Data-driven modeling of mitochondrial dysfunction in Alzheimer’s disease

Abstract

Intracellular accumulation of oligomeric forms of β amyloid (Aβ) are now believed to play a key role in the earliest phase of Alzheimer^′s disease (AD) as their rise correlates well with the early symptoms of the disease. Extensive evidence points to impaired neuronal Ca²⁺ homeostasis as a direct consequence of the intracellular Aβ oligomers. However, little is known about the downstream effects of the resulting Ca²⁺ rise on the many intracellular Ca²⁺-dependent pathways. Here we use multiscale modeling in conjunction with patchclamp electrophysiology of single inositol 1,4,5-trisphosphate (IP₃) receptor (IP₃R) and fluorescence imaging of whole-cell Ca²⁺ response, induced by exogenously applied intracellular Aβ₄₂ oligomers to show that Aβ₄₂ inflicts cytotoxicity by impairing mitochondrial function. Driven by patch-clamp experiments, we first model the kinetics of IP₃R, which is then extended to build a model for the whole-cell Ca²⁺ signals. The whole-cell model is then fitted to fluorescence signals to quantify the overall Ca²⁺ release from the endoplasmic reticulum by intracellular Aβ₄₂ oligomers through G-protein-mediated stimulation of IP₃ production. The estimated IP₃ concentration as a function of intracellular Aβ₄₂ content together with the whole-cell model allows us to show that Aβ₄₂ oligomers impair mitochondrial function through pathological Ca²⁺ uptake and the resulting reduced mitochondrial inner membrane potential, leading to an overall lower ATP and increased production of reactive oxygen species and H₂O₂. We further show that mitochondrial function can be restored by the addition of Ca²⁺ buffer EGTA, in accordance with the observed abrogation of Aβ₄₂ cytotoxicity by EGTA in our live cells experiments.

Introduction

Alzheimer^′s disease (AD) is associated with increased production and/or impaired clearance of self-aggregating forms of β-amyloid (Aβ) [1, 2]. Strong evidence indicates that soluble oligomeric Aβ aggregates represent the major toxic species in the etiology of AD, leading to uncontrolled elevation of cytosolic Ca²⁺ [3–11]. Proposed mechanisms of action of Aβ oligomers include formation of self-aggregating Ca²⁺-permeable pores in the plasma membrane (PM) [12–16], alteration of the physicochemical properties of membrane lipids and proteins [17–19], and direct interaction with endogenous Ca²⁺-permeable receptor/channels [20–22].

While most studies on Aβ toxicity to date have focused on the effect of extracellular Aβ oligomers, understanding the effect of intracellular Aβ on cell’s function is becoming increasingly relevant, substantiated by the following observations: (i) intracellular Aβ accumulation precedes extracellular deposition [23–25]; (ii) intracellular Aβ are likely to contribute in the earliest phase of the pathogenesis of AD [23, 26, 27]; (iii) the endoplasmic reticulum (ER) of neurons has been identified as the specific site of Aβ production [28]; (iv) accumulation of intracellular Aβ have been linked to profound deficits of long-term potentiation and cognitive dysfunction in AD mice models [29, 30]; and (v) the ER membrane is the site of action for fundamental Ca²⁺ release due to opening of inositol 1,4,5-trisphosphate (IP₃) receptors (IP₃Rs) and ryanodine receptors (RyRs), regulating numerous cell’s functions. Furthermore, there is compelling evidence that intracellular Aβ evokes the liberation of Ca²⁺ from intracellular stores [8, 31–33].

We have previously shown that intracellular injection of Aβ₄₂ oligomers stimulates Gprotein-mediated IP₃ production and consequently liberate Ca²⁺ from the ER through activation of IP₃Rs [32]. Here we synergistically combine multiscale modeling, patch-clamp electrophysiology of single IP₃R, and total internal reflection fluorescence (TIRF) microscopy of global Ca²⁺ signaling to show that exogenously applied intracellular Aβ₄₂ oligomers inflict cytotoxicity by impairing mitochondrial function. We achieve this by first developing a model for the kinetics of IP₃R, driven by single-channel patch-clamp data obtained from type 1 IP₃R in isolated nuclei of Xenopus laevis oocytes. This simple four state model accurately reproduces the gating of IP₃R as a function of its two ligands, Ca²⁺ and IP₃. The single channel model is used to build a preliminary model for whole-cell Ca²⁺ signaling. The whole-cell model is then fitted to fluorescence traces representing global Ca²⁺ signals recorded in Xenopus laevis oocytes in response to exogenously applied intracellular injection of Aβ₄₂ oligomers. This allowed us to quantify and model the overall G-protein-mediated IP₃ production and consequently the Ca²⁺ released from the ER as functions of intracellular Aβ₄₂ concentration ([Aβ₄₂]). The resulting data-driven models for IP₃ and cytosolic Ca²⁺ signals were then combined with a detailed model for mitochondrial Ca²⁺ signaling and bioenergetics to show that intracellular Aβ₄₂ impairs mitochondrial function due to pathological Ca²⁺ uptake and diminished mitochondrial inner membrane potential (△Ψ_m), resulting in overall lower ATP availability to the cell and increased production of reactive oxygen species (ROS) and [H₂O₂]. This dual theory-experiment approach allows us to investigate the extent of Ca²⁺ signaling disruption and mitochondrial dysfunction as functions of cell’s intracellular [Aβ₄₂]. We further show that mitochondrial function is restored by the addition of Ca²⁺ buffer ethylene glycol-bis(β-aminoethyl ether)-N,N,N’,N’-tetraacetic acid (EGTA), in agreement with our experimental observations where Aβ₄₂ cytotoxicity was abrogated by cytosolic injection of EGTA. In addition to unifying observations from two different experimental techniques across different spatiotemporal scales, from single channel to whole-cell, our study provides a testable hypothesis for Aβ₄₂ cytotoxicity. Moreover, this to our knowledge is the first demonstration of extracting parameters related to single channel kinetics from whole-cell Ca²⁺ signals.

Methods

Computational Model

In the following, we describe the steps to model the intracellular Ca²⁺ signaling pathways affected by intracellular Aβ₄₂ (Fig. S1). Intracellular Aβ₄₂ oligomers stimulate the production of IP₃ through PLC, which binds to IP₃Rs to release Ca²⁺ from the ER. Ca²⁺ is also released from the ER through leak channels, pumped back into the ER through sarco/ER Ca²⁺-ATPase (SERCA), and buffered by Ca²⁺ sensitive dye and other buffers. Ca²⁺ released from the ER is also buffered by mitochondria that affects the production of ATP and reactive species among other things. Models for mitochondrial function and Ca²⁺ buffers are based on our previous work [34, 35] and the functional forms for Ca²⁺ leak and SERCA fluxes are adopted from [36]. To complete the computational framework for all the pathways shown in Fig. S1, we first develop a model for the gating of type-1 IP₃R in Xenopus Laevis oocytes as a function Ca²⁺ and IP₃ concentrations, based on the observations from patch-clamp electrophysiology of the channel in the absence of Aβ₄₂. A kinetic scheme closely reproducing the gating of IP₃R is crucial for estimating the amount of IP₃ generated by Aβ₄₂ as the open probability (P_o) of the receptor encodes information about the available IP₃ concentration. Since the changes in cytosolic Ca²⁺ due to Aβ₄₂-induced IP₃ generation are observed as fluctuations in the fluorescence of Ca²⁺-sensitive dye through TIRF microscopy, we next quantify the amount of Ca²⁺ represented by the fluorescence changes. Once the relationship between the changes in fluorescence and cytosolic Ca²⁺ concentration is determined, we estimate the amount of IP₃ generated by Aβ₄₂. Finally, we put all these components together to estimate the amount of IP₃ generated by exogenously applied intracellular Aβ₄₂ and investigate its downstream effects.

Single Ca²⁺ channel model

Our single-channel IP₃R model is an extension of our previous work [35] and uses a previously developed method that ensures the law of mass action and detailed balance [37–39]. We refer the interested reader to these papers for details about the modeling procedure for the single IP₃R. The study in Ref. [35] modeled the gating of IP₃R as a function of Ca²⁺ only at fixed IP₃ concentration of 10μM. While we proposed a seven-state Markov chain that could model both the Ca²⁺ and IP₃ dependence of IP₃R gating, we did not derive the rate equations for the model in our previous study [35]. Since IP₃ in this study is a dynamic variable that depends on the intracellular concentration of Aβ₄₂, the model developed here takes into account the explicit dependence of the channel’s P_o on Ca²⁺ and IP₃ as seen in our patchclam experiments on type-1 IP₃R in Xenopus Laevis oocytes [40, 41] (Fig. 1). Furthermore, the four-state model developed in this work is a much simpler model that reproduces the experimental data with the same accuracy as the more complex models proposed previously. Note that each point in Fig. 1 is an average of multiple experiments for the same ligand concentrations of intracellular Ca²⁺ ([Ca²⁺]_i) and IP₃ ([IP₃]). We first write the P_o of IP₃R in terms of occupancies of gating states as

$${\text{P}}_{\text{o}}\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}},\left[{\text{IP}}_3\right]\right)=\frac{{\text{Z}}_{\text{O}}}{{\text{Z}}_{\text{O}}+{\text{Z}}_{\text{C}}},$$ (1)

where

$${\text{Z}}_{\text{O}}={\displaystyle \sum _{\text{m}=0}^{{\text{m}}_{\text{max}}}{\displaystyle \sum _{\text{n}=0}^{{\text{n}}_{\text{max}}}{\text{KO}}_{\text{mn}}}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^{\text{m}}{\left[{\mathrm{IP}}_3\right]}^{\text{n}},$$

and

$${\text{Z}}_{\text{C}}={\displaystyle \sum _{\text{m}=0}^{{\text{m}}_{\text{max}}}{\displaystyle \sum _{\text{n}=0}^{{\text{n}}_{\text{max}}}{\text{KC}}_{\text{mn}}}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^{\text{m}}{\left[{\text{IP}}_3\right]}^{\text{n}}$$

are the total occupancies of all open and all close states respectively. ${\text{KO}}_{\text{mn}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^{\text{m}}{\left[{\text{IP}}_3\right]}^{\text{n}}$ and ${\text{KC}}_{\text{mn}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^{\text{m}}{\left[{\text{IP}}_3\right]}^{\text{n}}$ are the unnormalized occupancies of an open and a close state with m Ca²⁺ and n IP₃ bound respectively. As shown below, KO_mn and KC_mn turn out to be functions of [IP₃]. We perform an exhaustive search, testing millions of combinations of states with 0 to 5 Ca²⁺ and 0 to 4 IP₃ bound, searching for the combination so that eq. (1) gives the best fit to the P_o of the channel under optimal [IP₃]=10 μM (Fig. 1, black circles) according to Bayesian information criterion [42]. A model with four states gives us the best fit to the data. These states are a rest state (R) with no Ca²⁺ bound, an active state (A) with 2 Ca²⁺ bound, an open state (O) with 2 Ca²⁺ bound, and an inhibited state (I) with 5 Ca²⁺ bound. Thus, ${\text{P}}_{\text{o}}\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}},\left[{\text{IP}}_3\right]\right)={\text{KO}}_{2\text{n}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2{\left[{\text{IP}}_3\right]}^{\text{n}}/({\text{KC}}_{0\text{n}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^0{\left[{\text{IP}}_3\right]}^{\text{n}}+{\text{KO}}_{2\text{n}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2{\left[{\text{IP}}_3\right]}^{\text{n}}+{\text{KC}}_{2\text{n}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2{\left[{\text{IP}}_3\right]}^{\text{n}}+{\text{KC}}_{5\text{n}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^5{\left[{\text{IP}}_3\right]}^{\text{n}})$. All states have the same number of IP₃ bound, which is a consequence of constant [IP₃]. For clarity, we drop the subscripts from occupancy parameters and represent KC_0n,KC_2n,KC_5n, and KO_2n by KR,KA,KI, and KO respectively. Where KR, KA, KO, and KI are proportional to the occupancies of state R, A, O, and I respectively. We take R to be a reference state with KR = 1. Thus,

$${\text{P}}_{\text{o}}\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}},\left[{\text{IP}}_3\right]\right)=\frac{\text{KO}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}{1+\text{KO}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2+\text{KA}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2+{\text{KI[Ca}}^{2+}{]}_{\text{i}}^5}.$$ (2)

We then fit eq. (2) to the P_o data taken at [IP₃] = 33nM (Fig. 1, blue), 20nM (Fig. 1, red), and 10nM (Fig. 1, green), leaving KA, KO, and KI free each time to obtain the values of these parameters as functions of [IP₃]. Thus, we get four values for each one of these three parameters corresponding to the four [IP₃] values. These values are then fitted with the following formulas to get the functional forms of the three parameters.

$$\text{KA}=\frac{{\text{a}}_1\left({\text{a}}_3^{{\text{a}}_2}\right)}{{\left[{\text{IP}}_3\right]}^{\text{a}2}+{\text{a}}_3^{{\text{a}}_2}}+{\text{a}}_4,$$ (3)

$$\text{KI}=\frac{{\text{a}}_5\left({\text{a}}_7^{{\text{a}}_6}\right)}{{\left[{\text{IP}}_3\right]}^{\text{a}6}+{\text{a}}_7^{{\text{a}}_6}}+{\text{a}}_8,$$ (4)

$$\text{KO}=\frac{{\text{a}}_9{\left[{\text{IP}}_3\right]}^{{\text{a}}_{10}}}{{\left[{\text{IP}}_3\right]}^{{\text{a}}_{10}}+{\text{a}}_{11}^{{\text{a}}_{10}}}.$$ (5)

Constants a₁ − a₁₁ given by the fits are listed in Table 1. We would like to remark that this approach of writing the rate constants (or parameters related to the transition rates) as functions of IP₃ and/or Ca²⁺ has been frequently used in literature (see for example [36, 43, 44]) as it significantly simplifies the Markov chain model for the channel. For example, without such simplification a model for IP₃R in Xenopus Laevis oocytes will have a minimum of seven states [35] while those in other cells could have anywhere from eight to eighteen states [37, 41, 45–47].

Fig. 1:

Equilibrium P_o of the single IP₃R channel in Xenopus laevis oocytes as a function of Ca²⁺ and IP₃ concentrations. P_o at [IP₃] = 10μM (black), 33nM (blue), 20nM (red), and 10nM (green) as we vary [Ca²⁺]_i. The symbols and lines represent experimental results [40, 41] and model fits respectively.

Table 1:

Constants involved in the functions for occupancy variables. There parameters were obtained from fits to the single channel P_o data recorded at different [Ca²⁺]_i and [IP₃] values using patch-clamp on type-1 IP₃R.

Parameter	Value
a1	76.87μM−2
a2	4.98
a3	0.013μM
a4	4.26μM−2
a5	1.45μM−5
a6	17.76
a7	0.025μM
a8	0.000086μM−5
a9	17.66μM−2
a10	1.59
a11	0.017μM

The P_o data can only be used to infer the number of states and their occupancies in the model. For the connectivity of these states and the related transition rates, we use the information from rapid perfusion experiments on type-1 IP₃R [48]. The kinetic data suggests a direct link between states O and I but no direct link between states R and O. These experiments also point to a direct link between states R and I. All considerations from the kinetic experiments lead to the network shown in Fig. 2. Further details about the connectivity between states and the derivation of transitions rates are given in the next section. Model fits to the experimental data as we vary [Ca²⁺]_i and [IP₃] are shown by solid lines in Fig. 1.

Fig. 2:

Schematic of the four-state model for single IP₃R Ca²⁺ channel. K_XY represents the transition rate from state X to Y where X,Y =, A, O, I , and R respectively.

Transition rates for the single-channel model

In the kinetic experiments, we measured the response of the channel to rapid jumps in both [IP₃] and [Ca²⁺]_i. To measure the inhibition latency (the time for the channel to go from actively gating to inhibited), we jumped [Ca²⁺]_i from optimal ([Ca²⁺]_i = 2μM where the P_o is maximal or near maximal ) to inhibiting ([Ca²⁺]_i = 300μM where P_o is near zero) at fixed [IP₃] = 10μM. The optimal and inhibiting values of [Ca²⁺]_i are estimated from the experimental observations about the P_o shown in this paper and our previous studies on type-1 IP₃R. If the channel had needed to traverse some sequence of states to get from the open state to the inhibited state (or states) there would not exist very short inhibition latencies. At least with the resolution afforded by our experiments, we found non-zero values in the distribution of times to inhibition (f_i(t_i)) at t_i = 0ms. Thus, there appears to be no states intermediate to O and I. We therefore conclude that the open state is directly connected to the inhibited state. There appears to be nothing in the data that demands a link between A and I. We did report a deficit of short activation latencies when we jumped [Ca²⁺]_i from < 10 nM to optimal. This implies that R is not directly connected to O.

In another set of experiments, we jumped [Ca²⁺]_i from near 0 ([Ca²⁺]_i <10 nM) to inhibiting value of 300μM and found that the channel failed to burst (open) in 9 out of 103 experiments. Similarly, in experiments where we jumped [Ca²⁺]_i from 300μM to < 10 nM, 88 out of 94 times the channel deactivated without bursting. Both these observations suggest a direct link between R and I states. Thus, the kinetic data on type-1 IP₃R appears to be consistent with a network topology shown in Fig. 2.

Next, we use our novel occupancy/flux formalism [49] and the information from these kinetic experiments to derive expressions for the mean transition times between different states. As discussed in detail previously [37], we prefer this approach over parameterizing the model with reaction rates because we find it simpler and more intuitive. It separates thermodynamic quantities (equilibrium occupancies) from kinetic quantities (reaction fluxes), or equivalently, it separates reaction rates from equilibrium constants. Occupancy and flux parameters are used also because occupancy determination is far more robust than state lifetime measurements. Failure to detect brief events (channel opening and closing) can have a large impact on estimates of mean lifetimes of the states but does not affect state occupancy estimates significantly. Furthermore, this approach ensures detailed balance and microscopic reversibility.

According to this formalism, the mean transition time between two states X and Y where the channel has m and n ligands ([L]) bound respectively is given as

$$\begin{gathered} {\text{T}}_{\text{XY}}=\text{KX}{\left[\mathrm{L}\right]}^{\text{m}}\left(\frac{1}{{\text{k}}_{\text{mn}}{\left[\mathrm{L}\right]}^{\mathrm{q}}}\right), \\ {\text{T}}_{\text{YX}}=\text{KY}{\left[\mathrm{L}\right]}^{\text{n}}\left(\frac{1}{{\text{k}}_{\text{mn}}{\left[\mathrm{L}\right]}^{\mathrm{q}}}\right). \end{gathered}$$

Where KX[L]^m and KY [L]ⁿ is occupancy of state X and Y respectively, $k_{mn}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^{\text{q}}$ is the probability flux between the two states, and q = max(m, n).

Note that in our model there are transitions that involves binding of more than one Ca²⁺. For example, the channel has 2 and 5 Ca²⁺ bound in states O and I respectively. However, the direct link between O and I does not necessarily mean that the channel binds 3 Ca²⁺ simultaneously. It rather means that the states with 3 and 4 Ca²⁺ bound have very low occupancy and are not required to be included in the model. However, these transition states act as speed-bumps for the probability flux and their effect can be incorporated in the mean transition times. Thus, if there is a low-occupancy transition state Z between states X and Y with o ligands bound, the mean transition times between states X and Y become

$$\begin{gathered} {\text{T}}_{\text{XY}}=\text{KX}{\left[\mathrm{L}\right]}^{\text{m}}\left(\frac{1}{{\text{k}}_{\text{mo}}{\left[\mathrm{L}\right]}^{\text{r}}}+\frac{1}{{\text{k}}_{\text{on}}{\left[\mathrm{L}\right]}^{\text{s}}}\right), \\ {\text{T}}_{\text{YX}}=\text{KY}{\left[\mathrm{L}\right]}^{\text{n}}\left(\frac{1}{{\text{k}}_{\text{mo}}{\left[\mathrm{L}\right]}^{\text{r}}}+\frac{1}{{\text{k}}_{\text{on}}{\left[\mathrm{L}\right]}^{\text{s}}}\right), \end{gathered}$$

where r = max(m, o) and s = max(o, n).

The transition rate from state X to Y (K_XY) is simply the inverse of mean time to transition from state X and Y and vice versa.

Using this formalism, the mean transition times between different states in our single-channel model (Fig. 2) are given as

$${\text{T}}_{\text{RA}}=\text{KR}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^0\left(\frac{1}{{\text{k}}_{01}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}+\frac{1}{{\text{k}}_{12}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}\right),$$ (6)

$${\text{T}}_{\text{AR}}=\text{KA}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2\left(\frac{1}{{\text{k}}_{01}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}+\frac{1}{{\text{k}}_{12}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}\right),$$ (7)

$${\text{T}}_{\text{AO}}=\text{KA}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2\left(\frac{1}{{\text{k}}_{22}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}\right),$$ (8)

$${\text{T}}_{\text{OA}}=\text{KO}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2\left(\frac{1}{{\text{k}}_{22}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}\right),$$ (9)

$${\text{T}}_{\text{OI}}=\text{KO}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2\left(\frac{1}{{\text{k}}_{23}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^3}+\frac{1}{{\text{k}}_{34}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^4}+\frac{1}{{\text{k}}_{45}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^5}\right),$$ (10)

$${\text{T}}_{\text{IO}}=\text{KI}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^5\left(\frac{1}{{\text{k}}_{23}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^3}+\frac{1}{{\text{k}}_{34}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^4}+\frac{1}{{\text{k}}_{45}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^5}\right),$$ (11)

$$\begin{gathered} {\text{T}}_{\text{RI}}= \\ \text{KR}\left(\frac{1}{{\tilde{\text{k}}}_{01}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}+\frac{1}{{\tilde{\text{k}}}_{12}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}+\frac{1}{{\tilde{\text{k}}}_{23}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^3}+\frac{1}{{\tilde{\text{k}}}_{34}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^4}+\frac{1}{{\tilde{\text{k}}}_{45}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^5}\right), \end{gathered}$$ (12)

$${\text{T}}_{\text{IR}}=\text{KI}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^5\left(\frac{1}{{\tilde{\text{k}}}_{01}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}+\frac{1}{{\tilde{\text{k}}}_{12}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^2}+\frac{1}{{\tilde{\text{k}}}_{23}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^3}+\frac{1}{{\tilde{\text{k}}}_{34}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^4}+\frac{1}{{\tilde{\text{k}}}_{45}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^5}\right).$$ (13)

We assume that the probability flux between two intermediate states is very high so that we can use relative large values for k₃₄, ${\tilde{k}}_{12},{\tilde{k}}_{23}$, and ${\tilde{k}}_{34}$. Thus, the terms involving these constants can be ignored without affecting the mean transition times.

The mean open time of type-1 IP₃R under fixed [Ca²⁺]_i of 2μM is 10ms, i.e. T_OA ∼ 10ms. Using eq. 9, we get

$${\text{k}}_{22}=\frac{\text{KO}}{10\text{ms}}.$$ (14)

The remaining six flux parameters are determined from the rapid-perfusion experiments described above. Observations from more than three thousands of these kinetic experiments are summarized as follows (see [48] for further details).

Mean activation time (changing [Ca²⁺]_i from < 10 nM to 2 μM) = 40 ± 3 ms.

Mean de-activation time (changing [Ca²⁺]_i from 2 μM to < 10nM) = 160 ± 20 ms.

Mean inhibition time (changing [Ca²⁺]_i from 2 μM to 300μM) = 290 ± 40 ms.

Mean inhibition-recovery time (changing [Ca²⁺]_i from 300 μM to 2 μM ) = 2.4 ± 0.2s. Changing [Ca²⁺]_i from < 10 nM to 300μM $\frac{9}{103}$ experiments failed to cause bursts. Changing [Ca²⁺]_i from 300 μM to < 10 nM, $\frac{6}{94}$ times the channel bursts before getting de-activated.

Thus we can write:

$${{\text{T}}_{\text{RA}}|}_{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}=2\mu \text{M}}=40\text{ms},$$ (15)

$${{\text{T}}_{\text{AR}}|}_{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}=10\text{nM}}=160\text{ms},$$ (16)

$${{\text{T}}_{\text{OI}}|}_{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}=300\mu \text{M}}=290\text{ms,}$$ (17)

$${{\text{T}}_{\text{IO}}|}_{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}=2\mu \text{M}}=2.4\text{s},$$ (18)

$${\frac{{\text{T}}_{\text{RA}}}{{\text{T}}_{\text{RI}}}|}_{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}=300\mu \text{M}}=\frac{9}{103},$$ (19)

$${\frac{{\text{T}}_{\text{IR}}}{{\text{T}}_{\text{IO}}}|}_{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}=10\text{nM}}=\frac{6}{94}.$$ (20)

Using identities (15) and (16), we extracted k₀₁ and k₁₂. Identities (17) and (18) give us k₂₃ and k₄₅. ${\tilde{\text{k}}}_{01}$ and ${\tilde{\text{k}}}_{45}$ are extracted from (19) and (20). The values of these six probability flux parameters are given in Table 2.

Table 2:

Probability flux parameters used in the transition rates between the four states of single-channel model. These parameters were estimated from the rapid perfusion experiments measuring the response of type-1 IP₃R to rapid switch of ligands Ca²⁺ and IP₃ between different values using patch-clamp electrophysiology [48].

Parameter	Value
k01	0.0162/μMms
k12	0.0127/μM2ms
k23	2.1651 × 10−4/ μM3ms
k45	3.5935 × 10−8/ μM5ms
${\tilde{\mathrm{k}}}_{01}$	0.0014127/μMms
${\tilde{\mathrm{k}}}_{45}$	5.6297 × 10−7/μM5ms

Thus, the kinetics of IP₃Rs in the cell, based on the scheme shown in Fig. 2, are given by the following rate equations that are nonlinear in [Ca²⁺]_i and [IP₃] (through rates K_XY).

$$\frac{dA}{dt}={\text{K}}_{\text{RA}}\text{R}+{\text{K}}_{\text{OA}}\text{O}-\left({\text{K}}_{\text{AR}}+{\text{K}}_{\text{AO}}\right)\text{A},$$ (21)

$$\frac{dO}{dt}={\text{K}}_{\text{IO}}\text{I}+{\text{K}}_{\text{AO}}\text{A}-\left({\text{K}}_{\text{OI}}+{\text{K}}_{\text{OA}}\right)\text{O},$$ (22)

$$\frac{dI}{dt}={\text{K}}_{\text{RI}}\text{R}+{\text{K}}_{\text{IO}}\text{O}-\left({\text{K}}_{\text{IO}}+{\text{K}}_{\text{IR}}\right)\text{I}.$$ (23)

The fraction of channels in state R is given by the conservation of total probability, i.e.

$$\text{R}=1-\text{A}-\text{O}-\text{I}.$$ (24)

Whole-cell Ca²⁺ signaling model

The single-channel model developed above is used to build the whole-cell model. IP₃R releases Ca²⁺ from the ER through IP₃R to the cytoplasm when gating in state O (J_ipr). The background Ca²⁺ leak (J_leak) from the ER is another source of Ca²⁺ release. Ca²⁺ is relocated to ER by Sarco-ER Ca²⁺-ATPase (J_s) and buffered by fluorescence dye Fluo-4 ([dye]). In some simulations, we also incorporate mobile Ca²⁺ buffer EGTA that can uptake cytosolic Ca²⁺. Some of these fluxes depend nonlinearly on [Ca²⁺]_i and/or [IP₃], resulting in a nonlinear set of rate equations [Ca²⁺]_i, dye ([dye]), and EGTA ([EGTA]) concentrations in the whole-cell model as given below.

$$\begin{gathered} \frac{d{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{dt}= \\ {\text{J}}_{\text{ipr}}+{\text{J}}_{\text{leak}}-{\text{J}}_{\text{s}}+({\text{k}}_{\text{dye}}^{\text{r}}\left({\text{B}}_{\text{dye}}-[\text{dye}]\right)-{\text{k}}_{\text{dye}}^{\text{f}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}[\text{dye}]) \\ +\left({\text{k}}_{\text{EGTA}}^{\text{r}}\left({\text{B}}_{\text{EGTA}}-[\text{EGTA}]\right)-{\text{k}}_{\text{EGTA}}^{\text{f}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}[\text{EGTA}]\right). \end{gathered}$$ (25)

Where B_dye and B_EGTA is the total concentration of Fluo-4 and EGTA, ${\text{k}}_{\text{dye}}^{\text{f}}$ and ${\text{k}}_{\text{dye}}^{\text{r}}$ are the binding and unbinding rates of Ca²⁺ to Fluo-4, and ${\text{k}}_{\text{EGTA}}^{\text{f}}$ and ${\text{k}}_{\text{EGTA}}^{\text{r}}$ are the binding and unbinding rates of Ca²⁺ to EGTA respectively. The functional form of the three fluxes are adopted from [36] and given as

$${\text{J}}_{\text{leak}}={\text{k}}_{\text{leak}}\left({\left[{\text{Ca}}^{2+}\right]}_{\text{ER}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\right),$$ (26)

$${\text{J}}_{\text{s}}=\frac{{\text{V}}_{\text{s}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^{\text{ns}}}{{\text{K}}_{\text{s}}^{\text{ns}}+{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}^{\text{ns}}},$$ (27)

$${\text{J}}_{\text{ipr}}={\text{k}}_{\text{ipr}}\text{O}\left(\left[{\text{Ca}}^{2+}\right]\text{ER}-{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\right).$$ (28)

Ca²⁺ concentration in the ER ([Ca²⁺]_ER) is given by the conservation of total Ca²⁺ ([Ca²⁺]_t) in the cell, i.e.

$${\left[{\text{Ca}}^{2+}\right]}_{\text{ER}}=\gamma \left({\left[{\text{Ca}}^{2+}\right]}_{\text{t}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\right).$$ (29)

Rate equations for Fluo-4 and EGTA are given as

$$\frac{d[\text{dye}]}{dt}={\text{k}}_{\text{dye}}^{\text{r}}\left({\text{B}}_{\text{dye}}-[\text{dye}]\right)-{\text{k}}_{\text{dye}}^{\text{f}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}[\text{dye}],$$ (30)

$$\frac{d[\text{EGTA}]}{dt}={\text{k}}_{\text{EGTA}}^{\text{r}}\left({\text{B}}_{\text{EGTA}}-[\text{EGTA}]\right)-{\text{k}}_{\text{EGTA}}^{\text{f}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}[\text{EGTA}].$$ (31)

Parameter values used in the whole-cell Ca²⁺ model are given in Table 3.

Table 3:

Parameter values for Ca²⁺ and buffer rate equations

Parameter	Description	Value	Reference
kleak	ER leak flux coefficient	0.0032 s−1	[36]
kipr	IP3R flux coefficient	0.2379 s−1	fit to the data
Γ	the cytoplasmic-to-ER volume ratio	5.405	[130]
Vs	Maximum capacity of SERCA	23.17 μM/s	fit to the data
ks	SERCA half-maximal activating [Ca2+]i	4.146 μM	fit to the data
ns	Hill coefficient for SERCA	1.084	fit to the data
[Ca2+]t	Total intracellular Ca2+	50 μM	Estimated
Bdye	Total fluorescent dye	40 μM	Our experimental value
BEGTA	Total buffer EGTA	0–3000 μM	Our experimental values
${\mathrm{k}}_{\mathrm{dye}}^{\mathrm{r}}$	reverse rate of dye	450 s−1	[131]
${\mathrm{k}}_{\mathrm{dye}}^{\mathrm{f}}$	forward rate of dye	150μM−1s−1	[131]
${\mathrm{k}}_{\mathrm{EGTA}}^{\mathrm{r}}$	reverse rate of EGTA	0.75 s	[132]
${\text{k}}_{\text{EGTA}}^{\text{f}}$	forward rate of EGTA	5 μM−1s−1	[132]
kdiff	Ca2+ diffusional flux coefficient	10 s−1	[36]
γ1	the cytoplasmic-to-microdomain volume ratio	100	[36]

Converting Ca²⁺-bound dye to fluorescence units

TIRF microscope measures changes in [Ca²⁺]_i in terms of fluorescence changes (ΔF/F_o) as Ca²⁺ binds and unbinds to indicator dye Fluo-4. Where F₀ and ΔF is background fluorescence and change in fluorescence in response to Ca²⁺ binding to Fluo-4. Thus, we convert Ca²⁺-bound dye from concentration units to ΔF/F_o, which is then used in fitting the model to fluorescence signals from TIRF microscopy.

A single channel opening on average increases the fluorescence over a 1.2μm × 1.2μm area around the channel relative to the background signal by 0.11 ± 0.01 (i.e. ΔF/F_o = 0.11 ± 0.01) in TIRF microscopy experiments using Fluo-4 [50]. We use this information to convert [Ca²⁺]-bound [dye] to ΔF/F_o in the model. We simulate a single channel, placed at the center of a 40μm × 40μm area (equal to the area scanned in the TIRF microscopy experiments in [50] as well as in our experiments) and allow it to open for 10ms (the mean open time of IP₃R). The [Ca²⁺]_i and [dye] equations described above are modified to include diffusion of these two species with the widely accepted diffusion coefficients of 223μm²/s and 200μm²/s for free Ca²⁺ (D_Ca) and Fluo-4 (D_dye) respectively. Furthermore, to side-step the requirement of using extremely small spatial grid and stay consistent with the spatial resolution of 0.3μm of the TIRF microscopy experiments in [50], we partially adopt the procedure in [36], which simulates Ca²⁺ release and uptake at the microdomain around the channel separately from the rest of the simulation area. The movement of Ca²⁺ from the microdomain to the nearest gird point (the central grid point in our case) and vice versa is given by diffusion. Also, to remain consistent with the TIRF microscopy experiments that estimated the mean fluorescence change during single channel opening, no EGTA was included in these simulations. With these modifications, the rate equation for [Ca²⁺]_i becomes

$$\frac{d{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{dt}={\text{D}}_{\text{Ca}}{\nabla }^2{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}+{\text{J}}_{\text{diff}}+{\text{J}}_{\text{leak}}-{\text{J}}_{\text{s}}+\left({\text{k}}_{\text{dye}}^{\text{r}}\left({\text{B}}_{\text{dye}}-[\text{dye}]\right)-{\text{k}}_{\text{dye}}^{\text{f}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}[\text{dye}]\right).$$ (32)

Where the first and second terms represent the diffusion of Ca²⁺ between neighboring grid points and the transfer of Ca²⁺ from the microdomain around IP₃R to the grid point at the center of simulating area respectively. The functional form of J_diff is adopted from [36] where it is modeled with a function similar to Fick’s first law. That is, J_diff = k_diff([Ca²⁺]_b−[Ca²⁺]_i). For all other grid points, J_diff = 0. [Ca²⁺]_b is Ca²⁺ concentration in the microdomain and is given by the rate equation

$$\frac{d{\left[{\text{Ca}}^{2+}\right]}_{\text{b}}}{dt}={\gamma }_1\left({\text{J}}_{\text{ipr}}-{\text{J}}_{\text{diff}}\right).$$ (33)

J_ipr for the single channel is governed by the current through the channel, i.e.

$${\text{J}}_{\text{ipr}}=\frac{\text{I}}{2\times \text{F}\times \delta \text{V}}.$$ (34)

Where I = 0.05pA is the observed current through IP₃R [51], F is Faraday’s constant, and δV is the volume of a hemisphere over the channel with a radius of 10nm [52, 53].

With the inclusion of the microdomain in the model, the expression for [Ca²⁺]_ER changes accordingly, i.e.

$${\left[{\text{Ca}}^{2+}\right]}_{\text{ER}}=\gamma \left({\left[{\text{Ca}}^{2+}\right]}_{\text{t}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}-{\left[{\text{Ca}}^{2+}\right]}_{\text{b}}/{\gamma }_1\right).$$ (35)

With the inclusion of diffusion, the rate equation for [dye] changes to

$$\frac{d[\text{dye}]}{dt}={\text{D}}_{\text{Ca}}{\nabla }^2[\text{dye}]+{\text{k}}_{\text{dye}}^{\text{r}}\left({\text{B}}_{\text{dye}}-[\text{dye}]\right)-{\text{k}}_{\text{dye}}^{\text{f}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}[\text{dye}].$$ (36)

Initial and boundary conditions used for different equations are given in section “Numerical and Experimental Methods” below. The above equations are simulated using forward difference method and the peak change in Ca²⁺-bound dye (Δ[dyeCa²⁺]) with respect to resting level ([dyeCa²⁺]_o) is recorded, where [dyeCa²⁺] = B_dye − [dye]. A single channel opening of 10ms resulted in Δ[dyeCa²⁺]/[dyeCa²⁺]_o = 0.1756 when averaged over 1.2μm × 1.2μm area around the channel. This together with the experimentally observed mean ΔF/F_o during a single IP₃R opening is used to convert Ca²⁺-bound dye to ΔF/F_o.

Estimating IP₃ production due to exogenously applied intracellular Aβ₄₂

Injection of different concentrations of Aβ₄₂ oligomers into the oocytes stimulates IP₃ production through a G-protein coupled mechanism [32]. IP₃ activates IP₃R on the ER membrane, releasing Ca²⁺ into the cytoplasm. The global [Ca²⁺]_i is then represented by ΔF/F_o, imaged through TIRF microscopy. As shown in Fig. 3, after injecting Aβ₄₂, [Ca²⁺]_i first increases to some fixed value and then decays slowly to base level. The rise time, peak, and decay rate of [Ca²⁺]_i depends on the amount of Aβ₄₂ injected into the cell. Guided by the shape of these traces we write the following function for IP₃ generated by exogenously applied intracellular Aβ₄₂.

$$\left[{\text{IP}}_3\right]={\text{p}}_1\left[\frac{1}{1+\exp \left(\frac{{\text{p}}_2-\text{t}}{{\text{p}}_3}\right)}\right]\exp (-\frac{\text{t}}{{\text{p}}_4}).$$ (37)

Note that in addition to generation by Aβ₄₂, eq. 37 also reflects the subsequent dephosphorylation and phosphorylation of IP₃ into inositol bisphosphate and inositol tetrakisphosphate respectively.

Fig. 3:

Intracellular injection of Aβ₄₂ oligomers leads to IP₃ generation and consequently Ca²⁺ release from the ER into the cytoplasm. Time-traces of ΔF/F_o indicating [Ca²⁺]_i rise from experiment (circles) and model fits (solid lines) due to injection of 30μg/ml (blue), 10μg/ml (black), 3μg/ml (red), and 1μg/ml (green) [Aβ₄₂] oligomers (A) and corresponding IP₃ production (B). The experimental traces represent the mean time course of Ca²⁺dependent fluorescence recorded from 5 (30 μg/ml), 4 (10 μg/ml), 4 (3 μg/ml), and 6 (1 μg/ml) oocytes respectively [32].

Next, the whole-cell model (eqs. 21–30 together with eq. 37 and Ca²⁺-bound [dye] to ΔF/F_o conversion) is fitted to the fluorescence trace recorded at Aβ₄₂ = 30μg/ml (blue circles in Fig. 3). To perform the fit, we write a Matlab code that computes the least squares error (χ²) as follows. The code first solves the rate equations for the whole-cell (Eqs. 21–30) and uses the conversion factor for Ca²⁺-bound dye to ΔF/F_o to get the whole-cell fluorescence signal given by the model (ΔF/F_o,_M). Using the whole-cell fluorescence signal from TIRF microscopy experiments (ΔF/F_o,_E) as observable, we can write χ² as

$${\chi }^2=\frac{1}{\mathrm{N}}{\displaystyle \sum _{\mathrm{i}=1}^{\text{N}}{\left(\mathrm{\Delta }\text{F}/{\text{F}}_{\text{o},\text{M}}-\mathrm{\Delta }\text{F}/{\text{F}}_{\text{o}},\text{E}\right)}^2},$$ (38)

where N is the number of data points in the observed fluorescence trace. χ² is minimized by leaving parameters p₁ −p₄, V_s, k_s, ns, and k_ipr free. Where the last four parameters represent maximum SERCA rate, SERCA half-maximal activating [Ca²⁺]_i, Hill coefficient for SERCA, and maximum flux through IP₃R.

We repeat the same procedure for fluorescence traces recorded at Aβ₄₂ = 10,3, and 1μg/ml, with the exception that V_s, k_s, ns, and k_ipr are fixed at the values given by the fit to the fluorescence trace recorded at Aβ₄₂ = 30μg/ml. This is due to the fact that we are not aware of any evidence showing that these parameters change as Aβ₄₂ is changed. Parameter values for p₁ −p₄ from these fits are given in Table 4 as functions of [Aβ₄₂] and can be closely approximated by the following equations.

$${\text{p}}_1=0.019\frac{{\left[\text{A}{\beta }_{42}\right]}^{1.15}}{\left[\text{A}{\beta }_{42}\right]{]}^{1.15}+{0.0025}^{1.15}},$$ (39)

$${\text{p}}_2=26.93\exp \left(-88.06\left[\text{A}{\beta }_{42}\right]\right),$$ (40)

$${\text{p}}_3=4.76\exp \left(-2.85\left[\text{A}{\beta }_{42}\right]\right)+1.73,$$ (41)

$${\text{p}}_4=3.96\exp \left(-2.28\left[\text{A}{\beta }_{42}\right]\right)+\mathrm{304.07.}$$ (42)

Where [Aβ₄₂] is in units of μM. In our experiments, 10nl of 30, 10, 3, and 1μg/ml of Aβ₄₂ were injected into the cell. Using the molecular weight of 4514.1 g/mol gives [Aβ₄₂] in units of μM (Table 4). We remark that converting [Aβ₄₂] unit from μg/ml to μM is not required for the model and the functions in Eqs. 39–42 are valid in any case with proper unit conversion.

Table 4:

Parameter values involved in IP₃ response function. These parameters were obtained by fitting the whole-cell model to the whole-cell Ca²⁺ signals recorded in our TIRF microscopy experiments.

[Aβ42] injected at 10nL	Values
30 μg/ml	p1=0.0279 μM
=	p2=7.869 s
0.0127 μM	p3=1.855 s
	p4=24.069 s
10 μg/ml	p1=0.023 μM
=	p2=19.986 s
0.00423 μM	p3=3.155 s
	p4=49.359 s
3 μg/ml	p1=0.006 μM
=	p2=27.203 s
0.00127 μM	p3=4.304 s
	p4=117428.967 s
1 μg/ml	p1=0.003 μM
=	p2=25.243 s
0.000423 μM	p3=5.948 s
	p4=1.507 s

We would like to point out that after expressing in units of nM, the Aβ₄₂ concentrations used in our experiments (Table 4) are well in the range normally used to investigate Aβ₄₂ toxicity [8, 33, 54, 55]. The numbers given in Table 4 are based on the molecular weight of Aβ₄₂ monomer. In oligomeric form, there can be tens of Aβ₄₂ monomer in a single oligomer. Thus the oligomer concentrations used in our experiments are in reality much smaller than the numbers give in Table 4. Furthermore, the intracellular accumulation of Aβ₄₂ can be two to three folds higher than the extracellular concentration [27, 56–61].

Details about the numerical scheme used to solve the rate equations and optimize χ² function are given in the “Numerical Methods” section below.

Mitochondrial function model

The rate equations modeling mitochondrial function are adopted from our previous work [34, 62] and are listed in Table S1. These equations are originally based on the model in [63, 64], which is an extension of the Magnus-Keizer model [65]. The model includes processes such as the tricarboxylic acid (TCA) cycle, electron transport chain (ETC), Ca²⁺ signaling pathways, reactive oxygen species (ROS), and [H₂O₂] production. The key components of this model are shown in Fig. S2 and described in SI Appendix, section S1. For further details of the mitochondrial model, we refer the interested reader to Ref. [34]. We would like to point out that different extensions of Magnus-Keizer model are applied to different cell types [66–75], indicating the robustness of the model and its suitability to represent many cell types.

The models for IP₃ generation by exogenously applied intracellular Aβ₄₂ and whole-cell Ca²⁺ signaling are coupled with the mitochondrial model through the rate equation for mitochondrial Ca²⁺ concentration ([Ca²⁺]_m in Table S1) (Fig. S1) to investigate Ca²⁺ signaling disruption as well as mitochondrial dysfunction as functions of intracellular Aβ₄₂ concentration, and to identify the conditions that would restore mitochondrial function. Specifically, we will show that an increase in [Ca²⁺]_i due to Aβ₄₂ increases Ca²⁺ influx to mitochondria through mitochondrial Ca²⁺ uniporter (V_uni in Table S1), which affects mitochondrial ATP and ROS production. The increase in mitochondrial Ca²⁺ concentration ([Ca²⁺]_m) resulting from intracellular injection of Aβ₄₂ can be counterbalanced and mitochondrial function restored by buffering excessive cytosolic Ca²⁺ using EGTA.

Numerical and Experimental Methods

Fits to the single IP₃R P_o data were performed in Mathematica using in-built function NonlinearModelFit. Akaike information criterion (AIC) and Bayesian information criterion were used for assessing the quality of fits and model selection. The model with smallest AIC and BIC scores was selected as the best model fitting the P_o data.

Simulations in section “Converting Dye to ΔF/F_o” were performed using forward difference method with a time-step of 50μs and a spatial grid size of 0.3μm, equal to the pixel size in the experiments in [50] that estimated the mean ΔF/F_o value during a single channel opening. Reducing the grid size and time-step did not change our estimates significantly. The initial value of [Ca²⁺]_i was set to 50nM, approximately equal to the resting cytosolic Ca²⁺ concentration. Based on our simulations, we observed that in resting state approximately 1μM of fluorescence dye was bound to Ca²⁺. We thus set the initial value of [dye] = 39μM. The boundaries were fixed accordingly at steady state values of [Ca²⁺]_i = 50nM and [dye] = 39μM.

Since there was no stimulus at the start of the experiments, IP₃Rs were initially considered to be in the resting state, that is, the initial values of R, A, O, and I were set at 1, 0, 0, and 0 respectively. In simulations involving EGTA, we observed that approximately 1μM of EGTA was bound to Ca²⁺ in the resting conditions. We thus set the initial value of [EGTA] = B_EGTA − 1μM, where B_EGTA is the total EGTA concentration. In all cases, simulations were allowed to reach steady state before applying any stimulus, so selecting slightly different initial conditions would not change the final results.

To fit the whole-cell model to the TIRF signals, the rate equations were solved in Matlab using the in-built function “ode15s”. χ² function was minimized using the in-built Matlab function “fminsearch”. The functions in Eqs. 3–5 and 39–42 were optimized by using inbuilt Matlab function “lsqcurvefit”. Using other Matlab optimization functions such as “fminsearch”, “fmincon”, or “nlinfit” did not change the quality of fits.

Numerical integration of the full model equations (21–24, 25, 30, 31, 37, and equations for mitochondrial Ca²⁺ dynamics and bioenergetics) was performed with Intel Fortran compiler (Intel Corporation, Santa Clara, CA). ODEs were solved using RK4 method. Code producing key results in the paper is available upon request from authors.

Details about the experimental methods are given in the SI Appendix, Section S2.

Results

Our main goal in this study is to use data-driven modeling to estimate the amount of IP₃ generated and Ca²⁺ released from the ER as a function of exogenously applied intracellular Aβ₄₂ concentration. The estimated [IP₃] and [Ca²⁺]_i result in a good model for IP₃ and Ca²⁺ dynamics in the cell that paves the way for investigating the cytotoxicity of intracellular Aβ₄₂. Specifically, we show that the observed cytotoxicity of intracellular Aβ₄₂ is due to mitochondrial dysfunction.

Exogenously applied intracellular Aβ₄₂ oligomers release Ca²⁺ from the ER into the cytoplasm through IP₃ generation

Previously reported findings [32] indicate that intracellular Aβ₄₂ oligomers but not monomers, fibrillar, or scrambled Aβ₄₂ peptides leads to significant increase in cytosolic Ca²⁺ signals after a few tens of seconds of injection. Fig. 3A shows representative traces illustrating increase in ΔF/F_o in response to 10 nl injection of Aβ₄₂ oligomers at concentrations of 1 (6 oocytes), 3 (4 oocytes), 10 (4 oocytes), and 30 (5 oocytes) μg/ml as indicated by the symbols. The amplitude of Ca²⁺ signals increases, while latency to the onset and peak of the responses shortens progressively with increasing concentration of Aβ₄₂.

To rule out any contribution due to extracellular Ca²⁺ influx, all these experiments were performed in a Ca²⁺-free bathing solution including 2 mM EGTA [32]. Intracellular injection of heparine, an IP₃R inhibitor, completely abolished the observed Ca²⁺ signals. Caffeine that acts as a reversible membrane-permeant antagonist of IP₃R [76–80], also blocked transient global Ca²⁺ signals. Aβ₄₂-induced Ca²⁺ signals were also suppressed by treatment of oocytes with PTX to inhibit G-protein mediated activation of PLC and lithium to deplete membrane inositol lipids. All these observations confirm that Aβ₄₂ induces Ca²⁺ release from the ER through IP₃R by stimulating PLC-dependent IP₃ generation. It’s worth mentioning that these and several other pathways could contribute to intracellular Ca²⁺ signals caused by Aβ₄₂ in neurons in vivo. Aβ₄₂ also form pores in the plasma membrane that are permeable to Ca²⁺ and would impact intracellular Ca²⁺ signals [12, 14–16, 81]. Furthermore, the Aβ₄₂induced Ca²⁺ signals in neurons could be significantly different since the Ca²⁺ signaling machinery in neurons is significantly different than oocytes.

To quantify the amount of IP₃ generated in response to different concentrations of injected Aβ₄₂, we fit the whole-cell model to the fluorescence traces. The lines in Fig. 3A represent the respective model fits at each concentration of Aβ₄₂ oligomers injected into the oocytes. Fig. 3B shows the time traces of [IP₃] from the fits that would evoke the respective ΔF/F_o signals in Fig. 3A. Much like the TIRF signals, the [IP₃] follows similar trends of increased amplitude and shortening of onset as the concentration of Aβ₄₂ oligomers is increased. These results suggest that more IP₃ is released with increasing concentrations of intracellular Aβ₄₂, leading to rapid changes in ΔF/F_o in the TIRF signals. Furthermore, the delayed onset of TIRF signals relative to the time of injection of Aβ₄₂ is not due to slow diffusional spread of Aβ₄₂ but rather reflects the time required for activation of IP₃ production and subsequent accumulation of IP₃ levels to evoke Ca²⁺-release through IP₃R since the model does not incorporate any lag due to diffusion of Aβ₄₂. Total [IP₃] generated over 60 s is shown in Fig. S3A. At 30 μg/ml and 10 μg/ml we observe comparable amounts of IP₃ produced over 60 s. This is because Ca²⁺ responses to higher doses of Aβ₄₂ display an increasingly rapid decay which could be either due to the desensitization of IP₃R or the depletion of IP₃ reserves in the cell. Both these effects are not incorporated in the model. As shown in Fig. 4C, we rule out ER depletion as a reason for the rapid decay as the ER holds significant Ca²⁺ at the peak of cytosolic Ca²⁺ response. The P_o of IP₃R as a function of Aβ₄₂ concentration from the fits is shown in Fig. S4, indicating that 10nM Aβ₄₂ leads to a significant P_o of the channel. We remark that this is the first demonstration of estimating the P_o of IP₃R as a function of intracellular Aβ₄₂. Note that Aβ₄₂ activate IP₃R by generating IP₃ and Fig. S4 does not imply a direct interaction of Aβ₄₂ with the channel.

Fig. 4:

High [Ca²⁺]_i in response to intracellular injection of Aβ₄₂ oligomers leads to impaired mitochondrial function. Time-traces of [Ca²⁺]_i (A), [Ca²⁺]_m (B), [Ca²⁺]_ER (C), △Ψ_m (D), [NADH] (E), [ATP] (F), ${\left[{\text{O}}_2^{-}\right]}_{\text{m}}$(G), and [H₂O₂] (H) responses due to injection of 30 μg/ml (blue), 10 μg/ml (black), 3 μg/ml (red), and 1 μg/ml (green) [Aβ₄₂] oligomers. No EGTA is added in these simulations.

Mitochondrial response to evoked Ca²⁺

Previously, we showed that enhanced Ca²⁺ release from ER, induced by the gain-of-function enhancement of IP₃R due to presenilin mutations found in the brains of patients with Familial AD [39, 82, 83], impaired mitochondrial function [34, 62]. To investigate whether intracellular Aβ₄₂ oligomers also alter mitochondrial function, we couple the whole-cell Ca²⁺ signaling model to the model for mitochondrial function. Results from these simulations are shown in Fig. 4. At 30 μg/ml, we observe a peak [Ca²⁺]_i of ∼ 4 μM during the 150s simulation that drops to ∼ 2 μM at 10 μg/ml and less than 1 μM at 3 μg/ml and 1 μg/ml. This change in [Ca²⁺]_i from the steady state leads to Ca²⁺ uptake by mitochondria through Ca²⁺ uniporter that affects mitochondrial function. Traces of [Ca²⁺]_m at different Aβ₄₂ concentrations are shown in Fig. 4B. Total [Ca²⁺]_i and [Ca²⁺]_m during 60 s are shown in Fig. S3B and S3C respectively. At 30 and 10 μg/ml Aβ₄₂ oligomers, the total amount of cytosolic Ca²⁺ exceeds 50 μM leading to an increase in [Ca²⁺]_m of well over 100 μM over 60 s. Such excessive [Ca²⁺]_m have been shown to impair mitochondrial function, leading to apoptosis [34, 62, 84–86].

Changes in key variables governing mitochondrial function in response to Ca²⁺ release from the ER (Fig. 4C) through IP₃R and entering the matrix through the uniporter are shown in Fig. 4C-H. Intracellular injection of 30μg/ml Aβ₄₂ oligomers leads to a decrease of 24 mV in mitochondrial membrane potential (△Ψ_m), thereby disrupting the proton motive force (△μ_H) formed by respiration pumps oxidizing NADH on the ETC. While our model incorporates different enzymes and dehydrogenases involved in the TCA cycle, we primarily focus on how excessive Ca²⁺ alters [NADH] production (Fig. 4E) and consequently affects the synthesis of ATP and toxic agents. The large influx of Ca²⁺ increases the rate of oxygen consumption (oxidation of NADH) that leads to lower [NADH] and the rates of three Ca²⁺-sensitive enzymes in the TCA cycle (α-ketoglutarate dehydrogenase, isocitrate dehydrogenase, and malate dehydrogenase) that stimulates NADH production (Fig. S5). However, the increase in the rate of NADH oxidation exceeds that of the three enzymes in the TCA cycle combined, thereby, lowering the rate of NADH production. Between lowered [NADH] and diminished △Ψ_m, [ATP] production is negatively affected as increasing Aβ₄₂ concentration leads to higher Ca²⁺ influx into the mitochondria (Fig. 4F). At 30 and 10 μg/ml Aβ₄₂ oligomers, [ATP] drops by 600 μM. Even at 3 μg/ml oligomers there is almost 100 μM decrease in [ATP], dropping to 12 μM at 3 μg/ml. For smaller concentrations of Aβ₄₂, the effect on mitochondrial function is negligible.

Fig. 4G and3H show the amount of ROS $\left({\left[{\mathrm{O}}_2^{-}\right]}_{\text{m}}\right)$ and [H₂O₂] produced by mitochondria as we increase Aβ₄₂ concentration. Enhanced and prolonged oxidation leads to significant ROS production and therefore [H₂O₂], both extremely toxic to the cell when present in excess [87–89]. At Aβ₄₂ > 3 μg/ml, we observe more than 80 μM change in ROS. Interestingly, in case of 3 μg/ml Aβ₄₂, ROS shows a steady rise over time. This is due to the Ca²⁺ response observed in our experiments that did not decay over the duration of the experiment. The amount of [H₂O₂] decreases due to diminished △Ψ_m but increases later and overshoots the resting value as the ROS has reached its peak values. The overshoot is significant and longlasting in case of 30 and 10 μg/ml Aβ₄₂. Our simulation results clearly show that intracellular Aβ₄₂ leads to compromised mitochondrial function.

EGTA restores mitochondrial function

The experiments in [32] also showed that no cells remained viable within 37h of Aβ₄₂ injection (Fig. 6). Interestingly, the cell viability was restored to more than 70% when injected with mobile Ca²⁺ buffer EGTA to a final concentration of 3 mM. We argue that EGTA significantly reduces the amount of [Ca²⁺]_i that the mitochondrial uniporter senses, which would restore mitochondrial function and consequently will improve cell viability. Fig. 5 shows the same simulation as in Fig. 4 except that here we add 1000 μM EGTA to the cell. We see a significant decrease in the peak and total [Ca²⁺]_i over 60 s evoked by Aβ₄₂ oligomers (Fig. 5A and Fig. S6A), leading to a slower rise and lower peak in [Ca²⁺]_m (Fig. 5B). The total increase in [Ca²⁺]_m over 60 s drops to less than a third of the concentration where no EGTA is present (Fig. S6B). Similar trends are observed in other key variables governing mitochondrial function (Fig. 5D and E). The decrease in △Ψ_m, [NADH], and [ATP] drops to less than 2 mV, 150 μM, and 200 μM, respectively at 30 μg/ml Aβ₄₂. A similar trend is observed for other Aβ₄₂ concentrations. ROS and [H₂O₂] production are also affected by the addition of EGTA in similar fashion and their peak values drop as compared to the case with no EGTA. The restoration of the mitochondrial function due to EGTA could explain the observations in Fig. 6.

Fig. 6:

Ca²⁺ buffer EGTA restores cell viability. No oocytes injected with 1 μg/ml of Aβ₄₂ oligomers remained viable after 37 hours of injection (black). Injecting 3mM of EGTA, restored the viability of oocytes to 70% [32]. Oocytes were bathed in a solution containing no added Ca²⁺ in order to eliminate influx from extracellular solution.

Fig. 5:

EGTA can reverse the affect of Aβ₄₂ oligomers on Ca²⁺ signaling and mitochondrial function. Time-traces of [Ca²⁺]_i (A), [Ca²⁺]_m (B), [Ca²⁺]_ER (C), 4Ψ_m (D), [NADH] (E), [ATP] (F), ${\left[{\text{O}}_2^{-}\right]}_{\text{m}}$(G), and [H₂O₂] (H) response due to injection of 30 μg/ml (blue), 10 μg/ml (black), 3 μg/ml (red), and 1 μg/ml (green) [Aβ₄₂] oligomers with 1000 μM [EGTA] added to the cell.

To estimate the amount of [EGTA] needed to rescue mitochondrial function in varying concentration of intracellular Aβ₄₂ oligomers, we show surface plots of the peak change in key variables from steady-state values as functions of [Aβ₄₂] and [EGTA] in Fig. 7. For example, 30μg/ml Aβ₄₂ oligomers leads to a 24 mV and 2 mV drop (with respect to the resting state) in △Ψ_m in the absence of EGTA and presence of 1000 μM EGTA respectively. Thus in the surface plot for Δ[△Ψ_m], a point at [Aβ₄₂] = 30μg/ml and EGTA = 0 and 1000 μM will be equal to 24 and 2 mV respectively. We vary [Aβ₄₂] and [EGTA] over a range of 0–30 μg/ml (0.0127 μM) and 0–3000 μM respectively and determine the peak change for various combinations of [EGTA] and [Aβ₄₂]. Fig. 7A-F shows the peak change in [Ca²⁺]_i, [Ca²⁺]_m, ${\left[{\text{O}}_2^{-}\right]}_{\text{m}}$, ΔΨ_m, [H₂O₂], and [ATP] respectively (note that panels (D) and (F) indicate a decrease (absolute value of change) with respect to resting values). Dependence of all key variables on [Aβ₄₂] is prevalent as the peak change increases with [Aβ₄₂]. Predictably, increasingly higher concentration of EGTA is needed to buffer the higher [Ca²⁺]_i due to increasing Aβ₄₂ content and restore mitochondrial function. The key observation that can be made from Fig. 7 is that for small and intermediate values of [Aβ₄₂], EGTA will restore mitochondrial function. However, for large concentrations of Aβ₄₂ oligomers, restoring mitochondrial function with EGTA alone is not possible. This could explain the partial restoration of cell viability (up to 70%) by 3mM EGTA in our experiments (Fig. 6). Nevertheless, a polytherapy involving EGTA with drugs such as pertussis toxin to inhibit G-protein, lithium to deplete membrane inositol lipids, and IP₃R antagonist heparin might restore mitochondrial function and hence fully abrogate cytotoxicity. Although not tested simultaneously, each one of EGTA, pertussis toxin, and lithium individually have been shown to restore cell viability to more than 70% of the control value [32]. Furthermore, higher resting [Ca²⁺]_i has been observed in primary cortical neurons from triple transgenic mouse model of AD that exhibits intraneuronal accumulation of Aβ oligomers as compared to non-transgenic neurons [90]. Higher base levels of [Ca²⁺]_i have been shown to significantly affect the average ATP production of mitochondria [72]. Thus a carefully devised polytherapy that would not only buffer the transient rise in Ca²⁺ concentration but also restore the physiological resting [Ca²⁺]_i would be more appropriate.

Fig. 7:

EGTA rescues mitochondrial function. Surface plots showing the dependance of Aβ₄₂ and [EGTA] on change (Δ) in values from steady state of [Ca²⁺]_i (A), [Ca²⁺]_m (B), ${\left[{\text{O}}_2^{-}\right]}_{\text{m}}$(C), △Ψ_m (D), [H₂O₂] (E), and [ATP] (F). Note that the change in panels (D) and (F) indicates a decrease with respect to resting values, while all other panels indicate an increase with respect to the resting levels.

Discussion

Understanding the downstream effects of intracellular Aβ oligomers accumulation in AD-affected neurons is crucial as they are shown to lead to profound deficits in memory and cognitive function [29, 30], and are likely to play a major role in the earliest phase of AD pathogenesis [23, 27]. As reported by different groups including us, the liberation of Ca²⁺ from intracellular stores is seen as a possible mechanism for the cytotoxicity of intracellular Aβ oligomers [8, 31–33]. In line with this notion, we have shown that buffering the Ca²⁺ released from the ER in response to intracellular injection of Aβ₄₂ oligomers restores cell viability and protects against cell death [32]. However, the intermediate pathways linking Aβ₄₂-induced Ca²⁺ rise to cytotoxicity are not well understood.

During normal cell functioning, transient intracellular Ca²⁺ rise is restored to base level by PM Ca²⁺ ATPase, Na⁺/Ca²⁺ exchange, and SERCA pumps located in the ER membrane [91]. Cytosolic Ca²⁺ is also buffered by mitochondria through Ca²⁺ uniporter [92]. Prolonged and frequent occurrence of high cytosolic Ca²⁺ has been suggested to compromise mitochondrial function [85, 86, 93, 94]. This together with the observed Ca²⁺ up-regulation and mitochondrial dysfunction in several neurodegenerative diseases could provide a link between Ca²⁺ rise in these diseases and cytotoxicity [95–103]. The studies in Refs. [99, 101] in particular show the association of intracellular Aβ with the extent of mitochondrial dysfunction both in AD and Down’s Syndrome where most patients develop AD neuropathology [104]. However, the molecular mechanism through which intracellular Aβ impairs mitochondrial function is not well understood. Moreover, other potential downstream effects of Aβ-triggered Ca²⁺ liberation from the ER have not been properly investigated and identified yet.

Our approach combines patch-clamp electrophysiology of single Ca²⁺ channel and fluorescent imaging of whole-cell Ca²⁺ signals into a multiscale model that allows us to closely reproduce key observations at both scales, and estimate the amount of IP₃ generated and consequently the Ca²⁺ content liberated from the ER as a function of Aβ₄₂ concentration during our experiments [32]. While inferring parameters related to the single-channel behavior (e.g. [IP₃] and channel flux) from localized Ca²⁺ signals (μm-sized events called puffs) has been demonstrated before [105, 106], this to our knowledge is the first exhibition of estimating such parameters from the whole-cell Ca²⁺ signals. We are also not aware of any other study directly fitting a whole-cell Ca²⁺ model to TIRF signals. The data-driven model is then used to investigate the effect of Aβ₄₂-induced Ca²⁺ liberation on mitochondrial function.

We show that intracellular Aβ₄₂ oligomers disrupt mitochondrial function by inducing global Ca²⁺ signals through IP₃ generation, leading to high [Ca²⁺]_m and ROS. The enhanced [Ca²⁺]_m also decreases △Ψ_m and subsequently 4μ_H to a point where ATP production is negatively affected (Fig. S5). These observations are well in line with the higher level of ROS, decreased △Ψ_m, and decreased ATP production observed in hippocampal and cortical mitochondria of AD transgenic mice as compared to non-transgenic mice [101, 107]. The decreased respiration control ratio observed in [101] was attribute to diminished △Ψ_m in different brain regions which is consistent with our results. The dependence of the level of mitochondrial dysfunction on Aβ₄₂ concentration (hence Ca²⁺ content) would also explain the relative severe impairment of synaptic mitochondria as compared to non-synaptic ones as the Ca²⁺ liberation in synapses would be compounded by the higher influx from the extracellular space through Ca²⁺ channels in the PM [101]. Our results are also consistent with the observed oxidative stress in AD and Down’s Syndrome [104, 108, 109].

We remark that the computational framework proposed here is different than our previous study [34], where we fused experimental time-traces for cytosolic [Ca²⁺]_i with the model for mitochondrial function. In other words, we only modeled mitochondrial function without modeling the pathways regulating [Ca²⁺]_i, [Ca²⁺]_ER, and [IP₃]. As a simplification, the model for these variables was replaced by the experimental time-trace for cytosolic calcium. While this made modeling the effect Ca²⁺ signaling disruptions on mitochondrial function a lot easier, the framework is specific to the cells studied as the input was coming directly from the cell observed experimentally. The framework developed here is more general as it models all components involved in the Ca²⁺ signaling disruption and its downstream effects. Furthermore, Ca²⁺ signaling disruptions and consequently the mitochondrial dysfunction studied previously was due to Familial Alzheimer’s disease-causing mutations in presenilin. However, more than 95% cases of Alzheimer’s disease are sporadic where intracellular Aβ₄₂, most likely through Ca²⁺ signaling disruptions, are believed to play a key role in the earliest phase of the disease.

Mitochondrial dysfunction due to intracellular Aβ in Ca²⁺-dependent manner makes a testable mechanism for neuronal toxicity in AD. For example, Simakova and Arispe demonstrated that neurons with low ATP levels exhibit higher affinity to Aβ₄₂ binding, and are consequently vulnerable to toxicity [110]. In healthy cells, PM asymmetry is maintained by the activity of aminophospholipid translocase flippase – an ATP-dependent transmembrane transporter that shuttle phosphatidylserine from the outer to the inner leaflet of the membrane. The reverse transport of phosphatidylserine from the inner to the outer surface of PM, induced by stress and lack of ATP, is believed to be the basis for the cell-selective neurotoxicity of Aβ in certain brain regions [110]. One can envisage that as a result of Ca²⁺ rise due to intracellular Aβ oligomers, nearby mitochondrial ATP production drops in parts of the cell where high level of ATP is constantly required to maintain PM asymmetry through steady flippase activity. The lack of ATP and rise in ROS would impair flippase activity, allowing spontaneous translocation to the outer leaflets of the PM and leaving the cell membrane more susceptible to extracellular Aβ oligomers binding. This would lead to the exacerbation of Ca²⁺ dyshomeostasis and the structural and functional disruption of neuronal networks in AD [110, 111].

In addition to biogenesis, several aspects of mitochondrial dynamics are also impaired in the early stages of AD, including slower anterograde and retrograde axonal transport, decreased number of large mitochondria, lower number of mitochondria overall, and disrupted balance between fusion and fission when compared to physiological conditions [112–118]. Mitochondrial transport [119–122], fission, fusion, fragmentation [123, 124], and morphology [125, 126] are all disrupted by increased ROS, reduced ATP, depolarized mitochondrial membrane, and up-regulated intracellular Ca²⁺, all of which we show are significantly influenced by intracellular Aβ₄₂. The fact that sustained increase of [Ca²⁺]_i triggers intracellular Aβ₄₂ production would further exacerbate the potential impairment of mitochondrial dynamics and function [11, 127].

The fluorescence signals in our TIRF experiments and variables in the model represent the average effect over the entire cell. That is, we treat the oocyte as a point cell and ignore the spatial aspects of various signaling pathways such as the arrangement of IP₃Rs into clusters, the realistic distribution of clusters in the cell [53], and the close contacts between mitochondrion and the ER leading to privileged entry of Ca²⁺ released through IP₃Rs into mitochondrion [128, 129]. Investigating the changes in the spatial aspects of Ca²⁺ signals in AD, their role in cell dysfunction, and numerous other questions related to the mitochondrial dynamics listed in the preceding paragraphs are beyond the scope of this study and the subject of our future research.

Highlights.

1. A simple four state model for IP₃R, driven by extensive single channel patch-clamp experiments.

2. Estimating Ca²⁺ and IP₃ from TIRF Signals.

3. Estimating the amount of IP₃ generated due to intracellular beta amyloid.

4. Evaluating mitochondrial dysfunction due to intracellular beta amyloid.

5. Estimating single channel parameters from whole-cell Ca²⁺ experiments.

6. The use of Ca²⁺ buffer EGTA to restore viability of cells with Alzheimer’s disease pathology.