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ISBN 978-951-42-9074-9 (Paperback)ISBN 978-951-42-9075-6 (PDF)ISSN 0355-3221 (Print)ISSN 1796-2234 (Online)
U N I V E R S I TAT I S O U L U E N S I S
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D 1011
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Topi Korhonen
OULU 2009
D 1011
Topi Korhonen
MATHEMATICAL MODELING OF THE REGULATION, DEVELOPMENT AND GENETICALLY ENGINEERED EXPERIMENTAL MODELS OF CARDIAC EXCITATION-CONTRACTION COUPLING
FACULTY OF MEDICINE,INSTITUTE OF BIOMEDICINE, DEPARTMENT OF PHYSIOLOGY,UNIVERSITY OF OULU;FACULTY OF SCIENCE,DEPARTMENT OF PHYSICAL SCIENCES, DIVISION OF BIOPHYSICS,UNIVERSITY OF OULU;BIOCENTER OULU, UNIVERSITY OF OULU
A C T A U N I V E R S I T A T I S O U L U E N S I SD M e d i c a 1 0 1 1
TOPI KORHONEN
MATHEMATICAL MODELING OFTHE REGULATION, DEVELOPMENT AND GENETICALLY ENGINEERED EXPERIMENTAL MODELS OF CARDIAC EXCITATION-CONTRACTION COUPLING
Academic dissertation to be presented with the assent ofthe Faculty of Medicine of the University Oulu for publicdefence in Auditorium F101 of the Department ofPhysiology (Aapistie 7), on 3 April 2009, at 12 noon
OULUN YLIOPISTO, OULU 2009
Copyright © 2009Acta Univ. Oul. D 1011, 2009
Supervised byDocent Pasi Tavi
Reviewed byAssistant Professor Eric A. SobieProfessor Juha Voipio
ISBN 978-951-42-9074-9 (Paperback)ISBN 978-951-42-9075-6 (PDF)http://herkules.oulu.fi/isbn9789514290756/ISSN 0355-3221 (Printed)ISSN 1796-2234 (Online)http://herkules.oulu.fi/issn03553221/
Cover designRaimo Ahonen
OULU UNIVERSITY PRESSOULU 2009
Korhonen, Topi, Mathematical modeling of the regulation, development andgenetically engineered experimental models of cardiac excitation-contractioncouplingFaculty of Medicine, Institute of Biomedicine, Department of Physiology, University of Oulu,P.O.Box 5000, FI-90014 University of Oulu, Finland; Faculty of Science, Department ofPhysical Sciences, Division of Biophysics, University of Oulu, P.O.Box 3000, FI-90014University of Oulu, Finland; Biocenter Oulu, University of Oulu, P.O.Box 5000, FI-90014University of Oulu, Finland Acta Univ. Oul. D 1011, 2009Oulu, Finland
AbstractExcitation-contraction coupling (ECC) is a process linking the electrical excitation of the musclecell (myocyte) membrane to the contraction of the cell. In this study the possibilities ofmathematical modeling were studied in current ECC research. Mathematical modeling wasemployed in two distinct ECC research areas, the enzymatic regulation of ECC and ECC duringcardiac myocyte development. Despite the distinction, both of these are extremely complexbiological systems characterized by diverse and partly contradictory reported experimental results,with a large part based on genetically engineered animal models.
Novel mathematical models were developed for both of these research areas. The model ofventricular myocyte ECC with calmodulin-dependent protein kinase II (CaMKII)-mediatedregulation faithfully reproduced the heart-rate dependent regulation of ECC. This regulation isthought to be the major effect of CaMKII-mediated regulation. The model of the embryonicventricular myocyte provided the first comprehensive system analysis of how the embryonicheartbeat is generated at the cellular level. A similar type of model was also developed to showthe notable differences between neonatal and adult ventricular myocyte ECC.
The mathematical models of ECC presented in this study were further used to simulate ECC ingenetically engineered myocytes. The cellular mechanisms of genetically engineered animalmodels could be better understood by employing mathematical modeling in parallel toexperimental characterization of the animal model. It was found in simulations that the indirectconsequences and the compensatory mechanisms induced by genetic modification may have amore significant effect on ECC than the direct consequences of the modification.
To understand the overwhelming complexity of biological systems including ECC, competentsystem analysis tools, such as mathematical modeling, are required. The purpose of mathematicalmodeling is not to replace the experimental studies, but to provide a more comprehensive systemanalysis based on the experimental data. This system analysis will help in planning subsequentexperiments needed to gain the most relevant information about the studied biological system.
Keywords: action potentials, calcium, cardiac myocytes, computer simulation, confocalmicroscopy, embryonic development, genetic engineering, ion channels, patch-clamptechniques, signal transduction, systems biology
5
Acknowledgements
This study was carried out in the research group of Docent Pasi Tavi at the
Department of Physiology, Institute of Biomedicine, Biocenter Oulu, University
of Oulu. I owe my deepest gratitude to Docent Tavi for the possibility to work
under his professional supervision. I highly appreciate the effort he put into
planning this study and guiding me whenever needed, but also his trust by giving
me the freedom and responsibility to do things and make decisions independently.
I wish to thank Professor Matti Weckström for the additional support during
this study and for the collaboration with his Biophysics division in the
Department of Physics, University of Oulu. Especially, I would like to thank
Professor Weckström for encouraging me in 2006 to continue my M.Sc. thesis
with this study.
I am very grateful to the official examiners, Assistant Professor Eric A. Sobie,
Mount Sinai School of Medicine, New York, and Professor Juha Voipio,
University of Helsinki, for their careful review and valuable comments on this
thesis.
During this work I have had a privilege to work in an innovative and friendly
research environment. I would like thank my co-authors Risto Rapila, Jussi
Koivumäki, Jouni Takalo and Sandra Hänninen for scientific collaboration.
Special thanks go to Sandra also for reviewing the language of this thesis and
several other written texts during this study and to Risto for teaching me the
secrets of the patch-clamp method. I am also grateful to Jarkko Ronkainen and
Veli-Pekka Ronkainen for their expert comments during this study and for their
help with experimental laboratory work, especially with confocal microscopy. Dr.
Perttu Niemelä is acknowledged for kindly providing the Matlab code for the
CaMKII model.
I would like to express my sincere gratitude to Special Laboratory
Technicians Anneli Rautio and Alpo Vanhala for the high quality laboratory and
technical work they have made for this study. I would also like to thank
Laboratory Manager Eero Kouvalainen, Secretary Marika Hämeenaho, Professor
Olli Vuolteenaho and the rest of the staff in the Department of Physiology for
their companionship and assistance in numerous issues. I would also like to thank
Professor and Vice-rector Heikki Ruskoaho and the Department of Pharmacology
for the joint Pharmacology-Physiology scientific seminars and events.
Last but not least, I would like to express my warm thanks to family
Korhonen (Jukka, Maija, Suvi, Janne, Lauha, and the little sunshine of family
6
Korhonen). I am also thankful to all my friends including the Siilinjärvi crew,
which has remained in contact although we headed to different directions several
years ago from the yard of the Siilinjärvi High School. Warm thanks go also to
the family Rukajärvi. Most of all I would like to thank my dear Anna-Leena. All
these people have given me a great deal of support and companionship during this
study and reminded me that there is also life outside of the laboratory.
This study was supported by the Finnish Heart Research Foundation,
Academy of Finland, Orion-Farmos Research Foundation, Instrumentarium
Science Foundation, Aarne Koskelo Foundation and Sigrid Juselius Foundation.
Oulu, March 2009 Topi Korhonen
7
Abbreviations
AC Adenylyl cyclase
AP Action potential
APDxx Action potential duration at xx % of repolarization
ATP Adenosine triphosphate
ATPase ATP consuming ion transporter
AV Atrioventricular node
β-AR β-adrenergic receptor
cAMP cyclic adenosine monophosphate
CaN Calcineurin
CaM Calmodulin
CaMKII Calmodulin-dependent protein kinase II
CICR Ca2+-induced Ca2+ release
Cm Capacitance of the cell membrane
CSQN Calsequestrin
DCa Diffusion coefficient for Ca2+
DHPR Dihydropyridine receptor
dt Time-step
Ex Equilibrium potential of ion x
ECC Excitation-contraction coupling
E9-11 Embryonic days 9–11
FDAR Frequency-dependent acceleration of relaxation
FFR Force-frequency relation
GHK Goldman-Hodgkin-Katz
Gs s-type G protein
H-H Hodgkin-Huxley
I-V Current-voltage
ICaL, ICaT The L- and T-type voltage-activated Ca2+ currents
If Hyperpolarization-activated current
INa The fast Na+ current
INCX Na+/Ca2+-exchanger current
IK1 Time-independent K+ current
Ito Transient outward K+ current
IP3R inositol-1,4,5-triphosphate receptor
JSR Junctional SR
Km Half-maximal activation concentration
8
KO Knockout
NCX Na+/Ca2+-exchanger
NSR Network SR
ODE Ordinary differential equation
Px Permeability of ion x
PDE Partial differential equation or phosphodiesterase
PLB Phospholamban
PP1 Protein phosphatase 1
PKA Protein kinase A
R Ideal gas constant
r.p. Resting membrane potential
RyR Ryanodine receptor
SA Sinoatrial node
SERCA Sarco/endoplasmic reticulum Ca2+ ATPase
SL Sarcolemma
SR Sarcoplasmic reticulum
T Temperature
TnC Troponin C, Ca2+-binding subunit
TnI Troponin I, Inhibitory subunit
TnT Troponin T, Tropomyosin-binding subunit
V Membrane voltage
VACC Voltage-activated Ca2+ channels
WT Wild-type
z Valence of ion
[x]y Free concentration of ion x; y = i/o intracellular or extracellular
9
List of original articles
This thesis is based on the following articles, which are referred to in the text by
their Roman numerals:
I Korhonen T & Tavi P (2008) Automatic time-step adaptation of the forward Euler method in simulation of models of ion channels and excitable cells and tissue. Simulation Modelling Practice and Theory 16: 639–644.
II Koivumäki J*, Korhonen T*, Takalo J, Weckström M & Tavi P (2009) Integrative Model of Excitation-Contraction Coupling in Mouse Cardiac Myocytes. Manuscript.
III Rapila R, Korhonen T & Tavi P (2008) Excitation-contraction coupling of the mouse embryonic cardiomyocyte. Journal of General Physiology 132(4): 397–405.
IV Korhonen T, Rapila R & Tavi P (2008) Mathematical model of mouse embryonic cardiomyocyte excitation-contraction coupling. Journal of General Physiology 132(4): 407–419.
V Korhonen T, Hänninen SL & Tavi P (2009) Model of excitation-contraction coupling of rat neonatal ventricular myocytes. Biophysical Journal 96(3): 1189–1209.
*equal contributions
10
11
Contents
Abstract Acknowledgements 5 Abbreviations 7 List of original articles 9 Contents 11 1 Introduction 15 2 Review of the literature 17
2.1 Excitation-contraction coupling (ECC) in cardiac muscle cells.............. 17 2.1.1 Excitability of a cardiac muscle cell............................................. 17 2.1.2 Ca2+ dynamics in the cardiac muscle cell ..................................... 20 2.1.3 Contraction of the cardiac muscle cell ......................................... 21 2.1.4 The ECC in different regions of the heart .................................... 23 2.1.5 Species-dependent differences in ventricular
cardiomyocyte ECC ..................................................................... 26 2.2 Development of cardiac ECC.................................................................. 29
2.2.1 ECC in pre- and postnatal ventricular cardiomyocytes ................ 30 2.2.2 Developing cardiomyocytes in cell culture .................................. 32
2.3 Regulation of cardiac ECC...................................................................... 32 2.3.1 Inotropic response during sympathetic stimulation...................... 32 2.3.2 Activation of Ca2+-dependent enzymes ........................................ 34 2.3.3 CaMKII-mediated regulation of ECC .......................................... 35 2.3.4 Faulty regulation of ECC.............................................................. 36
2.4 Studies of ECC based on genetic engineering......................................... 37 2.5 Mathematical modeling of cardiac ECC ................................................. 37
2.5.1 History of ECC models ................................................................ 39 2.5.2 Integrated models of ECC and CaMKII regulation...................... 40 2.5.3 ECC models of developing cardiomyocytes ................................ 41 2.5.4 Structure of ECC models.............................................................. 41 2.5.5 The models of ion channels and membrane voltage..................... 43 2.5.6 Chemical reaction models ............................................................ 45 2.5.7 Diffusion of ions........................................................................... 45 2.5.8 Simulation of excitable cell and tissue models............................. 46
3 Aims of the research 49 4 Summary of materials and methods 51
4.1 Experimental methods............................................................................. 51
12
4.1.1 Studied myocytes (III, IV, V)........................................................ 51 4.1.2 Electrophysiology and confocal microscopy laboratory............... 51 4.1.3 Electrophysiological recordings (III, IV, V) ................................. 52 4.1.4 Ca2+ imaging (III, V) .................................................................... 53
4.2 Mathematical modeling........................................................................... 54 4.2.1 Model of the mouse ventricular myocyte (II) ............................... 54 4.2.2 Models of developing cardiomyocytes (IV, V)............................. 56 4.2.3 Simulations of the models (I, II, IV, V) ........................................ 58
5 Results 61 5.1 Comparison and development of simulation methods (I) ....................... 61 5.2 Model of mouse ventricular ECC with CaMKII regulation (II).............. 62 5.3 ECC during embryonic and postnatal development (III, IV, V).............. 63
5.3.1 The dual parallel ECC modes in the embryonic
cardiomyocyte (III, IV) ................................................................ 64 5.3.2 The unique features of ECC in the rat neonatal
cardiomyocyte (V)........................................................................ 66 5.4 Simulations of phenotypes of genetically engineered myocytes
(II, IV) ..................................................................................................... 67 5.4.1 The simulation of altered NCX, If and CSQN expression
in embryonic mouse cardiomyocytes (IV) ................................... 67 5.4.2 The simulation of PLB knockout (KO) and CaMKII
overexpression in the adult mouse ventricular
myocyte (II) .................................................................................. 69 6 Discussion 73
6.1 Simulation methods................................................................................. 73 6.2 Model of CaMKII activation................................................................... 73 6.3 Regulation of ECC and genetic engineering ........................................... 74 6.4 ECC in developing cardiomyocytes ........................................................ 75
6.4.1 Common features of embryonic and neonatal
cardiomyocyte ECC...................................................................... 76 6.4.2 Developing cardiomyocytes as general experimental
models of cardiomyocytes ............................................................ 77 6.5 Limitations of the models........................................................................ 78
6.5.1 Variability of parameters .............................................................. 78 6.5.2 Extracellular regulation ................................................................ 79 6.5.3 Global vs. local signals................................................................. 79 6.5.4 The models of developing cardiomyocytes .................................. 80
13
6.6 Applicability of the models..................................................................... 81 7 Summary and conclusions 83 References 85 Appendix 99 Original articles 111
14
15
1 Introduction
Excitation-contraction coupling (ECC) is a process linking the electrical
excitation of the cardiac muscle cell (myocyte) membrane to contraction (for a
review, see e.g. (Bers 2001)). During ECC a large amount of Ca2+ is released from
intracellular stores to the cytosol activating contractile elements in the muscle cell.
The current understanding of ECC is based mostly on studies made after the
1950s (for a review, see e.g. (Dulhunty 2006)). In addition to experimental studies,
mathematical modeling has been used during these last five decades to present a
comprehensive system analysis of ECC based on currently available experimental
data (for a review, see e.g. (Puglisi et al. 2004)). Although the basic mechanism of
ECC is relatively well known, open questions exist regarding ECC during
development and the pathogenesis and regulation of ECC.
Calmodulin-dependent protein kinase II (CaMKII) is one of the important
regulators of ECC (for a review, see e.g. (Maier & Bers 2002)). The activation of
CaMKII depends on the free intracellular Ca2+ concentration ([Ca2+]i) and the
frequency of [Ca2+]i oscillations. CaMKII phosphorylates several proteins
involved in ECC and provides a feedback mechanism for the Ca2+ signaling in the
myocyte to regulate itself. CaMKII dysfunction is involved in cardiac arrhythmias
and the development of cardiac failure (Hoch et al. 1999, Wu et al. 2002).
However, a detailed system analysis of CaMKII activation by Ca2+ and feedback
to ECC by mathematical modeling has not been presented.
During development the cardiac myocytes differentiate towards the
phenotype they have in the adult animal. The ECC of developing cardiomyocytes
plays an important role in producing the contractions of the developing heart, but
also in regulating the differentiation of the cardiomyocytes by the Ca2+ signals
(Porter et al. 2003). Also, in contrast to adult ventricular myocytes, the early
embryonic cardiomyocytes are capable of generating the ECC spontaneously
without an external electrical trigger (Viatchenko-Karpinski et al. 1999). The
mechanisms underlying ECC in developing cardiomyocytes are not completely
understood.
Genetic engineering has been widely used to study ECC as well as the
CaMKII-mediated regulation and development of ECC (for example studies, see
e.g. (Maier et al. 2003, Stieber et al. 2003)). The genetically engineered animal
models are studied extensively with experimental methods, but detailed system
analyses of the underlying mechanisms in these models are rarely made. The
analysis of the relationship between the induced genetic alteration and the altered
16
ECC is, however, relatively complicated due to the non-linearity of ECC (Qu et al.
2007). The genetic modifications may also induce compensatory changes in the
expression of other genes further complicating the analysis (Maier et al. 2003).
The aim of the present study was to characterize the ECC and to develop new
mathematical models of ECC relating to the CaMKII-mediated regulation of ECC
and the development of ECC in cardiac myocytes. The developed models were
further employed to study the possibilities of mathematical modeling in the
design and analysis of genetically engineered experimental animal models.
17
2 Review of the literature
2.1 Excitation-contraction coupling (ECC) in cardiac muscle cells
Cardiac muscle tissue is formed from individual muscle cells, myocytes, bound to
each other. For the contraction of a single myocyte, the chemical energy available
in the myocyte in the form of adenosine triphosphate (ATP) has to be transformed
into mechanical work to produce cell shortening. To produce coordinated forceful
contraction of the whole tissue, this transformation must occur simultaneously in
each myocyte of the tissue. The electrical signal is recruited as a rapidly deviating
signal, which triggers the transformation of chemical energy into mechanical
energy simultaneously throughout the whole tissue. The electrical signal
propagates in cardiac tissue via gap junctions between cardiac myocytes. In each
myocyte, the electrical signal initiates a process referred to as excitation-
contraction coupling (ECC) (for a review, see e.g. (Bers 2001)). ECC uses the
electrochemical gradients of ions, generated with ATP-consuming ion transporters
(ATPases) and secondary active transporters, to produce a transient increase in
[Ca2+]i through a complex mechanism. This Ca2+ directly activates the
myofilaments in the myocyte, which use the ATP to produce conformational
changes resulting in a sliding of the filaments and shortening of the cell.
Consequently, the generation of contraction depends highly on the [Ca2+]i
dynamics. The analysis of ECC and the contraction of a single myocyte and the
cardiac tissue is focused mostly on how the transient increase in [Ca2+]i is
produced.
2.1.1 Excitability of a cardiac muscle cell
The ECC process begins with an action potential (AP), which is a process of rapid
increase (depolarization) and decrease (repolarization) of the voltage over the cell
membrane (see Fig. 5 on p. 27). In a myocyte the intracellular [Na+] is lower and
intracellular [K+] is higher than the concentrations outside the cell because of the
active ion transport via NaK-ATPase at the cell membrane (sarcolemma, SL, Fig.
1). The transported ions diffuse passively through the cell membrane via protein-
formed pores, ion channels, from a higher to lower concentration. As the ions are
charged particles, the diffusion of the ions produces an electrical field over the
cell membrane. The diffusion of single ion type x along its concentration gradient
18
produces an electrical field which drives the same ions x in the opposing direction.
When the concentration gradient-driven flux and the electrical field driven-flux
are equal, producing a net flux of zero for the electrodiffusion of ion x, the
voltage over the cell membrane (E) can be calculated from the Nernst equation
o
i
[x]ln
[x]x
RTE
zF
=
(1)
where R = the ideal gas constant, T = temperature, z = the valence of ion x, F =
Faraday’s constant, and [x]o and [x]i = the concentration of ion x outside and
inside the cell membrane, respectively (for a review, see e.g. (Keener & Sneyd
1998, Hille 2001)). With physiologically relevant concentrations, we can estimate
the magnitude of the ENa to be +70 mV (10 mM inside, 140 mM outside) and EK
to be −89 mV (140 mM inside, 5 mM outside) at 37 ºC (see e.g. (Bers 2001)).
Fig. 1. A schematic figure of the major features of excitation-contraction coupling
(ECC) in a cardiac myocyte. Abbreviations: IK, K+ current; ICa, voltage-activated Ca2+
current; INa, fast Na+ current; NaK, NaK-ATPase; NCX, Na+/Ca2+-exchanger; SERCA and
SL pump, Ca2+ ATPases; RyR, ryanodine receptor.
ICa
SR
Cytosol
Myofilaments
Ca2+ flow (relaxation)
Na+ flow
K+ flow
Ca2+ flow (activation)Extracellular space
SL[Ca2+]i[Na+]i[K+]i
[Ca2+]o[Na+]o[K+]o
[Ca2+]SR
RyR
RyR
SER
CA
NCX SL pumpNaK
INa
IK
19
Electrodiffusion changes the charge distribution at the inner and outer surfaces of
the SL, changing thus the voltage over the SL (V = Vin – Vout). The
electrodiffusion of ion x drives the V always towards Ex. In a system of many ions,
the permeability of the cell membrane to a certain ion defines how much the ion
contributes to the V. Consequently, by altering the permeabilities it is possible to
alter the V and produce pulses, such as AP. If only ions with z = ±1 are permeable,
the steady-state V, when no net current flows, can be presented with the
Goldman-Hodgkin-Katz (GHK) voltage equation
i o
1 1o i
1 1
ln i i j jz z
i i j jz z
Pc P cRTV
F Pc P c=− =
=− =
+= +
(2)
where ci, cj, Pi and Pj are the concentrations and permeabilities for ions i and j,
and superscripts of c define the intra- or extracellular (i or o) concentration (see
e.g. (Keener & Sneyd 1998), for references see also (Hille 2001)).
As the phospholipid bilayer of the cell membrane is impermeable to ions, the
permeability of an ion through the cell membrane depends on the permeability
properties of the ion channels. There are several different ion channel types in
cells, which usually have high selectivity (permeability) to a certain single ion
species. It is thus obvious that the single ion channel permeabilities, total number
of ion channels, and the fractional amounts of different ion channel types
contribute to the total permeability for each ion. However, in the time scale of an
AP, the major factor contributing to the permeability, and especially to changes in
it, is the gating of the ion channels. The gating means that a single ion channel
can rapidly change its molecular conformation, resulting in opening or closing of
the ion channel. Since the latter results in zero single ion channel permeability,
the total permeability is always affected by the fractions of the ion channel
population in the open and closed states (for a review, see e.g. (Hille 2001)).
The gating of ion channels is controlled by changes in V, the binding of
ligands, and mechanical deformations. In a resting myocyte, there is a moderate
open probability for K+ channels but low open probability for Na+ channels,
resulting in a resting V (r.p.) near the EK of −89 mV (Fig. 1 & see Fig. 5 on p. 27).
The electrical signal propagating along the cardiac tissue depolarizes the
membranes and consequently triggers action potentials in individual cells. The
gating of the fast-type Na+ channels is such that depolarization opens them. This
results in a large permeability and consequently a large current for Na+ (INa)
20
through the cell membrane, which depolarizes the cell membrane rapidly towards
the ENa of +70 mV. However, due to their gating properties, the fast Na+ channels
inactivate rapidly after opening. The depolarization also opens more K+ channels.
Thus, after the rapid depolarization, the K+ permeability is high and the V returns
to the resting value near EK. Again, near EK only the K+ channels have a moderate
open probability and the V is maintained near EK (for a review, see e.g. (Bers
2001, Nerbonne & Kass 2005)).
2.1.2 Ca2+ dynamics in the cardiac muscle cell
The transient increase in [Ca2+]i in the cytosol from diastolic ~0.1 μM to systolic
~1–2 μM is an essential part of ECC in the cardiac muscle cell as it is the link
between AP and contraction. The major source for this Ca2+ is the sarcoplasmic
reticulum (SR) (Fabiato & Fabiato 1975, Kirby et al. 1992). SR is a large
intracellular compartment where the Ca2+ is stored in bound form to the
calsequestrin protein (Fig. 1)(Campbell et al. 1983). The high Ca2+ content in the
SR is maintained with SR Ca2+-ATPase (SERCA), which pumps the Ca2+ from
the cytosol to the SR (Kirby et al. 1992). There are also Ca2+ extrusion
mechanisms at the SL: the SL Ca2+-ATPase and Na+/Ca2+-exchanger (NCX). The
NCX uses the Na+ gradient over the cell membrane to move Ca2+ against its
electrochemical gradient. The SL Ca2+ extrusion mechanisms have less extrusion
capacity than SERCA at the SR (Negretti et al. 1993).
The depolarization of AP activates the dihydropyridine receptors (DHPR). In
the case of the cardiac isoforms of DHPR, L-type voltage-activated Ca2+ channels
(VACC), activation results in significant Ca2+ flow, L-type Ca2+ current (ICaL), to
the cell (Fig. 1), which due to the ion current also shapes the AP (Lamb & Walsh
1987, Field et al. 1988, Han et al. 2002a)(for a review, see (Lamb 2000)). The
opening of the RyR cardiac isoform (RyR2) is sensitive to [Ca2+]i, which makes it
possible for the ICaL to trigger a larger Ca2+ release via RyR from the SR (Fabiato
& Fabiato 1975)(for a review, see (Xu et al. 1998)). This event is referred as Ca2+-
induced Ca2+ release (CICR).
The mechanism of termination of the SR Ca2+ release is not completely clear,
but it has been suggested that RyR inactivation and the depletion of the SR Ca2+
concentration via dependence of RyR opening on SR Ca2+ concentration would
contribute to it (Stern et al. 1999)(for a review, see e.g. (Bers 2001)). After the
Ca2+ release via RyR is inactivated, the net Ca2+ flux to the cytosol becomes
negative mostly due to SERCA and NCX Ca2+ extrusion, with a minor
21
contribution from the SL Ca2+ ATPase and mitochondrial Ca2+ uptake, and the
[Ca2+]i declines back to the diastolic steady-state value (Negretti et al. 1993). The
same amounts of Ca2+ are extruded to the SR and extracellular space, which came
from there correspondingly during activation. This Ca2+ homeostasis makes it
possible to continuously produce similar Ca2+ releases and transients and
consequent contractions (Trafford et al. 1997, Eisner et al. 2000).
When considering the Ca2+ fluxes to and from the cytosol and changes in
[Ca2+]i, it is essential to consider that Ca2+ is highly buffered in the cytosol (for a
review, see e.g. (Bers 2001)). For example, in the ventricular cardiac myocyte the
ratio of bound vs. free Ca2+ is ~100:1. The largest contributor to Ca2+ buffering
during ECC is troponin C (TnC) (see 2.1.3). Other important buffers considered
to contribute to Ca2+ signaling are SERCA, calmodulin (CaM), cell membranes,
ATP, myosin and creatine phosphate.
2.1.3 Contraction of the cardiac muscle cell
The contraction of the myocyte is based on the sliding movement of two types of
myofilaments relative to each other (Huxley 1953, Huxley & Niedergerke 1954,
Huxley & Hanson 1954)(for a review, see (Huxley 2000)). The two different
myofilament types are the thin filaments, consisting mainly of actin and with
lesser quantities of troponin and tropomyosin proteins, and the thick filaments,
consisting mainly of myosin protein. The thin and thick filaments overlap each
other within the myocyte (Fig. 2). The myosin heads bind and bend against the
thin filament resulting in the sliding of thin and thick filaments past each other
and consequent contraction of the myocyte. The biochemical reactions between
actin and myosin in this sliding process consume ATP, thus converting the
chemical energy to mechanical work (for a review, see e.g. (Geeves et al. 2005)).
22
Fig. 2. Schematic diagram of thin and thick filaments within the myocyte. Modified
from Pocock & Richards 2004.
To regulate the activation and relaxation of the muscle, the sliding process
described above must be able to turn on and off. The troponin protein complex
located in the thin filament is formed from TnT (tropomyosin-binding subunit),
TnI (inhibitory subunit) and TnC (Ca2+-binding subunit). During diastole, when
[Ca2+]i is low, the troponin complex inhibits the interaction of actin and myosin.
However, during SR Ca2+ release the released Ca2+ binds to the low-affinity Ca2+-
specific sites of TnC, and in this Ca2+-bound form the troponin protein complex
does not inhibit the interaction of actin and myosin. Consequently, the binding of
Ca2+ to TnC “switches” the contractile mechanism on (for a review, see e.g.
(Farah & Reinach 1995)).
Due to the role of Ca2+ described above, the strength of myocyte contraction
is related to [Ca2+]i , i.e. a larger Ca2+ release from the SR activates more actin-
myosin interactions and results in stronger contraction (Solaro et al. 1974, Fabiato
& Fabiato 1978, Gwathmey & Hajjar 1990). Several factors modulate the [Ca2+]i-
force relation, such as pH, temperature, phosphorylation status of TnI and several
chemicals (Fig. 3) (for a review, see e.g. (Bers 2001)). However, in ECC studies
this modulation is often neglected and the ECC process is studied from AP to the
dynamics of [Ca2+]i, and the rest is considered to be directly predictable based on
[Ca2+]i.
Thin filamentThick filament & myosin heads
23
Fig. 3. The dependence of the tension generated by skinned rat ventricular myocytes
from the [Ca2+]i in pCa units and from the pH in the intracellular solution.
pCa = −log10[Ca2+]i, where the unit of [Ca2+]i is M. Modified from Fabiato & Fabiato 1978.
2.1.4 The ECC in different regions of the heart
The heart consists of different cardiac myocyte types for different tasks. The
myocytes of the sinoatrial node (SA) generate spontaneous APs and determine the
heart beat frequency. The electrical signal propagates from SA node via atrial
tissue to the atrioventricular node (AV). The propagation is based on APs
triggering APs in neighboring cells. Consequently, contractions are triggered in
atrial myocytes that generate the force for pumping the blood out of the atriums.
The AV myocytes conduct the electrical signal further to the myocytes in Purkinje
fibers. The Purkinje fibers are located around the ventricles. Thus, the electrical
signal is finally conducted from Purkinje fibers to ventricles to trigger APs and
contractions in ventricular myocytes. Contraction of ventricular myocytes
generates the force for pumping the blood out of the ventricles.
In addition to SA myocytes, the AV and Purkinje fiber myocytes are also
capable of generating spontaneous APs, but with a lower frequency than the SA
myocytes. Thus, pacemaking normally originates from the SA, but AV and
24
Purkinje fiber pacemaking exist as backup mechanisms for pacemaking during
SA dysfunction or AV block (for reviews, see e.g. (James 2003, Mangoni &
Nargeot 2008)). The capability to generate spontaneous APs is a major difference
between the SA, AV and Purkinje fiber vs. atrial and ventricular myocytes, since
the latter remain quiescent without an external electrical stimulus.
The spontaneous AP generation in SA, AV and Purkinje fiber myocytes is
based on the activation of a hyperpolarization-activated cation current, the
pacemaker current (If) (for a review of If, see (Accili et al. 2002)). If is activated
by the hyperpolarization of the cell membrane after the AP is repolarized. If
produces a slow depolarization during diastole which finally activates the voltage-
activated depolarizing channels generating the AP. The ion channel generating
cardiac If is encoded by three different genes, HCN1, HCN2 and HCN4, which all
generate individually the ion channel, but with different voltage-dependence and
time constants for the activation of the If current (Altomare et al. 2001)(for
reviews, see e.g. (Accili et al. 2002, Biel et al. 2002)). The HCN isoforms are
differentially expressed in different types of cardiac myocytes, which in part
explains the varying intrinsic spontaneous AP frequencies (Shi et al. 1999). In
contrast to depolarization by If, it has also been reported that the underlying
mechanism of spontaneous APs might be more complex and involve interaction
between the SR and SL via cytosolic Ca2+ signals (Bogdanov et al. 2001). Due to
the stoichiometry of Na+ and Ca2+ transport by NCX (3 Na+ : 1 Ca2+ in opposite
directions (Hinata et al. 2002)), Ca2+ extrusion generates a depolarizing current.
Bogdanov et al. (2001) suggested that the initiation of the pacemaker AP also
involves a depolarization by NCX extruding Ca2+ released from the SR.
Most of the same major SL ion current types are found in all cardiac
myocytes in different regions of the heart (Fig. 1). The major voltage-activated
Ca2+ currents are the L- and T-type (ICaL and ICaT). The major K+ current types
contributing to AP repolarization are the transient outward (Ito) and different
delayed rectifier K+ currents. The time-independent K+ current (IK1) contributes to
the resting potential. Although the same main ion current types are present, the
shape and duration of APs differ among cardiac myocytes. This results from the
differences in the activation and inactivation properties and expression of ion
channel types generating these currents in myocytes from different regions of the
heart (for a review, see e.g. (Nerbonne & Kass 2005)).
Compared to myocytes from other regions of the heart, the ventricular
myocytes have a unique structural feature, T-tubules (Fig. 4) (Ayettey &
Navaratnam 1978)(for a review, see (Brette & Orchard 2003)). The T-tubules are
25
invaginations of SL, where the L-type VACCs are mostly localized in ventricular
myocytes. On the intracellular side of T-tubules, the SR is very near to the SL
forming a close localization of RyRs and VACC (Brette & Orchard 2003). The
small cytosolic volume between the SR and SL is referred to as the subspace (Fig.
4). Furthermore, the release site of the SR near the subspace is often referred to as
the junctional SR (JSR) and the rest of SR as the network SR (NSR). Although
the electrical properties of T-tubules are not completely understood, the
depolarization is generally assumed to deviate also to the T-tubules, when an
electrical stimulus initiating an AP is applied to the ventricular myocyte (Brette &
Orchard 2003). Due to the close localization of RyRs and VACCs, the CICR
initiated by the deviated depolarization occurs rapidly in the subspace from where
the Ca2+ diffuses to local areas of cytosol (Cannell & Soeller 1997). The T-tubules
and the RyR-VACC couplings cover the whole 3-dimensional structure of the
ventricular myocyte. Consequently, the Ca2+ is released near the contractile
elements and the activation of the cytosolic Ca2+ transient occurs simultaneously
in the whole myocyte, resulting in uniform cytosolic Ca2+ transients (Fig. 4)(Yang
et al. 2002).
Fig. 4. Schematic layout of the interaction between voltage-activated L-type Ca2+
channels generating the L-type voltage-activated Ca2+ current (ICaL) at the SL and
ryanodine receptors (RyR) at the SR in ventricular cardiac myocytes. The black arrows
indicate Ca2+ fluxes during activation of ECC. The figure is an enlargement of the gray
area in the inset.
ICaL RyR
RyR
ICaL RyR
RyR
ICaL RyR
RyRICaL
RyR
RyR
RyR
ICaL
RyR
RyR
RyR
ICaL
RyR
RyR
RyR
SL
SR
CytosolT-tubule
Subspace
Cytosol
Ventricular myocyte
26
The atrial, AV, SA and Purkinje fiber myocytes have less or lack completely
T-tubules (Ayettey & Navaratnam 1978). In these cells the Ca2+ enters from the
SL and diffuses towards the interior of the cell. In atrial cells this Ca2+ diffusion is
facilitated by local CICRs (Huser et al. 1996). Consequently in the AV, SA and
Purkinje fiber myocytes and, despite the facilitation, also in atrial cells, the
cytosolic Ca2+ transient is inhomogeneous as the activation of the Ca2+ transient at
the center of the cell is delayed from that in the SL region of the cell (Huser et al.
1996, Cordeiro et al. 2001, Yang et al. 2002, Blatter et al. 2003)(for a review, see
(Brette & Orchard 2003)).
2.1.5 Species-dependent differences in ventricular cardiomyocyte ECC
Although the basic ECC scheme is the same in all mammalian ventricular
myocytes, notable differences exist between different species. This is an
important issue when trying to build a general view of ECC based on experiments
with different species and when extrapolating the results from laboratory animals
to humans. In particular, the two most popular generally used laboratory animals,
rat and mouse (COM 2007), have several unique features compared to larger
mammalians, such as the human, canine and rabbit.
The frequency of the heart beat in mouse (~600 beats/min at rest (Desai et al.
1999)) and rat (~300 beats/min at sinus rhythm (Krandycheva et al. 2006)) is
significantly faster than in larger mammalians (for e.g. ~70 beats/min in humans).
To enable such a rapid heart beat, a shorter duration for ECC and AP is required
compared to large mammalians (~50–70 ms APD75 in rat (AP duration at 75%
repolarization) (Voutilainen-Myllyla et al. 2003) and ~50 ms APD90 in mouse
(Maier et al. 2003) vs. 336 ± 16 ms APD95 in humans (Li et al. 1996) and 307 ±
20 ms APD90 in rabbit (Bassani et al. 2004))(Fig. 5).
27
Fig. 5. Differences in shape and duration between APs in mouse and human
ventricular myocytes. The brace above the human AP indicates the plateau phase of
the AP, not found in the mouse AP, during which the V is positive and remains almost
constant. The (*) indicates the time-point of Ito-mediated repolarization. The figure is
modified from Maier et al. 2003 and Li et al. 1996
The short APD in rat and mouse ventricular myocytes is mostly explained by the
lack of the plateau phase, a general characteristic of ventricular APs in
mammalians (Fig. 5). During the plateau phase, the depolarizing currents, mostly
ICaL, and repolarizing K+ currents are equal for some time and the V remains
almost constant. The ICaL activates during the rapid depolarization by INa and
remains active at positive V. Before the plateau, an initial repolarization by Ito
occurs (* in Fig. 5). If the initial repolarization by Ito is large enough to reduce the
V below ~0 mV, the ICaL activated during depolarization is inactivated and no
plateau is generated (for a review, see e.g. (Nerbonne & Kass 2005)). This is the
case in rat and mouse ventricular myocytes as they have a large density of Ito
current (19 ± 4 pA/pF at V = +50 mV in mouse (Wang & Duff 1997) and
25.5 ± 1.3 pA/pF at V = +40 mV in rat (epicardial) (Volk et al. 2001) vs. for e.g.
10.6 ± 1.08 pA/pF at V = +40 mV in humans (subepicardial) (Nabauer et al.
1996)).
The non-plateau AP reduces the open time of L-type VACCs and thus the
amount of ICaL-mediated Ca2+ intrusion (Yuan et al. 1996). This can also be seen
from the fractional contribution of Ca2+ extrusion mechanisms. To maintain the
Ca2+ homeostasis during one cycle of ECC, the same amount of Ca2+ coming to
the cytosol via L-type VACC has to be removed by NCX, with a minor
contribution by the SL Ca2+-ATPase (Trafford et al. 1997, Eisner et al. 2000).
Mouse ventricular AP Human ventricular APplateau
* *
28
Similarly, the amount of Ca2+ released via RyR has to be removed by SERCA.
The ratio for NCX:SERCA Ca2+ extrusion (i.e. ~ICaL:RyR Ca2+ intrusion) is ~1:9
for mouse (Li et al. 1998) and rat (Negretti et al. 1993) compared to e.g. ~1:3 for
humans (Piacentino et al. 2003), suggesting that the ICaL Ca2+ intrusion has to be
indeed smaller during AP without plateau than during AP with plateau. However,
in rat and mouse the contribution of ICaL to SL Ca2+ intrusion may be even smaller.
In suitable conditions, the NCX can also operate in reverse mode in which it
intrudes Ca2+ and extrudes Na+ (Han et al. 2002b). The reversal potential of NCX
is determined by the Ca2+ and Na+ reversal potentials and consequently the
concentration gradients of these ions
3 2 .NCX Na CaE E E= − (3)
In most mammalians, the [Na+]i is low (4–8 mM) and the ENCX is higher than V
during the majority of the ECC and diastole and no significant NCX-mediated
Ca2+ intrusion occurs (for reviews, see e.g. (Bers 2001, Bers et al. 2003)). In rat
and mouse the [Na+]i is much higher (10–15 mM), resulting in significantly lower
values for ENCX (Bers et al. 2003). It has been estimated that with this high [Na+]i
~50% of the Ca2+ activating CICR originates via NCX and the rest ~50% via
L-type VACC (Han et al. 2002b). However, despite the route via which the Ca2+
enters through the SL, in rat and mouse the amount of entering Ca2+ is small, but
it is still obligatory for ECC as it is the trigger for the larger CICR required for
contraction.
The relationship between contractile force and the frequency of APs and
ECCs (force-frequency relation, FFR) is different in rat and mouse myocytes
compared to that in most mammalian species. In most mammalian species the
FFR is positive for most of the frequency range, i.e. contractile force increases
with an increasing frequency of contractions, but at very high frequencies FFR
usually turns negative (for a review, see e.g. (Endoh 2004)). The most evident
unique feature in rat and mouse myocytes is the negative FFR at very low
frequencies compared to the normal heart beat frequencies (below 1 Hz in mouse
(Redel et al. 2002) and below 0.3 Hz in rat (Schouten & Terkeurs 1991, Endoh
2004). At higher frequencies the behavior of FFR in rat and mouse myocytes
depends highly on the temperature and extracellular [Ca2+], resulting in either
negative, flat or positive FFR in different experimental conditions (Schouten &
Terkeurs 1991, Redel et al. 2002, Endoh 2004).
29
2.2 Development of cardiac ECC
Development of the mammalian embryo from fertilization to adult can be divided
into prenatal development before birth, which is further divided into
embryogenesis and fetal development, and postnatal development (Fig. 6). The
heart is first formed as a tube-shaped tissue during embryogenesis (for a review,
see e.g. (Moorman et al. 2003)). During pre- and postnatal development, the tube
forms into the shape of the adult heart. The first step is the looping of the tube,
after which the region for ventricular myocytes can be separated. Simultaneously,
the cardiomyocytes in different regions of the developing heart begin to
differentiate towards the corresponding adult cardiomyocytes. Notable is that the
cytosolic Ca2+ signals in embryonic cardiomyocytes have been shown to regulate
these changes in gene expression and structure of the heart during embryonic
development (Porter et al. 2003).
30
Fig. 6. Development of the mouse heart and ventricular myocytes. The data used to
construct the figure was taken from studies of mouse, or species that were as close
as possible such as rat, development (Nakanishi et al. 1988, Kirby et al. 1992, Ioshii et
al. 1994, Kaufman 1995, Bassani et al. 1998, Haddock et al. 1999, Mesaeli et al. 1999,
Viatchenko-Karpinski et al. 1999, Bers 2001, Yasui et al. 2001, Moorman et al. 2003,
Seki et al. 2003, Creazzo et al. 2004, Mery et al. 2005, Sasse et al. 2007).
2.2.1 ECC in pre- and postnatal ventricular cardiomyocytes
At the cellular level, ventricular myocytes undergo dramatic changes during
development (Fig. 6). In contrast to adult ventricular myocytes, the embryonic
ventricular myocytes are capable of generating spontaneous cytosolic Ca2+
transients required for contraction and spontaneous APs (Viatchenko-Karpinski et
al. 1999, Yasui et al. 2001). However, based on AP recordings only part of the
cells are spontaneously active during the later stage of embryonic development
(Yasui et al. 2001). In addition to the spontaneous activity, the embryonic
Postnatal/Neonatal dev.
Fertiliz
ation
Birth
AdultEmbryogenesis/Embryonic development Fetal development
7 8 9 10 sdklfsdkfölsk
Strong SR or SL Ca2+ signals? Strong SR Ca2+ signals
Strong SL Ca2+ signals
Cytosolic Ca2+ buffering capacity (TnC)
Ventricular cells form a spontaneously active culture
Heart tube Ventricles
Cell size
T-tubules
Homogeneous Ca2+ signalsHeterogeneous cytosolic Ca2+ signals
Controversial ECC & spontaneous activity mechanisms suggested Known ECC
Long diffusion distance between SL and SR
day 0 11 12 13 14 15 16 17 18 19 sdfhskjhfdksjhfksdjhfsdkjhfkjsdhfkjdshfkjdshkjfhsdkjfhfsdfsd
weeks
Cells divide and differentiate
SR Ca2+ nirtseuqeslaC niluciterlaC :reffub
Individual ventricular cells are spontaneously active (AP & Ca2+)
31
cardiomyocytes are also capable of synchronizing their electrical and contractile
activity similarly to adult cardiomyocytes (for a review, see e.g. (Kamino 1991)).
Based on previous studies, no clear conclusion can be made about the
underlying ECC mechanism in embryonic cardiomyocytes, since the explanations
of the generation of spontaneous activity are controversial. The electrical activity
might be based on spontaneous APs generated by If (Yasui et al. 2001) or a
cytosolic Ca2+ oscillation triggering depolarization via NCX (Viatchenko-
Karpinski et al. 1999, Sasse et al. 2007). Both mechanisms are also suggested to
exist in adult pacemaker myocytes (see 2.1.4). The cytosolic Ca2+ oscillation
might originate from SL sources (Nakanishi et al. 1988) or spontaneously from
SR sources with the contribution of the inositol-1,4,5-triphosphate receptor (IP3R)
and RyR (Viatchenko-Karpinski et al. 1999, Mery et al. 2005, Sasse et al. 2007).
In addition to the RyR, the IP3R is a SR Ca2+ release channel, but with different
Ca2+ sensitivity and slower dynamics for Ca2+ releases than the RyR (Streb et al.
1983, Kentish et al. 1990).
The embryonic and neonatal ventricular myocytes lack or have irregular
T-tubules (Fig. 6). Similarly as in adult atrial, SA, AV and Purkinje fiber
myocytes, this results in loose coupling, i.e. a long diffusion distance, between
Ca2+ channels at the SL and SR. Even though the embryonic and neonatal
ventricular myocytes are smaller than adult ventricular myocytes, this results in
heterogeneous cytosolic Ca2+ signals as in adult atrial, SA, AV and Purkinje fiber
myocytes (see 2.1.4)(Haddock et al. 1999, Seki et al. 2003). In embryonic
myocytes, the reports of the contributions of SL and SR Ca2+ sources (ICaL, ICaT
and NCX vs. RyR and IP3R correspondingly) to cytosolic Ca2+ signals are
controversial (Nakanishi et al. 1988, Viatchenko-Karpinski et al. 1999, Mery et al.
2005, Sasse et al. 2007). In neonatal and adult myocytes, this issue is more clear:
the contribution of SL sources is larger and the contribution of SR sources is
smaller in neonatal than adult ventricular myocytes (for a review, see e.g. (Bers
2001)). This is especially notable in species where the Ca2+ transient and
contraction in the adult myocyte depend practically completely on SR Ca2+
release, such as in the rat, but the neonatal myocyte can produce Ca2+ transients
with a disabled SR (Kirby et al. 1992, Seth et al. 2004).
Cytosolic Ca2+ is highly buffered in cardiac myocytes. However, in neonatal
and embryonic cells the cytosolic Ca2+ buffering is weaker compared to adult
cells due to the smaller amount of TnC (Bassani et al. 1998, Creazzo et al. 2004).
In the SR, Ca2+ is buffered by calreticulin in embryonic cells, but during
development the expression of it is downregulated (Mesaeli et al. 1999). At the
32
same time the expression of calsequestrin is upregulated and consequently, in the
neonatal and adult cells, the SR Ca2+ is buffered by calsequestrin (Ioshii et al.
1994).
2.2.2 Developing cardiomyocytes in cell culture
Adult cardiomyocytes are terminally differentiated and do not divide. The
developing cardiomyocytes, however, have the ability to divide and differentiate.
In contrast to adult cardiomyocytes, the developing cardiomyocytes can thus be
used for long-term cell culture applications. Although the individual isolated
neonatal ventricular myocytes are not spontaneously active, after a few days of
culturing they form a spontaneously beating confluent cell culture. Consequently,
the neonatal rat cardiomyocyte cell culture in particular, as one of the few cardiac
cell culture models, has become a popular experimental cell model in biochemical,
molecular biological and cellular signaling research (for example studies, see e.g.
(Gaughan et al. 1998, Zobel et al. 2002, Most et al. 2005)).
2.3 Regulation of cardiac ECC
The ECC, as biological systems usually are, is a complex non-linear system. To
maintain the production of contractile force in a standard rhythm with standard
magnitude, the performance of the system has to be monitored and dynamically
regulated (i.e. feedback regulation). On the other hand, the frequency of the ECC
and the contractile force has to be able to be modulated rapidly to produce more
cardiac output during exercise. It is obvious that the expression levels of
E-C-coupling proteins affect the performance of this system (Li et al. 1998, Yao
et al. 1998). Important regulation of cardiac gene expression indeed exists based
on e.g. Ca2+ signals (Molkentin et al. 1998, Porter et al. 2003, Tavi et al. 2004).
However, the change in protein expression levels requires more time than a few
heartbeats. Thus, in the time scale of beat-to-beat regulation other dynamic
regulatory mechanisms are also required.
2.3.1 Inotropic response during sympathetic stimulation
Sympathetic stimulation produces an increased heartbeat, due to its effects on SA
node pacemaking, and increased contractile force (for a review, see e.g. (Bers
2001)). The increased contractile force (inotropy) is mostly the result of the
33
activation of β-adrenergic receptors in cardiac myocytes. The β-adrenergic
receptors are activated by the hormones epinephrine and, with less effect,
norepinephrine, both secreted during sympathetic stimulation. Some contribution
to the inotropic response also comes from the activation of α-adrenergic receptors
by epinephrine and norepinephrine and from the stretch-induced changes in
E-C-coupling (Bers 2001, Tavi et al. 2001). The activation of β-adrenergic
receptors results in the activation of protein kinase A (PKA) within the
cardiomyocyte (Fig. 7). The increased contractile force is based on larger Ca2+
transients, which is based on the activation (phosphorylation) of RyR, ICa and
SERCA by PKA (Bers 2001, Tavi et al. 2005). The effect of larger Ca2+ transients
on contractile force is significant, although it is slightly compensated by the
desensitization of TnC to Ca2+ by PKA (Bers 2001).
Fig. 7. The activation schemes of PKA by β-adrenergic stimulation (blue) and CaMKII
by [Ca2+]i (red), as well as the interaction between these schemes. The two black lines
represent the cell membrane. The arrows represent activation and the circle blocks
inhibition. β-AR, β-adrenergic receptor; Gs, s-type G protein; AC, adenylyl cyclase;
cAMP, cyclic adenosine monophosphate; PKA, protein kinase A; PDE,
phosphodiesterase; CaM, Calmodulin; CaN, Calcineurin; PP1, protein phosphatase 1;
CaMKII, CaM-dependent protein kinase II. The figure is based on reviews (Bers 2001,
Bhalla 2005).
β-AR &Gs
epinephrine norepinephrine
AC
cAMP
PKA
CaM
[Ca2+]i
CaN
PP1
CaMKII
PDE
34
2.3.2 Activation of Ca2+-dependent enzymes
In addition to PKA, another dynamic regulator of ECC is the calmodulin (CaM)-
dependent protein kinase II (CaMKII), which is activated by Ca2+ through CaM
(Fig. 7). CaMKII was in fact discovered due to its dependence on Ca2+ and CaM
for activation (Schulman & Greengard 1978)(for a review, see (Anderson 2007)).
The primary difference between activation of PKA by β-adrenergic stimulation
and activation of CaMKII is that the first depends on an exogenous (extracellular)
stimulus (epinephrine and norepinephrine), while the latter depends on an
endogenous (intracellular) stimulus ([Ca2+]i). However, the activation of PKA and
CaMKII are not completely separate, since activated PKA supports the activation
of CaMKII by preventing the inhibition of CaMKII by protein phosphatase 1
(PP1) (Fig. 7). More importantly, the activation of PKA produces larger Ca2+
transients, which is the stimulus for CaMKII (for reviews, see e.g. (Bers 2001,
Bhalla 2005)).
Calmodulin (CaM) is a protein capable of binding four Ca2+ ions (Cheung
1970, Kakiuchi & Yamazaki 1970)(for a review, see (Schaub & Heizmann 2008)).
The affinity of CaM to Ca2+ is such that in diastole in a cardiomyocyte there is
little Ca2+ bound, but during systole the binding sites are occupied (for a review,
see e.g. (Maier & Bers 2002)). Consequently, CaM is a dynamic Ca2+ buffer (see
2.1.2) and contributes ~3% of the total Ca2+ buffering during ECC in the
ventricular myocyte (Bers 2001). In the Ca2+-bound form CaM interacts with
many proteins, causing inactivation or activation (for a review, see e.g. (Maier &
Bers 2002)). Although the activation of calcineurin (CaN) and CaMKII by the
Ca2+-CaM complex is highlighted here (Fig. 7), CaM has also been shown to
directly affect E-C-coupling proteins such as the voltage-activated Ca2+ channel
(Alseikhan et al. 2002).
The binding of Ca2+-CaM to CaMKII enables CaMKII to phosphorylate its
targets (for a review, see e.g. (Maier & Bers 2002)). It is notable that activation of
CaMKII by Ca2+-CaM is also balanced by negative regulation. The activation of
CaN by the Ca2+-CaM complex inhibits the activation of CaMKII, since the CaN-
activated protein phosphatase 1 (PP1) dephosphorylates (i.e. inactivates) the
active states of CaMKII (Fig. 7) (Schworer et al. 1986, Strack et al. 1997).
An important feature in the activation of CaMKII is its “memory” due to
autophosphorylation (for a review, see e.g. (Hudmon & Schulman 2002)). After
activation of CaMKII by Ca2+-CaM, CaMKII can phosphorylate its own Thr286
site. Due to this, the Ca2+-CaM complex is “trapped” in CaMKII and CaMKII
35
remains active for several seconds even though [Ca2+]i and consequently [Ca2+-
CaM]i declines (Meyer et al. 1992, Hudmon & Schulman 2002). In addition to
trapping Ca2+-CaM, the autophosphorylation also decreases the ability of the
autoinhibitory domain of CaMKII to inhibit itself (Colbran et al. 1989, Hudmon
& Schulman 2002). By both of these mechanisms, the concentration of active
CaMKII declines much slower than the Ca2+ transient, and during repeated Ca2+
transients CaMKII integrates the cytosolic Ca2+ signal. Consequently, in addition
to the amplitude and duration, the frequency of Ca2+ transients determines the
amount of active CaMKII (De Koninck & Schulman 1998).
Although the activation of CaMKII by Ca2+ proceeds schematically as
discussed above, it should be mentioned that each stage of this scheme involves
several chemical reactions. Consequently, quantitative description of CaMKII
activation requires the use of dozens of chemical reactions, which have been well
documented previously (Bhalla & Iyengar 1999, Bhalla 2005). It should also be
mentioned that the activation of CaMKII might be different within different
regions of the cardiac myocyte due to the different local Ca2+ signals (Saucerman
& Bers 2008).
2.3.3 CaMKII-mediated regulation of ECC
CaMKII phosphorylates, thereby regulating, proteins within the cell which are
involved in the handling of the [Ca2+]i, most importantly the L-type VACC, RyR
and SERCA (for a review, see e.g. (Maier & Bers 2002)). The frequency-
dependent increase in CaMKII activity seems to be recruited especially in the
frequency-dependent acceleration of relaxation (FDAR), although the underlying
mechanism is unresolved (Maier & Bers 2002). Phospholamban (PLB) is a
phosphoprotein inhibiting SERCA, but the inhibition is relieved when PLB is
phosphorylated with a proper protein kinase (James et al. 1989)(for a review, see
e.g. (Brittsan & Kranias 2000)). It has been shown that phosphorylation of PLB
with CaMKII only in increases SERCA’s sensitivity for Ca2+, thereby reducing
the half-maximal activation concentration (Km) for Ca2+ (Odermatt et al. 1996).
However, evidence also exist that active CaMKII directly phosphorylates SERCA,
increasing its maximum Ca2+ transport capacity (for a review, see e.g. (Maier &
Bers 2002)). The FDAR results in faster relaxation and narrowing of the Ca2+
transient (Tavi et al. 2005). This enables greater SR Ca2+ refilling and
consequently larger Ca2+ releases during positive FFR in a shorter diastolic time
at higher heart beat frequencies (Antoons et al. 2002).
36
In line with the positive FFR and larger SR Ca2+ releases, CaMKII
phosphorylation of RyR has been shown to contribute to the enhancement of the
SR Ca2+ release by sensitizing the RyR to Ca2+ (Li et al. 1997, Wehrens et al.
2004, Kohlhaas et al. 2006). However, controversial results also exist suggesting
that CaMKII desensitizes the RyR to the Ca2+ triggering CICR (Wu et al. 2001).
Interestingly, RyR phosphorylation has been proposed to be involved in cardiac
failure and cardiac arrhythmias (Hoch et al. 1999, Wu et al. 2002). CaMKII
expression increases in cardiac failure (Hoch et al. 1999), and the consequent
large phosphorylation of RyR would result in a notable opening of RyRs during
diastole. The increased diastolic Ca2+ leak from the SR depletes the SR and
results in weakened Ca2+ transients, characteristic to cardiac failure (Ai et al.
2005).
Less controversial results have been reported for the facilitation of ICaL by
CaMKII. The underlying mechanism is that CaMKII promotes longer single
channel openings of L-type VACC (mode-2 gating) (Dzhura et al. 2000). The
physiological role of this ICaL facilitation is not clear, but it results in a larger peak
value and slower inactivation of the current (Xiao et al. 1994)(for a review, see
(Maier & Bers 2002)).
2.3.4 Faulty regulation of ECC
Compromised regulation of ECC underlies the pathological function of the heart,
for example in arrhythmias and contractile dysfunctions (for a review, see e.g.
(Bers 2001)). It is difficult, however, to distinguish whether the compromised
ECC regulation is the cause, the result, or both of these in cardiac diseases, since
the ECC is also regulated by itself. One of the most common cardiac diseases is
heart failure, which is usually preceded by and coexistent with hypertrophy
(increase in cell and tissue size). There is lot of variation in the reported ECC
phenotypes and underlying ECC mechanisms of failing and hypertrophied
myocytes. However, Ca2+ transients with smaller amplitude and slower relaxation,
lower SR Ca2+ content and prolonged APs are usually found. To explain these
changes, altered gene expression levels and phosphorylation status of ECC
proteins such as RyR, SERCA, NCX and K+ ion channels have been reported
(Marx et al. 2000, Bers 2001, Pogwizd et al. 2001, Prasad et al. 2007). Increased
CaMKII expression is also found in cardiac failure (see 2.3.3).
37
2.4 Studies of ECC based on genetic engineering
Genetic engineering has been widely used to study ECC (e.g. (Wakimoto et al.
2000, Maier et al. 2003, Tavi et al. 2005)). In genetic engineering studies, the
expression level of a certain protein is modified. The induced change in the
expression level of the protein and the consequent changes in the ECC parameters
can be measured. Other features from the animal, heart, and the isolated myocytes
with altered gene expression can be also studied, such as the morphology, heart
rate and viability. Based on these data, analysis of the functional roles of different
proteins can be made. Several previously characterized, genetically engineered
experimental models have provided a large amount of information about ECC.
The analysis of these functional roles, however, is complicated by the
complexity of the system. The ECC is a highly non-linear system (Qu et al. 2007).
In a non-linear system the change in an input parameter, such as the expression
level of a protein, does not correlate linearly with the output of the system, such
as contractile force. Thus, estimation of input-output relationship (protein
expression level vs. features of the ECC) based on a few different gene expression
levels is difficult. On the other hand, the performance of the ECC system has also
been shown to effect the transcription of genes (Molkentin et al. 1998, Porter et al.
2003, Tavi et al. 2004). Thus, changes in ECC induced by genetic alteration to a
single protein may induce further changes in the expression levels of other
proteins (Maier et al. 2003). This complicates the estimation of the contribution
of the original genetic alteration to the ECC.
2.5 Mathematical modeling of cardiac ECC
The primary purpose of mathematical modeling and computer simulations of
mathematical models is to explain how the studied systems work. Using
mathematics it is possible to integrate experimental data from the studied system
and previous theories into a single context, which explains how the system works
(Fig. 8). With modeling it is also possible to extrapolate how the system would
function in a different environment and with different features. Part of the real life
experiments can thus be replaced with model simulations of the studied system.
Parameters and interrelationships, which are hard or impossible to measure from
the studied system, can also be estimated with modeling. Modeling becomes very
useful especially when the complexity of the system increases beyond our
capability to understand it directly:
38
“The mind has the capability of holding and analyzing perhaps five to ten
different sequential phenomena, each occurring at different rates and each
interrelated with the other phenomena by various cross-linkages… On the
other hand, the modern computer can handle literally thousands of such
crosslinking interrelationships at the same time and can develop answers that
the mind alone cannot achieve.” (Guyton et al. 1974)
Fig. 8. Mathematical modeling is used to explain and describe how systems work.
A good example of the advantages of modeling is the modeling of cell signaling
(Bhalla & Iyengar 1999). The complexity and the magnitude of the cell signaling
networks are way beyond the capabilities of the mind. However, a large amount
of experimental chemical reaction data can be represented in mathematical form
to build a model of the cell signaling pathways (for references, see (Bhalla &
Iyengar 1999)). The model can then be used to estimate the activities of all
interesting cell signaling components simultaneously in various conditions and
with various stimuli (Bhalla & Iyengar 1999, Tavi et al. 2003, Tavi et al. 2004,
Aydin et al. 2007), which would be an impossible experiment in vivo or in vitro.
Mathematicalmodel
of the system
Experimental data
Existing theories
System-natural phenomenon
-machine-economy
…
Guess
Translation into the
language of
mathematics
Explains the function
39
The downside of modeling, obviously, is that it is only a model, not the real
system. Often, when lacking proper theory or experimental data, models include
approximations and guesses, which result in limitations in the reliability and
usability of the models. However, when these limitations are kept in mind when
judging the simulation results, mathematical modeling has proven its capabilities.
Consequently, mathematical modeling is applied in several fields from
engineering and natural sciences to economics, etc. In biology and physiology
and furthermore in ECC studies, mathematical modeling has become an essential
part of the research methodology with an increasing trend (for reviews, see e.g.
(Keener & Sneyd 1998, Puglisi et al. 2004, Winslow et al. 2005)).
2.5.1 History of ECC models
The first step towards mathematical modeling of cardiac ECC was made in the
area of AP modeling, when Hodgkin and Huxley presented the mathematical
description of ion currents generating APs in the squid giant axon (Hodgkin &
Huxley 1952). The Hodgkin-Huxley (H-H) equations were later introduced to the
field of cardiac APs (Noble 1962), and the H-H formalism is still used as a part of
nowadays AP modeling (Han et al. 2002a, Puglisi et al. 2004, Shannon et al. 2004,
ten Tusscher et al. 2004). In addition to AP modeling, the work of Hodgkin and
Huxley stands out in biology as one of the most successful combinations of
experiment and theory (Keener & Sneyd 1998) – an approach endeavored to
emulate in this thesis to provide an understanding of biology.
The role of intracellular Ca2+ and its underlying mechanisms began to be
revealed in the 1970s and 1980s (for reviews, see e.g. (Puglisi et al. 2004,
Dulhunty 2006)). At the same time, the first cardiomyocyte models with
intracellular Ca2+ dynamics (Beeler & Reuter 1977) and with intracellular release
and uptake mechanisms (DiFrancesco & Noble 1985) were introduced. However,
modeling did not change completely from AP modeling to ECC modeling. In
addition to complete ECC models, sole AP models with no or very simplified
intracellular Ca2+ dynamics are still used nowadays. For example, when AP
propagation is simulated using a tissue model formed from a large number of
myocyte models, the simpler models for individual myocytes are used to reduce
the computational load (ten Tusscher & Panfilov 2006). But, even though this
kind of approximation is sometimes necessary, it limits the usability and
reliability of the model since intracellular Ca2+ signals are always linked to AP via
NCX and SL Ca2+ ion channels (Han et al. 2002a).
40
After the mid-1990s, modeling moved from general models towards species-
dependent models, which is reasonable due to the large differences in ECC
between different species (Han et al. 2002a)(for a review, see e.g. (Puglisi et al.
2004)). As mentioned, single cell ECC modeling has also been used as a sub-
model in generating tissue-level phenomena (ten Tusscher & Panfilov 2006). On
the other hand, local-control theory introduced a completely new accuracy scale
to the modeling and understanding of intracellular mechanisms in ECC (Stern
1992). Usually in ECC models, as in the present study, each ion channel or other
protein resembles the average behavior of all of those in the cell (e.g. 28000
L-type VACC in the guinea pig cardiomyocyte (Rose et al. 1992)). However, to
explain the graded dependence of SR Ca2+ release on the amount of L-type VACC
Ca2+ flux requires the individual VACC-RyR couplings to be considered as
separate local sub-systems (Stern 1992). Interestingly, the experimental evidence
of these separate local Ca2+ releases (Ca2+ sparks (Cheng et al. 1993)) was found
after the local-control theory was suggested, which emphasizes the importance of
modeling in increasing our understanding of ECC. The stochastic variation
between the function of individual ion channels and other proteins may result in
other significant global phenomena, such as early-after depolarizations causing
arrhythmias (Tanskanen et al. 2005). Unfortunately, the extremely high
computational load of modeling individual ion channels and protein functions
separately still limits severely the practical usability of this kind of modeling in
ECC models (Stern et al. 1999, Tanskanen et al. 2005).
2.5.2 Integrated models of ECC and CaMKII regulation
Despite the importance of CaMKII-mediated regulation, it has not been
completely integrated into any existing myocyte model. A detailed model exists of
the chemical reactions involved in the complex chain from Ca2+ to CaMKII
activation (Bhalla & Iyengar 1999, Tavi et al. 2003, Tavi et al. 2004). However,
the only attempt to integrate CaMKII activation and feedback regulation of ECC
proteins by CaMKII to the myocyte model uses only a qualitative description of
this chain and does not simulate the activities of important components between
Ca2+ and CaMKII, such as CaM, calcineurin and protein-phosphatase 1 (Hund &
Rudy 2004).
41
2.5.3 ECC models of developing cardiomyocytes
Currently, no model for ECC during embryonic or postnatal development exists.
The reason for this may be the lack of consistent experimental data, especially for
embryonic ECC (see 2.2.1). Previous attempts to model ECC in developing cells
include only a partial description of ECC such as AP solely without detailed Ca2+
signaling (Krogh-Madsen et al. 2005, Wang & Sobie 2008) or only cytosolic Ca2+
(Haddock et al. 1999).
2.5.4 Structure of ECC models
The ECC models are compartmental models (see e.g. (Bondarenko et al. 2004,
Shannon et al. 2004)). The intracellular structure is divided into compartments,
such as the SR, cytosol and subspace, and the ion concentrations in these
compartments are changed by the dynamic ion fluxes representing the ion
channels, receptors, pumps and other routes for ion transport (Fig. 9). The cell
membrane is modeled usually as a capacitor whose charge depends on the ion
transport through the cell membrane. These descriptions result in a system of
differential equations to be solved to simulate the ECC.
42
Fig. 9. Example of the structure of the ECC model (modified from (Bondarenko et al.
2004)). The compartments are the extracellular space, cytosol, subspace (ss), the
network SR (NSR), and the junctional SR (JSR). The transmembrane currents are the
NCX current (INCX), L-type Ca2+ channel current (ICaL), Ca2+ pump current (Ip(Ca)), NaK-
ATPase current (INaK), the background Ca2+ current (ICab), the rapidly inactivating
transient outward K+ current (IKto,f), the slowly inactivating transient outward K+ current
(IKto,s), the slow delayed rectifier K+ current (IKs), the rapid delayed rectifier K+ current
(IKr), the ultra rapidly activating delayed rectifier K+ current (IKur), the noninactivating
steady-state voltage-activated K+ current (IKss), the time-independent K+ current (IK1),
the Ca2+-activated Cl- current (ICl,Ca), the background Na+ current (INab), and the fast Na+
current (INa). The SR Ca2+ fluxes are flux via SR ATPase (Jup), leak flux from SR to
cytosol (Jleak), flux from NSR to JSR (Jtr), and Ca2+ release flux via RyR (Jrel). Jxfer is the
Ca2+ flux from the subspace to the cytosol. Ca2+ buffers in the model are calsequestrin
in the JSR and calmodulin and Troponin C in the cytosol. Dynamic ion concentrations
in the model are the Ca2+, Na+, and K+ concentrations in the cytosol and Ca2+ in the
subspace, JSR, and NSR. Extracellular Ca2+, Na+ and K+ concentrations are constants.
INCX
43
2.5.5 The models of ion channels and membrane voltage
To model the electrical activity of the cell membrane, the electrical circuit model
of the cell membrane is used in ECC models (Fig. 10) (Hodgkin & Huxley
1952)(for a review, see e.g. (Puglisi et al. 2004)). The cell membrane separates
charge and thus it can be viewed as a capacitor. The ion channels are conductors
in parallel to the capacitor. The conductivity of these conductors depends on the
amount of open individual ion channels. The membrane voltage depends on the
sum of currents through the ion channels
1
.xxm
dVI
dt C= − (4)
In the H-H electrical circuit model (Fig. 10), each ion channel has a voltage
source representing the equilibrium potential for the ion. The difference between
membrane voltage and equilibrium potential determines the driving force for the
ion. The resulting current-voltage (I-V) relationship is linear with a constant
conductance. However, in some models (see for e.g. (ten Tusscher et al. 2004))
the Goldman-Hodgkin-Katz (GHK) current equation is used to describe the
voltage dependence of individual currents:
2 2 exp
1 exp
i o
zFVc c
z F RTI P V
zFVRTRT
− − =− −
(5)
where I = current of the ion, P = permeability of the membrane to the ion,
z = valence of the ion, F = Faraday’s constant, R = the ideal gas constant,
T = temperature, V = membrane voltage, and ci and co = intra- and extracellular
concentration of the ion. The GHK equation is derived from the Nernst-Planck
electrodiffusion equation (for references, see (Hille 2001)) by assuming that the
electrical field over the cell membrane is constant. With a constant permeability
(conductance), the GHK equation describes the I-V-relationship as non-linear and
is in some cases more suitable to describe the voltage dependence of the ion
current (for a review, see e.g. (Keener & Sneyd 1998)). The non-linearity is a
consequence of different ion concentrations driving independent inward and
outward ion fluxes (if ci = co, the I-V is linear)(see e.g. (Hille 2001)).
44
Fig. 10. The electrical circuit model of a cell membrane (Hodgkin & Huxley 1952). E =
membrane voltage, Ex = equilibrium potential of ion x, Ix = current of ion x, Rx =
resistance/conductance for ion x, Cm = capacitance of the membrane. EL and RL are the
equilibrium potential and resistance for the total leak current (IL) carried by different
ions. The number of conductances can be varied depending on the modeled system.
As in real excitable cells, the I-V relationship for any given ion current in the
electrical circuit model of the cell membrane is further shaped by the time and
membrane voltage-dependent changes in conductance (Fig. 10). The modeled
conductance may also depend on the binding of ligands and mechanical
deformations (see e.g. (Tavi et al. 1998, ten Tusscher et al. 2004)). The traditional
way to model the dynamics of the conductance is to use H-H gating equations,
which are empirical descriptions of the total conductance of the ion channel
population (Hodgkin & Huxley 1952)(see e.g. (Hille 2001)). However, in some
recent studies Markov state models have been used to describe the changes in the
conductance (for an example model, see e.g. (Bondarenko et al. 2004)). The
Markov state models are said to recapitulate more accurately the conformational
changes of the individual ion channel proteins (Bondarenko et al. 2004). In
contrast, the use of H-H gating equations results in simpler and computationally
more efficient models, and the fitting of the gating parameters to
electrophysiological data is more straightforward (ten Tusscher et al. 2004).
Since H-H gating is usually associated with the membrane voltage, Markov
state models are favored especially when modeling ion channels which are not
45
located at the SL, such as RyR (Keizer & Levine 1996) and IP3R (Sneyd &
Dufour 2002). In these cases the ion flux depends solely on the permeability,
based on the state of the Markov model, and on the electrochemical gradient over
the ion channel. The electrochemical gradient, however, is usually approximated
as the concentration gradient (see e.g. (Sneyd et al. 2003, Bondarenko et al.
2004)).
2.5.6 Chemical reaction models
The ion channel modeling methods discussed in 2.5.5 are used to model passive
diffusion through an ion channel population with changing total permeability. A
different class of models, not restricted to ion transport, are those where the
binding of an ion and possibly movement of the bound ion is modeled, e.g. the
buffering of ions and active ion transport. The basic chemical reaction equations
are used accurately, or with Michaelis-Menten or Hill’s equation approximations
for these purposes (Michaelis & Menten 1913, Bondarenko et al. 2004)(for a
review, see e.g. (Keener & Sneyd 1998)). In addition, barrier models, in which the
conformational change of the protein is assumed to require the crossing of a free
energy barrier, are used to model exchangers and active ion transport (Luo &
Rudy 1994, Keener & Sneyd 1998, Bondarenko et al. 2004). Barrier models can
also be used to model ion channels by assuming that the ions have to jump over a
discrete number of energy barriers within the channels (for a review, see e.g.
(Keener & Sneyd 1998)).
2.5.7 Diffusion of ions
The diffusion of chemical species in homogenous isotropic media can be
presented with Fick’s law of diffusion
D c= − ∇J (6)
where J = diffusion flux, D = diffusion coefficient and c = concentration of the
chemical species (Fick 1855)(for reviews, see (Keener & Sneyd 1998, Hille
2001)). Based on this, in ECC models the diffusion flux between two
compartments is usually presented in a more simplified form as
46
c
JτΔ= (7)
where Δc = concentration difference between the compartments and τ = time-
constant for diffusion, representing the total contribution of D and the diffusion
distance to J (Bondarenko et al. 2004, Shannon et al. 2004). If Fick’s law of
diffusion (6) is implemented into the conservation law for material, the equation
2cD c f
t
∂ = ∇ +∂
(8)
can be obtained where f is the local production (or consumption) of the chemical
species (for a review, see e.g. (Keener & Sneyd 1998)). The equation (8) is
referred to as the reaction-diffusion equation or Fick’s second law of diffusion. In
ECC models it can be used to describe spatial differences of c within a
compartment, e.g. Ca2+ diffusion in the cytosol (Haddock et al. 1999).
2.5.8 Simulation of excitable cell and tissue models
The excitable cell models discussed here, including the ECC models, and the sub-
components of these models, e.g. the ion channel and chemical reaction models,
are systems of ordinary differential equations (ODEs) (Bhalla & Iyengar 1999,
Han et al. 2002b, Tavi et al. 2003, Bondarenko et al. 2004). However, combined
systems of ordinary and partial differential equations (ODEs and PDEs) also exist
when spatial phenomena, such as AP propagation with many cell models (ten
Tusscher et al. 2004), or diffusion is modeled (Haddock et al. 1999). The
combined ODE and PDE systems are reduced to a system of ODEs by
approximating the partial derivatives with a finite space step (ten Tusscher et al.
2004)(for a review, see e.g. (Kreyszig 1999)).
To simulate these models, numerical integration is applied with a single CPU
(i.e. a desktop PC) or in the case of large-scale problems, with several parallel
CPUs (Han et al. 2002b, ten Tusscher et al. 2004). Complex, higher than first-
order adaptive time-step methods are often applied for efficient numerical
integration (Shampine & Reichelt 1997, Kreyszig 1999, Han et al. 2002b, Tavi et
al. 2003). However, less efficient first-order methods, such as the forward Euler
method, are generally applied due to the easier implementation especially for
parallel computing (Kreyszig 1999, Sneyd & Dufour 2002, ten Tusscher et al.
47
2004, Fall & Rinzel 2006). The inefficiency of the Euler method is emphasized
particularly with the models discussed here. The models often consist of rapidly
changing components compared to the time scale of solution and the dynamics of
other variables. This results in stiffness of the problem, requiring a very small
fixed time-step with the Euler method (for a review, see e.g. (Haataja 2002)).
48
49
3 Aims of the research
Mathematical modeling has proven its usability in studying ECC in cardiac
muscle cells (for reviews, see e.g. (Puglisi et al. 2004, Winslow et al. 2005)).
However, two significant areas, the enzymatic regulation of the activity of ECC
components and embryonic development of ECC, lack proper mathematical
analysis. The purpose of the present study was to develop models of development
and regulation of ECC and to use ECC modeling to produce new information
with biological significance in these areas. Furthermore, the applicability of
mathematical modeling to analyze genetically engineered cardiomyocyte models
was studied with the developed models. The specific issues were:
1. To study the efficiency and usability of different numerical methods for
simulating the ECC models. Could the efficiency of the Euler method be
increased with a time-step adaptation? How does the efficiency of the Euler
method relate to more complex, higher order methods?
2. To integrate the model of CaMKII activation into a model of ECC with
feedback regulation of ECC. How does the model explain the pacing
frequency-dependent regulation of ECC?
3. To analyze and explain ECC during development with experimental
characterization of ECC features and mathematical modeling. The ECC is
studied in E9-11 mouse embryonic and neonatal rat ventricular myocytes.
4. Could the mathematical modeling of ECC be useful in the design and
analysis of genetically engineered animal models? The developed
mathematical models of ECC were used to study the ECC mechanisms of
previously reported genetically engineered experimental models.
50
51
4 Summary of materials and methods
Several different research methods were used in the original papers (I–V). The
presenting author was responsible for the mathematical modeling,
electrophysiological recordings and confocal Ca2+ imaging. The cell isolation and
culturing (III, IV and V) and the immunolabeling (III) methods were performed
by others. The latter methods are not discussed or are mentioned only briefly here.
4.1 Experimental methods
4.1.1 Studied myocytes (III, IV, V)
The E9-11 mouse embryonic ventricular myocytes were isolated and cultured
using a modified method (III, IV) based on (Sturm & Tam 1993, Liu et al. 1999,
Doevendans et al. 2000). The isolation and culturing of neonatal rat ventricular
cardiomyocytes (V) was performed as previously described (Ronkainen et al.
2007a, Ronkainen et al. 2007b). The experimental procedures were approved by
the Animal Use and Care Committee of the University of Oulu.
4.1.2 Electrophysiology and confocal microscopy laboratory
The majority of the experimental work in the present study was performed in a
custom-built combined electrophysiology (patch-clamp and intracellular
recordings) and confocal microscopy laboratory (Fig. 11). For list of the
equipment see the figure legend (Fig. 11).
52
Fig. 11. Combined electrophysiology and confocal microscopy laboratory (III, IV, V).
Clockwise from below left: Grass S44 electrical stimulator with platinum wires;
perfusion system with heater, pump and tubes from gas bottles (last not shown);
Olympus Fluoview 1000 confocal inverted microscope on an anti-vibration table with a
Faraday cage, two (one is visible in the picture) 3-D micromanipulators (Roe 200,
Sutter Instruments) for patch-clamp and intracellular microelectrode holders/pre-
amplifiers and a video camera; a control unit for the confocal microscope and two
desktop PCs for data acquisition and controlling the confocal microscope and
electrophysiological recording devices with the Olympus Fluoview 1000 and Clampex
9.2 (Axon Instruments) software; Digidata 1322A A/D-D/A converter and an
oscilloscope (right, top shelf); microinjector (Picopritser II, Parker Instrumentation);
Axopatch-1D and Axoclamp 2B electrophysiology amplifiers (Axon Instruments). Not
pictured were multi-line argon- and UV-lasers and a Sutter P-80 microelectrode puller.
4.1.3 Electrophysiological recordings (III, IV, V)
The patch-clamp technique (Sakmann & Neher 1984) was used to measure ion
currents and action potentials. The whole-cell voltage clamp was used to record
the ion currents (ICaL, ICaT, INCX, INa, IK1 and delayed rectifier K+ current), with the
exception that the perforated patch whole-cell voltage clamp was used to measure
the INCX simultaneously with intracellular Ca2+ (Ginsburg & Bers 2005). The
action potentials were recorded in current clamp mode (I = 0). The membrane
capacitances and series (i.e. access) resistances were measured by applying a 5
53
mV pulse to the holding potential and compensated using the compensation
electrical circuits in the patch-clamp amplifier, expect when recording action
potentials from a confluent cell culture (V). Electrodes were made from glass
capillaries with a Sutter P-80 microelectrode puller. AgCl wires were used as a
measurement electrode within glass capillaries and as a ground electrode in the
bath solution. Electrode resistances were 1–5 MΩ in whole-cell current
recordings and 10–15 MΩ in action potential recordings. The recording solutions
and temperatures are listed in original papers III, IV and V. The ion currents were
expressed as current densities by dividing them with the measured cell membrane
capacitances. Data analysis was made using Clampfit 9.2 (Axon Instruments),
Origin 7.5 (OriginLab Corporation) and Matlab 6.5 (The MathWorks) (III, IV, V).
4.1.4 Ca2+ imaging (III, V)
For Ca2+ imaging the cardiac myocytes were loaded in DMEM for 0.5–1 h at 37 ºC in an incubator with Fluo-4-AM-ester or Fluo-3-AM-ester (1–10 μM, dissolved
in Pluronic F-127 DMSO (dimethylsulfoxide); Molecular Probes). The solution
was changed two times and cells were incubated at room temperature (20–22 °C)
or at 37 °C at least 30 min for the dye to de-esterify. To measure cardiomyocyte
calcium signals, the Petri dishes with loaded myocytes were placed in the custom-
made perfusion system built into an Olympus Fluoview 1000 confocal inverted
microscope. The recording solutions and temperatures are listed in original papers
III and V. Myocytes were excited at 488 nm and the emitted light was collected in
the line-scan or frame-scan mode with a spectral detector from 520 to 620 nm
through a 20× or 60× objective lens. To electrically excite the cells, myocytes
were stimulated with 1 ms voltage pulses through two platinum wires located on
both sides of the Petri dish. The recorded Ca2+ signals are expressed as an
F/F0-ratio, where F is the background subtracted fluorescence intensity and F0 is
the background subtracted minimum fluorescence value measured from each cell
at rest. The cardiomyocyte Ca2+ signals data analysis was performed with
Olympus fluoview 1000, Origin 7.5 (OriginLab Corporation), ImageJ 1.36b
(http://rsb.info.nih.gov/ij/) and Matlab 6.5 (The MathWorks) software (III, V).
54
4.2 Mathematical modeling
4.2.1 Model of the mouse ventricular myocyte (II)
A mathematical model of the mouse ventricular myocyte was developed. The
model is partly based on the previously published model (Bondarenko et al. 2004).
However, several modifications and additions were made to the original mode.
The parameters of the existing model components were adjusted to gain better
correspondence between experimental mouse data and simulated results,
especially in Ca2+ signaling (II). The model of the contractile element (Cortassa et
al. 2006) was also integrated into the myocyte model to compute the absolute
contractile force based on the cytosolic Ca2+ transient. The parameters of
myofilament Ca2+ sensitivity were fitted according to Ca2+ transient and
contractile force characteristics in mouse ventricular myocytes (Ito et al. 2000,
Stuyvers et al. 2002).
The most notable development was the integration of a model of the Ca2+-
dependent CaMKII activation reaction scheme (Fig. 12). As previously, Ca2+-
dependent CaMKII activation was modeled with the Ca2+, CaM, CaN, PP1 and
CaMKII components from the previously published reaction scheme (Fig. 12)
(Bhalla & Iyengar 1999, Tavi et al. 2003, Aydin et al. 2007). The PP1 inhibition
by protein kinase A (PKA) was modeled with a constant value for active PKA
(0.014735 μM) and the original higher neuronal [CaMKII] was reduced to 1 μM
(Tavi et al. 2003). The total [CaM] was set to compare the value in cardiac
myocytes (6 μM) (Shannon et al. 2000, Tavi et al. 2004). The CaMKII reaction
scheme uses the simulated [Ca2+]i as an input. The simulated CaMKII activity
modulates the RyR, SERCA and ICaL as discussed in 2.3.3.
55
Fig. 12. Schematic diagram of the processes included in the developed mouse
ventricular myocyte model (II). In this figure INKA = NaK-ATPase current, IPMCA = SL Ca2+-
ATPase current, CaMK = calmodulin-dependent protein kinase II.
CaMKII regulation (CaMKIIreg) in the model is calculated using a Hill’s equation
based on the concentration of simulated active CaMKII (CaMKIIactive)
( )
3
3 3.
1.2 M active
reg
active
CaMKIICaMKII
CaMKIIμ=
+ (9)
The Hill’s equation parameters 1.2 μM and 3 were based on fitting the whole cell
model function to experimental data. The SERCA Ca2+ flux (Jup) is modeled as a
Hill’s equation (Jafri et al. 1998). The CaMKII modulation was implemented as
previously (Hund & Rudy 2004)
( )( )
1.82
maxmax 1.81.82 max
, ,
1 iup up reg
m up m PLB regi
CaJ J CaMKII V
Ca K K CaMKII
+
+
= Δ ⋅ + ⋅ ⋅ + − Δ ⋅
(10)
where the ( )max 1up regJ CaMKIIΔ ⋅ + resembles the direct phosphorylation of SERCA
and the ( )max, ,m up m PLB regK K CaMKII− Δ ⋅ resembles the PLB phosphorylation-
mediated reduction of Km for SERCA. The maximal CaMKII-mediated changes
56
in parameters were based on previously reported experimental values (Toyofuku
et al. 1994, Odermatt et al. 1996). The ICaL phosphorylation resulted in longer
openings (see 2.3.3). The fast (subspace) Ca2+-dependent inactivation transition
parameter (γ) of the ICaL Markov state model (Bondarenko et al. 2004) was set to
depend negatively on CaMKII to promote the longer openings with higher
CaMKII activity
( )2
,max 2,
1.01 0.61 .ssreg pc
pc half ss
CaCaMKII K
K Caγ
+
+
= − ⋅ ⋅ ⋅ +
(11)
where Kpc parameters are the Hill’s equation parameters. Based on the
experimental data, the CaMKII modulation was set to produce ~10% increase in
ICaL (II) (Li et al. 1997, Zuhlke et al. 1999, Gao et al. 2006, Kohlhaas et al. 2006).
The RyR Ca2+ sensitization by CaMKII (see 2.3.3) was implemented by setting
the Ca2+-dependent transition rate bk + to be CaMKII-dependent
( ) -3 -10.00405 9.18 0.82 μM ms .b regk CaMKII+ = + (12)
The bk + transition rate moves the model state from open state 1 to open state 2,
which prolongs the open time of the RyR model (Keizer & Levine 1996). In some
of the simulations, the role of CaMKII-induced diastolic SR leak flux was studied.
This was simulated according to the following equation (in JSR volume):
( ) ( )2 2 -1100 1 0.0025 ms .leak reg JSR SSJ CaMKII Ca Ca+ + = ⋅ + ⋅ ⋅ − (13)
4.2.2 Models of developing cardiomyocytes (IV, V)
A model of the embryonic mouse cardiomyocyte during embryonic days 9–11
(E9-11) and a model of a (cultured) neonatal rat ventricular cardiomyocyte were
developed. Both of these cell types have heterogeneous cytosolic Ca2+ signals due
to the lack of T-tubules. To simulate these Ca2+ signals, the common-pool cytosol
with one or a few additional compartments, which is generally used in adult
ventricular myocyte models (Pandit et al. 2001, Bondarenko et al. 2004, Shannon
et al. 2004), is not sufficient. The cytosolic Ca2+ can be easily simulated in 3-D
with Fick’s second law of diffusion (see 2.5.7). However, when this 3-D diffusion
is coupled with the SL and SR, the diffusion in parallel to SL and SR would
57
require separate calculation of Ca2+-dependent ion channels and Ca2+ fluxes in all
points of the SL and SR surface grid. This would require an extremely large
computational power compared to e.g. common-pool ECC models. Based on
microscopy observations, the shape of the developing cardiomyocytes was
approximated to be spherical with radial symmetry in diffusion and thus also in
the Ca2+ dynamics below the SR and SL (Fig. 13). With this approximation, it was
possible to simulate the ECC with a practical number of differential equations and
still faithfully reproduce the Ca2+ diffusion and the structural features of the
myocyte.
Fig. 13. Schematic diagram of the model of the embryonic cardiomyocyte.
Based on microscopy observations, the nucleus was located at the center of the
cell and the SR was modeled as a thin surface on the nucleus (Fig. 13). In the rat
neonatal model, the SR was divided into the uptake and release compartments (V).
The dimensions of the whole myocyte and the nucleus were estimated from the
microscopy images. The cell radius, related to the cell surface area, was also
estimated from membrane capacitances measured by us (IV) or reported in
literature (V) (Avila et al. 2007, Guo et al. 2007). The volume of the SR and the
amount of SR Ca2+ buffer (calreticulin (IV) (Mesaeli et al. 1999) or calsequestrin
(V) (Ioshii et al. 1994)) were based on our experimental estimates of the Ca2+
storing capacity of these cells. The cytosolic Ca2+ buffering capacity was reduced
from the values used in adult myocyte models (Creazzo et al. 2004).
Nucleus NC
X
ICaL
If
ICaT
ICab
INa
NaK-ATPase
ATP
IP3
Ca2+
SERCA
SL
Cytosol SR Nucleus
RyR
RyR
RyR
SR
Cytosol
Ca2+Ca2+
IKdr
IK1
58
The ion channel models to be included in the SL and the fitting of the
parameters of the ion channel models were based on our experimental
electrophysiological recordings (IV, V) and reported electrophysiological data
from literature (V). The basic formulations for ion channel models were taken
from previously published models (Luo & Rudy 1994, Dokos et al. 1996, Wang
et al. 1997, Pandit et al. 2001, Bondarenko et al. 2004, ten Tusscher et al. 2004).
In the embryonic mouse myocyte model, the SR included the models of RyR
(Keizer & Levine 1996), SERCA (Jafri et al. 1998) and IP3R (Sneyd & Dufour
2002) in line with our experiments (III). As in real cells, the SR of the neonatal rat
myocyte model included the models of RyR and SERCA, with a passive Ca2+ leak
(Shannon et al. 2000, Kapiloff et al. 2001, Sobie et al. 2002, Seth et al. 2004).
The parameters of the SR Ca2+ channels and pumps were fitted according to our
experimental data on SR Ca2+ releases (III, V).
4.2.3 Simulations of the models (I, II, IV, V)
Simulations were performed by numerically solving the ODE systems. The
combined ODE and PDE problems in original papers I, IV and V were also
solved as ODE systems by approximating the partial derivatives with a finite
space step (Kreyszig 1999, ten Tusscher et al. 2004). All models were
incorporated into the Matlab 6.5 (The Mathworks Inc.) computing environment
for simulations. A similar self-written Matlab program and code structure was
used in all models. An example of the Matlab code files (m-files) used to simulate
a myocyte model is shown in the Appendix. The Matlab code for the Ca2+-
dependent CaMKII activation model (Bhalla & Iyengar 1999) was kindly
provided by Dr. Perttu Niemelä.
In original article I, different numerical integration methods were studied. A
time-step adaptation algorithm suitable for ECC models was developed for the
widely used forward Euler method (Kreyszig 1999, Sneyd & Dufour 2002, ten
Tusscher et al. 2004, Fall & Rinzel 2006)(I). The Euler method and the developed
method were compared to Matlab’s more complex built-in ODE solvers
(Shampine & Reichelt 1997)(I). A model of INa, a one-dimensional cable of
cardiomyocyte models, and a simple stiff equation
2 3 4'( ) , (0) 10y t y y y −= − = (14)
were used as test problems (Bondarenko et al. 2004, ten Tusscher et al. 2004)(I).
59
The developed method was used in original article IV. In original articles II
and V, the Matlab’s ‘ode15s’ routine for stiff ODE systems was used. The
‘ode15s’ is a 5th order method based on numerical differentiation formulas
(Shampine & Reichelt 1997).
60
61
5 Results
5.1 Comparison and development of simulation methods (I)
The forward Euler method was found very inefficient in the simulations of the ion
channel model (INa), the 1-D cable of cardiomyocyte models (Bondarenko et al.
2004), and the simple stiff differential equation:
2 3 4'( ) , (0) 10 .y t y y y −= − = (15)
The use of an automatic time-step (dt) adaptation based on the estimate of local
relative error was found to significantly increase the efficiency of the forward
Euler method. However, the solving was still not optimal as the algorithm used
unnecessarily short time-steps at some points of the solution. The error estimate
neglects the O(dt3) terms from the Taylor expansion of the forward Euler solution,
which become significant in stiff systems. Consequently, faulty high estimate of
the error is used to calculate too small a value for proper dt.
To overcome this problem, we found that limiting the error estimate-based
reduction of the time-step was successful (limited method in Fig. 14):
max
0.99 if *
0.99 if *
i ii
i i
TOLdt error TOL B
errordt
dt error TOL BB
≤
= >
(16)
where dtmax = proper time-step, dti = test-step, TOL = error tolerance level,
errori = error estimate based on the test-step and B = parameter determining the
maximal reduction of dti. Based on our simulations, the parameter B can be
chosen robustly and a problem-independent parameter could exist
(B = 5000 − 108). Consequently, the developed method could be used generally
for stiff problems.
62
Fig. 14. Comparison of the accuracy and required CPU time of the forward Euler
method with a fixed time-step (Euler), without limitation of the automatic time-step
reduction (non-limited) and with limitation of the automatic time-step reduction
(limited) (I). The model of INa was used as a test model (Bondarenko et al. 2004).
Despite the improvements to the Euler method developed here, it was still not as
efficient as more complex methods. The differential equation solver algorithms in
Matlab software (The Mathworks, Inc.) required less CPU time. However, the
accuracy of the developed method was good compared to the Matlab solvers. The
developed method is thus a significantly better choice than the fixed-step Euler,
when the use of more complex solvers is not possible or desired (Sneyd & Dufour
2002, ten Tusscher et al. 2004, Fall & Rinzel 2006).
5.2 Model of mouse ventricular ECC with CaMKII regulation (II)
To create a mathematical model which could recapitulate some of the essential
physiological features of cardiac myocytes, the model of CaMKII activation was
integrated into a model of mouse ventricular ECC. The developed model
reproduces experimental AP parameters, intracellular ion concentrations, Ca2+
signaling parameters and contraction force (see II, Tables 1 and 2). The
implementation of the Ca2+-dependent enzymes and the phosphorylation targets
of CaMKII made it possible to reproduce realistically the pacing frequency-
dependent changes in the Ca2+ signaling of isolated myocytes (Fig. 15). It is
notable that the model also reproduces the negative frequency vs. [Ca2+]i transient
amplitude slope below 1 Hz (Ito et al. 2000). An interesting feature of the Ca2+-
dependent enzymes was the difference in the pacing-dependent activation range
(Fig. 15 right). CaN activates mostly at frequencies below 2–3 Hz, whereas
CaMKII is relatively constant below 2 Hz but activates significantly above 2 Hz.
0.01 0.1 1 1010
100
1000
CP
U ti
me
(s)
INa
peak value error %
Euler Non-limited Limited
63
Fig. 15. Left: Diastolic and systolic [Ca2+]i values as a function of pacing frequency.
Simulated values (black squares) are compared with the experimental results of (Ito et
al. 2000) (red circles) and (Antoons et al. 2002) (green triangles). Right: The simulated
concentrations of active calcineurin (CaN) and Ca2+/calmodulin-dependent protein
kinase II (CaMK) at pacing steady-state. The enzyme activities are shown in relation to
the maximum pacing-induced value (at 4Hz). The divergent pacing dependence of CaN
and CaMK is demonstrated even more clearly in the inset, which shows the enzyme
activities scaled to the maximum calcium-induced value. (II)
5.3 ECC during embryonic and postnatal development (III, IV, V)
Mathematical modeling was employed to present a system analysis of ECC based
on the experimental data from embryonic mouse and neonatal rat cardiomyocytes.
Both the embryonic mouse and rat neonatal ventricular myocyte models had a
similar structure due to the lack of T-tubules (see 4.2.2). The heterogeneous
cytosolic Ca2+ signals were faithfully reproduced with this structure (Fig. 16).
Other experimentally defined ECC features and parameters were also reproduced
with the models.
The simulations of the models revealed some common features of developing
cardiomyocyte ECC (see Discussion 6.4.1). However, the modeled cells are quite
different and also the purpose of the modeling was different. The neonatal model
is a more quantitative description of ECC in this certain cell type, whereas the
embryonic model is more qualitative and focuses more on the fundamental
question of the general ECC and pacemaking mechanism during the early
embryonic days.
64
Fig. 16. Spatiotemporal distribution of the cytosolic calcium during excitation in the
cultured rat neonatal ventricular myocyte and in the model. (A) Line-scan recording of
Ca2+ diffusion within the cytosol. The contour plot shows the radial distribution of
cytosolic Ca2+ in a single cardiomyocyte after electrical excitation of the cell. (B) The
experimental recording reproduced by the model. The contour plot shows the
simulated cytosolic Ca2+ after electrical stimulation of the model cell. The upper panels
in (A) and (B) show the traces of Ca2+ signals near the SR and nuclear region (SR,
black line) and near the cell membrane (SL, gray line). The dashed lines indicate the
time delay for the Ca2+ diffusion from sub-SL to sub-SR. (V)
5.3.1 The dual parallel ECC modes in the embryonic cardiomyocyte (III, IV)
Isolated embryonic cardiomyocytes generate spontaneous SR Ca2+ releases
rhythmically (~0.1–1.3 Hz). Both IP3Rs and RyRs are required for these releases
(III). Previously, only ~0.1–0.01 Hz Ca2+ oscillations, generated by RyR, IP3R or
A
B
0246
SL
F/F
0
SR
0.2 0.4 0.6 0.8
SL
SR
[Ca
2+ ] i ( µ
M)
n
0.0 0.5 Time (s)
0.0 0.5 Time (s)
5 μ m
Experiment
Model
Δt
Δ t
16000 F
5000 F
0.2 µ M
0.5 µ M
5 μ m n
65
both of these, have been modeled in other cell types than cardiac cells (Deyoung
and Keizer, 1992; Keizer and Levine, 1996; Sneyd et al., 2003). However, based
on the present modeling study rapid oscillations (~1 Hz) can also be theoretically
generated by RyR and IP3R. The IP3R acts as an SR leak which slowly increases
the local cytosolic [Ca2+]i below the SR and triggers the larger release via RyR.
The amount of IP3R Ca2+ leak is determined by the [IP3]. Thus as in real cells (III),
the [IP3] modulates the frequency of these oscillations and consequently the
frequency of spontaneous APs (see below) (IV).
The spontaneously released Ca2+ was shown to trigger APs and, on the other
hand, external electrical stimulus triggered APs and CICRs in the same cell (III).
The If was found not to contribute to the triggering of spontaneous APs (III). All
the major ion currents (INCX, If, INa, delayed rectifier K+ current, IK1, ICaL and ICaT)
were characterized with patch-clamp from E9-E11 mouse cardiomyocytes (IV).
Based on this data, corresponding mathematical models were developed of these
ion currents and implemented into the model of spontaneous Ca2+ oscillations.
With this developed model it was shown that the SL ion channels of the
embryonic mouse cardiomyocyte form a system which 1) triggers APs due to an
NCX-generated depolarizing current after spontaneous Ca2+ release, 2) an
external electrical stimulus induces APs and consequent CICRs, and 3) is
quiescent without triggers 1 or 2 (i.e. If does not trigger APs) (Fig. 17, IV).
Fig. 17. A) Simulated spontaneous and AP-triggered Ca2+ releases (top) and
corresponding APs (bottom). Note that the intracellular Ca2+ gradient between the SL
and SR reverses depending on the EC-coupling mode. B) The depolarizing current of
NCX-mediated Ca2+ extrusion (arrow), which triggers APs during spontaneous activity
but is not present during externally triggered activity (IV).
0.0
0.5
1.0
1.5
[Ca2+
] (μ M
)
500 ms
near SR (0.5 μm)
near SL (0.5 μm)
1 Hz pacing
-50
0
V (m
V)
1 Hz pacing
500 ms
500 ms
1 pA
/pF
spont
pacedINCX
A B
66
5.3.2 The unique features of ECC in the rat neonatal cardiomyocyte (V)
Several unique features were found that distinguish the neonatal rat ventricular
myocyte from the adult rat myocyte. The neonatal cells have long APs (~150–
300 ms (Gaughan et al. 1998, Motlagh et al. 2002, Guo et al. 2007),V). Due to
the high [Na+]i (Hojo et al. 1997, Hayasaki-Kajiwara et al. 1999) NCX produces a
large Ca2+ intrusion (32% of total SL intrusion) to the cell during AP. The total SL
Ca2+ intrusion estimated with the model is 57% of the total SL + SR Ca2+
intrusion to the cytosol. However, due to the higher Ca2+ buffering capacity of the
cytosol at lower [Ca2+]i, the Ca2+ transient amplitude with inhibited RyRs (only SL
Ca2+ intrusion present) is only 37 ± 4% in a real cell (n = 14) and 35% in the
model.
Based on experiments and modeling, the cytosolic Ca2+ diffusion seems to be
amplified in the rat neonatal cardiomyocytes with local CICRs as in adult atrial
myocytes (Huser et al. 1996, Sheehan & Blatter 2003). During initiation of AP,
the Ca2+ diffused through the cytosol with a velocity of 0.31 ± 0.07 μm/ms (n = 4)
(Fig. 16), which is comparable to the velocity in atrial myocytes (0.269 ± 0.015
μm/ms (Sheehan & Blatter 2003)). To reproduce this diffusion velocity in the
model an approximately 10-fold diffusion coefficient for Ca2+ (DCa) compared to
the coefficient in aqueous solution or cytosol is required (Kushmerick &
Podolsky 1969, Cussler 1997)(V). With inhibited RyRs the slope of the global
Ca2+ transient rise was reduced as in atrial myocytes (Sheehan & Blatter 2003)(V).
In the model, the experimental global cytosol Ca2+ signals with inhibited RyRs
were better reproduced with a lower DCa than in the control situation.
The rat neonatal cardiomyocytes have been previously used as an
experimental model in hypertrophy research (Gaughan et al. 1998, Barac et al.
2005, Prasad et al. 2007). The developed model was used to study the role of
increased cytosol size in the generation of the hypertrophic cardiomyocyte
phenotype. Based on simulations, the increase in cytosol size alone in neonatal
cardiomyocytes, without gene expression or any other changes in ECC proteins,
changes the ECC phenotype towards that seen in hypertrophic cardiomyocytes.
Slight prolonging of AP (APD50 prolonged 14% with a 2-fold diffusion distance in
the model vs. 42% in real hypertrophy (Gaughan et al. 1998)) was seen in the
simulations. More dramatic effects were seen in Ca2+ signals (Fig. 18). Even a
small increase in diffusion distance (from 4.5 μm to 5 μm, i.e. 11% increase) was
sufficient to change the Ca2+ transient properties (amplitude, diastolic
67
concentration and decay) and SR Ca2+ content in a physiologically significant
magnitude (~10–20%).
Fig. 18. The simulated effect of the increase of the diffusion distance between the SL
and SR on Ca2+ signaling and APs. The sub-SR (black) and sub-SL (gray) cytosolic
Ca2+ signals are shown in a normal cell (left, solid line) and in a cell with a 2-fold
diffusion distance (right, dashed line). The ∆t indicates the time delay in Ca2+ diffusion
from the sub-SL to sub-SR cytosolic region. (V)
5.4 Simulations of phenotypes of genetically engineered myocytes (II, IV)
Previously reported experimentally characterized phenotypes of genetically
engineered cardiac myocytes were simulated with the embryonic and adult mouse
cardiomyocyte models. Comparison of simulated and real genetically engineered
phenotypes further validated the developed mathematical models. On the other
hand, simulations were able to provide further insight into the underlying
mechanism of these phenotypes.
5.4.1 The simulation of altered NCX, If and CSQN expression in
embryonic mouse cardiomyocytes (IV)
The ablation of the cardiac NCX isoform (NCX1) results in embryonic lethality
before E10 in mouse (Wakimoto et al. 2000). However, the isolated
cardiomyocytes are still excitable and generate Ca2+ signals (Koushik et al. 2001).
In the simulations, the reduction of NCX to 25% of the control amount resulted in
the inability of NCX to trigger APs. Also, the spontaneous Ca2+ oscillation
n n
0.0 0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
SL
Time (s)
[Ca2+
] i (μM
)
SR∆t
∆t
0.0 0.2 0.4 0.6 0.8Time (s)
68
diminished and stopped due to the cytosolic Ca2+ accumulation. Based on these
results, the ablation of NCX disables embryonic cardiac pacemaking and thus
leads to embryonic lethality. However, since other SL ion channels remain intact,
the model cell was in line with real cells (Koushik et al. 2001), excitable and able
to generate small Ca2+ transients by Ca2+ through L-type VACCs.
The embryonic mice with ablated calreticulin have impaired heart function
(Mesaeli et al. 1999), whereas the overexpression of calreticulin results in a lower
heart rate (Nakamura et al. 2001). Both of these genetic modifications are lethal
at later stages of development (Mesaeli et al. 1999, Nakamura et al. 2001). In line
with both of these results, the spontaneous AP frequency vs. calreticulin
expression level was a bell-shaped curve, with the highest frequency near the
control value. At 0.25-fold expression and below, the spontaneous APs were
completely abolished.
Embryonic pacemaking does not require If (III). However, reduction of
functional If to 25% of control with genetic ablation of one of the If coding genes
(HCN4) has been reported to result in ~35% reduction in embryonic heart beat
rate and lethality between E9.5-E11.5 (Stieber et al. 2003). This discrepancy was
explained with the simulations of the developed model (Fig. 19). The If is partly
carried by Na+ ions and thus reduced If reduces [Na+]i. On the other hand, reduced
[Na+]i amplifies Ca2+ extrusion via NCX, which consequently disturbs the
spontaneous Ca2+ oscillation between the SR and cytosol. The If is also a
depolarizing current which thus affects the r.p. and excitability of the membrane.
Although If is not directly responsible for pacemaking, by these mechanisms the
reduced If reduces the frequency of spontaneous Ca2+ oscillation and the APs
triggered by it. It is thus rational that If affects the phenotype of HCN4−/− mice.
69
Fig. 19. The simulated effect of altered If expression in the embryonic mouse
cardiomyocyte model. Left: The [Na+]i (red) and frequency of APs (blue) are shown as
relative values. Right: The INCX during a twitch is shown at 0.25-, 1- and 2-fold
If expression levels. (IV)
5.4.2 The simulation of PLB knockout (KO) and CaMKII overexpression in the adult mouse ventricular myocyte (II)
The simulation of PLB-KO enhanced SERCA function. In line with previously
reported experimental data, the simulated SR Ca2+ content and Ca2+ transient
amplitudes were increased (over 2-fold and 1.6-fold, correspondingly), whereas
the decay of the Ca2+ transient decreased 28% at 1 Hz pacing (Chu et al. 1996,
Huser et al. 1998)(II). Since the contractile function is enhanced dramatically at
lower frequencies, less resources are left for frequency-dependent amplification.
In fact, FFR turns negative in PLB-KO and FDAR is reduced. The genetic
ablation of PLB has been reported to induce decreased RyR expression (Chu et al.
1996). Based on the simulations, this compensates slightly the phenotype (Ca2+
transients, CaMK and CaN activities) of PLB-KO towards WT. The
compensatory gene expression changes are quite mild compared to those in
CaMKII overexpression mice (see below). At higher frequencies (near the normal
mouse heart rate) the simulated WT and PLB-KO have similar CaN activity. This
might explain in part the mild PLB-KO induced compensatory gene expression
changes (Chu et al. 1996, Li et al. 1998, Tavi et al. 2004).
The compensatory gene expression changes induced by the 3-fold CaMKII
overexpression (Maier et al. 2003) were found to significantly contribute to the
phenotype of the CaMKII overexpression mouse (II). The pure 3xCaMKII
0.0 0.5 1.0 1.5 2.0
0.2
0.4
0.6
0.8
1.0
x If expression
rela
tive
[Na+ ] i a
nd fr
eque
ncy
2 x If
1 x If
0.5
pA/p
F
200 ms
INCX 0.25 x If
Ca2+ extrusionincreases
70
simulation produced a phenotype similar to PLB-KO (Fig. 20, 117% increase in
the amplitude and 41% reduction in the decay of the Ca2+ transient). However,
with the previously reported compensatory changes in SERCA, PLB, RyR and
NCX expressions (Maier et al. 2003), the phenotype changes towards a different
direction from the control (Fig. 20)(II). In line with experimental data, the Ca2+
transient amplitude decreased 62% without a significant change in the decay
(Maier et al. 2003). The simulated pure 3-fold CaMKII model had 4.3-fold
CaMKII activity compared to WT at 1 Hz. The compensatory mechanisms in real
cells, however, seem to reduce this activity increment towards that in WT. With
the compensations, CaMKII activity was reduced from 4.3-fold to 2.5-fold
compared to WT.
Fig. 20. Comparison of experimental (Maier et al. 2003) and simulated (II) Ca2+
transients of WT (black), CaMKII overexpression with previously reported
compensations in SERCA, PLB, RyR and NCX expression levels (red), and CaMKII
overexpression with no compensations (green). The corresponding test cases without
and with CaMK-induced SR leak are depicted with continuous lines and dashed lines,
respectively.
CaMKII overexpression has been suggested to increase the diastolic Ca2+ leak
through RyR (Maier et al. 2003). The model with a CaMKII-dependent RyR leak
was simulated as a separate case (Fig. 20). The SR Ca2+ content in the 3xCaMKII
model with compensations compared better to experimental data of Maier et al.
with CaMKII-dependent RyR leak than without (II). The other model output
parameters were the same with and without the CaMKII-dependent leak in WT
and in 3xCaMKII with compensations (Fig. 20). In the pure 3xCaMKII model,
however, the CaMKII-dependent RyR leak significantly compensated for the
71
changes in the Ca2+ transient (Fig. 20), force- and Ca2+-frequency relationship,
and CaMKII and CaN activities. In line with these findings, the effect of acute
CaMKII overexpression on contractility in rabbit cardiomyocytes has been
reported to be completely compensated by the consequences of increased RyR
leak (Kohlhaas et al. 2006).
72
73
6 Discussion
6.1 Simulation methods
The disadvantages of the Euler method are well known in simulations of
differential equation systems (for a review, see e.g. (Haataja 2002)). Although
several more efficient methods exist, the Euler method is still widely used in the
simulations of excitable cell and tissue models due to its easy implementation (Sneyd & Dufour 2002, ten Tusscher et al. 2004, Fall & Rinzel 2006). When the
Euler method is applied, the efficiency of the simulation could be easily improved
(I). The standard automatic step-size adaptation of the Euler method based on a
local relative error estimate (see e.g. (Kreyszig 1999)) was found to increase the
efficiency significantly (I). Furthermore, by limiting the reduction of time-steps in
the case of overestimation of the local error, the efficiency of the Euler method
could be improved even further (I).
An additional advantage to the faster simulations with the developed method
is that the error of the solution is always estimated and controlled in contrast to
the fixed-step Euler method. The solutions with the developed method are thus
more reliable as an error level to the solution is always determined. Also,
compared to other adaptive methods used to simulate excitable cells, such as the
Rush and Larsen scheme (Rush & Larsen 1978) where the time-step depends on
the membrane voltage derivative, the limited method is more versatile as the
time-step depends on the errors of all differential variables (I). When it is possible, more efficient methods than the Euler method should be
applied as in original papers II and V. However, when the Euler method is applied
to simulate cell and tissue models, or any other (stiff) differential equation system,
it should be applied with step-size control as described here (I). In this study, the
embryonic mouse model was simulated with the developed method (IV).
6.2 Model of CaMKII activation
The model of CaMKII activation (Bhalla & Iyengar 1999) was found suitable to
be included in the ECC model (II). As previously, CaMKII and CaN were found
to be activated in a work intensity-dependent manner (De Koninck & Schulman
1998, Maier & Bers 2002, Tavi et al. 2004, Aydin et al. 2007)(II). CaMKII
activity in the cardiac myocyte model was regulated gradually with 2–4 Hz
74
pacing frequencies, but below 2 Hz the activity remained almost constant (Fig.
15)(II). With the feedback from CaMKII to RyR, SERCA and ICaL in the model,
the frequency relationship of Ca2+ signaling in isolated cardiomyocyte (Ito et al.
2000, Antoons et al. 2002) was simulated realistically (II).
6.3 Regulation of ECC and genetic engineering
Two major features seem to characterize ECC regulation. Firstly, the ECC is a
relatively robust system for small changes. In the embryonic cardiomyocyte, the
spontaneous Ca2+ oscillation was maintained with ± 25% changes in RyR, IP3R
and SERCA expression (IV). Also, the same magnitude of changes in calreticulin
and If had very little effect on the performance of the whole system (Fig. 19) (IV).
The robustness is beneficial as the biological systems always have some
variability in their components. Especially during development, the robustness
helps to keep the system functional while it is modified continuously (Yasui et al.
2001, Porter et al. 2003, Creazzo et al. 2004). Secondly, the ECC is regulated
non-linearly (Qu et al. 2007). Larger changes, for example several fold changes in
the expression levels of ECC proteins, dramatically and non-linearly alter the
performance of the whole system (Fig. 19, II & IV)(Mesaeli et al. 1999,
Nakamura et al. 2001, Maier et al. 2003).
These features make the characterization of ECC and the role of its different
components difficult. With small interventions the end-effect easily diminishes
under the natural variation. Consequently, large interventions are used to obtain a
measurable effect in the experiments (see e.g. (Wakimoto et al. 2000, Maier et al.
2003, Stieber et al. 2003)). However, due to the non-linearity of the system, large
intervention of almost anything can produce significant effects, such as lethality
(Mesaeli et al. 1999, Wakimoto et al. 2000, Nakamura et al. 2001, Stieber et al.
2003). Consequently, the role of the particular component may be misunderstood,
such as the If in the embryonic cardiomyocyte (Fig. 19 & III, IV)(Stieber et al.
2003). The analysis of the interventions is further complicated by the induced
compensatory mechanisms (II) and by the induced changes in the physical
structure of the cell, such as in hypertrophy, which also affect the ECC (V).
The compensatory mechanisms can sometimes compensate for drastic
changes, but it depends on the target of the intervention (e.g. PLB-KO vs.
3xCaMKII, see 5.4.2) and whether the intervention is acute or chronic (Chu et al.
1996, Huser et al. 1998, Li et al. 1998, Maier et al. 2003, Kohlhaas et al. 2006).
For example, the dramatic difference in the compensated end phenotype between
75
chronic PLB-KO and 3xCaMKII models might be explained by the different roles
for PLB and CaMKII in the regulatory system (Fig. 12). CaMKII detects the
regulated signal (Ca2+) and phosphorylates several targets affecting this signal
(Maier & Bers 2002). PLB is only one of these regulated targets (Odermatt et al.
1996), and when it is disturbed the myocyte still has several other targets for
CaMKII which can be used to control the Ca2+ signal.
6.4 ECC in developing cardiomyocytes
The embryonic ECC mechanism suggested here (III, IV) explains how the
individual embryonic ventricular myocytes can produce electrical and contractile
activity spontaneously alone as well as synchronize these activities with other
myocytes. The major reason for several controversial previous experimental
reports on embryonic ECC (Nakanishi et al. 1988, Viatchenko-Karpinski et al.
1999, Yasui et al. 2001, Mery et al. 2005, Sasse et al. 2007) may be that instead
of one ECC mechanism there are in fact two parallel modes for the ECC to
operate (III, IV). Identifying and explaining these two complex parallel
mechanisms would not have been possible without the integration of
mathematical modeling and experimental methods. During the experimental
characterization of embryonic cardiomyocyte ECC, several different experiments
were performed (III). The experiments provided several individual findings of the
ECC mechanisms and parameters of ECC components. With mathematical
modeling it was possible to integrate these components together and show that
these mechanisms could indeed exist in a single cardiomyocyte. Furthermore, a
significant amount of additional information was produced with modeling
concerning the phenotypes and underlying mechanisms of genetically engineered
embryonic cardiomyocytes (Mesaeli et al. 1999, Wakimoto et al. 2000, Koushik
et al. 2001, Nakamura et al. 2001, Stieber et al. 2003)(III, IV).
The spontaneous Ca2+ oscillations between the SR and cytosol were found to
initiate the [Ca2+]i transients and electrical activity in mouse E9-E11
cardiomyocytes (III, IV). The spontaneous Ca2+ oscillation is a fundamental
physiological phenomenon as it is found from several adult cell types and also
from the fertilized oocyte (see e.g. (Sneyd et al. 2003, Whitaker 2006,
Lavrentovich & Hemkin 2008)). It seems thus rational to build the embryonic
cardiomyocytes upon a fundamental pre-existing mechanism. On the other hand,
it has been suggested that the pacemaking ECC mechanism characterized here
76
from embryonic cardiomyocytes might exist still in adult pacemaking
cardiomyocytes (Bogdanov et al. 2001).
The role of the spontaneous Ca2+ oscillation as a trigger of the embryonic
heart beat suggests that all the issues regulating this oscillation should regulate the
embryonic heart beat. The effect of [IP3] concentration on Ca2+ oscillation was
shown in the experiments and in the model (III, IV). Thus, regulators of [IP3]
could be further investigated as embryonic heart beat regulators. Also, it is easy to
assume that regulation of SERCA and RyR by PKA via β-adrenergic stimulation
and CaMKII based on Ca2+-dependent activation should have an impact on the
embryonic heartbeat.
The individual embryonic cardiomyocytes were capable of continuously
producing the Ca2+ signals and APs spontaneously (mode 1), but the Ca2+ signals
and APs could also be triggered with external electrical stimuli (mode 2). It is
thus possible that in embryonic cardiomyocyte tissue, the cell with the highest
frequency of spontaneous activity synchronizes the activity of the other cells to its
own to form a uniform rhythmic contraction of the whole tissue. Although modes
1 and 2 can operate simultaneously, the spontaneous activity is favored at more
depolarized r.p., whereas the externally triggered mode is favored at more
hyperpolarized r.p. than in control. The r.p. hyperpolarizes during development
(Reppel et al. 2007). Consequently, the cells could better maintain their activity
alone during cell migration in the course of early development (for a review, see
e.g. (Buckingham et al. 2005)). On the other hand, later in development when the
tissue is formed, the cells begin to acquire the phenotype of adult ventricular
cardiomyocytes requiring an external trigger (for a review, see e.g. (Kamino
1991)).
6.4.1 Common features of embryonic and neonatal cardiomyocyte
ECC
The most characteristic feature in the developing cardiomyocytes studied here
was the heterogeneous Ca2+ signals and the Ca2+ diffusion delay between the sub-
SL and sub-SR (III, IV, V). This feature was reproduced with the models. Due to
the smaller Ca2+ buffering in the cytosol, the amounts of simulated released Ca2+
(concentration in cytosol volume) required for Ca2+ transients were smaller than
in adult cells (24% in rat neonatal and 35% in mouse embryonic compared to
adult rat and mouse (Delbridge et al. 1997, Maier et al. 2003)(IV, V). The fraction
of SL Ca2+ intrusion from the total Ca2+ intrusion to the cytosol was large. The
77
amplitude of the AP-induced Ca2+ transient with inhibited RyR was (model vs.
experiment) 42.3% vs. 32.5 ± 2.3% in mouse embryonic and 35 vs. 37 ± 4% in rat
neonatal myocytes (III, IV, V). The SR Ca2+ content was estimated by exposing
RyRs to caffeine, which results in the release of the whole SR Ca2+ content to the
cytosol, and measuring the amplitude of the consequent Ca2+ transient. The
experimental and simulated SR Ca2+ content in cytosol volume was ~1.25-fold in
rat neonatal and ~1.5-fold in mouse embryonic cardiomyocytes compared to
normal twitch Ca2+ transient amplitude (III, IV, V). In contrast, the adult rat
ventricular myocyte has ~2–3 fold higher SR Ca2+ content in cytosol volume
(Delbridge et al. 1997, Diaz et al. 1997, McCall et al. 1998).
The electrophysiological features of both developing cardiomyocytes were
implemented into the models based on our own recordings and data from the
literature. The developing cardiomyocytes have both L- and T-type VACCs (Avila
et al. 2007)(III, IV). Based on simulations, however, the T-type VACC does not
seem to contribute to APs or Ca2+ signaling. The NCX has a significant role in
both types of developing cardiomyocytes. It has to be strong enough to
compensate for the large Ca2+ influx through the SL. However, in neonatal rat
NCX also contributes significantly to Ca2+ intrusion in the reverse mode (32%)
(V). In the embryonic cardiomyocyte, NCX has an important role in triggering the
AP in pacemaking (III, IV).
6.4.2 Developing cardiomyocytes as general experimental models of cardiomyocytes
In this study the ECC and several of its unique features were identified in
embryonic mouse and neonatal rat ventricular myocytes (III, IV, V). The latter in
particular is a popular experimental model of the cardiac cell (for example studies,
see (Gaughan et al. 1998, Zobel et al. 2002, Most et al. 2005)). These
characterized features can be useful when considering the applicability of the
neonatal cardiomyocyte as an experimental model or when interpreting the
obtained experimental results from the neonatal cardiomyocytes in the context of
cardiomyocytes in general. Most notably, the Ca2+ signaling in cultured neonatal
cells was more similar to atrial than ventricular myocytes (Huser et al. 1996,
Sheehan & Blatter 2003)(V). Consequently, the neonatal culture is not very useful
to study for example the adult ventricular type CICR due to the loose coupling
between L-type VACC and RyR. In contrast, the system of SL ion channels in
neonatal cardiomyocytes has features usable as an experimental model. The AP is
78
long and changes in ion currents should be seen more easily compared to the short
spike in adult rat (Gaughan et al. 1998, Motlagh et al. 2002, Voutilainen-Myllyla
et al. 2003, Guo et al. 2007)(V). The L-type VACC generates a large current and
large Ca2+ intrusion (V)(Xiao et al. 1997, Pignier & Potreau 2000). Consequently,
the effects of ICaL modulation should be easily seen in the electrophysiology and
[Ca2+]i of rat neonatal cardiomyocytes.
6.5 Limitations of the models
The principles of the model development were: 1) the function of the individual
components of the model (e.g. ion channels) must be based on experimental data;
2) the simulated functions must be in line with experimentally defined functions;
and 3) due to the available resources and to maximize the further use of the
models, they must be able to run with a normal workstation computer. Obviously,
some compromises between these principles had to be made concerning the
complexity of the models or in the case of lack of experimental data. On the other
hand, these limitations are also possible aspects for future development of the
ECC models.
6.5.1 Variability of parameters
The major factor limiting the accuracy of the models is the large variability in the
experimental data used in the model development. The experimental data for
model development is usually collected from several different experiments and
laboratories (II, V (Bondarenko et al. 2004, ten Tusscher et al. 2004)). Different
recording environments (temperature, solutions) and devices and different sources
and isolation methods of the samples all generate large variability in the data
between different laboratories. For example, the reported peak current of the ICaL
in cultured rat neonatal ventricular myocytes varies from −13 ± 1 pA/pF (Walsh
& Parks 2002) to −3.23 ± 0.96 pA/pF (Li & Shi 2004). This problem can be
overcome to some extent by adjusting model parameters, such as temperature and
extracellular ion concentrations, depending on the currently used data set. A better
option is to use data measured in similar conditions. In the model of the
embryonic cardiomyocyte, this issue was well handled. Although there was some
variation in experimental solutions and in temperature, most of the data used in
the model development was recorded by us in a single laboratory with the same
equipment and from similarly treated samples (III, IV).
79
However, despite the identical and accurate nature of the experimental
conditions, there is always natural variability from cell to cell, which is the final
limit for the accuracy of the experiments and the model. Considering all this
variation, requiring extremely exact fits between the model and data during model
development is unnecessary (II–V). The models should be also robust for
moderate parameter variation (at least ~10–20%) to explain the maintaining of the
basic function upon natural variability.
6.5.2 Extracellular regulation
The ECC is studied in isolated cardiomyocytes to understand the generation and
regulation of contraction in the heart. In experimental studies, however, the
myocyte is usually isolated from its natural environment from tissue to a lower
temperature and/or different extracellular solution (see e.g. (Gaughan et al. 1998,
Motlagh et al. 2002, Bassani et al. 2004)(III, IV, V)). The cells have to be isolated
since experimental studies, such as electrophysiological voltage-clamp recordings
of ion currents or subcellular confocal Ca2+ imaging, are not possible from the
whole heart. Furthermore, by using unphysiological solutions and temperature the
kinetics and magnitude of e.g. ion currents can be modified to be more easily
recorded (Gaughan et al. 1998, Motlagh et al. 2002). The studied cells also lack
extracellular stimuli or receive them artificially, such as electrical pacing or
β-adrenergic stimulation (Tavi et al. 2005). Especially in studies related to pacing
frequency-dependent ECC (e.g. (Antoons et al. 2002)(II)), the lack of
β-adrenergic stimulation is problematic as it is coupled with the increment of
heart beat frequency in tissue.
The data obtained as discussed above is used to explain ECC and its
regulation and also to develop models of ECC. Consequently, our current
understanding of ECC and also the models of ECC are rather related to the ECC
in isolated cells in an artificial environment than ECC in cells in tissue. The size
of this gap in knowledge might be interesting and useful to study in the future.
6.5.3 Global vs. local signals
In ECC models the transmembrane currents and fluxes and intracellular ion
concentrations represent the average values over the whole cell (II, IV, V)
(Shannon et al. 2004). This obviously diminishes the consequences of variation
80
between single functional components such as individual ion channels (Stern
1992, Tanskanen et al. 2005).
In original paper II the activation of Ca2+-dependent enzymes was calculated
based on the global cytosol Ca2+ signal. However, the local concentrations of Ca2+
activating the enzymes are significantly different, for example below the SR Ca2+
release channel and in the bulk cytosol (Soeller & Cannell 1997). Also, the Ca2+-
dependent enzymes can have different local concentrations and can be pre-bound
to their targets (for a review, see e.g. (Maier & Bers 2002)), further complicating
the use of a global reaction scheme for Ca2+-dependent enzymes in the model (II).
The differences between CaN and CaMKII activation in different cytosol
compartments without feedback to ECC have been recently estimated with
mathematical modeling (Saucerman & Bers 2008). By further implementing the
local feedback regulations into these different compartments, based on for
example the formalism in (II), the total physiological effect of the separate local
Ca2+-dependent feedback regulation systems could be estimated.
6.5.4 The models of developing cardiomyocytes
The long free Ca2+ diffusion distance seen in developing cardiomyocytes in
experimental Ca2+ imaging required novel approaches for modeling ECC in these
cells (Fig. 16)(III, V). Fick’s law of diffusion was integrated into the “standard”
ECC model to model Ca2+ propagation in the cytosol (IV, V). In rat neonatal
cardiomyocytes, the propagation was found to be amplified with local Ca2+
releases (V). A fire-diffuse-fire model (Dawson et al. 1999) could thus have been
used in original paper V, but it was considered unreasonable due to the lack of
proper data for fitting such model (V). The contribution of mobile Ca2+ buffers
(CMDN) to the facilitation of diffusion was not studied here, since based on
diffusion coefficients and previous calculations (Kushmerick & Podolsky 1969,
Wagner & Keizer 1994, Cussler 1997, Slaughter et al. 2004) they would explain
only a minor part of the measured high diffusion velocity. The propagation might
also be amplified in the embryonic cardiomyocytes, since the delay between sub-
SL and sub-SR Ca2+ signals was about the same magnitude in these cells than in
neonatal cells (III, V).
The shape of the developing cardiomyocytes on the culture plate is relatively
flat in the Z-direction and round-like in the X-Y direction (III, IV, V)(Delcarpio et
al. 1989). To model the Ca2+ diffusion in all of these directions is not problematic.
However, the consequent concentration differences in parallel to the surfaces of
81
the inner SL and outer SR would require division of these surfaces into a grid,
where each square includes Ca2+-dependent ion channel and pump models.
Compared to homogenous membranes in adult myocyte models (Tavi et al. 1998,
Bondarenko et al. 2004, Shannon et al. 2004), the computational load would
probably be between 20 × 20 = 400 to 100 × 100 = 10000-fold, depending on the
grid accuracy. This was beyond the computing power of a normal workstation
computer.
To be able to use homogenous SR and SL membrane dynamics, a cylinder-
shaped cytosol with radial symmetry has been used previously (Haddock et al.
1999, Michailova et al. 2002). However, the spherical shape was considered to be
closer to the shape of a real cell and so it was used in this study (IV, V). Both a
cylinder and a sphere are rough approximations. In the embryonic cardiomyocyte
model, the shape of the cytosol was a bit problematic as there was no variation in
the SR-SL distance or in the cytosol shape, which both affect the activation delay
between spontaneous Ca2+ release and AP (IV). This resulted in too long a delay
between the SR Ca2+ release and initiation of AP (IV).
6.6 Applicability of the models
Despite the limitations, the ECC models have proven their usability in this and
previous studies (see e.g. (Puglisi et al. 2004, Shannon et al. 2004, Winslow et al.
2005)). However, modeling is not an alternative to traditional experimental
research work. Modeling should be used in parallel with experimental studies to
gain more information from the experimental results and to help plan further
experimental studies. The development of the models is always limited by the
available experimental information of the studied cells. On the other hand, the
simulated results are only estimates until they are validated by performing the
same experiments in real cells.
Modeling can be used to integrate large amounts of experimental data into a
single context and to explain the performance of the modeled system with normal
or altered parameters. With modeling, the causal relationships within the system
can also be assessed, which is often difficult or impossible to achieve in
experiments. The development of the embryonic cardiomyocyte model and
explaining the two parallel modes of ECC is a good example of integration of
experimental data to a model to produce an understandable conclusion based on
the data (III, IV). In contrast, the simulations of the ECC in cardiomyocytes from
genetically engineered animals are good examples of how modeling could explain
82
the system performance in altered conditions (II, III, IV). When considering that
the action potential was already explained with the interplay of experiments and
modeling in the 1950s, it is surprising that researchers still endeavor to explain
such complex issues as ECC in genetically engineered myocytes without
modeling (Hodgkin & Huxley 1952, Maier et al. 2003, Stieber et al. 2003).
However, there is an increasing trend to use ECC models (Winslow et al. 2005),
and hopefully this trend will continue among the unexplored features of ECC.
83
7 Summary and conclusions
1. In the simulations of ECC models, the efficiency of the forward Euler method
can be increased significantly using the developed time-step adaptation
method. However, when possible more efficient higher order methods should
be used instead of the forward Euler method (I).
2. An integrated model of Ca2+-dependent enzymes and ECC in mouse
ventricular myocytes was developed. With the CaMKII-mediated feedback
regulation of Ca2+ signaling, the model faithfully reproduces the pacing
frequency dependence of Ca2+ signaling in isolated mouse ventricular
myocytes (II).
3. Two models with a similar novel general structure were developed to model
cardiomyocyte ECC during early embryonic and postnatal development (IV,
V). The more heterogeneous, SL-dominated cytosolic Ca2+ signals and longer
APs distinguish the neonatal from the adult rat ventricular myocytes and
result in limitations in the usability of neonatal cardiomyocytes as general
experimental models (V).
4. The E9-E11 embryonic mouse cardiomyocytes have two parallel ECC modes.
In the spontaneous mode the RyR and IP3R generate spontaneous rhythmic
Ca2+ releases which trigger APs via NCX. The spontaneous activity can be
synchronized with electrical pacing, which produces voltage-activated Ca2+
influx and CICR. With these modes the same embryonic cardiomyocytes can
maintain their activity alone and also form a uniformly contracting tissue (III,
IV).
5. The ECC models were able to explain underlying mechanisms of
compromised ECC in previously reported genetically engineered
cardiomyocytes. In some cases, such as If-KO in the embryonic
cardiomyocyte and 3-fold CaMKII overexpression in the adult cardiomyocyte,
the compromised ECC is caused by the compensatory mechanisms and
indirect effects induced by the intervention rather than the intervention itself
(II, IV).
84
85
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99
Appendix 1 Example of m-files for an ECC model
A similar code structure was used in all models. The following m-files were used
in original paper V. The ‘main_neon.m’ file solves the problem placed in file
‘dy_neon.m’ and stores the data to the file. The ‘stimcreator.m’ function generates
the stimulus current trace used to stimulate the cell model. The simulation can be
run by typing in Matlab for e.g.:
stim_1Hz = stimcreator(1, 0.5, 5000);
main_neon(stim_1Hz, 'y0.dat', 'outputfile.dat', 5000, 1, 0);
which simulates 1 Hz pacing of the model cell for 5000 ms starting from the
initial conditions in file ‘y0.dat’.
The m-files are not included in the electrical version of the dissertation.
110
111
Original articles
I Korhonen T & Tavi P (2008) Automatic time-step adaptation of the forward
Euler method in simulation of models of ion channels and excitable cells and
tissue. Simulation Modelling Practice and Theory 16: 639–644.
II Koivumäki J, Korhonen T, Takalo J, Weckström M & Tavi P (2009)
Integrative Model of Excitation-Contraction Coupling in Mouse Cardiac
Myocytes. Manuscript.
III Rapila R, Korhonen T & Tavi P (2008) Excitation-contraction coupling of the
mouse embryonic cardiomyocyte. Journal of General Physiology 132(4):
397–405.
IV Korhonen T, Rapila R & Tavi P (2008) Mathematical model of mouse
embryonic cardiomyocyte excitation-contraction coupling. Journal of General
Physiology 132(4): 407–419.
V Korhonen T, Hänninen SL & Tavi P (2009) Model of excitation-contraction
coupling of rat neonatal ventricular myocytes. Biophysical Journal 96(3):
1189–1209.
Reprinted with permission from Elsevier (I), the Rockefeller University Press (III,
IV) and the Biophysical Society (V).
The original articles are not included in the electronic version of the dissertation.
112
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MATHEMATICAL MODELING OF THE REGULATION, DEVELOPMENT AND GENETICALLY ENGINEERED EXPERIMENTAL MODELS OF CARDIAC EXCITATION-CONTRACTION COUPLING
FACULTY OF MEDICINE,INSTITUTE OF BIOMEDICINE, DEPARTMENT OF PHYSIOLOGY,UNIVERSITY OF OULU;FACULTY OF SCIENCE,DEPARTMENT OF PHYSICAL SCIENCES, DIVISION OF BIOPHYSICS,UNIVERSITY OF OULU;BIOCENTER OULU, UNIVERSITY OF OULU
