Regulation of 1, 4, 5-triphosphate receptor channel gating dynamics by mutant presenilin in Alzheimer’s disease cells

Abstract

In neurons of patients with Alzheimer’s disease, the intracellular Ca²⁺ concentration is increased by its release from the endoplasmic reticulum via the inositol 1, 4, 5-triphosphate receptor (IP₃R). In this paper, we discuss the IP₃R gating dynamics in familial Alzheimer’s disease (FAD) cells induced with presenilin mutation PS1. By fitting the parameters of an IP₃R channel model to experimental data of the open probability, the mean open time and the mean closed time of IP₃R channels, in control cells and FAD mutant cells, we suggest that the interaction of presenilin mutation PS1 with IP₃R channels leads the decrease in the unbinding rates of IP₃ and the activating Ca²⁺ from IP₃Rs. As a result, the increased affinities of IP₃ and activating Ca²⁺ for IP₃R channels induce the increase in the Ca²⁺ signal in FAD mutant cells. Specifically, the PS1 mutation decreases the IP₃ dissociation rate of IP₃R channels significantly in FAD mutant cells. Our results suggest possible novel targets for FAD therapeutic intervention.

Graphical abstract

Keywords

Ca²⁺ signal / channel / neuron / Alzheimer’s disease

1 Introduction

Alzheimer’s disease (AD) is a common chronic disease involving the slow destruction of neurons, resulting in serious cognition disorders. Most AD is sporadic and occurs after the age of 60 years. About 10% of AD develops before the age of 60 years, and is termed autosomal dominant familial Alzheimer’s disease (FAD). Although familial and sporadic ADs are different in their age of onset, they have the same symbolic characteristics, including extracellular age-related pigment deposition by the β amyloid protein and neurofibrillary tangles in neurons by Tau protein hyperphosphorylation [1-4], indicating the same etiology for both familial and sporadic AD. According to the β-amyloid (Aβ) cascade hypothesis of AD pathogenesis, mutations of amyloid precursor protein (APP) and presenilin lead to abnormal secretion and abundance of Aβ. The Aβ proteins deposited in the brain tissue are toxic to the neurons and synapses, and can destroy synaptic membranes, causing neuronal death [5, 6]. It has been suggested that APP and presenilin mutations PS1 and PS2 are the main causes of FAD [1].

While extracellular Aβ can influence the steady state concentrations of intracellular Ca²⁺ both in vitro and in vivo, FAD presenilin mutations can also affect intracellular Ca²⁺ concentrations via an intracellular mechanism [7, 8]. In AD neurons, the increased Ca²⁺ concentration is caused mainly by an increase in Ca²⁺ release from the endoplasmic reticulum (ER) [9-13]. In response to the second messenger inositol 1, 4, 5-triphosphate (IP₃), release of Ca²⁺ from the ER through the IP₃ receptor channels (IP₃R) is a common mechanism and has been observed in many cells [14]. A proper increase in the cytosolic Ca²⁺ concentration plays a pivotal role in regulating many intracellular functions, for example in learning and memory [15].

Several molecular mechanisms have been suggested for the increased cytosolic Ca²⁺ in AD neurons. It might be related to the enhanced ability to store Ca²⁺ in the ER lumen, such as via the enhanced sarcoendoplasmic reticulum Ca²⁺-ATPase (SERCA) pump [16]. Other hypothesized mechanisms for the increased cytosolic Ca²⁺ concentration include the increased release of Ca²⁺ through IP₃Rs [10, 12] or ryanodine receptors (RyR) [11, 17, 18], an increased abundance of channels, and an enhanced affinity of IP₃R for its ligand IP₃. The increased cytosolic Ca²⁺ signals might be also caused by a decrease of the ER-mitochondrial distance [19, 20]. In fact, the increased Ca²⁺ concentration is used to diagnose FAD [21, 22].

Recently, Cheung et al. studied how the biochemical interaction of FAD mutant PS1(M146L)/PS2 with the IP₃Rs affects the intracellular Ca²⁺ signal in response to IP₃ stimulation [23, 24]. They indicated that, compared with the wild-type PS, the interaction of FAD mutant PS with the IP₃R channels profoundly enhanced the gating activity of the IP₃R, resulting in the increased intracellular Ca²⁺ levels. However, the details of how the mutant PS influences the gating dynamics of Ca²⁺ and IP₃ binding to and unbinding from IP₃Rs are unclear and remain a challenging experimental problem. Understanding the modulation of the ligand binding and unbinding properties of IP₃Rs by mutant PS is important for the design of novel targets for therapeutic intervention. In this study, we used computational modeling of IP₃Rs to investigate how the IP₃R gating dynamics are influenced by FAD mutant PS1.

Various mathematical models of IP₃R have been used to study IP₃R gating dynamics. DeYoung and Keizer put forward a model (the DYK model) where the entire channel is made up of three identical and independent subunits [25], each of which have binding sites for IP₃ and Ca²⁺ (see below for more detail). Sneyd and Dufour considered a specific kinetic model for the type-2 IP₃R by asserting that the channel functions as one entity rather than as four subunits [26]. Mak et al. proposed an allosteric four-plus-two conformation model in which the IP₃R comprises four identical and independent IP₃R monomers [27]. Shuai et al. suggested a modified De Young–Keizer model, taking into account the four independent subunits and considering IP₃R opening through a configuration change [28]. Further IP₃R models are discussed and compared by Shuai et al. [29]. Ullah et al. established a data-driven model, comprising a Markov chain with nine closed states and three open states, to discuss the modal gating behavior [30]. More recently, a model with four independent subunits was proposed [31] to account for equilibrium channel properties, transient responses to changing ligand conditions, and modal gating statistics [32, 33], showing that cooperative binding or concerted conformational changes are not required for IP₃R modal gating.

In this study, we adopted a simulated annealing algorithm to optimize the parameters in a modified DYK model by fitting the experimental results of open probability P_O, mean open time {\tau }_O, and mean closed time {\tau }_C in wild-type PS1-expressing Sf9 cells (control cells) and in FAD mutant PS1-M146L-expressing cells (M146L cells). The results of comparing the parameter distributions between IP₃R models for the control and M146L cells suggested that the affinities of IP₃R channels for IP₃ and the activating Ca²⁺ both increase in the cell with mutated PS1, leading to the increased Ca²⁺ release.

2 Method and model

2.1 Experimental data for IP₃R

The experimental data of the IP₃R gating dynamics influenced by FAD mutations in PS were obtained at various Ca²⁺ concentrations (C) and IP₃ concentrations (I) in Spodoptera frugiperda (Sf9), DT40 cells, human B lymphoblasts, and mouse cortical neurons [23, 24]. Using the membrane patch clamp technique, the IP₃R gating parameters, including open probability P_O, mean open time {\tau }_O, and mean closed time {\tau }_C, were obtained. Compared with the IP₃R gating dynamics in wild-type PS1-expressing cells, the open probabilities P_O of IP₃R channels in FAD mutant PS1-M146L-expressing cells increase significantly at all IP₃ concentrations. The corresponding mean open time {\tau }_O increases too, while the mean closed time {\tau }_C decreases.

As an example for SF9 cells, at C=1 μM and I=10 μM, P_O=0.76\pm 0.05 in the control cells, while P_O=0.86\pm 0.03 in the M146L cells [23]. At an intracellular calcium concentration of C=1 μM and an IP₃ concentration of I=33 nM, the relevant open probabilities are P_O=0.27\pm 0.01 for the control cells, and P_O=0.75\pm 0.06 for the M146L cells. Interestingly, the open probabilities at low IP₃ concentrations in the M146L cells were nearly equal to those at saturating IP₃ concentration in the control cells. Subsequently, by fitting the experimental data to a modified DYK IP₃R model, we discuss in detail how the mutant PS1 affects the binding and dissociation dynamics of Ca²⁺ and IP₃ for an IP₃R channel.

2.2 The IP₃R model

To date, many different IP₃R channel models have been suggested [25-31]. In the present study, we adopted a modified version of the DYK model [28], in which each parameter has a direct biological meaning related to Ca²⁺ or IP₃ ligand binding or association with the channel. The modified DYK IP₃R model comprises four subunits. In each subunit, there are three binding sites: an IP₃ binding site, an activating Ca²⁺ binding site, and an inhibitory Ca²⁺ binding site. The state of each subunit is denoted by the triplet index (ijk), where index i represents the state of the IP₃ binding site, j the state of the activating Ca²⁺ binding site, and k the state of the inhibitory Ca²⁺ binding site. An occupied site is represented by 1, and a non-occupied site by 0, giving rise to eight possible states for each subunit (Fig. 1). The subunit is active when it is bound with IP₃ and activating Ca²⁺, but not inhibitory Ca²⁺, i.e., if it is in the state (110). The channel is open when there are three or four subunits in the active state (110).

As indicated in Fig. 1, the transitions between the eight states are governed by the second-order rate constants a_i for the binding processes and by the first-order rate constant b_i for the unbinding processes. The dissociation constants, i.e., d_i=b_i/a_i, should satisfy d_1{d}_2=d_3{d}_4 because of the detailed balance. In Fig. 1, C and I represent the concentrations of Ca²⁺ and IP₃ in the cytoplasm, respectively.

According to the schematic diagram, we can calculate the unnormalized equilibrium probabilities q_{ijk} of each state, which are described by the ratio of binding and dissociation constants along the shortest binding/unbinding path relative to the state (000), i.e., q_{000}=1, q_{001}=\frac{C}{d_4}, q_{010}=\frac{C}{d_5}, q_{011}=\frac{C^2}{d_4{d}_5}, q_{100}=\frac{I}{d_3}, q_{101}=\frac{IC}{d_2{d}_3}, q_{110}=\frac{IC}{d_3{d}_5}, and q_{111}=\frac{I{C}^2}{d_2{d}_3{d}_5}[29]. The normalized equilibrium probability for state (110) then reads

\begin{array}{}\text{(1)}& w_{110}=\frac{q_{110}}{Z},\end{array}

where Z is the normalization factor defined by the sum Z={\sum }_{ijk}{q}_{ijk}.

Because the channel opens when at least three of the four subunits are in state (110), the open probability P_O of IP₃R channel is given by

\begin{array}{}\text{(2)}& P_O=P_{4O}+P_{3O}=w_{110}^4+4w_{110}^3(1-w_{110}).\end{array}

The closed states (010, 100, 111) are the only states that directly connect to the active states, the equilibrium probability flux J can be written as follows [29]:

\begin{array}{}\text{(3)}& J=3P_{3O}(b_3+b_5+a_{2}C).\end{array}

The mean open time {\tau }_O, and mean close time {\tau }_C are then given by

\begin{array}{}\text{(4)}& {\tau }_O={\displaystyle \frac{P_O}{J}},\end{array}

\begin{array}{}\text{(5)}& {\tau }_C={\displaystyle \frac{1-P_O}{J}}.\end{array}

As a result, the open probability P_O is only related to all dissociation constants d_i. While the mean open time {\tau }_O and mean closed time {\tau }_C of the channel are further related to the binding constant a₂, and the unbinding constants b₃ and b₅. Thus, the parameters d₁, d₂, d₃, d₄, d₅, a₂, b₃, and b₅ of the IP₃R model can be determined by the experimental data for P_O,{\tau }_O, and {\tau }_C. Considering d_1{d}_2=d_3{d}_4, we calculated d_1=d_3{d}_4/d_2 and then searched for the seven parameters d₂, d₃, d₄,d₅, a₂, b₃, and b₅.

2.3 Annealing algorithm

We adopt a simulated annealing algorithm to optimize these seven model parameters by fitting the experimental data of P_O, {\tau }_O, and {\tau }_C of the control and M146L cells given in Refs. [23, 24]. In our simulated annealing algorithm, the optimization functions are defined as

\begin{array}{}\text{(6)}& E_{P_O}={\displaystyle \frac{{\mathrm{\Sigma }}_{I,C}|P_O^{\mathrm{expt}}(I,C)-P_O^{\mathrm{mod}}(I,C)|}{{\mathrm{\Sigma }}_{I,C}{P}_O^{\mathrm{expt}}(I,C)}},\end{array}

\begin{array}{}\text{(7)}& \begin{array}{ll}{E}_{\tau }=\hfill & \frac{{\text{Σ}}_{I,C}|\log ({\tau }_O^{\exp \text{t}})-\log ({\tau }_O^{\mathrm{mod}})|}{{\text{Σ}}_{I,C}\log ({\tau }_O^{\exp \text{t}})}\hfill \\ \hfill & +\frac{{\text{Σ}}_{I,C}|\log ({\tau }_C^{\exp \text{t}})-\log ({\tau }_C^{\mathrm{mod}})|}{{\text{Σ}}_{I,C}\log ({\tau }_C^{\mathrm{expt}})},\hfill \end{array}\end{array}

in which the superscript “expt” indicates the experimental data, and superscript “mod” indicates the modeling data.

Before optimizing the seven parameters, we had to define the search ranges of these parameters. Considering that the channel opens above a minimal concentration of I=10 nM, we set the search range of d₃ as between {10}^{-3} to {10}^{-1} μM. The channel can open when C is lower than 0.1 μM; therefore, we set the search range of d₅ to 0.01 to 1 μM. In the model, there are two dissociation constants for inhibitory Ca²⁺ binding, i.e., d₂ and d₄. It has been suggested that the binding rate of inhibitory Ca²⁺ without bound IP₃ is slower than that with bound IP₃[28], indicating that the parameter d₂ should be larger than d₄. Thus the search ranges for d₂ and d₄ were 1–100 μM and 0.1–100 μM, respectively. Inhibitory Ca²⁺ is bound only when the Ca²⁺ concentration is high; therefore, the binding rate a₂ of inhibitory Ca²⁺ was set as 10^-3–10 μM⋅s^-1, while the unbinding rates b₅ and b₃ of activating Ca²⁺ and IP₃ were set in the biologically reasonable range of 10^-3–10³ s^-1.

With the decreasing temperature in the ensuing simulated annealing algorithm, the seven model parameters were updated iteratively, resulting in decreasing values of the optimization functions. Once the values of the optimization functions became smaller than a critical value, the current values of the model parameters were then identified as the optimal parameters. In our annealing algorithm, we first applied Eq. (6) with the experimental data of P_O to determine the optimal dissociation constants d₂, d₃, d₄, d₅, and then applied Eq. (7) with the data of {\tau }_O and {\tau }_C to determine the optimal rates a₂, b₃, b₅.

3 Results and discussion

3.1 Efficiency of the annealing algorithm to fit the model parameters

A set of optimal parameters of d_i can be obtained using our annealing algorithm. As an example, shown in Fig. 2, P_O is plotted versus C at different levels I with d_2=27.0 μM, d_3=0.01 μM, d_4=0.9 μM, and d_5=0.28 μM (which are typical parameters given in Table 1). The experimental data for the control cells are indicated by symbols. Clearly, the experimental data can be fitted well with the IP₃R parameters obtained by the simulated annealing algorithm, although the quality of the fit is not completely perfect, especially for the low IP₃ concentration of 0.01 μM. These experimental data are statistical values averaged with various IP₃R channels in different cells. In our simulation, 27 experimental data of P_O at different C and I are fitted by four parameters of d_i. Here we are not trying to find out a set of best parameters to fit the experimental data, but to find many sets of good parameters with a small, but not too small, critical values of E_{P_O} in Eq. (6), and then to discuss the distributions of these optimal parameters. We assume that the distributions of these optimal parameters obtained with the model reflect, to a certain extent, the variability of biological parameters of various IP₃R channels in different cells.

3.2 Range of optimal dissociation constants

Two thousand independent simulated annealing runs were carried out, generating 2000 sets of optimal parameters \{d_i\}. Figure 3(a) shows scatter plots of all obtained optimal values of d_1\mathrm{\dots }{d}_5. Note that d₁ is calculated with d_1=d_3{d}_4/d_2. Blue symbols indicate parameter values obtained from control cells, while red symbols indicate parameter values of the M146L cells. For parameter d₃, which is related to IP₃ binding, the scatter ranges have no overlap between the control and M146L cells. In contrast, the ranges of d₄ are almost identical in the two kinds of cells. In addition, the ranges of d₂ and d₅ in the two types of cells have significant overlaps.

Besides the scatter ranges, we further studied the probability distributions of these optimal parameters, as shown in Figs. 4 (a)–(d) for d_i (i=2, 3, 4, 5). Figure 4(a) shows that the tri-modal probability distributions of d₂ are quite similar for the control and mutant cells. The probability distributions of d₃ for the control (black) and the mutant cells (red) are disjointed [see Fig. 4(b)]. The distributions of optimal values for d₄ are almost identical for the control and mutant cells [see Fig. 4(c)]. Although the scatter ranges of d₅ overlap strongly for the control and mutant cells [Fig. 3(a)], Figure 4(d) indicates that their probability distributions are almost disjointed. In conclusion, FAD mutant PS1 expression influences IP₃Rs mainly by affecting the values of d₃ and d₅, rather than d₂ and d₄.

Figure 4 also shows that the probability distributions of the optimal parameter values can be quite small for wide ranges of parameters, especially for d₄ and d₅. Considering the robustness of the model parameters, we removed the values with small probabilities and retained those values with large probabilities. Each set of parameters was obtained within a group using our annealing algorithm; therefore, we deleted all sets of optimal parameters where one parameter value has a probability of less than 0.05. Using the remaining parameters, we re-plotted the scatter ranges of the parameters in Fig. 3(b) and the probability distributions in Figs. 5(a)–(d).

Figure 3(b) indicates that the scatter ranges of the filtered parameters become narrower for the two types of cells. The differences between the two cell types are more pronounced for the pruned probability distribution shown in Fig. 5. The two dissociation constants for Ca²⁺-induced inhibition, d₂ with IP₃-bound and d₄ without IP₃-bound, display almost the same distributions for the control and the mutant cells. However, the dissociation constants d₃ and d₅ for IP₃-binding and calcium binding to the inhibitory binding site are quite different for the control and M146L cells, which are both systematically smaller in the mutant cells.

As a result of selecting the parameters from their pruned probability distributions, a set of typical parameters of d_1=0.3 μM, d_2=27.0 μM, d_3=0.01 μM, d_4=0.9 μM, and d_5=0.28 μM are suggested for the control cells (Fig. 2); and d_1=0.045 μM, d_2=27.0 μM, d_3=0.0015 μM, d_4=0.9 μM, and d_5=0.18 μM for the M146L cells (see Table 1).

3.3 Dissociation constants d₃ and d₅ are both modulated by mutant PS1

In this section, we show that parameters d₃ and d₅ have to decrease to fit the experimental data of P_O in M146L cells. We investigated the dependence of P_O on d₃ and d₅ by setting the other parameters to their typical values (d_2=27.0 μM and d_4=0.9 μM), according to the results shown in Fig. 5, with C=1 μM.

First, we studied the dependence of P_O on d₃ and d₅ at a high concentration of IP₃, I=10 μM. As plotted in Fig. 6(a), the almost vertical color bars indicate that P_O is mainly modulated by d₅, and is not sensitive to d₃. Thus, the experimental results of P_O=0.75 for control cells and 0.86 for the M146L cells at I=10 μM are located in the yellow and orange vertical bands, respectively.

We then studied the dependence of the open probability P_O of the channel on d₃ and d₅ at a low IP₃ concentration of I=33 nM. As plotted in Fig. 6(b), P_O is sensitive not only to d₅, but also to d₃ at low I. Accordingly, the experimental results of P_O=0.27 for the control cells and 0.76 for the M146L cells at I=33 nM are located in the green and orange bands in Fig. 6(b), respectively.

Thus, the possible distributions of values of d₃ and d₅ represent the overlap areas of those specific bands in Figs. 6(a) and (b), represented by the black squares and circles, respectively. These black areas are exactly the high probability regions for parameters d₃ and d₅ shown in Figs. 5(b) and (d). The analysis of Fig. 6 showed that both d₃ and d₅ have changed in the M146L cells. The increased open channel probability P_O in the M146L cells mainly results from the change in the affinities of both IP₃ binding and activating Ca²⁺ binding.

3.4 Selection of optimal binding and unbinding rates

We next considered the channel binding and unbinding rates of a₂, b₃, and b₅, by applying the simulated annealing algorithm with Eqs. (3)–(5) and (7) to fit experimental data of {\tau }_O and {\tau }_C. Again, 2000 sets of the optimal parameters were obtained through 2000 independent runs of the simulated annealing algorithm. The resulting probability distributions of the fitted rates are plotted in Fig. 7. The distributions of a₂ for the control cells and mutant cells were almost identical and mainly concentrated in the interval 0–0.5 μM^-1⋅s^-1. However, Figs. 7(b) and (c) show that the most probable values of b₃ and b₅ are concentrated on two branches, both with one branch around 0.

As discussed above, the M146L mutation influences IP₃Rs by decreasing d₃ and d₅, leaving d₂ unchanged. Thus, we concluded that parameters a₂ and b₂ are unchanged. Thus, a typical value of a_2=0.05 μM^-1⋅s^-1 is suggested.

We investigated how to determine b₃ and b₅ in the control and M146L cells. According to the experiment, for C=1 μM and I=33 nM, the values of {\tau }_O and {\tau }_C were {\tau }_O=11.5 s and {\tau }_C=41.7 s in the control cells, and {\tau }_O=31.9 s and {\tau }_C=10.0 s in the M146L cells. For C=1 μM and I=10 μM, the values of {\tau }_O and {\tau }_C were {\tau }_O=14.5 s, {\tau }_C=4.1 s in the control cells, and {\tau }_O=26.7 s, {\tau }_C=3.6 s in the M146L cells. Inserting these values into Eqs. (1)–(5) with the typical values for d_i selected above, we observed that b_3+b_5 in the probability flux J should be around 35 s^-1 in the control cells and 23 s^-1 in the M146L cells. Because b₃ and b₅ might assume small values with large probability [see Figs. 7(b) and (c)], there are two simple ways to change b₃ and b₅. The first way is to change b₃ from approximately 35 s^-1 in control cells to approximately 23 s^-1 in the M146L cells, while keeping b₅ at approximately 0 for both cell types. The second way is to change b₅ from approximately 35 s^-1 in the control cells to approximately 23 s^-1 in the M146L cell, while keeping b₃ at approximately 0 for both cell types. Considering the relation d_i=b_i/a_i, and the fact that d₃ is much smaller than d₅ [see Figs. 5(b) and (d)], the natural choice is the second option, i.e., keeping the parameters a₃ and a₅ unchanged. This choice is also supported by Figs. 7(b) and (c) which also show that the parameter values of b_5=35 s^-1 in the control cells and b_5=23 s^-1 in the M146L cells, and b₃ at around 0 in both cell types, correspond to high distribution probabilities. Thus, the typical parameters were selected as a_2=0.05, b_3=0.15, and b_5=35.4 in the control cells, and as a_2=0.05, b_3=0.02, and b_5=22.75 in the M146L cells.

Finally, the typical values of all parameters are listed in Table 1. Table 1 indicates that the PS1 mutation in Sf9 cells mainly causes a decrease in the dissociation rates of Ca²⁺ and IP₃ from IP₃Rs, resulting in an increase in the affinities of IP₃ and activating Ca²⁺. Figure 8 plots the curves of P_O, {\tau }_O, and {\tau }_C as a function of the calcium concentration for the control cells (black) and the M146L cells (red) at I=10 μM and 33 nM, respectively. In the M146L cells, a large open probability was obtained, even at a small IP₃ stimulus of I=33 nM, with a large mean open time.

4 Conclusions

In this study, we used a simulated annealing algorithm to find changes in the kinetic rate constants for a well-established mathematical model of the IP₃ receptor for M146L cells and SF9 control cells. Experimental data for the control and mutant cells were characterized by channel open probability P_O the mean open time {\tau }_O, and the mean closed time {\tau }_C. We discuss how the interaction of the presenilin mutation PS1 with the IP₃R channels modulates the gating dynamics of Ca²⁺ and IP₃ binding to and unbinding from IP₃Rs.

By comparing the typical model parameters for Sf9 control cells and the M146L mutant cells, we concluded that the PS1 mutation in Sf9 cells typically causes an increase in the channel’s affinities for IP₃ and activating Ca²⁺ (i.e., a decrease in d₃ and d₅). More specifically, rather than modulating the binding rates, both the dissociation rates of IP₃ and activating Ca²⁺ become decreased in the M146L cells. Compared with the small decrease of dissociation rate b₅ from 35.4 s^-1 in control cells to 22.76 s^-1 in M146L cells, the decrease of b₃ from 0.15 s^-1 in control cells to 0.025 s^-1 in M146L cells is notable. Thus, the PS1 mutation decreases the IP₃ dissociation rate of IP₃R channels significantly in the M146L mutant cells.

Such decreases in dissociation rates mean that, once IP₃ messengers bind to the IP₃R or Ca²⁺ ions bind to the activating site of IP₃R, it is more difficult for IP₃ or activating Ca²⁺ to unbind from the IP₃R in the M146L mutant cells than in the control cells, allowing the channel subunit to stay in the active state (110) for longer (see Fig. 1). Thus, the channels in M146L cells show a longer open time {\tau }_O and a shorter closed time {\tau }_C with a larger open probability P_O, as shown in Fig. 8. The more easily opened IP₃Rs in M146 mutant cells will release more Ca²⁺ ions into the intracellular space from the ER pool, resulting in a high intracellular calcium concentration, as observed experimentally [23, 24].

In this study, the IP₃R channel gating dynamics were simulated using a modified DYK model. Such an IP₃R model was developed to account only for the equilibrium behaviors of IP₃R dynamics of P_O, {\tau }_O, and {\tau }_C. Other behaviors of channel gating, including modal gating [32] and the kinetic response to changes in ligand concentrations [33], have not been considered here because the larger number of parameters in more complex models cannot be constrained sufficiently by the limited experimental data (only the three parameters of P_O, {\tau }_O and {\tau }_C were available for the M146L mutant cells). We suggest, however, that the binding/dissociation rates of the more complex models should be similar, because they describe the same processes.

Ullah et al. considered some Markov chain models for ion channel gating of Aβ pores associated with Alzheimer’s disease pathology [34]. They demonstrated the upregulation of gating of various Ca²⁺ release channels because of Aβ pores and showed that the extent and spatial range of such upregulation increased as Aβ pores with a low open probability and Ca²⁺ permeability transitioned into those with a high open probability and Ca²⁺ permeability. The present study was the first to study how a presenilin mutation in cells changes the gating dynamics of IP₃R channels to modulate the calcium signaling in Alzheimer’s cells using a theoretical model. Our study offers a theoretical basis to explore the channel modulation mechanism of presenilin mutations in the cells of patients with Alzheimer’s disease and may provide insights for the prevention and treatment of AD.

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