Composite mathematical modeling of calcium signaling behind neuronal cell death in Alzheimer’s disease
Abstract
Background
Alzheimer’s disease (AD) is a progressive neurological disorder, recognized as the most common cause of dementia affecting people aged 65 and above. AD is characterized by an increase in amyloid metabolism, and by the misfolding and deposition of β-amyloid oligomers in and around neurons in the brain. These processes remodel the calcium signaling mechanism in neurons, leading to cell death via apoptosis. Despite accumulating knowledge about the biological processes underlying AD, mathematical models to date are restricted to depicting only a small portion of the pathology.
Results
Here, we integrated multiple mathematical models to analyze and understand the relationship among amyloid depositions, calcium signaling and mitochondrial permeability transition pore (PTP) related cell apoptosis in AD. The model was used to simulate calcium dynamics in the absence and presence of AD. In the absence of AD, i.e. without β-amyloid deposition, mitochondrial and cytosolic calcium level remains in the low resting concentration. However, our in silico simulation of the presence of AD with the β-amyloid deposition, shows an increase in the entry of calcium ions into the cell and dysregulation of Ca ^{2+} channel receptors on the Endoplasmic Reticulum. This composite model enabled us to make simulation that is not possible to measure experimentally.
Conclusions
Our mathematical model depicting the mechanisms affecting calcium signaling in neurons can help understand AD at the systems level and has potential for diagnostic and therapeutic applications.
Keywords
Calcium signaling Neuronal cell death Alzheimer’s disease Mathematical modelingBackground
Alzheimer’s disease (AD) is characterized by the deposition of β-amyloid (A β) oligomers in and around neurons in the brain accompanied by dysfunctional neuronal calcium homeostasis. Autophagy is generally an efficient mechanism for removing amyloids. During the onset of AD, autophagy is increased but the transfer of autophagic vesicles to the lysosomes is blocked [1]. This may contribute to the accumulation of amyloids. There is increasing evidence to support the hypothesis that A β induces an up-regulation of intracellular Ca ^{2+} and leads to AD. Multiple studies on AD mouse models have shown that Ca ^{2+} dysregulation leads to increased Ca ^{2+} entry into the cytoplasm resulting in neuronal cell death and AD [2, 3].
The observations about the effect of A- β oligomers on neuronal calcium signaling led to the formulation of the calcium hypothesis of AD [4]. The basic argument behind the hypothesis is that the activation of the amyloidogenic pathway results in a remodeling of the neuronal calcium signaling pathway. The up-regulation of Ca ^{2+} distorts the normal neuronal Ca ^{2+} signaling by increasing the amount of Ca ^{2+} being taken up by the mitochondria. A sustained increase in the mitochondrial Ca ^{2+} may activate the mitochondria to initiate the intrinsic pathway of Ca ^{2+}-induced apoptosis, as described in the calcium hypothesis of Alzheimer’s disease [4].
According to Berridge, the increased output of Ca^{2+} due to the hypersensitivity of the Ca^{2+} signaling system may activate the mitochondria to initiate the intrinsic pathway of Ca^{2+}-induced apoptosis by opening up the mitochondrial permeability transition pore (PTP), causing collapse of the mitochondrial membrane potential and releasing cytochrome c and other factors that activate the caspase cascade responsible for apoptosis.
Ichas and Mazat [5] demonstrated that the mitochondrial PTP operates at the crossroads of 2 distinct physiological pathways i.e. the Ca^{2+} signaling network during the life of the cell and the effector phase of the apoptotic cascade during Ca^{2+}-dependent cell death. It has 2 open conformations correspondingly. The low-conductance state, which allows the diffusion of small ions like Ca^{2+}, is pH-operated, promoting spontaneous closure of the channel. A high-conductance state, which allows the unselective diffusion of big molecules, stabilizes the channel in open conformation [5].
Mitochondria in open high-conductance state can no longer maintain a proton gradient, and thus cannot sustain oxidative phosphorylation, resulting in an arrest of aerobic ATP synthesis (necrotic cell death). This also results in an oncotic imbalance in the mitochondria causing it to swell up. The cristae formed by the inner membrane unfold, leading to rupture of the outer membrane that brings into direct communication the former intermembrane space and the extra-mitochondrial medium. Soluble components like cytochrome c and Apoptosis Inducing Factor (AIF), which are normally trapped in the intermembrane space, are released into the cytosol, thereby inducing cell apoptosis [5].
Thul [6] described the use of Ordinary Differential Equations (ODEs) to model intracellular Ca^{2+} oscillations, assuming intracellular Ca^{2+} concentration to be spatially homogeneous. The use of ODEs is widespread among modelers because: (a) the study of ODEs is computationally well-supported, with a large body of techniques available to investigate ODEs in great detail, and (b) lack of sufficient experimental data to develop a spatially extended model.
- (1)
When there is no abnormality in β-amyloid folding and deposition, the initiation of an action potential does not lead to long-term sustained oscillations (an extension of the result detailed in [7]). This will be explained further in the Results section.
- (2)
When β-amyloid misfolding affects calcium signaling within the neuron, the action potential is prevented from dying down immediately (a more complex expression of the qualitative trend represented by [9]). As β-amyloid deposition increases with time, the rate of entry of calcium ions into the cell increases, thus allowing the calcium ion oscillations to continue by maintaining high calcium ion levels in the cytosol. As this rate of entry continues to increase, the cytosol and mitochondria attempt to release excess calcium ions to one another more frequently and in smaller amounts, thus resulting in smaller but more rapid oscillations.
We used time-course simulation to relate the occurrence of events to biological processes, thereby verifying our model. We also quantified certain findings from the time-course simulations, with special emphasis on the time taken for the PTP to open in high conductance state.
Methods
The Fall-Keizer model
The Fall-Keizer model is an integrated model depicting mitochondrial Ca ^{2+} handling and metabolic function. It integrates the Magnus-Keizer model [10, 11] and the De Young-Keizer model [12]. The Magnus-Keizer model is a comprehensive mitochondrial model with six proton transfer mechanisms that affect Ca ^{2+} signaling.
A key motivation for using the Fall-Keizer model was to improve the original Magnus-Keizer model, by modifications to the Ca ^{2+} uniporter so that the prediction of Ca ^{2+} signaling can be more accurate.
The inclusion of the De Young-Keizer model for inositol-1,4,5-triphosphate (IP3)-mediated Ca ^{2+} release along with appropriate scaling and provisions for accommodating different cell types allows a modeler to easily shape this model to his or her objective. Furthermore, the comprehensive nature and the modularity of the model that have been maintained by Fall and Keizer make this model an ideal choice for the purposes of this paper.
Equation (1) was obtained from the Fall-Keizer model [7]. CAC is directly proportional to the total volume of mitochondria in the cell (M) multiplied by the rate of transfer of calcium ions into the cytosol from the mitochondria, which is dependent on the sodium-calcium ion exchanger, the calcium uniporter and the PTP. CAC is also directly proportional to the total volume of ER in the cell (E) multiplied by the rate of transfer of calcium ions into the cytosol from the ER, which is affected by the SERCA pump and the leakage of calcium ions from the ER.
Since our focus is on mitochondria-induced cell apoptosis, this model provides us with the appropriate foundation to build upon.
The Mitochondrial PTP model
The model proposed by Oster et al. [8] provides a representation of the mitochondrial permeability transition pore (PTP) behavior. In our model, we have focused on the high conductance state of PTP. In its high-conductance conformation, PTP opening induces unselective solute fluxes that dissipate the concentration gradients of relatively big molecules. However, since most proteins remain trapped in the matrix, the resulting oncotic imbalance (at least in vitro) causes high amplitude swelling of the organelle. The subsequent unfolding of the inner membrane causes rupture of the outer membrane, which results in the release of soluble components (mainly cytochrome c and Apoptosis Inducing Factor (AIF)) that are normally located in the intermembrane space [13, 14, 15].
The transition of the pore to a high-conductance state requires prolonged levels of high mitochondrial Ca ^{2+}. Once the pore opens to this state, it remains open, leading to cell death. The model proposed by Oster et al. [8] assumes that whether or not the pore enters the high-conductance state depends on a secondary slow process, which in turn depends on the overall mitochondrial Ca ^{2+} load.
The Amyloid metabolism model
It is widely known that amyloids perturb Ca ^{2+} homeostasis, and β-amyloids perturb the balance between Ca ^{2+} entry in and extrusion out of the cytoplasm. In healthy neurons, these processes equilibrate, leading to a basal Ca ^{2+} level in the range of 50-100 nM [16]. Studies on the cortical neurons of AD stricken animals found a basal Ca ^{2+} level of around 250 nM, i.e. around twice that found in controls [17].
This equation shows the effect of amyloids on intracellular calcium concentration. It was formulated based on the assumption that amyloids increase intracellular calcium concentration by increasing the permeability of plasma membranes. This assumption is also made in our model. Here, the rate at which calcium enters the cytoplasm and the rate of eliminating calcium ions from the cytoplasm are assumed to be constant. Furthermore, calcium ion concentration in the cytoplasm is assumed to have first order kinetics.
Composite model
We used XPPAUT [19], an.ode model file as in Additional file 1 and MATLAB to plot our time-course simulations. Details of the composite model constructed based on the afore-mentioned three models are given below. A schematic diagram of the composite model is illustrated in Fig. 2. The parameters used in our composite model were obtained from the individual models.
Parameters | Value | Biological significance |
---|---|---|
V | 1 ml | Total volume |
uMmM | 1000 | Converts μM to mM |
τ _{ min } | 60 | Converts minutes to seconds |
p _{ cytosol } | 0.5 | Proportion of volume occupied my cytosol |
d _{ cytosol } | 75 mg/ml | Density of cytosolic protein mg/ml |
p _{ mito } | 0.05 | Proportion of volume occupied my mitochondria |
d _{ mito } | 1000 mg/ml | Density of mitochondrial protein mg/ml |
p _{ er } | 0.1 | Proportion of volume occupied my ER |
d _{ er } | 1000 mg/ml | Density of ER protein mg/ml |
c _{ mito } | 0.0725 \(\frac {nmol}{mV*mg}\) | Mitochondrial calcium concentration |
ρ _{ uni } | 300 \(\frac {nmol}{mg*min}\) | Maximum rate of transport through mitochondrial uniporter |
\(\rho _{Na}^{Ca} \) | 3 \(\frac {nmol}{mg*min}\) | Maximum rate of transport through Na^{+}/Ca^{2+} exchanger |
ρ _{ res } | 0.4 | Mitochondrial respiration co-efficient |
ρ_{ F }1 | 0.7 \(\frac {nmol}{mg*min} \) | Mitochondria - Fo/F1 ATPase |
[P_{ i }]_{ m } | 20 mM | Concentration of free phosphates |
ρ _{ leak } | 0.2 \(\frac {nmol}{mg*min} \) | Mitochondrial membrane proton leak |
J _{red,basal} | 20 \(\frac {nmol}{mg*min}\) | NADH reduction rate |
J _{max,ANT} | 900 \(\frac {nmol}{mg*min}\) | ATP/ADP antiport flux |
glc | 1 mM | Glucose concentration in cytosol |
J _{hyd,max} | 30.1 mM | Cytosol hydrolysis of ATP |
V _{IP3} | 3000 μM | IP_{3} receptor volume |
V _{ leak } | 0.1 | proportion of leakage from IP_{3} receptor |
J _{ leak } | 0.1 | ER leak |
d _{IP3} | 0.25 μM | IP_{3} receptor sensitivity |
d _{ ACT } | 1 μM | IP_{3} receptor Ca^{2+} activation constant |
d _{ INH } | 1.4 μM | IP_{3} receptor Ca^{2+} inhibition constant |
τ | 4 s | IP_{3} receptor inhibitory time constant |
V _{ serca } | 110 \(\frac {nmol}{mg*min}\) | SERCA pump flux |
k _{ serca } | 0.4 μM | SERCA pump Ca^{2+} sensitivity |
V _{1} | 0.0065 nM/s | Constant rate of β-amyloid synthesis |
V _{ α } | 0.05 nM/s | Maximal rate of β-amyloid synthesis |
K _{ α } | 120 nM | Half-saturation constant |
k _{ β } | 0.2 nM^{3}/s | Rate constant of increased Ca^{2+} entry |
K _{1} | 0.01 s^{−1} | Rate constant of β-amyloid elimination |
n | 2 | Hill coefficient for activation of β-amyloid synthesis |
m | 4 | Cooperativity coefficient |
f _{ m } | 0.0003 | Mitochondrial Ca^{2+} buffering coefficient |
f _{ i } | 0.01 | Cytosolic Ca^{2+} buffering coefficient |
Parameters | Value | Biological significance |
---|---|---|
b a s e l i n e | 0.3 μM | Base concentration |
a m p l i t u d e | 0.3 μM | Oscillation amplitude |
i n i t | 10 ms | Initial time |
d u r a t i o n | 100 ms | Duration of oscillations |
Parameters | Value | Biological significance |
---|---|---|
C A M ^{∗} | 4 μM | Threshold Mitochondrial Ca^{2+} |
y ^{∗} | 0.8 | Secondary process threshold |
\(f_{H_{M}} \) | 1.28 x 10 ^{−6} | Fast buffering constant for protons in mitochondria |
p _{1} | 0.022 | Parameter p_{1} |
p _{2} | 0.0001 | Parameter p_{2} |
p _{3} | 0.0231 | Parameter p_{3} |
p _{4} | 0.0001 | Parameter p_{4} |
a m p _{ τ } | 26000 | Amplitude for time constant |
p _{6} | 0.001 | Parameter p_{6} |
\(perm_{l}^{H}\) | 3.0 | PTP Permeability to protons |
p e r m _{ Ca } | 0.4 | PTP Permeability to calcium ions |
postptp | 2 | PTP opening indication constant |
Equation | Biological significance |
---|---|
dNADH_{ m }/dt=(J_{ red }−J_{ o })∗M/(uMmM∗V_{ m }∗minute) | ODE for mitochondrial NADH concentration change (mM/s) |
dADP_{ m }/dt=(J_{ ANT }−J_{p,TCA}−J_{p,F1})∗M/(uMmM∗V_{ m }∗minute) | ODE for mitochondrial ADP concentration change (mM/s) |
dADP_{ i }/dt=(−J_{ ANT }∗M+(J_{ hyd }−J_{p,gly})∗C)/(uMmM∗V_{ c }∗minute) | ODE for cytosolic ADP concentration change (mM/s) |
\(dPSI/dt= -\left (-J^{H}_{res} + J_{H,F1} + J_{ANT} +J^{H}_{PTP} +J^{H}_{L} +2*J_{uni} +2 * J^{Ca}_{PTP}\right) * M/\left (c_{mito} * minute \right)\) | ODE for mitochondrial inner membrane voltage |
dh/dt=(d_{ INH }−(CAC+d_{ INH })∗h)/τ | ODE for change in percentage of closed channels |
Parameters | Value | Biological significance |
---|---|---|
PSI(0) | 164 mV | Base potential at time 0 |
CAM(0) | 0.05 μM | Mitochondrial Ca^{2+} concentration at time 0 |
CAC(0) | 0.05 μM | Cytosolic Ca^{2+} concentration at time 0 |
CAER(0) | 11 μM | ER Ca^{2+} concentration at time 0 |
ADP_{ m }(0) | 4.46 mM | Mitochondrial ADP concentration at time 0 |
ADP_{ i }(0) | 0.028 mM | Cytosolic ADP concentration at time 0 |
NADH_{ m }(0) | 0.16 mM | Mitochondrial NADH concentration at time 0 |
h(0) | 95% | Percentage of closed channels at time 0 |
a(0) | 0 mM | β-amyloid concentration at time 0 |
y(0) | 0 | Secondary slow process involved in PTP opening at time 0 |
PTP_{ h }(0) | 0 | PTP closed at time 0 |
PTP_{ l }(0) | 0 | PTP closed at time 0 |
Equation | Biological significance |
---|---|
M=V∗p_{ mito }∗d_{ mito } | Calculation of mitochondrial protein amount |
C=V∗p_{ cytosol }∗d_{ cytosol } | Calculation of cytosolic protein amount |
E=V∗p_{ er }∗d_{ er } | Calculation of ER protein amount |
Equation | Biological significance |
---|---|
V_{ m }=(V∗p_{ mito }) | Calculation of mitochondrial compartment volumes |
V_{ c }=(V∗p_{ cytosol }) | Calculation of cytosolic compartment volumes |
V_{ e }=(V∗p_{ er }) | Calculation of ER compartment volumes |
Equation | Biological significance |
---|---|
ATP_{ m }=(12∗d_{ mito }/uMmM)−ADP_{ m } | Mitochondrial ATP concentration |
NAD=(8∗d_{ mito }/uMmM)−NADHM | Mitochondrial NAD concentration |
List of model equations used in calculation of proportion of free nucleotides (obtained from [11])
Equation | Biological significance |
---|---|
ADP_{ mf }=0.8∗ADP_{ m } | Unbound mitochondrial ADP concentration |
ADP_{ if }=0.3∗ADP_{ i } | Unbound cytosolic ADP concentration |
List of model equations used in calculation of proportion of charged, free nucleotides (obtained from [11])
Equation | Biological significance |
---|---|
[ADP^{3−}]_{ m }=0.45∗ADP_{ mf } | Unbound, 3- charged mitochondrial ADP concentration |
[ADP^{3−}]_{ i }=0.45∗ADP_{ if } | Unbound, 3- charged cytosolic ADP concentration |
[MgADP^{−}]_{ i }=0.55∗ADP_{ if } | Unbound, 1- charged cytosolic MgADP^{−} concentration |
[ATP^{4−}]_{ i }=0.05∗ATP_{ i } | Unbound, 4- charged cytosolic ATP concentration |
[ATP^{4−}]_{ m }=0.05∗ATP_{ m } | Unbound, 4- charged mitochondrial ATP concentration |
List of model equations used in mitochondrial Ca^{2+} handling (obtained from [10])
Equation | Biological significance |
---|---|
Mitochondrial uniporter equations | |
MWC_{ num }=(CAC/6)∗((1+(CAC/6))^{3}) | MWC numerator |
MWC_{ denom }=((1+(CAC/6))^{4})+(50/((1+(CAC/0.38))^{2.8})) | MWC denominator |
MWC=MWC_{ num }/MWC_{ denom } | MWC fraction value |
\( V^{D}_{uni} = \left (PSI - 91\right)/13.35 \) | Uniporter potential exponent |
\( J_{uni} = \left (\rho _{uni} * V^{D}_{uni} * \left (MWC - CAM * exp\left (-V^{D}_{uni}\right)\right)/\left (1 - exp\left (-V^{D}_{uni}\right)\right)\right) * \left (1-PTP_{h}\right) \) | Rate of transport through uniporter considering PTP in high conductance state |
Na^{+}/Ca^{2+} exchanger equations | |
\( V^{D}_{nc} = exp\left (\left (PSI - 91\right)/53.4\right) \) | Na^{+}/Ca^{2+} exchanger potential generated |
\( J_{nc} = \left (\rho _{nc} * V^{D}_{nc} * \left (1/\left (1 + \left (9.4/30\right)**2\right)\right) * \left (1/\left (1 + \left (0.003 * Dmito/CAM\right)\right)\right)\right) * \left (1-PTP_{h}\right) \) | Rate of Na^{+}/Ca^{2+} exchange |
List of model equations used in calculation of mitochondrial respiration equations (obtained from [11])
Equation | Biological significance |
---|---|
A_{ res }=(1.35e18)∗NADHM^{0.5}/(NAD)^{0.5} | A_{ res } = affinity bracketed expression |
\( V^{D}_{res} = exp\left (0.191 * PSI\right) \) | Respiration potential generated |
Proton pump equations | |
r_{1}=7e−7 | Variable r_{1} |
r_{2}=(2.54e−3)∗A_{ res } | Variable r_{2} |
\( r_{3} = 0.639 * V^{D}_{res} \) | Variable r_{3} |
r_{4}=7.58e13+(1.57e−4)∗A_{ res } | Variable r_{4} |
\( r_{5} = \left (1.73 + A_{res} * 1.06e-17\right) * V^{D}_{res} \) | Variable r_{5} |
\( J^{H}_{res} = 360 * \rho _{res} * \left (\left (r1 + r2 - r3\right)/\left (r4 + r5\right)\right)\) | Rate of transport through proton pump during respiration |
Oxygen consumption rate equations | |
o_{1}=A_{ res }∗2.55e−3 | Variable o_{1} |
o_{2}=A_{ res }∗2.00e−5 | Variable o_{2} |
\( o_{3} = 0.639*\left (V^{D}_{res}\right) \) | Variable o_{3} |
\( o_{4} = \left (V^{D}_{res}\right) * A_{res} * 8.63e-18 \) | Variable o_{4} |
o_{5}=(1+A_{ res }∗2.08e−18)∗7.54e13 | Variable o_{5} |
\( o_{6} = \left (1.73 + 1.06e-17 * A_{res}\right) * V^{D}_{res} \) | Variable o_{6} |
J_{ o }=30∗ρ_{ res }∗(o1+o2−o3+o4)/(o5+o6) | Rate of oxygen consumption during respiration |
List of model equations used in calculation of mitochondrial Fo/F1-ATPase equations (obtained from [11])
Equation | Biological significance |
---|---|
A_{F1}=(1.71e9)∗(ATP_{ m })/(ADP_{ mf }∗pim) | A_{F1} = affinity bracketed expression |
\( V^{D}_{F1} = exp(0.112 * PSI) \) | ATPase potential generated |
F_{0}/F_{1} ATPase phosphorylation of ADP_{ m } | |
f_{1}=10.5∗A_{F1} | Variable f_{1} |
\( f_{2} = 166 * V^{D}_{F1} \) | Variable f_{2} |
\( f_{3} = (4.85e-12) * A_{F1} * V^{D}_{F1} \) | Variable f_{3} |
f_{4}=(1e7+0.135∗A_{F1})∗275 | Variable f_{4} |
\( f_{5} = (7.74 + (6.65e-8) * A_{F1}) * V^{D}_{F1} \) | Variable f_{5} |
J_{p,F1}=−60∗ρ_{F1}∗((f_{1}−f_{2}+f_{3})/(f_{4}+f_{5})) | Rate of F_{0}/F_{1} ATPase phosphorylation |
\( J_{H,F1} = -180 * \rho _{F1} * \left (0.213 + f_{1} - 169 * V^{D}_{F1}\right)/\left (f_{4} + f_{5}\right) \) | Proton flux due to ATPase |
J_{H,leak}=ρ_{ leak }∗(PSI+24.6) | Mitochondrial membrane proton leak |
f_{ PDH }=1/(1+(1.1∗(1+(15/(1+(CAM/0.05))^{2})))) | Fraction of activated pyruvate |
J_{ red }=J_{red,basal}+6.3944∗f_{ PDH }∗J_{gly,total} | NADH reduction rate |
ATP/ADP antiport flux | |
ant_{1}=([ATP^{4−}]_{ i }/[ADP^{3−}]_{ i })∗([ADP^{3−}]_{ m }/[ATP^{4−}]_{ m })∗exp(−PSI/26.7) | Variable ant_{1} |
ant_{2}=1+([ATP^{4−}]_{ i }/[ADP^{3−}]_{ i })∗exp(−PSI/53.4) | Variable ant_{2} |
ant_{3}=1+([ADP^{3−}]_{ m }/[ATP^{4−}]_{ m }) | Variable ant_{3} |
J_{ ANT }=J_{max,ANT}∗((1−ant_{1})/(ant_{2}∗ant_{3})) | Rate of Adenine Nucleotide Translocator (ANT) activity |
Phosphorylation of ADP_{ m } from TCA cycle | |
J_{p,TCA}=(J_{red,basal}/3)+0.84∗f_{ PDH }∗J_{gly,total} | Unbound, 3- charged mitochondrial ADP concentration |
List of model equations used in calculation of cytosolic components (obtained from [11])
Equation | Biological significance |
---|---|
Glycolytic rate based on hexokinase | |
gly_{ num }=(123.3∗(1+1.66∗glc)∗(glc∗ATP_{ i }))∗0.0249 | Glycolytic rate numerator |
gly_{ denom }=1+(4∗ATP_{ i })+((1+2.83∗ATP_{ i })∗1.3∗glc)+((1+2.66∗ATP_{ i })∗0.16∗glc^{2}) | Glycolytic rate denominator |
J_{gly,total}=gly_{ num }/gly_{ denom } | Glycolytic rate |
J_{p,gly}=2∗J_{gly,total} | Phosphorylation of ADPi from glycolysis |
J_{ hyd }=41∗(ATP_{ i })+J_{hyd,max}/(1+(8.7/glc)^{2.7}) | Cytosolic hydrolysis of ATP |
List of model equations used in calculation of ER Ca^{2+} handling (modified from [21])
Equation | Biological significance |
---|---|
J_{er,out}=(V_{IP3}∗((IP_{3}/(IP_{3}+d_{IP3}))^{3})∗((CAC/(CAC+d_{ ACT }))^{3})∗(h^{3})+v_{ leak })∗(CAER−CAC) | IP_{3} receptor and leak |
\( J_{serca} = V_{serca} * CAC^{2}/\left (k_{serca}^{2}+CAC^{2}\right) \) | SERCA pump |
List of model equations used in IP_{3} step function (modified from [21])
Equation | Biological significance |
---|---|
stepupf=heav(t−init) | Heaviside step up function |
stepdownf=heav(t−(init+duration)) | Heaviside step down function |
IP_{3}=baseline+amplitude∗(stepupf−stepdownf) | IP_{3} step function |
List of model equations used in PTP Integration (modified from [21])
Equation | Biological significance |
---|---|
τ_{ y }=1000∗((1000/cosh(CAM/0.1))+0.1) | Time constant for secondary slow process |
τ_{ h }=τ_{ y }/8 | Time constant for PTP high conductance state |
\( PTP_{h}^{\infty } = heav\left (y - y^{*}\right) \) | Heaviside step function for PTP max value |
y^{ ∞ }=heav(CAM−CAM^{∗}) | Heaviside step function for y threshold value |
dy/dt=(y^{ ∞ }−y)/τ_{ y } | Secondary slow process involved in opening of PTP high conductance state |
\( dPTP_{h}/dt = \left (PTP_{h}^{\infty } - PTP_{h}\right)/\tau _{h} \) | PTP high conductance state dynamics |
\( J^{H}_{PTP} = perm_{l}^{H} * PTP_{l} * PSI * \left (H_{M} - 0.0000000398 * exp\left (-37.434 * PSI\right)/\left (1-exp\left (-37.434 * PSI\right)\right)\right) \) | Proton flux through PTP in high conductance state |
\( J^{Ca}_{PTP} = perm_{Ca} * PTP_{l} * J_{uni} * \left (1-postptp * PTP_{h}\right) \) | Rate of Ca^{2+} ion transport across PTP |
\( dH_{M}/dt = \left (f_{H_{M}}/\tau _{h}\right)*\left (J^{H}_{L} + J^{H}_{F1} - J^{H}_{res}+J^{H}_{PTP}\right) \) | Change in mitochondrial proton concentration |
τ_{ l }=p_{6}+amp_{ τ }/cosh((H_{ M }−p_{3})/p_{4}) | Time constant for PTP low conductance state |
\( PTP_{l}^{\infty } = 0.5 * \left (1 + tanh\left (\left (p1 - H_{M}\right)/p_{2}\right)\right)\) | Rate of change of polling function |
\( dPTP_{l}/dt = \left (PTP_{l}^{\infty } - PTP_{l}\right)/\tau _{l} \) | PTP low conductance state dynamics |
Results
In the absence of pathology
In this case, the neuron responds as it would normally do to a stimulus, with the Ca ^{2+} ions being released rapidly from the ER via the IP_{3} channels to maintain calcium homeostasis across the cell. This sudden spurt results in the onset of oscillations of Ca ^{2+} ion concentration in the mitochondria, cytosol and ER. There is no further activity, and the concentration of Ca ^{2+} is sustained at a low equilibrium value.
In the presence of pathology
As this continues, oscillation of calcium levels resumes, except this time the mitochondrial calcium ion concentration continuously increases, indicating an increase in sequestration of calcium ions by the mitochondria. As this constant rise is maintained for a period of time, as detailed in Oster et al. [8], a slower secondary process is activated, which results in the PTP opening in the high-conductance state. The opening of PTP in high-conductance state detailed by Oster et al. [8] follows exactly the same pattern, with the high, sustained levels of mitochondrial calcium ion concentration resulting in the activation of a slower, secondary process. This secondary slow process, on completion, results in the PTP opening in high conductance state.
When the PTPh opens in the high conductance state, calcium ions rush out of the mitochondria causing the CAM to level off. As the mitochondria stop functioning due to the PTP opening in high conductance, its calcium ion concentration is frozen, with CAC and CAER skyrocketing due to uncontrollable amounts of calcium ions flowing into the cell. This may be because the amyloid concentration has also increased over time (although in a much more regulated fashion), causing the plasma membrane to become more permeable and allowing more calcium ions into the neuron. Consequently, a jump occurs in cytosolic calcium ion concentration, which oscillates for a while and once stabilized begins to increase at an accelerated rate.
Conclusion and discussion
This paper presents a mathematical model of the biological processes involved in the deposition of β-amyloids in and around the neurons and its effect on neuronal calcium signaling homeostasis. Using a detailed mitochondrial model of calcium signaling [7], the paper relates this change in calcium signaling in the cytosol to the calcium level in the mitochondrial matrix. Moreover, the introduction of the permeability transition pore and its characteristics allowed the model to depict the irreversible onset of apoptosis. By combining the models and features regarding β-amyloid deposition, mitochondrial calcium signaling and permeability transition pore activity, we simulated the opening of the permeability transition pore using amyloid deposition as the trigger. We found that, using the parameters and model equations above, high-conductance state is reached at around 865 ms after amyloid deposition begins.
The lack of comprehensive models of subcellular dynamics in Alzheimer’s disease poses a challenge for computational scientists to explore. We envisage a need for integrated modeling, i.e. selecting individual and specialized models and finding ways to combine them, to formulate a composite model of a neuron’s cell fate in AD.
In this paper, we have modeled only a single neuron undergoing mitochondrial PTP-related apoptosis, which is one of the mechanisms of cell death in AD. We have not studied necrosis due to Ca ^{2+} excitotoxicity, or looked at the effect of β-amyloid deposition and misfolding on cognitive functions, both of which can substantially extend the scope of this research. Furthermore, the effect of the neuron on its neighboring neurons in such a scenario has not been considered in this paper, as we assumed the neuron to exist in isolation. These issues are pertinent, and extending our composite model to address them could significantly improve our understanding of the disease.
In summary, we constructed a composite model by integrating three individual models to recapitulate a sequence of events and their repercussions that eventlead to neuronal death. Through this model, we are able to shed new light on a single sequence of processes starting from the deposition and misfolding of β-amyloid in and around the neuron all the way to neuronal apoptosis. Compared with existing models, this model provides a more comprehensive view of these molecular processes occurring in Alzheimer’s disease. It represents a step towards building more realistic models to facilitate the diagnosis and treatment of this aging-related disease.
Notes
Acknowledgements
We would like to thank Dr. Andrew Oster for his support in this project in answering our numerous questions and providing supplementary material for his work. We also wish to acknowledge the funding support for this project from Nanyang Technological University under the Undergraduate Research Experience on CAmpus (URECA) programme.
Funding
We wish to acknowledge the funding support for this project from Nanyang Technological University under the Undergraduate Research Experience on CAmpus (URECA) programme. This work was also supported by the MOE AcRF Tier 1 grant (2015-T1-002-094), Ministry of Education, Singapore. The funding for publication of this article was provided by the MOE AcRF Tier 1 grant (2015-T1-002-094), Ministry of Education, Singapore.
Availability of data and materials
Not applicable.
About this supplement
This article has been published as part of BMC Systems Biology Volume 12 Supplement 1, 2018: Selected articles from the 16th Asia Pacific Bioinformatics Conference (APBC 2018): systems biology. The full contents of the supplement are available online at https://bmcsystbiol.biomedcentral.com/articles/supplements/volume-12-supplement-1.
Authors’ contributions
Experimental concept and design: BR, CKH, ZJ; Sample collection and data contribution: BR; Data analysis and interpretation: BR, CKH, ZJ; Computational support: BR, CKH; Manuscript writing and figures: BR, ZJ; Manuscript review: All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Supplementary material
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