A Model of Mitochonrial Calcium Induced Calcium

Home , Proton pump

A MODEL OF MITOCHONRIAL CALCIUM INDUCED

CALCIUM RELEASE

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By Balbir Thomas, The Ohio State University

2007

Dissertation Committee: Approved by

David Terman, Adviser

Douglas R. Pfeiffer Adviser

Edward Overman Biophysics Graduate Program

Christopher P. Fall ABSTRACT

Cytoplasmic calcium plays a dual role in cellular physiology. On one hand it acts as a second messenger in intra-cellular signalling, and on the other hand it is also the trigger for calcium dependent apopotosis. A mechanistic explanation of this dual role of cytoplasmic calcium was proposed by Ichas and Mazat. Their hypothesis involved the permeability transition pore was based on the observation that the permeability transition pore can exist in multiple conductance states. Speciﬁcally there exist a persistent high conductance state and a transitory low conductance state. Ichas et.al. also observed that the low conductance state is opened by a rise in mitochondrial matrix pH, in contrast to what was already know about the high conductance state, which opens in response to prolonged elevation of mitochondrial calcium. In this dissertation we build a detailed, physiological model of the mitochondrial switch between calcium signalling and cell death based on a simple three state model of the permeability transition pore. This model agrees with the substance of the Ichas and Mazat hypothesis and provides a substrate for further modeling to study the spatial and temporal dynamics of mitochondrial involvement in intracellular calcium signalling, and the interaction of mitochondria and endoplasmic reticulum during this process.

ii °c Copyright by

Balbir Thomas

2007 This dissertation is dedicated to Mr. Tajinder Singh

iv ACKNOWLEDGMENTS

I am deeply obliged to Dr. David Terman and the Biophysics graduate program for having given me the oppertunity and support in conducting this research. During this process I have had many fruitfull disccusssions with Dr. Douglas R. Pfeiffer, Dr. Edward

Overman, Dr. Christopher P. Fall Dr. Abdoul Kane and Dr. Andrew Oster. I am also grateful to Cindy Bernlohr for her advise and help on many adminstrative issues. Much of the production of this dissertation, owes a debt to the multitude of open source software developers, who have made my task, considerably easier. I am also indebted to my mother for her paitence during the long years of my absence and to Marta Wojciechowska for her continual encouragement and support.

v VITA

May14,1970 ...... Born-Baroda, India

1990 ...... B.Sc.(Hons.), Human Biology

1992 ...... M.Sc.,Biophysics

1993-1994 ...... Technical services executive,

Industrial enzyme division, Biocon,

India 1995-1997 ...... JuniorResearch Fellow,

Council of Scientiﬁc and Industrial Re- search, India October 1997 - August 1998 ...... Software Developer and Trainer, Wintech Computers,

India May1998-August1998 ...... AssistantEditor, Software Review Magazine,

India August1998-present ...... GraduateAssociate,

Biophysics program, Ohio State University,

U.S.A vi PUBLICATIONS

Instructional Publications

B. Thomas, “The Markup Language Primer : SGML, XML and HTML” Software Review Magazine, August, 1998.

B. Thomas “Modulo Arithematic and Public Key Cryptography” Software Review Maga- zine, August, 1998.

FIELDS OF STUDY

Major Field: Biophysics

Studies in:

Mathematical Biology Prof. David Terman Numerical Analysis Prof. Edward Overman

Mitochondrial Physiology Prof. Douglas R. Pfeiffer

vii TABLE OF CONTENTS

Page

Abstract...... ii

Dedication...... iv

Acknowledgments...... v

Vita ...... vi

ListofFigures...... xi

Chapters:

1. INTRODUCTION ...... 1

2. BACKGROUND ...... 3

2.1 Introduction...... 3 2.2 Generation of reducing equivalents ...... 5 2.3 Adenine Nucleotide Transporter ...... 10 2.4 CalciumUniporter ...... 10 2.5 Sodium-CalciumExchanger ...... 13 2.6 IP3Receptorchannel...... 13 2.7 SERCApump ...... 14 2.8 SummaryofODEs...... 15

3. MITOCHONDRIAL PROTON CURRENTS ...... 18

3.1 Introduction...... 18 3.2 pH sensitive Proton currents ...... 20

viii 3.2.1 A pH sensitive respiratory Proton Pump ...... 20

3.2.2 A pH sensitive F0-F1 ATPase ...... 23 3.3 Steadystaterateequations ...... 25 3.3.1 Respiratoryrates...... 26 3.3.2 Oxidative phosphorylation rates ...... 27 3.3.3 CycleFluxes...... 27 3.3.4 CycleForces...... 29 3.3.5 RateConstants...... 45 3.4 Electro-neutralweakacidﬂux ...... 49

4. PERMIABILITYTRANSITIONPORE ...... 51

4.1 Introduction...... 51 4.2 Threestatemodel...... 52

5. PORECURRENTS...... 55

5.1 Introduction...... 55 5.2 Calcium current through the pore ...... 56 5.3 Proton current through the pore ...... 57

6. SIMPLIFICATION ...... 58

6.1 introduction...... 58 6.2 Phenomenological rate equations ...... 58

7. RESULTS...... 61

7.1 pH dependent respiratory ﬂuxes ...... 61 7.2 Simpliﬁcation...... 65 7.3 Mitochondrial response to cytoplasmic calcium ...... 65 7.4 BehaviourofthePTP...... 82 7.4.1 Low conductance pore ...... 82 7.4.2 High conductance pore ...... 85

8. DISCUSSION...... 89

Appendices:

A. PARTIALDIAGRAMS...... 91

ix B. UNSIMPLIFIEDTERMSINRATEEXPRESSIONS ...... 92

C. MODELPARAMETERS...... 100

C.1 Thermodynamicconstants ...... 100 C.2 ConversionConstants ...... 100 C.3 Compartmentalization parameters ...... 103 C.4 Metabolicparameters...... 103 C.5 Rateconstants(ProtonPump) ...... 103 C.6 Rateconstants(ATPase) ...... 104 C.7 Electroneutralweakacidparameters ...... 104 C.8 Permiability transistion pore parameters ...... 105 C.9 IP3receptorandleakageparameters ...... 105 C.10SERCApumpparameters ...... 106 C.11Inputs...... 106

D. MODELEQUATIONS ...... 107

D.1 Compartmental equations ...... 108 D.2 Nucleotideconservation ...... 109 D.3 Cytosolic components ...... 110 D.4 Mitochondrial components ...... 111 D.5 Endoplasmic Reticulum components ...... 113 D.6 DifferentialEquations ...... 114 D.7 Initialconditions ...... 115

Bibliography ...... 116

x LIST OF FIGURES

Figure Page

2.1 Components of the MCICR model ( IP3R = IP3 Receptor, SERCA = SERCA Pump, PTP = Permeability Transition Pore, UNI = Calcium uniporter, NaCaE = Sodium Calcium exchanger, ETC = Electron transport Chain,

F0F1 = F0-F1 ATPase)...... 6

2.2 Kinetic diagram of Adenine Nucleotide Transporter ...... 11

2.3 Allosteric diagram of Calcium Uniporter ...... 12

2.4 Kinetic diagram of Sodium-Calcium exchanger ...... 13

3.1 Kinetic diagram of hypothetical proton pump ...... 21

3.2 Possible cycles in the six-state kinetic diagram ...... 22

3.3 Kinetic diagram of hypothetical F0-F1 pump...... 24

3.4 Surface and phase boundary potentials of inner mitochondrial membrane . . 36

4.1 Three state permeability transition pore ...... 52

7.1 Comparison of pH dependent Jhres ﬂux with that of Magnus and Keizer . . 61

7.2 Comparison of pH dependent Jhres ﬂux with that of Magnus and Keizer . . 62

7.3 Comparison of pH dependent Jhf1 ﬂux with that of Magnus and Keizer . . 62

7.4 Comparison of pH dependent Jhf1 ﬂux with that of Magnus and Keizer . . 63

xi 7.5 Comparison of pH dependent Jo ﬂux with that of Magnus and Keizer . . . 63

7.6 Comparison of pH dependent Jo ﬂux with that of Magnus and Keizer . . . 64

7.7 Comparison of pH dependent Jpf1 ﬂux with that of Magnus and Keizer . . 64

7.8 Comparison of pH dependent Jpf1 ﬂux with that of Magnus and Keizer . . 65

7.9 Jhres in simpliﬁed and unsimpliﬁed forms ...... 66

7.10 Jhres in simpliﬁed and unsimpliﬁed forms ...... 67

7.11 Jhres in simpliﬁed and unsimpliﬁed forms ...... 68

7.12 Jo in simpliﬁed and unsimpliﬁed forms ...... 69

7.13 Jo in simpliﬁed and unsimpliﬁed forms ...... 70

7.14 Jo in simpliﬁed and unsimpliﬁed forms ...... 71

7.15 Jhf1 in simpliﬁed and unsimpliﬁed forms ...... 72

7.16 Jhf1 in simpliﬁed and unsimpliﬁed forms ...... 73

7.17 Jhf1 in simpliﬁed and unsimpliﬁed forms ...... 74

7.18 Jpf1 in simpliﬁed and unsimpliﬁed forms ...... 75

7.19 Jpf1 in simpliﬁed and unsimpliﬁed forms ...... 76

7.20 Jpf1 in simpliﬁed and unsimpliﬁed forms ...... 77

7.21 Change in mitochondrial pH after a cyptoplasmic calcium pulse at t=250 . . 78

7.22 Increase in proton pumping by ETC in response to elevated matrix calcium 79

7.23 Elevated baseline cytoplasmic calcium (without ER) ...... 79

7.24 Elevated baseline mitochondrial calcium (without ER) ...... 80

7.25 Mitochondrial calcium response in presence of ER ...... 80

xii 7.26 Mitochondrial pH response in presence of ER ...... 81

7.27 CalciumuptakebyER ...... 81

7.28 Response of the PTP to fast cytoplasmic calcium pluses in the presence of theEndoplasmicReticulum...... 83

7.29 Response of the PTP to fast cytoplasmic calcium pluses in the presence of theEndoplasmicReticulum...... 84

7.30 Response of the PTP to slow buildup of cytoplasmic calcium in the absence oftheER ...... 85

7.31 Response of the PTP to slow buildup of cytoplasmic calcium in the pres- enceoftheER...... 86

7.32 Mitochondrial calcium induced calcium release ...... 87

A.1 Partial Diagrams for a six state mechanism ...... 91

xiii CHAPTER 1

INTRODUCTION

Ionised Calcium is one of the most common “second messengers” in intra-cellular sig- nal transduction [7]. Not only does it play a vital role in the normal physiology of the cell, but is also capable of initiating programmed cell death (Apoptosis), under conditions of prolonged elevation. Hence its intra-cellular concentrations are tightly regulated by compartmentalization and buffering. The endoplasmic reticulum and mitochondria are the two principal organelles that sequester calcium. The role of endoplasmic reticulum in calcium signalling has long been recognised. However it has become increasingly clear that the mitochondria play an active part in modulating cytoplasmic calcium signalling [32, 31, 28, 21, 19, 20]. Information is encoded in these calcium signals in a frequency modulated manner, both spatially and temporally [66]. Calcium waves are generated in the cytoplasm by Inosito-tri-Phosphate (IP3) mediated stimulation of the endoplasmic reticulum. The characteristics of these waves are seen to be modulated by the spatial distribution of mitochondria [63] and their physiological state.

In explaining cytoplasmic calcium’s dual role as a messenger in intracellular calcium signalling, and the trigger for calcium dependent apopotosis Ichas and Mazat proposed a simple hypothesis involving the permeability transition pore [30]. This hypothesis was

1 based on the observation that the permeability transition pore can exist in multiple conductance states. Speciﬁcally there exist a persistent high conductance state and a transitory low conductance state [69]. Ichas et.al. also observed that the low conductance state is opened by a rise in mitochondrial matrix pH [28], in contrast to what was already know about the high conductance state, which opens in response to prolonged elevation of mitochondrial calcium.

In this dissertation we build a detailed, physiological model of the mitochondrial switch between calcium signalling and cell death based on a simple three state model of the permeability transition pore. This model agrees with the substance of the Ichas and Mazat hypothesis and provides a substrate for further modeling to study the spatial and temporal dynamics of mitochondrial involvement in intracellular calcium signalling.

2 CHAPTER 2

BACKGROUND

2.1 Introduction

The principal function of mitochondria is the generation of adenosine triphospate (ATP)

using the free energy from oxidation of nicotinamide adenine dinucleotide (NADH) and

ﬂavin adenine dinucleotide (FADH). This free energy is stored in the form of a trans-

membrane electro-chemical potential, across the inner mitochondrial membrane, that is

generated by pumping protons out of the mitochondrial matrix. However the free energy of

the proton gradient is also dissipated by active transport and calcium signaling [64, 35, 32,

29, 42] and the mitochondria are now recognised to be important effectors in intra-cellular

calcium signalling [32, 29, 42, 59, 58, 60, 10]. The permeability transition pore (PTP) and

the calcium uniporter play important roles in this process [27, 25, 26, 22, 35]. Calcium en-

ters the mitochondria chieﬂy through the calcium uniporter, as indicated by the ruthenium

red sensitivity of the process [18, 1]. And it leaves the mitochondria, by a Cyclosporin-

A sensitive pathway, which is a potent inhibitor of the permeability transition pore [1].

These observations led to the idea that under normal physiological conditions, mitochon-

dria, much like the endoplasmic reticulum can be involved in calcium induced calcium

3 release (mCICR) [29, 28, 9, 24]. However it was known for a long time that calcium in-

duced opening of the permeability transition pore was cytotoxic [61]. In order to explain

how cellular calcium could effect two different outcomes, Ichas and Mazat proposed, that

it was the rate of calcium entry into the mitochondria that determined the behaviour of the

permeability transition pore [30, 28]. As per their proposal rapid calcium entry depolarised

the mitochondria, and hence resulted in enhanced respiratory proton ejection. This lead

to an alkalanisation of the mitochondrial matrix. High matrix pH induces an open state

of the permeability transition pore but with low conductance (< 500 pS and < 300 Da

cut-off) [3, 65, 30]. This low conductance state of the pore equilibrates small ions such

as, proton and calcium across the inner mitochndrial membrane and consequently lowers

matrix pH. Lower matrix pH closes the low conductance pore and the whole process may

potentially repeat itself leading to oscillatory mitochondrial calcium induced calcium re-

lease. This process is often termed as “pore ﬂickering”. On the other hand if calcium entry

into the mitochondria is sufﬁciently slow, gradual alkalanisation of the matrix may be off-

set by an equally slow electroneutral redistribution of weak acids. Hence the pore does not

open in its low conductance state. This leads to calcium buildup in the matrix. High levels

of mitochondrial calcium reversibly induce an open state with high conductance (< 1200

pS and < 1500 Da cut-off), in the pore [27, 25, 26, 22]. This leads to cytotoxic effects and mitochondrial apoptosis [33] and the pore is said to have “popped”.

In order to obtain a physiological model of the two modes of mitochondria’s calcium response, it is necessary to quantitatively describe calcium and proton ﬂuxes that control this process and construct a model of the permeability transition pore that shows both pH and calcium gating. Figure 2.1 depicts all those aspects of the cellular machinery that is

4 relavent. This involves the mitochondrial, endoplasmic reticulum and cytoplasmic compartments. IP3 induces the oscillatory release of calcium from the endoplasmic reticulum via the IP3 receptor. It is subsequently taken up by the mitochondria (via the calcium uniporter) and the endoplasmic reticulum (via the SERCA pump). This process is relatively fast and results in mitochondrial calcium induced calcium release, by opening the pore in the low conductance state. Such “flickering” can only occur in the presence of energized mitochondrial that actively maintain and proton gradient and electrical potential across their inner membrane [14, 37]. The energised state of the mitochondria is maintained by the electron transport chain which uses the free energy of NADH and FADH oxidation to pump protons out of the mitochondrial matrix. Also involved in mitochondrial pH homeostasis are the Fo − F1 ATPase which phosphorylates ADP by dissipating the proton gradient, and a small non specific membrane proton leakage. Further transfer of charged species across the mitochondrial membrane also affects the potential across it. While there are potentially many such electrogenic fluxes, the exchange of ADP and ATP plays and important role and is key to normal mitochondrial function in cellular physiology. Reducing equivalents

(NADH and FADH) that are fed to the electron transport chain are produced by Glycolysis in the cytosol and by the Tri-Carboxylic Acid cycle (TCA) in the matrix.On the other hand any other small extraneous source of calcium release in the endoplasmic reticulum that causes a slow build up of cytoplasmic calcium leads to “popping” of the pore.

2.2 Generation of reducing equivalents

Reducing equivalents used by the electron transport chain (ETC) are generated by glycolysis and the TCA cycle. These are in the form of NADH and FADH2. The former is oxidised by complex I of the ETC and the latter by complex II. In addition the glycerol

5 2+ Ca UNI

F0F1 IP3R

+ 2+ Ca H PTP + TCA H

SERCA Endoplasmic Reticulum

ETC Mitochondria NCaE

2+ Ca

Cytoplasm

Figure 2.1: Components of the MCICR model ( IP3R = IP3 Receptor, SERCA = SERCA Pump, PTP = Permeability Transition Pore, UNI = Calcium uniporter, NaCaE = Sodium Calcium exchanger, ETC = Electron transport Chain, F0F1 = F0-F1 ATPase )

6 phosphate shunt transfers reducing equivalents to complex II. The rates of these processes

have been previously measured [6]. Based on these measurements Magnus and Keizer [46]

were able to formulate expression for the rate of reducing equivalent generation by all the

three processes. They expressed these rates in terms of the rate of D-glucose utilization.In

doing so they assumed quasi steady state conditions for D-glucose metabolism, disregard-

ing any complication due to glycolytic oscillations.

Each molecule of pyruvate on conversion to acetyle co-enzyme A yeilds one molecule of NADH and each molecule of acetyle co-enzyme A generates 3 molecules of NADH on transiting the TCA cycle. Since each molecule of D-glucose produces 2 molecules of pyruvate, after undergoing glycolysis

∆Jred,I = 2∆JPDH + 6∆JTCA (2.1)

Where Jred,I is the rate at which reducing equivalents of NADH are presented at complex I, ∆JPDH is the rate of conversion of pyruvate to acetyl co-enzyme A (the primary substrate for the TCA cycle) and JTCA is rate of the TCA cycle. The ∆ is used to indicate

substrate induced rather than basal rates. It must be noted that ∆JPDH is effectively the

rate of D-glucose utilisation as glycolysis is an unbranched pathway (effect of the glycerol

phosphate shunt being considered separately)

Boscheror et. al. [6] showed that ∆JTCA was about 41-43% of ∆JPDH for varying

levels of D-glucose utilisation. Hence

∆Jred,I =(2+6 · 0.42)∆JPDH = 4.52∆JPDH (2.2)

7 Thus the rate of NADH presentation at complex I is completely expressed in terms of the rate of D-glucose utilisation. Likewise the rate at which reducing equivalents (FMNH2,

FADH2) are presented to complex II (∆Jred,II ) are expressed in terms of the rate of succinate dehydrogenase activity (∆JSDH ) and the rate of the glycerol phosphate shunt.

∆Jred,II = ∆JSDH + ∆Jshunt (2.3)

Since a FMNH2 from the glycerol phosphate shunt and another FADH2 from succinate dehydrogenase (TCA cycle) are presented to complex II. Further each molecule of glucose that is metabolised by the glycerol phosphate shunt is ﬁrst split into 2 molecules of

D-glyceraldehyde-3-phosphate and then oxidised by glyceraldehydephosphate dehydrogenase which produces 2 NADH molecules. Magnus and Keizer [46] assume that at steady state NADH produced in the cytosol either reduces pyryvate to lactate (anaerobic glycolysis) or enters the glycerol phosphate shunt. Hence

∆Jshunt = 2∆Jgly,total − 2∆Jgly,anaerobic (2.4)

where ∆Jgly,total is the rate of glycolysis and ∆Jgly,anaerobic is the rate of lactate dehydrogenase activity. Magnus and Keizer also assume that all D-glucose is metabolised either to pyruvate (and enters the TCA cycle after oxidation by pyruvate dehydrogenase) or to lactate by lactate dehydrogenase (anaerobic glycolysis) . Hence

∆Jgly,total = ∆Jgly,anaerobic + ∆JPDH (2.5)

8 Expressing the succinate dehydrogenase rate in terms of they TCA cycle rate as ∆JSDH =

2∆JTCA (assuming not loss of pyruvate) and rewriting ∆JTCA in terms of ∆JPDH , which

yeilds

∆Jred,II = 2.84∆JPDH (2.6)

The experimental results of Boscheror et. al showed that ∆JPDH increases from about

0.28∆Jgly,total to about 0.48∆Jgly,total for an increase of glucose concentration ([Glc]) from

2.8 mM to 16.7 mM. Magnus and Keizer proposed that this increase was due to calcium activation of pyruvate dehydrogenase and expressed it as a function “f([Glc])” of D-glucose concentration.

1 f([Glc]) = (2.7) −2 [Ca2+]m 1+ u2 1+ u1 1+ KCa2+ · ³ ´ ¸ They could then express the pyruvate dehydrogenase rate as

∆JPDH = f([Glc])∆Jgly,total (2.8)

Hence yeilding

∆Jred,I = 4.52f([Glc])∆Jgly,total (2.9)

and

∆Jred,II = 2.84f([Glc])∆Jgly,total (2.10)

9 Finally, based on the observation that D-glucose transport into cells is not the rate lim-

iting step of glycolosis [48, 49] and that the in vivo kinetic properties of glucokinase are

similar to those of D-glucose utilisation, Magnus and Keizer equated the rate of glucoki-

nase [13] with the rate of glycolosis obtaining the expression

βmax (1 + β1[Glc]) [Glc][AT P ]i ∆Jgly,total = 2 1+ β3[AT P ]i +(1+ β4[AT P ]i) β5[Glc]+(1+ β6[AT P ]i) β7[Glc] (2.11)

2.3 Adenine Nucleotide Transporter

The adenine nucleotide transporter plays a key role in cellular metabolism in the electrogenic exchange of ADP 3− and AT P 4− across the inner mitochondrial membrane. A mechanism for this process was proposed by by Bohnensack [5]. In this mechanism sites on opposite sides of the mitochondrial membrane bind ADP 3− and AT P 4−, and only after

both sites are bound the carrier (ANT) isomerises to exchange the position of the bind-

ing sites. The kinetic diagram of this scheme is show in ﬁgure 2.2. Based on this kinetic

scheme Bohnensack also obtained the following a rate law for exchange of nucleotides by

the ANT

− − [ATP 4 ]i[ADP 3 ]m −F ∆Ψ − − − 1 [ADP 3 ]i[ATP 4 ]m exp RT JANT = Jmax,ANT − − (2.12) [ATP 4 ]i −fF ∆Ψ [ADP 3 ]m − ¡ ¢ − 1+ [ADP 3 ]i exp RT 1+ [ATP 4 ]m h ¡ ¢i h i

10 4 5 ADPm ATPm

ADPi 1 ATPi

ADPi ATPi

ATPm ADPm 2 3

Figure 2.2: Kinetic diagram of Adenine Nucleotide Transporter

2.4 Calcium Uniporter

The calcium uniporter is one of the two dominant mechanisms by which calcium tran- sits the mitochondrial membrane [16, 15, 17]. The other being the sodium-calcium exchanger. Magnus and Keizer proposed an allosteric mechanism for the calcium transport, that takes into account the dependence on cytoplasmic calcium concentration and inner membrane potential. This mechanism was based on the observation that the uniporter showed a Hill coefﬁcient in the range 2 to 3.5 and that the uniporter has two binding sites for calcium, one which “activates” it and another that is the transport site [39, 38, 41, 40].

This mechanism is illustrated in ﬁgure 2.3

The Magnus and Keizer rate law for the calcium uniporter is a function of the maximal rate of transport (Jmax,uni), fractional saturation of the transport site, the mitochondrial

11 2+ Ca i L K trans

2+ 2+ 2+ Ca i 2+ Ca i Ca Ca i i 2+ Ca i K act Ca Ca Ca Ca

K trans L Relaxed Taut L

Figure 2.3: Allosteric diagram of Calcium Uniporter

membrane potential and follows from Monod-Wyman-Changeux theory of allosteric regulation [52].

3 ∗ [Ca2+]i [Ca2+]i 2F (∆Ψ−∆Ψ ) Ktrans 1+ Ktrans RT J = J (2.13) uni max,uni 4 ∗ [Ca2+]i ³ L ´ −2F (∆Ψ−∆Ψ ) 1+ + na 1 − exp Ktrans [Ca2+]i RT 1+ K ³ ´ „ act « h ³ ´i

A problem with the Magnus and Keizer uniporter model was that it would allow for unbounded uptake of cytosolic calcium. Fall and Keizer [10] were able to correct this and obtain the following expression

12 ∗ 2+ 2F (∆Ψ−∆Ψ∗)/RT 2F (∆Ψ − ∆Ψ ) MWC − [Ca ]m exp J = J ∗ (2.14) uni max,uni RT 1 − exp2F (∆Ψ−∆Ψ )/RT · ¸ · ¸

where

3 [Ca2+]i [Ca2+]i Ktrans 1+ Ktrans MWC = 4 (2.15) ³ [Ca2+]´i ³ L ´ 1+ + na Ktrans [Ca2+]i 1+ K ³ ´ „ act « 2.5 Sodium-Calcium Exchanger

Magnus and Keizer proposed a three state model for the sodium-calcium uniporter that

is in ﬁgure 2.4. The exchanger is assumed to operate in an electroneutral manner exchang-

ing two sodium ions for calcium ion transported to the extra-mitochondrial space. They

further assumed the binding and unbinding equilibriums of sodium and calcium with the

exchanger to be rapid. Based on these assumptions and the fact that calcium uptake by

the sodium-calcium exchanger is not observed experimentally (implying that the rate of

transition from state 1 to state 3 in the kinetic diagram is zero), they were able to obtain the

following rate law for calcium eﬂux by the exchanger

bF (∆Ψ−∆Ψ∗) exp RT JNa+/Ca2+ = Jmax,Na+/Ca2+ n (2.16) KNa³ K´na 1+ [Na+]i 1+ [Ca2+]m ³ ´ ³ ´ 2.6 IP3 Receptor channel

The endoplasmic IP3 receptor is a calcium sensitive calcium release channel that is

shown to have a biphasic response to cytoplasmic calcium concentrations [12, 4, 68]. This

13 2 3

+ + 2+ 2+ Ca + 2 Na Ca + 2 Na 1 i m m i

Figure 2.4: Kinetic diagram of Sodium-Calcium exchanger

channel consists of four identical subunits [11, 50] and it can open with four distinct conductance levels [67]. De Young and Keizer were able to model this channel by assuming that it consists of three identical subunits, each of which had three binding sites – one for

IP3, one calcium binding site for calcium activation and another calcium binding site for calcium transport [8]. Their model was able to reproduce all the key observations on IP3 receptor equilibrium kinetics and show cytoplasmic calcium oscillations. The model however was elaborate consisting of nine variables. Li and Rinzel were able to subsequently simplify the De Young - Keizer model to a two variable model by analysing the different time scales of the three distinct processes of IP3 binding, calcium activation and inactiva- tion [44]. These equations in the form reported by Fall and Keizer [10] are

[IP 3] 3 [Ca2+] 3 J = J i h3 + J [Ca2+] − [Ca2+] erout max,IP 3 [IP 3] + d [Ca2+] + d leak ER i " µ IP 3 ¶ µ i ACT ¶ # ¡ (2.17) ¢

14 2.7 SERCA pump

Along with their model of the IP3 receptor channel De Young and Keizer also utilised

the following expression for the rate of inward pumping of calcium (into ER) by the ATP

dependent SERCA pump [8]

[Ca2+] J = J i (2.18) SERCA max,SERCA k2 + [Ca2+] · SERCA i ¸

2.8 Summary of ODEs

Calcium enters the cytoplasm from the endoplasmic reticulum (Jerout) and from the mitochondria (via the sodium-calcium exchanger Jnc). On the other hand calcium in the cytoplasm is taken up by the endoplasmic reticulum (via the SERCA pump Jserca) and the mitochondria (via the uniporter Juni). In addition to provide for pathological build up

(slow) of mitochondrial calcium a parameter JCa,in has been incorporated. Under normal physiological conditions the value of this parameter is zero. To model gradual elevation of cytoplasmic calcium, the parameters needs to be set to a small posative value.

dCAC f [M (J − J ) − E (J − J )+ J M + J ] = i nc uni serca erout ptplca cain (2.19) dt Vc minute

dH d − (CAC + d ) H = inh inh (2.20) dt τ

Balance of calcium in the endoplasmic reticulum is determined by two currents, the

SERCA pump (inward) and the IP3 receptor (outward, Jerout).

15 dCAER f [E (J − J )] = i serca erout (2.21) dt Ve minute

Calcium enters the mitochondria via the uniporter and leaves though the sodium-calcium

exchanger and the permeability transition pore (Jptplca)

dCAM f [J − J − J ] M = m uni nc ptplca (2.22) dt Vm minute

Electrogenic ﬂuxes that determine mitochondrial membrane potential are, respiratory proton pumping (Jhres), ATPase (Jhf1), adinine nucleotide transporter (Jant), proton leak

(Jhl), calcium uniporter (Juni), proton and calcium ﬂuxes through the pore (respectively

Jptplh, and Jptplca).

d∆Ψ [−J + J + J + J − J + 2J − 2J ] M = − hres hf1 ant hl ptplh uni ptplca (2.23) dt Cmito minute

Mitochondrial NADH balance is determined by the rate of NADH oxidation (Jo) and the rate of NAD reduction (Jred)

dNADH [J − J ] M m = red o (2.24) dt µ Vm minute

Mitochondrial ADP concentrations are determined by the rate of entry (Jant), the rate of oxidative phosphorylation (Jpf1) and the rate of substrate level phosphorylation (Jptca).

dADP [J − J − J ] M m = ant ptca pf1 (2.25) dt µ Vm minute

16 Cytoplasmic ADP concentrations are determined by the rate of entry (Jant), the rate of cytosolic ATP hydrolysis (Jpgly) during glycolysis

dADP [−J M +(J − J ) C] i = ant hyd pgly (2.26) dt µ Vc minute

Mitochondrial proton concentrations are determined by the rate of uptake by ATPase

(inward, Jhf1), the proton leak (inward, Jhl), the rate of respiratory pumping (outward,

Jhres), the weak acid ﬂux (inward, Jah), and the rate of entry via the pore (inward, Jptplh)

dH Hm [J + J − J + J − J ] m = r hf1 hl hres ah ptplh (2.27) dt minute

The low and high conductance gates of the permeability transition pore are respectively described as perturbed Hodgkin-Huxley type channels, gated by proton and calcium.

dPTP P T P − [P T P − P T P ] P T P ∞ L = − L T H L (2.28) dt τPTPL

dPTP P T P − [P T P − P T P ] P T P ∞ H = − H T L H (2.29) dt τPTPH

17 CHAPTER 3

MITOCHONDRIAL PROTON CURRENTS

3.1 Introduction

One of the principle functions of the mitochondria is to maintain a proton gradient

across its inner mitochondrial membrane. This transmembrane proton gradient plays a key

role in metabolism. Chemiosmotic theory [51] linked the generation of transmembrane pro-

ton gradients to ADP phosphorylation. It has been recognised that the free energy of this

gradient is used for other tasks other than oxidative phosphorylation [64]. This dissipation

of free energy leads to incomplete coupling of NADH oxidation and ADP phosphoryla-

tion. Rate laws for chemiosmotic electron transfer and proton ejection, by mitochondrial

inner membrane proteins, have been previously described based on a hypothetical six-state

proton pump model [54]. A modiﬁed version of this model does take into account the in-

complete coupling due to non oxidative dissipation of the proton gradient [54], [55], [56].

The modiﬁed model was successfully employed, by Gerhard Magnus, in quantitatively

elucidating the mechanism of D-Glucose induced bursting of pancreatic β-cell membrane

potential [45], [45], [46], [47]. The Magnus model employed biochemical units, in terms

of which experimental results are frequently cited. It also used the simplifying scheme of a

18 single site for all proton translocations associated with NADH oxidation. This model and

the underlying six state mechanism is taken as the starting point for further development

into pH dependent ﬂuxes. This six state mechanism is already seen to be in good agree-

ment with experimental data and has been tested in conjunction with other components of

the present model.

The central role of the proton gradient in mitochondrial function suggests that fluxes involved in the generation and maintenance of this gradient would be the dominant proton currents across the mitochondrial membrane Apart from respiratory proton efflux, and uptake of protons during oxidative phosphorylation which are the predominant proton flux, across the inner mitochondrial membrane, a small non-linear leakage current is also usually assumed to be another key determinant of mitochondrial proton concentration. However the mitochondrial membrane is known to contain a large family of proton symports or an- tiports that are involved in cyclic transfer of free or bound protons. Although the net result of such transfer may have little effect on the much larger respiratory fluxes, never the less these currents may play a vital role in making mitochondrial pH homeostasis more robust based on a redundancy of fluxes that can compensate small patho-physiological perturbations of the transmembrane gradient, and its generative currents. It is possible that the unidentified “electro-neutral” weak acid flux described by Ichas and Mazat may actually be constituted as the net result of such cyclic transfer of protons, and weak acids.

Clearly the ﬂuxes contributing to the generation and maintenance of chemisosmotic proton gradients are also important determinants of mitochondrial pH homeostasis and are hence involved in regulating the opening/closing of the permeability transition pore. It should be expected that the association/dissociation of protons with the electron transport

19 chain and F0 − F1 ATPase would be sensitive to local proton concentrations in the mitochondrial matrix. None of the previous models take into account the possible variation of respiratory fluxes in response to mitochondrial pH changes. They assume cytosolic and mitochondrial pH to be fixed at 4.2 and 4.6 respectively. Modeling of these pH sensitive respiratory fluxes is described later in this chapter. Apart from these the only other proton currents of relevance, in the Ichas and Mazat hypothesis are the “electro-neutral weak acid flux” and proton conduction by the low conductance state of the pore. The latter is described in chapter 4 and the former is described herein.

3.2 pH sensitive Proton currents

3.2.1 A pH sensitive respiratory Proton Pump

The approach taken here, to model chemiosmotic proton ejection by the mitochondria,

closely follows that of Magnus, Pietrobon and Caplan and uses the method of Altman and

King [34] and Hill [23]. This involves regarding the multi-protein complex known to be

involved in proton ejection, as a hypothetical machine that operates cyclically through six

states. It must be emphasised that though not based on our current understanding of the

molecular mechanisms of proton translocation, the approach has been shown to be in good

agreement with experimental measurements of proton ejection rates [54], [55], [56] and

hence provides an appropriate simpliﬁcation of an otherwise complex process.

The six states of the hypothetical proton pump are shown in ﬁgure 3.1.The vertices of the diagram represent, hypothetical states of the proton pump and its edges are transitions between these states. The edges of the diagram are placed so that all transitions associated with the mitochondrial matrix side are placed in one vertical column and likewise with all

20 transitions on the cytoplasmic (inter-membrane space) space side. Each edge is labeled

by the subscripted (α) unimolecular rate constant for the corresponding transition. The

subscript is an ordered pair representing the starting and ending states of the transition.

1 6

+ + 6 H 6 H Cytoplasm Matrix 2 5 1/2 NADHm + + H

+ 1/2 NAD

Figure 3.1: Kinetic diagram of hypothetical proton pump

As seen in ﬁgure 3.1 a single cycle of the six-state proton pump, passing through states 1 through 6 (counterclockwise) results in the net ejection of 6 protons from the mitochondrial matrix side to the cytoplasmic side using free energy from the oxidation of 1/2 NADH.

This is not the only cycle through with the pump may operate. As shown in ﬁgures 3.2 it can also operate by cycling through states 1-2-5-6, or 2 through 5, which are respectively called the “proton slip” and “reaction slip” cycle ( collectively they are often referred to

21 as “futile cycles” ). The former results in the translocation of protons across the inner mitochondrial membrane without the expenditure of free energy of NADH oxidation and the latter results in the dissipation of this free energy without any proton translocation.

Any of these three cycles can potentially proceed in either the clockwise or anti-clockwise direction under the right conditions. This may be represented in ﬁgure 3.1 by labeling each edge by two unimolecular rate constants corresponding to the two directions. These constants have been previously constrained by experimental measurements by Pietrobon and Caplan and are used here in the form reported by Gerhard Magnus. However some of the unimolecular rate constants, did include a factor corresponding to the matrix or cytoplasmic proton concentrations (assumed to be constants). These were factored out and the proton concentrations are now variables. Hence the unimolecular rate constants are products of bimolecular rate constants and variable concentrations terms (such as that for proton). As a convention the bimolecular rate constants will be written with a superscript asterisk (α∗).

a b

Figure 3.2: Possible cycles in the six-state kinetic diagram

22 The methods of King and Altman in conjuntion with ﬁgure 3.1 is used to construct

the steady state rate expressions of respiratory proton translocation across the mitochon-

drial membrane as a function of mitochondrial pH, mitochondrial NADH concentration

and inner mitochondrial membrane potential. They also yield and expression for the mito-

chondrial Oxygen consumption rate which is related to the rate of respiration implied by

the rate of proton translocation. The details of this construction are elaborated in section 3.3

and the resultant unsimpliﬁed rate expressions are provided in appendix B.

3.2.2 A pH sensitive F0-F1 ATPase

Pietrobon and Caplan also constructed a six state model of the mitochondrial ATPase.

Oxidative phosphorylation ensues when the F0-F1 ATPase moiety represented by the six

state model cycles through these states in the clockwise direction as can be seen in ﬁg-

ure 3.3. However for the sake of consistency with the respiratory proton pump, the coun-

terclockwise direction is taken to be posative. This also ensure that the signs of expressions

involving proton translocations are consistent with their directions. All other conventions

regarding labeling of the rate constants are same as the case of the proton pump except that

the rates constants for the ATPase will henceforth be designated by the letter β. Just as in

the case of the proton pump the F0-F1 ATPase can work bidirectionally and involve “futile”

cycles that result in the dissipation of free energy stored in the proton gradient across the

inner mitochondrial membrane. Because they share the same basic six state mechanism,

obtaining steady state rate equations for the ATPase is analogous to obtaining them for

the proton pump and are detailed in section 3.3 as well. The key difference between the

two expressions are the functions that represent the rate constants. The rate constants in

case of the proton pump are potentially functions of mitochondrial proton concentration

23 (“Hm”), mitochondrial NADH concentration (“NADHm”) and mitochondrial membrane potential(“Ψ”). While the ATPase rate constants are potentially functions of mitochondrial proton concentration, mitochondrial ATP (“ATPm”) concentration (or equivalently, by a conservation condition, mitochondrial ADP concentration - “ADPm”) and mitochondrial membrane potential. No contradiction in terms is implied in describing functions as “rate constants” as they are really pseudo ﬁrst order rate constants that are indeed products of second order rate constants and concentration terms. Further as in the case of the proton pump experimentally constrained values of the second order rate constants were obtained from the work of Pietrobon and Caplan in the units used by Gerhard Magnus. Further details regarding each of these pseudo ﬁrst order rate constants are given in section 3.3.5

1 6

+ + 6 H 6 H Cytoplasm Matrix 2 5 1/2 NADHm + + H

+ 1/2 NAD

Figure 3.3: Kinetic diagram of hypothetical F0-F1 pump

24 3.3 Steady state rate equations

The six state kinetic diagram, of both the proton pump and ATPase, embodies all information required to compute the rate of proton transfer by the former and the rate of ADP phosphorylation by the latter. Typically one would write out differential equations for each of the states of this diagram using fundamental kinetic principles such as mass action kinetics and algebraically or numerically solve them, under steady state conditions to obtain expressions for the “state probabilities” (probability of occupancy of a state). Then the rate of transition from state ‘i’ to state ‘j’ (“transition ﬂux”), would be given as

Jij = rijpi − rjipj (3.1)

where rij is the rate constant for the i to j transition and pi is the state probability of

state i. However the presence of cycles and a moderately large number ( six! ) of states

makes this at best a cumbersome process. Besides what is required, is actually the rate

of proton translocation or ATP synthesis. Both these process are consequences of cyclic

activity of the six state mechanism. Hence in order to compute the steady state rates of these

processes, what is required, is the rate of such cyclic transitions, in which the corresponding

operational ﬂux (proton ejection or ATP synthesis) take place. More particularly since

these ﬂuxes are present in more than one cycle, they result from the sum of those cycle

rates. Such steady state rate equations are more easly obtained by the diagramatic method

of King, Altman and Hill [34], [23] as described below.

25 3.3.1 Respiratory rates

In ﬁgures 3.1 (also see ﬁgure 3.2) only cycles a and c contribute to a net transfer

of electrons through the electron transport chain, hence the rate of respiration (electron

transfer) denoted by “Jres” must be given as

Jres = Cres(Jres,a + Jres,c) (3.2)

where Ji is the rate of counterclockwise cycling of cycle i and Cres is a scaling constant

that is 1 when considering the rate of a single energy conserving site but will involve the

density of such sites, and dimensional constants when expressing Jres in terms of biochem-

ical units. Similarly the rate of proton ejection (“JH,res”)is given by

JH,res = 6Cres(Jres,a + Jres,b) (3.3)

where the factor of six is present because for each molecule of NADH that is oxidised, six protons are translocated. And the rate of oxygen consumption is given by

J 6C (J + J ) J = res = res res,a res,b (3.4) o 2 2

since two electrons need to be transfered to reduce a single Oxygen atom.

In order to complete these equations, all that is required are expressions for the “net

cycle ﬂuxes” (Jres,a, Jres,b, and Jres,c) in terms of the unimolecular rate constants and other

physiological variables in the model.

26 3.3.2 Oxidative phosphorylation rates

From ﬁgures 3.3 and 3.2 the rate of H+ ejection is the sum of the rates of cycle a and b. The stoichiometry of H+ ejection is 3 per ATP hydrolysed. Hence this rate may be expressed as

Jh,F 1 = −3CF 1 (JF 1,a + JF 1,b) (3.5)

where CF 1 is a scaling and dimensional constant.

Keeping with the conventions of the Magnus-Keizer model all differential equations will be written in a form such that, all ﬂuxes are positive for their natural direction ( here clockwise for ATP synthesis and proton uptake) under physiological conditions. Hence the negative sign is required to ensure that the negative (counterclockwise direction) yields a positive ﬂux.

Similarly from the above mentioned ﬁgures the ATP hydrolysis rate is given by the rate of state 3 to state 4 transition is a sum of cycles a and c (in the counterclockwise direction).

Hence an expression for this rate may be written as

Jp,F 1 = −CF 1 (JF 1,a + JF 1,c) (3.6)

3.3.3 Cycle Fluxes

It has been show [34] [23] that for any cycle k the net cycle ﬂuxes Jk can be expressed as

− − Σk J = k+ k (3.7) k Σ ¡Q Q ¢

27 where k+, k−, Σ and Σk are sums of terms that are products of the (psuedo-) first order rate constants.Q Q The procedure for deriving these is know as the diagramatic method of King, Altman and Hill and is as follows. For any given kinetic diagram ‘D’ construct a set of all possible diagrams, taking only a subset of the edges of D, such that, any diagram in this set does not contain a cycle. Further the diagrams must be such that adding any missing edge to any of the diagrams in the set would produce a cycle. Also every diagrams in the set must connect all vertices with at least one edge. The diagrams in this set are know as the “partial diagrams”. The partial diagrams for a six state mechanism are show in appendix A.1. Now, for each vertex ( state ), construct a set of “direction diagrams” by adding arrows to the edges of each of the partial diagrams so that all the arrows “flow” towards the particular vertex. Each direction diagram is a graphical notation for the product of first order rate constants corresponding to the arrows in it. Now is the sum of all the P directional diagrams for all the vertices ( states ). For any given cycle k, k+ and k− are products of the first order rate constants respectively in the positive ( hereinQ counterclock-Q wise) and negative ( clockwise ) direction. To compute k, it is necessary to construct “flux diagrams”. For any given cycle k, flux diagrams areP a set of graphs that include the cycle itself, along with a set of arrow flowing into it, which connect every vertex that is not part of the cycle, with the cycle. These arrows flowing into the cycle represent transition streams that “feed” the cycle, and are algebraically represented by the product of the corresponding first order rate constants. k is the sum of all these “arrows” for cycle k. The flux diagrams themselves are representedP by the product of the rate constants associated with the arrows multiplied by k+ − k− Hence the net cycle flux can equivalently be expressed as Q Q

28 ( sum of all ﬂux diagrams for cycle k ) jk = (3.8) P P 3.3.4 Cycle Forces

Analogous to the concept of force in Newtonian physics, a thermo-chemical ﬂux, such

as a chemical reaction or a transport process, is driven by a conjugate “thermodynamic

force”. This thermodynamic force is nothing but a difference in chemical potential (molar

free energy) that determines the direction of spontaneous change. The thermodynamic

force coupled to the inward ﬂux of protons across the mitochondrial membrane, is the

difference in chemical potential of oxidised and reduced states of NADH ( and the electron

transport chain). Likewise the thermodynamic force driving ADP phosphorylation is the

difference in chemical potential of protons across the inner mitochondrial. It shall be show

that each cycle in a kinetic diagram is associated with some thermodynamic force that

determines the direction of “spontaneous” cycling. In fact the net cycle ﬂuxes shall always

be expressed in terms of their associated forces in the form

χk − exp − 1 Σk J = k RT (3.9) k Σ Q £ ¡ ¢ ¤

where χk is the thermodynamic force associated with cycle k, which is related to the ratio k+ / k− as will be shown later. These thermodynamics forces which are functions of stateQ variablesQ such as metabolite concentrations and the membrane potential, constrain the rate constants of each cyclic ﬂux. It is these constraints that contain the dependencies of cycle ﬂuxes on thermodynamic forces that are themselves function of other physiological variables such as metabolite concentrations and membrane potentials. All such relationships for the proton pump and the ATPase are elucidated below.

29 Thermodynamic forces of respiratory cycles

Cycle b Under isothermal and isobaric conditions the electrochemical potential µi of a

chemical species “i” is given by

o µi =(µi )T,P + ziF Ψ+ RT ln(ci) (3.10)

o where (µi )T,P is the standard electrochemical potential of molecule “i”, zi its valency, F

is Faraday’s constant, R is gas constant, T is the absolute temperature, ci is its concentration

and Ψ is its electrical potential [2]. In applying this relation to a single proton pump the concentrations terms “ci” are replaced by their equivalent state probabilities “pi” (which is nothing but that fraction of an ensemble of proton pumps, which are in state i ).

In ﬁgure 3.1 state 1 transitions to state 2 on binding 6 protons. The electrochemical potential of the system in state 1 is the sum of the chemical potential of state 1, six protons in the bulk phase and the electrical potential of the surface of the inner mitochondrial membrane relative to the bulk phase (“surface boundary potential”). Likewise the electro- chemical potential of the system in state 2 may be computed noting that it would lack the term for the six bulk phase protons which are now bound to the proton pump. At equilibrium the electro-chemical potential of the system in the two states must be equal. Hence

o e 1 o e 2 µ1 + RT ln(p1) + 6F ΨB,m + µ6H+ = µ2 + RT ln(p2) + 6F ΨB,m (3.11)

where

o + µ6H+ = µH+ + 6RT ln[H ] (3.12)

30 i and ΨB,s is the surface boundary potential of state i on the s side (m for matrix and i for cytoplasmic) of the inner mitochondrial membrane. Collecting the electrical potential terms we have

o e o e µ1 + RT ln(p1) − 6F ∆ΨB,m + µ6H+ = µ2 + RT ln(p2) (3.13)

2 1 where ∆ΨB,m = ΨB,m − ΨB,m . Rearranging

e o o o p µ + µ + − µ −6F ∆Ψ 2 = exp 1 H 2 exp B,m [H+]6 (3.14) pe RT RT m 1 µ ¶ µ ¶ where the superscript e for the state probabilities designate equilibrium conditions. Un- der equilibrium conditions, detailed balance (i.e. equality of each one way transition ﬂux in a reversible reaction) requires that

e e rijpi = rjipj (3.15)

where r’s are rate constants. In particular, for the transition between states 1 and 2 of the proton pump, where the rate constants are designated by α we obtain

o o o α µ + µ + − µ −6F ∆Ψ 12 = exp 1 H 2 exp B,m [H+]6 (3.16) α RT RT m 21 µ ¶ µ ¶

Pietrobon and Caplan [54] choose values for ΨB,i and ΨB,m such that proton unbinding was favoured to a much greater extent on the matrix side relative to the cytoplasmic side. This ensures that the proton pump does not spontaneously transport protons out of the mitochondria and its resistance to proton ejection increases with membrane potential.

Their choice of ΨB,i = −50 and ΨB,m = 0 is adopted in this model too. The negative

31 cytosolic phase boundary potential favours surface binding by positively charged protons.

∗ + 6 Substituting the value of ΨB,m = 0 in equation 3.16 noting α12 = α12[H ] we obtain

∗ o o o α µ + µ + − µ 12,0 = exp 1 H 2 (3.17) α RT 21,0 µ ¶

where αij,0 is used to designate the rate constant for the i to j transition at zero potential.

The above relation is an example, whereby rate constants are constrained by thermodynamic forces ( chemical potentials differences ). We continue to obtain such relations as they will be used later in expressing the cycle ﬂuxes in terms of state variables ( such as metabolite concentrations and membrane potential ). It must be noted that equation 3.17 is not dependent on any thermodynamic state variable (containing only intrinsic constants of the system ) and hence is valid even under non-equilibrium conditions. Hence combining equations 3.17 and 3.16 we obtain

∗ α α −6F ∆Ψ 12 = 12,0 exp B,m [H+]6 (3.18) α α RT m 21 21,0 µ ¶ The transition between states 5 and 6 of the proton pump also involve a proton association/dissociation step. Hence we may proceed as in the case of transitions 1 and 2, noting that this transition occurs on the cytoplasmic side.

o 6 o 5 µ6 + RT ln(p6) + 6F ΨB,i + µ6H+ = µ5 + RT ln(p5) + 6F ΨB,i (3.19)

where ΨB,i is the phase boundary potential on the cytoplasmic side. Hence

o o o α µ + µ + − µ −6F ∆Ψ 65 = exp 6 H 5 exp B,i [H+]6 (3.20) α RT RT i 56 µ ¶ µ ¶

32 5 6 where ∆ΨB,i = ΨB,i − ΨB,i As before we may rewrite equation 3.20 as

α α∗ −6F ∆Ψ 65 = 65 exp B,i [H+]6 (3.21) α α RT i 56 56,0 µ ¶ where

∗ o o o α µ + µ + − µ 65 = exp 6 H 5 (3.22) α RT 56,0 µ ¶ The transition between state 1 and 6 involves a reorientation of a charged binding site

(for protons) from one face (cytoplasmic or matrix )of the mitochondiral membrane to the other. There exists a potential difference between these two faces, and hence there must be an associated free energy change when the pump switches between them. Let the potential on the s face (i for cytoplasmic and m for matrix) be ΨM,s then at equilibrium the electrochemical potential the system in the two state must be equal. So

o e o e µ1 + RT ln p1 − 6F ΨM,m = µ6 + RT ln p1 − 6F ΨM,i (3.23)

since the valency of the pump without any protons bound is taken to be −6. Using detailed balancing as before and rearranging

α µo − µo 6F ∆Ψ 16 = exp 1 6 exp M,m (3.24) α RT RT 61 µ ¶ µ ¶

where ∆ΨM,m = ΨM,i − ΨM,m

Under conditions of zero potential (i.e. when ∆ΨM = 0)

33 α µo − µo 16,0 = exp 1 6 (3.25) α RT 61,0 µ ¶ Combining equations 3.24 and 3.25

α α 6F ∆Ψ 16 = 16,0 exp M,m (3.26) α α RT 61 61,0 µ ¶ For a fully deenergised mitochondrial system, at equilibrium with the cytosol, each

e e transition i ⇔ j in cycle b is at equilibrium, which implies αijpi = αjipj. Keeping in mind

∗ that under such conditions many of the αij’s are really either αij or αij,0 Multiplying all the four such relations for cycle b we obtain

∗ e e e e α12α25α56,0α61 p2p5p6p1 ∗ = e e e e = 1 (3.27) α21,0α16α65α52 p2p1p6p5

Using equations 3.18, 3.21 3.26 and 3.27

∗ α α α α α −6F ∆Ψ 12 25 56 61 = 12,0 exp B,m [H+]6 (3.28) α α α α α RT m 21 16 65 52 21,0 µ ¶ α 6F ∆Ψ − 56,0 exp B,i [H+] 6 α∗ RT i 65 µ ¶ α −6F ∆Ψ 61,0 exp M,m α RT 16,0 µ ¶ −6F ∆Ψ = exp B,m [H+]6 (3.29) RT m µ ¶ 6F ∆Ψ − exp B,i [H+] 6 RT i µ ¶ −6F ∆Ψ exp M,m RT µ ¶

34 By deﬁnition of Πb+ and Πb−

Πb+ −6F ∆ΨB,m + 6 = exp [H ]m (3.30) Π − RT b µ ¶ 6F ∆Ψ − exp B,i [H+] 6 RT i µ ¶ −6F ∆Ψ exp M,m RT µ ¶ 6F (∆Ψ − ∆Ψ − ∆Ψ ) [H+]6 = exp B,i B,m M,m m (3.31) RT [H+]6 · ¸ i

The total potential difference between the matrix and inter-membrane compartments of the mitochondrial “∆Ψ” is the net result of phase boundary potentials and the potential difference between the two surfaces (∆ΨM ). As show in ﬁgure 3.4 let the surface potentials on the matrix and cytoplasmic side of the inner mitochondrial membrane be “Ψ2”and“Ψ3” respectively. Further let the bulk phase potential of the matrix and inter-membrane space by “Ψ1” and “Ψ4” respectively. There are two phase boundary potentials corresponding to the two surfaces, one measures the difference between the surface potential and the bulk phase potential on the matrix side “∆ΨB,m” and the other does likewise for the cytoplasmic side “∆ΨB,i” where

∆ΨB,m = Ψ2 − Ψ1 (3.32)

and

∆ΨB,i = Ψ3 − Ψ4 (3.33)

The total phase boundary potential is now deﬁned as

35 ψ 1 ψ 2ψ 3 ψ 4

∆ Ψ M ∆ Ψ ∆ Ψ B,i B,m

1 2 3 4 matrix inter− cytosol membrane

Figure 3.4: Surface and phase boundary potentials of inner mitochondrial membrane

36 ∆ΨB = −∆ΨB,i + ∆ΨB,m (3.34)

and hence the mitochondrial (inner) membrane potential “∆Ψ” i.e. the difference between the bulk phase potentials of the matrix and cytoplasm is given by

∆Ψ = ∆ΨM + ∆ΨB (3.35)

The proton motive force (PMF, ‘∆p’) across the inner mitochondrial membrane is de-

ﬁned as

RT ∆p = ∆Ψ + ∆pH (3.36) F

where [H+] ∆pH = ln i (3.37) [H+] µ m ¶ From equations 3.30, 3.34, 3.35 and 3.36 we obtain

Π −6F ∆p b+ = exp (3.38) Π − RT b µ ¶ The thermodynamic force that drives cycle b is

χb = −6F ∆p (3.39)

That −6F ∆p is indeed the thermodynamic that drives cycle b can be checked by adding all the free energy changes that result in each of its transitions. The negative sign indicate that the favoured direction of this cycle is clockwise.

37 Now in view of equations 3.9 and 3.38 we may write the ﬂux for cycle b in the six state proton pump mechanism as

− − Σk J = b+ b (3.40) res,b Σ ¡Q Q −6¢F ∆p − exp − 1 Σb = b RT (3.41) Σ Q £ ¡ χb ¢ ¤ − exp − 1 Σb = b RT (3.42) Σ Q £ ¡ ¢ ¤

Cycle c As before under equilibrium conditions, detailed balance in cycle c yields

α α α α 23 34 45 52 = 1 (3.43) α32α25α54α43

However α23 and α43 are pseudo ﬁrst order rate constants since from the kinetic diagram of ﬁgure 3.1 it is clear

∗ + α23 = α23 [NADH]m [H ]m (3.44) p p and

∗ − α43 = α43 [NAD ]m (3.45) p − + where [NADH]m, [NAD ]m and [H ]m are respectively mitochondrial matrix

concentrationsp of NADHp, NAD− and protons.p

Hence

∗ α23α34α45α52 ∗ = Kres (3.46) α32α25α54α43

38 where Kres is the equilibrium constant for NADH oxidation (including a factor for the partial pressure for Oxygen equal to 0.2 atm.) and is given by

− e [NAD ]m Kres = (3.47) e + e [NADHp ]m [H ]m where the superscript e refers to equilibriump concentrations.p

The afﬁnity of this oxidation reaction is deﬁned as

RT K [NADH] [H+] A = ln res m m (3.48) res F − Ã p [NAD ]pm ! p By deﬁnition of c+ and c− Q Q α α α α c+ = 23 34 45 52 (3.49) α α α α Qc− 32 25 54 43 Using equation 3.46 this reducesQ to

+ c+ Kres [NADH]m [H ]m = − (3.50) Qc− p [NAD ]pm Q p and by deﬁnition of Afﬁnity ‘Ares’

FA c+ = exp res (3.51) RT Qc− µ ¶ Hence the thermodynamic forceQ that drives cycle c is

χc = FAres (3.52)

Analogous to equation 3.42 the net cycle ﬂux, for cycle c of the proton pump my now be written as

39 FAres − exp − 1 Σc J = c RT (3.53) res,c Σ Q £ ¡ ¢ ¤

Cycle a From ﬁgure 3.2, intuitively one would expect that the thermodynamic force that

drives cycle a would be the sum of those for cycles b and c. This is in fact the case since

detailed balancing at equilibrium, for cycle a implies

α α α α α α 12 23 34 45 56 61 = 1 (3.54) α16α65α54α43α32α21

Then as before replacing pseudo ﬁrst order rate constants by true ﬁrst order rate constants and replacing ratios of rate constants by equivalent terms from equations 3.18,3.21,3.26,

3.46,3.34 and 3.35 yields

F (A − 6∆p) c+ = exp res (3.55) RT Qc− µ ¶ wherein the thermodynamicQ force associated with cycle a is

χa = F (Ares − 6∆p) (3.56)

Now the net cycle ﬂux for cycle a can be written as

F (Ares−6∆p) a− exp RT − 1 Jres,a = (3.57) Q h ³ Σ ´ i

Since there are no paths “feeding into” cycle a, the product of rate constants of such paths is taken to be 1.

40 Thermodynamic force of oxidative phosphorylation

Cycle c At equilibrium detailed balancing of the “reaction slip” cycle (cycle c) of the

ATPase yields

β β β β 23 34 45 52 = 1 (3.58) β25β54β43β32

Here β23 and β43 are pseudo ﬁrst order rate constants and are expanded as

∗ β23 = β23[AT P ]m (3.59)

and

∗ β43 = β43[ADP ]m[Pi] (3.60)

Substituting these expressions into equation 3.58 we get

∗ β23β34β45β52 ∗ = KF 1 (3.61) β25β54β43β32

where KF 1 is the equilibrium constant for ATP hydrolysis given by

[ADP ]m[Pi]m KF 1 = (3.62) [AT P ]m

Now the ratio of cyclic rate constants for cycle c is

K [AT P ] c+ = F 1 m (3.63) [ADP ] [P ] Qc− m i m In terms of the afﬁnity of ATPQ hydrolysis given by

41 RT K [AT P ] A = ln F 1 m (3.64) F 1 F [ADP ] [P ] µ m i m ¶ equation 3.63 may be rewritten as

χ c+ = exp c (3.65) c− RT Q ³ ´ Q where χc given by

χc = FAF 1 (3.66)

is the thermodynamic force associated with cycle c of the ATPase. Hence the net cycle

ﬂux is

FAF 1 − exp − 1 Σc J = c RT (3.67) F 1,c Σ Q £ ¡ ¢ ¤

Cycle b In switching from state 1 to state 2 the ATPase binds three protons. At equilibrium the electro-chemical potential of the system in these two states must be equal. Using the same notation as in the proton pump this implies

o e 1 o e 2 µ1 + RT ln(p1) + 3F ΨB,m + µ3H+ = µ2 + RT ln(p2) + 3F ΨB,m (3.68)

Combining and rearranging terms as before this yields

o o o β µ + µ + − µ −3F ∆Ψ 12 = exp 1 H 2 exp B,m [H+]3 (3.69) β RT RT m 21 µ ¶ µ ¶

At zero potential (∆ΨB,m = 0)

42 o o o β µ + µ + − µ 12,0 = exp 1 H 2 (3.70) β RT 21,0 µ ¶ Hence

β β −3F ∆Ψ 12 = 12,0 exp B,m [H+]3 (3.71) β β RT m 21 21,0 µ ¶ The transition between state 5 and 6 also involves the binding/unbinding of 3 protons, and reasoning in the same way, we may write

o e 6 o e 5 µ6 + RT ln(p6) + 3F ΨB,i + µ3H+ = µ5 + RT ln(p5) + 3F ΨB,i (3.72)

o o o β µ + µ + − µ −3F ∆Ψ 65 = exp 6 H 5 exp B,i [H+]3 (3.73) β RT RT i 56 µ ¶ µ ¶

β β −3F ∆Ψ 65 = 65,0 exp B,i [H+]3 (3.74) β β RT i 56 56,0 µ ¶ Further, just as in the case of the proton pump the transition between states 1 and 6 involve the movement of charged binding site, across a electrical ﬁeld (∆ΨM,m). The only difference is that here the valency of the binding site is taken to be −3 since it binds 3 protons, in this case. Hence proceeding as before

o e o e µ1 + RT ln p1 − 3F ΨM,m = µ6 + RT ln p1 − 3F ΨM,i (3.75)

β µo − µo 3F ∆Ψ 16 = exp 1 6 exp M,m (3.76) β RT RT 61 µ ¶ µ ¶

43 β β 3F ∆Ψ 16 = 16,0 exp M,m (3.77) β β RT 61 61,0 µ ¶ The conditions for fully denenergised equilibrium in cycle b is

∗ β12β25β56,0β61 ∗ = 1 (3.78) β21β16β65β52

The product rate constants ratios of cycle b, for the two directions may now be written using equations 3.71,3.77,3.74, the equilibrium condition, and the deﬁnition of proton motive force (‘∆p’).

−3F ∆Ψ b+ = exp B,m [H+]3 (3.79) RT m Qb− µ ¶ 3F ∆Ψ Q exp B,i [H+]−3 RT i µ ¶ −3F ∆Ψ exp M,m RT µ ¶ 3F (∆Ψ − ∆Ψ − ∆Ψ ) [H+]3 = exp B,i B,m M,m m (3.80) RT [H+]3 · ¸ i −3F ∆p = exp (3.81) RT µ ¶

wherein the thermodynamic force driving cycle b is

χb = −3F ∆p (3.82)

Hence the net cycle ﬂuxe for cycle b is

−3F ∆p − exp − 1 Σb J = b RT (3.83) F 1,b Σ Q £ ¡ ¢ ¤

44 Cycle a The thermodynamic force for cycle a is the sum of forces for cycles b and c,

just as in the case of the proton pump. While it possible to derive it following an identical

procedure to that of cycles b and c, this is not done here to avoid repetition of the work already done in those two cycles. So the net cycle ﬂux for cycle a is

F (AF 1−3∆p) a− exp RT − 1 JF 1,a = (3.84) Q h ³ Σ ´ i

wherein the thermodynamic force associated with cycle a is

χa = F (AF 1 − 3∆p) (3.85)

3.3.5 Rate Constants

Section 3.3.4 detailed thermodynamic constraints on ratios of rate constants for the six

state mechanism of the proton pump and ATPase. These constraints were taken into con-

sideration by Pietrobon and Caplan, while choosing values of rate constants, for obtaining

proton transfer rates commensurate with experimental results. Values of these constants

are given in appendix C.5 and C.6 as reported by Magnus, with modiﬁcations to allow for

variable pH. This section details, expressions for the pseudo ﬁrst order rate constants that

are used in computing terms of the net cycle ﬂuxes.

Proton Pump States 1 and 6 of the proton pump have no protons bound, and hence have

a total charge of −6. As a result transitions, between these two states, that represent a con-

formational change (or equivalently a tranmembrane movement), are inﬂuenced by the po-

tential difference between the two surfaces of the inner mitochondrial membrane “∆ΨM ”.

45 Hence the transition probabilities between states 1 and 6 varry in response to changes in mi-

tochondrial membrane potentials, (∆Ψ), resulting from bulk phase, ionic ﬂuxes, across the

inner mitochondrial membrane. On assuming the conformational change involved in this

transition encounters only a single, symmetrical, sharp energy barrier, Eyring’s reation-rate

theory implies [43], that the rates constants must be of the form

zβF ∆Ψ α = α exp M (3.86) 0 RT

where α0 is the rate at zero potential (∆Ψ = 0), z is the net charge (z = −6), β is the fractional distance from the surface of the membrane to the peak of the energy barrier and must be 0.5 by the assumption of symmetry. Also F , R, and T are respectively, Faraday’s constant, gas constant, and absolute temperature. Hence the unimolecular rate constants

α16 and α61 are given by

3βF ∆Ψ α = α exp M (3.87) 16 16,0 RT −3βF ∆Ψ α = α exp M (3.88) 61 61,0 RT

The transitions 1 → 2 and 6 → 5 involves diffusion controlled binding of 6 protons from the bulk phase, so the corresponding rate constants are of the form

∗ + 6 α12 = α12[H ]m (3.89)

∗ + 6 α65 = α65[H ]i (3.90)

46 The corresponding deprotonation transitions are assumed to depend on the respective phase boundary potentials and thereby having rate constants of the form, implied by transition state theory

6F ∆Ψ α = α exp B,m (3.91) 21 21,0 RT 6F ∆Ψ α = α exp B,i (3.92) 65 65,0 RT

where α16,0, α61,0, α12,0, and α65,0 are the rates at corresponding zero potentials.

The rate constants corresponding to NADH oxidation/reduction involve concentration terms for the associated ligands as in

∗ α23 = α23 [NADH]m (3.93) p and

∗ − + α43 = α23 [NAD ]m [H ]m (3.94) p p Other rate constants not mentioned here are true ﬁrst order rate constants whose value

has not been changed from those of Magnus and are hence not discussed here but are

reported in the table of parameter values.

ATPase The transitions between states 1 and 6, involve reorientation of a charged (va-

lency = -3) binding site of the ATPase in the electrical ﬁeld of the mitochondrial membrane

(∆ΨM ). Hence the form of these rate constants are dictated by transition state theory and are

47 3F ∆Ψ β = β exp M (3.95) 16 16,0 RT µ ¶

−3F ∆Ψ β = β exp M (3.96) 61 61,0 RT µ ¶ The proton unbinding transtions 2 → 1 and 5 → 6 are assumed to depend on their respective phase boundary potentials and have rate constants as per transition state theory.

3F ∆Ψ β = β exp B,m (3.97) 21 21,0 RT µ ¶

3F ∆Ψ β = β exp B,i (3.98) 56 56,0 RT µ ¶ On the other hand the proton binding transitions 1 → 2, and 6 → 5 are assumed to be diffusion controlled and have rates of the form

∗ + 3 β12 = β12[H ]m (3.99)

∗ + 3 β65 = β65[H ]i (3.100)

ATP binding rate is, as expected dependent on ATP concentration and given by

∗ β23 = β23[AT P ]m (3.101)

while the ADP and phosphate binding rate is

48 ∗ β43 = β43[ADP ]m,free[Pi]m (3.102)

Other rate constants not mentioned here are true ﬁrst order rate constants whose value has not been changed from those of Magnus and are hence not discussed here but are reported in the table of parameter values.

3.4 Electro-neutral weak acid ﬂux

Selivanov et.al. had suggested that the electro-neutral weak acid ﬂux may be a linear

function of the proton gradient across the mitochondria [62]

Jah = kah (Hc − Hm) (3.103)

However this expression is found to be unsuitable [53] as it permits the weak acid ﬂux

to increase without bound with increasing transmembrane proton gradient. Weak acids

such as succinate (suggested by Ichas and Mazat) are shuttled across the membrane by

speciﬁc transporters that have a maximal turnover number that eventually limits the rate of

transport.

Hence an alternative, phenomenological, Michaelis-Menten type rate expression is used for the weak acid transport rate. Further the fundamental role played by the weak acid is to slowly restore mitochondrial acid balance after small perturbations. Hence rather then make the weak acid ﬂux a function of the proton gradient across the membrane , it is chosen to be a function of the displacement of the proton concentration from its steady state value.

∆H J = V (3.104) ah max,ah ∆H + K µ ah ¶ 49 ∆H = Hs − Hm (3.105)

where kah is the proportionality constant and Hs and Hm are respectively mitochondrial steady state proton concentration and mitochondrial proton concentrations.

50 CHAPTER 4

PERMIABILITY TRANSITION PORE

4.1 Introduction

The permeability transition pore was identiﬁed as a voltage dependent anionic channel

(VDAC) in the inner mitochondrial membrane. While its molecular identity is yet to be

established, it is thought to be a “channel complex” rather than a single molecule or multi-

mer. The Adenine nucleotide transporter (ANT) is thought to be an important component

of this complex, based on the fact that bonkrekic acid and atractyloside which are ANT

ligands, also inhibit and activate the mitochondrial permeability transition. In addition a

large voltage dependent anionic channel (VDAC) and cyclophilin D ( Cyp-D ) are consid-

ered to be the other key contituents of the PTP. However recent gene knock out studies

have shown a functional PTP, despite the absence of ANT [36]. Similar results were also

obtained in Cyp-D and VDAC knockouts. Hence the molecular identity of the permeability

transition pore is as yet unresolved. Electrophysiological studies of mitochondrial extracts

have shown that the permeability transition pore can exist in multiple conductance states.

Mitochondrial calcium induced calcium release requires at least two conductances states

51 that are differentially regulated by pH and calcium. With this in view a model of the permeability transition pore, as a three state ligand gated ion channel may be constructed. As shown in ﬁgure 4.1 the three states are 1) a pH gated low conductance state, 2) a calcium gated high conductance state and a closed state. While there is some indication that the higher conductance state may develop out of the lower conductance states, in this model this assumption is not made. The resultant equations are in the form of perturbed Hodgkin-

Huxley channels and thus ease the choice of parameter values.

4.2 Three state model

klb khf

PTPl PTPc PTPh

klf khb

Figure 4.1: Three state permeability transition pore

From ﬁgure 4.1 using mass action kinetics, we obtain

dPTP L = k P T P − k P T P (4.1) dt LF C LB L

dPTP H = k P T P − k P T P (4.2) dt HF C HB H

and

52 dPTP C = k P T P − k P T P + k P T P − k P T P (4.3) dt LB L LF C HB H HF C

This implies

P T PC + P T PL + P T PH = P T PT = Constant (4.4)

where P T PC , P T PL, P T PH and P T PT are respectively the concentrations of the closed, low conductance, high conductance states and the total amount of pore in the inner mitochondrial membrane.

Using equation 4.4 the above system of ODEs may be rewritten as

dPTP P T P − [P T P − P T P ] P T P ∞ L = − L T H L (4.5) dt τPTPL

and

dPTP P T P − [P T P − P T P ] P T P ∞ H = − H T L H (4.6) dt τPTPH

where

kLF P T PL∞ = (4.7) kLF + kLB kHF P T PH∞ = (4.8) kHF + kHB 1 τPTPL = (4.9) kLF + kLB 1 τPTPH = (4.10) kHF + kHB

53 By assuming that the forward and backward rate constants, for the low and high conductance pore are respectively exponential functions of mitochondrial proton and calcium concentrations, these expression may be put in standard Hodgkin-Huxley form.

54 CHAPTER 5

PORE CURRENTS

5.1 Introduction

Both the endoplasmic reticulum and the mitochondria are known to function as store houses of large amounts of calcium. Unlike the mitchondria the endoplasmic reticulum is capable of generating IP3 induced cytoplasmic calcium transients (oscillations). It has been shown that mitochondria is able to sequester and release a signiﬁcant amount of such cytosolic calcium pulses [1]. Entry of calcium takes place through the calcium uniporter

(ruthenium red sensitive mechanism) and exit via a Cyclosporin A sensitive pathway. Cy- closporin A is a well known potent inhibitor of the permeability transition pore. To a lesser extent the sodium calcium exchanger is also involved in bidirectional calcium transport between the mitochondria and cytoplasm. Likewise the endoplasmic reticulum is able to release and take up signiﬁcant amounts of calcium. Calcium is usually released by the endoplasmic reticulum though the IP3 (rynodine) receptor, whose conductance shows a biphasic dependence on cytoplasmic calcium concentration. Uptake of calcium by the ER occurs chieﬂy via the SERCA pump. It has been shown that the biphasic properties of

55 the IP3 receptor is key to the generation of self-sustaining IP3 induced cytoplasmic os-

cillations. In summary there are four currents that are essential to mitochondrial calcium

induced calcium release.

1. IP3 receptor mediated release from the ER

2. SERCA pump mediated uptake by the ER

3. Calcium uniporter mediated uptake by the mitochondria

4. Permeability transition pore mediated release

The IP3 receptor mediated calcium ﬂux is well studied and has been previously modeled

by De Young and Keizer [8] and has been simpliﬁed by Li and Rinzel [44] and are used

here in the form reported by Fall and Keizer [10]. The calcium uniporter has also been

previously described [17, 16, 15, 45] and is used here, again, in the form reported by Fall

and Keizer [10]. The dependence of the rate of calcium uptake by the SERCA pump has

also been modeled by De Young and Keizer [8] and is used here as such.

5.2 Calcium current through the pore

The molecular weight cut off of the low conductance pore is about 500 KDa. It thus presents a fairly large channel as far as ionic species such as calcium are concerned. Even with it hydration shell a calcium ion should be free to diffuse through the low conductance pore, down its electro-chemical gradient. Since we are concerned with transport through the inner mitochondrial membrane, assuming the inter-membrane space to be continuous with the cytoplasm, passive electro-chemical diffusion of calcium should be adequately described by Goldman, Hodgkin, Katz equations of the form

56 CAM − CAC exp ziF ∆Ψ J = P RT (5.1) ptplca ca − ziF ∆Ψ 1 exp RT ¡ ¢ ¡ ¢ 5.3 Proton current through the pore

Similarly the proton ﬂux is described as

+ ziF ∆Ψ [H ]m − Hc exp J = P RT (5.2) ptplh h − ziF ∆Ψ 1 exp RT¡ ¢ ¡ ¢

57 CHAPTER 6

SIMPLIFICATION

6.1 introduction

The steady state proton ﬂuxes derived in chapter are cumbersome and complex. The also make simulations difﬁcult by introducing more than moderate stiffness into the system of differential equations. This is because of the very large difference (1050) in orders of

magnitude of terms contained in them. For the same reason continuation routines, tend to

become unstable when used with them. Hence it is important to obtain simpliﬁed forms

of these ﬂuxes. Our effort is to build a a physiological model with a well deﬁned range

within which,the value of state variables lie, even under pathological conditions. Hence it

is sufficient to find a reasonably good approximation of the fluxes that does not perturb the

behaviour of the model over physiological ranges for each of the variables.

6.2 Phenomenological rate equations

Each of the concerned rate equations is a function of at most 3 variables, of physiologi-

cal interest. Hence the approach adopted here in is to choose suitable functions that capture

the dependence of the rates on each of the physiological variables. Then by a judicious

58 choice of shifting and scaling parameters the so obtained functions of a single variable

are combined to obtain multivariable functions that have the same level surfaces as the

original rates. This provides the approximate form, of a reasonable choice for simpliﬁed

expressions of the steady state rate equations obtained by the diagrammatic method. This

approximation is then reﬁned using the Nedler and Meade simplex marching algorithm.

Jhres By examining ﬁgure 7.9 it seems reasonable to assume that within the range of

interest Jhres is a sigmoid function of membrane potential ‘∆Ψ’, and and exponential func-

+ tion of mitochondrial proton concentration ‘[H ]m’. A little thought and some exploration with a graphing tool such as gnuplot or matlab/octave will show that an equations of the form

exp([H+]m) − exp(∆Ψ) J = (6.1) hres 1+ exp(∆Ψ)

yields the same shape of a surface as can be obtained by plotting Jhres as a function of the same two variables. Also as seen in ﬁgure 7.10. Jhres is almost a linear function of

NADHm. Again by exploring possible forms of a three variable function using a plotting

tool, the following simpliﬁed expression for Jhres was obtained

+ − r [NADHm − r ] + exp([H ]mr4) − exp(∆Ψ r5) r 2 3 6 (6.2) Jhres = r1 (∆Ψ−r ) [1 + exp 7 ] r8

where the parameters r1 to r8 are obtained by ﬁtting and given in appendix C

Jo The respiratory oxygen consumption ‘Jo’, rate is closely related to the respiratory proton ﬂux ‘Jhres’. Infact they are linearly related and the form of the phenomenological

59 equations for the two are identical, differing only in parameter values (o1 to o8, also given

in appendix ).

+ − o (NADHm − o ) + exp([H ]mo4) − exp(∆Ψ o5) o 2 3 6 (6.3) Jo = o1 (∆Ψ−o ) [1 + exp 7 ] o8

Jhf1 The mitochondrial proton uptake rate ‘Jhf1’, is also a sigmoid function of membrane

+ potential ‘∆Ψ’, and an exponential function of mitochondrial proton concentration ‘[H ]m’

as seen in ﬁgures 7.16 and 7.15. However in this case the form of a function that was found

to best approximate its dependence on these two variables was found to be

− + exp(∆Ψ [H ]m) −1 Jhf1 = (6.4) 1 + exp(∆Ψ−[H+]m)

Noting, from ﬁgure 7.16 that Jhf1 has an almost linear dependence on mitochondrial

ATP concentration ‘AT Pm’, the following expression was found to yield a good ﬁt.

− exp(∆Ψ h2) h − ([H+] − h ) h − 1 3 m 4 5 (6.5) Jhf1 = h1 (∆Ψ−h7) + [1 + h6AT Pm exp ] h8 − [[H ]m − h9] h10 + h11

and the parameter values are in appendix C

Jpf1 The mitochondrial ATP hydrolysis rate ‘Jpf1’ is related to the rate of ejection of protons by the F0-F1 ATPase, and depends on the same three variables. In fact, except for

the parameter values, the form of a function that gives a good ﬁt is also similar.

− exp(∆Ψ p2) p − ([H+] − p ) p − 1 3 m 4 5 (6.6) Jpf1 = p1 (∆Ψ−h7) + [1 + p6AT Pm exp ] p8 − [[H ]m − p9] p10 + p11

The results of ﬁtting these equations is shown in ﬁgures 7.9 to 7.20.

60 CHAPTER 7

RESULTS

7.1 pH dependent respiratory ﬂuxes

The pH dependent respiratory ﬂuxes, “Jhres, Jo, Jhf1 and Jpf1”, obtained from the six

state model of the electron transport chain and the F0-F1 ATPase, are in good agreement

with those obtained by Magnus and Keizer as show in ﬁgures 7.1 to 7.8.

Jhres vs. PSI 700 Thomas Magnus 600

500

400

300

200 Jhres [nmol/(mg-protein min)] 100

-100 140 145 150 155 160 165 170 175 180 psi

Figure 7.1: Comparison of pH dependent Jhres ﬂux with that of Magnus and Keizer

61 Jhres vs. NADHm 1200 Thomas Magnus 1100

1000

900

800

700

600 Jhres [nmol/(mg-protein min)]

500

400

300 0 1 2 3 4 5 6 7 nadhm

Figure 7.2: Comparison of pH dependent Jhres ﬂux with that of Magnus and Keizer

Jhf1 vs. PSI 1000 Thomas Magnus 900

800

700

600

500

400

Jhf1 [nmol/(mg-protein min)] 300

200

100

0 140 145 150 155 160 165 170 175 180 psi

Figure 7.3: Comparison of pH dependent Jhf1 ﬂux with that of Magnus and Keizer

62 Jhf1 vs. ATPm 350 Thomas Magnus

300

250

200

150 Jhf1 [nmol/(mg-protein min)]

100

50 3 4 5 6 7 8 9 10 11 aptm

Figure 7.4: Comparison of pH dependent Jhf1 ﬂux with that of Magnus and Keizer

Jo vs. PSI 60 Thomas Magnus

20 Jo [nmol/(mg-protein min)]

0 140 145 150 155 160 165 170 175 180 nadhm

Figure 7.5: Comparison of pH dependent Jo ﬂux with that of Magnus and Keizer

63 Jo vs. NADHm 100 Thomas Magnus 90

50 Jo [nmol/(mg-protein min)]

20 0 1 2 3 4 5 6 7 nadhm

Figure 7.6: Comparison of pH dependent Jo ﬂux with that of Magnus and Keizer

Jpf1 vs. PSI 350 Thomas Magnus

300

250

200

150

Jpf1 [nmol/(mg-protein min)] 100

0 140 145 150 155 160 165 170 175 180 psi

Figure 7.7: Comparison of pH dependent Jpf1 ﬂux with that of Magnus and Keizer

64 Jpf1 vs. ATPm 110 Thomas Magnus 100

50 Jhf1 [nmol/(mg-protein min)]

20 3 4 5 6 7 8 9 10 11 aptm

Figure 7.8: Comparison of pH dependent Jpf1 ﬂux with that of Magnus and Keizer

7.2 Simpliﬁcation

Figures 7.9 through 7.20 show that simplified version of the respiratory fluxes are in good agreement with their unsimplified counterparts. These figures are plots of each of the respiratory fluxes and its corresponding simplified version, with respect to two (out of three) of its independent variables.

7.3 Mitochondrial response to cytoplasmic calcium

In response to a single pulse of cytoplasmic calcium mitochondria rapidly take up calcium though the calcium uniporter (see ﬁgure 7.24 ). There is a concomitant membrane potential change which leads to alkalanisation of the matrix due to increased proton pumping (ﬁgure 7.22). In the absence of the endoplasmic reticulum, a single pulse of calcium

65 Simplified and Unsimplified J hres

12000

10000

8000

6000

4000 nmol/mg−protein min

hres 2000 J

180 −2000 3.5 160 3 140 2.5 2 120 −5 x 10 1.5 1 100 H ∆ Ψ m

Figure 7.9: Jhres in simpliﬁed and unsimpliﬁed forms

66 Simplified and Unsimplified J hres

1000

800

600

400

200 nmol/mg−protein min hres J 0

−200 100 0.1 0.15 120 0.2 140 0.25 160 0.3 0.35 180 ∆ Ψ NADH m

Figure 7.10: Jhres in simpliﬁed and unsimpliﬁed forms

67 Simplified and Unsimplified J hres

8000

7000

6000

5000

4000

3000

nmol/mg−protein min 2000 hres J 1000

−1000 0.1 1 0.15 1.5 2 0.2 2.5 0.25 3 0.3 −5 x 10 3.5 0.35 NADH m H m

Figure 7.11: Jhres in simpliﬁed and unsimpliﬁed forms

68 Simplified and Unsimplified J o

1000

800

600

400

200 nmol/mg−protein min o J 0 180 170 −200 160 3.5 150 3 140 2.5 130 −5 2 x 10 120 1.5 1 110 ∆ Ψ H m

Figure 7.12: Jo in simpliﬁed and unsimpliﬁed forms

69 Simplified and Unsimplified J o

30 nmol/mg−protein min o J 20

0.5 0 0.4 0.3 110 120 130 140 0.2 150 160 170 180 0.1 NADH m ∆ Ψ

Figure 7.13: Jo in simpliﬁed and unsimpliﬁed forms

70 Simplified and Unsimplified J o

700

600

500

400

300

200 nmol/mg−protein min o

J 100

−100 1 1.5 0.1 2 0.15 2.5 0.2 −5 0.25 x 10 3 0.3 3.5 0.35 NADH H m m

Figure 7.14: Jo in simpliﬁed and unsimpliﬁed forms

71 Simplified and Unsimplified J hf1

1600

1400

1200

1000

800

600

nmol/mg−protein min 400 hf1 J 200

0 3.5 −200 3 110 120 2.5 130 140 2 −5 150 x 10 160 1.5 170 180 1 H m ∆ Ψ

Figure 7.15: Jhf1 in simpliﬁed and unsimpliﬁed forms

72 Simplified and Unsimplified J hf1

800

700

600

500

400

300

nmol/mg−protein min 200 hf1 J 100

−100 8 110 9 120 130 10 140 150 11 160 170 12 180 ∆ Ψ ATP m

Figure 7.16: Jhf1 in simpliﬁed and unsimpliﬁed forms

73 Simplified and Unsimplified J hf1

700

600

500

400

300

200

nmol/mg−protein min 100 hf1 J 0

−100 8

10 3.5 3 11 2.5 2 −5 1.5 x 10 12 1 ATP H m m

Figure 7.17: Jhf1 in simpliﬁed and unsimpliﬁed forms

74 Simplified and Unsimplified J pf1

600

500

400

300

200

nmol/mg−protein min 100 pf1 J

−100 100

120 3.5 140 3 2.5 160 2 −5 1.5 x 10 180 1 H ∆ Ψ m

Figure 7.18: Jpf1 in simpliﬁed and unsimpliﬁed forms

75 Simplified and Unsimplified J pf1

300

250

200

150

100 nmol/mg−protein min

pf1 50 J

−50 8 180 170 9 160 150 10 140 130 11 120 110 12 ATP m ∆ Ψ

Figure 7.19: Jpf1 in simpliﬁed and unsimpliﬁed forms

76 Simplified and Unsimplified J pf1

250

200

150

100

50 nmol/mg−protein min pf1 J 0

−50 8

11 3.5 3 2.5 2 12 1.5 −5 ATP 1 x 10 m H m

Figure 7.20: Jpf1 in simpliﬁed and unsimpliﬁed forms

77 leads to elevated mitochondrial and cytoplasmic calcium (ﬁgure 7.23). The elevated mitochondrial calcium levels result in a higher value of the steady state mitochondrial pH

(ﬁgure 7.21).

In the presence of the endoplasmic reticulum, mitochondria rapidly take up calcium but release it slowly to the endoplasmic reticulum via the cytosol (ﬁgures 7.25, and 7.27). This enables the restitution of baseline mitochondrial calcium and pH levels (ﬁgures 7.25 and

7.26).

Mitochondrial pH response to cytoplasmic calcium 7.696 Mito. pH

7.694

7.692

7.69

7.688 pH

7.686

7.684

7.682

7.68 0 50 100 150 200 250 300 350 400 time (sec)

Figure 7.21: Change in mitochondrial pH after a cyptoplasmic calcium pulse at t=250

78 Increased proton pumpuing after cytoplasmic calcium pulse 240 Jhres

220

200

180

160

140

120

Jhres [nmol/(mg-protein min)] 100

40 0 50 100 150 200 250 300 350 400 time (sec)

Figure 7.22: Increase in proton pumping by ETC in response to elevated matrix calcium

Elevated cytoplasmic calcium 0.7 CaC

0.6

0.5

0.4

0.3 CaC [micro M]

0.2

0.1

0 0 50 100 150 200 250 300 350 400 time (sec)

Figure 7.23: Elevated baseline cytoplasmic calcium (without ER)

79 Elevated mitochondrial calcium 0.55 CaM

0.5

0.45

0.4

0.35

0.3 CaM [micro M]

0.25

0.2

0.15

0.1 0 50 100 150 200 250 300 350 400 time (sec)

Figure 7.24: Elevated baseline mitochondrial calcium (without ER)

Calcium uptake and release by Mitochondria 0.4 CaM

0.35

0.3

0.25

0.2 CaM [micro M] 0.15

0.1

0.05

0 0 50 100 150 200 250 300 350 400 time (sec)

Figure 7.25: Mitochondrial calcium response in presence of ER

80 Mitochondrial pH response 7.69 pH

7.689

7.688

7.687

7.686

7.685

7.684 Mitochondrial pH

7.683

7.682

7.681

7.68 0 50 100 150 200 250 300 350 400 time (sec)

Figure 7.26: Mitochondrial pH response in presence of ER

Calcium uptake and release by ER 10 CaER

5 CaER [micro M]

1 0 50 100 150 200 250 300 350 400 time (sec)

Figure 7.27: Calcium uptake by ER

81 7.4 Behaviour of the PTP

Behaviour of the three state pore used in this dissertation, is determined by 8 parameters

— permeability to protons, permeability to calcium, maximal rate of weak acid inﬂux, max-

imal rate of change in the weak acid inﬂux, and the forward and backward rate constants

(or equivalently the Hodgkin-Huxley parameters). Together these determine the differen-

tial response of the low and high conductance states of the pore to rapid and slow calcium

entry into the mitochondria.

7.4.1 Low conductance pore

The low conductance state of the permeability transition pore opens in response to elevation in mitochondrial pH. This is dependent on the rate of calcium entry into the mitochondria. A single pulse of cytoplasmic calcium does not yield sufﬁcient matrix pH build up, to open the pore. On the other hand a rapid sequence of calcium pulses does result in matrix pH elevation and concomitant opening of the low conductance pore (ﬁgure 7.28).

Subsequent to opening of the pore there is a net calcium efﬂux from and proton uptake by the mitochondria. This resetting of the mitochondrial calcium pH results in closing of the pore and the whole process can potentially repeat. It is pertinent to note that a rapid sequence of three cytoplasmic pulses results and delayed opening of the pore. This is in agreement with the observations of Ichas et.al. [28].

In the absence of the endoplasmic reticulum, calcium released into the cytoplasm by the

mitochondria accumulates there and disrupts further mitochondiral calcium induced cal-

cium spiking (ﬁgure 7.29). Based on a simple estimation of the calcium loading needed to

initiate mitochondrial calcium release Ichas and Mazat had speculated that the phenomenon

82 Response to fast calcium pulses (with ER) 1.4 CAC PTPL PTPH 1.2

0.8

molar), PTPL/PTPH (%) 0.6 µ

Calcium ( 0.4

0.2

0 50 55 60 65 70 75 80 85 90 95 100 time (sec)

Figure 7.28: Response of the PTP to fast cytoplasmic calcium pluses in the presence of the Endoplasmic Reticulum

83 “is most likely to occur speciﬁcally at the level of the mitochondria tightly associated with

the calcium release channels of the ER”. Our model reinforces this assumption with the

observation that rapid reuptake of cytoplasmic calcium is required for a sustained mito-

chondrial calcium induced calcium release.

Response to fast calcium pulses (without ER) 0.9 CAC PTPL 0.8 PTPH

0.7

0.6

0.5

0.4 molar), PTPL/PTPH (%) µ

0.3 Calcium (

0.2

0.1

0 0 10 20 30 40 50 60 70 80 90 100 time (sec)

Figure 7.29: Response of the PTP to fast cytoplasmic calcium pluses in the presence of the Endoplasmic Reticulum

84 7.4.2 High conductance pore

Slow buildup of cytoplasmic calcium, in the absence of ER, leads to an equilibrium

distribution of the states of the pore that favours the high conductance pore, for sufﬁciently

high matrix calcium (ﬁgure 7.30). Prolonged elevation of mitochondrial calcium beyond

a threshold value, results in the opening of the high conductance pore. In vivo this would

lead to mitochondrial calcium apoptosis.

Response to slowly rising calcium (without ER) 1 CAC PTPL 0.9 PTPH

0.8

0.7

0.6

0.5 molar), PTPL/PTPH (%)

µ 0.4

0.3 Calcium (

0.2

0.1

0 0 100 200 300 400 500 600 700 800 time (sec)

Figure 7.30: Response of the PTP to slow buildup of cytoplasmic calcium in the absence of the ER

85 Slow buildup of cytoplasmic calcium in the presence of the ER generates a transient endoplasmic reticulum’s calcium induced calcium response. However the presence of the

ER does not effect the ﬁnal outcome of slow build up of cytoplasmic calcium and the PTP still opens into its high conductance state (ﬁgure 7.31).

Response to slowly rising calcium (with ER) 1.8 CAC PTPL 1.6 PTPH

1.4

1.2

0.8 molar), PTPL/PTPH (%) µ

0.6 Calcium (

0.4

0.2

0 0 100 200 300 400 500 600 700 time (sec)

Figure 7.31: Response of the PTP to slow buildup of cytoplasmic calcium in the presence of the ER

The above results were obtained by directly injecting calcium into the cytoplasm. How- ever the same behaviour (mitochondrial calcium induced calcium release) is also observed in response to IP3 induced calcium release by the endoplasmic reticulum (see ﬁgure 7.32).

86 It is interesting to note that in this case mitochondrial calcium induced calcium release can

not be continued for an arbitrary length of time due to a gradual buildup in the fraction

of pore open in its low conductance state. However such calcium spiking can occur for

prolonged lengths of time if it is interspersed with brief periods of inactivity.

Mitochondrial Calcium Induced Calcium Release 1.4 CAC PTPL PTPH 1.2

0.8

molar), PTPL/PTPH (%) 0.6 µ

Calcium ( 0.4

0.2

0 150 200 250 300 350 400 time (sec)

Figure 7.32: Mitochondrial calcium induced calcium release

The present model is hence seen to show the delayed response to rapid calcium spiking in the cytoplasm. In addition we find that a three state model of the pore and a small, saturating weak acid flux are sufficient to replicate the behaviour of the Ichas and Mazat

87 theory.In summary our model corroborates, all key observations of the Ichas and Mazat hypothesis, that explains mitochondrial transition from calcium signalling to cell death.

88 CHAPTER 8

DISCUSSION

Fine character of chemiosmotic flows The Pietrobon-Caplan and Magnus-Keizer models show that the sum of proton fluxes due to ETC pumping, Oxidative Phosphorylation and leakage are non zero, except perhaps for a few isolated values of physiological variables. Clearly there must be fluxes which, in physiological context, ensure mitochondrial pH homeostatsis. Pokhilko et.al. [57] have demonstrated three modes of calcium uptake by mitochondria, under conditions of steady state oxidative phosphorylation, and in the presence of seven proton fluxes. These fluxes are generated by ETC proton pumping, proton leak, Phosphate/hydroxide transporter, Potassium/Hydrogen transporter, Cal- cium/Hydrogen transporter and fluxes through the permeability transition pore. Mitochon- dria have evolved as organelles that maintain proton gradients across their inner membrane and use its free energy to generate the high energy bonds of adeniosine triphosphate. The three fluxes involved in this process are proton pumping by the ETC, proton uptake via the

F0-F1 ATPase, and a small leak. Of these the first two fluxes are of the order of several hundred nano moles per mill gram protein per minute. They are much larger than any of the other proton fluxes, and hence any complete description of mitochondrial acid balance must take them into consideration. Our model shows that the three chemiosmotic fluxes

89 with a small saturating proton ﬂux is sufﬁcient for mitochondrial calcium induced calcium

release while having stable steady state behaviour. As Pokhilko et.al [57] have already

show that the “weak acid” ﬂux need not necessarily have a single point of origin, we take

the saturating proton ﬂux to be the net result of multiple such currents, that may possibly

serve redundant roles. One would expect the robustness concomitant with such redundancy

in what is (pH homeostasis), a central part of mitochondrial function. If there are multiple

mechanisms (sets of ﬂuxes) that maintain pH balance, despite the deﬁcit in chemiosmotic

currents, then disruption of any single one of them, will only be a small perturbation of pH

homeostasis. It would hence be very instructive to experimentally delineate, all such pH

control systems in the mitochondria.

Weak acids Further while it has increasingly become clear that the mitochondrial are involved in intracellular calcium signalling, their precise role in this process remains to be delineated. This model now provides a basis, to further build spatial models that can inves- tigate the interaction of mitochondria and endoplasmic reticulum in intracellular calcium signalling. The Pokhilko et.al model does not take into consideration the endoplasmic reticulum but does provide insight into a possible basis of the weak acid flux. Hence it would be instructive to examine closely to what extent the fluxes suggested by their model reproduce the results of the more phenomenological weak acid flux used herein.

Spatial model This model in its current state describes a point process. However the real

signiﬁcance of mitochondria’s role in intracellular calcium signalling can only be under-

stood in a spatial context, that takes into consideration the spatial distribution and extent

of mitochondrial and endoplasmic reticulum, withing a cell. Towards this end it is now

possible to construct spatial reaction-diffusion systems using this model.

90 APPENDIX A

PARTIAL DIAGRAMS

Figure A.1: Partial Diagrams for a six state mechanism

91 APPENDIX B

UNSIMPLIFIED TERMS IN RATE EXPRESSIONS

The pseudo ﬁrst order rate constants represented as γ here correspond to either those in the six state electron transport chain (α) or those in the six state ATPase (β). With the exception of this substitution the expressions given below are identical in the two cases.

However it must be stressed that the expressions for the rate constants themselves are not identical and have been described in detail in the thesis.

92 = γ61γ56γ25γ45γ34 1 X + γ61γ56γ25γ32γ45

+ γ61γ56γ25γ32γ43

+ γ21γ61γ56γ45γ34

+ γ32γ21γ61γ56γ45

+ γ43γ32γ21γ61γ56

+ γ34γ45γ52γ21γ61

+ γ32γ21γ52γ45γ61

+ γ43γ32γ21γ52γ65

+ γ21γ52γ65γ45γ34

+ γ21γ32γ52γ65γ45

+ γ21γ32γ43γ52γ65

+ γ23γ34γ45γ56γ61

+ γ21γ32γ43γ54γ61

+ γ21γ32γ43γ54γ65

93 = γ16γ65γ52γ45γ34 2 X + γ16γ65γ52γ45γ32

+ γ16γ65γ52γ32γ43

+ γ12γ61γ56γ45γ34

+ γ12γ61γ56γ45γ32

+ γ12γ61γ56γ32γ43

+ γ61γ12γ52γ45γ34

+ γ61γ12γ32γ52γ45

+ γ61γ12γ52γ32γ43

+ γ12γ52γ65γ45γ34

+ γ12γ32γ52γ65γ45

+ γ12γ52γ65γ32γ43

+ γ16γ65γ54γ43γ32

+ γ61γ12γ32γ43γ54

+ γ12γ32γ43γ54γ65

94 = γ16γ65γ25γ54γ43 3 X + γ16γ65γ52γ23γ45

+ γ16γ65γ52γ23γ43

+ γ21γ16γ65γ54γ43

+ γ45γ56γ61γ12γ23

+ γ56γ61γ12γ23γ43

+ γ61γ12γ25γ54γ43

+ γ61γ12γ23γ52γ45

+ γ61γ12γ23γ52γ43

+ γ12γ25γ65γ54γ43

+ γ12γ23γ52γ65γ45

+ γ12γ23γ52γ65γ43

+ γ16γ65γ54γ43γ23

+ γ61γ12γ23γ43γ54

+ γ12γ23γ43γ54γ65

95 = γ16γ65γ25γ54γ34 4 X + γ16γ65γ25γ32γ54

+ γ16γ65γ52γ23γ34

+ γ21γ16γ65γ54γ34

+ γ32γ21γ16γ65γ54

+ γ56γ61γ12γ23γ34

+ γ61γ12γ25γ54γ34

+ γ61γ12γ25γ32γ54

+ γ61γ12γ52γ23γ34

+ γ12γ25γ65γ54γ34

+ γ12γ25γ65γ32γ54

+ γ12γ23γ34γ52γ65

+ γ16γ65γ54γ23γ34

+ γ61γ12γ23γ34γ54

+ γ12γ23γ34γ54γ65

96 = γ16γ65γ25γ45γ34 5 X + γ16γ65γ25γ32γ45

+ γ16γ65γ25γ32γ43

+ γ21γ16γ65γ45γ34

+ γ32γ21γ16γ65γ45

+ γ43γ32γ21γ16γ65

+ γ61γ12γ25γ45γ34

+ γ61γ12γ25γ32γ45

+ γ61γ12γ25γ32γ43

+ γ12γ25γ65γ45γ34

+ γ12γ25γ32γ65γ45

+ γ12γ25γ43γ32γ65

+ γ16γ65γ45γ34γ23

+ γ61γ12γ23γ34γ45

+ γ12γ23γ34γ45γ65

97 = γ16γ56γ25γ34γ45 6 X + γ16γ56γ45γ25γ32

+ γ16γ56γ25γ32γ43

+ γ21γ16γ56γ45γ34

+ γ32γ21γ16γ56γ45

+ γ16γ21γ32γ43γ56

+ γ16γ21γ52γ45γ34

+ γ16γ21γ32γ52γ45

+ γ16γ21γ52γ32γ43

+ γ12γ25γ56γ45γ34

+ γ12γ25γ56γ32γ45

+ γ12γ25γ56γ32γ43

+ γ16γ56γ45γ34γ23

+ γ16γ21γ32γ43γ54

+ γ12γ23γ34γ45γ56

98 = + + + + + 1 2 3 4 5 6 X X X X X X X = γ43γ32 + γ34γ45 + γ32γ45 Xb = γ61γ12 + γ16γ65 + γ12γ65 c X = γ16γ65γ54γ43γ32γ21 a− Y = γ16γ65γ52γ21 − Yb = γ25γ54γ43γ32 c− Y

99 APPENDIX C

MODEL PARAMETERS

C.1 Thermodynamic constants

Name units Symbol Xpp name Value

Gas Constant milli-J/(mol k) R R 8314.472 Temperature K T Temp 310 Faraday’s Constant Coulumb/mol F F 96484.6

C.2 Conversion Constants

Name units Symbol Xpp name Value Minute to Seconds none minute minute 60

µ Molar to m Molar none µMmM uMmM 1000

100 Name units Symbol Xpp name Value Total volume ml V V 1.0 Fractional cytosolic Volume none pcytosol pcytosol 0.53 Fractional mitochondrial volume none pmito pmito 0.05 Fractional ER volume none per per 0.10 Cystosolic protein density mg/ml dcytosol dcytosol 90 Mitochondrial protein density mg/ml ddmit dmito 1250 ER protein density mg/ml der der 1000 Mitochondrial membrane capacitance nmol/(mV*mg) cmito cmito 0.0725 Cytoplasmic phase boundary potential mV ΨB,i psibi -50

101 Mitochondrial phase boundary potential mV ΨB,m psibm 0 Cytoplasmic pH none pHc pHc 7.2 Fitting factor none g g 0.85 Fitting factor none ff ff 2.5 Total Proton Pump Concentration nmol/mg-protein ρres rhores 0.4 Total ATPase Concentration nmol/mg-protein ρf1 rhof1 0.7 Total Proton Leak Channel Concentration nmol/mg-protein ρleak rholeak 0.2 Total Calcium Uniporter Concentration nmol/mg-protein ρuni rhouni 300 Total Sodium Calcium Exchanger Concentration nmol/mg-protein ρnc rhonc 3 Cytosolic calcium buffering none fi ﬁ 0.01 Mitochondrial calcium buffering none fm fm 0.0003 Mitochondrial pH scaling constant none fh fh 1e-7 Name units Symbol Xpp name Value Glucose concentration mill Molar glc glc 1 −1 Maximal cytosolic ATP hydrolysis rate sec Jhydmax Jhydmax 30.1 102 Mitochondrial matrix phosphate concentration mill Molar none Pim 20 −1 21 Equilibrium constant of NADH hydrolysis lit mol Kres Kres 8.5179 × 10 6 Equilibrium Constant of ATP Hydrolysis milli molar Kf1 Kf1 1.71 × 10 −1 −1 Basal NADH Reduction Rate nmol mg min Jred,basal Jredbasal 20 C.3 Compartmentalization parameters

C.4 Metabolic parameters

C.5 Rate constants (Proton Pump)

Name units Symbol Xpp name Value

−1 ∗ 30 1 → 2 sec α12 als12 6.0746 × 10

−1 −1 ∗ 6 2 → 3 mg nmol sec α23 als23 7.5958 × 10

1/2 −1/2 −1 ∗ 4 → 3 mg nmol sec α43 als43 159

6 −6 −1 ∗ 33 6 → 5 mg nmol sec α65 als65 1.5187 × 10

−1 2 → 1 sec α21,0 alz21 5.0

−1 6 5 → 6 sec α56,0 alz56 7.535 × 10

−1 1 → 6 sec α16,0 alz16 130.0

6 −6 −1 12 6 → 1 mg nmol sec α61,0 alz61 1.0 × 10

−1 4 3 → 2 sec α32 al32 8.0 × 10

−1 3 → 4 sec α34 al34 400

−1 4 → 5 sec α45 al45 40

−1 5 → 4 sec α54 al54 0.4

−1 −17 2 → 5 sec α25 al25 6.75 × 10

−1 −4 5 → 2 sec α52 al52 1.0 × 10

103 C.6 Rate constants (ATPase)

Name units Symbol Xpp name Value

3 −3 −1 16 1 → 2 mg nmol sec β12 bts12 4.9294 × 10

−1 −1 2 → 3 mg nmol sec β23 bts23 5000.0

3 −3 −1 ∗ 6 → 5 mg nmol sec β65 bts65 2.9022e+12

2 −1 ∗ 4 → 3 (millMolar) sec β43 bts43 50

−1 2 → 1 sec β21,0 btz21 40.0

−1 5 → 6 sec β56,0 btz56 2.7452e+05

−1 1 → 6 sec β16,0 btz16 100.0

−1 6 → 1 sec β61,0 btz61 4.98e7

−1 3 → 2 sec β32 bt32 5.0e3

−1 3 → 4 sec β34 bt34 100

−1 4 → 5 sec β45 bt45 100

−1 5 → 4 sec β54 bt54 100

−1 2 → 5 sec β25 bt25 1.17e-12

−1 5 → 2 sec β25 bt52 2.0

C.7 Electroneutral weak acid parameters

Name units Symbol Xpp name Value

−1 Maximal weak acid ﬂux nmol mg min Vmax,ah Vmaxah 1.14e7

pseudo Michaelis-Menten rate constant none kah kah 0.5

104 C.8 Permiability transistion pore parameters

Name units Symbol Xpp name Value

Total pore amount none P T PT ptpt 1.0

−1 −1 pH gating parameter nmol mg min p1 p1 1.77e-5

−1 −1 pH gating parameter nmol mg min p2 p2 1e-7

−1 −1 pH gating parameter nmol mg min p3 p3 1.77e-5

−1 −1 pH gating parameter nmol mg min p4 p4 1.0e-3

Calcium gating parameter µM p5 p5 0.5

Calcium gating parameter µM p6 p6 1e-2

Calcium gating parameter µM p7 p7 0.5

Calcium gating parameter µM p8 p8 1.0 Proton permeability mg min nmol−1(mV )−1 permlh permlh 1e2 Calcium permiability (µM)−1(mV )−1 permca permca 0.8

C.9 IP3 receptor and leakage parameters

Name units Symbol Xpp name Value

vIP3 µM vIP 3 vIP3 3000

vLEAK µM vleak vLEAK 0.1

dIP3 µM dIP 3 dIP3 0.25

dinh µM dinh dinh 1.4

dact µM dact dact 1 tau sec τ tau 4

105 C.10 SERCA pump parameters

Name units Symbol Xpp name Value

vserca µM vserca vserca 110

kserca µM kserca kserca 0.4

C.11 Inputs

Name units Symbol Xpp name Value

Baseline IP3 concentration µM baseline baseline 0.3

Amplitude of IP3 input pulse µM amplitude amplitude 0.8 Start time for IP3 input sec init init 10

Duration of IP3 input sec duration duration 200

−1 −1 Applied calcium current nmol mg min JCa,in Jcain 0

106 APPENDIX D

MODEL EQUATIONS

107 D.1 Compartmental equations

M = V pmito dmito

C = V pcytosol dcytosol

E = V per der

Vc = V pcytosol

Vm = V pmito

Ve = V per

Ψm = ∆Ψ + Ψb,i − Ψb,m

6 pHm = − log10 Hm10 dmito − 10( pHc¡)106 ¢ Hc = dcytosol

∆pH = pHc − pHm 2.303 R T Z = g Fg 2.303 R T Z = F

∆pg = Ψ − Zg pHg

∆p = Ψ − Z pH

108 D.2 Nucleotide conservation

12 d AT P = mito − ADP m µ m 8 d NAD = mito − NADH µ m

AT Pi = 2 − ADPi

ADPm,free = 0.8 ADPm

ADPi,free = 0.3 ADPi

3− ADPm = 0.45 ADPm,free

3− ADPi = 0.45 ADPi,free

− MgADPi = 0.55 ADPi,free

4− AT Pi = 0.05 AT Pi

4− AT Pm = 0.05 AT Pm

109 D.3 Cytosolic components

Jgly,num = 0.0249 [123.3(1+1.66 glc)(glc AT Pi)]

Jgly,dena = 1+4AT Pi + 1.3 glc (1+2.83AT Pi)

2 Jgly,denb = 0.16 glc (1+2.66AT Pi)

Jgly,den = Jgly,dena + Jgly,denb

Jgly,num Jgly,total = Jgly,den

Jp,gly = 2 Jgly,total

Jhyd,max Jhyd = 41 AT Pi + 8.7 2.7 1+( glc )

stepup = heav(t − init)³ ´

stepdown = heav(t − (init + duration))

IP 3 = baseline + amplitude (stepup − stepdown)

110 D.4 Mitochondrial components

Ca Ca 3 MWC = c 1+ c num 6 6 µ ¶ µ 4 ¶ Cac 50 MWCdenom = 1+ + 6 Cac 2 µ ¶ 1+ 0.38 .8 MWC h i MWC = num ¡ ¢ MWCdenom (∆Ψ − 91) VD = uni 13.35 (−VDuni) VDuni MWC − Cam exp Juni = ρuni − (−VDuni) £ 1 exp ¤ PSI−91 ( . ) VDnaca = exp 53 4 1 1 Jnc = ρnc VDnaca 2 9.4 1+(0.003 dmito/Cam) 1+ 30 NADm = 8 − NADHm ¡ ¢ + ([H ]mr4) (∆Ψ−r5) r2 [NADHm − r3] + exp − exp r6 Jhres = r1 (∆Ψ−r ) [1 + exp 7 ] r8 + ([H ]mo4) (∆Ψ−o5) o2 (NADHm − o3) + exp − exp o6 Jo = o1 (∆Ψ−o ) [1 + exp 7 ] o8 (∆Ψ−p2) + exp p3 − ([H ]m − p4) p5 − 1 Jpf1 = p1 (∆Ψ−h7) + [1 + p6AT Pm exp ] p8 − [[H ]m − p9] p10 + p11 (∆Ψ−h2) + exp h3 − ([H ]m − h4) h5 − 1 Jhf1 = h1 (∆Ψ−h7) + [1 + h6AT Pm exp ] h8 − [[H ]m − h9] h10 + h11

Jhl = ρleak ∆p 1 fpdh = 15 1 + 1.1 1+ Cam 2 (1+ . ) µ 0 05 ¶ Jred = Jred,basal + 6.3944 fpdh Jgly,total

111 Jmax,ANT = 900 4− 3− AT Pi ADPm −Ψ ant1 = exp ADP 3− AT P 4− 26.7 µ i ¶ m µ ¶ 4− AT Pi −Ψ ant2 = 1+ exp ADP 3− 53.4 µ i ¶ µ ¶ ADP 3− ant = 1+ m 3 AT P 4− µ m ¶ 1 − ant Jant = J 1 max,ANT ant ant µ 2 3 ¶ J J = red,basal + 0.84 f J ptca 3 pdh gly,total

Jah = kah (Hc − Hm) p1 − x ptpl∞(x) = 0.5 1+ tanh p · µ 2 ¶¸ 1 τ (x) = ptpl − cosh x p3 2 p4

³ ´ x − p5 ptph∞(x) = 0.5 1 + tanh p · µ 6 ¶¸ 1 tau (x) = ptph − cosh x p7 2 p8 + ziF ∆Ψ [H³ ]m −´Hc exp J = P RT ptplh h − ziF ∆Ψ 1 exp RT¡ ¢ CAM − CAC exp ziF ∆Ψ J = P ¡ ¢ RT ptplca ca − ziF ∆Ψ 1 exp RT ¡ ¢ ¡ ¢

112 D.5 Endoplasmic Reticulum components

IP 3 3 Ca 3 J = v c H3 + v erout IP 3 IP 3+ dIP 3 Ca + dact leak Ã µ ¶ µ c ¶ ! (CaER − Cac) 2 Cac Jserca = vserca 2 2 kserca + Cac

113 D.6 Differential Equations

dCac M (Jnc − Juni) − E (Jserca − Jerout)+ Jptplca M + Jcain = fi dt Vc minute dH d − H (Ca + d ) = inh c inh dt τ dCaER E (Jserca − Jerout) = fi dt Ve minute dCam Juni − Jnc − Jptplca = fm M dt Vm minute dΨ −(−J + J + J + J − J + 2 J − 2 J = M hres hf1 ant hl ptplh uni ptplca dt cmito minute dNADHm J − J = M red o dt µMmM Vm minute dADP J − J − J m = M ant ptca pf1 dt µMmM Vm minute dADP −Jant M +(Jhyd − Jpgly) i = C dt µMmM Vc minute dH J + J − J + J − J m = f hf1 hl hres ah ptplh dt h minute dPTPl − (PTPl − (PTPt − PTPh) PTPl∞(hm)) = dt τptpl(hm) PTPh − (PTPh − (PTPt − PTPl) PTPh∞(cam)) = dt τptph(cam)

114 D.7 Initial conditions

Cac(0) = 0.087

CaER(0) = 7.73

Cam(0) = 0.12

H(0) = 0.95

Ψ(0) = 161

NADHm(0) = 0.11

ADP M(0) = 9.4

ADPi(0) = 0.213

Hm(0) = 1.67e − 5

PTPl(0) = 0.0

PTPh(0) = 0.0

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