Cross Talking Calcium, IP₃ and Buffer Dynamics Alters ATP and NADH Level in Obese and Normal Hepatocyte Cell

Abstract

The cross talk between calcium (Ca²⁺), IP₃ and buffer dynamics regulate various mechanisms in hepatocyte cells. The study of independent systems of calcium, IP₃, and buffer signaling provides limited information about cell dynamics. In the current study, coupled reaction-diffusion equations are used to design a cross-talk model for IP₃, buffer, and calcium dynamics in a hepatocyte cell. The one-way feedback of calcium, buffer, and IP₃ in ATP production, ATP degradation, and NADH production rate is incorporated into the model. Numerical simulation has been done using the Finite Element Method (FEM) along the spatial direction and the Crank-Nicolson (C-N) method along the temporal direction. The numerical results are analysed to determine the effects of alterations in processes of cross-talking dynamics of IP₃, buffer, and calcium on ATP and NADH production and degradation rate of ATP in a hepatocyte cell under normal and obesity conditions. The comparative analysis of these findings unveils notable distinctions induced by obesity in calcium dynamics, ATP and NADH synthesis, and ATP degradation kinetics.

Introduction

Metabolism, detoxification, and homeostasis regulation in the body are mainly controlled by the liver. The parenchyma of the liver is arranged into lobules, which are hexagonal in shape, each consisting of a central vein surrounded by a portal triad. Each portal triad consists of a hepatic vein, bile duct, and hepatic artery. The hepatocytes are the predominant cells, comprising 70% of the liver. In the liver, cholangiocytes, endothelial, stellate, kupffer, and oval cells are the non-parenchymal cells. These cells perform many important functions in the liver [1].

Various activities of the cells have been regulated using Ca²⁺ which acts as an intracellular messenger in the cells. Through two different types of channels, Ca²⁺ is released from the intracellular stores (endoplasmic reticulum (ER)) in the cytosol. One of these is the family of inositol 1,4,5-triphosphate receptors (IP3R). IP₃ is produced by receptor activation of phosphoinositide-specific phospholipase C (PLC), causing the activation of IP3Rs [2]. Ryanodine receptors (RyR) which are the second family of intracellular Ca²⁺ channels, were named due to their great affinity for the plant alkaloid ryanodine [2]. Hepatocyte’s canalicular region contains a majority of the type II IP3R, while other cytosolic regions include more of the type I IP3R and less of the type II receptor [3].

Many researchers have investigated calcium signaling in diverse cell types such as neurons, myocytes, oocytes, pancreatic acinar cells, fibroblasts, astrocytes, and hepatocytes, among others [4,5,6,7,8,9,10,11].

Kotwani et al. [7, 12, 13]; Hemant et al. [14, 15]; Naik et al. [8, 16,17,18,19,20,21,22]; Pathak et al. [10, 23] and Jha et al. [24, 25] studied calcium distribution involving various parameters such as excess buffer, RyR, voltage-gated calcium channels, etc. for fibroblasts, lymphocytes, oocytes, myocytes, and astrocytes, respectively. Panday et al. [9, 26] devised a model for Ca²⁺ dynamics in oocytes that took into account Ca²⁺ advection within the cell. Amrita et al. [4, 5, 27] employed a finite element method to investigate how source geometry, NCX, SERCA pump, leak, and other factors influence Ca²⁺ distributions in neuron cells and dendritic spines. Manhas et al. [28, 29] reported the impact of mitochondria on Ca²⁺ signaling while studying calcium fluctuation in pancreatic acinar cells. Tewari et al. [11, 30] and Jagtap et al. [6, 31, 32] studied calcium dynamics in neurons and hepatocytes, respectively. Vedika et al. [33, 34] studied Ca²⁺ dynamics in obese and normal hepatocyte cells. The interdependent effect of calcium and buffer dynamics was also examined. Kothiya et al. [35,36,37,38] studied the system dynamics of calcium in a fibroblast cell. Vaishali et al. [39, 40] studied calcium dynamics in pancreatic β cells.

Hajnoczky et al. [41] discussed that the predominant elevation of calcium concentration ([Ca²⁺]) in non-excitable cells is via the second messenger IP₃. Politi et al. [42] formulated a model aimed at explaining the calcium-triggered stimulation of phospholipase C and IP₃ 3-kinase. This model captures the mechanisms through which calcium orchestrates both the enhancing and diminishing influences on IP₃ metabolism. Jean et al. [43] found that the liver’s IP3R was co-purified with markers of the plasma membrane. Thurley et al. [44] discovered that a stochastic calcium spike is produced by the interaction of IP3R clusters. Wagner et al. [45] modeled fertilization of a xenopus egg’s Ca²⁺ wave in one dimension using cartesian coordinates. Handy et al. [46] did the bifurcation simulations and analysis for the study of IP₃ dependent calcium variation. To investigate nonlinear IP₃ dependent calcium dynamics in cardiac myocytes, Singh et al. [47] devised a mathematical model for myocytes. Pawar et al. [48,49,50,51,52,53] studied the interdependence of Ca²⁺ on IP₃, NO, dopamine, β-amyloid etc. for neuron cells using mathematical models. Pawar et al. [54, 55] have also discussed calcium and the system dynamics of calcium with other ions using a fractional reaction-diffusion equation.

Neher et al. [56] investigated the calcium gradient and buffer in chromaffin cells derived from cows and concluded that out of total Ca²⁺ that enters the cell, 98–99% Ca²⁺ bind with endogenous Ca²⁺ buffer. Smith et al. [57,58,59,60] considered the three-parameter regimes given by the rate of response and dimensionless diffusion coefficients of Ca²⁺ and buffer with regard to each other. Martin Falcke [61] found that the concentration profile of a fast buffer around an open channel is more localized than that of a slow buffer. Through, the construction of a mathematical model, M.D. Stern [62] showed that the stabilization of rapid fluctuations in Ca²⁺ fluxes requires a buffer with rapid kinetics. According to Prins et al. [63], one thing that all organellar Ca²⁺ buffers have in common is their multifunctionality, as evidenced by the variety of Ca²⁺ binding and reactions they display. In addition to serving as an inactive Ca²⁺ breakdown within intracellular organelles, Ca²⁺ buffering proteins also modulate the Ca²⁺ release pathway, fold proteins, and regulate apoptosis. Gabso et al. [64, 65] highlighted the significant role of cellular calcium buffers in modulating the intensity and diffusion of calcium signals within neurons.

The literature review suggests a notable gap in research concerning the complex interplay of calcium, IP₃, and buffering dynamics within hepatocyte cells. Previous studies have predominantly modeled buffering dynamics as static, failing to account for their dynamic nature. This oversight simplifies the intricate relationships between calcium dynamics and other cellular processes, such as the interaction of calcium with IP₃, nitric oxide (NO), and dopamine, where buffers were assumed to be constant. However, recognizing that buffer concentrations are indeed variable can lead to more accurate and insightful models. The effects of changes in the mechanisms governing independent Ca²⁺ dynamics on the synthesis and degradation of ATP as well as the synthesis of NADH in normal and obese hepatocyte cells have received very little attention in previous studies. But no attempt is reported in the literature for the study of the effects of changes in the mechanisms governing cooperative Ca²⁺, IP₃, and buffer dynamics on the synthesis and degradation of ATP as well as the synthesis of NADH in normal and obese hepatocyte cells. Thus, a new model is developed in this paper to explore better insights into biophysical mechanisms and their effects on calcium, buffer, and IP₃ dynamics, as well as their consequential impacts on the synthesis and degradation of ATP as well as the synthesis of NADH in normal and obese hepatocyte cells. The model integrates the coupled reaction-diffusion equations for Ca²⁺, IP₃ and buffers through their mutual fluxes, aims to provide a deeper understanding of these processes. Numerical simulations were conducted using the FEM combined with the C-N Method to explore these dynamics further.

Problem Formulation

The reaction-diffusion equation proposed by Wagner et al. [45] is modified by incorporating calcium buffering fluxes and reaction term for Ca²⁺ profile in one dimensional unsteady state case which is expressed as:

$$\frac{\partial [C{a}^{2+}]}{\partial t}={D}_{Ca}\frac{{\partial }^{2}[C{a}^{2+}]}{\partial {x}^{2}}+\frac{[{J}_{IP3R}+{J}_{LK}-{J}_{SERCA}]}{{V}_{c}}-{J}_{on}+{J}_{off}.$$

(1)

Here, [Ca²⁺] represents calcium concentration in the cytosol, D_Ca represents diffusion coefficient of calcium in hepatocyte cell, J_IP3R is flux due to IP₃ receptor, J_LK denotes leak from ER, J_SERCA is efflux from cytosol to ER, J_on and J_off represent Ca²⁺ buffering fluxes.

The different fluxes are modeled as,

$${J}_{IP3R}={V}_{IP3R}{m}^{3}{h}^{3}({[C{a}^{2+}]}_{ER}-[C{a}^{2+}]),$$

(2)

Where, V_IP3R is IP₃ receptor flux rate constant, ${[C{a}^{2+}]}_{ER}$ is calcium concentration in ER [45].

$$m=\frac{V}{{K}_{IP3R}+V}\frac{[C{a}^{2+}]}{[C{a}^{2+}]+{K}_{Act}},$$

(3)

Here, V represents IP₃ concentration in the cytosol [45]. The proportion of subunits that Ca²⁺ has not yet inactivated is represented by the variable h,

$$\frac{dh}{dt}=\frac{{h}_{\infty }-h}{\tau },$$

(4)

Here,

$${h}_{\infty }=\frac{{K}_{Inh}}{{K}_{Inh}+[C{a}^{2+}]},$$

(5)

$${J}_{LK}={V}_{Leak}({[C{a}^{2+}]}_{ER}-[C{a}^{2+}]),$$

(6)

Here, V_Leak is rate of ER leakage.

$${J}_{SERCA}={V}_{SERCA}\frac{{[C{a}^{2+}]}^{2}}{{K}_{SERCA}^{2}+{[C{a}^{2+}]}^{2}},$$

(7)

Here, SERCA’s half-maximal activating cytosolic Ca²⁺ concentration is K_SERCA and its maximum outflow from the bulk cytosol is V_SERCA.

$${J}_{on}={k}_{j}^{+}[C{a}^{2+}]b,$$

(8)

$${J}_{off}={k}_{j}^{-}\frac{{b}_{tot}[C{a}^{2+}]}{K+[C{a}^{2+}]},$$

(9)

Here, association and dissociation rate of buffer are represented by ${k}_{j}^{+}$ and ${k}_{j}^{-}$ respectively [57, 66].

The following provides cytosolic IP₃ concentration(V)’s reaction-diffusion equation [45],

$$\frac{\partial V}{\partial t}={D}_{I}\frac{{\partial }^{2}V}{\partial {x}^{2}}+\frac{{J}_{Production}-\lambda ({J}_{Kinase}+{J}_{Phosphatase})}{{V}_{c}}.$$

(10)

Here, the diffusion coefficient of IP₃ is D_I, production of IP₃ due to Ca²⁺ is J_Production, J_Kinase and J_Phosphatase are IP₃ degradation due to 3-kinase and 5-phosphatase given as [45],

$${J}_{Production}={V}_{Production}\frac{{[C{a}^{2+}]}^{2}}{{[C{a}^{2+}]}^{2}+{K}_{Production}^{2}},$$

(11)

$${J}_{Kinase}=(1-\theta ){V}_{1}\frac{V}{V+2.5}+\theta {V}_{2}\frac{V}{V+0.5},$$

(12)

$${J}_{Phosphatase}={V}_{3}\frac{V}{V+30},$$

(13)

Here, θ is given by,

$$\theta =\frac{[C{a}^{2+}]}{[C{a}^{2+}]+0.39},$$

(14)

The nullcline-a surface along which the variable does not change is obtained by solving for the variable while setting each reaction-diffusion equation’s right side to zero. The result of h nullcline is h=h_∞. Therefore, it gives,

$$h=\frac{{K}_{Inh}}{{K}_{Inh}+[C{a}^{2+}]},$$

(15)

The buffer(b) dynamics has been modeled using the following diffusion equation [66],

$$\frac{\partial b}{\partial t}={D}_{b}\frac{{\partial }^{2}b}{\partial {x}^{2}}-{J}_{on}+{J}_{off}.$$

(16)

Where, J_on and J_off are given in equations (8) and (9)

Based on the presumption that the Ca²⁺, buffer and IP₃ concentrations in cell are 0.1, 0 and 0.16 μM respectively, at rest, the following initial conditions are imposed [6].

$$({[C{a}^{2+}]}_{t = 0})=0.1\quad \mu M,$$

(17)

$${V}_{t = 0}=0.16\quad \mu M,$$

(18)

$$({b}_{t = 0})=0\quad \mu M,$$

(19)

To arrive at the solution, the following boundary conditions are used [6].

$${\lim }_{x\to 0}\left(-{D}_{Ca}\left(\frac{\partial [C{a}^{2+}]}{\partial x}\right)\right)={\sigma }_{Ca},$$

(20)

Where, σ_Ca represents source influx.

$${\lim }_{x\to 15}([C{a}^{2+}])={C}_{\infty }=0.1\quad \mu M,$$

(21)

$${\lim }_{x\to 15}(V)=0.16\quad \mu M,$$

(22)

$${\lim }_{x\to 0}(V)={V}_{\infty }=3\quad \mu M,$$

(23)

$${\lim }_{x\to 0}{D}_{b}\left(\frac{\partial b}{\partial x}\right)=0,$$

(24)

$${\lim }_{x\to 15}(b)={b}_{\infty }=\frac{K{b}_{tot}}{K+{C}_{\infty }},$$

(25)

Here, total buffer concentration is b_tot and K=$\frac{{k}^{-}}{{k}^{+}}$ is buffer’s dissociation constant [32, 34].

ATP degradation rate-calcium dependent is calculated using [67],

$${J}_{HYD}=\frac{{J}_{SERCA}}{2}+{K}_{HYD}\frac{{[ATP]}_{c}}{{[ATP]}_{c}+{K}_{h}}.$$

(26)

J_SERCA is calculated using Eq. (7).

For ATP hydrolysis, the Michaelis-Menten constant is K_h and the maximal rate is K_HYD and cytosolic ATP concentration is [ATP]_c.

Ca²⁺-dependent ATP production is computed by [68],

$$ATP(C)=\frac{\frac{{F}_{o}}{{F}_{o}-1}-2\frac{[C{a}^{2+}]}{{\left[C{a}^{2+}\right]}_{max}}}{\frac{1}{{F}_{o}-1}-{\left(\frac{[C{a}^{2+}]}{{\left[C{a}^{2+}\right]}_{max}}\right)}^{2}}.$$

(27)

IP₃-dependent ATP production is computed by [68],

$$ATP(V)=\frac{\frac{{G}_{o}}{{G}_{o}-1}-2\frac{V}{{V}_{max}}}{\frac{1}{{G}_{o}-1}-{\left(\frac{V}{{V}_{max}}\right)}^{2}}.$$

(28)

NADH production rate is calculated by [67],

$${I}_{AGC}={V}_{AGC}\frac{[C{a}^{2+}]}{{K}_{AGC}+[C{a}^{2+}]}\frac{{q}_{2}}{{q}_{2}+{c}_{m}}\exp ({p}_{4}{V}_{m}).$$

(29)

Where, K_AGC and V_AGC, respectively, represent the rate constants for the synthesis of NADH and calcium’s dissociation from the aspartate-glutamate carrier (AGC).

The Taylor’s approximation method is used to linearize the proposed problem approximately where the IP₃ concentration is 3 μM, buffer concentration is 5 μM and [Ca²⁺] is 0.1 μM. Since the Ca²⁺, buffer and IP₃ concentrations are constrained to a restricted range, the nonlinear parts of the Taylor’s series become insignificant.

After linearisation, Eq. (1) can be rewritten as follows:

$$\frac{\partial u}{\partial t}={D}_{Ca}\frac{{\partial }^{2}u}{\partial {x}^{2}}+{A}_{1}u+{B}_{1}V+{C}_{1}b+{D}_{1},$$

(30)

The constants A₁, B₁, C₁ and D₁ were found using Taylor’s approximation approach (TAA) and u stands for [Ca²⁺].

Equation (10) can be redefined as, in a similar way for IP₃ concentration as V.

$$\frac{\partial V}{\partial t}={D}_{I}\frac{{\partial }^{2}V}{\partial {x}^{2}}+{A}_{2}u+{B}_{2}V+{C}_{2}b+{D}_{2},$$

(31)

The constants A₂, B₂, C₂ and D₂ were found using TAA.

It is possible to rewrite Eq, (16) as,

$$\frac{\partial b}{\partial t}={D}_{b}\frac{{\partial }^{2}b}{\partial {x}^{2}}+{A}_{3}u+{B}_{3}V+{C}_{3}b+{D}_{3},$$

(32)

Where A₃, B₃, C₃ and D₃ were found using TAA.

The numerical results are derived employing the variational finite element methodology applied to the cytoplasmic domain of the hepatocyte cell, segmented into 80 linear elements. The discretized variational functional of Eq. (30) is depicted as follows:

$$\begin{array}{r}{J}^{(e)}=\frac{1}{2}\int\nolimits_{{x}_{m}}^{{x}_{n}}\left[{u}^{{(e)}^{{\prime} 2}}+\frac{1}{{D}_{Ca}}\frac{{\partial {u}^{(e)}}^{2}}{\partial t}-{A}_{1}{{u}^{(e)}}^{2}-2{B}_{1}{V}^{(e)}{u}^{(e)}-2{C}_{1}{b}^{(e)}{u}^{(e)}-2{D}_{1}{u}^{(e)}\right]dx\\ -{\mu }^{(e)}\left(\frac{{\sigma }_{Ca}}{{D}_{Ca}}{u}_{(x = 0)}^{(e)}\right),\end{array}$$

(33)

μ^(e) is equal to 1 for the first element and 0 for every further element.

The shape function for calcium concentration is given as the following linear fluctuation due to the small size of the elements.

$${u}^{(e)}={c}_{1}+{c}_{2}x,$$

(34)

The Eq. (34) can be expressed as

$${u}^{(e)}={Q}^{T}{C}^{(e)},$$

(35)

Here,

$${Q}^{T}=\left[\begin{array}{cc}1&x\end{array}\right]\,\&\, {C}^{(e)}=\left[\begin{array}{c}{c}_{1}\\ {c}_{2}\end{array}\right]$$

Values of u^(e) at nodes x_m and x_n are given by,

$${u}^{(e)}({x}_{m})={c}_{1}+{c}_{2}{x}_{m},$$

(36)

$${u}^{(e)}({x}_{n})={c}_{1}+{c}_{2}{x}_{n},$$

(37)

Using above equations, it is obtained as,

$${\overline{u}}^{(e)}={Q}^{(e)}{C}^{(e)},$$

(38)

Here, ${Q}^{(e)}=\left[\begin{array}{cc}1&{x}_{m}\\ 1&{x}_{n}\end{array}\right]$ & ${\overline{u}}^{(e)}=\left[\begin{array}{c}{u}_{m}\\ {u}_{n}\end{array}\right]$

From Eqs. (36)–(38) we obtain,

$${u}^{(e)}={Q}^{T}{S}^{(e)}{\overline{u}}^{(e)},$$

(39)

Here, ${S}^{(e)}={{Q}^{(e)}}^{-1}=\frac{1}{{x}_{n}-{x}_{m}}\left[\begin{array}{cc}{x}_{n}&-{x}_{m}\\ -1&1\end{array}\right]$

$${J}^{(e)}={J}_{k}^{(e)}+{J}_{m}^{(e)}-{J}_{l}^{(e)}-{J}_{r}^{(e)}-{J}_{p}^{(e)}-{J}_{q}^{(e)}-{J}_{s}^{(e)},$$

(40)

Where,

$${J}_{k}^{(e)}=\frac{1}{2}\int\nolimits_{{x}_{m}}^{{x}_{n}}\left[\left({{Q}_{x}}^{T}{S}^{(e)}{{\overline{u}}^{(e)}}^{2}\right)\right]dx,$$

(41)

$${J}_{m}^{(e)}=\frac{1}{2}\int\nolimits_{{x}_{m}}^{{x}_{n}}\frac{1}{{D}_{Ca}}\frac{\partial }{\partial t}\left[\left({Q}^{T}{S}^{(e)}{{\overline{u}}^{(e)}}^{2}\right)\right]dx,$$

(42)

$${J}_{l}^{(e)}=\frac{1}{2}\int\nolimits_{{x}_{m}}^{{x}_{n}}{A}_{1}\left[\left({Q}^{T}{S}^{(e)}{{\overline{u}}^{(e)}}^{2}\right)\right]dx,$$

(43)

$${J}_{r}^{(e)}=\int\nolimits_{{x}_{m}}^{{x}_{n}}{B}_{1}\left[\left({Q}^{T}{S}^{(e)}{\overline{u}}^{(e)}{{Q}^{T}{S}^{(e)}\overline{V}}^{(e)}\right)\right]dx,$$

(44)

$${J}_{p}^{(e)}=\int\nolimits_{{x}_{m}}^{{x}_{n}}{C}_{1}\left[\left({Q}^{T}{S}^{(e)}{\overline{u}}^{(e)}{{Q}^{T}{S}^{(e)}\overline{b}}^{(e)}\right)\right]dx,$$

(45)

$${J}_{q}^{(e)}=\int\nolimits_{{x}_{m}}^{{x}_{n}}{D}_{1}\left[\left({Q}^{T}{S}^{(e)}{\overline{u}}^{(e)}\right)\right]dx,$$

(46)

$${J}_{s}^{(e)}={\mu }^{(e)}\left[\left(\frac{{\sigma }_{Ca}}{2{D}_{Ca}}{Q}^{T}{S}^{(e)}{\overline{u}}_{(x = 0)}^{(e)}\right)\right],$$

(47)

Minimizing J^(e) with respect to ${\overline{u}}^{(e)}$,

$$\frac{d{J}^{(e)}}{d{\overline{u}}^{(e)}}=0,$$

(48)

That is,

$$\frac{d{J}^{(e)}}{d{\overline{u}}^{(e)}}=\frac{d{J}_{k}^{(e)}}{d{\overline{u}}^{(e)}}+\frac{d{J}_{m}^{(e)}}{d{\overline{u}}^{(e)}}-\frac{d{J}_{l}^{(e)}}{d{\overline{u}}^{(e)}}-\frac{d{J}_{r}^{(e)}}{d{\overline{u}}^{(e)}}-\frac{d{J}_{p}^{(e)}}{d{\overline{u}}^{(e)}}-\frac{d{J}_{q}^{(e)}}{d{\overline{u}}^{(e)}}-\frac{d{J}_{s}^{(e)}}{d{\overline{u}}^{(e)}},$$

(49)

Which can be written as,

$$\frac{dJ}{d{\overline{u}}^{(e)}}=\mathop{\sum }\limits_{e=1}^{80}{\overline{M}}^{(e)}\frac{d{J}^{(e)}}{d{\overline{u}}^{(e)}}{\left({\overline{M}}^{(e)}\right)}^{T}=0,$$

(50)

Where,

$${\overline{M}}^{(e)}={\left[\begin{array}{cc}0&0\\ .&.\\ 0&0\\ 1&0\\ 0&1\\ 0&0\\ .&.\\ 0&0\end{array}\right]}_{81\times 2}\begin{array}{l}{m}^{th}row\\ {n}^{th}row\end{array},{\overline{u}}^{(e)}={\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\\ {u}_{3}\\ {u}_{4}\\ .\\ .\\ {u}_{80}\\ {u}_{81}\end{array}\right]}_{81\times 1}$$

In similar manner Eqs. (31) and (32) are solved using linear elements leading again a 81 × 81 system respectively.

Consequently, the system of linear algebraic equations illustrated below is obtained.

$${[\overline{K}]}_{(243\times 243)}\overline{U}+{[\overline{N}]}_{(243\times 243)}{\frac{\partial \overline{U}}{\partial t}}_{(243\times 1)}={[\overline{F}]}_{(243\times 1)}.$$

(51)

Here, $\overline{U}$ is given by $\left[\begin{array}{c}\overline{u}\\ \overline{V}\\ \overline{b}\end{array}\right]$, system matrices are shown as $\overline{K}$ and $\overline{N}$ and the characteristic vector $\overline{F}$.

Utilizing the MATLAB program, the Crank-Nicolson method is employed to solve the system.

Table

For the purpose of resolving the proposed problem (51), these physiological variables are employed.

Findings and Discussion

Calcium, buffer and IP₃ fluctuation is depicted in Fig. 1. Calcium fluctuation relative to space is seen in Fig. 1A. The calcium concentration is first high near the source and then starts to decline as one moves along the spatial dimension until it achieves its equilibrium concentration. The maximum calcium content is approximately 0.8 μM. Graph shows non-linear behavior of the calcium concentration pattern between x = 0 to 10 μm. This may be due to major imbalances among biophysical processes like diffusion, buffering, efflux and influxes which is clear from the major difference in calcium concentration at x = 0 and x = 10 μm. Calcium variation over time is shown in Fig. 1B. The graph shows that the calcium concentration initially rises abruptly up to around 80 ms, then rises steadily and smoothly until it reaches steady state at approximately 80 ms. The spatial variation of IP₃ concentration is depicted in Fig. 1C. It has been noted that the graph initially behaves nonlinearly before eventually changing to linear behavior. As one moves through space away from the source, the concentration of IP₃ decreases until it reaches ≈ 0.16 μM. Close to the source, the concentration of IP₃ is high. The IP₃ concentrations can reach a maximum value of ≈ 3 μM. The fluctuation of IP₃ concerning time is shown in Fig. 1D. According to the graph, the initial IP₃ concentration increases swiftly for the first 100 ms or so before rising steadily and gradually until it reaches steady state at around 100 ms. Figure 1E displays the variation in free buffer concentration over space. Because too much calcium is toxic for cells, calcium-bound buffer is formed when free buffer and free calcium bond together. Since calcium concentrations are higher close to the source, more buffer is required to lower calcium concentrations there. The free buffer value is therefore lowest close to the source. Buffer diffuses to the calcium source, whereas calcium diffuses to the other end of the cell. Therefore, it is seen that the buffering process is dominated by the source influx of free calcium close to the source, whereas the buffering process is dominated by the calcium signals at the opposite end of the boundary. The fluctuation in free buffer concentration over time is depicted in Fig. 1F. Free buffer concentration first rises steadily and gradually for 500 ms before reaching steady state.

Fig. 1

Ca²⁺, IP₃ and buffer concentration dynamics along space and time

Figure 2 shows [Ca²⁺] variations for different b_tot values along space and time. Figure 2A depicts a map of the variation in [Ca²⁺] with regard to space. As the levels of total buffer concentration rise, a drop in calcium concentration is observed. Calcium concentration falls as a result of an increase in the amount of calcium-bound buffer caused by an increase in buffer’s overall value. [Ca²⁺] is highest close to the source, and it reaches equilibrium as one moves away from it. The fluctuation in [Ca²⁺] with respect to time is depicted in Fig. 2B. The concentration of calcium first rises quickly until 350 ms, at which point it stabilizes. The fluctuation in calcium concentration exhibits the same behavior as that depicted in Fig. 1.

Fig. 2

Ca²⁺ concentration dynamics along space and time for various values of total buffer concentration

Figure 3 illustrates [Ca²⁺] variation over time and space for various association rates of buffers which are BAPTA, EGTA and Triponin C. Figure 3A shows [Ca²⁺] variation over space. It is demonstrated by comparing the [Ca²⁺] at the source that it is maximum in the presence of EGTA buffer. Presence of BAPTA buffer changes the behavior of [Ca²⁺] dynamics and Triponin C reduced [Ca²⁺] by 90% as association rate for BAPTA and Triponin C are very high. Fig. 3B illustrates the variation in calcium content over time. The results show that the addition of EGTA causes the calcium concentration to rise sharply and gradually up to 350 ms before stabilizing. Triponin C and BAPTA buffers allow to achieve a steady state in 30 ms. The presence of BAPTA and triponin C buffers causes oscillations.

Fig. 3

Ca²⁺ concentration dynamics along space and time for various buffer association rates

Figure 4 illustrates [Ca²⁺] variation over space and time for different buffers’s diffusion coefficient (D_b). Figure 4A illustrates how the [Ca²⁺] varies along space. It has been demonstrated that with increase in D_b values, [Ca²⁺] decreases, leading to an increase in the generation of calcium-bound buffers. Calcium has its maximum concentration near the source and approaches equilibrium farther away. The calcium concentration fluctuation with time is shown in Fig. 4B. Prior to reaching a steady state at 1800 ms, the concentration of calcium first increases swiftly. The small oscillation in the curves are due to formation of Ca²⁺ bound buffer’s binding time and diffusion of free buffer and Ca²⁺ towards each other.

Fig. 4

Ca²⁺ concentration dynamics along space and time for various values of diffusion coefficient of buffer

Figure 5 illustrates variation in ATP degradation rates across space and time. Figure 5A demonstrates that the ATP degradation rate is highest at the source and diminishes progressively with distance as. This spatial trend in ATP degradation mirrors the non-linear pattern due to major imbalances among biophysical processes like diffusion, buffering, efflux and influxes as observed in Fig. 1A. Figure 5B details the temporal dynamics of ATP degradation, showing a sharp initial increase up to 200 ms, followed by a more gradual increase to reach steady state after 200 ms. This temporal pattern of ATP degradation shares similarities with the temporal fluctuations of calcium observed in Fig. 1B.

Fig. 5

ATP degradation dynamics along space and time

Figure 6 depicts production rate of NADH along space and time. Figure 6A, the spatial distribution of NADH production rate shows a trend where higher calcium concentrations correspond to elevated NADH production rates, gradually stabilizing to around 0.38 μM/sec as distance from the source increases. This non-linear relationship due to major imbalances among biophysical processes like diffusion, buffering, efflux and influxes mirrors the behavior of Ca²⁺ variation as depicted in Fig. 1A. In Fig. 6B, the temporal dynamics of production rate of NADH reveal an initial sharp increase up to 200 ms, followed by a more gradual and smooth rise, eventually reaching a steady state by 200 ms. This temporal pattern of NADH production rate closely resembles the temporal behavior of Ca²⁺ variation illustrated in Fig. 1B.

Fig. 6

NADH production dynamics along space and time

Figure 7 displays spatial and temporal Ca²⁺-dependent ATP production rate. Figure 7A shows variation in calcium-dependent production rate of ATP along space. Near source Ca²⁺-dependent production rate of ATP is high and decreases upto 0.18 μM/sec moving away from the source. Maximum calcium-dependent ATP production rate is ≈ 0.7 μM/sec. The non-linear behavior due to major imbalances among biophysical processes like diffusion, buffering, efflux and influxes mirrors the behavior of Ca²⁺ variation as depicted in Fig. 1A. Figure 7B shows the rate of Ca²⁺-dependent ATP synthesis over time. The graph shows that, at first, the Ca²⁺-dependent ATP production rate increases quickly up to ≈ 200 ms, after which it grows steadily and smoothly to reach steady state at almost 200 ms. The calcium variation’s nonlinear behavior in Fig. 1 and the ATP generation rate in Fig. 7 exhibit synergistic effects.

Fig. 7

Ca²⁺ dependent ATP production dynamics along space and time

Figure 8 shows spatial and temporal IP₃-dependent ATP production rate. Figure 8A shows variation in IP₃-dependent production rate of ATP along space. Near the source IP₃-dependent ATP production rate is high and decreases upto 0.18 μM/sec moving away from the source. Maximum IP₃-dependent ATP production rate is ≈ 1 μM/sec. The non-linear behavior is similar to Fig. 1A initially as time increases such as 500 ms behavior of the curves changes and becomes concave. This change in behavior of the curves is due to role of increase in calcium concentration on IP₃ and ATP production. Initially when calcium concentration is low, the IP₃ concentration is also lower and ATP production rate is also lower and with the passage of time, calcium concentration increases and its effect is visible on IP₃ profiles in Fig. 1C and on ATP production in Fig. 8A. Also after sometime such as 500 ms, the calcium concentration becomes high enough to play significant role which is visible by change in behavior of curves in Figs. 1A and 8A. Figure 8B represents IP₃-dependent production rate of ATP with respect to time. The graph demonstrates how the initial IP₃-dependent ATP production rate grows abruptly up to ≈ 20 ms, then rises gradually and smoothly until it takes ≈ 20 ms to reach steady state.

Fig. 8

IP₃ dependent ATP production dynamics along space and time

The variations in calcium concentration for both normal and obese hepatocyte cells across time and space are shown in Fig. 9. The fluctuations in [Ca²⁺] with respect to space is shown in Fig. 9A. It has been noted that increased ER leakage causes an increase in calcium concentration in obese hepatocyte cells. [Ca²⁺] is highest close to the source and decreases to the background level (0.1 μM) as one moves away. Moreover, the temporal fluctuation in calcium concentration for normal and obese hepatocyte cells is shown in Fig. 9B. The fluctuations in the calcium concentration exhibit behavior that is in line with the observations shown in Fig. 1A and B. A notable variation in the calcium content is seen from the curves in Fig. 9 for the two distinct scenarios.

Fig. 9

Ca²⁺ dynamics for normal and obese cell along space and time

The change in the rate of ATP breakdown for normal and obese hepatocyte cells over time and space is shown in Fig. 10. The spatial variation in ATP degradation rate for normal and obese hepatocyte cells is shown in Fig. 10A. It is demonstrated from the figure that how the rate of ATP breakdown increases in tandem with the amounts of calcium in the obese hepatocyte cell as the ER becomes leaky. The rate of ATP degradation peaks at the source and falls away from it to a constant value of 4 μM/sec. The variation in the rate of ATP breakdown for both normal and obese hepatocyte cells over time is shown in Fig. 10B. Figure 10 illustrates a large variation in the rate of ATP breakdown for two distinct scenarios. The ATP degradation rate variation behaves similarly to Fig. 9.

Fig. 10

ATP degradation rate dynamics for normal and obese cell with respect to space and time

The variation in the rate of NADH synthesis in a obese and normal hepatocyte cell over time and space is depicted in Fig. 11. The spatial variation in the rate of NADH synthesis for normal and obese hepatocyte cells is shown in Fig. 11A. The figure shows that as compared to a normal hepatocyte cell, the rate of NADH synthesis rises in the obese state as the ER becomes leaky. The decrease in the rate of NADH generation for both normal and obese hepatocyte cells over time is shown in Fig. 11B. Two different scenarios with different rates of NADH generation are depicted in Fig. 11. According to Fig. 9, the NADH production rate fluctuation acts similarly.

Fig. 11

Production rate of NADH dynamics for normal and obese cell along space and time

The calcium-dependent ATP generation rate variation for normal and obese hepatocyte cells over time and space is shown in Fig. 12. Figure 12A illustrates the variation in the rate of Ca²⁺-dependent ATP synthesis in relation to space in a obese and normal hepatocyte cells. Figure shows how the rate of calcium-dependent ATP production increases in an obese hepatocyte cell in tandem with an increase in calcium concentration when the ER becomes leaky. The rate at which calcium-dependent ATP is produced reaches its maximum near the source and falls away from it to a stable value of 0.2 μM/sec. The calcium-dependent ATP generation rate changes with time for both normal and obese hepatocyte cells are shown in Fig. 12B. The ATP generation rates for two distinct circumstances are displayed in Fig. 12, each of which exhibits a notable variance. The behavior of the Ca²⁺-dependent ATP production rate fluctuation is similar to that of Ca²⁺, as seen in Fig. 9.

Fig. 12

Ca²⁺- dependent production rate of ATP dynamics for obese and normal cell over space and time

Error Analysis

For time =0.1, 0.2, 0.3, 0.4, and t = 0.5 sec at x=0, error analysis is carried out Tables 1 and 2. According to Tables 3, 4 and 5, the greatest percentage error between nodes 80 and 90 for calcium, IP₃ and buffer dynamics is 0.048172601%, 0.22712485% and 0.001948211%, respectively. The accuracy of the finite element approach with 80 linear elements is found to be 99.95% for the calcium profile, 99.77% for the IP₃ profile and 99.99% for buffer profile in this problem.

Table 1 Ca²⁺, IP₃ and buffer variation parameters [32, 57]

Table 2 Comparative study [70]

Table 3 Calcium profile error analysis with 80 & 90 elements

Table 4 IP₃ profile error analysis with 80 & 90 elements

Table 5 Buffer concentration profile’s error analysis with 80 & 90 elements

Stability Analysis

The finite element approach has a spectral radius of 0.9994 which is below one, indicating that it is stable.

Validation

The results are similar with earlier study by Wagner and Pawar et al. [45, 48] at time t = 50 sec, as shown in Tables 6 and 7. The concentration profiles of [Ca²⁺] and IP₃ that were obtained for the parameter at x = 0, 3, 6, 9, 12 and 15 μm are compared to these results.

Table 6 [Ca²⁺] profiles compared with Wagner and Pawar et al. [45, 48]

Table 7 IP₃ concentration profiles compared with Wagner and Pawar et al. [45, 48]

Conclusion

A mathematical model of cross talking dynamics of calcium, buffers and IP₃ is proposed and successfully implemented to study impacts of spatio temporal variations in one of them on the other two signaling molecules. The outcomes of the simulations lead to the following fundamental conclusions:

• (i) [Ca²⁺] increases with increase in source influx and consequently IP₃ increases.

• (ii) Calcium concentration decreases with increasing buffer value and consequently IP₃ decreases.

• (iii) Calcium has the maximum concentration near the source and approaches equilibrium when one travels farther away from it in space. For IP₃, comparable outcomes are attained.

• (iv) According to temporal study, the calcium concentration initially rises abruptly for about 450 ms before rising gradually and smoothly until it reaches a steady state. Similar results are obtained for IP₃.

The above basic conclusions are the same as obtained by earlier researchers [9, 45] which indicate that the proposed model is effective in prediction of experimental results.

The following novel findings are reached by analyzing numerical results:

• (i) The free buffer concentration falls till x= 10 μm as a result of strong buffering activity which lowers the free [Ca²⁺] at those places (x= 0 to 10 μm) and thus the spatial regions with high calcium concentrations also have low free buffer concentrations.

• (ii) The concentration of free buffer rises slowly in the spatial places (x=1 to 10 μm) where the concentration of calcium rises the fastest because the majority of free buffer binds to free calcium. The calcium concentration rises slowly in the same way that it does wherever the buffer concentration is rising quickly.

• (iii) Depending on where each of the calcium, IP₃ and buffers dominates at different times, it leads to the variations in concentration levels of the other two signaling molecules. Thus these three signaling systems work in a synergistic manner to balance each others fluctuations to achieve the homeostasis.

• (iv) The dynamic relationship between the fluctuations of free buffer and free calcium is determined by the rate of a modest increase in buffer activity.

• (v) Source, buffer, SERCA pump and other parameters that affect calcium profiles are synergistically conveyed to IP₃, ATP degradation, ATP synthesis and NADH production rate. Therefore, changes in these parameters can result in considerable variations in the rates of ATP generation, ATP degradation and NADH production which can result in a variety of liver illnesses like obesity, diabetes etc.

• (vi) The significant discrepancy in calcium concentration, ATP production, degradation and NADH production rate in a obese and normal hepatocyte cell has provided us with new insight into the severity of the disorder in the form of obesity, among other things.

• (vii) As ER calcium content drops by 50% in case of obesity, ER calcium release rises by 50%. According to Arruda et al. [69], mitochondria receive 80% of the inflow from the ER. This led to the prediction that a hepatocyte cell’s cytoplasmic calcium concentration would only rise by 10%, as seen in Fig. 9.

The finite element method (FEM) showcases remarkable versatility, yet grapples with mesh dependency, complexity in 3D scenarios, and challenges in handling discontinuities. Meanwhile, the Crank-Nicolson method, lauded for its stability, contends with constraints in stability, computational overhead, and accuracy, particularly in nonlinear domains. Both methods demand finesse in implementation, with FEM demanding meticulous meshing and Crank-Nicolson necessitating careful consideration of system dynamics. Despite these limitations, their combined prowess remains a cornerstone in numerical simulations, driving innovation across diverse fields of science and engineering. Consequently, the proposed model demonstrates a commendable ability to accurately estimate calcium concentration levels in both normal and obese cellular conditions. Additionally, the model demonstrates the ability to forecast the impacts of diverse variable alterations on calcium, buffer and IP₃ concentrations, alongside NADH generation, ATP biosynthesis and ATP degradation rate within hepatocyte cells, across both normal and obese states. This predictive capability enhances our understanding of the intricate cellular dynamics and provides valuable insights into the physiological implications of these changes in calcium and energy-related processes.