.The numerical solutions obtained above can be used to study the relationships among different types of parameters in the normal and abnormal conditions which can be useful to biomedical scientists for developing new protocols for the diagnosis and treatment of various neurological diseases.

Figure 10
Figure 10(b) depicts variation of ER calcium concentration with cytoplasmic buffer concentration and time.As the cytoplasmic buffer increases more calcium ions inside the cytosol get bound.The refilling by SERCA pump is proportional

Figure 1 . 2010 Figure 5 .Figure 7 .
Figure 1.Variation of calcium with time and cytoplasmic buffer concentration