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The mitochondrial permeability transition pore confersexcitability and CICR wave propagation

Andrew M. Oster1, Balbir Thomas2, David Terman1,2, and Christopher P. Fall1,3

1. Mathematical Biosciences Institute, Ohio State University2. Department of Mathematics, Ohio State University

3. Department of Anatomy and Cell Biology, University of Illinois at Chicago

December 20, 2008

Abstract

Mitochondria have long been known to sequester cytosolic Ca2+ and even to shape intracellu-lar patterns of endoplasmic reticulum -based Ca2+ signaling. Accumulating evidence suggeststhat the mitochondrial network is an excitable medium which can demonstrate Ca2+ inducedCa2+ release via the mitochondrial permeability transition. The role of this excitability re-mains unclear, but mitochondrial Ca2+ handling appears to be a crucial element in diversediseases as diabetes, neurodegeneration and cardiac dysfunction. In this report, we extend

the modular Magnus-Keizer computational model for respiration-driven Ca2+

handling toinclude a transition pore and we demonstrate both excitability and Ca2+ wave propagationthat is accompanied by depolarizations similar to those reported in cell preparations. Thesewaves depend on the energy state of the mitochondria, as well as other elements of mito-chondrial physiology. Our results support the concept that mitochondria can transmit statedependent signals about their function in a spatially extended way.

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MBI Technical Report: work in progress

Introduction

Calcium (Ca2+) plays a critical role in the modulation of intracellular functions as diverse

as fertilization, metabolism and programmed cell death. Furthermore, Ca2+

dysregulationis implicated in human health problems including diabetes, cardiac disease and Parkinsonsneurodegeneration. One of the hallmarks of intracellular Ca2+ signaling is the rich dynamicbehavior that has been observed. These range from frequency modulated oscillations insmall cells in response to extracellular hormone to the bistable and excitable spatiotemporalpatterns seen in very large frog oocytes and eggs.

Ca2+ signaling has been a model system for mathematical and computational biology foryears. Numerous theoretical and computational models have explored Ca2+ signaling in theendoplasmic reticulum (ER) as an oscillatory or excitable system and many of these modelingstudies have involved spatiotemporal simulations which replicate the rich behavior of theseintracellular Ca2+ signals. The fundamental principle involved in the dynamics of ER-basedCa2+ signaling is so-called Ca2+-induced Ca2+ release (CICR). This paradigm involves fastactivation and positive feedback onto ER Ca2+ release channels that interact with slowerinactivation to set up a system capable of supporting excitability, oscillations and relateddynamical behaviors. Other factors such as second messenger modulation of channels canbe thought of as excitability parameters or as additional slow variables in the system.

Mitochondria have long been known to sequester cytosolic Ca2+ and even to shape intracel-lular patterns of ER-based Ca2+ signaling. In a remarkable series of observations, Ichas andcolleagues demonstrated that mitochondria themselves constitute an excitable Ca2+ signal-ing system that can support waves of Ca2+ release. The central component in this case isa mitochondrial permeability (MPT) that both releases Ca2+ and is directly or indirectly

responsive to Ca2+. The positive feedback by Ca2+ on the transition appears to conferexcitability when examined in cell-free experiments.

The MPT has been the nexus of a large experimental effort principally because it appearsthat permanent activation of the transition and subsequent depolarization of mitochondriaare an obligate initiating step to most non-receptor mediated programmed cell death. Anextraordinary body of work has led to the biophysical characterization of the properties ofthis permeability transition, but to date, the precise molecular players comprising the MPTremain unknown. It has been hypothesized that the MPT is the result of action of a porecomplex, which has been termed the permeability transition pore (PTP). Players known toaffect the probability of an MPT event are many of the same factors that are implicated in

cell death in both health and disease: Ca2+, reactive oxygen species, altered membrane lipids,electron transport chain anomalies and altered gene expression. Computational modelingof mitochondrial Ca2+ dynamics may therefore aid in the understanding of how to tip thebalance of cell death towards acceleration in the case of cancer therapies or towards protectionin the case of neurodegenerative diseases.

From a theoretical and computational perspective, Ca2+ handling in cells is a complicateddynamical system comprised of two separate but interacting excitable media: the endoplas-mic reticulum and mitochondria. Only recently have mitochondria been incorporated intobiophysical models of intracellular Ca2+ signaling and, to our knowledge, only two studies

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have ever considered the PTP. Neither of these models is spatially explicit, and they donot consider wave propagation. Selivanov et al. first modeled mitochondrial CICR using aminimal model and replicated their experimental results showing frequency dependence ofPTP opening and excitability. A more recent study by Holmuhamedov et alutilized a muchmore complicated model incorporating elements of respiration and other details. This workis especially interesting because it suggests that mitochondria might parametrically switchbetween dynamical modes due to the overall Ca2+ load in the system; however, this studydid not address the phenomenon of excitability.

A primary goal of this paper is to better understand the role of the PTP in the propaga-tion of waves of depolarization and Ca2+ release seen in experiments [8]. Our approach liessomewhere between the two previous modeling efforts: We start with the detailed biophysi-cal model for mitochondrial Ca2+ handling due to Magnus and Keizer [15, 16, 17]and thenincorporate a biophysical mechanism for the PTP, a pH variable similar to those used bySelivanov et al. [23, 22], and a weak-acid term that is essential for robust mitochondrial

pH-homeostasis. We do not incorporate additional mechanisms or currents, as was done byHolmuhamedov et al., in order to minimize the computational overhead in our spatial sim-ulations. These calculations represent the first spatiotemporal simulations of mitochondrialfunction of which we are aware. (Note to reviewers: we would be grateful for any assistancein giving credit to prior work.)

Another goal of this work is to begin to address the complexity of modular models of phys-iological processes. The modular approach involves incorporating individual mechanismstuned separately with biophysical data into a meta-model for a particular system. Thisapproach is unavoidable if one seeks to perform computational experiments on ever moreinterlinked physiological processes; however, the result is extraordinary complexity and of-

ten a less than satisfactory understanding of the underlying mathematical structure and itsproperties. Therefore, we have built and tested a relatively complete model of mitochon-drial Ca2+ handling, including the additional mechanisms related to the PTP, and then wehave begun to reduce this model to the essential elements necessary for excitability and thepropagation of waves of depolarization and CICR. We feel that such collaborations betweencomputational cell biologists and mathematical biologists will be necessary going forward touncover the basis of the complex spatiotemporal phenomena observed in experiments andreplicated in computer simulations.

Biological background: mitochondria as an excitablemedium

Here we will review some of the basic concepts necessary for nonspecialists to understand thework that follows. It must be emphasized that the field and literature of mitochondrial phys-iology is vast and quite detailed, and therefore it is impossible to cover this material in anydepth here. We will build from commonly accepted concepts to evidence for mitochondrialexcitability and a hypothesis for the mechanism that we use for our modeling effort.

The first concept is that mitochondria are quite polarized organelles that maintain a elec-

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MitochondrialMatrix

IntermembraneSpace

OuterMembrane

Porin

Innermembrane

NADH

NAD

ADP+P

ATP H

+

H+

ATPADP

Electron TransportChain

ATP Synthase

Adenine NeuclotideTranslocator

Figure 1: A schematic diagram of basic mitochondrion structure. The outer membrane has large channels (called porin)making it highly permeable, rendering the intermembrane space to be a close approximation to the cytosol in terms of ionicconcentrations. Within the mitochondria is the matrix. Separating the matrix and the intermembrane space is the innermem-brane, along which respiration transpires, that is, the chemosmotic process takes place. As such, it is folded into numerouscristae to greatly increase its surface area.

trochemical gradient () much like neurons, and the interiors of mitochondria are on theorder of -150mV relative to the cytosol. This gradient is generated by the active pumpingof protons (H+) across the mitochondrial membrane by the electron transport chain (ETC)during aerobic respiration. The generation of and the dissipation of the proton com-ponent by the Fo/F1 ATPase to phosphorylate ADP to ATP form the basis of Mitchellschemiosmotic hypothesis for the transformation of glucose and oxygen substrates to energyforms usable by cellular processes.

The second concept is that mitochondria sequester and release Ca2+. Ca2+ introduced acrossthe plasma membrane or released from the ER is sequestered into mitochondria through theCa2+ uniporter, a process driven by the mitochondrial potential, [21, 7]. Physiologicalinflux of Ca2+ into the mitochondria causes a measurable concurrent mitochondrial depolar-

ization [13], which indicates a tight coupling of Ca2+ intake to the polarization state of themitochondria. There are several means of Ca2+ release from mitochondria. The two princi-pal pathways of Ca2+ efflux are the Na+ dependent and the Na+ independent transporters[21]. These two pathways transport Ca2+ against and thus require the expenditure ofenergy; the Na+ dependent Ca2+ exchanger requires the maintenance of an appropriate Na+gradient and the Na+ independent Ca2+ pump may directly utilize ATP. In the general case,the sequestration of Ca2+ is quite rapid and the release occurs on a much slower time scale.Therefore, mitochondria have been considered as damper for cytosolic Ca2+ signals.

The third important concept is that mitochondria exhibit a stereotypic property known asthe mitochondrial permeability transition (MPT). A variety of stressors such as excessive

Ca2+, reactive oxygen species, shifts in gene expression and other factors have been shown tocause mitochondria to undergo a catastrophic depolarization. Interestingly, this permeabilitytransition can occur in two modes: mitochondria in situ have been observed to undergotransient depolarizations with no apparent ill effect (ichas paper mentions no swelling), andmitochondria can undergo permanent depolarization which leads to apoptotic cell death.Because of the known transition inducers and because the permeability transition has beenlinked with cell death in a variety of disorders, the machinery associated with the transitionhas been described as a sensor of mitochondrial health. One useful analogy is the weightedregulator for a pressure cooker, which resonates to maintain an appropriate internal pressure

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but which locks open when the pressure is too great. It may be that flickering of thepermeability transition has a physiological role in preventing damage to highly active orstressed mitochondria. For the purposes here, what is important is that the permanentactivation of the MPT has been shown definitively to open gaping holes in the mitochondrialinner membrane sufficient to pass anything smaller than 1500DA. Transient opening of thePTP is thought to be to a low conductance state limited to approximately 350DA. In eithermode ion gradients are dissipated, resulting in the rapid release of mitochondrial Ca2+ intothe cytosol.

What we have discussed thus far is reasonably accepted. Going forward as we build andexplore our model, we move from consensus to hypotheses with somewhat less experimen-tal support. For example, a central problem in the permeability transition field is that achannel has not been definitively isolated, although it may be the same as the electrophysio-logically characterized mitochondrial megachannel. A functional pore instead of some phasetransition or other phenomenon in the membrane is most likely and it has been termed the

mitochondrial permeability transition pore (PTP, as opposed to MPT). It is a matter offurther controversy as to whether the pore might comprise a single channel or a complexof proteins. Consistent with the transition physiology, this pore may have two open statessensitive to different stressors which result in a transient, low conductance state (PTP l) andan high conductance state with irreversible opening (PTPh).

Mitochondria have been demonstrated in vitro to be Ca2+ - excitable organelles capable ofsupporting CICR, and there is supporting evidence from in-situ recordings. The work wedescribe now is central to our interest in this problem, and the resulting hypotheses arecentral to our computational investigations. The key in vitro evidence comes from work byIchas, Holmuhamedov, Mazat and colleagues. They were able to quantitatively demonstrate

Ca2+ induced excitability in isolated mitochondrial suspensions, and, in an elegant experi-ment using mitochondria embedded in a planar gel, to demonstrate that mitochondria canpropagate a wave of CICR. They were further able to demonstrate that mitochondrial CICR(mCICR) is blocked by cyclosporin A (CsA), which implicates the PTP. In situ, mitochon-drial flickering and coordinated depolarizations have been shown to depend on both Ca 2+

and the PTP. In the Xenopus oocyte, which is large enough for the visualization of complexspatiotemporal Ca2+ waves, energization of mitochondria has been shown to amplify ratherthan dampen ER-based Ca2+ waves, consistent with increased excitability. Such a role mayhelp to explain the close apposition of the extended syncytial webs of endoplasmic reticulumand mitochondria in cells.

Holmuhamedov et al. attribute mCICR to PTP l. Notably, PTPlis not thought to be directlyCa2+ sensitive. Instead, in their model, pH is the initiator of PTPlopening. Ca

2+ rushingin through the Ca2+ uniporter results in a depolarization, which then causes the electrontransport chain to compensate. Heavy ETC activity results in an increase in pH, whichtriggers the PTPl. Excitability implies a threshold effect, and, in this case, the threshold isdetermined by the ability of weak acidic inorganic phosphates to buffer the pH and compen-sate for ETC activation. Thus large or repetitive cytosolic Ca2+ transients can overwhelmpH buffering and cause PTPlactivation, while the ETC response to low or slow cytosolicCa2+ transients is adequately compensated for.

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80

0

40

170

90

150

60 s

7.65

7.62

7.64

CaC

(M)

pH

CaM

Ca2+

CaC

CaC

60 s 60 s60 s

180 s 180 sWith CsA

(a) (b)

max

min

max

min

Figure 2: Key experiments, performed by Ichas et al. [8], demonstrating mitochondrial excitability and its dependence uponpH (redrawn from [8]). In (a), three pulses of Ca2+ are introduced at different rates with the same amount of Ca2+ introduced.When Ca2+ is introduced at 30s intervals, the Ca2+ is efficiently sequestered away in the mitochondria, leading to a rise inthe mitochondrial calcium load. The pH rises transiently, yet returns to a state of homeostasis. When the Ca2+ is introducedat 10s intervals, the pH rises precipitously and this corresponds with PTPl. The pH then rapidly drops, the innermembranepotential drops, and Ca2+ leaves the mitochondria and enters the cytosol. The Ca2+ is slowly reabsorbed by the mitochondrionas the pore closes, so that the system eventually returns to a homeostatic state. In (b), a bulbous of calcium is introduced in agel containing isolated mitochondria that initiates traveling waves of Ca2+ and potential. By previously dousing the gel withcyclosporin A (CsA), an agent which obstructs pore opening, both the Ca2+ and potential changes do not propagate the length

of the gel. We suggest that wave propagation is due primarily to diffusion, however the pore itself is necessary for continuedpropagation.

A model of respiration driven mitochondrial Ca2+ handling

with a PTP module

Modeling background

There are two principal areas in which Ca2+ in relation to mitochondria have received theo-

retical treatments. One is in the context of ER based Ca2+ release in the cytosol, and thesemodels have been presented with varying degrees of biophysical detail. Beginning in 1998,Marhl, Schuster and colleagues have explored the consequences of mitochondrial function onintracellular Ca2+ signaling using a phenomenological model of mitochondrial Ca2+ handling,arguing that mitochondria may serve to normalize the amplitude of intracellular Ca2+ os-cillations [19]. This initial work was followed by a similarly phenomenological, but spatiallyexplicit, model of ER-mitochondrial Ca2+ handling in the Xenopus oocyte [18] which soughtto explore the paradoxical increase in amplitude and frequency seen in periodic Ca2+ signalswith enhanced mitochondrial function [10]. Fall and Keizer combined a reduced ER Ca2+

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model with a biophysical mitochondrial Ca2+ handling model (discussed below) to show thatmitochondria move the combined dynamics of Ca2+ signaling into an oscillatory mode undercertain parameter regimes.

The other area is models of mitochondria themselves focused on Ca2+

handling and its conse-quences. The pioneers in this area were Magnus and Keizer, who developed a quite detailedbiophysical model of mitochondrial Ca2+ handling and, in a series of papers [15, 16, 17], ex-plored the role of mitochondria in pancreatic -cell Ca2+ oscillations. This work has servedas an important basis for several subsequent efforts to examine the role of Ca2+ stimulationof mitochondrial metabolism at ATP production and the interplay of the mitochondria withER Ca2+ signaling [4, 3, 20]. Two efforts bear more directly on the role of the PTP in mCICRthat we explore here. The first model to explore the role of the PTP was a minimal modelby Selivanov, Ichas, Holmuhamedov, Mazat and colleagues which replicated their mCICRexperimental findings discussed above [23]. Holmuhamedov and colleagues followed up onthis work with a far more biophysical model that used elements of the Magnus and Keizer

model along with additional mechanisms such as a K+ flux and changes in mitochondrialvolume. The authors proposed that the cellular Ca2+ load can serve as a bifurcation param-eter for the two MPT behaviors (transient PTPland sustained PTPh) and demonstrated thiscompelling hypothesis in their model which included a a two state PTP mechanism.

We were faced with choosing between the Magnus-Keizer model and the Holmuhamedovmodel as the basis of our investigations and we have chosen the more accepted Magnus-Keizer model for several reasons. For example, volume changes, which are included in theHolmuhamedov model, do not appear to be relevant to PTPL and wave propagation. Inaddition, our goal is to arrive at a more minimal model. Finally, we are interested primarilyin excitability.

The modular Magnus-Keizer model

The Magnus-Keizer model forms that basis of our exploration of depolarization and wavepropagation, and here we briefly describe only those details of the model that will be usedin our study. We note that the original description of the Magnus-Keizer model requiredthree separate papers and these papers were condensed from a nine volume dissertation byMagnus [14]. For a description of the individual mechanisms, we refer the reader to theoriginal papers.

The full Magnus-Keizer model describes details of mitochondrial Ca2+

handling and ATPproduction, and includes respiratory proton pumping, ADP phosphorylation by the Fo/F1ATPase, a proton leak, the Adenine Nucleotide Transporter, the mitochondrial Ca2+ uni-porter and the associated Na/Ca2+ exchanger. Metabolic pathways and controls are alsoexplicitly modeled, including glucose-dependent glycolytic and TCA-cycle production ofNADH (under Ca2+ control), glycolytic phosphorylation and hydrolysis of ATP. Finally,metabolic function is linked to membrane behavior with plasma membrane Katp channelsinteracting with voltage gated Ca2+ and K+ channels. The resulting system of 12 dynamicvariables captures the bursting behavior of the pancreatic -cell in response to various levelsof glucose.

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In our version of the original Magnus-Keizer model, we have eliminated the plasma mem-brane mechanisms. We have also incorporated two modifications that have been publishedpreviously [4]: First, the uniporter mechanism was reverted to an earlier formulation that al-lows for saturation in Ca2+ uptake [7]. Second, the scaling and compartmentalization schemewas modified so that this modular model could more easily be incorporated with existingCa2+ signaling models. With these choices of parameters, the dynamics of the original modelare unchanged.

A complete description of the differential equations model used in this paper is given in theAppendix. We have slightly altered the notation used in the previous work of Magnus-Keizerfor clarity, and use Js to represent fluxes; however, superscripts now track the type of ionor proton being trafficked and subscripts signify the process or channel through which theflux passes. For example, JCauni represents the calcium flux through the Ca

2+ uniporter.

In order to study the role of the PTP in excitability and wave propagation, we have added

several features to the Magnus-Keizer model. These include a mechanism for the PTP basedon the hypothesis of Ichas and colleagues. Because the PTP relies upon dynamic pH, whichwas constant in the Magnus Keizer model, we have also included a mechanism to track pH.Finally, we have incorporated an electroneutral inorganic phosphate flux term which servesas a weak acid as described by Ichas and colleagues. This flux provides for a mechanismof mitochondrial pH buffering and allows the mitochondrial proton concentration to returnto homeostasis. These additional mechanisms are described in detail in the following sub-sections. In Figure 3, we show the structure of the Magnus-Keizer model with additionalcomponents colored in blue.

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H+

ADP

Ca2+

CytosolMitochondrial

Matrix

Respiration

Uniporter

Na Exchanger

PTP

Adenine NucleotideTranslocator

H+

H+

PTP H+

H+

H+

Weak Acid

Proton Leak

F0F1 ATP-ase

ATP ADP

ATP

ADP

Na

ATP ADP + P

hydrolysis

glycolysis

Ca2+

Ca2+

Ca2+

TCA cycle

F0F1 ATP-ase

NADH NAD+

reduction

oxidation

3-

4-

+

Figure 3: Schematic of the model currents and compartments. The mitochondria compartment is colored in green and thecytosol in yellow. Arrows show fluxes and processes relating to H+, Ca2+ and ADP, which are displayed in the three dashedboxes, and each process is tracked by a differential equation. Additions to the Magnus-Fall-Keizer model are colored in blueand the Ca2+ current from the pore shows arrows in both direction as Ca2+, yet Ca2+ being ejected from the mitochondrionto the cytosol is particularly of interest.

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Elements of the mitochondrial permeability transition pore model

Our model for the hypothesized PTP is based on the properties outlined by Ichas andcolleagues [9]. Briefly, the PTP displays three states: a closed state, a low-conductance openstate (PTPl), and a high-conductance open state (PTPh). Evidence by Ichas [8] suggests thatthe PTP enters the low-conductance state in a pH dependent manner, see Figure 2(a). Thetransition of the pore to the high-conductance state requires prolonged levels of mitochondrialCa2+, and once the pore has opened to this state it remains open, ultimately leading to celldeath.

Cytosol

Mito

Mito

H +

High Conductance State

Mito

Low Conductance State

Closed

Open

pH M

Ca M

Cytosol

CytosolCa

2+

Ca2+

Ca2+

Ca2+

H +

H +

H +

H +

H +

Ca2+

PTPl

PTPh

Figure 4: The Ichas-Mazat model of the PTP: closed, PTPland PTPh. The transition to the low-conductance state dependsupon the pH in the matrix and the transition to the high-conductance state depends on the mitochondrial Ca 2+ load. Withpore opening, protons and Ca2+ rush into the mitochondria, dissipating , and a transition occurs leading to the expulsionof Ca2+ into the cytosol.

In this section, we outline new additions to the Magnus-Keizer model including the currentsflowing through the PTP, the contribution of the weak acid flux, and the mitochondrialmatrix proton dynamics. The pore gating variable, P T Pl , in its low and high conductancestate will determine if the PTP is open or not.

We assume that protons and Ca2+ travel through the pore while in its open states, as would

other ions not considered in this model. The proton flux is taken to be of the Goldman-Hodgkin-Katz form multiplied by the state of the pore:

JHPTP =i=,h

Hi P T Pi

HMHC

exp(FRT

)

1 exp(FRT

)

(1)

where i = , h indicate whether the pore is in the low or high conductance state with anassociated permeability of i.

For consistency, the Ca2+ flux is taken to be proportional to the flux through the uniporter

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yet dependent upon the state of the pore.

JCaPTP = CaPTPP T PlJuni(CaC, CaM) (2)

where CaC and CaM denote cytocolic and mitochondrial Ca2+ levels, respectively, and theconstant CaPTP is given in the Appendix. Alternatively, other models (e.g., [23, 22]) take theuniporter and flow through the pore to be a modified Goldman-Hodgkin-Katz flow with acorrection term.

A module for the weak acid flux

Here we describe an electroneutral inorganic phosphate flux () which serves as a weak acidand provides a mechanism for mitochondrial pH buffereing. There is little easily applicabledata on the dynamics of and pH buffering on which to base a modular mechanism. Seli-

vanov [23] took the weak acid flux to be linear; however, our computational studies suggestthat a linear weak acid flux is not sufficiently robust to account for observed excitability andwave-like dynamics. Pokhilko et al. [22] developed an extensive model of which involvesseveral differential equations and accounts for phosphate levels within both the matrix andthe cytosol. Their model treated the weak acid flux as a hydroxide(OH)/phosphate ex-changer which exported hydroxide from the matrix. Due to basic chemistry, we remind thereader that

pH+ pOH = 14. (3)

So that a decrease in OH would be tied to a rise in H+. Over a wide range of parameters,the weak acid flux due to Pokhilko et al. effectively reduces to a sigmoidal function of pH. We

have, therefore, adopted a simplified sigmoidal form for this restorative flux of pH buffering.

It is unclear what occurs to the weak-acid flux during a pore opening event. One couldaccount for such behavior by introducing an nonlinearity to the the Michaelis-Menten-likeweak acid flux or we could consider a very sharp weak acid flux as shown below in Figure, which would turn off when the pore opens and the pH drops below its usual homeostaticvalue.

JHa (HM) = kah

1 + tanh(

(HhomeoM

HM)10 )

2

(4)

with kah = 70, HhomeoM to correspond to a pH value of 7.619, and is a measure of the

steepness of the weak acid flux.

A module for proton dynamics

In the Magnus-Keizer model, mitochondrial pH was not considered to be a dependent variableand was assumed to be at steady-state. Because we are basing our investigations on thepremise that PTPL is pH gated, we have had to add a differential equation to track changesin mitochondrial pH. The Magnus-Keizer model contained three H+ fluxes: respiration, theFoF1 ATPase, and a -dependent leak term. To these we have added the H+ flux through

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7.65 7.63 7.61 7.59

0

20

40

60

80

pHnmol/[(mgprotein)(min)]

Figure 5: Phenomenological weak acid component that becomes stronger in a sigmoidal fashion when the mitochondrial pHis disrupted from its homeostatic value. The strength of this proton flux is on the order of one fourth the size of the dominantproton flux due respiration.

the PTP and the electroneutral weak acid flux . Hence, we can write a differential equation

for changes of the mitochondrial proton concentration asdHM

dt=

fHMM

VMmin

(JHL + J

HF0F1 JHres + JHa + JHPTP), (5)

where fHM is the fast buffering constant for protons within the mitochondria. Note that thecurrent due to respiration and the ejection of protons out of the mitochondria has a differentsign than the other currents (proton leak, F0F1 ATP-ase, weak acid, and the proton currentthrough the pore).

A module for channel representation of the PTP

We model the opening and closing of the pore in a standard Hodgkin-Huxley-like fashion.Denote the number of channels that are in the closed, low-conductance and high-conductancestates as P T Pclosed, P T P and P T Ph, respectively. We assume conservation so P T Pclosed +P T P + P T Ph is constant. The dynamics of the low-conductance state is taken to dependupon pH and is governed by the equation

dPTPdt

=[P T P (Hm) P T P]

(Hm)(6)

where both the opening rate, P T P , and the time constant, , depend upon the mitochon-

drial proton concentration Hm, i.e., mitochondrial pH. In particular, we take

P T P (u) = 0.5

1 + tanh

p1 u

p2

(7)

and(u) =

p5

cosh( (up3)(2p4) )+ p6 (8)

where the constants pi are given in the Appendix. We display P T P and in Figure .

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7.61 7.62 7.63 7.64 7.65 7.66 7.67

0

0.2

0.4

0.6

0.8

1

t=0

PTPl

pH

10x10

5

5

0

Figure 6: The equilibrium values of the low-conductance open state of the pore as a function of pH with the associatedtime constant for the low-conductance state. Superimposed upon this is a trajectory from a triggered pore opening. Note that

once the pore opens, the pH is rapidly reduced due to the influx of protons, thus the gating variable does not reach its maximalvalue of 1.

Results

Addition of the PTP confers excitability to the Magnus-Keizer

model

A critical proof of concept of our introduced model is the reproduction of the dynamics

outlined by Ichas et al. [8] where the low-conductance state of the PTP was induced byrapidly stimulating with Ca2+. The pore flickered depending upon whether the pH of themitochonodrial matrix, which rises transiently with Ca2+ stimuli, was sufficiently high. Thepore does not flicker and the mitochondrion returns to homeostasis if the simulation is ata low frequency. To replicate this behavior, we numerically simulate the system using aprotocol which stimulates the cytosol with three Ca2+ pulses for a duration of 2s over aperiod of 25s and 90s. During slow stimulation, i.e., 90s stimulation protocol, the pH risestransiently but ultimately returns to a homeostatic value, slightly elevated in relation to itsbeforehand value since Ca2+ is conserved. However, with the faster stimulation, the pH risesabove a threshold and the pore flickers, that is, the pore transitions to its low-conductancestate.

In its open state, protons rush into the matrix, rapidly returning the pH to its homeostaticvalue. In effect, PTPlacts as a pressure release where the cell rescues the proper protondistribution. During this rapid return to pH homeostasis, the potential across the innermem-brane has dramatically dropped. With this lower potential difference, the Ca2+ uniporterejects rather than takes in Ca2+. Likewise, when the pore is open and the potential differenceis reduced, Ca2+ is expelled through the pore.

It is evident that the potential drop induces the the expunging of the Ca 2+ from the mito-chondria by following the trajectories in time, as shown in Figure , where with pore opening

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6

0 100 200 300

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7.68

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0

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0.8

1

pHm

CaM

PTP

Pore Remaining Closed Pore Flickering

CaC

0 100 200 300

0 100 200 300 0 100 200 300

0 100 200 300

0 100 200 300

0 100 200 300 0 100 200 300

0 100 200 300

0 100 200 300

Figure 7: A simulation demonstrating that the model captures the transient PTP behavior as in [8]. By stimulating withthree pulses of Ca2+over a span of 90s, no pore opening occurs, yet when one stimulates with the same amount of Ca2+ in theshorter span of 25s, the pore gating variable for the low conductance state becomes non-zero. See Figure 2 to compare withthe experimental observations.

in (a) the potential rapidly drops and in (b) we note that when the potential is sufficientlylow and the pore is open, cytosolic Ca2+ dramatically rises.

The Ca2+

pulses cause for a rise in the cytosolic Ca2+

concentration and in turn change thepotential difference across the inner membrane via flow the the Ca 2+ uniporter. Changesin then affect the proton fluxes, as evident in Figure . The proton flux that undergoesthe most dramatic change is that associated with the ATP-ase. It precipitously falls withdecreases of , in turn H+ falls, which is equivalent to a rise in pH. Associated with thisperiod, the mitochondrion expels protons through the ATP-synthase to restore the electro-chemical gradient. While in this state, the mitochondrion relies solely on the TCA cycleto produce ATP. The rapid changes in the pH due to the Ca 2+ pulses, lead to the poretransitioning to its low-conductance state, which allows a large influx of protons through

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0

2

4

6

8

10

12

14

ADPm

ADPc

NADHm

0

0.5

1

1.5

2

0

1

2

3

4

5

0

2

4

6

8

10

12

14

0

0.5

1

1.5

2

0

1

2

3

4

5

Pore Remaining Closed Pore Flickering

0 100 200 300

0 100 200 300 0 100 200 300

0 100 200 3000 100 200 300

0 100 200 300

Figure 8: Secondary variables during stimulation.

7.617.62

7.637.64

7.657.66

7.67

00.2

0.40.6

0.81

110

120

130

140

150

160

PTPl

pH

t=0

Pore

Opening

7.617.627.637.647.657.667.67

00.2

0.40.6

0.81

0

1

2

3

4

5

6

7

Ca

pHPTPl

t=0Pore

Opening

C

Figure 9: Trajectories of flickering event with the initial point labeled and an arrow indicates the direction of the trajectory.The x-axis and y-axis are the gating variable PT P and mitochondrial pH, respectively. In (a) we plot on the z-axis the potential and in (b) the z-axis tracks the cytostolic Ca2+ concentration. If no Ca2+ is applied, the plot would remain at its initial pointin the upper left hand corner. We infuse the cytosol with three Ca2+ pulses. Each Ca2+ pulse drops the potential , seen in(a), and an upswing in the cytostolic calcium, shown in (b). After the first two pulses, the potential recovers to approximatelyits initial value, but the pH has risen, i.e., [H+ ]M has dropped. During the third Ca

2+ pulse, the pH is sufficiently high toelicit a PTPlevent. The pore opens. H

+ rushes into the mitochondria causing the potential to drop and in turn the uniporterejects calcium into the cytosol, demonstrated in (b).

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the pore (i.e., the JHPTP flux). This greatly lowers the membrane potential, , and causesthe uniporter and pore to expel Ca2+ from the mitochondrial matrix, resulting in the wave-crest of cytostolic Ca2+ pictured in Figure . In effect, the mitochondria can dampen slowCa2+ signals and also amplify fast Ca2+ signals by this mechanism. We demonstrate nu-merically that this outward rush of Ca2+ can lead to the emergence of a traveling wave inthe potential and in the cytostolic Ca2+.

0

50

100

150

200

250

300

0 50 100 150 200

Jhres

Jhf1

Jptph

Jah

Jhl

Figure 10: Proton fluxes leading to PTPh. The flux due to respiration, JHres, is the only outward flux of protons from thematrix. Note that at each time the cytosol is stimulated with calcium, the flux of protons through the ATP-synthase, JH

F0F1,

dramatically drops, suggesting that the changes in this flux are the principle reason for the drops in mitochondrial protonconcentration (and dually the rises in the matrix pH) which occur with each application of Ca2+. After time t = 125, the poreopens and a flood of protons enter the matrix through the PTP and help to return the mitochondrion to pH homeostasis. Thisinflux of protons destroys the electrical potential and the ATP-synthase actually ejects protons until the potential eventuallyrecovers (hundreds of seconds later.

Cytosolic glucose level affects mitochondrial excitability

The Magnus-Keizer model included a treatment of the metabolic pathways and controls,including glucose-dependent glycolytic and TCA-cycle production of NADH (under Ca2+

control). The resulting model captures the dynamic behavior for various levels of glucose.In the Magnus-Keizer model, the glucose levels is treated as a parameter. The amountof glucose directly affects the availability of NADH, the principle substrate of respiration.That is, a low glucose level corresponds to a low level of NADH and a high glucose levelis associated with a high level of NADH. Since NADH levels play an integral part of therespiration process, the glucose level affects the mitochondrions respiration capability. Assuch, since changes in the pH occur due to increases in the respiration, a mitochondrionsdegree of excitability also is tied to the glucose level. We consider the same parameters as

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previously used in the above simulation, but now vary glucose. We find that for lower valuesof the glucose levels, the pore opening event is diminished.

0

1

2

3

4

5

6

0 50 100 150 200 250 300 350

CaC

glc=1.0 glc=3.0glc=2.0

0

1

2

3

4

5

6

0 50 100 150 200 250 300 3500

1

2

3

4

5

6

0 50 100 150 200 250 300 350

Figure 11: In a), we perform the Ca2+ stimulation procedure to induce flickering event with a set glucose level of mM.We note that no pore opening o ccurs and the system returns to approximately the same steady-state. In b), we repeat theCa2+ stimulation procedure, but with a higher level of glucose (2mM). We found a pore opening event but to a lesser extent.When we raise the glucose level to 3mM, the Ca 2+ stimulation procedure induces a large pore opening even and there is adramatic out rush of Ca2+ into the cytosol.

Impaired pH buffering can lead to PTP oscillations

The pH buffering capability in the matrix is in part due to the transport of phosphatecarriers across the inner membrane. This effective flux of protons into the matrix is termedthe weak acid flux. The precise form of the weak acid flux is not well understood, so weconsider a phenomenological, sigmoidal representation of the weak acid flux that dependsupon the pH of the matrix. We consider a range of shapes for the weak acid flux that vary

in steepness and in absolute magnitude. For strong and sharp weak acid fluxes, we findthat the mitochondrial dynamics go to a stable steady-state. However if we weaken and/ortake a less sharp representation of the weak acid flux, that is, we reduce the parameter in Equation 4, oscillations in the matrix pH, calcium level, and potential arise. Theseoscillations are due to a periodic opening (flickering) of the PTP and differ are from theoscillations as studied by Bertram et al. where the energetics enter into oscillations. In ourmodel, the recovery pathway of the weak acid flux is not sufficiently strong to keep the pHat a healthy level. Instead the pH rises above the threshold and the pore transiently opensin order to regulate the pH level.

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6.5 7 7.5 8 8.5 9

0

20

40

60

80

nmol/(mg-prot

min)

=5

=2

=1.5

Figure 12: By decreasing in Equation 4, the strength and sharpness of the weak acid flux changes in kind. Here we plotthree representative plots with a fixed kah but with taking on the values of 5,2, and 1.5.

0

1 0.021

0.02580

180

=5.0 =1.0=2.5

PTP Hm

PTP Hm

PTP Hm0

1 0.021

0.02580

180

0

1 0.021

0.025

80

180

a) c)b)

Figure 13: By decreasing the strength and sharpness of the weak acid flux also decrease. In a), = 5 and the weak acidflux is strong and sharp. There exists a stable steady state with the PTP remaining closed. When is decreased to = 2.5,the stable steady of the pore remaining closed is lost, with the pore opening and then relaxing to a permanently open state,pictured in b). However if is furthered decreased to a value of 1.0, the trajectories enter into an oscillatory state and thereare oscillations in pH, Ca2+, potential, and the other dynamic variables (not all pictured).

The mitochondrial excitable medium supports a traveling wave

In a preparation using mitochondria isolated from Ehrlich cells and suspended in a gel, Ichaset al. found traveling depolarization and Ca2+ waves [8]. This experiment demonstrated thatmitochondrial populations constitute an autonomous Ca2+-excitable medium for propagating

electrical and Ca2+

signals in space [8]. The most obvious source of spatial coupling in thesystem is diffusion of Ca2+ in the gel. We extend our model to describe this behavior byextending the system in space (one-dimension) and include a term for the diffusion of Ca 2+ inthe gel. In this section, we use the cytosol notation for the gel, yet take into account thegels lack of buffering capability.

In the case of a gel, no buffering is present. We can explore the effects of buffers on thepropagation of a wave by considering an effective diffusion rate of unbound Ca 2+ with no

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mobile buffers, which is reasonably approximated (see Equation (24) in [24]) by

Deff KC

KC + BTDCa = fCDCa (9)

where KC is the disassociation constant for binding and unbinding of Ca2+ to stationarybuffers and BT is the total concentration of stationary buffer binding sites [24]. Typicalmeasured values for fC range between 0.01 and 0.05 (See chapter 5 of [5]). Since the con-centration gradient is large when a calcium stimulus is applied, we include a diffusive termfor cytostolic calcium using Ficks law, so that our dynamics for cytosolic calcium are

tCaC(x, t) = fC

DCa

x2CaC(x, t) +

1

Vc min[M(JCana,ex JCauni JCaPTP) + JCastim]

(10)

where fC = 1 in the case where the mitochondria are suspended in a gel with no calciumbuffering. In the cytoplasm, the diffusion constant of unbound Ca2+has been estimated tobe about 225-3002/s [1, 25].

Without the inclusion of the permeability transition pore, it rapidly diffuses until it is ab-sorbed by the underlying mitochondrial network. However, with the inclusion of the PTP,we find excitable behavior: traveling waves of calcium and potential that can propagatethe length of the domain or in certain instances become damped and eventually becomeextinguished.

250

0

50

100

150

200

0 50 100 150 200

250

0

50

100

150

200

0 50 100 150 200

t

x x

CaC

Figure 14: A 5 micron strip at x = 0 is stimulated at t = 15s with a bulbous of Ca2+

. The Ca2+

rapidly diffuses inthe gel until the majority of it is absorbed by the underlying mitochondrial network. The diffusion constant for calcium is setat DCa = 250

2/s. We show the results of a simulation with the PTP included and a mitochondrial proton buffering rapidbuffering coefficient of fHM = 1.8x10

6. At x 50, the speed of propagation drastically changes; diffusion has run its courseand the pore flickering excitability propagates the wave, but at a slower rate. We plot cytosolic Ca2+, with an altered color scalewhere all values over a concentration of 2M are truncated. We also plot the potential difference across the innermembrane.

We find that wave propagation depends on the rapid buffering constant, fHM (See Equation5). For strong proton buffering in the matrix, Ca2+ rapidly propagates at a speed approx-imately that of diffusion in the gel, but ultimately dies as the calcium is absorbed by themitochondrial network (See below ).

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However, the experiments by Ichas et al. suggest that the wave propagates at a rate slowerthan diffusion in the gel. In the span of 180s the potential and calcium wave travels ap-proximately 250. The speed of propagation is slower than 1/s, that is, the wave speedwas not determined by the rate of diffusion of free calcium in gel, rather was limited by thedynamics of the mitochondrial network. With the inclusion of the permeability transitionpore and weaker proton buffering, the rapid influx of Ca2+ results in continual pore openingevents (See Figure ). Where once the signal died, the signal propagates to the end of thedomain by the opening of the pores in a process mediated by the diffusion of Ca2+ in thegel. Furthermore, the traveling waves of depolarization and gel Ca2+travel at speeds whichdepend upon the dynamics of the pore. Thus, we see an elbow in the space time plots forthe wave: A fast wave that is diffusion dominant and an angled component, associated withslower wave speed, when pore flickering sustains the wave.

We now explore the effects of different diffusion rates on the propagation speed of theCa2+ wave. We consider DCa to vary from 350

2/s down to 502/s. We expect that as

DCa decreases the wavespeed would scale as DCa.

D = 50 /sCa

2D = 350 /s

Ca

2D = 200 /s

Ca

2

Figure 15: Plots of Ca2+ waves, similar to those in Figure (a,d) except we vary the diffusion constant from 50 up to 3502/s.We note that the wavespeeds slow with diffusion but not in a linear fashion. We calculate wavespeed by examining the timethe wave reaches 160 and when it reaches 170. We find that the wavespeeds vary from c50 1.25/s, c200 1.852/s, andc350 2.222/s.

50 100 150 200 250 300 350

1.2

1.4

1.6

1.8

2

2.2

DCa

c

Figure 16: A plot of wavespeed as a function of DCa. We note that the wavespeed scales as the

DCa .

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Buffering capability of the matrix

In this work, we have used parameters in order to obtain the qualitative behavior of theobserved dynamics. We have used a rapid buffering approximation for the proton dynamicswith a buffering coefficient of fHm. The value of fHm is not known and would likely vary indifferent cell types as well as possibly vary in the life of the individual mitochondrion. We testthe role of the matrix buffering capability on the generation and propagation of Ca2+ andpotential traveling waves. The preparation is similar to the previous stimulation at timet = 15 of a 5 micron strip about the origin with a consistent Ca 2+ stimulus. As seen in Figure, we vary the buffering capability of the mitochondria and find that proton buffering has aprofound effect on the generation of the traveling wave and its wavespeed. For preparationswith high buffering capability (that is fHm is small), the matrix effectively buffers transientrises in pH due to the influx of Ca2+ into the cytosol and the sequence of pore opening eventsfails to take place and the wave is never generated. As the proton buffering capability of the

matrix decreases (fHm rises), a traveling wave is generated and propagates. With furtherdecreases of the matrix pH buffering capability the resulting wavespeed increases. In Figure, we see wave failure for strong pH buffering and the wave emerging for weaker pH bufferingwith the wavespeed increasing linearly with fHm.

300

0

100

200

t

2001000

x2001000

x2001000

x2001000

x

CaCa) b) c) d)CaCCaCCaC

Figure 17: We vary the proton buffering coefficient from low to high. In a), the proton buffering capability is greatand fHm=2x10

7. The traveling wave fails to terminate. In b), we decrease the proton buffering capability (that is, weraise the proton buffering coefficient) so that fHm=2x10

6. We note that a wave is generate and propagates at a wavespeedapproximately 1 /s. By increasing fHm further we note increased wavespeeds evident in c) and d), with fHm=1x10

6 andfHm=2x10

5, respectively. In all of these simulations, the calcium diffusion constant was set to DCa = 2002/s.

High Conductance PTP Opening

The progression to an unhealthy state in vivo is likely a slow process where the level ofcytostolic calcium remains elevated for an extended period of time before a pore poppingevent occurs. We propose a model that captures the qualitative behavior of the pore poppingevent. Making the ansatz, that pore entering the high-conductance state to depends notdirectly on mitochondrial calcium load, but rather a secondary slow process that dependsupon mitochondrial Ca2+ load, e.g., reactive oxidative species (ROS) or phosphate carriers.In effect, mitochondrial Ca2+ must remain elevated for an extended period of time in orderto lead to a PTPhevent.

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Let P T Ph denote the gating variable of the pore opening to its high conductance state. Thedynamics are taken to be akin to a Hodgkin-Huxley-like gating variable

dPTPh

dt =

[(P T Pclosed)P T Ph ()

P T Ph(t)]

h(CaM) , (11)

where P T Ph is a function that depends upon a secondary process necessary for pore opening,which we call . For simplicity, we take

P T Ph () = ( ) (12)

where is the Heaviside function and is a threshold value that the secondary processmust to initiate opening to the high-conductance state. Similarly, we take the secondaryprocess, , to be governed by

ddt

= (CaM CaM) (CaM)

. (13)

The constant CaM is the threshold at which time the secondary process is activated. Thefunction could take on various forms: sigmoidal, Heaviside, or piecewise-linear. Taking to be sigmoidal would mean that the time that calcium being above some threshold inorder to lead to pore popping would depend on the extent that the mitochondrial calciumload is above the threshold. For simplicity, we consider to be the Heaviside function,.The key feature of this formulation is that past the threshold for mitochondrial calcium load,CaM, the secondary processes activates. In turn this slow secondary process must attain athreshold whereby the pore activates and transitions to its high conductance state.

The time constants for these processes are taken to depend upon the mitochondrial calciumload, whereby both changes in the secondary process and the transition to PTPhare slow forlow levels of CaM. In particular, we consider

(CaM) = 1000

5x106

cosh

CaM1e1

+ 1

. (14)

The time constant is relatively large since this is to be the slower of the two processesand take

h(CaM) = 100(CaM). (15)

In effect, once PTPhoccurs, the PTP remains open because the mitochondrial calcium hasall been expunged into the cytosol.

Previously, we explored a stimulation protocol that induced PTPl- a reversible openingthat eventually results in the pore closing again. However, under longer slow subthresholdCa2+ stimulation, mitochondria can attain a breaking point whereby the pore pops and en-ters into its open high-conductance state. Below, in Figure , are the results of a numericalsimulation of the pore transitioning to PTPh. A protocol for calcium stimulation that mim-ics the slow degrade in Ca2+ sequestering in the cell is used. Calcium is infused a constantrate that is sufficiently small so that the weak-acid flux can compensate and the pH remains

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approximately at its original homeostatic value. The cytostolic calcium slowly rises, and themitochondrial calcium load in turn rises as it sequesters Ca 2+ out of the cytosol. Eventually,the mitochondrial load exceeds the threshold value of CaM and the secondary process be-comes activated. As Ca2+ infusion continues, the secondary process slowly rises, during thistime the mitochondrial Ca2+ load is elevated for an extended period of time. When the sec-ondary process it attains its threshold vale , the PTP transitions to its high-conductancestate. The mitochondrial Ca2+ rushes out of the pore into the cytosol, where Ca2+ is weaklybuffered in comparison to the mitochondrial matrix. Thus the cytosolic Ca2+ is highly ele-vated with no other mechanism for clearing Ca2+, the cell would undergo Ca2+ cytotoxicitywhereby elevated cytosolic Ca2+ concentrations can activate cysteine proteases that mediateplasma membrane breakdown through the cytoskeletal and plasma membrane proteins [12](in review by [6]).

The model compensates to preserve its diminished electrical potential by utilizing ATP toeject H+ through the ATP-synthase. As protons are being ejected rather than admitted, the

mitochondrion enters a state where it relies solely on the TCA cycle for ATP production.The cell cannot live in this state of limbo for an extended period of time, because themitochondrion is now an ineffectual a calcium store and calcium cytotoxicity will ensue.Though we point out that, in this model, osmotic effects of the pore popping are neglectedand the mitochondria would likely swell and burst with the pore in its popped state. Themodel of [22] tracks the K+ and Na+ fluxes which would be involved in swelling.

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t

t

t

t

t

CCa

PTPH

t

MCa

pH

0

10

20

30

40

0

2

4

6

8

7.4

7.45

7.5

7.55

7.6

7.65

7.7

0

0.2

0.4

0.6

0.8

0

20

40

60

80

100

120

140

160

0

0.2

0.4

0.6

0 1000 2000 3000 4000 50000 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 50000 1000 2000 3000 4000 5000

MCa*

*

t1

t2

t1

t2

t2

t2

t2

Figure 18: Calcium is continually infused into the cytosol at a sufficiently slow rate so that the pH remains at homeostasis.In turn, the mitochondrial calcium load increases and eventually exceeds the threshold value of 4m at time t1 1800s,activating the secondary process necessary for PTPh. Approximately 1700s later, the pore opens into PTPhat time t2 3500.The electric potential and the mitochondrial Ca2+ plummet at time t = t2, and the cytostolic Ca2+ becomes incredibly elevated.

Discussion

As one reviews the modeling literature related to mitochondria and Ca 2+ handling and thesupporting experimental work, several themes emerge. The first is that highly simplifiedphenomenological models are unlikely to be sufficient as a starting point. This is because ofthe complexity of mitochondrial physiology and the fact that coupling between the ER andmitochondria appears to be significant. Second, we have to rely on modular formulations bydefault. The myriad mechanisms involved have been characterized in isolation with mito-chondria from different sources. Mitochondrial function almost certainly varies dramaticallybetween cell types, which is an unavoidable source of problems for interpretation. If we do

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use a modular approach, then the parameter space is vast and the system is highly complexand difficult to understand; in particular, because we are interested in nonlinear phenomenalike oscillations and waves which would be lost in linearized formulations. Therefore ourapproach is to start with the complete and accepted modular model available (Magnus andKeizer), to add the additional modules necessary for exploring wave propagation, and thento reduce the model as far as we can in order to determine the necessary components.

At this stage, our model allows us to capture the novel behavior of the experimentally ob-served mitochondria driven Ca2+ and potential waves in culture. Our work is a steppingstone to understanding a myriad of issues pertaining to in vivo propagation of Ca2+ signals.As mentioned previously, the study of Ca2+ dynamics has centered upon the trafficking ofCa2+ between the ER and the cytosol. We aim to link our simplified model of mitochondria-cytosol interaction to the Li-Rinzel ER-cytosol model [11] to study the interaction betweenthese two excitable media. We postulate that the mitochondria network will act in a mannerthat will amplify or dampen calcium signals depending upon its strength. For simplicity

this could be done in point model then extended to the spatial case. Since the mitochondiadistributions vary depending upon cell type, we intend to explore the effects of hetero-geneous spatial distributions of the mitochondrial network interacting with the ER. Thespatial interplay of the ER and the mitochondria will likely uncover traveling waves thathave non-constant wavespeeds, that is, it is a likely arena where a lurching wave exists.

While much remains elusive, it is clear is that the MPT is implicated in numerous diseases,(e.g., Ca2+ dysregulation is implicated in human health problems including diabetes, cardiacdisease and Parkinsons neurodegeneration [2]. This permeability transition is the nexus ofa large experimental effort principally because it appears that permanent activation of thetransition and subsequent depolarization of mitochondria are an essential initiating step

to most non-receptor mediated programmed cell death. At this point we have considereda model for the pore-popping event whereby the pore opens to a high-conductance stateand remains open, ultimately leading to apoptosis or necrosis due to calcium cytotoxicityor bursting due to osmotic effects. This model displays pore popping behavior under thecondition of high mitochondrial calcium load. The current model is phenomenological innature, as such a more detailed model that describes the pore forming process would beof value. It would also be intriguing to study the roles of other factors associated withMPT: reactive oxygen species, altered membrane lipids, electron transport chain anomaliesand altered gene expression. Computational modeling of mitochondrial Ca2+ dynamics maytherefore aid in the understanding of how to tip the balance of cell death towards accelerationin the case of cancer therapies or towards protection in the case of neurodegenerative diseases.

Acknowledgments

Funding was provided by NSF DMS 0718558 and NIH MH-64611 to CPF, and NSF 0514356to DT. The authors thank the Mathematical Biosciences Institute at The Ohio State Uni-versity for hosting much of this project under NSF agreement 0112050. CPF is grateful forthe support of Joel Keizer and the discussions with him that informed much of this work.

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Computational Methods

Point model calculations

The numerics for the point model were predominately carried out using the software packageXPPAUT developed by Bard Ermentrout.

Spatiotemporal caculations

We numerically simulate the system of partial differential equations with the method of linesand an adaptive step algorithm using the software package MATLAB. The model containsa wide range of scales making the problem numerically sensitive, that is, stiff. As such, thespatial calculations are computationally expensive and time consuming.

Ca2+ Stimulation

In this work, we are principally concerned with two phenomena: the transient pore flickeringevents and the permanent pore popping events. In order to induce that behavior we considertwo distinct Ca2+ stimulation protocols: (1) rapidly injecting pulses of Ca2+ to the cytosolin order to evoke a flickering of the PTP and (2) a slow infusion of Ca2+ that eventuallyleads to the pore popping that is, opening to its high conductance state. This is meantas an approximation of an ever increasing calcium load which could in part be due to adecrease in the cells buffering capabilities (whether by the ER or the other mitochondria).

Mathematically, we represent the stimulation procedures (infusion and pulses) using thefollowing current:

JCastim = Ainf(t inf) + ApulseN

j=1

[(t tj)(t tj pulse)] (16)

where represents the Heaviside function; that is, (t) = 0 if t < 0 and (t) = 1 ift > 0.Note that a baseline level of calcium, Ainf, is infused for inf seconds. Pulses of amplitudeApulse and duration pulse are applied at times tj .

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Nucleotide conservation

NAD NAD = 8 dmito NADHMmitochondrial ATP ATPM =

12 dmito ADPM

cytostolic ATP ATPC = 2ADPCmitochondrial free ATP ADPM,free = 0.8ADPMcytostolic free ATP ADPC,free = 0.3ADPCcharged nucleotides ADP3(C,M) = 0.45ADP(C,M),freecharged nucleotides MgADPC = 0.55ADPC,freecharged nucleotides ATP4(C,M) = 0.05ATP(C,M)

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REFERENCES MBI Technical Report: work in progress REFERENCES

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