An Efficient Numerical Scheme for Fractional Order Mathematical Model of Cytosolic Calcium Ion in Astrocytes

Abstract

The major aim of this article is to obtain the numerical solution of a fractional mathematical model with a nonsingular kernel for thrombin receptor activation in calcium signals using two numerical schemes based on the collocation techniques. We present the computational solution of the considered fractional model using the Laguerre collocation method (LCM) and Jacobi collocation method (JCM). An operational matrix of the fractional order derivative in the Caputo sense is needed for the recommended approach. The computational scheme converts fractional differential equations (FDEs) into an algebraic set of equations using the collocation method. The technique is used more quickly and successfully than in other existing schemes. A comparison between LCM and JCM is also presented in the form of figures. We obtained very good results with a great agreement between both the schemes. Additionally, an error analysis of the suggested procedures is provided.

1. Introduction

Our body needs calcium to function properly. The body uses calcium oscillations for a variety of purposes. Calcium functions as an intermediate in the processing of information. Calcium dynamics is a term entailing the rigorously regulated spatiotemporal fluctuations of the intracellular calcium concentration, which enable the use of calcium ions for diverse signaling purposes. The enzyme phospholipase C (PLC), which is attached to the membrane, is activated by G protein. The complicated balance of cellular calcium homeostasis is caused by several factors. To respond to agonist stimuli, the cell can mobilize intracellular calcium from its reserves. The activation of the inositol phospholipid cascade results in an increase in cytosolic calcium levels, which mediates many signal transduction pathways [1,2]. Phosphatidylinositol 4,5-bisphosphate [PtdIns(4, 5)$P_2$] is hydrolyzed by the activated form of PLC into inositol 4,5-trisphosphate [Ins(I, 4, 5)$P_3$], and diacylglycerol (DAG). Ins(1, 4, 5)$P_3$ stimulates the release of endogenous calcium from the endoplasmic reticulum. The quantity of Ins(I, 4, 5)$P_3$ in the cytosol affects the release of calcium from ER reserves. Ins(I, 4, 5)$P_3$ is produced at a rate inversely correlated with the amount of activated cell surface receptors.

The current model’s ligand is thrombin, a multifunctional serine protease that is known to cause a calcium transient in endothelial cells [3]. One of thrombin’s effects on endothelial cells is the activation of protease-activated receptors (PARs), and Kim et al. [4] describe how this happens. A computational model was created by Lenoci et al. [5] to forecast the platelet behavior in response to thrombin-activated PAR1 (protease-activated receptor) activation. These receptors form a network that results in stable platelet aggregation. The amino terminal extension of the receptors is cut, resulting in a new receptor amino terminus that functions as a tethered ligand and activates the receptors. Thrombin supports this cleavage, and Vu et al. [6] describe the process.

When creating models for problems in the actual world, fractional calculus is helpful. Numerous researchers have conducted studies in this field [7,8,9,10] and discussed its importance in several scientific, engineering, and financial fields. Fractional calculus includes crucial concepts like the fractional derivative and integrals. Djordjevic et al. [11] created a rheological model of the smooth muscle cells that line the airways utilizing a procedure involving least squares data fitting and fractional calculus. For the calcium concentration profile, Agarwal et al. [12] investigated the impact of the fractional advection-diffusion equation. For use in nanotechnology, an arbitrary order model utilizing a differential-difference equation was offered by Kumar et al. [13]. Kumar et al. [13,14] solved fractional Navier–Stokes equation and derived an analytical solution. A fractional model was considered by Agarwal et al. [15] to determine the concentration of contaminants in flowing groundwater. Khader [16] introduced the space-fractional diffusion equation solution using the Chebyshev collocation method. A time–space spectral approach and a hybrid scheme were created by Li and Xu [17] to solve the time-fractional diffusion equation. Atangana and Alkahtani [9] studied the Keller–Segel model with a kernel that lacks singularity and a derivative of arbitrary order. Kumar et al. [18] studied the numerical and computational analysis of fractional order mathematical models for chemical kinetics and carbon dioxide absorbed into phenyl glycidyl ether.

In the sequence of research in fractional calculus, we use the Caputo fractional derivative for mathematical modeling of receptor activation for calcium signaling and present a computational solution of the proposed model using the Jacobi collocation method (JCM) and Laguerre collocation method (LCM). It is worth mentioning that the proposed model is handled in this study using the collocation strategy.

2. Preliminaries

2.1. Fractional Calculus

Here, we provide a brief overview of a few fractional calculus concepts, traits, and results for developing the computational algorithm required to examine fractional differential equations.

Definition 1.

For a function ϕ, the Riemann–Liouville (RL) fractional integral operator of order $\theta >0$ is expressed as:

$$\begin{array}{cc}\hfill I^{\theta }\varphi \left(y\right)& =\frac{1}{\Gamma \left(\theta \right)}{\int }_0^y{(y-x)}^{\theta -1}\varphi \left(x\right)dx,\theta >0\hfill \\ \hfill I^0\varphi \left(y\right)& =\varphi \left(y\right)\hfill \end{array}$$

Definition 2.

The following is the definition of the Caputo operator of the fractional derivative:

$$D^{\theta }\varphi \left(y\right)=\frac{1}{\Gamma (m-\theta )}{\int }_0^y\frac{{\varphi }^{\left(m\right)}\left(x\right)}{{(y-x)}^{\theta +1-m}}dx,\theta >0,y>0,$$

where $m-1<\theta \le m,m\in \mathbb{N}$, and $\varphi \in C^m[0,1]$. So, the Caputo operator follows:

$$D^{\theta }{y}^l=\left\{\begin{array}{cc}0,\hfill & l\in 0,1,2,\dots ,\lceil \theta \rceil -1\hfill \\ \frac{\Gamma (1+l)}{\Gamma (1+l-\theta )}{y}^{l-\theta },\hfill & l\in \mathbb{N}\wedge l\ge \lceil \theta \rceil \hfill \end{array}\right.$$

To discover more about the properties and definitions of fractional derivatives, see [19,20].

2.1.1. Laguerre Polynomial

The definition and symbol for the Laguerre polynomials of degree q are [21,22,23]:

$$L_m\left(y\right)=\frac{1}{m!}{e}^y{\delta }_y^m\left(y^m{e}^{-y}\right),m=0,1,\dots$$

Analytical outlines of the rth-degree Laguerre polynomials are expressed as follows

$$L_r\left(y\right)=\sum _{\chi =0}^r\frac{r!{(-1)}^{\chi }}{(-\chi +r)!{(\chi !)}^2}{y}^r,r=0,1,\dots$$

(1)

The Laguerre polynomial has the following orthogonal properties:

$${\int }_0^1{L}_m\left(y\right)L_n\left(y\right)\eta \left(y\right)dy={\delta }_{m,n}$$

(2)

${\delta }_{m,n}$ is kronecker delta function, $\eta \left(y\right)$ is weight function defined as:

$$\eta \left(y\right)=e^{-y}.$$

2.1.2. Laguerre Operational Matrix for Fractional Differentiation

Theorem 1.

Assuming that $\nu >0$ and H(y) = $[L_0\left(y\right),L_1\left(y\right),\dots ,L_n\left(y\right)]$ be Laguerre vector; then

$$D^{\nu }H\left(y\right)=D^{\left(\nu \right)}H\left(y\right)$$

$D^{\left(\nu \right)}$ is the operational matrix of the Caputo definition’s $(n+1)\times (n+1)$ order for ν order differentiation.

$$D^{\left(\nu \right)}=\left(\begin{array}{cccc}0& 0& \cdots & 0\\ ⋮& ⋮& \cdots & ⋮\\ 0& 0& \cdots & 0\\ P_{\nu }(\lceil \nu \rceil ,0)& P_{\nu }(\lceil \nu \rceil ,1)& \cdots & P_{\nu }(\lceil \nu \rceil ,n)\\ ⋮& ⋮& \cdots & ⋮\\ P_{\nu }(j,0)& P_{\nu }(j,1)& \cdots & P_{\nu }(j,n)\\ ⋮& ⋮& \cdots & ⋮\\ P_{\nu }(n,0)& P_{\nu }(n,1)& \cdots & P_{\nu }(n,n)\end{array}\right)$$

(3)

where

$$P_{\nu }(h,\varphi )=\sum _{i=\lceil \nu \rceil }^h\frac{{(-1)}^{i}h!\Gamma (\varphi -i+\nu )}{\varphi !(h-i)!i!\Gamma (-i+\nu )}$$

(4)

Proof. 

Please, see the work [21,22,23]. It is seen in the expression that in $D^{\left(\nu \right)}$, the first $\lceil \nu \rceil$ rows all are zero. □

2.1.3. Jacobi Polynomial

The shifted Jacobi polynomial is prescribed as [21,24,25]:

$$a_n^{(w,z)}\left(y\right)=\sum _{i=0}^n{(-1)}^{n-i}\frac{\Gamma (n+z+1)\Gamma (n+i+w+z+1)}{\Gamma (i+z+1)\Gamma (n+w+z+1)(n-i)!i!}{y}^i,$$

where w and z are Jacobi polynomial parameters, outlined in [21].

The following orthogonal characteristics hold to Jacobi polynomials:

$${\int }_0^1{a}_n^{(w,z)}\left(y\right)a_m^{(w,z)}\left(y\right){\vartheta }^{(w,z)}\left(y\right)dy={\eta }_n^{w,z}{\delta }_{nm},$$

${\delta }_{nm}$ Kronecker delta function and ${\vartheta }^{(w,z)}\left(y\right)$ weight function and defined as

$${\vartheta }^{(w,z)}\left(y\right)={(1-y)}^w{y}^z$$

and

$${\eta }_n^{w,z}=\frac{\Gamma (n+w+1)\Gamma (n+z+1)}{(2n+w+z+1)n!\Gamma (n+w+z+1)}.$$

2.2. Operational Matrix for JACOBI Polynomials

Theorem 2.

Suppose the shifted Jacobi vector $A_m\left(y\right)={[a_0^{(w,z)},a_1^{(w,z)},\dots ,a_m^{(w,z)}]}^T$ and $\nu >0$, then

$$D^{\nu }{a}_n^{(w,z)}\left(y\right)=D^{\left(\nu \right)}{A}_m\left(y\right),$$

$D^{\left(\nu \right)}=\left(O(n,i)\right)$ is the operational matrix of $(m+1)\times (m+1)$ order and ν represents order of fractional derivative, whose entries are offered as follows:

$$O(n,i,w,z)=\sum _{q=\left[\nu \right]}^n{(-1)}^{n-q}\frac{\Gamma (n+z+1)\Gamma (n+q+w+z+1)}{(n-q)!\Gamma (q+z+1)\Gamma (n+w+z+1)\Gamma (q-\nu +1)}$$

$$\times \sum _{r=0}^i{(-1)}^{i-r}\frac{\Gamma (w+1)\Gamma (i+r+w+z+1)\Gamma (q+r-\nu +z+1)(2i+w+z+1)i!}{(i-r)!\left(r\right)!\Gamma (i+w+1)\Gamma (r+z+1)\Gamma (q+r-\nu +w+z+2)}.$$

Proof. 

Please, refer the work [21,24,25]. □

Function Approximation

Having $\left|{\psi }^{″}\right(y\left)\right|\le C$, the function $\psi$ defined in $L^2[0,1]$ can be extended as an infinite sum of the shifted Laguerre polynomials:

$$\psi \left(y\right)=\underset{k\to \infty }{lim}\sum _{r=0}^k{c}_r{L}_r\left(y\right)$$

(5)

where $L_r\left(y\right)$ is provided by Equation (1)

$$c_r={\int }_0^1\psi \left(y\right)L_r\left(y\right)\eta \left(y\right)dy;r=0,1,2,\dots$$

(6)

Using assumptions for finite dimensions in Equation (5), we discover

$$\psi \left(y\right)\cong \sum _{r=0}^m{c}_m{L}_m\left(y\right)=C^{T}H\left(y\right)$$

(7)

where the $(m+1)\times 1$ matrices C and $H_m\left(y\right)$ are denoted as:

$$C=[c_0,c_1,\dots ,c_m],H_m\left(y\right)=[L_0\left(y\right),L_1\left(y\right),\dots ,L_m\left(y\right)]$$

(8)

3. Thrombin Receptor Activation Mechanism Computational Model

To explain the kinetics of cytosolic calcium, Wiesner et al. [3] created mathematical models. This model provides a numerical representation of calcium-mediated signaling in cells of the endothelial system. In calcium homeostasis, the amount of Ins(1, 4, 5)$P_3$ in the cytosol determines the release of calcium and the quantity of activated cell surface receptors ($\zeta$) affects the rate of synthesis of Ins(1, 4, 5)$P_3$, which when bound to a ligand, creates a receptor–ligand complex (Υ), which, when broken down, creates activated receptors ($\chi$). According to the following set of differential equations, the balances for these diverse species can be described mathematically:

$$\frac{d\zeta }{dt}=-\beta \zeta \sigma +\lambda \mathsf{{\rm Y}}$$

(9)

$$\frac{d\mathsf{{\rm Y}}}{dt}=\beta \zeta \sigma -\lambda \mathsf{{\rm Y}}-\xi \mathsf{{\rm Y}}$$

(10)

$$\frac{d\chi }{dt}=\xi \mathsf{{\rm Y}}$$

(11)

$\zeta \left(0\right)=\mathsf{\Lambda }$ and $\mathsf{{\rm Y}}\left(0\right)=\chi \left(0\right)=0$ are the initial conditions.

Here, $\mathsf{\Lambda }$ is the overall quantity of receptors, the cell surface receptors are called $\zeta$, $\mathsf{{\rm Y}}$ is a combination of a receptor and ligand, $\chi$ stands for activated receptors, for binding of thrombin, $\beta$ is the on rate constant, the off rate constant is $\lambda$, and $\sigma$ represents the amount of thrombin present on a cell surface.

When PtdIns(4, 5)$P_2$ is hydrolyzed into Ins(1, 4, 5)$P_3$ and DAG, the PLC enzyme acts as a catalyst. Activated receptors ($\chi$) through G-protein begin the process of the PLC enzyme functioning. The production rate of Ins(1, 4, 5)$P_3$ improves as the number of cleaved receptors ($\chi$) rises.

Here, we take into account the model above with the Caputo derivative in place of the integer order derivative to investigate the activation of the thrombin receptor [26].

$$D^{\nu }\zeta \left(t\right)=-\beta \zeta \left(t\right)\sigma +\lambda \mathsf{{\rm Y}}\left(t\right),$$

(12)

$$D^{\nu }\mathsf{{\rm Y}}\left(t\right)=\beta \zeta \left(t\right)\sigma -\lambda \mathsf{{\rm Y}}\left(t\right)-\xi \mathsf{{\rm Y}}\left(t\right),$$

(13)

$$D^{\nu }\chi \left(t\right)=\xi \mathsf{{\rm Y}}\left(t\right).$$

(14)

Here, $D^{\nu }$ is the Caputo derivative. The relevant initial conditions: $\zeta \left(0\right)=\mathsf{\Lambda }$, $\mathsf{{\rm Y}}\left(0\right)=\chi \left(0\right)=0$.

4. Outline of Technique

Here, we examine the algorithm for creating the FDE solution utilizing the operational matrix and collocation method. Using the approximation that follows:

$$\psi \left(y\right)\cong \sum _{r=0}^m{c}_m{L}_m\left(y\right)=C^T{H}_m\left(y\right)$$

(15)

where C and $H_m\left(y\right)$ are provided by Equation (8)

Then, by using the order one derivative of Equation (15), we arrive at

$$D^{\prime }\psi \left(y\right)=C^T{D}^{\prime }{H}_m\left(y\right)=C^T{D}^{\left(1\right)}{H}_m\left(y\right)$$

(16)

here $D^{\left(1\right)}$ is operational differentiation matrix of order 1.

We now take $\nu$ order fractional derivative of Equation (15):

$$D^{\nu }\psi \left(y\right)=C^T{D}^{\nu }{H}_m\left(y\right)=C^T{D}^{\left(\nu \right)}{H}_m\left(y\right)$$

(17)

here $D^{\left(\nu \right)}$ is the operational differentiation matrix of order $\nu$.

We can write from Equation (15)

$$\psi \left(0\right)=C^T{H}_m\left(0\right)$$

(18)

5. Computational Process by LCM

Equations (15) and (17) are used in Equations (12)–(14) to arrive at the following equations:

$$C_1^T{D}^{\left(\nu \right)}{H}_m\left(t\right)+\beta C_1^T{H}_m\left(t\right)\sigma -\lambda C_2^T{H}_m\left(t\right)=0,$$

(19)

$$C_2^T{D}^{\left(\nu \right)}{H}_m\left(t\right)-\beta C_1^T{H}_m\left(t\right)\sigma +\lambda C_2^T{H}_m\left(t\right)+\xi C_2^T{H}_m\left(t\right)=0,$$

(20)

$$C_3^T{D}^{\left(\nu \right)}{H}_m\left(t\right)-\xi C_2^T{H}_m\left(t\right)=0.$$

(21)

where $D^{\left(\nu \right)}$ is the Laguerre operational matrix for fractional derivative provided by Equation (3).

Equations (19)–(21) residuals are stated as follows:

$$R_{1m}\left(t\right)=C_1^T{D}^{\left(\nu \right)}{H}_m\left(t\right)+\beta C_1^T{H}_m\left(t\right)\sigma -\lambda C_2^T{H}_m\left(t\right),$$

(22)

$$R_{2m}\left(t\right)=C_2^T{D}^{\left(\nu \right)}{H}_m\left(t\right)-\beta C_1^T{H}_m\left(t\right)\sigma +\lambda C_2^T{H}_m\left(t\right)+\xi C_2^T{H}_m\left(t\right),$$

(23)

$$R_{3m}\left(t\right)=C_3^T{D}^{\left(\nu \right)}{H}_m\left(t\right)-\xi C_2^T{H}_m\left(t\right).$$

(24)

Equations (22)–(24), when collocated at m locations, are now expressed as $t_k=\frac{k}{m},$

$k=0,1,\dots ,m-1$.

$$R_{1m}\left(t_k\right)=C_1^T{D}^{\left(\nu \right)}{H}_m\left(t_k\right)+\beta C_1^T{H}_m\left(t_k\right)\sigma -\lambda C_2^T{H}_m\left(t_k\right),$$

(25)

$$R_{2m}\left(t_k\right)=C_2^T{D}^{\left(\nu \right)}{H}_m\left(t_k\right)-\beta C_1^T{H}_m\left(t_k\right)\sigma +\lambda C_2^T{H}_m\left(t_k\right)+\xi C_2^T{H}_m\left(t_k\right)$$

(26)

$$R_{3m}\left(t_k\right)=C_3^T{D}^{\left(\nu \right)}{H}_m\left(t_k\right)-\xi C_2^T{H}_m\left(t_k\right).$$

(27)

Equation (18)’s form serves as a representation of the initial conditions.

$$C_1^T{H}_m\left(0\right)-\mathsf{\Lambda }=0,$$

(28)

$$C_2^T{H}_m\left(0\right)=0,$$

(29)

$$C_3^T{H}_m\left(0\right)=0.$$

(30)

Combined with the initial conditions (28)–(30), Equations (25)–(27) constitute a system of 3(m + 1) equations. By solving the attained system, we obtain values of $C_1^T$, $C_2^T$, and $C_3^T$. After obtaining the value of unknown matrices, i.e., $C_1^T$, $C_2^T$, and $C_3^T$ we easily obtain the solution of the fractional differential Equations (12)–(14).

6. Analysis of the Computational Scheme

Theorem 3.

Let ϑ: [0, 1] ⟶ R, ϑ∈$C^{(m+1)}$ [0, 1], and the $m^{th}$ approximation discovered using Laguerre polynomials is denoted by ${\vartheta }_m$, then

$$S_{\vartheta ,m}^{\psi }=\left|\right|\vartheta -{\vartheta }_m{\left|\right|}_{N_{\psi }^2[0,1]}$$

(31)

error vector $S_{\vartheta ,m}^{\psi }$ tends to zero as m tends to ∞.

Proof. 

For the proof, refer the work by Rivlin [27], Kreyszig [28] and Behroozifar and Sazmand [29]. □

Theorem 4.

For ν order operational matrix differentiation, the error vector is $S_{D,m}^{\nu ,\psi }$, which is achieved by utilizing (m + 1) Laguerre polynomials, then we have

$$S_{D,m}^{\nu ,\psi }=D^{\left(\nu \right)}{H}_m\left(t\right)-D^{\nu }{H}_m\left(t\right)$$

(32)

Equation (32)’s error vector $⟶0$ as m ⟶∞.

Proof. 

Please see the work [30,31]. □

Theorem 5.

Let us take a functional G into account, then we have

$$\underset{m\to \infty }{lim}{\varphi }_m\left(t\right)=\varphi \left(t\right)=\underset{t\in [0,1]}{inf}G\left(t\right).$$

(33)

Proof. 

For proof, see the research work by Ezz-Eldien [32]. □

The functional G for Equation (12) is provided as

$$G\left(t\right)=D_t^{\nu }+\beta \zeta \left(t\right)\sigma -\lambda \mathsf{{\rm Y}}\left(t\right),$$

(34)

Equations (15) and (17) are used to obtain

$$G^{\left(S\right)}\left(t\right)=C_1^T{D}^{\left(\nu \right)}{H}_m\left(t\right)+S_{D,m}^{\nu ,\psi }+(\beta C_1^T{H}_m\left(t\right)\sigma +S_{\vartheta ,m}^{\psi })-(\lambda C_2^T{H}_m\left(t\right)+S_{\vartheta ,m}^{\psi })$$

(35)

where

$$S_{\vartheta ,m}^{\psi }=C_1^{T}H\left(t\right)-C_1^T{H}_m\left(t\right)$$

(36)

$$S_{D,m}^{\nu ,\psi }=C_1^T{D}^{\left(\nu \right)}{H}_m\left(t\right)-C_1^T{D}^{\nu }{H}_m\left(t\right)$$

(37)

For Equation (37), residual is

$$R_m^{\left(G\right)}\left(t\right)=C_1^T{D}^{\left(\nu \right)}{H}_m\left(t\right)+S_{D,m}^{\nu ,\psi }+(\beta C_1^T{H}_m\left(t\right)\sigma +S_{\vartheta ,m}^{\psi })-(\lambda C_2^T{H}_m\left(t\right)+S_{\vartheta ,m}^{\psi })$$

(38)

the m points at which Equation (38) collocates are provided by $t_k=\frac{k}{m},k=0,1,\dots ,m-1$, we attain

$$R_m^{\left(G\right)}\left(t_k\right)=C_1^T{D}^{\left(\nu \right)}{H}_m\left(t_k\right)+S_{D,m}^{\nu ,\psi }+(\beta C_1^T{H}_m\left(t_k\right)\sigma +S_{\vartheta ,m}^{\psi })-(\lambda C_2^T{H}_m\left(t_k\right)+S_{\vartheta ,m}^{\psi })$$

(39)

A system of algebraic equations is created by combining Equations (39) and (28). We find the value of the $C_1^T$ by solving the attain system. Solving Equation (34) is the next stage. Let the resulting answer be denoted by ${\varphi }_m^{*}\left(t\right)$

The limit $k⟶\infty$ and the Theorems 3 and 4 will now be applied, and we obtain

$${\varphi }_m^{*}\left(t\right)⟶{\varphi }_m\left(t\right)$$

(40)

Equation (40) and Theorem 5 allow us to deduce that

$$\underset{m\to \infty }{lim}{\varphi }_m\left(t\right)=\varphi \left(t\right)$$

(41)

It is possible to assemble the same evidence for FDEs (13) and (14).

Similarly, we can solve the fractional model of thrombin receptor activation in calcium signaling by JCM which requires a Jacobi operational matrix for fractional derivative.

7. Numerical Results and Discussions

In this section, we compute the numerical results for $\zeta \left(t\right),\mathsf{{\rm Y}}\left(t\right)$, and $\chi \left(t\right)$ at different values of $\nu$ with respect to time. LCM is applied to obtained the numerical results. There is an estimated numerical solution provided and a graphical presentation has been developed. Regarding using the Laguerre collocation method to solve the current model and simulating receptors as a function of time, we see that the way receptors behave during the activation process, which begins with ligand binding, the fractional order of the derivative, which is used to mathematically construct the model, controls the receptor’s activation once it forms a complex with the ligand and receptor. From the figures showing the fractional model of receptor activation by using thrombin as a ligand, there is no doubt that using fractional values rather than integral values can better explain the behaviors of the receptors throughout the activation process, such as the cleavage of complexes to generate active receptors. The values of various parameters are taken [3] as $\xi$ = 0.12 s⁻¹, $\beta$ = 0.0005 M⁻¹s⁻¹, $\lambda$ = 142.8 s⁻¹, $\mathsf{\Lambda }$ = 4.4 $\times {10}^4$ No./cell, and the agonist concentration $\sigma$ = 1 unit/mL, m = 4. Figure 1 shows that the cell surface receptors $\left(\zeta \right(t\left)\right)$ that are prepared for thrombin binding, decrease as $\nu$ decreases from 1 to $0.9$. The quantity of th receptor–ligand complex $\left(\mathsf{{\rm Y}}\right(t\left)\right)$ is seen to rise over time in Figure 2. Whereas $\mathsf{{\rm Y}}\left(t\right)$ decreases as $\nu$ decreases from 1 to $0.9$.

Figure 3 shows that there were initially no activated receptors $\left(\chi \right(t\left)\right)$, but that as the process progresses, the quantity of these complexes increased over time. It is also noticeable that as the value of $\nu$ increased, the number of complexes poised to cleave to a proteolytic fragment and the activated receptor $\left(\chi \right(t\left)\right)$ body increased.

8. Conclusions

This paper uses the Caputo derivative to study the fractional dynamical framework for the activation of receptors. By using LCM and JCM, the fractional-order model’s solutions have been computed. To illustrate the impact of fractional order, a simulation has been conducted on the provided data [3]. One advantage of the proposed mathematical technique is that it simplifies the problems into a set of simple algebraic equations. We also show a comparison of two techniques (LCM and JCM) through Figure 4 and we obtain very good results. All of the figures also show the effect of an order of fractional derivative on solution profiles $\zeta \left(t\right)$, $\Gamma \left(t\right)$, and $\chi \left(t\right)$. All of the numerical results demonstrate how accurate, quicker, reliable, and successful the recommended technique is. We used Mathematica to compute numerical results. We found that our approach led to more effective outcomes. The computed solution of the proposed model by using the collocation approach shows that this technique can be applied to explain the fractional order problems occurring in chemistry. We can solve more complex fractional calculus problems by using the collocation technique that arises in real word applications.