Impact of Interdependent Ca²⁺ and IP₃ Dynamics On ATP Regulation in A Fibroblast Model

Abstract

The vital participation of Ca²⁺ in human organ functions such as muscular contractions, heartbeat, brain functionality, skeletal activity, etc, motivated the scientists to thoroughly research the mechanisms of calcium (Ca²⁺) signalling in distinct human cells. Ca²⁺, inositol triphosphate (IP₃), and adenosine triphosphate (ATP) play important roles in cell signaling and physiological processes. ATP and its derivatives are hypothesized to be important in the pathogenic process that leads to fibrotic illnesses like fibrosis. Fluctuations in Ca²⁺ and IP₃ in a fibroblast cell influence ATP production. To date, no evidence of coupled Ca²⁺ and IP₃ mechanics regulating ATP generation in a fibroblast cell during fibrotic disease has been found. The current work suggests an integrated mechanism for Ca²⁺ and IP₃ dynamics in a fibroblast cell that regulates ATP generation. Simulation has been carried out using the finite element approach. The mechanics of interdependent systems findings vary dramatically from the results of basic independent system mechanics and give fresh information about the two systems’ activities. The numerical results provide new insights into the impacts of disturbances in source influx, the serca pump, and buffers on interdependent Ca²⁺ and IP₃ dynamics and ATP synthesis in a fibroblast cell. According to the findings of this study, fibrotic disorders cannot be attributed solely to disruptions in the processes of calcium signaling mechanics but also to disruptions in IP₃ regulation mechanisms affecting the regulation of calcium in the fibroblast cell and ATP release.

Introduction

Ca²⁺ is an essential signaling ion in many biological processes and activities across various tissues and animals. Ca²⁺ signaling is crucial for a number of cellular processes in fibroblast cells, including proliferation, migration, myofibroblast differentiation and extracellular matrix (ECM) production. Dysregulation of Ca²⁺ signaling may contribute to the development of fibrotic diseases [1].

G-protein-coupled receptors (GPCRs) are proteins found on the surface of fibroblast cell [2]. When a neurotransmitter or signal molecule connects to a GPCR, the GPCR’s shape changes. This modification activates a G protein that is linked to a GPCR. This G protein serves as a messenger, allowing the signal to be sent farther. The activated G protein subsequently interacts with a phospholipase C (PLC) effector enzyme. This interaction causes the synthesis of IP₃. IP₃ serves as a backup messenger [3]. It goes throughout the cell in search of its target, the endoplasmic reticulum (ER), a compartment within the cell that stores calcium ions. IP₃ binds to particular receptors on the ER membrane when it enters the ER. This binding leads the ER to release calcium ions that have been held in the cell’s cytoplasm [4, 5]. In fibroblast cells, IP₃ has a significant impact in mediating various cellular processes including differentiation, migration cell, proliferation and survival. Specifically, IP₃-mediated Ca²⁺ signaling regulates fibroblast migration and proliferation, which are important processes for wound healing and tissue repair [5].

ATP is a molecule that significantly impacts cellular energy metabolism. ATP has also been linked to the pathogenesis of fibrotic disorders. Fibrosis is a pathological disease characterized by an abnormal accumulation of ECM proteins leading to tissue scarring and dysfunction [6]. In the context of fibrosis and wound healing, the production and degradation of ATP play important roles in cellular processes that influence tissue repair and remodeling. During tissue injury or inflammation, cells often experience increased metabolic demands and stress. This can lead to overproduction of ATP through processes like glycolysis and oxidative phosphorylation in mitochondria. Overproduction of ATP can be influenced by factors such as increased cellular activity, inflammation, and hypoxia [7]. ATP is essential for cell migration and proliferation, which are critical for wound healing and tissue repair [8]. ATP acts as a signaling molecule and can attract immune cells to the site of injury or inflammation, aiding in the resolution of tissue damage. Overproduction of ATP can lead to excessive inflammation and oxidative stress, which can contribute to tissue damage and exacerbate fibrosis [9]. High levels of extracellular ATP can activate purinergic receptors on fibroblasts and other cells, promoting fibrosis by stimulating collagen production and fibroblast differentiation [10]. ATP degradation leads to the production of adenosine, which has vasodilatory and anti-inflammatory properties. Adenosine can promote tissue protection and healing by reducing inflammation and enhancing blood flow to the site of injury [11]. Adenosine can modulate immune responses and promote tissue repair by promoting the differentiation of fibroblasts into myofibroblasts, which are important for wound contraction and matrix deposition. Inefficient ATP degradation can result in the accumulation of ATP in the extracellular space, leading to sustained activation of purinergic receptors and perpetuating inflammation and scarring in multiple organs, including the lung, heart, kidney and liver [10,11,12]. Therefore, a surplus of ATP and inadequate removal of ATP insufficient degradation can leads various illnesses of the fibroblast, including wound healing and fibrosis. As a result, ATP can trigger excitotoxicity, which can harm fibroblasts or even kill them. ATP pathogenic mechanisms and their roles in fibrotic diseases are still unknown. Therefore, it is essential to make models that show the movements of Ca²⁺ and IP₃ affecting the production of ATP.

Modeling keratinocyte-fibroblast interactions during the normal and abnormal wound recovery process is described by Shakti et al. [13]. Droniou et al. [14] investigated a two-dimensional tumor development model with shifting boundaries that were numerically solved. Using the idea of advection-diffusion, Simpson et al. [15] investigated the contributions of directed and random motility in a fibroblast cell migration test induced by an electric field. Michell [16] proposed a link between high intracellular Ca²⁺ and PIP₂ catabolism. Berridge and Irvine [3] discovered IP₃ to be a second messenger that mobilizes Ca²⁺ from the ER. Regarding cellular transmitter release, Fogelson and Zucker [17] proposed the model of presynaptic diffusion of Ca²⁺ that includes inflow, cytoplasmic binding and surface pumps. Jafri and Keizer [18] examined Ca²⁺ transport in the cell cytosol and Ca²⁺ handling by the ER in their study of IP₃-induced Ca²⁺ wave propagation.

Smith et al. [19] developed a reaction-diffusion model describing Ca²⁺ transport with diffusion, source influx, and buffer mechanisms. The reaction model for calcium buffering has been described in fibroblast cells [20, 21]. Wagner and Keizer [22] studied the effects of fast buffers on Ca²⁺ transport and fluctuation. Kotwani and Adlakha [23] developed a reaction-diffusion model to study effect of buffer binding affinity, diffusion and source amplitude in fibroblast cells. Kotwani and Adlakha [24] investigated a computational model of calcium distributions with source fluxes, serca, and buffer approximations in a fibroblast cell. Sun et al. [25] constructed a dynamic model for intracellular calcium responses in a fibroblast cell induced by electrical stimulation.

Manhas and Pardasani [26] proposed a probable mechanism for Ca²⁺ fluctuations based on the interaction of the cell’s three Ca²⁺ stores like mitochondria, cytosol and ER. Manhas and Pardasani [27] created a mathematical model to investigate the IP₃ kinetics-based Ca²⁺ fluctuations in pancreatic acinar. Jha and Adlakha [28] created a two-dimensional unsteady state model of Ca²⁺ concentration change in neurons. Joshi and Jha [29] analyze the physiological involvement of calbindin-D28k and VGCC in Parkinson’s disease using a mathematical model that incorporates all key factors such as diffusion coefficient and fluxes. Joshi and Jha [30] created a mathematical model based on the Hilfer fractional reaction-diffusion equation to investigate Ca²⁺ transport in cells. Jha and Adlakha [31] conducted two-dimensional finite element simulations of unsteady state Ca²⁺ diffusion in neurons with a ER leak and a serca pump. Jha and Adlakha [32] sought to investigate the impact of sodium-Ca²⁺ exchangers and source geometry on the dynamics of Ca²⁺ in a neuron cell. Tewari and Pardasani [33] investigated the oscillations in Ca²⁺ caused by dynamically changing membrane potential, focusing on the sodium pump. Pathak and Adlakha [34] created a two-dimensional model of Ca²⁺ distribution in myocytes, including excess buffer approximation, diffusion and point source. In a one-dimensional instance, Naik and Pardasani [35] investigated the influence of the serca pump and ryanodine receptor (RyR) on the Ca²⁺ profile in oocytes. For a one-dimensional unstable state situation, Naik and Pardasani [36] created a finite element model of Ca²⁺ distribution in oocytes. Naik and Pardasani [37] used a three-dimensional finite element model to investigate the spatiotemporal Ca²⁺ distribution in oocytes. Kotwani et al. [38] created a two-dimensional unsteady state Ca²⁺ distribution model in fibroblast cells. Kothiya and Adlakha [39] developed a model to investigate the function of Ca²⁺ signaling in the production of IP₃ and ATP in fibroblast cells. Jha et al. [40] use FEM to investigated the cytosolic Ca²⁺ concentration distribution in astrocytes in a two-dimensional steady-state situation with excess buffer. Jagtap and Adlakha [41] used finite volume and Crank Nicolson simulations to investigate the advection-diffusion of Ca²⁺ in a hepatocyte cell. ATP interacts with P₂Y purinergic receptors in the cell membrane, activating PLC, which catalyses IP₃ synthesis from PIP₂. IP₃ interacts synergistically with Ca²⁺ in receptors on the surface of the endoplasmic reticulum (ER) and opens channels that allow Ca²⁺ ions to be secreted into the cytoplasm. Ca²⁺ promotes and inhibits its own release. The huge Ca²⁺ concentration gradient also causes a Ca²⁺ leak. An ATP-dependent pump returns Ca²⁺ to the ER. As a result of this cascade, ATP is released into the extracellular space. The study of Stamakis and Mantzaris [42] provides two ideas concerning ATP release: Ca²⁺-regulated or IP₃-regulated ATP synthesis.

Let’s go further into the physiological dynamics of ATP synthesis, concentrating on the interaction between adenosine generation via de novo purine and salvage pathways, as well as the mitochondrial ETC. De Novo Purine Pathway and Adenosine Synthesis: The de novo purine process entails the progressive synthesis of purine bases from simple precursors such as aspartate, carbon dioxide, glutamine, and glycine. The route begins with the creation of the purine ring structure, which is followed by a series of enzymatic activities that result in the production of inosine monophosphate (IMP). This is subsequently converted into adenosine monophosphate (AMP) through a series of reactions, ultimately leading to ATP synthesis during oxidative phosphorylation [43]. Salvage Pathway for Adenosine Production: Cells use the salvage process to recycle purine bases (adenine and guanine) from nucleotides and nucleosides produced during DNA and RNA degradation. This recycling conserves energy and resources by reusing purine components to build new nucleotides. This enzyme catalyzes the cleavage of nucleosides (e.g., adenosine) into their respective purine bases (adenine). Adenine is then converted into adenosine monophosphate (AMP) through the addition of a phosphoribosyl group. AMP can subsequently undergo phosphorylation to form adenosine diphosphate (ADP) and, ultimately, ATP through the process of oxidative phosphorylation in the mitochondria [44]. Mitochondrial Electron Transport Chain (ETC) andATP Synthesis: The mitochondrial ETC is a crucial component of ATP synthesis, occurring within the inner mitochondrial membrane. It involves a series of protein complexes that facilitate electron transport and proton pumping. Electrons obtained from molecules like glucose and fatty acids are shuttled through these complexes, leading to the transfer of electrons along the ETC. As electrons move through the complexes, protons (H+ ions) are pumped across the inner mitochondrial membrane, creating a proton gradient. This enzyme complex, also known as Complex V, is embedded in the membrane and harnesses the proton gradient to synthesize ATP from ADP and inorganic phosphate (Pi) through a process called chemiosmotic coupling. As protons flow back into the mitochondrial matrix through ATP synthase, the energy generated is used to drive ATP synthesis [45].

In mouse fibroblasts, Bryan et al. [46] investigated the dual function of ATP in maintaining volume-regulated chloride channels and in regulating the release of Ca²⁺. Miyoshi et al. [47] looked at how cells die as they age and what role ATP plays in apoptosis and necrosis caused by hydrogen peroxide. Astrocytic cells were explored by Stamatakis and Mantzaris [42], who investigated the link between Ca²⁺-regulated and IP₃-regulated ATP production mechanisms. Chen et al. [6] demonstrated that ATP activates the P₂Y₂ receptors in fibroblasts of human cardiac muscle and raises their proliferation via increasing cell cycle progression. It also has the additional effect of increasing the migration of cells. These actions of ATP may assist in modifying the structure of damaged hearts. Kotwal et al. [48] investigated the essential function ATP plays in macrophages as they expedite wound healing and regeneration. One-dimensional mathematical models of T-lymphocyte Ca²⁺ distribution were constructed by Kumar et al. [49] using a finite element technique. Source inflow, buffers and RyR receptor effects were also acquired for unusual circumstances, and the serca pump effects were found for the temporal case. A one-dimensional fractional reaction-diffusion model for RyR, VGCC and calbindin-D28k was explored by Joshi and Jha [29]. They found analytical answers for Parkinson’s brain physiology. Jagtap and Adlakha [50] used the finite volume method to be looked into the cytosolic Ca²⁺ distribution in hepatocyte cells. Joshi et al. [51] investigated intracellular calcium dynamics in fibroblasts using an exponential kernel rule.

Glycolysis is a key metabolic pathway that happens in the cytoplasm of cells and acts as the first step in the breakdown of glucose to create ATP. Glycolysis begins with the activation of glucose, a six-carbon sugar. Two molecules of ATP are initially invested to convert glucose into fructose-1,6-bisphosphate. The six-carbon sugar molecule is split into two three-carbon molecules: glyceraldehyde-3-phosphate and dihydroxyacetone phosphate. Only glyceraldehyde-3-phosphate continues through the pathway. Glyceraldehyde-3-phosphate undergoes a series of reactions, leading to the production of two molecules of NADH (a high-energy electron carrier) and four molecules of ATP. However, since two ATP molecules were initially invested, there is a net gain of two ATP molecules per glucose molecule. The final steps of glycolysis convert each glyceraldehyde-3-phosphate into pyruvate, a three-carbon compound. During this process, an additional two molecules of NADH are produced [52]. The Tricarboxylic Acid (TCA) Cycle, often known as the citric acid cycle or Krebs cycle, is a major metabolic system that occurs in eukaryotic cell mitochondria and prokaryotic cell cytoplasm. It is an essential component of respiration in cells and helps to generate energy in the form of ATP. The TCA cycle begins when acetyl coenzyme A (Acetyl-CoA) enters the cycle. Acetyl-CoA is derived from various sources, including the breakdown of carbohydrates, fats, and amino acids. The acetyl group from Acetyl-CoA is combined with a four-carbon compound called oxaloacetate to form citrate. Citrate undergoes a series of enzyme-catalyzed reactions, including isomerization and dehydration, leading to the regeneration of isocitrate. It is further oxidized by the enzyme isocitrate dehydrogenase, resulting in the release of carbon dioxide (CO₂) and the formation of α-ketoglutarate. This step also generates reduced nicotinamide adenine dinucleotide (NADH), a carrier molecule that stores high-energy electrons [53]. α-Ketoglutarate is then oxidized by the enzyme α-ketoglutarate dehydrogenase complex, producing another molecule of CO₂ and another molecule of NADH. The resulting molecule, succinyl-CoA, also captures high-energy electrons. Succinyl-CoA undergoes a series of reactions that ultimately lead to the formation of succinate. This step releases a molecule of guanosine triphosphate (GTP), which is a molecule similar to ATP and can be readily converted into ATP. Succinate is further oxidized by the enzyme succinate dehydrogenase, which is embedded in the inner mitochondrial membrane and also serves as a part of the ETC. This oxidation generates another molecule of FADH2, another carrier molecule storing high-energy electrons. The oxidation of succinate produces fumarate. This step involves the addition of water to the double bond of succinate. Fumarate is then enzymatically converted to malate. Malate is oxidized by the enzyme malate dehydrogenase, producing another molecule of NADH and regenerating oxaloacetate. Oxaloacetate can then combine with another acetyl group from Acetyl-CoA to restart the cycle. The reduced electron carriers, NADH and FADH2, generated during the TCA cycle, play a crucial role in ATP production. These carriers carry high-energy electrons to the ETC located in the inner mitochondrial membrane. As these electrons move through the ETC, they establish a proton gradient across the membrane, which creates a flow of protons (H⁺) back into the mitochondrial matrix through ATP synthase [54]. This flow of protons drives the ATP synthase enzyme to synthesize ATP from inorganic phosphate (Pi) & ADP in a process known as oxidative phosphorylation. The coupling of electron transport and proton pumping in the ETC with ATP synthesis is termed chemiosmotic coupling. The exact number of ATP molecules generated from one round of the TCA cycle varies based on factors such as the availability of oxygen and the efficiency of electron transport. However, in total, one round of the TCA cycle can contribute to the production of multiple molecules of ATP [55].

Several researchers have investigated the interdependence Ca²⁺ and IP₃ dynamics in cells, such as hepatocytes [56], myocyte [57], etc. Some one-way feedback models of Ca²⁺ with other ions like nitric oxide, β-amyloid and dopamine are proposed by Pawar and Pardasani [58,59,60]. The literature contains models of Ca²⁺ dynamics in fibroblast cells that assume IP₃ as a constant. To make the model more realistic, examine the coupling of Ca²⁺ with other systems’ dynamics.

The role of interdependent Ca²⁺ and IP₃ kinetics a in the control of distinct signalling molecules such as nitric oxide, β-amyloid and ATP in neurons have been studied by Pawar and Pardasani [61,62,63]. Recently, feedback model of NO generation is regulated by Ca²⁺ and IP₃ in fibroblast cells was developed by Kothiya and Adlakha [64]. Singh and Adlakha [65] investigated the influence of source geometry on the system kinetics of calcium and IP₃ in myocytes of the heart. Joshi et al. [51] established a model to study the impact of buffer and calcium oscillation in fibroblast cells. The effects of buffer, serca pump and source influx, on coupled calcium and IP₃ dynamics with insulin secretion in beta cells have been studied by Vaishali and Adlakha [66]. Kothiya and Adlakha [67] developed a two-way feedback model of the system kinetics model of PLC and Ca2⁺ incorporating the buffer mechanism, diffusion parameters, source influx, and serca rate in a fibroblast cell.

There has been no attempt to investigate the influence of coupled calcium and IP₃ kinetics on the control of ATP synthesis. Given the regulatory functions of IP₃, ATP and Ca²⁺ ions in fibroblasts and cancer-associated fibroblast (CAF), there is a need for knowledge about the effects of IP₃ and Ca²⁺ dynamics on ATP synthesis and the cytoprotective and cytotoxic effects of ATP. In the past, no such theoretical investigation in this area has been recorded. The goal of this study is to look at the regulatory and deregulatory effects of system mechanics of calcium and IP₃ in fibroblasts that regulate ATP synthesis. The findings were derived with the use of the FEM. The impact of numerous factors on the paired dynamics of calcium and IP₃, as well as their impacts on ATP synthesis, have been investigated. A model is provided here that incorporates a two-way coupling between the Ca²⁺ and IP₃ systems in fibroblast cells to regulate ATP levels. It is simulated to disclose information on changes in ATP synthesis induced by changes in the amounts of Ca²⁺ and IP₃ in fibroblasts. The coupled dynamics of calcium and IP₃ with a buffer mechanism and ATP release propose to study the role of constitutive processes like buffer, source influx, association rate, etc., in normal and fibrotic conditions.

Mathematical Formulation

By adding buffer terms to the model of Wagner et al. [68] the Ca²⁺ transport in fibroblasts in the presence of IP₃ is expressed as

$$\begin{array}{l}\frac{\partial [C{a}^{2+}]}{\partial t}={D}_{c}\left[\frac{{\partial }^{2}[C{a}^{2+}]}{\partial {x}^{2}}\right]+\left[\frac{{J}_{I{P}_{3}R}-{J}_{serca}+{J}_{leak}}{{F}_{c}}\right]\\ \qquad\qquad -{K}^{+}{[B]}_{\infty }\left[[C{a}^{2+}]-{[C{a}^{2+}]}_{\infty }\right].\end{array}$$

(1)

In this case, [B]_∞ represents the steady-state buffer concentration, while ${[C{a}^{2+}]}_{\infty }$ represents the steady-state Ca²⁺ ion concentration.The notation D_c denotes the diffusion coefficient of Ca²⁺. The variable for time is denoted by the letter ‘t’, while the variable for the position is denoted by the letter ‘x’. The symbol K⁺ represents the rate of buffer association. The influx and outflux terms of equation (1) are given by Keizer et al. [69] as given below,

$${J}_{I{P}_{3}R}={V}_{I{P}_{3}R}{m}^{3}{h}^{3}\left[{[C{a}^{2+}]}_{ER}-[C{a}^{2+}]\right],$$

(2)

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

(3)

$${J}_{leak}={V}_{leak}\left[{[C{a}^{2+}]}_{ER}-[C{a}^{2+}]\right].$$

(4)

Where ${J}_{I{P}_{3}R},{J}_{leak}$ and J_serca each represent a different flow associated with the IP₃ receptor, the leaking pump and the serca pump, respectively. V_leak and ${V}_{I{P}_{3}R}$ are the flux rate constants for leak and IP₃ respectively.The maximum rate and the Michaelis constant for the serca pump are represented by the variables V_leak and ${V}_{I{P}_{3}R}$, respectively.

The definitions of m and h can be found in Keizer and Young’s work [69].

$$m=\left[\frac{[I{P}_{3}]}{[I{P}_{3}]+{K}_{I{P}_{3}}}\right]\left[\frac{[C{a}^{2+}]}{[C{a}^{2+}]+{K}_{Ac}}\right],$$

(5)

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

(6)

${K}_{I{P}_{3}},{K}_{Ac}$ and K_Inh are the dissociation constants of IP₃ that activate Ca²⁺ and inhibit Ca²⁺ successively. Here, IP₃ dynamics in the presence of Ca²⁺ described by Wagner et al. [68] is used and might be represented in fibroblasts as;

$$\frac{\partial [I{P}_{3}]}{\partial t}={D}_{I}\left[\frac{{\partial }^{2}[I{P}_{3}]}{\partial {x}^{2}}\right]+\left[\frac{{J}_{production}-\lambda ({J}_{kinase}+{J}_{phosphates})}{{F}_{c}}\right].$$

(7)

Where [IP₃] shows the cytosolic concentration of IP₃. D_I is IP₃ diffusion coefficient. Wagner et al. [68] gives the Ca²⁺ dependent IP₃ release terms as given below;

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

(8)

Where K_production represents the Michaelis constant for calcium activation, J_production represents the flux term for IP₃ production and V_production represents the maximal IP₃ generation rate. Bugrim et al. [70] and Sims and Allbrittont [71], as noted below, supply the additional flux terms, namely the IP₃ degradation by J_Kin and J_phosphatase.

$${J}_{kin}=(1-\theta 1){V}_{1}\left[\frac{[I{P}_{3}]}{[I{P}_{3}]+2.5}\right]+\theta 1{V}_{2}\left[\frac{[I{P}_{3}]}{[I{P}_{3}]+1/2}\right],$$

(9)

$${J}_{phosphatase}={V}_{3}\left[\frac{[I{P}_{3}]}{[I{P}_{3}]+30}\right].$$

(10)

$$\theta 1=\left[\frac{[C{a}^{2+}]}{[C{a}^{2+}]+(39/100)}\right].$$

(11)

Where V₁, V₃ and V₂ are the maximum rate constants for low calcium (3-kinase), phosphatase and high calcium (3-kinase) respectively. A variable parameter (λ = 30) estimates the degradation rate [68].

The cell calcium level calculates the $C{a}_{ER}^{2+}$ via conversation relationship.

$${[C{a}^{2+}]}_{T}={F}_{E}{[C{a}^{2+}]}_{ER}+{F}_{C}{[C{a}^{2+}]}_{C}.$$

(12)

Initial Conditions

The following two initial conditions are framed by Brown et al. [72] and Dupont et al. [73] based on the assumption that the cell is initially at rest, having an initial concentration of IP₃ and Ca²⁺.

$${[I{P}_{3}]}_{t = 0}=0.16\,\mu M.$$

(13)

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

(14)

Boundary Conditions

For Ca²⁺, the boundary conditions framed by Brown et al. [72] as mentioned below;

$$\left[-{D}_{c}\frac{\partial [C{a}^{2+}]}{\partial x}\right]\to \sigma ,\,as\,x\to 0\,\mu m,$$

(15)

where σ represents the source amplitude.

The furthest border from the source is indicated by [74, 75]

$$\mathop{\lim }\limits_{x\to 10}[C{a}^{2+}]={[C{a}^{2+}]}_{\infty }=0.1\,\mu M.$$

(16)

Also, for IP₃ kinrtics, the boundary circumstances given by Fink et al. [76] and Brown et al. [72] as given below.

$$\mathop{\lim }\limits_{x\to 0}[I{P}_{3}]=3\,\mu M,t \,>\, 0;$$

(17)

$$\mathop{\lim }\limits_{x\to 10}[I{P}_{3}]=0.16\,\mu M.$$

(18)

Regulation of C a ²⁺ and I P ₃ Dependent A T P

The Ca²⁺ and IP₃ regulated ATP synthesis is given by Stamatakis and Mantzaris [42] as,

$${J}_{release}=\beta P([C{a}^{2+}])Q([I{P}_{3}]).$$

(19)

Where, β is the maximum synthesis rate and P([Ca²⁺]) and Q([IP₃]) are the ATP release dependent on Ca²⁺ and IP₃ respectively. The Ca²⁺ and IP₃ dependent ATP release can be characterized respectively as given below;

$$P([C{a}^{2+}])=\left[\frac{\left(\frac{{a}_{0}}{{a}_{0}-1}\right)-2\left(\frac{[C{a}^{2+}]}{{[C{a}^{2+}]}_{max}}\right)}{\left(\frac{1}{{a}_{0}-1}\right)-{\left(\frac{[C{a}^{2+}]}{{[C{a}^{2+}]}_{max}}\right)}^{2}}\right].$$

(20)

$$Q([I{P}_{3}])=\left[\frac{\left(\frac{{b}_{0}}{{b}_{0}-1}\right)-2\left(\frac{[I{P}_{3}]}{{[I{P}_{3}]}_{max}}\right)}{\left(\frac{1}{{b}_{0}-1}\right)-{\left(\frac{[I{P}_{3}]}{{[I{P}_{3}]}_{max}}\right)}^{2}}\right].$$

(21)

Where a₀ and b₀ are constants for ATP feedback and ${[C{a}^{2+}]}_{max}$ and ${[I{P}_{3}]}_{max}$ are the cell’s maximum Ca²⁺ and IP₃ values. The finite element and Crank Nicholson procedure followed for the solution are given in Appendix.

Results and Discussion

A quantitative solution of the partial differential equations representing the coupled dynamics of IP₃ and Ca²⁺ was achieved using MATLAB software. Graphs are employed to visually represent the numerical findings generated by solving the equations. Table 1 shows the numerical measurements of the various parameters used in this study. Figure 1 depicts the systematic pathways of calcium and IP_3 dynamics with the regulation of neurotransmitter ATP production.

Table 1 Physiological variables and their values are listed below [58, 59, 61, 62, 68]

Fig. 1

The schematic diagram reprents the calcium and IP₃ mechanism with ATP release

Figure 2 depicts the geographic and temporal distributions of Ca²⁺ in fibroblast cells at various periods in time and space. Because the source channel delivers many Ca²⁺ ions into the cell, there may be a high Ca²⁺ distribution at the source x = 0 μm. The spatial Ca²⁺ profile lowers from x = 0 μm to x = 10 μm and eventually convergence occurs to the cellular background Ca²⁺ profile (0.1 μM). Because the source channel opens, the temporal concentration of Ca²⁺ rises with time in Fig. 2B and the Ca²⁺ level rises in the cytosol of fibroblast cells due to increasing intracellular Ca²⁺ concentration coming from the source. The temporal Ca²⁺ profile achieved equilibrium within 150 ms regardless of the cell’s location.

Fig. 2: The Ca²⁺ profile in a fibroblasts at different periods and places for [B] = 5 μm, D_c = 16 μm²/s and σ = 15pA.

A The spatial distribution of calcium and B the temporal distribution of calcium

Figure 3 displays the ATP release which is Ca²⁺-dependent for the different time points and positions. The impact of Ca²⁺ concentration on ATP release is significant in situations where Ca²⁺ is high. The ATP release is also high wherever Ca²⁺ profile is high in the fibroblast cell. The Ca²⁺ distribution is generally high near the source. The ATP release is also high at x = 0 μm. In Fig. 2B, it has been seen that Ca²⁺ concentration gets a steady state in 150 ms. In Fig. 3B, the ATP release achieves a steady state in 150 ms. This implies that the ATP release directly depends on Ca²⁺ concentration in the cell.

Fig. 3: Ca²⁺ dependent ATP release for various position and time for σ = 15pA.

A The spatial profile of calcium-dependent ATP release and B the temporal profile of calcium-dependent ATP release

Figure 4 displays the spatiotemporal profile of IP₃ profile at various times and places in fibroblasts. In Fig. 4A, the profile of IP₃ goes down (x = 0 to x = 10) and ends up at a concentration of 0.16 μM in the cell. The IP₃ particles stick together near the IP₃ receptor and achieve it’s border value (3 μM at x = 0 μm) for time t = 0, 5, 10, 20, 50, 100 and 1000 ms in the fibroblast cell. In Fig. 4A it is observed that IP₃ dynamics is nonlinear initially and approaches a linear behavior after the passage of time reaching a steady state condition. This nonlinearity is due to an imbalance among the various processes of the cell and as these processes achieve equilibrium, the behavior of the profile becomes linear. Figure 4B shows the profile of time of IP₃ distribution which increases with time for various places x = 0.25, 0.50, 0.75, 1, 2, 5 & 10 μm synergetically with the increase in Ca²⁺ with time in Fig. 2B.

Fig. 4: IP₃ distribution for various time and position for D_c = 16 μm²/s and σ = 15pA.

A The spatial distribution of IP₃ and B the temporal distribution of IP₃

Figure 5 displays the Ca²⁺ distribution at x = 5 μm for [B] = 400 μM under fibrotic conditions in cell. The fluctuation in the temporal profile of Ca²⁺ is observed for greater buffer levels in the cell at the initial time. This damping fluctuation is due to an imbalance among the various processes of the cell and the buffer binds the Ca²⁺ ions and lowers the level of Ca²⁺ concentration in the cell. In Fig. 5A, we found substantial differences in the Ca²⁺ profiles of normal fibroblasts and CAF, though the behavior of the curves is similar in both cases. The oscillation in the Ca²⁺ profile in Fig. 5A is due to the substantial imbalances among the regulatory processes that exist up to 20 ms and then achieve equilibrium. Similarly, in Fig. 5B, the synergetic changes in ATP level are observed like those in Fig. 5A. The spatiotemporal Ca²⁺ concentrations are higher (30%) in the case of cystic fibrosis than in normal fibroblasts [77], As a result, an increased level of Ca²⁺ concentration is hazardous and can lead to fibrosis [78].

Fig. 5: Ca²⁺ concentration and Ca²⁺ dependent ATP release in a fibroblasts for the buffer concentration [B] = 400 μM at x = 5 μm with the fibrotic conditions.

A The temporal distribution of calcium in normal and cystic fibrosis fibroblast and B the temporal distribution of calcium-regulated ATP in normal and cystic fibrosis fibroblast

Figure 6 shows the IP₃ dependent ATP release for different times and positions. In Fig. 4A, it was seen that IP₃ is highest at x = 0 μm and decreases as we move away from x = 0 μm to x = 10 μm. Similarly in Fig. 6A, we observe that ATP release is highest at x = 0 μm and reduces as it departs from the source. We observed that the behavior of the IP₃ profile changed from nonlinear to linear with the increment of time steps. Also from Fig. 6A, it is observed that the nonlinear behavior of IP₃-dependent ATP has changed. This is due to an imbalance among the various processes of the cell. The same patterns are observed in Fig. 4B and Fig. 6B for the temporal profile of IP₃ and ATP release. The IP₃ and ATP release profiles reach steady-state simultaneously at 150 ms.

Fig. 6: IP₃ dependent ATP release for various position and time for D_c = 16 μm²/s and σ = 15pA.

A The spatial distribution of IP₃-dependent ATP release and B the temporal distribution of IP₃-dependent ATP release

Figure 7 depicts the spatiotemporal Ca²⁺-dependent ATP release for the different source amplitude values along with the buffer concentration 5 μM and diffusion coefficient 16 μm²/s under normal and cystic fibrosis conditions. In Fig. 7, it can be seen that the distribution of Ca²⁺-dependent ATP is higher for greater source amplitude values. In Fig. 7A, the ATP concentration is higher at x = 0 μm, reduces along the spatial axis and converges on 0.1 μM at x = 10 μm. From Fig. 7B, it is clear that the ATP level achieves a steady state at time t = 120 ms. Further, there is a substantial difference in ATP levels between normal and cystic fibrosis patients. The spatiotemporal ATP fluxes are greater in cystic fibrosis patients because the Ca²⁺ concentration is 30% higher in CF fibroblasts than in normal [77]. The excess ATP is dangerous since it can cause fibrotic diseases [9].

Fig. 7: Ca²⁺ dependent ATP release for various source influx for [B] = 5 μM and D_c = 16 μm²/s

A The spatial distribution of calcium-dependent ATP release in normal and CF fibroblasts and B the temporal distribution of calcium-dependent ATP release in normal and CF fibroblasts

The computation has been carried out for Ca²⁺ concentration with varying buffer concentration values in the range t = 25 ms to t = 100 ms for ATP release in fibroblast cells. The consequences of buffer concentration on ATP distribution profile in fibroblast cells concerning time for various buffer values, i.e., B = 15, 30, 60, 120 μM, have been displayed in Fig. 8. For t = 25, 50, 75 & 100 ms, we observe that the increase in buffering reduces the ATP concentration in the cell. Thus, in all cases, the fall in ATP is lower for the low buffer and higher for the high buffer. Also, ATP concentration converges to a minimum of 0.1 μM.

Fig. 8: Ca²⁺ dependent ATP distribution at the time t = 15, 30, 60, 120 ms with different buffer concentration for D_c = 16 μm²/s and σ = 15pA.

A–D The spatial distribution of calcium-dependent ATP release with different times for the different buffer concentrations. For a better understanding of the spatiotemporal calcium-regulated ATP synthesis profile

Figure 9 depicts the spatial and temporal Ca²⁺ dependent ATP release under normal and cystic fibrosis for a range of buffer associations rates 5 μM. As seen in Fig. 9, the Ca²⁺ dependent ATP decreases as the buffer binding rate increases. Higher binding rates bind higher amounts of free calcium, thereby reducing calcium levels in the cell, which signals the ATP system to reduce the production of ATP in the cell. In Fig. 9A, we observe the alteration in the curves’ nonlinear nature. When the association rate of the buffer is high, we observe a fall in ATP production at the source and a steep fall in the ATP production curves near the source from x = 0 to x = 2 μm due to the dominance of calcium-reducing mechanisms weakening the calcium signal feedback to the ATP production system and prompting it to reduce ATP production. But when the buffer association rates are low, the ATP production is higher at the source due to a strong calcium signal caused by a reduction in the dominance of calcium-reducing mechanisms leading to more effect of calcium-elevating mechanisms. There is a substantial difference in ATP levels in normal and CF fibroblasts. Ca²⁺ dependent ATP is 30% higher in CF fibroblasts than in normal. This process occurs if the rate of association is high.

Fig. 9: Ca²⁺ dependent ATP release for various association rate for [B] = 5 μM, D_c = 16 μm²/s and σ = 15 pA.

A The spatial distribution of calcium-dependent ATP release and B the temporal distribution of calcium-dependent ATP release

Error Estimation

The absolute comparative error and its percentage in a fibroblast cell have been estimated and shown in Tables 2 and 3. The model’s accuracy for Ca²⁺ and IP₃ is 99.84019761%, 99.84070305%, 99.84070305 and 99.167848%, 99.18513997%, 99.18514701% for the time (t = 50, 100, 200, 500 & 1000) ms respectively. As a result, the absolute worst-case scenario for the error rate of this model is 0.16%, while 99.84% is the utmost best-case scenario for its accuracy. Because the grid sensitivity is essentially nonexistent, this strongly argues that the solution is not grid-reliant.

Table 2 [Ca²⁺] profile error analysis at x = 0 μm

Table 3 [IP₃] profile error analysis at x = 0.25 μm

Validation

For varied parameters, the profiles of Ca²⁺ and IP₃ levels were produced. For the Ca²⁺ profile, the amount of buffer was set at 0.0 μM, while the source inflow was set at 2.46 pA. The results of the Ca²⁺ and IP₃ in the current model for t = 50 s are compared with the findings of Pawar and Pardasani [61, 62] & Wagner et al. [68] as depicts in Tables 4 and 5. It appears that the proposed study’s results are very similar to those derived from previous studies.

Table 4 Comparison of Ca²⁺ profile with Wagner et al. [68] & Pawar and Pardasani [61, 62] at t = 50 s

Table 5 Comparison of IP₃ profile with Wagner et al. [68] & Pawar and Pardasani [61, 62] at t = 50 s

Stability Analysis

We determined the spectral radius for the stability investigation, and if the radius of the spectrum is equal to or below one, the system is considered stable [79]. In this specific case, the spectral radius was determined to be 0.9853, so it’s <1, indicating that the calculated result is stable.

Conclusion

A model is developed and successfully implemented to explore the consequences of disruptions in coupled IP₃ and Ca²⁺ dynamics on ATP generation in a fibroblast cell.

Disruptions in the interdependent IP₃ and Ca²⁺ levels can be induced by the failure of mechanisms such as buffering and the serca pump. As a result of some disorder in these processes, high amounts of source input, association rate and higher buffer values were investigated. These perturbations significantly impacted Ca²⁺ and IP₃ dynamics regulating ATP generation. For obtaining numerical solutions under various situations, the finite element and Crank Nicholson were used. In light of the unexpected results, the following conclusions are reached:

• 1. When the source of the inflow is substantial and the serca and buffer are low, the calcium concentration rises, resulting in increased IP₃ concentrations in fibroblast cells.

• 2. High buffer levels cause irregularities because of oscillations in the Ca²⁺ distributions in fibroblast cells.

• 3. An rise in Ca²⁺ concentration results from an increase in IP₃ levels, which in turn boosts ATP synthesis.

The fundamental findings described above are supported by biological evidence. In addition, the results of the stability analysis, error analysis, and validation procedures used in the current study suggest that the suggested model is appropriate and effective for the inquiry at hand. Furthermore, the following innovative deductions and inferences drawn from the numerical data are provided:

• 1. The large volumes of source input result in higher Ca²⁺ levels, which boosts the ATP release in normal fibroblast and cystic fibrosis fibroblast. These changes in ATP signalling can result in fibrotic disorders like fibrosis [46].

• 2. The presence of buffers decreases Ca²⁺ and IP₃ levels in normal fibroblast and cystic fibrosis fibroblast and lowers the rate of selective death of fibroblasts. insufficient quantities of EGTA and calmodulin buffers can lead to cystic fibrosis [77].

• 3. When buffer is low, Ca²⁺ levels rise, increasing ATP release in fibroblasts as well as cystic fibrosis fibroblast. The increase in Ca²⁺ and IP₃ concentrations within the cell promotes the loss of fibroblast cells, which leads to cancer [80].

• 4. Increased ATP production is risky since it might result in fibrotic diseases. One more novelty of the results is the spatiotemporal ATP production are greater in cystic fibrosis patients because the Ca²⁺ concentration is 30% higher in CF fibroblasts than in normal fibroblasts [77].

Disruptions in numerous generative processes of IP₃ and Ca²⁺dynamics might result in altered ATP synthesis, leading to various fibrotic diseases. Lowered buffering capacity, impaired buffer association performance, high levels of source inflow, and fluctuations in IP₃ levels can induce increases in free Ca²⁺ levels and ATP generation, which is the cause of cystic fibrosis when there is a discrepancy between adenosine triphosphate production and its metabolism.

The results concur with biological realities. In the current work, the FEM technique is extremely successful. The model proposed sheds light on the underlying processes of Ca²⁺ and IP₃ dynamics that cause an increase in ATP production, which has been identified as a factor causing cytotoxicity and contributing to the development of cystic fibrosis. The findings highlight the challenge of determining the correct constitutive procedure of IP₃ and Ca²⁺ dynamics, crucial in ATP production alterations in fibrosis. Although many questions remain unanswered, the results from studying the relationships between the constitutive processes of IP₃ and Ca²⁺ dynamics and ATP synthesis can pave the way to address the aforementioned challenges.

Appendix

The shape function is assumed

$$m{0}^{(e)}={q}_{1}^{(e)}+{q}_{2}^{(e)}x.$$

(A1)

$$n{0}^{(e)}={r}_{1}^{(e)}+{r}_{2}^{(e)}x.$$

(A2)

$$m{0}^{(e)}={X}^{T}\times {q}^{(e)},\,n{0}^{(e)}={X}^{T}\times {r}^{(e)}.$$

(A3)

Where,

$${S}^{T}=[1\,\,\,x],\,{q}^{{(e)}^{T}}=[{q}_{1}^{(e)}\,{q}_{2}^{(e)}]\,and\,{r}^{{(e)}^{T}}=[{r}_{1}^{(e)}\,{r}_{2}^{(e)}].$$

(A4)

In equation (A3), substitute the nodal conditions, we obtain

$${\bar{m}}0^{(e)}={X}^{(e)}\times {q}^{(e)},\,and\,{\bar{n}}0^{(e)}={X}^{(e)}\times {r}^{(e)}.$$

(A5)

Where,

$${\bar{m}}0^{(e)}=\left[\begin{array}{c}m{0}_{i}\\ m{0}_{j}\end{array}\right],\,{\bar{v}}^{(e)}=\left[\begin{array}{c}n{0}_{i}\\ n{0}_{j}\end{array}\right]\,and\,{X}^{(e)}=\left[\begin{array}{cc}1&{x}_{i}\\ 1&{x}_{j}\end{array}\right].$$

(A6)

By the equations (A5), we have

$${q}^{(e)}=({Y}^{(e)})\times {\bar{m}}0^{(e)}\,\,\& \,{r}^{(e)}=({Y}^{(e)})\times {\bar{n}}0^{(e)}.$$

(A7)

Where,

$${Y}^{(e)}=\frac{1}{{X}^{(e)}}.$$

(A8)

Substituting the values of q^(e) & r^(e) in equation (A3), we obtained

$$m{0}^{(e)}={X}^{T}\times {Y}^{(e)}\,{\bar{m}}0^{(e)}\,and\,n{0}^{(e)}={X}^{T}\times {Y}^{(e)}\,{\bar{n}}0^{(e)}.$$

(A9)

The partial differential equations (1) and (7) can be written as,

$${I}_{1}^{(e)}={I}_{ao1}^{(e)}-{I}_{bo1}^{(e)}+{I}_{co1}^{(e)}-{I}_{do1}^{(e)}+{I}_{eo1}^{(e)}-{I}_{fo1}^{(e)}-{I}_{go1}^{(e)}.$$

(A10)

Where

$${I}_{ao1}^{(e)}=\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[{\left[\frac{\partial m{0}^{(e)}}{\partial x}\right]}^{2}\right]dx,$$

(A11)

$${I}_{bo1}^{(e)}=\frac{d}{dx}\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[\frac{m{0}^{(e)}}{{D}_{c}}\right]dx,$$

(A12)

$${I}_{co1}^{(e)}=\frac{{V}_{I{P}_{3}R}}{{D}_{c}{F}_{c}}\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[\alpha m{0}^{(e)}+\beta n{0}^{(e)}+\gamma \right]dx,$$

(A13)

$${I}_{do1}^{(e)}=\frac{{V}_{serca}}{{D}_{c}{F}_{c}}\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[km{0}^{(e)}-\eta \right]dx,$$

(A14)

$${I}_{eo1}^{(e)}=\frac{{V}_{leak}}{{D}_{c}{F}_{c}}\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[{[C{a}^{2+}]}_{ER}-m{0}^{(e)}\right]dx,$$

(A15)

$${I}_{fo1}^{(e)}=\frac{{K}^{-}}{{D}_{c}}\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[m{0}^{(e)}-{[C{a}^{2+}]}_{\infty }\right]dx,$$

(A16)

$${I}_{go1}^{(e)}={f}^{(e)}{\left[\frac{\sigma }{{D}_{c}}\right]}_{x = 0}.$$

(A17)

$${I}_{2}^{(e)}={I}_{ao2}^{(e)}-{I}_{bo2}^{(e)}+{I}_{co2}^{(e)}-{I}_{do2}^{(e)}.$$

(A18)

$${I}_{ao2}^{(e)}=\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[{\left[\frac{\partial n{0}^{(e)}}{\partial x}\right]}^{2}\right]dx,$$

(A19)

$${I}_{bo2}^{(e)}=\frac{d}{dx}\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[\frac{n{0}^{(e)}}{{D}_{I}}\right]dx,$$

(A20)

$${I}_{co2}^{(e)}=\frac{{V}_{prod}}{{D}_{I}{F}_{c}}\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[\mu m{0}^{(e)}+\tau \right]dx,$$

(A21)

$${I}_{do2}^{(e)}=\frac{\lambda }{{D}_{I}{F}_{c}}\int\nolimits_{{x}_{i}}^{{x}_{j}}\left[\delta m{0}^{(e)}+\zeta n{0}^{(e)}+{\gamma }_{2}\right]dx.$$

(A22)

Linearizing IP₃ and Ca²⁺ dynamics yields the quantity of δ, γ, k, η, μ, ζ, γ₂, α, β and τ. The equations after calculation and boundary circumstances are as follows.

$$\frac{d{I}_{1}}{d{\bar{m}}0^{(e)}}=\mathop{\sum }\limits_{e=1}^{N}\left[{\bar{{Q}}^{(e)}}\frac{d{I}_{1}^{(e)}}{d{\bar{m}}0^{(e)}}{\bar{Q}}^{{(e)}^{T}}\right]=0.$$

(A23)

$$\frac{d{I}_{2}}{d{\bar{n}}0^{(e)}}=\mathop{\sum }\limits_{e=1}^{N}\left[{\bar{Q}}^{(e)}\frac{d{I}_{2}^{(e)}}{d{\bar{n}}0^{(e)}}{\bar{Q}}^{{(e)}^{T}}\right]=0.$$

(A24)

Where,

$$\begin{array}{l}\frac{d{I}_{1}^{(e)}}{d{\bar{m}}0^{(e)}}=\frac{d{I}_{ao1}^{(e)}}{d{\bar{m}}0^{(e)}}+\frac{d}{dt}\left[\frac{d{I}_{bo1}^{(e)}}{d{\bar{m}}0^{(e)}}\right]+\frac{d{I}_{co1}^{(e)}}{d{\bar{m}}0^{(e)}}\\ \qquad \qquad +\,\frac{d{I}_{do1}^{(e)}}{d{\bar{m}}0^{(e)}}-\frac{d{I}_{eo1}^{(e)}}{d{\bar{m}}0^{(e)}}-\frac{d{I}_{fo1}^{(e)}}{d{\bar{m}}0^{(e)}}.\end{array}$$

(A25)

$$\frac{d{I}_{2}^{(e)}}{d{\bar{n}}0^{(e)}}=\frac{d{I}_{ao2}^{(e)}}{d{\bar{n}}0^{(e)}}+\frac{d}{dt}\left[\frac{d{I}_{bo2}^{(e)}}{d{\bar{n}}0^{(e)}}\right]+\frac{d{I}_{co2}^{(e)}}{d{\bar{n}}0^{(e)}}+\frac{d{I}_{d2}^{(e)}}{d{\bar{n}}0^{(e)}}-\frac{d{I}_{eo2}^{(e)}}{d{\bar{n}}0^{(e)}}.$$

(A26)

$${\overline{M}}^{(el)}=\left[\begin{array}{cc}0&0\\ .&.\\ 0&0\\ 1&0\\ 0&1\\ 0&0\\ .&.\\ 0&0\end{array}\right],\overline{u}=\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\\ {u}_{3}\\ .\\ .\\ .\\ {u}_{30}\\ {u}_{31}\end{array}\right]and\,\overline{v}=\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\\ .\\ .\\ .\\ {v}_{30}\\ {v}_{31}\end{array}\right].$$

(A27)

$${\left[\begin{array}{c}A\end{array}\right]}_{62\times 62}\left[\begin{array}{c}{\left[\begin{array}{c}\frac{\partial \bar{u}}{\partial t}\end{array}\right]}_{31\times 1}\\ {\left[\begin{array}{c}\frac{\partial \bar{v}}{\partial t}\end{array}\right]}_{31\times 1}\end{array}\right]+{\left[\begin{array}{c}B\end{array}\right]}_{62\times 62}\left[\begin{array}{c}{\left[\begin{array}{c}\bar{u}\end{array}\right]}_{31\times 1}\\ {\left[\begin{array}{c}\bar{v}\end{array}\right]}_{31\times 1}\end{array}\right]={\left[\begin{array}{c}F\end{array}\right]}_{62\times 1}.$$

(A28)

We utilize the statistically stable Crank Nicholson Method on the matrices A and B to determine the time derivative using the Finite Element Approach, and the force vector is F. To solve the resultant system of equations, the Gaussian Elimination Method is utilized.