Modeling the non-linear dynamics of calcium inchromaffin cells

Ines Filipa Completo Guerreiro

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisors: Prof. Maria Teresa Ferreira Marques PinheiroProf. David Gall

Examinatiom CommitteeChairperson: Prof. Luıs Filipe Moreira Mendes

Supervisor: Prof. Maria Teresa Ferreira Marques PinheiroMember of the Committe: Prof. Hugo Fernando Santos Tercas

September 2015

Acknowledgments

I would like to thank my supervisors, Prof. Maria Teresa Pinheiro from Instituto Superior Tecnico

and Prof. David Gall from the Universite Libre de Bruxelles, for all the patience and availability to

answer all my questions, and for their guidance throughout all the work. I also want to thank Prof.

Genevieve Dupont from the Universite Libre de Bruxelles, who was a precious help in the development

of this work.

Finally, I would like to thank my family and close friends for all the support, not only during the time

I worked on this project, but throughout the entire course.

i

Resumo

Em resposta ao medo ou stress, celulas cromafins (celulas encontradas principalmente na medula

das glandulas supra-renais, localizadas acima dos rins) secregam catecolaminas como a adrenalina

na nossa corrente sanguınea. A secrecao e controlada por oscilacoes de calcio intracelular, que

podem ser medidas experimentalmente.

Medicoes experimentais de calcio em celulas de cromafina de ratos demonstram que diferentes

celulas pertencentes a uma mesma populacao, quando submetidas a um mesmo estımulo, apresen-

tam diferentes padroes oscilatorios. A origem desta heterogeneidade nos padroes oscilatorios e uma

questao em aberto.

Esta tese apresenta um novo modelo matematico que descreve a dinamica nao-linear do calcio

em celulas cromafins, onde oscilacoes de calcio sao produzidas pela interacao entre a atividade

eletrica da membrana celular e a libertacao de calcio do retıculo endoplasmatico. Este novo modelo

foi construıdo juntando e adaptando dois modelos existentes: o modelo de Gall-Susa, que descreve

as oscilacoes de calcio intracelular em celulas pancreaticas tendo em conta a atividade eletrica da

celula, e o modelo de Li-Rinzel, que descreve as oscilacoes de calcio tendo em conta o fluxo de

calcio vindo a partir do retıculo endoplasmatico para o citosol atraves dos receptores IP3 presentes

na sua membrana. A validade do nosso modelo foi estudada reproduzindo algumas observacoes ex-

perimentais. Nomeadamente, estudou-se o comportamento do modelo na presenca de agentes far-

macologicos a que as celulas foram submetidas experimentalmente. Para alem disso, reproduziram-

se ambos os padroes oscilatorios observados experimentalmente, apresentando-se uma explicacao

plausıvel e razoavel para a existencia dos mesmos: os dois diferentes padroes oscilatorios observa-

dos resultam da existencia de celulas de diferentes tamanhos na populacao analisada.

Palavras-Chave

Celulas cromafins, oscilacoes de calcio intra-celular, receptor IP3, dinamica nao-linear

iii

Abstract

In response to fear or stress, chromaffin cells (cells found mostly in the medulla of the adrenal

glands, located above the kidneys) secrete catecholamines like adrenaline into our bloodstream. Se-

cretion is controlled by intracellular calcium concentration oscillations, which can be measured exper-

imentally.

Experimental measurements of calcium in rat chromaffin cells show that different cells from the

same population display different oscillatory patterns. The origin of this heterogeneity remains an

open question.

In this thesis we construct a new mathematical model that describes the non-linear dynamics of

intracellular calcium in chromaffin cells where calcium oscillations are produced by the interplay be-

tween the electrical activity of the cell membrane and the release of calcium from the endoplasmic

reticulum. We do that by coupling and adapting two existing models: Gall-Susa model, which de-

scribes intracellular calcium oscillations in pancreatic β-cells having into account the electrical activity

of the cell and Li-Rinzel model, which describes calcium oscillations having into account a flux cal-

cium coming from the endoplasmic reticulum into the cytosol through IP3 receptors present in its

membrane. We study the validity of our model by reproducing experimental observations. Namely,

we study the behavior of the model in the presence of pharmacological agents and we reproduce

both oscillatory patterns observed.

We find that a difference in the radius of the cells where the intracellular calcium concentrations

were measured provides a reasonable testable explanation for the existence of different types of

oscillations observed in chromaffin cells.

Keywords

Chromaffin cells, intracellular calcium oscillations, IP3 receptor, non-linear dynamics

iv

Contents

1 Introduction 3

1.1 Intracellular Calcium Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Calcium oscillations in chromaffin cells . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Regulation of intracellular calcium . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.3 Models for calcium oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Electrical Properties of the Cell Membrane . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Chromaffin cells are excitable cells . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.2 The origin of membrane potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.2.A Ion channels and transporters . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 Equivalent circuit of the cell membrane . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.4 Models for membrane potential oscillations . . . . . . . . . . . . . . . . . . . . . 17

1.3 Aim of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Dynamics of theoretical models 23

2.1 Li-Rinzel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Gall-Susa model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 A new model for Ca2+ oscillations in chromaffin cells 35

3.1 New model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Study of the behavior of the model in the presence of pharmacological agents . . . . . . 45

3.3 Influence of the cell capacitance on the Ca2+ oscillatory patterns . . . . . . . . . . . . . 47

4 Conclusions and Perspectives 49

Bibliography 53

Appendix A Appendix A A-1

A.1 XPPAUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

A.2 Equations and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

A.3 *.ode Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-4

v

List of Figures

1.1 Experimental measurements of calcium in rat chromaffin cells provided by Dr. A. Mar-

tinez, Faculty of Medicine of Universidad Autonoma de Madrid . . . . . . . . . . . . . . 5

1.2 Schematic representation of the one-pool model for signal-induced calcium oscillations. 7

1.3 Schematic representation of the channel gating kinetics of the De Young-Keizer model

with the possible states of a subunit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Steady state open probability of the IP3 receptor as a function of cytosolic calcium

concentration from the De Young-Keizer model. . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Schematic view of the cell membrane and the different mechanisms of ionic transport. . 12

1.6 Permeable selectivity of the cell membrane. . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 A single-channel recording using the patch-clamp technique . . . . . . . . . . . . . . . . 14

1.8 Measurement of channel currents in response to different voltage pulses . . . . . . . . . 14

1.9 Comparison of the cell membrane and its equivalent electrical circuit. . . . . . . . . . . 15

1.10 Overall current produced by an ionic specie. . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.11 Equivalent electrical circuit of the cell membrane (Hodgkin and Huxley, 1952) . . . . . . 17

2.1 Reproduction of the time course of the intracellular calcium concentrations and of the

variable p for the Li-Rinzel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Bifurcation diagram of the Li-Rinzel model as a function of the inositol 1,4,5-trisphosphate

concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Reproduction of the time courses of the electrical activity, V, the gating variable, n, and

calcium concentrations in the cytosol and in the ER for the Gall and Susa model. . . . . 29

2.4 Overlap of the graphs for the membrane potential and the calcium concentration in the

ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Time course of V for the reduced 2-variable fast subsystem of Gall-Susa model. . . . . 31

2.6 Bifurcation diagram of the reduced 2-variable fast subsystem of Gall-Susa model . . . 31

2.7 Sketch of the bifurcation diagram of the Gall-Susa model . . . . . . . . . . . . . . . . . 32

3.1 Experimental observations where the electrical activity of cell membrane is measured

in parallel with the variation of cytosolic calcium . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Scheme of fluxes and currents in the cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Reproduction of the time courses of all the variables of the final model. . . . . . . . . . . 40

3.4 Comparison of predict results with experimental results . . . . . . . . . . . . . . . . . . 41

vii

3.5 Time course of V for the reduced 3-variable fast subsystem of the new model. . . . . . . 42

3.6 Bifurcation diagram of the reduced 3-variable fast subsystem of the new model . . . . . 42

3.7 Sketch of the bifurcation diagram of the final model with calcium concentration in the

ER as a bifurcation parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.8 Overlap of the graphs for the membrane potential and the calcium concentration in the

ER for the new model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.9 Bifurcation diagram of final model with v1 as a bifurcation parameter . . . . . . . . . . . 45

3.10 Comparison of the predict results with experimental results when the IP3 channels are

inhibited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.11 Time course of the intracellular calcium concentrations for the new model with C = 4800

fF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.12 Time courses the intracellular calcium concentrations for the final model for two different

sets of parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

viii

List of Tables

A.1 Parameter Values of final model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

ix

Abbreviations

IP3: inositol (1,4,5)-trisphosphate

ER: Endoplasmic reticulum

ATP : Adenosine triphosphate

CICR: Calcium-induced calcium release

LV A: Low voltage activated

HV A: High voltage activated

HB: Hopf bifurcation

APB: Aminoethoxydiphenye bonate

1

1Introduction

Calcium concentration oscillations in chromaffin cells were measured and it was observed that

cells from a same population, submitted to the same external stimulus, present two distinct patterns

of calcium oscillations. It is not known why this happens and, until the moment, none of the existing

models that describes calcium oscillations in chromaffin cells is able to reproduce both patterns ob-

served experimentally. Given that calcium oscillations control the secretion of adrenaline and other

substances into our bloodstream [1] and, consequently, the functional response of our body to danger,

different patterns of calcium oscillations are most likely going to result in different functional responses.

The work is divided into four chapters. In this first chapter we explain how the intracellular calcium

concentration is regulated in chromaffin cells. Namely, we study how IP3 receptors present in the

membrane of the endoplasmic reticulum (ER) and the electrical activity of the cell membrane control

the intracellular calcium concentrations. We also introduce the mechanisms underlying the models

that will be used in the construction of our new model, the Li-Rinzel and the Gall-Susa model. In

the second chapter, the Li-Rinzel and Gall-Susa model are analyzed in more detail; the third chapter

is where the new model is presented, as well as its results, and finally the fourth chapter where

conclusions and perspectives for future work are presented.

3

1.1 Intracellular Calcium Dynamics

Calcium takes part in several physiological processes important for the cell, one of them being

signaling. Variations in the cytosolic concentration of calcium, induced by an external stimulus, act

as a signal to the cell, controlling a diverse range of cellular processes, such as gene transcription,

muscle contraction and hormonal secretion.

In most cells, calcium has its major signaling function when it is elevated in the cytosolic compart-

ment as a consequence of calcium release from internal stores and/or calcium entry through plasma

membrane channels. The most common pattern of calcium signaling is a temporal pattern of peri-

odic discharges and elevations of cytosolic calcium concentrations[2] that are the result of numerous

channels and pumps that allow calcium to enter and exit cells and move between the cytosol and

intracellular stores.

1.1.1 Calcium oscillations in chromaffin cells

In chromaffin cells, calcium plays an essential role in coupling chromaffin cell stimulation to the

secretory response. Chromaffin cells from the adrenal medulla synthesize, store and secrete cate-

cholamines like adrenaline, contributing to the cardiovascular and metabolic adaptations of the body

to stressful and/or dangerous situations. The secretory response is triggered by an increase in the

free cytosolic concentration of calcium ions, Ca2+.

Rat chromaffin cells in culture exhibit oscillations of cytosolic calcium concentration when 100 µM

of methacholine, a synthetic drug that stimulates the production of IP3 in the cytosol, are adminis-

trated to the cells. Typical examples of such oscillations are shown in Fig.1.1.

The intracellular calcium measurements were made using Fura 2, a ratiometric fluorescent dye

that binds to free intracellular calcium. Fura 2 is excited at 340 nm and 380 nm of light and, regardless

of the presence of calcium, at 510 nm. Calculating the ratio between the fluorescent intensities

detected at 340 nm and 380 nm and the intensities detected at 510 nm it is possible to estimate the

level of intracellular Ca2+ [3].

4

(a)

(b)

Figure 1.1: Experimental measurements of calcium in rat chromaffin cells provided by Dr. A.Martınez, from the Faculty of Medicine of Universidad Autonoma de Madrid, using a ratiometric fluo-rescent dye that binds to free intracellular calcium, Fura 2. The concentration of cytosolic calcium fora sample of chromaffin cells extracted from the adrenal gland of a rat were measured. Throughout allthe measurements the cells were submitted to a dose of 100 µM of methacholine, a synthetic sub-stance that acts as an external stimulus provoking the synthesis of the IP3 molecule in the cytosol.Two different oscillatory patterns were observed, (a) and (b).

Despite the development of several models that attempt to describe intracellular calcium oscilla-

tions, it remains to be established the basic mechanisms by which these Ca2+ oscillations arise in

chromaffin cells.

Clearly, the mechanisms by which a cell controls its Ca2+ concentration are of central interest.

In particular, one would like to understand why some cells display large Ca2+ peaks (with smaller

variations superimposed) as in Figure 1.1 (a), while others show a faster, base-line spiking, as in

Figure 1.1 (b). It is important to emphasize that in both cases the cells were submitted to the same

stimulus (100 µM of methacholine) and were measured under the same conditions.

5

1.1.2 Regulation of intracellular calcium

There are a number of Ca2+ control mechanisms designed to ensure that Ca2+ is present in the

cytosol in sufficient amount to allow the cell to perform its necessary functions without becoming toxic

to the cell.

Calcium flux into the cytosol can occur via two principal pathways: inflow from the extracellu-

lar medium through Ca2+ channels present in the cell membrane and release from internal stores,

namely from the endoplasmic reticulum (ER).

In chromaffin cells, extracellular calcium concentration is ∼ 1mM and Ca2+ concentration in the

ER is ∼ 0.5mM . The Ca2+ concentration in the cytosol (∼ 100nM ) is much lower than either the

extracellular concentration or the concentration inside the ER [2]. This disparity gives rise to a con-

centration gradient from the outside of the cell to the inside and from the interior of the ER to the

cytosol. This gradient can lead to a massive and rapid flux of calcium into the cytoplasm from the

external medium and from the ER, through channels present in the cell and in the ER membrane.

The calcium channels present in the cell membrane are voltage-gated channels which, as their

name indicates, open in response to depolarization of the cell membrane, allowing the entry of calcium

in the cell. These channels are only present in electrically-excitable cells.

Calcium release from internal stores such as the ER is the second major Ca2+ influx pathway, and

this is mediated principally by the IP3 receptor, a Ca2+ channel presence in the ER membrane that

opens in the presence of the IP3 molecule. The IP3 receptor is found predominantly in non-muscle

cells, like chromaffin cells.

The water-soluble IP3 molecule is free to diffuse through the cell cytoplasm and binds to IP3

receptors situated on the ER membrane, leading to the opening of those receptors and subsequent

release of Ca2+ from the ER into the cytoplasm. The activity of IP3 receptors is modulated by the

cytosolic Ca2+ concentration, with Ca2+ both activating and inactivating Ca2+ release, but at different

rates (inactivation has a slower rate). There are evidences that in many cell types, including chromaffin

cells, calcium oscillations occur at constant cytosolic IP3 concentration[4, 5].

The concentration of calcium in the cytoplasm increases in response to an external stimulus that

triggers the synthesis of IP3 in the cytosolic compartment. Calcium oscillations usually occur only

when the IP3 concentration is greater than some critical value and disappear again when this value

gets too large. Thus, there is an intermediate range of IP3 concentration that generates oscillations.

As mentioned earlier, large quantities of calcium in the cell become toxic. So, cells have mecha-

nisms to remove cytosolic calcium, controlling its concentration and keeping it from reaching harmful

levels. Calcium is removed from the cytosol in two principal ways: it is pumped out of the cell, and it

is sequestered into internal stores such as the ER. Since Ca2+ concentration in the cytosol is much

lower than either the extracellular concentration and the concentration inside the ER, both methods

of Ca2+ removal require expenditure of energy. Removal of Ca2+ from the cytosol is then mediated

by a Ca2+ ATPase pump that uses chemical energy stored in ATP to pump Ca2+ against its gradi-

ent. There are ATPase pumps on the cell membrane that pump Ca2+ out of the cell and on the ER

membrane that pump Ca2+ into the ER [6].

6

1.1.3 Models for calcium oscillations

In the field of calcium dynamics, modeling was promoted by the fact that cytosolic Ca2+ was

initially the only measurable variable of the system [2]. During the last few years, the molecular

mechanisms underlying Ca2+ oscillations have been increasingly investigated. As a consequence,

some theoretical models have been proposed.

One of the earliest model conceived predicted that the self-amplification of Ca2+ release from the

ER into the cytoplasm lies at the basis of intracellular Ca2+ oscillations. This regulation is a result of

a positive feedback exerted by cytosolic Ca2+ on its release from intracellular stores, known as Ca2+

-induced Ca2+ release (CICR). Experimental observations indicate that CICR underlines oscillations

of the cytosolic calcium concentration [Ca2+]i in a variety of cells, namely in chromaffin cells.

In its original version, CICR assumed the existence of two types of pools - one sensitive to IP3

and other sensitive to Ca2+. According to this model, IP3 is synthesized in response to external

stimulation and binds to receptors located in the ER membrane, provoking the liberation of Ca2+ into

the cytosol. Through CICR, the latter increase triggers the release of Ca2+ from the Ca2+ sensitive

store. However, the existence of two-pools turned out to not be necessary as the IP3 receptor is itself

sensitive to IP3 and Ca2+. Thus, sustained Ca2+ oscillations can still arise from a one-pool model,

where the same Ca2+ channel is sensitive to both IP3 and Ca2+ behaving as co-agonists [8].

Figure 1.2: Schematic representation of a one-pool model for signal-induced Ca2+ oscillations. Im-age reproduced from [8].

The first theoretical model based on microscopic kinetics of IP3 and Ca2+ gating on the IP3

receptor is the De Young-Keizer model for Ca2+ oscillations. This model takes into account that

fast CICR is followed by a slow inhibition of Ca2+ release through the IP3 receptor at high Ca2+

concentration.

The De Young-Keizer model for agonist-induced [Ca2+]i oscillations describes the kinetics of IP3

and Ca2+ gating of the IP3 receptor. In this model, release of Ca2+ is mediated by the IP3 receptor

while Ca2+ uptake is accomplished through (Ca2+)-ATPase pumps present in the ER membrane.

7

One IP3 receptor is composed by 4 subunits. The gating model assumes the existence of three

binding sites on each subunit of the IP3 receptor: one for IP3 and two for Ca2+ which include an

activation and an inactivation site. Thus, each IP3 receptor subunit, Sijk, is assumed to have three

distinct binding sites: one activation binding site for IP3, one activation binding site for Ca2+ and one

inactivation binding site for Ca2+ which are labeled, respectively, by the first (i), the second (j) and

the third subscript (k). An empty site is denoted by ′0′ and an occupied site by ′1′. A subunit is in

an open state when it has the activation binding site of the IP3 and of the Ca2+ occupied and the

inactivation binding site of the Ca2+ empty, and it is said to be in the state S110. When analyzing the

dynamics of the IP3 receptor, the different subunit states can be represented in terms of xijk =[Sijk]ST

(with∑

i,j,k xijk = 1), which is the fraction of IP3 receptors in the state characterized by the subscript′ijk′. ST is the total amount of subunits.

Figure 1.3: Schematic representation of the channel gating kinetics of the De Young-Keizer modelwith the possible states of one subunit, Sijk. The rate at which the activation binding site of IP3, i,changes from occupied (empty) to empty (occupied) is α, the changing rate of the activation, j, andinactivation binding site of Ca2+, k, is β and γ. Image reproduced from [9]

The model assumes that Ca2+ passes through the IP3 receptor only when three subunits are in

8

the state x110. Thus, the opening probability of the receptor is x3110. One of the key properties used

in formulating models of the IP3 receptor is the experimental analysis of the open channel probability

as a function of [Ca2+]i. Bezprozvanny et al. [10] showed that this open probability is a bell-shaped

function of cytosolic Ca2+. Thus, at low [Ca2+]i an increase in [Ca2+]i increases the open probability

of the receptor, while at high [Ca2+]i an increase in [Ca2+]i decreases the open probability (Figure

1.4).

Figure 1.4: Steady state open probability of the IP3 receptor as a function of cytosolic calciumconcentration, [Ca2+]i, from the DYK model for three different IP3 concentrations. The receptor isactivated quickly by Ca2+ but inactivated by Ca2+ on a slower time scale. This characteristic isincorporated in the magnitude of the rate constants β and γ. Image reproduced from De Young andKeizer, 1992.

The De Young-Keizer model is a nine-variable model that, while unique in giving detailed gating

kinetics, depends on a relatively high number of variables. On the construction of a new model that

describes the non-linear dynamics of calcium in chromaffin cells, a simplified version of the De Young-

Keizer model will be used, the Li-Rinzel model. By exploiting the time scale difference between the

fast activation and slow inactivation of the IP3 receptor by Ca2+ as found in experiments [11], Li and

Rinzel reduced the nine-variable De Young-Keizer model to a two-variable system [9].

This means that, using a multiple time scales method, the De Young-Keizer model is reduced to

a two-variable system, the Li-Rinzel model. The Li-Rinzel model consists of a two equations model

that describes the dynamics of IP3 receptor-mediated [Ca2+]i oscillations, and it is represented by

the following equations

d[Ca2+]idt

= (c1v1w3∞p

3([Ca2+]ER − [Ca2+]i)︸ ︷︷ ︸receptor flux

+ c1v2([Ca2+]ER − [Ca2+]i)︸ ︷︷ ︸leakage flux

− v3[Ca2+]2ik23 + [Ca2+]2i︸ ︷︷ ︸

pump flux

(1.1)

dp

dt=p∞ − p

τp(1.2)

9

where c1 is volume ratio between the ER and the cytosol, v1, v2 and v3 are, respectively, the

maximum IP3-gated, an IP3 - independent leakage permeabilities, and maximum pump rate; p is the

fraction of channels not yet inactivated by Ca2+ (= x000 + x100 + x010 + x110 ), k3 is the dissociation

constant of Ca2+ to the pump, and where

w∞ =

([IP3]

[IP3] + d1

)([Ca2+]i

[Ca2+]i + d5

)τp =

1

a2(Q2 + [Ca2+]i)

p∞ =Q2

Q2 + [Ca2+]i

Q2 = d2[IP3] + d1[IP3] + d3

where d1 is the threshold value for activation of IP3 receptor by IP3, d2 is the threshold inactivation

by Ca2+ of IP3 receptor when IP3 is bound, d3 is the threshold inactivation by Ca2+ when there is

no IP3 bound to the receptor, d5 is the threshold activation by Ca2+ and a2 is the binding rate of

Ca2+ to the inhibiting site of the IP3 receptor. The first term in equation 1.1 describes the Ca2+

flux through the IP3 receptor, and it is proportional to the concentration difference between the ER

and the cytosol. The fraction of open channels is given by w3∞p

3, conserving the same form as the

original model (x3110). Then, the bell-shaped channel opening curve at steady-state (Figure 1.4) can

be understood as the ’weighing’ between a sigmoid activation w3∞ and inactivation curve p3∞. The last

term of equation 1.2 describes the action of Ca2+ ATPase pumps that pump Ca2+ from the cytosol

into the ER. It is important to mention that this model considers that the IP3 receptors are uniformly

distributed along the ER membrane which, even though is an acceptable explanation, it does not

correspond to the true: the IP3 receptors are localized in the membrane of the ER. The Li-Rinzel

model is described and analyzed in more detail in section 2.

1.2 Electrical Properties of the Cell Membrane

Every living cell has a plasma membrane. The cell membrane is selectively permeable, only

allowing certain ions to pass through it, and it controls the flux of ions in and out of the cell, maintaining

a difference of ions concentration between the extra and intracellular medium. This disparity gives

rise to a potential difference across the membrane. Under resting conditions, this potential remains

approximately constant, around -70 mV. This voltage, characteristic to each cell type, is called resting

potential. However, in some cells that are called ’electrically excitable’, the potential can change when

the cell is submitted to an external stimulus.

As mentioned before, an external stimulus may provoke the opening of Ca2+ channels in the

cell membrane, which grants the entry of Ca2+ from the extracellular medium into the cell. This

phenomenon can be better understood by studying the electrical properties of the cell membrane and

the models for membrane potentials.

10

1.2.1 Chromaffin cells are excitable cells

All cells have a resting potential that arises from an electrical charge gradient across the plasma

membrane when the cell is under no external stimulation. A resting potential of - 70 mV means that

the potential inside the cell membrane is about - 70 mV relative to the surrounding medium (which

is defined to be 0 mV), and the cell is said to be polarized. This situation changes when the cell

is subjected to an external stimulus. Depending on how cells respond to this stimulus they can be

considered excitable or non-excitable cells.

If a current is applied for a short period of time in a non-excitable cell, the membrane potential

changes, but it returns directly to its equilibrium value after the applied current is removed. On the

other hand, for excitable cells, if the applied current is sufficiently strong, the membrane potential

exhibits a large excursion towards higher voltages called action potential, in a non-linear manner.

Chromaffin cells are excitable cells [12].

Many cells use the membrane potential as a signal to operate different functions. In the case

of chromaffin cells, the electrical behavior of the cell membrane influences Ca2+ entry and, as a

consequence, catecholamines release, among which adrenaline.

The electrical activity of excitable cells has its origin in the property of proteins composing the

cell membrane - which include channels and transporters - to control the passage of ions across the

membrane.

1.2.2 The origin of membrane potential

The cell membrane provides a boundary separating the interior of the cell from its external envi-

ronment. This membrane consists essentially of phospholipids and proteins. The molecular structure

of phospholipids is characterized by the existence of a polar hydrophilic portion connected to two

hydrophobic chains. This particular structure implies that in an aqueous medium these molecules

spontaneously form a bilayer of about 50-100A [13], associating the hydrophobic portions and having

the hydrophilic parts facing the outside. The highly hydrophobic interior of the bilayer is a barrier to

the diffusion of solvated ions. This low diffusion of ions through a phospholipid bilayer results in an

intrinsic high electrical resistivity of the cell membrane. Thus, while some lipid-soluble material with a

low molecular weight can easily pass through the hydrophobic lipid core of the membrane by simple

diffusion, polar substances cannot do so and additional transport mechanisms are necessary to allow

the passage of such substances across the plasma membrane. As a result, the cell membrane is

said to be selectively permeable.

The transport across cell membranes is orchestrated by specific protein domains embedded in

the cell membrane, such as ion channels and transporters. Ion channels provide mechanisms of

entry/exit of ions following concentration gradients without energy input, whereas transporters can

move ions against concentration gradients with or without energy expenses.

11

Figure 1.5: Schematic view of the cell membrane and the different mechanisms of ionic transport.Some ions are passively transported through ion channels and transporters according to their con-centration gradient, or they are actively transported through transporters against their concentrationgradient, resorting to the use of energy. Image reproduced from [14]

If the membrane is selectively permeable to a certain specie of positively charged ions, these

ions can cross the cell membrane according to their gradient concentration, leaving behind uncom-

pensated negative charges (Figure 1.6). This separation of charges and consequent concentration

gradient gives raise to the membrane potential. This means that now the ions are not only in the

presence of a concentration gradient, but also of an electric field.

Figure 1.6: Permeable selectivity of the cell membrane. The system is divided in two homogeneouscompartments, I and II, occupied by a solution of different concentrations (CI < CII ). The twocompartments are separated by a membrane permeable to a certain specie of positively charged ionsbut impermeable to a certain specie of negatively charged ions. Due to the concentration difference,the positive ions diffuse through the membrane into compartment I. Compartments I and II start tobe, respectively, positively and negatively charged. The two solutions behave as two conductors inequilibrium with electrical potentials VI and VII , and their respective excess charges will be locatedon either side of the membrane, giving rise to a potential VI − VII > 0. Image reproduced from [14]

12

In the presence of an electric field, ions are submitted to a force. The current intensity per surface

unit that results from the movement of ions that are in the presence of a concentration gradient and

an electric field, traversing the membrane is given by

J = − zFDdc(x)

dx︸ ︷︷ ︸simple diffusion

− z2F 2Dc(x)

RT

dΦ(x)

dx︸ ︷︷ ︸diffusion under the effect of an electric field

(1.3)

where c(x) is the concentration of ions, D is the diffusion coefficient, z is the ion charge, F is

the Faraday’s constant, R is the gas constant, T is the ionic temperature and V (x) is the potential

associated with the electric field in question. The first term of equation 1.3 corresponds to the current

that results from the movement of ions due to the existence of a gradient concentration; the second

term of the equation is the ionic current that results from the movement of ions due to the presence

of an electric field. This phenomenon is called electrodiffusion,

Taking the example of Figure 1.6, as the concentration gradient provokes the migration of positively

charged ions to compartment II, an excess of positive charge in this compartment starts to appear,

repelling other positive ions that are trying to cross the membrane. At equilibrium, the net flux of

positive ions is null and we have

0 =dc(x)

dx+Fzc(x)

RT

dΦ(x)

dx(1.4)

which leads to

dΦ(x)

dx= − RT

Fzc(x)

dc(x)

dx= −RT

Fz

d

dx(lnc(x)) (1.5)

Integrating equation 1.5 we have that, in equilibrium, the voltage difference across the membrane

for a certain ionic specie is given by the Nerst equation (equation 1.6).

∆Φ = VI − VII =RT

zFln

(CII

CI

)(1.6)

1.2.2.A Ion channels and transporters

Ion channels are molecules that control the flow of ions across the cell membrane by opening

and closing in response to different types of stimulus. For example, voltage-gated channels open in

response to a change in the potential of the membrane, and mechanically-gated channels open in

response to a mechanical stress.

A channel opens and closes in an apparently random way (Figure 1.7), allowing a fixed amount of

current (on the order of picoamperes) to flow through it when it is in the open state.

This means that it is not possible to know at any given time whether a channel will be open or

closed. However, the laws of probability allow us to make a prediction of the average behavior of the

channel. Thus, by measuring the time during which a channel is open, topen during a total time of t,

we can characterize the channel’s behavior and define its open probability, PO.

13

Figure 1.7: A single-channel recording using the patch-clamp technique. The patch-clamp techniqueis a laboratory technique in electrophysiology that allows the record of currents of single ion channels.From Sanchez, Dani, Siemen, and Hille (1986).

PO =topent

(1.7)

This open probability is going to depend on modulating factors of the channels’ activity, like the

transmembrane potential or the presence of a ligand.

When subjecting the membrane to different electrical stimulus and measuring the amplitude of the

currents when the channel opens (Figure 1.8 (a)) we see that the relation between the single channel

current and membrane voltage is a straight line (Figure 1.8 (b)). This means that an open channel

behaves like an ohmic conductor.

(a) (b)

Figure 1.8: Measurement of channel currents in response to different voltage pulses. (a) Recordof a current produced by a channel formed by two molecules inserted in an artificial lipid bilayer inthe absence of transmembrane concentration gradient. We see that the current varies when varyingthe transmembrane potential difference. (b) The current through the open channel obeys Ohm’s law.Image reproduced from [15].

Ion channels have pores with a diameter of 5-8 A that are ion selective. Due to the variety of

pores’ molecular dimension and to the existence of charge distribution on the interior of the walls of

the channels, the pore size will allow to discriminate between different types of ions based on their

size and charge. Some ion channels become active or inactive with changes in membrane potential.

The Ca2+ voltage-gated channels can be low voltage activated (LVA) or high voltage activated (HVA),

14

based on the membrane potential threshold value.

On the other hand, transporters are highly selective proteins, which have binding sites for specific

ions or molecules. Substrate binding is associated with a conformational change such that the bound

substrate molecule, and only that molecule, is transported across the membrane. Transport through

this type of membrane protein domains can be either passive, facilitating the diffusion of a molecule

following its electrochemical gradient, or active, assisting transport against an electrochemical gradi-

ent (Figure 1.6). In the later, work done has energy costs obtained from phosphate release from ATP

by hydrolysis. This mechanism maintains the activity of Ca2+ ATPase pump, that extorts Ca2+ from

the cytoplasm.

The activity of ion channels and transporters generates differences in ion transmembrane concen-

tration originating an accumulation of charge from both sides of the membrane, by electrodiffusion.

The lipid bilayer may therefore be regarded as a capacitor. Moreover, one can think of the cell

membrane as a capacitor in parallel with ionic currents, and can describe the cell membrane as an

electrical circuit.

1.2.3 Equivalent circuit of the cell membrane

The electrical properties of excitable cells can be described in terms of electrical circuits, with each

component of the cell membrane acting as a basic component of an electrical circuit.

The cell membrane forms an insulating barrier separating electrically charged ions, and can be

thought of as a capacitor C in parallel with ionic currents.

Figure 1.9: Comparison of the cell membrane and its equivalent electrical circuit. The ion chan-nels inserted the cell membrane, when open behave as an ohmic conductor. Due to its selectivepermeability, the cell membrane acts as a capacitor.

Due to charge conservation, the electrical activity of the cell membrane can be described by

equation 1.8.

CdV

dt+ I = 0 (1.8)

where I represents the combination of all ionic currents passing through the cell membrane.

15

On the cell membrane there are several channels of different types, that are ion selective for

different species of ions. All the channels present in the membrane can be seen as a combination of

resistors connected in parallel (Figure 1.10). The current of each individual channel is therefore add

up to form the total ionic current I.

(a) (b)

Figure 1.10: Overall current produced by an ionic specie. (a) All channels present in the membrane,selective for a given ionic specie, can be seen as a combination of resistors connected in parallel,where γ is the channels individual conductance and i the current passing through one channel, gand I are, respectively, the conductance and the current that result from the combination of all theindividual channels. (b) The current produced by the opening of sodium channels voltage dependentfollowing a depolarizing potential. The central profile plot gives the total current corresponding to thesum of the current of 300 individual records, of which 9 are shown in the lower part of the figure.Image reproduced from [14]

This current is given by

I = g(t)(V − Vi) (1.9)

where g is the conductance that results from the combination of all the ion channel of a certain ion

specie, V the potential difference across the cell membrane, and Vi the Nerst equilibrium potential of

the ion specie i.

In terms of an electrical circuit, the different ion selective channels work in parallel with the cell

membrane, as illustrated in Figure 1.11.

16

Figure 1.11: Equivalent electrical circuit of the cell membrane (Hodgkin and Huxley, 1952). Eachbranch of the electrical circuit represents the contribution to the total transmembrane current from aspecific ion specie channel. In this case, three different type of ion channels are embedded in the cellmembrane, Na+, K+ and a leakage L.

The ion selective permeability of the cell membrane is represented by an electrical circuit com-

posed of a resistance in series with a voltage source equal to the ion equilibrium potential. In turn,

the representative branches of a specific ion specie channel are in parallel with a capacitance.

1.2.4 Models for membrane potential oscillations

The most important and complete work when it comes to the study of generation and propagation

of electrical signals in excitable cells is the work of Alan Hodgkin and Andrew Huxley. They developed

the first quantitative model of the propagation of an electrical signal along an axon, where the ionic

currents involved in the propagation of an action potential in an axon were revealed (K+ and Na+),

as well as the dynamics of its conductances.

They hypothesized the existence of selective ion channels whose opening is dependent on the

membrane potential. This means that the probability of a certain ion channel to be open varies with

the potential of the cell membrane. For example, the opening probability of the K+ channels varies

according to equation 1.10.

dn

dt=n∞(V ) − n

τn(V )(1.10)

where n∞ is the open probability in the stationary state and τn is the characteristic time. The

variable n is also know as the activation variable for the potassium channels. Hodgkin and Huxley

also noticed that, from an experimental point of view, the conductances of the K+ channels vary

according to equation 1.11.

gK(V ) = gKn4 (1.11)

17

where gK is the maximal conductance of the potassium channels, which corresponds to the sit-

uation where all the channels are open. A similar formalism is used to describe the dynamics of the

conductance of the Na+ channel, where we have that the activation and inactivation variables, m and

h, vary according to equation 1.12 and 1.13.

dm

dt=m∞(V ) −m

τm(V )(1.12)

dh

dt=h∞(V ) − h

τh(V )(1.13)

and the conductance gNa varies accordint to equation 1.14

gNa(V ) = gNam3h (1.14)

where gNa is the maximal conductance of the ion channels. The activation and deactivation vari-

ables m and h satisfy a first order transition of the same type as n, and one can write the following

differential equations,

Reminding that an ionic current is of the form I = g(V −Vi), we have that the membrane potential

oscillations in an excitable cell is given by equation 1.15

CdV

dt= −gKn4(V − VK) − gNam

3h(V − VNa) − gL(V − VL) (1.15)

where the last term represents a leakage current, referring to the influence of other currents pass-

ing through the membrane of the cell that are not significant enough to be considered individually.

Equation 1.15 and equations 1.13, 1.12 and 1.10 form the set of the Hodgkin-Huxley equations.

The Hodgkin-Huxley model describes the repetitive firing of action potentials in an axon but it can

be adapted to describe more complex behavior exhibited by other excitable cells.

Many cells’ behavior is characterized by brief bursts of oscillatory activity interspersed with qui-

escent periods during which the membrane potential changes only slowly; this behavior is called

bursting.

Calcium oscillations are modulated inside an electrical model of the cell membrane. Thus, the

Hodgkin-Huxley model can be adapted to describe intracellular Ca2+ oscillations. In particular, the

Gall-Susa model that investigates the role of plasma membrane Na+/Ca2+ in pancreatic β-cells. For

that, it describes intracellular Ca2+ oscillations having into the account the electrical activity of the cell

membrane and its potential variations, which is given by

CdV

dt= −IK − ICa − IK(Ca) (1.16)

dn

dt=n∞(V ) − n

τn(V )(1.17)

18

with

IK = gKn(V − VK)

ICa = gCam∞(V − VCa)

IK(Ca) = gKCa[Ca2+]i(V − VK)

70 + [Ca2+]i

where IK is the delayed rectifier K+ current, ICa is the voltage-dependent L-type (a subtype of

HVA) Ca2+ current, IK(Ca) is the Ca2+ -activated K+ current, and n is the gating variable for the

delayed-rectifier K+ channel. During the bursts, when the L-type Ca2+ channels are open, the Ca2+

is pumped from the cytosol in the ER. This Ca2+ accumulated in the ER is then gradually released in

the cytosol. Equation 1.18 gives us the variation of cytosolic Ca2+ concentration.

d[Ca2+]idt

= −αICa − kCa[Ca2+]i︸ ︷︷ ︸membrane flux

+ krel([Ca2+]ER − [Ca2+]i)︸ ︷︷ ︸

ER flux

− kpump[Ca2+]i︸ ︷︷ ︸ER pump flux

(1.18)

where α is a factor that converts current into concentration changes, kCa is the Ca2+ removal rate

by mechanisms other than sequestration in the ER, krel is the rate of calcium release from the ER

and kpump is the pump activity of the Ca2+ -ATPase in the ER. The last two terms in equation 1.18

are due to the presence of the ER. The evolution equation of calcium concentration in the ER is given

by

d[Ca]retdt

= −krel([Ca]ret − [Ca]i)︸ ︷︷ ︸ER flux

+ kpump[Ca]i︸ ︷︷ ︸ER pump flux

(1.19)

The Gall-Susa model will be described and analyzed in more details in section 2.

19

1.3 Aim of the Work

A diverse range of cellular activities is regulated through calcium signaling. Calcium is an im-

portant intracellular messenger that regulates cellular processes such as gene transcription, muscle

contraction and hormonal secretion. In the case of chromaffin cells, intracellular oscillations of calcium

control the secretion of catecholamines like adrenaline. Because of their wide availability, their ease

of isolation and preparation in primary cultures, and the fact that intracellular Ca2+ oscillations in chro-

maffin cells are produced by the same mechanisms that other secretory cells, chromaffin cells have

been widely used in biochemical and electrophysiological. Plus, chromaffin cells are structurally very

similar to pos-synaptic sympathetic neurons, being also used neuropharmacological studies [17, 18].

Thus, findings on catecholamine release in chromaffin cells can be extrapolated to basic mechanisms

in the central and peripheral nervous system [19]. It may even help to better understand the origin of

certain neurodegenerative diseases, since deregulation of Ca2+ -mediated signaling has been impli-

cated in many neurodegenerative diseases like Alzheimer’s disease [20]. In fact, the ER appears to

be a focal point when it comes to neuronal alterations that result in Alzheimer [21].

It was experimentally observed that the Ca2+ oscillations in chromaffin cells underlying the se-

cretion of catecholamines present two distinct patterns, fast oscillations imposed in a slow oscillating

pattern and fast spiking, which no existing model can explain. Given that all the cells analyzed where

submitted to the same stimulus (100µM of methacoline, a substance that triggers the synthesis of

IP3) and that different patterns of Ca2+ oscillations result in secretion of varying amounts of cat-

echolamines and, consequently, in different functional responses [22]. This puts into evidence the

importance of constructing a new mathematical model that describes the non-linear dynamics of in-

tracellular calcium in chromafin cells and that it is able to reproduce both patterns of oscillations found

experimentally, possibly explaining their existence. The already existing models that describe intra-

cellular Ca2+ oscillations in chromaffin cells fail to reproduce both pattern of oscillations and thus, to

provide an explanation for their existence.

Our aim is to build a biologically-plausible mathematical model able to reproduce the experimental

observations about calcium oscillations in rat chromaffin cells in order to help deciphering why similar

cells under the same conditions present different oscillatory patterns. For that, we will combine two

existing models: Li-Rinzel and Gall-Susa models.

Gall-Susa model describes calcium oscillations taking into account the electrical activity of the

cell membrane, but not the role of the IP3 receptor. On the other hand, Li-Rinzel model describes

calcium oscillations considering the contribution of calcium release from the ER only. Knowing that

both mechanisms play a major role in the regulation of [Ca2+]i, we will incorporate the flux of calcium

passing through the IP3 receptors and ER pumps described in the Li-Rinzel model, into the Gall-Susa

model, thus having a complete mathematical model that describes the non-linear dynamics of Ca2+

in chromaffin cells.

The fact that the Gall-Susa model structure resemblances the structure of Li-Rinzel encourages

the incorporation of one model into the other (both are based on the Hodgkin-Huxley model). Also,

20

both Gall-Susa and Li-Rinzel are simple models, having a small number of variables, which makes it

easy to incorporate one into the other.

We will also compare the behavior of this new model with experimental results of Ca2+ oscillations

in chromaffin cells in the presence of various pharmacological agents that interfere with Ca2+ chan-

nels, and propose some experiments to study the influence of certain parameters in the obtainment

of the different patterns of oscillations observed experimentally.

The new mathematical model we propose to build innovates by its simplicity. We retain only what

we know to be the essential mechanisms to produce intracellular Ca2+ oscillations (the electrical

activity of the cell membrane and the dynamics of the IP3 receptor in the ER). This makes the model

easy to study and to manipulate, and it facilitates future improvements that it may need. Also, it makes

it easier to adapt in order to study other related characteristics of chromaffin cells, serving as a tool for

future studies. As mentioned earlier, there are mathematical models for intracellular Ca2+ oscillations

in chromaffin cells. These models however, don not reproduce the two oscillatory patterns observed

experimentally by Dr. Martınez, and their complexity makes it hard to make the necessary study and

consequent changes to reproduce both patterns and provide an explanation for their existence.

21

22

2Dynamics of theoretical models

The model that will be constructed to study the Ca2+ dynamics in chromaffin cells results from the

coupling between two previously published models: one model that describes the plasma-membrane

Ca2+ dynamics [23] - Gall-Susa model - and another that describes the Ca2+ exchanges between

the endoplasmic reticulum and the cytosol [9] - Li-Rinzel model.

In this section, these two theoretical models - Li-Rinzel and Gall-Susa - will be analyzed. In

particular, we will reproduce some published results that result from the integration of the equations

that describe each model, and present some new ones that will give us a better understanding about

the origin of Ca2+ oscillations according to each model.

23

2.1 Li-Rinzel model

The Li-Rinzel model is a two-variable model that describes the cytosolic Ca2+ dynamics, d[Ca2+]idt ,

taking into account the activation/deactivation dynamics of the IP3 receptors, dpdt . Using the software

XPP and XXP-AUTO (see Appendix A), we reproduced the results published by Li and Rinzel in 1994

(Figure 2.1 and 2.2).

The Li-Rinzel model is composed by the following set of equations:

d[Ca2+]idt

= (c1v1w3∞p

3([Ca2+]ER − [Ca2+]i)︸ ︷︷ ︸receptor flux

+ c1v2([Ca2+]ER − [Ca2+]i)︸ ︷︷ ︸leakage flux

− v3[Ca2+]2ik23 + [Ca2+]2i︸ ︷︷ ︸

pump flux

(2.1)

dp

dt=p∞ − p

τp(2.2)

The first equation describes the Ca2+ exchange between the cytosol and the ER. When the con-

centration of calcium in the endoplasmic reticulum, [Ca2+]ER, is greater than the concentration in the

cytosol, [Ca2+]i, a flux of Ca2+ flows from the interior of the ER into the cytosol through IP3 receptors

present in the ER membrane, described by the first term of equation 2.1. While v1 represents the

rate/permeability of the IP3 receptor, w3∞p

3 represents the probability of the IP3 receptor being open;

more precisely, w3∞ is the probability of the IP3 receptor having the activation binding site of IP3 and

of Ca2+ occupied and p3 is the probability of having the inactivation binding site of Ca2+ empty. This

last one varies according to equation 2.2, where p∞ is the fraction of not inactivated IP3 receptor in

the stationary state.

Besides the flux of Ca2+ that passes through the IP3 receptor, there is also a leakage flux released

from the ER, described by the second term of equation 2.1.

Finally, the last term of equation 2.1 reflects the activity of the ATPase pumps present in the ER

membrane that takes Ca2+ from the cytosol into the ER, against its concentration gradient, at a rate

v3.

This set of equations bears some resemblance with the Hodgkin-Huxley model of the plasma

membrane electrical excitability: instead of the membrane voltage V , we have the cytosolic concen-

tration [Ca2+]i has a major regulator, and instead of the voltage deviation from the Nerst potential

V − Vi we have the concentration gradient [Ca2+]ER − [Ca2+]i as the driving force. Plus, in both

models channel activation and inactivation appear as separate factors with first-order gating kinetics

(dndt , dmdt and dh

dt in the Hodgkin-Huxley model and dpdt in the Li-Rinzel model).

Analyzing Figures 2.1 (a) and (b) we see that the concentration of calcium in the cytosol [Ca2+]i

reaches its maximal value shortly after the maximum of p, which is consistent with what is expected

since the more IP3 receptors are in a non-inactivated state, the bigger is the probability of these

receptors to be open allowing Ca2+ to flow through it and increasing the concentration of calcium in

the cytosol [Ca2+]i.

24

(a)

(b)

Figure 2.1: Reproduction of the time course of p, (a), and of [Ca2+]i, (b), for the Li-Rinzel model.Computed solutions of equations 2.1 and 2.2[9].

25

The periodic oscillations observed in Figure 2.1 do not occur for all concentration of IP3 in the

cytosol [IP3]. The bifurcation diagram of the Li-Rinzel model as a function of [IP3] is shown in Figure

2.2.

Figure 2.2: Bifurcation diagram of the Li-Rinzel model as a function of [IP3]. Reproduction of theresults obtained for the two-variable Li and Rinzel’s model, where we have good agreement of thebifurcation diagram, and thus of the oscillatory response, for the two versions of the model - theoriginal 9-variable and the reduced 2-variable model. The numerical simulations represented in figure2.1(a) and (b) are situated, in the bifurcation diagram, in the zone denoted by the red-dotted line. Wecan see that we are in a stable periodic orbit and we have calcium oscillations. Parameter valuesused for the construction of the bifurcation diagram are given in [9].

For small and big values of [IP3] the system is in a stable steady state and we do not have calcium

oscillations. There is however a stable periodic zone, limited by two Hopf bifurcation points, HB1 and

HB2. In our simulation, [IP3] = 0.4 (red-dotted line in figure 2.2 ) and the system is in a stable

periodic orbit and we have sustained Ca2+ oscillations. It only makes sense to study the behavior

of the system as [IP3] changes since all the other parameters that compose the Li-Rinzel model are

parameters characteristic of the cell and thus, do not change for a same cell, while the the value of

[IP3] can change according to the intensity and frequency of an external stimulus that generates its

production.

26

2.2 Gall-Susa model

The mathematical description of Gall-Susa model is an adapted and expanded version of the

Hodgkin-Huxley model. It takes into consideration the electrical activity of the cell membrane and the

intracellular calcium dynamics. The model is described by the following set of equations:

CdV

dt= −IK − ICa − IK(Ca) (2.3)

dn

dt=n∞(V ) − n

τn(V )(2.4)

d[Ca2+]idt

= −αICa − kCa[Ca2+]i︸ ︷︷ ︸membrane flux

+ krel([Ca2+]ER − [Ca2+]i)︸ ︷︷ ︸

ER flux

− kpump[Ca2+]i︸ ︷︷ ︸ER pump flux

(2.5)

d[Ca]ER

dt= −krel([Ca]ER − [Ca]i)︸ ︷︷ ︸

ER flux

+ kpump[Ca]i︸ ︷︷ ︸ER pump flux

(2.6)

Equation 2.3 describes how the potential of the cell membrane varies as different ion currents

pass through it. Even though there are more currents crossing the cell membrane, this model only

takes into account the potassium current passing through voltage-gated K+ channels and through

Ca2+- activatedK+ channels - IK and IK(Ca), respectively -, and the calcium current passing through

voltage-gated Ca2+ channels.

Similarly to what was seen in the Hodgkin-Huxley model, each ion channel has a certain probability

of being in the activated and in the inactivated state. As already explained in the Introduction, this

probabilities vary with the cell membrane potential. The probability corresponding to the dynamics

of the voltage-gated calcium channels and to the Ca2+-activated K+ channels is very fast when

compared to the one of the voltage-gated potassium channels. Thus, we only have an evolution

equation for the gating variable for the K+ current, n, and the gating variables for the other currents

are considered instantaneous.

Equation 2.5 describes how the concentration of free cytosolic calcium varies as Ca2+ passes

through the cell membrane and through the ER membrane. Through the cell membrane, we have the

calcium current passing through voltage-gated Ca2+ channels that it is going to reflect in an increase

of [Ca2+]i (αICa); on the other hand, we have Ca2+ being pumped out of the cell by ATPase pumps

present in the membrane that work against its concentration gradient (kCa[Ca2+]i), contributing to

the decrease of [Ca2+]i. Besides this, there are Ca2+ exchanges between the cytosol and the ER -

last two terms of equation 2.5 and equation2.6. When [Ca2+]ER is bigger than [Ca2+]i, a leakage of

Ca2+ starts to flow according to its concentration gradient, krel([Ca2+]ER − [Ca2+]i), while there is

Ca2+ constantly being pumped from the cytosol into the ER by ATPases present in the ER membrane,

kpump[Ca2+]i.

The original Gall-Susa model, published in 1999, also considered the present of aNa+-dependent

Ca2+ current - INa/Ca - that we chose not to use since this current caused no significant difference

in the bursting mechanism and thus, in the electrical activity of the cell [23]. This way, we have a

simple model that with a small number of variables is able to describe the bursting that results from

27

the electrical activity of the cell.

Using XPP to numerically integrate equations 2.3- 2.6, we reproduced the results obtain by Gall

and Susa (model III of [23]) in terms of cytosolic calcium dynamics. The results are illustrated in

Figure 2.3.

(a)

(b)

28

(c)

(d)

Figure 2.3: Reproduction of the time courses of the gating variable for voltage-gated K+ channels, n,the electrical activity, V , and calcium concentrations in the cytosol, [Ca2+]i, and in the ER, [Ca2+]ER,for the Gall and Susa model. Numerical solutions of equations 2.3, 2.4, 3.7 and 2.6[23].

The time course of the membrane potential V (Figure 2.3 (b)) exhibits spikes of approximately con-

stant amplitude (∼ 25 mV). This fast spiking is followed by a silent phase where V slowly increases,

and we have bursting.

The dynamics of [Ca2+]i (Figure 2.3 (c)) is almost as fast as the dynamics of V , following the

29

same pattern of fast spiking followed by a slow increase of the variable; the bursting oscillations in the

voltage are mirrored by bursting oscillations in cytosolic Ca2+ concentration.

On the other hand, [Ca2+]ER follows a much slower dynamic, slowly increasing during the active

phase and slowly decreases during the silent phase. This becomes clear when we overlap the two

graphs, V and [Ca2+]ER.

Figure 2.4: Overlap of the graphs for the membrane potential V and the calcium concentration in theER [Ca2+]ER. The concentration of Ca2+ in the ER slowly increases during the active phase of V ,and slowly decreases during the silent phase.

Exploiting the time scale difference that exists between the dynamics of the variables of the system

will help us to describe and interpret, in a simple manner, the behavior of the full model.

Rinzel (1985) used this approach to exploit the control mechanism of bursting in excitable cells

by studying how the slow behavior of intracellular Ca2+ influences the generation of fast spikes [24].

Mathematically, this is equivalent to consider a fast subsystem with the slow variable as a parameter,

which in the case of the model in study is [Ca2+]ER. We can reduce Gall-Susa 4-variable model

(V, n, [Ca2+]i, [Ca2+]ER) to a 3-variable fast subsystem (V, n, [Ca2+]i) with a slow variable [Ca2+]ER.

Our analysis, begins by treating the slowly varying quantity [Ca2+]ER as a parameter. We start

by studying the simple case in which no spikes occur during the active phase. We generate such

behavior in our model by imposing the instantaneous variation of n with V . This means that n reaches

its steady state n∞ instantaneously. To simulate this behavior, we consider n = n∞ and the evolution

equation for n disappears (n is now treated as a parameter), reducing Gall-Susa fast subsystem to a

2-variable system (V, [Ca2+]i).

30

Figure 2.5: Time course of V for the reduced 2-variable fast subsystem of Gall-Susa model(V ,[Ca2+]i) with the gating variable n and [Ca2+]ER as parameters. By considering the instantaneousadaption of n to the voltage V , the spikes in the active phase of bursting disappear.

Figure 2.6: Bifurcation diagram of the reduced 2-variable fast subsystem of Gall-Susa model(V ,[Ca2+]i) with the slow variable [Ca2+]ER as a bifurcation parameter

The Z-shaped curve in Figure 2.6 is the V-nullcline of the fast subsystem. In other words, it

corresponds to the values of V for which the total current passing through the cell membrane I =

IK + IK(Ca) + ICa is null, which corresponds to a steady state. The upper and lower branches of this

31

curve correspond to stable steady states, while the intermediate branch (dashed line) corresponds to

unstable steady states. We can see that the slow dynamics of [Ca2+]ER causes a switch between the

upper and lower steady state.

If we now take into consideration the delay in the adaptation of n to V , we have a fast subsystem

of 3-variables (V, n, [Ca2+]i) with [Ca2+]ER as a variable. The bifurcation diagram of such a system is

plotted in Figure 2.7.

Figure 2.7: Sketch of the bifurcation diagram of the Gall-Susa model for β cells, with [Ca2+]ER as thebifurcation parameter. The Z-shaped curve denotes the steady states of V as a function of [Ca2+]ER.A solid line indicates a stable steady state, and a dashed line indicates an unstable steady state. Thetwo branches denote the maximum and minimum of V over one oscillatory cycle. HB denotes a Hopfbifurcation point. A burst cycle of the full model (equations 1.31-1.34) is projected, in blue, on the (V,[Ca2+]ER) plane. (Results obtained with XPP-AUTO).

For increasing values of [Ca2+]ER, a stable steady state looses its stability through a Hopf bifur-

cation. The stable limit cycle then looses its stability through an homoclinic bifurcation. After this

bifurcation, a stable steady state corresponding to a low voltage is established.

If now we couple the dynamics of the fast subsystem to the slower dynamics of [Ca2+]ER, we

obtain the burst cycle for the full model (represented by the blue line of Figure 2.7). The behavior of

the full system can be understood as slow variations of the fast subsystem.

When the system is in a periodic stable state, it is in repetitive spiking mode (active phase). At

this point, the voltage-gated Ca2+ channels are open and calcium is entering the cell, which makes

the cytosolic Ca2+ concentration increase and, consequently, the Ca2+ concentration in the ER. The

entry of Ca2+ in the cytosol increases the number of open Ca2+ -activated K+channels. The opening

of this channels will provoke the exit of positively charged ions (K+) from the intracellular medium,

which is going to repolarize the cell membrane and the resting potential is restored (low branch of

stable steady state in Figure 2.7). At the resting potential, the Ca2+ channels are closed, the entry of

32

Ca2+, which are positively charged ions, into the cell stops. The membrane potential starts to slowly

increase as Ca2+ is release into the cytosol from the ER (silent phase). The membrane potential

increases until a value for which the voltage-gated Ca2+ channels open again completing the cycle.

It is now clear that the slow dynamics of [Ca2+]ER causes the switch between spiking (active

phase) and a steady state behavior (silent phase). The length of the burst is determined by the am-

plitude of the [Ca2+]ER oscillations (the burst cycle is limited by [Ca2+]ER = 154µM and [Ca2+]ER =

167µM ).

This allows us to conclude that there are two oscillatory processes interacting to make bursting: a

fast oscillation in V imposed on a slower oscillation in [Ca2+]ER.

33

34

3A new model for Ca2+ oscillations in

chromaffin cells

As mentioned before, our goal is to construct a new model for the intracellular calcium dynamics

in chromaffin cells, by coupling the Gall-Susa and the Li-Rinzel models. This requires an adaptation

of the models as they are put together, to ensure that their dynamics remain approximately the same

as new variables are added. Namely, Gall-Susa model will have to undergo some changes as terms

of Li-Rinzel model are added to it.

In this section, it will be explained in more detail what changes to the models we had to make and

why. We will also present an analysis and qualitative viewpoint of bursting in chromaffin cells using

the same method as for Gall-Susa model, and we will exploit the behavior of the new constructed

model by trying to reproduce some experimental observations; particularly we will study how the

model behaves in the presence of pharmacological agents that inhibit the activity of the IP3 receptor.

Finally, we will propose a reasonable testable explanation for the existence of different types of

oscillations observed in chromaffin cells.

35

3.1 New model

Gall-Susa model describes the dynamics of intracellular Ca2+ having into account the exchange

of Ca2+ between the extracellular medium and the cytosol. The voltage-gated Ca2+ channels open

and close in response to the electrical activity of the cell membrane.

Figure 3.1: Experimental observations where the electrical activity of the cell membrane V is mea-sured in parallel with the variation of cytosolic Ca2+. All the significant variations of cytosolic Ca2+

concentration coincide with variations of the membrane potential. The measurement of cytosolic Ca2+

concentration and of the membrane potential where made simultaneously. For this experiment, theonly stimulus to which the cell is submitted are the variations of potential of the cell membrane. Thus,this does not describe a situation where we have Ca2+ oscillations and the secretion of adrenaline.

Figure 3.1 illustrates the relation between the membrane potential of the cell and the cytosolic

concentration of Ca2+.

Gall-Susa model also describes the Ca2+ exchange between the cytosol and the ER, but not

through the IP3 receptor. Instead, it only considers a leakage flux of Ca2+ that passes through

channels present on the ER membrane that play a minor role. It doesn’t account, however, the Ca2+

passing through the IP3 receptors nor their complex dynamics.

Experiments performed by Dr. A. Martınez from the Universidad Autonoma de Madrid show that

in the presence of a pharmacological agent that inhibits the activity of the IP3 receptor, the Ca2+

oscillations disappear. This leads us to the conclusion that the IP3 receptors play a fundamental role

in the generation of cytosolic Ca2+ oscillations.

Unlike Gall-Susa, Li-Rinzel model describes the exchange of Ca2+ between the cytosol and the

ER having into account the flux of Ca2+ that passes through the IP3 receptors. Thus, in order to have

a model that fully describes the intracellular Ca2+ dynamics of chromaffin cells we need to have a

model that describes not only the electrical activity of the cell membrane and the flux of Ca2+ passing

through it but also the flux of Ca2+ passing through the IP3 receptors present in the ER membrane.

36

Figure 3.2: Scheme of fluxes and currents in the cell according to the new model. Present on the cellmembrane are ion channels that allow the entry and exit of ions, namely the entry of Ca2+ into thecell, and ATPases that pump the intracellular Ca2+ outside the cell. This ATPases are also present inthe ER membrane, preventing the cell to reach toxic values of Ca2+. Embedded in the ER membranethere are also the IP3 receptors that, when open allow the passage of Ca2+ from the ER into thecytosol. Image reproduced from [14].

Such model can be built by incorporating the flux of Ca2+ passing through the IP3 receptors of

Li-Rinzel model and the evolution equation describing its dynamics into Gall-Susa model.

In Li-Rinzel model, the flux of calcium through the IP3 receptors is given by

f = c1v1w3∞p

3([Ca2+]ER − [Ca2+]i) (3.1)

while the dynamics of the IP3 receptors is described by the following equation

dp

dt=p∞ − p

τp(3.2)

Adding this two components of the Li-Rinzel model to the Gall-Susa model, we obtain a model

described by the following set of equations

37

dV

dt= − 1

C

(IK + ICa + IK(Ca)

)(3.3)

dn

dt=n∞ − n

τ(3.4)

dp

dt=p∞ − p

τp(3.5)

d[Ca2+]idt

= −αICa − kCa[Ca2+]i︸ ︷︷ ︸membrane flux

+ krel([Ca2+]ER − [Ca2+]i)︸ ︷︷ ︸leakage flux

− c1kpump[Ca2+]i︸ ︷︷ ︸ER pump flux

(3.6)

+ c1v1w3∞p

3([Ca2+]ER − [Ca2+]i)︸ ︷︷ ︸flux trhough the IP3 receptors

(3.7)

d[Ca2+]ER

dt= − 1

c1krel([Ca

2+]ER − [Ca2+]i)︸ ︷︷ ︸leakage flux

+ kpump[Ca2+]i︸ ︷︷ ︸ER pump flux

− c1v1w3∞p

3([Ca2+]ER − [Ca2+]i)︸ ︷︷ ︸flux through the IP3 receptors

(3.8)

Because this set of equations describes variations of Ca2+ concentrations, they must depend on

the volume of the ER and the cytosol. Gall-Susa model assumes that the volume of the ER and the

volume of the cytosol are approximately the same, which is not accurate. The ratio between the ER

and the cytosol is ≈ 20%. Because of this, we also had to add a variable c1(= V olER

V olcytosol) to the terms

of the model corresponding to the Gall-Susa model.

Also, the values of some parameters had to be adjusted so that the order of magnitude of the

fluxes and the dynamics of the channels involved in the exchange of Ca2+ remain roughly the same

in our new model as in the Gall-Susa and Li-Rinzel models.

In particular, we had to change the threshold value of activation of the IP3R by IP3 d1 (from

d1 = 0.13 to d1 = 1), the threshold value for activation of the IP3R by Ca2+ d5 (from d5 = 0.08234 to

d5 = 0.2), the maximum IP3-gated permeability v1 (from v1 = 6s−1 to v1 = 0.45s−1). Although these

values are very different from the ones of the original model (Li-Rinzel model), there is no agreement

on the realistic values for this parameters and so, there is no reason to exclude the values chosen by

us and to not consider them valid.

The value for the rate of Ca2+ pumped out of the cell, kpump, originally used in the Gall and

Susa’s model (kpump = 0.2ms−1) has been modified in agreement with experimental data. We had

to decrease the value of this parameter conserving the behavior of the original model. For that,

the change of kpump (from kpump = 0.2ms−1 to kpump = 0.0589) led to a change of the rate of

Ca2+ release from the ER krel (from krel = 0.0006ms−1 to krel = 0.0002ms−1), of the maximal

conductance of the Ca2+ -activated K+ channels gK(Ca) (gK(Ca) = 30000pS to gK(Ca) = 29000pS),

and the maximal conductance of the K+ channels gK (from gK = 2700pS to gK = 1600pS). As

already referred, the conductance depends on the number of channels existing on the cell membrane.

Considering the order of magnitude of the conductances in study, the decreases performed to their

original values do not reflect on a dramatic change of the number of channels, and thus remain valid.

However, it is important to note that this is merely an estimate having into consideration the values

for the conductances of the ion channels found in literature [23], since there is no way to measure the

conductance of one single channel.

38

All the other values remain the same and were already validated for the publication of the respec-

tive original models, [9] and [23]. The values of all the parameters that are going to incorporate the

new model are listed in Table A.1 (Appendix).

The set of equations 3.3 - 3.8 forms a new mathematical model that describes the non-linear

dynamics of intracellular Ca2+ in chromaffin cells. Equation 3.3 describes the electrical activity of the

cell membrane having into account the contribution of the potassium current K+ from voltage-gated

potassium channels IK and from Ca2+ -activated potassium channels IK(Ca), and the contribution

of the Ca2+ current, ICa. Equation 3.4 and 3.5 describe the dynamics of the gating variable of the

voltage-gated K+ channels and the dynamics of the IP3 receptors, respectively. Equation 3.7 and

3.8 describe the concentration variations of Ca2+ in the cytosol and in the ER, where all the terms of

the evolution equations retain their original significance.

A numerically computed solution was developed for the final cell model, which is shown in figure

3.3.

(a)

39

(b)

(c)

Figure 3.3: Reproduction of the time courses of all the variables of the final model.(a) Gating variableof the K+ channels. (b) Fraction of IP3 receptors not yet inactivated by Ca2+. (c) Membrane’s poten-tial of the cell. (d) Cytosolic calcium concentration. (e) Calcium concentration in the ER. Computedsolutions of equations 3.3 - 3.8(consult Table A.1 for parameters’ values)

The results obtained, for this model with the set of parameters described in Table A.1, exhibit

bursts that bear a qualitative resemblance to one of the pattern of oscillations obtained experimentally.

Comparing figures 3.4 (d) and (e) we see that in both cases we have fast spikes during the active

40

phase followed by a gradual decrease in [Ca2+]i.

(d)

(e)

Figure 3.4: Comparison of predict results with experimental results. (d) Intracellular calcium concen-trations, [Ca2+]i, for the final model. (e) Experimental measurements of calcium in rat chromaffin cellsprovided by Dr. A. Martınez, Faculty of Medicine of Universidad Autonoma de Madrid

One can wonder if [Ca2+]ER is the slow variable that controls burst in this full model as well.

The calcium variations in the ER are much slower than the variation of cytoplasmic Ca2+, that varies

almost as quickly as the voltage of the cell membrane.

We start by evaluating the role of this slow variable in bursting by studying the behavior of the

fast subsystem (V ,p,[Ca2+]i), considering n = n∞, like what was done in section 2.2 for the study of

Gall-Susa model.

41

Figure 3.5: Time course of V for the reduced 3-variable fast subsystem of the new model. Byconsidering the instantaneous adaption of n to the voltage V , there are no spikes in the active phaseof bursting.

Like for the Gall-Susa model, by considering the instantaneous adaption of n to the voltage V in

the new model, there are no spikes in the active phase of bursting (Figure 3.5).

In order to study the influence of the slow variable [Ca2+]ER in the passage from the active to the

silent phase in bursting, we plot the bifurcation diagram of the fast subsystem (V, p, [Ca2+]i) with the

slow variable [Ca2+]ER as a bifurcation parameter (Figure 3.6).

Figure 3.6: Bifurcation diagram of the reduced 3-variable fast subsystem of the new model.

42

The Z-shaped curve is the V-nullcline of the fast subsystem (values of V for which the total current

I = IK +IK(Ca)+ICa is null). The upper and lower branches of this curve correspond to stable steady

states, while the intermediate branch correspond to unstable steady states. We can see that the slow

dynamics of [Ca2+]ER causes the switch between the upper and lower steady state.

If we now take into consideration the delay in the adaptation of n to V , we have a fast subsystem

of 4-variables (V, n, p, [Ca2+]i) with [Ca2+]ER as a variable. The bifurcation diagram of such a system

is plotted in Figure 3.7.

Figure 3.7: Sketch of the bifurcation diagram of the final model with [Ca2+]ER as the bifurcationparameter. The Z-shaped curve denotes the steady states of V as a function of [Ca2+]ER. A burstcycle of the full model (equations 3.3 - 3.8) is projected, in blue, on the (V, [Ca2+]ER) plane. Resultsobtained with XPP-AUTO.

Coupling the dynamics of the fast to the slow subsystem, we obtain the burst cycle for the full

model represented in Figure 3.7 by a blue line.

When the system is in a high-V state, the system is in repetitive spiking mode and [Ca2+]ER

increases (active phase); when the system is in a low-V steady state, we are in the silent phase and

[Ca2+]ER decreases. We see that once again the length of the burst is determined by the amplitude

of [Ca2+]ER oscillations and so this variable works as a control mechanism of bursting. In fact, Figure

3.8 makes it clear that the active phase is when [Ca2+]ER slowly increases and the silent phase when

[Ca2+]ER slowly decreases.

43

Figure 3.8: Overlap of the graphs for the membrane potential and the Ca2+ concentration in theER. During the active phase, [Ca2+]ER is slowly increasing, while during the silent phase is slowlydecreasing.

While having a similar shape, the graph of [Ca2+]i according to our new model differs from the

one obtain with Gall-Susa model in the way the silent phase is processed. In the new model, the

silent phase runs for a longer time than in Gall-Susa, which is a consequence of the dynamics of

the IP3 receptors. When the voltage-gated Ca2+ channels close, preventing the entrance of Ca2+

into the cell from the extracellular medium and putting an end to the active phase, there is still a

considerable amount of Ca2+ being release into the cytosol from the ER, passing through the IP3

receptors that remain open for as long as there is Ca2+ enough to bind to the activation binding sites.

Thus, the decrease of [Ca2+]i in the silent phase is less abrupt than in the Gall-Susa model and the

branch corresponding to the silent phase has a small declive, being more similar to the experimental

observations.

44

3.2 Study of the behavior of the model in the presence of phar-macological agents

In one of the experiments performed by Dr. A. Martınez, the calcium dynamics was measured in

the presence of an IP3 channel inhibitor pharmacological agent (2-Aminoethoxydiphenyl borate). It

was observed that after the introduction of the agent in the system, the Ca2+ oscillations ceased.

The equivalent on our mathematical model to inhibiting the IP3 channels is to change the value

of the parameter v1 that scales the number of IP3 channels in the ER membrane. We can study

the effect of variation of v1 in the system through a bifurcation diagram, using v1 as a bifurcation

parameter.

Figure 3.9: Bifurcation diagram of final model with v1 as a bifurcation parameter. For small values ofv1, the system is in a steady state and we have no oscillations of [Ca2+]i.

For small values of v1, the system is in a steady state. From the Hopf bifurcation point HB beyond,

the system is in a periodic oscillatory state, where the amplitude of the oscillations increase as we

increase the value of v1.

Now that we have constructed a mathematical model that describes the non-linear dynamics of in-

tracellular Ca2+ in chromaffin cells, we intend to reproduce some experimental observations. Namely,

we want to know how does our system behave when we inhibit the activity of the IP3 receptors. This

is accomplished by decreasing the value of v1. Figure 3.10 shows what happens when the permeabil-

ity of the IP3 receptors is reduced from 0.45s−1 to 0.1s−1. We see that at the time the value of v1 is

changed, the oscillations cease, which is in qualitative agreement with the experimental observations.

45

(a)

(b)

Figure 3.10: Comparison of the predict results with experimental results when the IP3 channels areinhibited. (a) Experimental measures of intracellular calcium oscillations in the presence of a pharma-cological agent that inhibits the activity of the IP3 channels. We see that when the IP3 channels areinhibited, the calcium oscillations stop. 2-APB 100 µM concerns the pharmacological agents usedto inhibit the IP3 channels. (b) Time evolution of final model when v1 is decreased. At t = 30s, thevariable that regards the number of IP3 channels in the ER is reduced from v1 = 0.45 to v1 = 0.1. Wesee that when we make this change, the calcium oscillations disappear.

46

3.3 Influence of the cell capacitance on the Ca2+ oscillatory pat-terns

As mentioned before, two patterns of oscillations for calcium concentrations were observed exper-

imentally. One type of oscillations was already obtained (Figure 3.4) with the parameters values of

Table A.1.

The main differences between the two oscillation patterns are the frequency and the amplitude

of the oscillations. Comparing the two patterns obtained experimentally we see the second pattern

(Figure 1.1 (b)) has a higher frequency and a smaller amplitude relative to the first (Figure 1.1 (a)). In

an attempt to reproduce the second pattern of oscillations, we changed the values of parameters that

could be the cause for the existence of the two different types of oscillations; namely the capacitance

of the cell membrane (from C = 5310fF to C = 4800fF ).

Figure 3.11: Time course of the intracellular calcium concentrations for the new model with C = 4800.

By decreasing the value of the capacitance, the new model exhibits single spikes with a higher

frequency and smaller amplitude than the pattern previously obtained, in section 3.1. Thus, it retains

the main characteristics of the second pattern obtained experimentally (Figure 3.12).

A decrease of the cell capacity from C = 5310fF to C = 4800fF translates in a decrease of the

cell radius from r = 6.5µm to r = 6.18µm (CA = c where A is the area of the cell 4πr2 and c the

specific capacitance that is 1× 10−6F ). This presents a possible explanation for the different patterns

of oscillations observed: the sample of chromaffin cells analyzes may have been composed of cells

of different sizes.

47

(a)

(b)

(c)

Figure 3.12: Time evolution of Ca2+ of the final model for two different sets of parameters. (a) Timeevolution of the final model with the value of parameters as described in the Tabel A.1. (b) Timeevolution of the final model with C = 4800fF . When the value of the capacitance C is decreasedfrom 5310fF to 4800fF we obtain an oscillatory pattern with single spikes and, in comparison withthe previous oscillatory pattern obtain, with a higher frequency and a lower amplitude of oscillations.(c) Experimental measurements of calcium in rat chromaffin cells provided by Dr. A. Martınez, Facultyof Medicine of Universidad Autonoma de Madrid

48

4Conclusions and Perspectives

The work developed in this master thesis was set out to build a new mathematical model that

describes the non-linear dynamics of intracellular calcium in chromaffin cells. The motivation behind

the construction of this new model came from the experimental observations of two distinct oscillatory

patterns in chromaffin cells from the same population and submitted to the same conditions, and the

non-existence of a good explanation for this phenomenon. Our goal was to build a model able to

provide such an explanation by reproducing both types of intracellular calcium oscillations observed.

Not only were we able to do that, but we were also able to reproduce some other experimental results

concerning chromaffin cells and their intracellular calcium concentrations, in order to validate our

model.

Calcium oscillations in chromaffin cells are produced by the interplay between the electrical ac-

tivity of the cell membrane and the release of calcium from the endoplasmic reticulum. Thus, our

model needs to have into consideration this two components. To build the new mathematical model

describing the non-linear dynamics of chromaffin cells, we coupled two existing models: one that

describes calcium oscillations cells taking into account the electrical activity of the cell membrane -

Gall-Susa model - and other based on IP3 receptor-mediated calcium oscillations - Li-Rinzel model.

Both these models are an adapted version of the Hodgkin-Huxley model. The Gall-Susa model de-

scribes intracellular calcium variations of concentration using the model of the plasma membrane

electrical excitability of Hodgkin and Huxley to describe the electrical activity of the cell membrane

in pancreatic β-cells. The Li-Rinzel model, even though it does not describe the electrical activity of

the cell has the same structure of the Hodgkin-Huxley model: [Ca2+]i, instead of the voltage V , has

the role of the major regulator of the system, and the concentration difference [Ca2+]ER − [Ca2+]i,

instead of the voltage deviation V − Vi, is the driving force. Thus, both models follow the same struc-

ture, which facilitates the incorporation of one into the other. Plus, Gall-Susa model already takes into

49

account a flux of calcium coming in and out of the ER, which makes it more straight forward to add the

terms from Li-Rinzel corresponding to the IP3 receptor. All this, plus the fact that the Gall-Susa model

describes the calcium concentration in the cytosol as a fast variable, which is in agreement to what

happens in chromaffin cells, makes the coupling of the Li-Rinzel and Gall-Susa models an efficient

way to construct a new mathematical model able to describe the non-linear dynamics of calcium in

chromaffin cells.

There are models that describe the intracellular calcium oscillations in neurons taking into account

the electrical activity of the cell membrane and the dynamics of the IP3 receptors [25,26,27]. The pa-

rameters of these models could be adapted to describe intracellular calcium oscillations in chromaffin

cells. However, these models are very complex, having a high number of variables, which makes it

hard to study them and make any modifications. Plus, they fail to reproduce both patterns of oscil-

lations observed by Dr. Martınez in chromaffin cells and the intracellular calcium dynamics that they

describe does not correspond to the one observed in chromaffin cells. For example, they describe a

chaotic bursting patterns of intracellular calcium and they describe the concentration of calcium in the

cytosol as a slow variable.

In order to couple Gall-Susa and Li-Rinzel models, we added the dynamics of the IP3 receptor

h described in the Li-Rinzel model to the Gall-Susa, as well as the Ca2+ flux through it. To do this

without changing its order of magnitude (the Ca2+ flux through IP3R in the new model can not be too

different than the Ca2+ flux through IP3R in the Li-Rinzel model), we had to adapt the value of some

parameters of the original models. Namely, we had to increase the threshold value of activation of the

IP3R by the IP3, d1, increase the threshold value of activation of the IP3R by Ca2+, d5, and decrease

the maximum IP3-gated permeability, v1. There are no experimental evidences of the real values of

these parameters, which makes our values as valid as the ones used in the Li-Rinzel model.

Another adaptation of the model that we had to make concerns the volume ratio of the ER and the

cytosol. Li-Rinzel model takes into account the fact that the ER and the cytosol have different volume

by considering a volume ratio between the ER and the cytosol c1 smaller than one. On the contrary,

in Gall-Susa model the volumes of the ER and the cytosol are considered to be approximately equal.

Moreover, Gall-Susa uses a value for the activity of the ATPase pump in the cell membrane that it is

bigger than what it is considered a realistic value [6]. To make the terms of the Gall-Susa model take

into account the difference in volumes between the ER and the cytosol, and to have a more realistic

value for the pump activity, we had to change some other parameters. Namely, we had to decrease

the rate of Ca2+ pumped out of the cell (kpump), the rate of Ca2+ release from the ER (krel), the

maximal conductance of the Ca2+ -activated K+ channels (gK(Ca)) and the maximal conductance of

the K+ channels (gK). There is no agreement in literature about the real value of the conductances.

Even so, we tried to deviate as less as possible from the values describe in Gall-Susa.

The new model presents, for the values describes in Table A.1 (Appendix), bursting of the cell

membrane potential. The variations in cytosolic Ca2+ concentration are almost as fast as the vari-

ations in potential. On the contrary, the Ca2+ concentration in the ER varies more slowly; it slowly

increases during the active phase and slowly decreases during the silent phase. This dynamical be-

50

havior can be described on an analysis based on the two different time scales occurring in the system:

the fast subsystem composed by V , n, h and [Ca2+]i, and a slow subsystem composed by [Ca2+]ER.

The behavior of the full system can be understood as slow variations of the fast subsystem, which

means that the slow dynamics of the calcium in the ER is what determines the time scale of the bursts.

With the parameters’ values described in Table A.1, we are able to reproduce one of the patterns

- rapid spikes followed by a slow decrease of the calcium concentration. The second pattern of oscil-

lations, simple oscillations with smaller amplitude and higher frequency, was obtained by decreasing

the value of the cell capacitance. The decrease of a cell capacitance is equivalent to a decrease of

its radius. Then, we can predict that the reason why cells of a same population of chromaffin cells

presents two different types of oscillations is because it is composed by cells of different sizes.

Comparing the oscillatory patterns obtained with the new model and the ones obtain experimen-

tally, we see that the time scale of the experimental measurements is much bigger than the one of

our model. This does not invalidate our model given that the time scale of our new mathematical

model is more realistic than the one obtain experimentally. This may be a result of the difficulties

that underlie these type of experimental measurements. Having access to the data corresponding to

these experimental measurements would help to better understand and interpret these experimental

results.

In summary, we have constructed a new model able to reproduce important experimental obser-

vations and that provides a reasonable testable explanation for the existence of the different types

of oscillations observed. However, this model is only a first attempt to reproduce this experimental

results and it has some limitations. In our model we only take into account the Ca2+ being pumped

into the ER and a flux of Ca2+ that it is released into the cytosol through IP3 receptors (besides a

small leakage flux). However, there is an extra Ca2+ flux that we did not take into account: the flux of

Ca2+ being released into the cytosol through the ryanodine receptors present in the ER membrane

[28]. We did not take this extra flux into account in our model because it did not reveal to be essential

to reproduce the two oscillatory patterns, and indeed we manage to reproduce qualitatively both types

of intracellular Ca2+ oscillations and provide a reasonable explanation for their co-existence, which

was our main purpose. Moreover, this model is based on simple existing models for calcium dynamics

and thus, does not take into account the presence of Na+ channels that play a role in the variation

of membrane’s potential in chromaffin cells [29] by allowing the passage of a sodium current through

the cell’s membrane. But once again, we did not yet take this current into consideration because we

wanted to construct a simple model able to describe our observations concerning the intracellular

calcium oscillations so that now it is easier to improve it and manipulate. This way this model serves

not only for the study of intracellular Ca2+ oscillations but may be eventually used, and after making

the necessary adaptations, to study other related phenomenon in chromaffin cells. For example, to

study the secretion of catecholamines in chromaffin cells and how our sympathetic nervous system

responds to it. Chromaffin cells are neuroendocrine cells that are structurally and functionally in close

proximity to post-synaptic sympathetic neurons, which are a part of the sympathetic nervous system.

Thus, a better understanding the mechanisms underlying calcium oscillations in chromaffin cells and

51

how different oscillatory patterns (and their origin) of intracellular calcium influence catecholamines

secretion can contribute to the study of the sympathetic nervous system and it may even give some

enlightenment about the source of nervous system related diseases.

It is important to mention that, in both the Gall-Susa and the Li-Rinzel model, and consequently on

our new model, it is assumed that the cells are equipotential, which is not true. However, we cannot

predict how this would change the intracellular calcium oscillations described by our model.

As mention before, the intracellular Ca2+ oscillations are directly related to the secretion of cat-

echolamines like adrenaline into our bloodstream. In fact, there are several models that study the

relation between different patterns of intracellular calcium (that are obtained when the cells are sub-

mitted to stimulus of different intensities and duration) and the catecholamine release. Thus, the cells

in study, even though they appear to have identical characteristics (apart, according to our conclu-

sion, from the size) provoke different catecholamine releases and consequently different functional

responses. For a future work, it would be interesting to study this aspect in more detail. For example,

find out how different the functional responses that result from the two oscillatory patterns are. Plus,

for the cell population studied 80% of the cells analyzed presented the second type of oscillations,

simple fast spikes with small amplitude, while only 20% presented bursting. It would also be interest-

ing to study what is the balance between the two types of oscillations in a population of cells from

different individuals and, in case it is not approximately the same for all individuals (80% of one type

oscillations and 20% of others like in this case) study how differently their system responds (change

of the hearth beat and of blood pressure, for example) to a same external stimulus.

52

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56

AAppendix A

A-1

A.1 XPPAUT

All numerical simulations where made using the software XPPAUT that is freely available at

http://www.math.pitt.edu/ bard/xpp/xpp.html. XPPAUT is an interactive package for numerically solv-

ing and analysing differential equations using a fourth order Runge-Kutta method. It also provides an

interface to most the most common features of bifurcation analysis called AUTO [32].

In this Appendix, we list the XPPAUT codes (*.ode files) used to produce some of the graphs in

Chapters 2 and 3. Namely, all the time series of the different models presented and the respective

bifurcation diagrams.

The equations of the different models analyzed where integrated using a Runge-Kutta method.

The bifurcation diagrams where made with AUTO, a program built into XPPAUT[32].

A.2 Equations and Parameters

New model

dV

dt= − 1

C

(IK + ICa + IK(Ca)

)dn

dt=n∞ − n

τdp

dt=p∞ − p

τp

d[Ca2+]idt

= −αICa − kCa[Ca2+]i + krel([Ca2+]ER − [Ca2+]i) − c1kpump[Ca2+]i

+ c1v1w3∞p

3([Ca2+]ER − [Ca2+]i)

d[Ca2+]ER

dt= − 1

c1krel([Ca

2+]ER − [Ca2+]i) + kpump[Ca2+]i − c1v1w3∞p

3([Ca2+]ER − [Ca2+]i)

with

IK = gKn(V − VK)

ICa = gCam∞(V − VCa)

IK(Ca) = gKCa[Ca2+]i(V − VK)

70 + [Ca2+]i

n∞ =1

1 + exp(−16−V5.6 )

p∞ =q2

q2 + [Ca2+]i

τp =1

a2(q2 − [Ca2+]i)

and

A-2

q2 = d2I + d1I + d3

w∞ =I

I + d1

[Ca2+]i[Ca2+]i + d5

m∞ =1

1 + exp(−20−V12 )

Table A.1: Parameter Values of final model

Parameter Meaning Value Reference

gK (pS)maximal conductance of thedelayed-rectifier K+ channels 1600 -

gCa (pS)maximal conductance of the Ca2+

voltage-dependent channels 1000 [23]

gKCa (pS)maximal conductance of the Ca2+

activated K+ channels 29 000 -

kpump (s−1) rate of calcium through the Ca2+

-ATPase pump in the ER 0.0589 [5]

krel (s−1) rate of leakage calcium release fromthe ER 0.0002 -

kCa (s−1)Ca2+ removal rate by mechanismsother than sequestration in the ERhaving buffering into account

0.0136 [23]

C (fF ) cell membrane capacitance 5310 [23]τ (s) characteristic time 20 [23]

α (mol/[(µm)3C]) factor that converts current intoconcentration changes 1 ×10−5 [23]

VK (mV ) Nerst potential for K+ -75 [23]VCa (mV ) Nerst potential for Ca2+ 25 [23]I (µM) IP3 concentration in the cytoplasm 0.4 [9]

k3 (µM)dissociation constant of the Ca2+

pump in the ER 0.1 [9]

d1threshold value for activation of theIP3R by IP3

1 [34]

d2threshold value for inactivation byCa2+ of the IP3R when IP3 is bound 1.049 [9]

d3

threshold value for inactivation ofIP3R by Ca2+ when there is no IP3

bound to the receptor0.9434 [9]

d5threshold value for activation of theIP3R by Ca2+ 0.2 -

a2binding rate of Ca2+ to the inhibitingsite of IP3R

0.2 [9]

c1volume ratio between the ER and thecytosol 0.25 [5]

v1 (s−1) maximum IP3-gated permeability 0.45 -

v2 (s−1) IP3 -independent leakagepermeability 0.11 [9]

A-3

A.3 *.ode Files

Gall and Susa’s model

#Initial Conditions

init V=-70,Cai=0.5,Caret=100,n=0.5

#Parameters

par gk=2700

par gca=1000

par gkca=30000

par kpump=0.2

par krel=0.0006

par kca=0.64

par C=5310

par lambda=0.85

par tau=20

par alpha=6e-5

par f=0.02

par Vk=-75

par Vca=25

par Kd=70

#Evolution Equations

dCaret/dt=-krel*(Caret-Cai)+kpump*Cai

dV/dt=(-Ik-Ica-Ikca)/C

dn/dt=lambda*(ninf-n)/tau

dCai/dt=f*(-alpha*Ica-kca*Cai)+krel*(Caret-Cai)-kpump*Cai

#Currents

Ik=gk*n*(V-Vk)

Ica=gca*minf*(V-Vca)

Ikca=gkca*Cai*(V-Vk)/(Kd+Cai)

#Functions

minf=1/(1+exp((-20-V)/12))

ninf=1/(1+exp((-16-V)/5.6))

A-4

@ total=40000, bound=10000

done

Li and Rinzel’s model

#Initial Conditions

init C=0.2 h=0.5

#Parameters

par I=0.4

par c0=2

par c1=0.185

par v1=6

par v2=0.11

par v3=0.9

par d1=0.13

par d2=1.049

par d3=0.9434

par d5=0.08234

par a2=0.2

par k3=0.1

#Evolution Equations

dC/dt=-c1*v1*minf^3*h^3*(C-Caer)-c1*v2*(C-Caer)-(v3*C^2)/(k3^2+C^2)

dh/dt=(hinf-h)/tau

#Functions

minf=(I/(I+d1))*(C/(C+d5))

tau=1/(a2*(q2+C))

hinf=q2/(q2+C)

q2=((I+d1)/(I+d3))*d2

Caer=(c0-C)/c1

done

A-5

New model

#Initial Conditions

init V=-70,Cai=0.5,p=0.5,Caret=100 ,n=0.5

#Parameters

par gk=1600

par gca=1000

par gkca=29000

par kpump=0.0589

par krel=0.0002

par kca=0.68

par C=5310

par lambda=1

par tau=20

par alpha=1e-5

par f=0.02

par Vk=-75

par Vca=25

par Vn=-16

par Sn=5.6

par Vm=-20

par Sm=12

par Kd=70

par I=0.4

par d1=1

par d2=1.049

par d3=0.9434

par a2=0.2

par k3=0.1

par c1=0.25

par v1=0.45

par v2=0.11

par d5=0.2

#Evolution Equations

dCaret/dt=-krel*(Caret-Cai)/c1-c1*v1*winf^3*h^3*(Caret-Cai)+kpump*Cai

A-6

dV/dt=(-Ik-Ica-Ikca)/C

dn/dt=lambda*(ninf-n)/tau

dCai/dt=f*(-alpha*Ica-kca*Cai)+krel*(Caret-Cai)+c1*v1*winf^3*h^3*(Caret-Cai)-c1*kpump*Cai

dp/dt=(pinf-p)/taup

#Currents

Ik=gk*n*(V-Vk)

Ica=gca*minf*(V-Vca)

Ikca=gkca*Cai*(V-Vk)/(Kd+Cai)

#Other functions

minf=1/(1+exp((-20-V)/12))

ninf=1/(1+exp((-16-V)/5.6))

q2=((I+d1)/(I+d3))*d2

winf=(I/(I+d1))*(Cai/(Cai+d5))

taup=1/(a2*(q2+Cai))

pinf=q2/(q2+Cai)

@ total=60000, bound=100000000

done

A-7

A-8