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Thesis Consent Form

A lYlathematical StudY of Calcium Oscillations and -Waves

Krasimira Todorova Tsane\ia-Atanasova

A th€sts submitted in partial futfilnnent of the

requirements tor the degree of Doctor of P'hilos'ophy in Matlematies,

The University of Auckland, 2004

Contents

Abstract

Ae[<notl,edger.lents

Pr,eface

I Introduetisn

1.1 eazt

as an intracellul&r rtress€nter

I.2

Pancreatie Secretion

1-3

The Pancreatic Acinar Cell

X.4

Modelling of Oaz+ qgcillatiqns and ^wa;rres

1.4.1

Freviolrs inodelling work

1.5

Summa.tnt

2 Model Conetructisn

2,L

The homogeneous single cell model

i

xul

1

2

l2

T7

stt

38

cOArTEN"S

2,L.1 Ca?+dynamics . . . . i r .

^!

2.1.2

IP3 dynamies

Tlte spatial m.qdel

2.2,1 \{itoehoudria !...ir!!.r....t

⁴⁰

2;t.2 Oa+ bufferiug

^4T

2.2.,3

Introduction of opntiat bnte-rogeueiff

2,3.4 Pa,rauetervalws . r. ; .. .. .

⁴⁹

2;,3

The

nodel

of tbree ooupled eells

.

⁴⁹

2J.1 Thepointmodel-I...!::i

⁵⁰

2.3;,.2

Thespatialmodel-Il ..i. ... 5f

3' Plasnra

rnembra.ne

fluxes and

Gaz+ oscilXatiohs

3.1

The whole-cell model

3,2 Theealeiunload .. . .;. .r

i 2.2

3E

M

45

EA

^The

closdeell

^model

3.4

^{ThE open eel[ rnodel}

55

57 58

59

62

67

'72

78

3.5

Dynamic rnodu'lafriorl of caleiun oseillations

3.6

The effeets of ehangiog nembrane

traosport . . !

¹

r . :

ⁱ

9.7

^tuIodel predicf,ions

CONTENTS

3.8

Elperimental vetifleation

CId+ oscillations and

uraves

in a pancreatic

acinus

+.L

Model -

I

4.1.1

D. escniptiou of the model

4.7:2

Nr,rmerieal method

4.1.3

Results

4.2

Model

- tI

4-2.1

Descdpti,on of thc'model

4.2.2

Numerical method

4-2.3

Results

5 Conclusions

.4,

Sunrrnrar5l

of the,nrodel

_equat'ions

4.1 ModeI-I. ..

?

i..

A.2 l\dode-l-U ..

B The Sneyd et aL rnodel

iii

81

85

88

88 88 88

104

104

104

106

L77

L21 191

122

[25

CI:

'The Li-Rinzel norlel lzg

lv

D The Atri et al. model

E The Finite Element Method

F Glossary

Bibliography

CONTEAITS 131

133

L45

L47

List of Figures

L.2

1.3

1.4

1.5

1.6

L.7

I.8

1.9

1.10

1,,1.1

r..14

1.1 Prineipal pattniways imrolved in ealciuro sigpeniug mediated by produc-

tion

of

IP3 . .

4

B'picalqJrtffiotie0#+o-scillations. . i

J

i ! ! i,.. r

⁵

Erperimenta,l data ftom physiological agpqi$t stimulation

ffi. ;

^a

, i

⁶

&elierimenba,l data from pbysiological agouist stimulation

pll.

V Facperimental data from phSrsiol,ogicail agonist stimulation

[143]. I

Pancreas I

Panereatic

Aeinus

¹⁰

Pancreatic,.teinar CeU.

.

^Lz

Experiroeutal data,

Clwter

of Aeinar Clells

.

^1S

Adapted from DeYorrng et aL model _[301.

. .

^2;3

Bifr:lcat'ion dia,gpm f,or

fhe

reference p,ararneters,given

in

Tahle 1.1.

DeYoung ed aL

ffi

²⁵

1'1t

1.1S

Bifrncati.qs

iagram

for a =

^{0.97 (Ieft panel).}

c -

^{0.$7 Qrrd za}

-

2.8 s-1, solid

line

denstes de.ustes

Ffs]

^(right pa,nel)" DeY,bung et,

aI. p}l

I1gT

OF FI,GUMES Feriodic solutions for

p{*|a.nd

broken lioe

2.1

2.2

2.3

2"4

2.5

2.,6

1.tr4 Eif.ureation diqg,ra,ms of Dupont; ef

al nodel

_[33].

I.15 Tlpical

oscillationE

in

Du.pont ^e*

d"

urodel

ffi

1.16 Adapted

fron

LeBea-u ei af. model _[661.

1.17 Bifurcation diagrams of LeBeau ef aL

uodel

[66].

Sehcnatiq di*gna@ of ^0he modcl fluxes SinpEfied ^diagramr of the IP3R model _[111J.

Sr,henoetic diasrarn of the RyR modol

tGE.

^.

Schematic diagrarn of the model of an acinar celi Schematic diagram o.f a eluqter of tlrree eeltrs

The

ryodd

^{chuter of thiree} panereatic acinar

cells

⁶²

3.1

Sshenratic diagram of the uo-del

S.2

Bi'ffncaJioq

diagta4

sf the etrosed-cell model.

3.3

Tvr.o-pararneter biturcat'ion ^diagr.a,m

in

^{(c1, p:)'space:}

g.a

Bifurcatioq diagraune

ofthe

open ecll urodel-

3.5

Two-pa,rarneter continuation

in

(p,6)-space of the Hopf bifurcations

of

theopen-cellmod.el , . ., .: i

^!

27 30

31

32

4l

42

45

56

60

61

63

LTS;T @'ETGURES

8.6

Typical oscillations,

obtaiad

by nimerical infegration of the op€n-cell model, for d

:0.1

_and difierent rra,lues of

p.

.

3,I

Typieal oseillations, obtained by numerieal iutegration of the o.pet-eell lnodel,

forp:

¹⁰ and difienent values of

d.

^.

3A

Experimentally observed osci[ations in motrge pancreatic apina.r,eetrls. 67

S.9

Dynamie mo-dulation of

tls

eytosolic ealeiqu

oseillatiols

^O-A

3,10

A

spjkiag cycle projected onto

the

_{(e eJ} plaoe for vaXue of p

=

^10.

. .

^6€

$. 1

The efiects of

cbngins

membrane

transport.

⁷¹

,S.12 Biftueation diagferq of the open'cqll ms.de-[ s.uperiaposed on

t}e

tw.e- pa,rameterbiftraatioudiagra,mof thee.lOed-cellmodel.

, . , . ; r . !

^V4

3.18 Twepar-a,uoeter bifir. eatiou diagra,m of the open-cell model

i" (or -p)

plane. .'.r .i.iti,s

7'5

3.14 Analogue of Figures 3.L2 and 3.13 for the Li-Rinzel

model.

⁷⁶

3.15 Ana,logue of Figures 8.12 and 3.13 for the

.{tii

et ot"

podel

3,16 trvlodol

siinulatims

80

3.U

Eqsperimenta,l ftra,ees

-

model predictioo 1.

. . .

⁸²

3.18 Experimental traces

-

model prediction 2-

. . .

⁸³

S.h9Experfunentaltraces .,-i

⁸⁴

Biftucation diagram of thesingle eeII mo.del. 92 95

4.1l,.

4-2:, Bifur,catiion diagrii.rqE of _the homogeneous nrodel

I,

IIST

OF PTGURES

4.3

^Typical oscillations

in

the hornogeneorrs

nodel L . . "

⁹⁷

44

Two-para,ueter

biftrqation

diagf-qrros-

io

(p*, e).+ ace (identieal

cells).

⁹⁸

4'6

Two.psrauoetler

bifu

eation dlagrains:i[ (F"t,c)-space (uoa"'identieal cells.100

4.6 Tlpical

syaehronised sscillsti@,1$ rn the mqdel in the case of n'eak con pling: t..ii.r.

4.7

Typieal synclrr.onised eiseilkitions

in

the,model

in the

case of strong

couplirtg.

^I03

4.8

Experimental image of a cluster of three pancreatie

acina cells.

¹⁰⁵

,[.9

TVepararneter diagra,m

in

(p'yrcy].seap€,

" .

¹⁰⁹

4.t0

Intereellular oseillations in the

triplet.

4'11

Typiea,l intercellUtrar qqeillations

io'a

heteroge.neous in

tergg

of stirnrr-

Iationcluster",,.i.r,. !,:d

LLz

4.1-2

ftperimental traees.,

¹¹³

413

T],'pieal oscillations in a structurally heterogeneous

elwter.

¹¹⁵

The

FEM

Mesh of Three Cells

Nwrberine Sehene of the lvlaster ^lUlemeut

ThclntedaceBotndary' ;!i.

⁷⁴³

ts,1

8.2 E,3

14il

List of Thbles

P,a,raureter

'*a,lues of Delfoung e,f al.

nodel tAq . "

²⁴

Values of the pa,r6,&eters ctraracterisirry InFl.,4,FPs eynthesis and

netaboliq usediuDupoote,tol;in-odell36] ... .. i i .. . .. . . i ..

29 1.1

1.2

A1

N2

123

8.1 128

$teadystatevaftlesof the,systemvruiables, . - -..,. i i i.

]v[ode] para,meters

P,araineter rnalrrlies:of the $neyd eC ol. model

[tfA;

^.

C.1

Pararrreter nq,trues sf thc Lj-Rinaetr model

V?l .

^.

D.,1

Paranreter vriluoe of the

Atri

et ul. mpdel

[6] ,

^.

Etl

^Gause points aud quadratlue weighfts

r . !

^{. L42}

IX

LIST OF TABLES

Abstract

In this thesis ^we study theoretically the dynamics of the free cytosolic Ca2+ concentra-

tion.

We construct a mathematical model of the Ca2+ dynamics in pancreatic acinar cells. Although this model refers

to

a particular _cell type,

it

also allows us

to

study some aspects of Ca2+ signatling

in

general. We begin

by

analysing the dependence

of the

Ca2+ oscillations on the plasma membrane

transport.

F\rrther we study the propagation of intercellular Ca2+ waves

in

a pancreatic acinus.

It

has been observed experimentally

that, in

many cell t1pes, calcium fluxes across

the plasma rnembrane a.ffect inositol trisphosphate IP3-induced calcium oscillations.

Since lP3-induced calcium oscillations involve the cycling of calcium

to

and from the endoplasrnic reticulum,

it

is not well understood how they can be so strongly ^a.ffected by membrane fluxes. We use a mathematical rnodel to answer this questionl a model

that

relies on

the

introduction

of a

slow variable,

the

Ca2+ load

of the cell.

Our model predictions a.re confirmed

by

experimental results. Since similar behaviour is observed in two other models of IP3-induced Ca2+ oscillations,

it

is possible

that

this bifurcation structure is a generic feature of Caz+ oscillation models.

The effect

of

intercellular coupling on

the

oscillatory dynamics

is

investigated the-

oretically. It

is dernonstratecl

that

junctional calciurn diffusion can account

for

the co-ordination and synchronisation of cytosolic calcium oscillations in a coupled

triplet

of cells under the assumption of constant IP3 concentration

in

each indiviclual cell.

Ftuthermore

a

bwo dirnensional version

of tliat

model, where Ca2+ and IP3 are as-

suned

to

diffuse

within

as well a^s between the cells, has been studied numerically.

Cornpared

to

the results frour

the

analysis of the ODE rnodel, the results from the xl

xii Abstract analysis of the PDE model (in tw'o spatial clirnensions) reveal some interesting spatial effects of the cliffusion, and of the geornetry of the cells on the collective oscillatory behaviour of the

systen.

Based on this combined approach, a suggestion about the specific role of both Ca2+ and IP3 in the intercellular Ca2+ signalling has been made.

Acknowledgements

I

would like

to

especially thank my supervisor

Prof.

James Sneyd without whom this thesis would

not exist.

He has helped me through extremely

difficult

times over the course of the analysis and the

writing

of

the

dissertation.

I

sincerely thank

him

for his confidence

iu

me. He also gave me the opportunity to jump into a new field

with

a rare chance

to

do real science.

Not

everyone would give such a chance

to

someone so new

to

the Mathernatical Biosciences.

I

wish

to thank Dr. Vivien Kirk for

sharing

with

rne her wealth

of

knowledge

in

the area of dynamical systems and bifurcation analysis. Her generous assistance and

the

productive scientific discussions have helped me

a lot

during

the

course

of

my research.

My

appreciation goes

to Dr.

David Yule

for inviting

me

to visit the

University of Rochester Medical Center.

I

should emphasise how much the opportunity to perform real experimeuts has contributed

to

a better understanding of the non mathematical

part of my work. I

^owe

a

great deal

to him, Prof. Tlevor

Shuttleworth and the many student workers

in their

labs.

Without

them,

I

would

not

have had any data

to

analyse and from which

to

develop theories about cytosolic calcium signalling.

I

started ntv graduate degree

at

lVlassey University, so

I

would like

to thank

some people

flom IIMS at Albany. Prof.

Robe.rt NlcKibbin,

Merrill

Bowers,

tnd

my col- leagues and friencls Cynthia Wang. ivlaha, Shakir, Amal

Al-Dujaily

were arnong those who kept me going a,t the beginning.

I

also appreciate the assistance

I

received from the students and fa,cnlty from

the

Departrnent

of

lvlathernatics

at

the University of

xt1r

xlv Aucklancl.

Acknowledgements

Finally,

I

would like to thank my parents. for their absolute conficlence in me. IUost of all

I

would like to acknowledge the great level of understanding and endless patience given

to

me by rny husband Atanas and my datrghter ^Deyana.

Preface

The work ptesented

in this

thesis ilttrstxat€q ou one haad. how ngthematies oari be r:sed

to

arlsrftrer physiological questioas, and on the other hand

it

ie an qncil.nple

of

hovu physiologiel questious may p.ose

vey

lnteresting math'smatical

problens. It

is an iuterdiseipliua,ry

stndy

which requires solid mathematical lnrowledge as well as

a vel].

^good understandiag

of the

physiologieal ^pnoc€sges underlying

the

problenis under

investigation.

Al'tb-ougl comiug,from pure matbematics baekground

I

^barae

always been interested

in

learning more

about,its

applicatious. The opportunity

to

wsnk

with Prol

James Srreyd rilds Bn exeellent ahianee

to

do thisn and

I

re;ally eqioyed

ir!

T,his thesis consists

of4

ehapters:

Ghapter 1

^{gives a ge-neral} introductio.n

to

thE phy.siologlo of tJre p,ancreetic acinar cells and tbe caleium sigaaning.

It

alqo cnntains a bdef qrreffiew of previous modelling wotk done in

this

ffeld.

Chapter 2

e;plaius the modelliqg details.

Mathmatical

analyses of the

nodel

de- sc,ribed in

this

chapter have been published

in

_[113, ^115-].

Chapter 3

addrEsses' a peudcular physiological qumtion about the role of the plasrna membrane transport for the calcium ose-illatione w,hieh are based onrfl,uxes:a,eroes

the endoptrasrnie

reticultrn.

This work has appeared

in

_[114, 129].

GhaBter 4

eontains

a

qtathc:natieal

study o{

ealoium oecillations and wavw

in

a 'cv

xvi

^Preface

triplet

of pancreatic acinar cells. The results from this study have been submit- ted for pLrblication

to

the Biophysical

Jounal.

All

the work presented

in this

thesis has been done

in

close collaboration

rvith

Dr.

David Yule and his colleagues from the University of Rochester Medical Center. USA.